Table of contents. Table of contents List of figures List of tables Introduction... 15

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5 Table of contents Table of contents... 1 List of figues... 5 List of tables Intoduction Spatial Patten Analysis Intoduction Methods Quadatcountsanalysis Quadatcountindices Quadat counts method applied to study site Lagoa A Fist-odeanalysis Neaestneighbomethods NeaestneighbomethodsappliedtostudysiteLagoaA Second-odeanalysis Ripley sk-function Simulationsinteval Edgeeffects Aeabasededgeeffectcoectionmethod Results Timepocessing Simulationenvelopewidth AGuadAea RealDataset Discussion Spatial Patten Analysis - an Application Intoduction MangoveFoest Avicennia geminans Rhizophoa mangle Lagunculaia acemosa

6 2 Table of contents Mangovefoestevolution Results Lagoa A Lagoa B Discussion Object Patten Analysis Intoduction Method UnivaiateAnalysis BivaiateAnalysis SimulationsEnvelope Results Lagoa A Lagoa B Discussion Wavelet Tansfom applied to Ecology Intoduction Methods KenelDensityEstimationMethod VaiogamAnalysis WaveletTansfom ContinuousWaveletTansfom DisceteWaveletTansfom InhomogeneousPoissonPocess DensityMapGeneation Results Heteogeneitydetection Lagunculaia acemosa - Lagoa A Lagunculaia acemosa - Lagoa B DeadTees-LagoaB Spatial-ScaleResolution Spatial elationship between Lagunculaia acemosa and Avicennia geminans - Lagoa A Spatial elationship between Lagunculaia acemosa and Avicennia geminans - Lagoa B SimulatinganInhomogeneousPoissonPocess Lagunculaia acemosa - Lagoa B Dead tees - Lagoa B Discussion

7 Table of contents 3 6 Conclusion Ackowledgements Refeences Appendix

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9 List of figues Figue 1.1 Inteaction between stand spatial stuctue, tee local envionment and natual pocesses. Adapted fom (Goeaud et al. 1998) Figue 1.2 Thesis stuctue Figue 2.1 Thee point pattens with 100 points epesenting a (a) CSR patten, (b) egula patten and (c) cluste patten Figue 2.2 Study aea Lagoa A with 560 tees. (black dot) Avicennia geminans and (blue dot) Lagunculaia acemosa. The axis x and y ae given in m Figue 2.3 (a) Regula patten with 50 points, (b) cluste patten with 50 points and (c) CSR patten with 50 points and (d), (e) and (f) ae thei espective L-function (black) and 99% confidence inteval (dashed ed). The confidence inteval was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations Figue 2.4 Study site Lagoa A. (a) Avicennia, (b) Lagunculaia, (c) Avicennia (blue) and Lagunculaia (ed). (d) and (e) ae the univaiate L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) fo the point patten in (a) and (b) espectively. The univaiate simulation envelope was calculated via Monte Calo method (Besag 1977) with 1000 simulations. (f) Bivaiate K-function (black) and its simulation envelope fo independence hypothesis (dashed ed) calculated fo the point patten contained in (c). The bivaiate simulation envelope was calculated via andom shifting method (Lotwick & Silveman, Diggle 1983) with 1000 simulations Figue 2.5 Study aea Ω with dimensions [0, a] [0, b] and a seach cicle c i () with adius centeed on a point p i within this egion. A i () and A i + () ae the aea of the egion of c i () outside and inside Ω espectively Figue 2.6 The fou possibilities of intesection between the seach cicle c i () and the edges of the study egion Ω Figue 2.7 Time pocessing fo the Aea s method (dashed) and Ripley s method (filled) Figue 2.8 Results obtained fo a CSR patten simulated within a study egion Ω=[0, 1] [0, 1]. 99% simulation envelope fo the CSR model fo n = 50 (a) and n = 200 (c) with simulations using the Aea method (filled) and Ripley method (dashed) and (b) and (d) shows the espective simulation envelope width Figue 2.9 Confidence inteval width of a egula model with paametes (a) n = 50 and = 0.01, (b) n = 50 and = 0.03, (c) n = 50 and = 0.05, (d) n = 200 and = 0.01, (e) n = 200 and = 0.03, (f) n = 200 and = 0.05 obtained by the Aea method (filled) and Ripley method (dashed). The simulation envelope width was obtained via Monte Calo method (Besag & Diggle 1977) with simulations Figue 2.10 Confidence inteval width fo a clumped model with paametes (a) n = 50 and = 0.05, (b) n = 50 and = 0.08, (c) n = 50 and = 0.1, (d) n = 200 and = 0.05, (e) n = 200 and = 0.08 and (f) n = 200 and = 0.1 obtained by the Aea method (filled) and Ripley method (dashed).the simulation envelope width was obtained via Monte Calo method (Besag & Diggle 1977) with simulations Figue 2.11 A whole study egion Ω=[0, 2] [0, 2] divided as a guad aea Ω g =[0.5, 1.5] [0.5, 1.5] (gay egion) and an buffe aea Ω b (hatched egion) suounding Ω g Figue 2.12 Study aea Lagoa A with 560 tees: 118 tees inside the guad aea Ω g (points) and 442 tees inside the buffe aea Ω b (cosses) Figue 3.1 Coastal zone of nothe Bazil whee the study sites Lagoa A and Lagoa B (black dots) ae located

10 6 List of figues Figue 3.2 Black mangove o Avicennia geminans Figue 3.3 Red mangove o Rhizophoa mangle Figue 3.4 White mangove o Lagunculaia acemosa Figue 3.5 All tees at stand site Lagoa A. (black coss) dead tee, (blue dot) Lagunculaia acemosa and (ed dot) Avicennia geminans. The size of dot is popotional to the dbh of Avicennia and Lagunculaia (thee s no infomation about the dbh of the dead tees). (scale in metes) Figue 3.6 Lagoa A - Histogams showing the size class distibution of the mean stem diamete in beast height (dbh) in cm obtained fo (a) all tees (excluding dead tees), (b) Avicennia geminans and (c) Lagunculaia acemosa.(scaleincm) Figue 3.7 (left) Spatial point patten elative to all tees of the stand Lagoa A and its (ight) espective L-function (black) and 99% simulation envelope (dashed ed). The simulation envelope was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations Figue 3.8 (left) Spatial point patten elative to the species Avicennia geminans within stand Lagoa A. Its (ight) espective L-function (black) and 99% simulation envelope (dashed ed). The simulation envelope was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations.. 56 Figue 3.9 (left) Spatial point patten elative to the species Lagunculaia acemosa within stand Lagoa A. Its (ight) espective L-function (black) and 99% simulation envelope (dashed ed). The simulation envelope was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations.. 57 Figue 3.10 (left) Spatial point patten elative to dead tees within stand Lagoa A. Its (ight) espective L-function (black) and 99% simulation envelope (dashed ed). The simulation envelope was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations Figue 3.11 The point pattens (a) and (c) epesents espectively the big tees (n=232) and small tees (n=328). (b) and (d) epesents thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesent the lage tees (blue) and small tees (ed). (f) epesents its bivaiate L-function (black) and 99% simulation envelope fo spatial independence hypothesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations Figue 3.12 The point pattens (a) and (c) epesents espectively the lage Avicennia (n=100) and small Avicennia (n=209). (b) and (d) epesent thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesents the big Avicennia (blue) and small Avicennia (ed). (f) epesents its bivaiate L-function (black) and 99% simulation envelope fo spatial independence hypothesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations Figue 3.13 The point pattens (a) and (c) epesents espectively the lage Lagunculaia (n=132) and small Lagunculaia (n=132). (b) and (d) epesents thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesents the big Lagunculaia (blue) and small Lagunculaia (ed). (f) epesents its bivaiate L-function (black) and 99% simulation envelope fo spatial independence hypothesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations Figue 3.14 The point pattens (a) and (c) epesents espectively the dead tees (n=252) and living tees (n=560). (b) and (c) epesents thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesents the dead tees (blue) and living tees (ed). (f) epesents its K 1 () K 2 () (black) and 99% simulation envelope fo andom labeling hyphotesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations Figue 3.15 The point pattens (a) and (c) epesents espectively the species Avicennia (n=309) and Lagunculaia (n=251). (b) and (c) epesents thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesents the species Avicennia (blue) and Lagunculaia (ed). (f) epesents its bivaiate L-function (black) and 99% simulation envelope fo spatial

11 List of figues 7 independence hypothesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations Figue 3.16 All tees of the stand Lagoa B. (black coss) dead tee, (blue dot) Lagunculaia acemosa and (ed dot) Avicennia geminans and (geen dot) Rhizophoa mangle. The size of dot is popotional to the dbh of Avicennia and Lagunculaia (thee s no infomation about the dbh of the dead tees) Figue 3.17 Lagoa B - Histogams showing the size class distibution of mean stem diamete in beast height (dbh) in cm obtained fo (a) all tees (excluding dead tees), (b) Avicennia geminans and (c) Lagunculaia acemosa Figue 3.18 (left) Spatial point patten elative to all tees of the stand Lagoa B and its (ight) espective L-function (black) and 99% simulation envelope (dashed ed). The simulation envelope was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations Figue 3.19 (left) Spatial point patten elative to the species Avicennia geminans within stand Lagoa B. Its (ight) espective L-function (black) and 99% simulation envelope (dashed ed). The simulation envelope was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations.. 67 Figue 3.20 (left) Spatial point patten elative to species Lagunculaia acemosa within stand Lagoa B. Its (ight) espective L-function (black) and 99% simulation envelope (dashed ed). The simulation envelope was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations.. 68 Figue 3.21 (left) Spatial point patten elative to dead tees within stand Lagoa A. Its (ight) espective L-function (black) and 99% simulation envelope (dashed ed). The simulation envelope was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations Figue 3.22 The point pattens (a) and (c) epesents espectively the lage tees (n=179) and small tees (n=255). (b) and (d) epesents thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesents the big tees (blue) and small tees (ed). (f) epesents its bivaiate L-function (black) and 99% simulation envelope fo spatial independence hypothesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations Figue 3.23 The point pattens (a) and (c) epesents espectively the lage Avicennia (n=46) and small Avicennia (n=210). (b) and (d) epesents thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesents the big Avicennia (blue) and small Avicennia (ed). (f) epesents its bivaiate L-function (black) and 99% simulation envelope fo spatial independence hypothesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations Figue 3.24 The point pattens (a) and (c) epesents espectively the lage Lagunculaia (n=129) and small Lagunculaia (n=42). (b) and (d) epesents thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesents the big Lagunculaia (blue) and small Lagunculaia (ed). (f) epesents its bivaiate L-function (black) and 99% simulation envelope fo spatial independence hyphotesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations Figue 3.25 The point pattens (a) and (c) epesents espectively the dead tees (n=116) and living tees (n=434). (b) and (d) epesents thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesents the dead tees (blue) and living tees (ed). (f) epesents its K 1 () K 2 () (black) and 99% simulation envelope fo andom labeling hyphotesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations Figue 3.26 The point pattens (a) and (c) epesents espectively the species Avicennia (n=256) and Lagunculaia (n=171). (b) and (d) epesents thei L-function (black) and 99% simulation envelope fo CSR hyphotesis (dashed ed) espectively. The point patten (e) epesents the species Avicennia (blue) and Lagunculaia (ed). (f) epesents its bivaiate L-function (black) and 99% simulation envelope fo spatial independence hyphotesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method

12 8 List of figues (Besag 1977) with simulations Figue 3.27 The point pattens (a) and (c) epesents espectively the species Lagunculaia (n=171) and dead tees (n=171). (b) and (d) epesents thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesents the species Lagunculaia (blue) and dead tees (ed). (f) epesents its bivaiate L-function (black) and 99% simulation envelope fo spatial independence hypothesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations Figue 4.1 Steps of a tansfomation of a tidimensional object (tee) tee into a point. (left) Real tees, (middle) bidimensional abstaction of a eal tee with cown (geen) and stem (bown) and (ight) point patten epesenting the bidimensional abstaction Figue 4.2 The cicle-to-point tansfomation T 2 indicates egulaity at lowe scales, instead of small-scale aggegation Figue 4.3 Two cicula objects c i ( i ) and c j ( j ) and a seach cicle c i () inside a study egion Ω Figue 4.4 Estimating the expected specific aea within a distance of an abitay cicula object of the study egion Ω Figue 4.5 Estimating the expected specific aea inside a ing Figue 4.6 Histogam calculated fo the R ZOI distibution elative to (a) all tees, (b) small tees (dbh<5 cm), (c) lage tees (dbh 5 cm), (d) Avicennia geminans, (e) smallavicennia (dbh<5 cm), (f) lage Avicennia (dbh 5 cm), (g) Lagunculaia acemosa,(h) smalllagunculaia (dbh<5 cm) and (i) lage Lagunculaia (dbh 5 cm) Figue 4.7 (a) Object patten elative to all tees within study site Lagoa A. (b) Object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model I hypothesis with 200 simulations Figue 4.8 (a) Object patten elative to Avicennia geminans within study site Lagoa A. (b) Object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model I hypothesis with 200 simulations Figue 4.9 (a) Object patten elative to Lagunculaia acemosa within study site Lagoa A. (b) Object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model I hypothesis with 200 simulations Figue 4.10 (a) Object patten elative to small (dbh<5 cm) (ed) and lage (dbh 5 cm) (blue) tees within study site Lagoa A. (b) Bivaiate object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model II hypothesis with 200 simulations.. 92 Figue 4.11 (a) Object patten elative to Avicennia geminans (ed) and Lagunculaia acemosa (blue) tees within study site Lagoa A. (b) Bivaiate object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model II hypothesis with 200 simulations Figue 4.12 Object patten elative to small (dbh<5 cm) (ed) and lage (dbh 5cm) (blue) Avicennia tees within study site Lagoa A. (b) Bivaiate object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model II hypothesis with 200 simulations.. 93 Figue 4.13 Object patten elative to small (dbh<5 cm) (ed) and lage (dbh 5 cm) (blue) Lagunculaia tees within study site Lagoa A. (b) Bivaiate object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model II hypothesis with 200 simulations Figue 4.14 Histogam calculated fo the R ZOI distibution elative to (a) all tees, (b) small tees (dbh<5 cm), (c) lage tees (dbh 5 cm), (d) Avicennia geminans, (e) smallavicennia (dbh<5 cm), (f) lage Avicennia (dbh 5 cm), (g) Lagunculaia acemosa,(h) smalllagunculaia (dbh<5 cm) and (i) lage Lagunculaia (dbh 5 cm) Figue 4.15 (a) Object patten elative to all tees within study site Lagoa B. (b) Object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the

13 List of figues 9 Model I hypothesis with 200 simulations Figue 4.16 (a) Object patten elative to Avicennia geminans within study site Lagoa B. (b) Object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model I hypothesis with 200 simulations Figue 4.17 (a) Object patten elative to Lagunculaia acemosa within study site Lagoa B. (b) Object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model I hypothesis with 200 simulations Figue 4.18 (a) Object patten elative to small (dbh<5 cm) (ed) and lage (dbh 5 cm) (blue) tees within study site Lagoa A. (b) Bivaiate object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model II hypothesis with 200 simulations.. 98 Figue 4.19 (a) Object patten elative to Avicennia geminans (ed) and Lagunculaia acemosa (blue) tees within study site Lagoa A. (b) Bivaiate object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model II hypothesis with 200 simulations Figue 4.20 Object patten elative to small (dbh<5 cm) (ed) and lage (dbh 5cm) (blue) Avicennia tees within study site Lagoa A. (b) Bivaiate object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model II hypothesis with 200 simulations.. 99 Figue 4.21 Object patten elative to small (dbh<5 cm) (ed) and lage (dbh 5 cm) (blue) Lagunculaia tees within study site Lagoa A. (b) Bivaiate object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model II hypothesis with 200 simulations Figue 5.1 Repesentation of the steps of ou methodology. (left) Point Patten (middle) Density Map (ight) Multiesolution Analysis Figue 5.2 The Kenel Density Estimation method tansfoms a spatial point patten (left) into a density map (ight). The scale of the study site is povided in m and of the unity of the density map is points/m Figue 5.3 Chaacteistics of a Vaiogam Figue 5.4 The Multiesolution Decomposition Analysis pefomed to decompose a density map (left) in diffeent scales (ight). The unit of the study site is povided in m and of the density map is points/m Figue 5.5 The gaphic (a) epesents a time seie that contains a supeposition of a low fequency signal (sin10t) and a high fequency signal (sin20t). The gaphic (b) epesents a time seie that contains a low fequency signal (sin10t) in the fist half and a signal with high fequency signal (sin20t) in the second half. The gaphics (c) and (d) epesent the esponse of the FT to the time seies epesented at the gaphics (a) and (b) espectively. The gaphics (e) and (f) aeepesenttheesponseofthewt to the time seies epesented at gaphics (a) and (b) espectively Figue 5.6 The gaphics (a),(b) and (c) epesent the same mothe wavelet with paamete λ =1, λ = 0.5 and λ = 0.25 espectively Figue 5.7 The gaphics (a) and (b) epesent the same mothe wavelet with paamete t =0 and t = 0.25 espectively Figue 5.8 MDA applied to a signal f(t). A 1 is the appoximation at level 1 and D 1 is the detail at level Figue 5.9 MDA applied to A 1. A 2 is the appoximation at level 2 and D 2 is the detail at level Figue 5.10 MDA at level n applied to a signal f(t)=a Figue 5.11 MDA applied to an image (density map). The oiginal image (a density map) was decomposed at its highe, intemediate and lowe scale components Figue 5.12 (a) Density map obtained fom spatial point patten elative to Lagunculaia acemosa within stand Lagoa A. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a)

14 10 List of figues Figue 5.13 (a) Appoximation A1 elative to the density map at the Figue (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a) Figue 5.14 (a) Appoximation A2 elative to the density map at the Figue 5.13a. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a) Figue 5.15 (a) Appoximation A3 elative to the density map at the Figue (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a) Figue 5.16 (a) Appoximation A4 elative to the density map at the Figue 5.13a. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a) Figue 5.17 (a) Point patten elative to Lagunculaia acemosa whit-in stand Lagoa A, now divided at two egions (ed and yellow). (b) and (c) epesents the the L-function (black) and 99% simulations inteval (dashed ed) calculated fo the point patten inside the ed and yellow espectively. The simulations inteval was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations. 120 Figue 5.18 (a) Density map obtained fom spatial point patten elative to Lagunculaia acemosa within stand Lagoa B. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a) Figue 5.19 (a) Appoximation A4 elative to the density map at the Figue (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a) Figue 5.20 (a) Point patten elative to Lagunculaia acemosa whit-in stand Lagoa B, now divided at two egions (ed and yellow). (b) and (c) epesents the the L-function (black) and 99% simulations inteval (dashed ed) calculated fo the point patten inside the ed and yellow espectively. The simulations inteval was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations. 122 Figue 5.21 (a) Density map obtained fom spatial point patten elative to dead tees within stand Lagoa A. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a) Figue 5.22 (a) Appoximation A4 elative to the density map at the Figue 5.25a. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a) Figue 5.23 (a) Point patten elative to dead tees whit-in stand Lagoa B, now divided at two egions (ed and yellow). (b) and (c) epesents the the L-function (black) and 99% simulations inteval (dashed ed) calculated fo the point patten inside the ed and yellow espectively. The simulations inteval was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations Figue 5.24 (a) Density map obtained fom spatial point patten elative to Avicennia geminans within stand Lagoa A. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a) Figue 5.25 (a) Density map obtained fom spatial point patten elative to Lagunculaia acemosa within stand Lagoa A. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a) Figue 5.26 Avicennia geminans (a) Appoximation A4 elative to the density map at the Figue (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a). 125 Figue 5.27 Lagunculaia acemosa (a) Appoximation A4 elative to the density map at the Figue (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a). 126 Figue 5.28 (a) Appoximation A4 elative to the density map obtained fo the Avicennia geminans and (b) Appoximation A4 elative to the density map obtained fo the Lagunculaia acemosa within Lagoa A Figue 5.29 (a) Density map obtained fom spatial point patten elative to Avicennia geminans within stand Lagoa B. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a) Figue 5.30 (a) Density map obtained fom spatial point patten elative to Lagunculaia acemosa within stand Lagoa B. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a)

15 List of figues 11 Figue 5.31 Avicennia geminans (a) Appoximation A4 elative to the density map at the Figue (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a). 128 Figue 5.32 Lagunculaia acemosa (a) Appoximation A4 elative to the density map at the Figue (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a). 128 Figue 5.33 (a) Appoximation A4 elative to the density map obtained fo the Avicennia geminans and (b) Appoximation A4 elative to the density map obtained fo the Lagunculaia acemosa within Lagoa B Figue 5.34 Inhomogeneous L-function (black) and 90% simulations inteval (dashed ed) calculated fo the Lagunculaia tees inside Lagoa B. The simulations inteval was calculated via Monte Calo method (Besag 1977) fo the inhomogeneous Poisson pocess hypothesis with 1000 simulations Figue 5.35 Inhomogeneous L-function (black) and 90% simulations inteval (dashed ed) calculated fo the dead tees inside Lagoa B. The simulations inteval was calculated via Monte Calo method (Besag 1977) fo the inhomogeneous Poisson pocess hypothesis with 1000 simulations

16

17 List of tables Table 1.1 Desciption of methods fo analysis of spatial data. Adapted fom Pey et al. (2002) Table 2.1 Summay analysis using quadat count methods applied to Lagoa A Table 2.2 Summay analysis using neaest neighbo methods applied to Lagoa A Table 2.3 Main chaacteistics of the null hypothesis of spatial independence and andom labeling.. 29 Table 2.4 Summay of K-function analysis applied to study site Lagoa A. The scale is mete Table 2.5 Edge coection facto w i () fo the 4 possibilities pesented in Figue Table 2.6 Relationship between the time pocessing fo the Ripley s method (t 1 )andfotheaea s method (t 2 ). The time is given in seconds Table 2.7 CSR model with 200 points epesenting a low density scenaio and 800 points epesenting a high density scenaio. Summay of the statistical esults fo the deviance factos D R and D A obtained fom simulations Table 2.8 Summay of the statistical esults of the deviance facto D R and D A calculated fo the egion Ω g obtained fom simulations. A egula model with 200 points and paamete = 0.01, 0.03 and 0.05 epesenting a low density scenaio and a egula model with 800 points and paametes =0.01,0.03and0.05 epesenting a high density scenaio Table 2.9 Summay of the statistical esults of the deviance facto D R and D A calculated fo the egion Ω g obtained with simulations. A clumped model with 200 points and paamete c = 0.05, 0.08 and 0.01 epesenting a low density scenaio and a egula model with 800 points and paametes =0.01,0.03and0.05 epesenting a high density scenaio Table 2.10 Deviance facto D R and D A obtained fo the study site Lagoa A Table 3.1 Summay of undelying pocess and an possible associated spatial point patten. (+) Positive and (-) negative inteaction Table 3.2 Chaacteistics of popagules of thee mangove species (adapted fom Rabinowitz 1978). (S) Salt wate conditions and (F) Fesh wate conditions Table 3.3 Ecological Chaacteistics of vaious mangove species. Salinity in ( 0 / 00 ).MS=Maximumpoe wate salinity measued in the fields at sites whee the species was gowing, OG = Salinity fo optimum gowth based on cultue studies. Adapted fom Smith III (1991) Table 3.4 A peliminay model of stand development in mangoves. The foest collapses with senescence when the cycle esumes with e-colonization. Adapted fom Duke (2001), Silvetown & Doust (1993) and Menezes et al. (2004) Table 3.5 Shot statistical summay of the mean stem diamete in beast height (dbh) fo Lagoa A site. (- ) Thee s no infomation about the dead tees s dbh. (*) Excluding the dead tees. (dbh in cm) Table 3.6 Summay of the univaiate L-function analysis obtained fo the site Lagoa A. (*) tendency and (**) in paticula. LS (Lowe Scales), IS (Intemediate Scales) and HS (Highe Scales) Table 3.7 Summay of the univaiate and bivaiate L-function analysis obtained fo the site Lagoa A. small = (dbh 5 cm) and lage = (dbh>5 cm).(*) tendency and (**) in paticula. LS (Lowe Scales), IS (Intemediate Scales) and HS (Highe Scales) Table 3.8 Shot statistical summay of the mean stem diamete in beast height (dbh) fo Lagoa B site. (- ) Thee s no infomation about the dead tees s dbh. (*) Excluding the dead tees. (dbh in cm) Table 3.9 Summay of the univaiate L-function analysis obtained fo the site Lagoa B. (*) tendency and (**) in paticula. LS (Lowe Scales), IS (Intemediate Scales) and HS (Highe Scales)

18 14 List of tables Table 3.10 Summay of the univaiate and bivaiate L-function analysis obtained fo the site Lagoa B. small = (dbh 5 cm) and lage = (dbh>5 cm). (*) tendency and (**) in paticula. LS (Lowe Scales), IS (Intemediate Scales) and HS (Highe Scales) Table 4.1 Intepetation of the K c () function Table 4.2 Intepetation of the L c () function Table 4.3 Intepetation of the function R ε () Table 4.4 Intepetation of the function R ε() Table 4.5 Intepetation of the function R B ε() Table 4.6 Basic statistics fo Lagoa A. R ZOI, min(r ZOI)andmax(R ZOI) inm and va(r ZOI) inm Table 4.7 Basic statistics fo Lagoa B. R ZOI, min(r ZOI)andmax(R ZOI)inmand va(r ZOI) inm

19 Chapte 1 Intoduction The spatial stuctue of a foest stand is of fundamental impotance fo foest dynamics, because the local envionment detemines competition among the tees, gowth, death and egeneation (see Figue 1.1). Futhemoe, the spatial configuation of individual tees in a foest stand may povide an indiect indication of the undelying ecological pocesses that ae occuing at the site (Malkinson et al. 2003). Analysis of the spatial configuation of the tees in a stand is of pime impotance, fo example, fo the development of individual tee gowth models (Goeaud et al. 1996) and to gain an undestanding of ecological systems and the dynamics of plant communities (Geig-Smith 1964, Young et al. 1999). In summay, the spatial configuation of individuals could be a esult of the natual pocesses occuing in the stand. In the liteatue, a lage numbe of spatial statistical methods have been descibed that could be applied to spatial patten analysis. The pincipal aim of this study is to apply, modify and impove these methods in ode to detemine and to analyze the spatial stuctue of tee individuals in a foest stand. In Chapte 2, I pesent some classical spatial statistical methods cuently applied to plant ecology, such as: quadat count methods, fist ode methods and second ode methods (in paticula, the well-known Ripley K-function). They ae suitable fo the analysis of spatial configuations and distibution pattens of individuals, such as plants o tees. Additionally, I pesent some explicit fomulas fo an aea-based edge effect coection method (heeinafte Aea method). Thee ae some efeences to this method in the liteatue, but the explicit fomulas ae absent o incompletely descibed. The edge effect coection facto is an essential pat of those spatial statistical methods that equie the counting of neighbos within a given distance (Goeaud & Pelissie 1999). An appopiate edge coection can impove the stability of the esults obtained fom spatial statistical methods and incease the sensitivity of the spatial statistical tests (Yamada & Rogeson 2003). A numbe of edge coections methods ae discussed in liteatue; I compae the Aea method which is the pincipal focus of this study with anothe widely used method, the so called Ripley edge coection method (heeinafte Ripley method). 15

20 16 Intoduction Figue 1.1. Inteaction between stand spatial stuctue, tee local envionment and natual pocesses. Adapted fom (Goeaud et al. 1998) The aim is to find out which edge coection method poduces the bette esults and is easie to implement. The aim is to find out which edge coection method poduces the best esults and is easiest to implement. The Chapte 3 is a diect application of the methods pesented in Chapte 2. Iapply the Ripley K-function in combination with the Aea method in ode to analyze the spatial configuation of the tees at two study sites located in Nothwesten Bazil. This dataset is used though the study. It is hoped by this means to obtain indiect infomation about the undelying ecological pocesses which might be giving ise to the spatial patten of the tees obseved in these sites. It is impotant to list at least two limitations of the methods pesented in Chapte 2 and Chapte 3. Fist of all, they equie the homogeneity of the spatial point patten being analyzed. The hypothesis of homogeneity means that the point patten is stationay (invaiant unde tanslation) and isotopic (invaiant unde otation). Howeve, it is wellknow that heteogeneity is geneally common in the natue. In summay, the second-ode featues of the point patten depends only on the distance between the points, but not on diection o location (Goeaud et al. 1996, Wiegand & Moloney 2004). Additionally, the classical methods conside a thee dimensional tee as a point. This abstaction is also

21 Intoduction 17 a limitation, because it can lead to misintepetation of the ecological pocesses that ae occuing in the stands. To ovecome this limitation, I develop a new method fo the spatial analysis of objects in Chapte 4, which appoximates each individual tee as a cicle, instead as a point. The idea of this method is to minimize the bias of the classical methods pesented in Chapte 2, which conside a eal tee as a point. I then test this method by applying it to the same dataset analyzed in Chapte 3. It is impotant to note that all methods pesented in Chapte 2 and also in Chapte 4 have a common limitation. They neglect infomation about the spatial configuation of the individuals. These methods (unde the hypothesis of homogeneity) ae able to distinguish whethe a point patten o a patten of objects tends towads complete spatial andomness, o towads a clumped o egula distibution and at which scale these chaacteistics occu, but they ae not able to povide infomation about the spatial location of these featues. Finally in Chapte 5, I popose a new methodology that povides spatial-scale infomation (subject to cetain estictions) about the spatial pocesses occuing in a foest stand. The main idea is to adapt the Wavelet Tansfomation method (heeinafte WT)so that this can be used fo the spatial analysis of point pattens and apply this methodology to plant ecology. In summay, the methodology consists in tansfoming a point patten into a density map using a Kenel density estimation method (heeinafte KDE) and decompose this map at diffeent scales using Multiesolution Decomposition Analysis (heeinafte MDA) obtained via WT method. Finally, I compae these esults with the esults obtained in Chapte 3 and Chapte 4 in ode to demonstate the powe of fo studying vegetation pattens. In Table 1.1 I pesent a shot summay of the pincipal chaacteistics of the statistical methods applied in this study. Additionally, in Figue 1.2 I pesent a geneal desciption of the stuctue of the thesis. Method Data Oiginal Scale Spatial Hypothesis Type Use Resolution Resolution Test Quadat Counts (x, y) Plant Ecology no no no Neaest Neighbo z Plant Ecology no no yes Ripley K-function (x, y) Plant Ecology yes no yes Vaiogam (x, y,z) Eath Sciences yes no no Wavelet Tansfom (x, y,z) Mathematics yes yes possible Table 1.1. Desciption of methods fo analysis of spatial data. Adapted fom Pey et al. (2002). All gaphic figues pesented in this wok wee ceated using the softwae R, a language and envionment fo statistical computing and gaphics. It is a GNU Poject which is simila to the S language and envionment which was developed at Bell Laboatoies (fomely AT&T, now Lucent Technologies) by John Chambes and colleagues. The algoithms wee implemented using the applications R, Scilab (a scientific softwae package fo numeical computation poviding a poweful open computing envionment fo engineeing and scientific applications) and Intel Fotan Compile 9.1 fo Linux.

22 18 Intoduction Figue 1.2. Thesis stuctue.

23 Chapte 2 Spatial Patten Analysis 2.1 Intoduction Spatial point patten analysis is a set of tools used to study the distibution of discete points. It is a statistical method applied to obtain and to analyze infomation about the spatial stuctue of individuals dispesed within a study aea. The idea is to distinguish between point pattens which tend towad complete spatial andomness (heeinafte CSR), clumping o egulaity (see Figue 2.1) and at which scale these chaacteistics occu. a b c y y y x x x Figue 2.1. Thee point pattens with 100 points epesenting a (a) CSR patten, (b) egula patten and (c) cluste patten. Stand spatial stuctue is a complicated concept that includes both hoizontal and vetical use of space by tees. In ode to simplify this appoach, I conside only on the hoizontal location of stems in the stand. Thus, the study aea is epesented by a pat of the hoizontal plane bounded by the stand bodes, and each individual plants is mapped as a point in whose position is shown by the Catesian coodinates (x, y). This simplification o abstaction of the study aea educes it to a finite set of points, called point patten. This point pocess epesentation of the stand pemits us to descibe and analyze point pattens, with the aim of detemining global popeties (laws) in the andom locations of tees in the stand (Goeuad 1997). 19

24 20 Spatial Patten Analysis Apoint pocess P is a andom pocess, a mathematical object simila to andom vaiable, whose ealizations ae point pattens. Its geneates andom point pattens that shae the same spatial stuctue, such as CSR, egula and clusteed pattens. The popeties of the pocess define constaints on its ealizations (fo example, in tems of density, distance between neighboing points, and stuctue). The main idea is to assume that thee exists an undelying pocess P and to use the popeties of that pocess to descibe the stuctue of the patten (Goeaud 1999). 2.2 Methods Now I pesent some spatial statistical methods in ode to analyze the spatial configuation of the tees within a study site named Lagoa A. It is located on the coast of the notheasten Bazilian State of Paa, nea the town of Bagança at a latitude and longitude of appoximately (01 03 S, W), (Mehlig 2001). This aea is located on the Acaao Peninsula on the estuay of Caete Rive, 150 km south of the Amazon delta in nothen Bazil (Bege et al. 1999). It foms pat of the wold s second lagest continuous aea of mangove foest, estimated to cove a total aea of 1.38 million ha along a coastline of ca km (Kjeve et al. 1993). Applying the point pocess epesentation to the study site Lagoa A, each tee individual is epesented by its spatial position inside the stand (see Figue 2.2), species and adius at beast height in cm (exception fo the dead tees). To avoid misundestanding, it is impotant to explain the notation used in this chapte. Location means a position (x, y) inside the study aea and point means an abitay point p i =(x i,y i ) of the point patten within the aea.

25 2.2 Methods 21 Lagoa A y x Figue 2.2. Study aea Lagoa A with 560 tees. (black dot) Avicennia geminans and (blue dot) Lagunculaia acemosa. The axis x and y ae giveninm Quadat counts analysis The quadat count method is one of the ealy methods of spatial point patten analysis. The basic pocedue is to sample the study site using andomly located quadates (e.g., a seach cicle of adius centeed inside the study aea) and to count the numbe of points (epesenting each individual of the stand) that lies inside each one. Unde a hypothesis of CSR, the distibution of the numbe of points inside a quadat with aea a is given by a Poisson seies with mean λa, wheeλ is the intensity of the point patten inside the whole study aea and it is estimated as λ = n A, (2.1) whee A and n ae espectively the aea and the numbe of points of the study aea.

26 22 Spatial Patten Analysis The pobability of encounteing n points inside a quadat with aea a is given by P(N(a)=n)=e λa(λa)n. (2.2) n! To compae the obseved values with the expected values, I need to pefom a χ 2 test of significance. In this case, point clusteing would be evidenced by counts that occued moe oftenthanexpectedandegulaitywouldbeevidencedbycountsthatoccuedlessthan expected Quadat count indices Thee ae a numbe of indices descibed in the liteatue that could be used with the quadat count method to detect a significant deviation fom a CSR patten (a Poisson distibution). The simplest and pobably the oldest of these was developed by Fishe et al. (1922). It is defined as I 1 = V X, (2.3) whee V and X ae the sample vaiance and the sample mean of the quadat counts espectively. The method is based on fact that the mean and the vaiance of a Poisson distibution ae the same, thus the expected value of the indices unde CSR hypothesis is I 1 = 1.0, I 1 >1.0 if I have a clumped patten and I 1 < 1.0 if I have a egula o a CSR patten. A futhe index was developed by Douglas (1975) and it is defined as I 2 = X 2 V 2 X, (2.4) whee V and X ae defined as above. If I conside that the numbe of points inside a quadat has a Poisson distibution with mean μ = λa (whee a is the aea of the quadat), then I 2 is equal to μ fo lage sample sizes.

27 2.2 Methods 23 A numbe of othe indices have been descibed (David & Mooe 1954, Lloyd 1967, Moisita 1959). Fo a eview, see Cessie (1993) Quadat counts method applied to study site Lagoa A I apply the quadat count method to the study site Lagoa A to quadats with dimensions 1 x 1 m (analysis at small scale), 3 x 3 m (analysis at intemediate scale) and 5 x 5 m (analysis at lage scale) andomly dispesed inside this study aea. The esults of the analysis obtained foeachscaleaesummaizedintablebelow. Quadat Counts Analysis Lagoa A (λ = 0.622) Quadat Scale a I 1 I 2 μ = λa Intepetation 1 x 1 m small clusteing 3 x 3 m intemediate clusteing 5x5m lage clusteing Table 2.1. Summay analysis using quadat count methods applied to Lagoa A. The esults pesented in Table 2.1 shown that I 1 > 1.0 and I 2 >μat all scales analyzed. It means that the point patten shows clusteing at diffeent scales Fist-ode analysis Intensity λ (o density) is the simplest fist ode popety of a point pocess. If I conside a homogeneous point patten (if it is invaiant unde tanslation and otation), λ is a constant and can be estimated by λˆ= n/a, wheen and A ae espectively the numbe of points and the aea of the study site. To intoduce the concept of local density, I define c (x,y) () as a cicle with adius centeed on a location (x,y) inside of the study site. Then the local density in a neighbohood of a location (x, y) of the study site can be estimated by whee N[c x,y ()] is the numbe of points inside c x,y (). λˆx,y()= N[cx,y()] π 2, (2.5)

28 24 Spatial Patten Analysis Neaest neighbo methods Neaest neighbo methods ae based on the fist-ode popety of a point pocess. They ae based on the distance between a point and its neaest neighbo. The basic pocedue of these methods is to estimate the mean point density λˆ (points pe unit aea) using infomation about the mean point-to-point distance. Then this estimated point density λˆ is compaed with the expected point density λ to classify the point patten as clumped, egula o CSR. An advantage of these methods in compaison to quadat count methods is that they make use of pecise infomation about the locations of the points and do not depend on the size o shape of the quadats (Cessie 1993). The simplest index attibuted to Fische et al (1922) is defined as I 3 = Va(d), (2.6) d whee d is the neaest neighbo distance ove all points of the point patten and d is the mean neaest-neighbo distance defined as d = 1 n d i, (2.7) n i=1 whee n is the numbe of points inside the study site and d i is the is the neaest neighbo distance fo point i inside the study site. The expected value of the index is 1 fo a andom patten, I 3 > 1 indicates clusteing and I 3 < 1 indicates egulaity. Anothe index was developed by Clak and Evans (1954) and it is defined as I 4 = d, (2.8) 1 2 λ whee d is as defined in equation (2.6), the denominato is the expected mean neaestneighbo distance unde CSR assumption and λ isthedensityofthepointsinsidethewhole study site. Futhe indices can be founded in the liteatue, see Cessie 2 λ (1993) Neaest neighbo methods applied to study site Lagoa A 1 I now apply the neaest neighbo method to the study site Lagoa A. The esults of the analysis obtained fo each index ae summaized in table below. Neaest Neighbo Analysis Lagoa A (λ = 0.622) Indices Result Intepetation I clusteing I clusteing Table 2.2. Summay analysis using neaest neighbo methods applied to Lagoa A.

29 2.2 Methods 25 The esults pesented in Table 2.2 show that I 3 and I 4 > 1.0. This indicates that the point patten pesents clusteing Second-ode analysis The second ode popety of a point patten is elated to the density of occuence of two points within a given distance fom each othe (Ripley 1977, Diggle 1983). This popety chaacteizes the numbe of points found in the neighbohood of an abitay point of the patten and pemits the spatial stuctue of these points to be descibed in tems of inteaction pocesses: aggegation, epulsion, etc... (Pelissie & Goeuad 2001). To calculate second ode local neighbo density, I define c i () as a cicle with adius centeed on a point p i. Then the second ode local neighbo density in a neighbohood of an abitay point p i of the patten can be estimated by λˆi()= N[ci()] π 2, (2.9) whee N[c i ()] coesponds to the numbe of points within c i () Ripley s K-function The Ripley K-function is a second ode method based on distances between all pais of points of the patten. The advantage of this method in compaison to othes (Quadat Count Methods and Fist Ode Analysis) is that it peseves infomation about distances between all points in the patten. It can be used to analyze a point patten at a ange of scales and to detemine at which scales these points tend to be egula, clumped o CSR. It can also be used to descibe the elationship between one, two o moe types of points contained inside the point patten. The geneal definition of the Ripley s K-function fo a cetain distance is K()=λ 1 E[], (2.10) whee E[] is the expected numbe of points within a distance fom an abitay point of the study egion Ω and λ is the density of points inside this aea estimated as λˆ= n A, (2.11) whee n and A ae the numbe of points inside and the aea of the study egion Ω, espectively.

30 26 Spatial Patten Analysis The K-function is defined so that λk() is the expected numbe of points contained at distance fom an abitay point of the patten inside Ω. In pactice, the univaiate K- function (whee only one type of point is being consideed) is estimated as Kˆ ()= A n n w n 2 ij ()δ ij (), (2.12) i=1 j i whee w ij () is an edge effect coection facto, δ ij () is an indicato function which defines whethe a point p j is inside a neighbohood of a point p i o not and is defined as with δ ij ()= { 1, if dij 0, othewise, (2.13) d ij = (x i x j ) 2 +(y i y j ) 2, (2.14) being the Euclidian distance between the points p i =(x i,y i ) and p j =(x j,y j ) within the study egion Ω. An advantage of the K-function is that calculated values ae independent of the shape of the study egion, poviding that adequate adjustments ae made fo edge effects (Cessie 1991). Futhemoe, an appopiate edge effect coection facto can impove the sensibility of the statistical esults in the sampling data and can incease the powe of the statistical tests (Yamada & Rogeson 2003). Because of its hypebolic behavio, the intepetation of K-function is not staightfowad. Fo this eason, a modification called L-function has been poposed to nomalize it (Besag 1977), Lˆ()= Kˆ (). (2.15) π The expected value of the univaiate L-function unde CSR is 0 fo all, positive when the patten tends to be clusteed and negative when the pattens tends to be egula The bivaiate K-function is used to analyze the spatial elation between two o moe diffeent type of points. Fist, I have to define Kˆ12()= A n 1 n 2 w ij()δ ij(), (2.16) n 1 n 2 i=1 j i whee w ij () is an edge effect coection facto, δ ij () is an indicato function which defines whethe a point p j of type 2 is inside a neighbohood of a point p i of type 1 o not and it is defined as { 1, if dij δ ij ()= 0, othewise, (2.17) with d ij = (x i x j ) 2 +(y i y j ) 2, (2.18)

31 2.2 Methods 27 being the Euclidian distance between the points p 1 i =(x i,y i ) of type 1 and p 2 j =(x j,y j ) of type 2 within the study egion Ω. The associated L-function is defined as Kˆ12 () Lˆ12()=. (2.19) π The expected value of the bivaiate L-function unde spatial independence is 0 fo all, positive when the two point pocesses tends to be aggegated and negative when the two point pocesses tends to be epulsive. Similaly, I define the function Kˆ21()= A n 2 n 1 w ij()δ ij(), (2.20) n 1 n 2 i=1 j i whee w ij () is an edge effect coection facto, δ ij () is an indicato function which defines whethe a point p j of type 1 is inside a neighbohood of a point p i of type 2 o not and it is defined as { 1, if dij δ ij ()= 0, othewise, (2.21) with d ij = (x i x j ) 2 +(y i y j ) 2, (2.22) being the Euclidian distance between the points p 1 i =(x i,y i ) of type 1 and p 2 j =(x j,y j ) of type 2 within the study egion Ω, and the its associated L-function is defined as Kˆ21 () Lˆ21()=. (2.23) π Then the bivaiate K-function is defined as KˆB = n1kˆ21 + n 2Kˆ12, (2.24) n 1 + n 2 anditslineaizationisdefinedas KˆB () LˆB =. (2.25) π The bivaiate K-function is defined so that λkˆb() is the expected numbe of points of type 2 contained at the distance of an abitay point of type 1 of the point patten Simulations inteval

32 28 Spatial Patten Analysis The estimatos of the second ode functions ae andom vaiables with cetain vaiance and in ode to test the null hypothesis of a CSR patten using eal data, I have to take this uncetainly into account. In ou case, I used the Monte Calo method to estimate these vaiations (Besag & Diggle 1977) and to geneate the confidence inteval. The methods applied to calculate the confidence inteval fo the univaiate K-function and the bivaiate K-function ae completely diffeent and will be explained below. Univaiate K-function In ode to test the deviation fom andomness (egulaity o clusteing) of the point patten using the univaiate K-function, I computed a 99% confidence inteval of L() using the Monte Calo method (Besag & Diggle 1977) fom 1000 simulated CSR pattens with the same numbe of points contained inside a egion with the same geomety. If the L-function intecepts the lowe bounds of confidence inteval and/o the uppe bounds of the confidence inteval I have an indication of egulaity and/o clusteing espectively. In Figue 2.3 I show examples of the intepetation of the univaiate K-function applied to thee diffeent point patten models, each containing 50 points: a CSR patten, a egula patten and a clumped patten. (a) (b) (c) n = 50 (d) L() n = 50 (e) L() n = 50 (f) L() Figue 2.3. (a) Regula patten with 50 points, (b) cluste patten with 50 points and (c) CSR patten with 50 points and (d), (e) and (f) ae thei espective L-function (black) and 99% confidence inteval (dashed ed). The confidence inteval was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations.

33 2.2 Methods 29 Bivaiate K-function The geneation of the simulation envelopes fo the bivaiate case is moe complicated; fo details of the methodology, see Goeaud & Pelissie (2003). Basically, in ode to geneate the coect simulation envelope fo the bivaiate case, I have to choose one of two diffeent null hypotheses: independence o andom labeling. The independence hypothesis must be chosen if the location of the points of type 1 and type 2 esults fom two apioiindependent spatial point pocesses. In this case the location of the type 1 points is independent fom the location of the type 2 points. In ode to geneate the simulation envelope that coesponds to the hypothesis of spatial independence, I have to hold the point patten of the points of type 1 and type 2 unchanged and andomize thei elative position in each Monte Calo simulation. Fo moe details, see Lotwick & Silveman (1982), Diggle (1983) and Goeuad & Pelissie (2003). The andom labeling hypothesis must be chosen if the location of the points of type 1 and type 2 is esult of events affecting a posteioi individuals of a single population. It means that the pobability that one event occus is the same fo all points and does not depends on neighbo the identity of the neighboing point (Goeaud & Pelissie 2003). In ode to geneate the simulation envelope that coesponds to the hypothesis of andom labeling, I have to simulate point pocesses with the same obseved spatial stuctue consideing all points without type distinction. Then I hold the simulated patten and simulate the point types in the same popotion as that obseved in the study aea. Fo moe details, see Diggle (1983) and Goeaud & Pelissie (2003). In Table 2.3 I pesent a summay of the main chaacteistics of the null hypothesis applied to bivaiate K-function. Geneal Famewok Independence Random labeling Null hypothesis L B ()=0 L B ()=L() Aggegation L B () > 0 L B () >L() Repulsion L B () < 0 L B () <L() Simulation pocedue Random shifting of type 1 points Random attibutions of maks Biological example Between species o cohots inteaction Disease attack o distubances Table 2.3. Table 2.3 Main chaacteistics of the null hypothesis of spatial independence and andom labeling. Figue 2.4 show an example of the an application of the bivaiate K-function applied to the stand Lagoa A. In Table 2.5 I pesent a summay of the esults pesented in Figue 2.4.

34 30 Spatial Patten Analysis K-function analysis Point Patte Regulaity Clusteing Lagunculaia - 1 <<9 Avicennia - 1 Lagunculaia & Avicennia <4 - Table 2.4. Summay of K-function analysis applied to study site Lagoa A. The scale is mete. The univaiate analysis of the point patten that epesents Lagunculaia tees shows clusteing at diffeent scales, 1 m <<8 m. The univaiate K-function analysis applied to Avicennia shows clumping at scale 1 mete. But it is impotant to notice that a epulsion patten exists between the Lagunculaia and Avicennia tees at scale <2 m. In summay, the tees of the same species tend to occupy the same aea and the tees of diffeent species tends to avoid each othe. (a) (b) (c) y x (d) L2() y x (e) L1() y x (f) L12() Figue 2.4. Study site Lagoa A. (a) Avicennia, (b) Lagunculaia, (c) Avicennia (blue) and Lagunculaia (ed). (d) and (e) ae the univaiate L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) fo the point patten in (a) and (b) espectively. The univaiate simulation envelope was calculated via Monte Calo method (Besag 1977) with 1000 simulations. (f) Bivaiate K-function (black) and its simulation envelope fo independence hypothesis (dashed ed) calculated fo the point patten contained in (c). The bivaiate simulation envelope was calculated via andom shifting method (Lotwick & Silveman, Diggle 1983) with 1000 simulations.

35 2.2 Methods Edge effects The edge effect poblem usually occus when it is necessay to count the numbe of points within a seach cicle c i () that intecepts the edges of the study aea (see Figue 2.5). This seach cicle has a adius and is centeed at a point p i located inside a study aea Ω. It has two distinct pats c + i () and c i (), that espectively mak the egions of the seach cicle which belong o do not belong to Ω. Usually, thee is no infomation about the numbe of points within c i (). Howeve, if these points ae not consideed, c i () contains fewe points than expected. The pupose of edge effect coection facto is to minimize this effect. An altenative is to estimate the numbe of points within c i () using the infomation of the numbe of points within c + i (). Figue 2.5. Study aea Ω with dimensions [0,a] [0,b] and a seach cicle c i () with adius centeed on a point p i within this egion. A i () and A i + () ae the aea of the egion of c i () outside and inside Ω espectively. In what follows I pesent some explicit fomulas fo an aea based edge effect coection method (heeinafte Aea method). Some aticles efe to this method (Getis & Fanklin 1987, Besag 1977, Dale & Powell 2001), but details of the coesponding fomulas ae missing o incomplete. Let define the total numbe of points within c i () as N i ()=N i + ()+N i () (2.26) whee N i + () and N i () ae the numbe of points within c i + () and c i () espectively. By similaity, the total aea of the cicle c i () can be defined as A i ()=A i + ()+A i ()=π 2 (2.27)

36 32 Spatial Patten Analysis whee A + i () and A i () ae the aeas of the egions c + i () and c i () espectively (see Figue 2.5). Futhemoe, I suppose that the point density within c i () is equal to the point density within c + i (). Using this agument and the density definition, it follows that N i () A i () = N i + () A + i () N i ()= A i () A + i () N i + (), (2.28) and combining the equations (1.26), (1.27) and (1.28), N i () can be estimated as N i ()= A i() A i + () N i + ()=w i ()N i + (), (2.29) with w i () being an aea based edge effect coection facto (heeinafte aea coection facto). The aea coection facto depends on the elative position of the point p i inside the study egion Ω andontheadiusoftheseachciclec i (). This featue pemits us to edefine the K-function in equation (2.3) as K()= A n w n 2 i () n i=1 j i δ ij (). (2.30) This alteation significantly educes the numbe of opeations needed fo the calculation of the K-function and consequently educes the time needed to cay out the analysis, but without compomising its pecision Aea based edge effect coection method The explicit fomulae fo the Aea method ae when the study egion Ω is with ectangula and the adius of the seach cicle c i (), used fo the calculation of the K-function, is up to half the length of the shote side of the study site Ω. Fou diffeent cases need to be distinguished and thei w i () fomulas ae pesented in Table 2.5. Case 1. The seach cicle c i () does not intecept the edges of the study egion Ω (see Figue 2.6 and Table 2.5). Case 2. The seach cicle c i () intecepts one edge of the study egion Ω (see Figue 2.6 and Table 2.5)and α = accos(d/) (2.31) e = 2 d 2

37 2.2 Methods 33 whee d is the distance between the cente of the seach cicle c i () to the intecepted edge of the study aea Ω. Case 3. The seach cicle c i () intecepts two edges of the study egion Ω and 2 > d d 2 2 (see Figue 2.6 and Table 2.5) Case 4. The seach cicle c i () intecepts two edges of the study egion Ω and 2 d d 2 2 (see Figue 2.6 and Table 2.5). In both cases 3 and 4 the following equation is valid α i = accos(d i /),i=1, 2 e i = 2 2 d i,i=1, 2 (2.32) whee d 1 and d 2 ae the distance fom the cente of the seach cicle c i () to the intecepted edges of the study aea Ω. Figue 2.6. The fou possibilities of intesection between the seach cicle c i () and the edges of the study egion Ω. Case Condition w i () 1 no intesection 1 2 one intesection π 2 (ed +(π α) 2 ) 1 3 two intesections with 2 >d d 2 2 π 2 (d 1 d (e 1 d 1 + e 2 d 2 )+(0.75π 0.5α 1 0.5α 2 ) 2 ) 1 4 two intesections with 2 d d 2 π 2 (e 1 d 1 + e 2 d 2 +(π α 1 α 2 ) 2 ) 1 Table 2.5. Table 2.5 Edge coection facto w i () fo the 4 possibilities pesented in Figue 2.6.

38 34 Spatial Patten Analysis 2.3 Results Time pocessing To compae the time equied to calculate the Ripley K-function using the two methods pesented above I simulated a CSR patten vaying the numbes of point inside a study aea with squae geomety [0, 1] [0, 1]. Then I obtained the time necessay to calculate the Ripley s K-function fo these point pattens using Ripley method (t 1 ) and the Aea method (t 2 ). The gaph in Figue 2.7 shows that the Aea method in elation to the Ripley method is at least 8.50 times faste (see also Table 2.6). Time Pocessing(s) Numbe of points Figue 2.7. Time pocessing fo the Aea s method (dashed) and Ripley s method (filled). n t t t 1 /t Table 2.6. Table 2.6 Relationship between the time pocessing fo the Ripley s method (t 1 )and fo the Aea s method (t 2 ). The time is given in seconds.

39 2.3 Results Simulation envelope width The simulation envelope width is the diffeence between the uppe and lowe bounds of the simulation envelope obtained by the Monte Calo method (Besag & Diggle 1977). Intuitively, an edge coection method with a wide simulation envelope width has lowe statistical powe to detect clumped and/o egula point pattens. An edge coection method with a naowe simulation envelope width is moe stable unde statistical fluctuations and has a highe statistical powe (Yamada & Rogeson 2003). In this wok, a 99% simulation envelope was obtained fo diffeent point pattens inside astudyaeaω=[0, 1] [0, 1] by pefoming simulations. Fo each data set the simulation envelope width was calculated using eithe Ripley method othe Aea method. The models used to geneate the point patten ae listed below: 1. Complete spatial andomness o CSR model. I simulated a low density scenaio with 50 points and a high density scenaio with 200 points. 2. Regula model. This model was simulated using a sequential spatial inhibition pocess (Kaluzny et al. 1997) with paamete c=0.01, 0.03 and 0.05 (The paamete c specifies minimal distance between the points). Fo each paamete c, I simulated a low density scenaio with 50 points and a high density scenaio with 200 points. 3. Clumped model. This model was simulated using an algoithm pesented by Yamada & Rogeson (2003) with paamete c = 0.05, 0.08 and 0.01 (the paamete c specifies the mean adius of the cluste). Fo each paamete c, I simulated a low density scenaio with 50 points and a high density scenaio with 200 points. Figue 2.8 shows the simulation envelope and its espective width calculated fo the CSR model. Figue 2.9 and Figue 2.10 show only simulation envelope widths obtained fo the egula and clumped models espectively. Fo both cases, the simulation envelope width obtained by the Aea method is moe stable than those of Ripley method.

40 36 Spatial Patten Analysis a c L() L() b d L() width L() width Figue 2.8. Results obtained fo a CSR patten simulated within a study egion Ω=[0, 1] [0, 1]. 99% simulation envelope fo the CSR model fo n = 50 (a) and n = 200 (c) with simulations using the Aea method (filled) and Ripley method (dashed) and (b) and (d) shows the espective simulation envelope width. a b c L() width d L() width L() width e L() width L() width f L() width Figue 2.9. Confidence inteval width of a egula model with paametes (a) n = 50 and = 0.01, (b) n = 50 and = 0.03, (c) n = 50 and = 0.05, (d) n = 200 and = 0.01, (e) n = 200 and = 0.03, (f) n = 200 and = 0.05 obtained by the Aea method (filled) and Ripley method (dashed). The simulation envelope width was obtained via Monte Calo method (Besag & Diggle 1977) with simulations.

41 2.3 Results 37 a b c L() width d L() width L() width e L() width L() width f L() width Figue Confidence inteval width fo a clumped model with paametes (a) n = 50 and = 0.05, (b) n = 50 and = 0.08, (c) n = 50 and = 0.1, (d) n = 200 and = 0.05, (e) n = 200 and = 0.08 and (f) n = 200 and = 0.1 obtained by the Aea method (filled) and Ripley method (dashed).the simulation envelope width was obtained via Monte Calo method (Besag & Diggle 1977) with simulations A Guad Aea A vitual expeiment can be caied out to evaluate the quality of the edge coection factos descibed above. The idea is to compae the estimated numbe of points outside the study egion with the eal numbe of points occuing in A i (). Fo this, the study aea Ω is now divided in two egions: a guad aea Ω g which contains the measuable points and abuffeaeaω b that that contains the points to be estimated (see Figue 2.11). TheK-functionfotheguadaeaΩ g canbedefinedas with the associated L-function K G ()= A n g n 2 δ ij () (2.33) n g i=1 j i L G ()= K G(), (2.34) π whee n g and n ae the numbe of points inside the egions Ω g and Ω espectively, wheeas A is the aea of Ω. Notice that K G () includes all points in the entie study egion. It is, theefoe, not necessay to use an edge coection facto in this case, because all necessay

42 38 Spatial Patten Analysis infomation is aleady included. In ode to evaluate the eo associated with each edge coection factos, K G () is compaed with the K-functions calculated fo Ω g without using the infomation inside Ω b. The K-function calculated fo Ω g using the Aea method is and its associated L-function is K A ()= Ag n g 2 n g w i () n g δ ij() (2.35) i=1 j i L A ()= KA(). (2.36) π The K-function calculated fo Ω g using the Ripley method is and its associated L-function is K R ()= Ag n g 2 n g n g w ij()δ ij() (2.37) i=1 j i L R ()= KR(), (2.38) π whee n g and A g ae the numbe of points inside and the aea of Ω g espectively. Figue A whole study egion Ω=[0, 2] [0, 2] divided as a guad aea Ω g =[0.5, 1.5] [0.5, 1.5] (gay egion) and an buffe aea Ω b (hatched egion) suounding Ω g.

43 2.3 Results 39 To measue the deviation of the functions L A () and L R () fom the efeence function L G (), deviance factos can be defined as and whee max is the maximal scale of the analysis. max D A = [LG () L A ()] 2 d (2.39) 0 max D R = [LG () L R ()] 2 d, (2.40) 0 Fo this test, I simulated a point patten within a study egion Ω=[0, 2] [0, 2], sepaated into a guad aea Ω g =[0.5,1.5] [0.5, 1.5] located in the cente and a suounding buffe aea Ω b. I pefomed simulations fo each point patten model descibed above. In contast to the analyses of the simulation envelope width, fo each model two density scenaios wee consideed: a low density scenaio with 200 points inside Ω and a high density scenaio with 800 points inside Ω. These settings mean that an aveage of 50 and 200 points occu inside Ω g, in the low and high density scenaios espectively. Table 2.7 and Table 2.9 show the mean deviance facto (D A and D R )andthecoesponding vaiances fo all CSR and clumped scenaios. The mean of the deviance facto obtained shows that the pefomance of the Aea method is bette than o equivalent to the Ripley method. A Student T-test calculated fo the esults pesented fo the egula scenaios in Table 2.8 (LD 2.1 and HD 2.2 )showsthatd A and D R ae equivalent. LD scenaio mean vaiance D R D A HD scenaio mean vaiance D R D A Table 2.7. CSR model with 200 points epesenting a low density scenaio and 800 points epesenting a high density scenaio. Summay of the statistical esults fo thedeviancefactosd R and D A obtained fom simulations t=-0.016, df=10, p-value=0.987 and 95% confidence inteval = (-3.49,3.44) t=-0.031, df=10, p-value=0.976 and 95% confidence inteval = (-2.95,3.03).

44 40 Spatial Patten Analysis LD Scenaio =0.01 =0.03 =0.08 Deviance mean vaiance mean vaiance mean vaiance D R D A HD Scenaio =0.01 =0.03 =0.08 Deviance mean vaiance mean vaiance mean vaiance D R D A Table 2.8. Summay of the statistical esults of the deviance facto D R and D A calculated fo the egion Ω g obtained fom simulations. A egula model with 200 points and paamete = 0.01, 0.03 and 0.05 epesenting a low density scenaio and a egula model with 800 points and paametes = 0.01, 0.03and0.05 epesenting a high density scenaio. LD scenaio c=0.05 c=0.08 c=0.1 Deviance mean vaiance mean vaiance mean vaiance D R D A HD scenaio c=0.05 c=0.08 c=0.1 Deviance mean vaiance mean vaiance mean vaiance D R D A Table 2.9. Summay of the statistical esults of the deviance facto D R and D A calculated fo the egion Ω g obtained with simulations. A clumped model with 200 points and paamete c = 0.05, 0.08 and 0.01 epesenting a low density scenaio and a egula model with 800 points and paametes = 0.01, 0.03 and 0.05 epesenting a high density scenaio Real Dataset The vitual expeiment descibed above is a poweful tool fequently applied in ecology to evaluate the quality of ecological paametes (see e.g. Bege et al. 1999b o Pene & Scüle 2004). In this simulation expeiment, the guad aea methodology applied to a eal data set obtained fom a mangove foest. The method was applied to the data set obtained fom the study site Lagoa A. The site has geomety: Ω=[0, 30] [0, 30]. Fo the analysis, it was sepaated into two egions: a guad aea Ω g =[7.5, 22.5] [7.5, 22.5] and a buffe aea Ω b suounding the aea Ω g (see Figue 2.12). The study site Lagoa A contains 560 tees: 118 tees inside the egion Ω g and 442 tees inside the egion Ω b.

45 2.3 Results 41 Site Lagoa A y x Figue Study aea Lagoa A with 560 tees: 118 tees inside the guad aea Ω g (points) and 442 tees inside the buffe aea Ω b (cosses). A compaison of the deviance factos D A and D R shows that the pefomance of the Aea method is bette than to Ripley method (see Table 2.10). Study aea D R D A Lagoa A Table Deviance facto D R and D A obtained fo the study site Lagoa A.

46 42 Spatial Patten Analysis 2.4 Discussion The esults pesented in this Chapte shown some chaacteistics of the QC and NN Methods, Fist Ode Methods (Fische et al. 1922, Douglas 1975, Clak & Evans 1954, Cessie 1993) based on the Fist Ode Popety (mean numbe of points pe unit aea at any abitay location in the study egion). The QC Method (Fische et al. 1922, Douglas 1975) has at least two main limitations. This method educes all spatial infomation of the point patten into a single one-dimensional index. This chaacteistic implies that: (1) The spatial patten analysis can be pefomed only at a single scale. Because the method povides infomation about the intensity (numbe of events pe unit aea) of the spatial point patten using only a single paticula size quadat. Howeve, the choosing of the quadat size can be quite complicated, because it depends geneally on the scale of inteest in the spatial patten and/o the changing patten at the changing scales. (2) The analysis povides no infomation about the elative spatial position of the tee individuals inside the study site. In summay, the QC Method shows only the type of deviation fom point andomness (i.e. egulaity and/o clusteing) and all othe spatial infomation of the individuals is completely lost. In the same way, the esults show that the Neaest Neighbo Methods (heeinafte NN Method) (Fische et al. 1922, Clak & Evans 1954, Cessie 1993) also have impotant limitations. All infomation about individual point-to-point distances is lost and summaized in a mean. These methods conside only on the closest points (i.e. distance to the neaest neighbo fo each point in the patten) and the spatial infomation of the individuals at lage scales is completely lost. Futhemoe, while the indices can show the diection of deviation fom andomness (i.e., towad clumping and/o egulaity), the numeical behavio of many of these indices emains lagely unexploed. In summay, the QC and NN Methods show that the tees within the study site Lagoa Aaeclumped(seeTable 2.1 and Table 2.2 espectively), but the methods do not povide the detection of the scale at which this patten occus. This is a common limitation of both QC and NN methods. All infomation about the spatial localization of the tee individuals is completely lost. Recent studies in plant ecology eveal that positive and negative inteactions between individual tees may occu togethe at diffeent scales and detemine simultaneously the hoizontal and vetical stuctue of the plant community (Malkinson et al. 2003). Thus, to educe all spatial infomation of all scales into one single index, is a citical limitation of the QC and NN methods. Fo this, the Ripley K-function (Ripley 1977), a second ode method, was developed to ovewhelm some of these limitations. The main advantage of the Ripley K-Function in compaison the Fist Ode Methods (Fische et al. 1922, Douglas 1975, Clak & Evans 1954, Cessie 1993), such as QC and NN Methods is to peseve distance infomation and it pemits to analyze a point patten at a

47 2.4 Discussion 43 ange of scales and to detemine at which scales these points tend to be egula, clumped o CSR. In this expeiment, the analysis obtained by Ripley K-function confim that the spatial configuation of tees in Lagoa A ae clusteed but show in addition that clusteing occus at lowe and intemediate scales. This infomation was not possible to obtain using the QC and NN Methods. In this chapte I also povide a complete desciption of an Aea Based Edge Effect Coection Facto (heeinafte Aea Method) (Getis & Fanklin 1987, Besag 1977, Dale & Powell 2001) in ode to impove the sensibility of the Ripley K-Function to detect deviation fom andomness in spatial point pattens. Momentay, this method is only suitable to analyze the spatial configuation of a point patten within a ectangula study aea. Additionally, the maximal scale of the spatial analysis is esticted to a half of the shotest side of the study site (see Table 2.6). Howeve, the time pocessing simulation expeiment shows that the Aea Method is about eight times faste (see Table 2.6) than the Ripley Method fo edge coection (Ripley 1977, Diggle 1983). The bette time pefomance of the Aea Method in compaison to the Ripley method can be explained by compaing equations (2.3) and (2.30). Fo a fixed scale and n points, the calculation of the Ripley K-function using the Ripley edge coection method needs at n 2 opeations. In contast, the Aea Method needs only n opeations. This chaacteistic of the Aea Method enables theefoe moe simulations to be pefomed in calculating the simulation envelope. It thus impoves the statistical powe and the sensibility of the Ripley K-function in ode to detect clumping and egulaity (Yamada & Rogeson 2003). This advantage is also paticulaly impotant fo the analysis of lage data sets, educing the time pocessing of the analysis. The compaison of the simulation envelope widths shows that the pefomance of both methods is simila until =0.25 (consideing max=0.5), but the Aea Method pefoms bette pefomance fo >0.25 (see Figue 2.8, Figue 2.9 and Figue 2.10). This esult povides additional evidence of the geate statistical powe and sensibility of the Aea Method in ode to analyze spatial point pattens. The esults obtained with the guad aea expeiment show that spatial analysis pefomed with the Aea Method is bette o equivalent in compaison to the Ripley edge coection method (see Figue 2.7, Figue 2.8, Figue 2.9 and Figue 2.10). Both vitual expeiments (with computed and eal field data) demonstate, futhemoe, the geate pecision of the Aea Method in elation to the Ripley Method. Thus, I conclude that it is wothwhile applying this method when the study site is ectangula. Futue studies will povide a complete set of equations also suitable fo iegula study sites and no scale analysis constaints. Additionally, it is impotant to note that the AeaMethodcanbeappliedincombinationwithany spatial statistical method that equies the use of an edge effect coection facto. In the next Chapte, I pefom a moe detailed analysis of the study sites Lagoa A and Lagoa B, using the Ripley K-function, with the intention to detect the undelying ecological pocess detemining the spatial configuation of tee individuals hee.

48 44 Spatial Patten Analysis The algoithm fo the calculation of the K-function and the Aea Method utilized in this chapte was implemented in FORTRAN 95 and can be found in the Appendix.

49 Chapte 3 Spatial Patten Analysis - an Application 3.1 Intoduction In this chapte, I pefom a moe complete analysis of two eal data sets obtained fom two mangove foest stands Lagoa A and Lagoa B. They ae located on the coast of the notheasten Bazilian State of Paá (fo moe details, see Chapte 2). The two foest study sites ae located about 15 m apat nea the lagoon in the cental pat of the peninsula (see Figue 3.1). Site Lagoa A contains two species: Lagunculaia and Avicennia. The foest at Lagoa B is fomed by lage sized Lagunculaia, and a few Rhizophoa and Avicennia tees.. Both aeas ae inundated once a month duing vey high tides unde the influence of Caete Rive tidal egime. The inundation egime is not pecisely known but the fequency of inundation is pesumably lowe at Lagoa A than at Lagoa B, due to its lowe basin (Haum 2004). The pincipal aim of this chapte is to analyze the spatial configuations of the tees in these stands. I intepet these to make infeences about the undelying ecological pocesses which ae likely to be occuing within the study sites, such as competition o/and facilitation between the tees, seed dispesal (Stene et al. 1986, Baot et al. 1999), nuse-plant effects (Tielböge & Kadmon 2000), intaspecific competition (Kenkel 1998), intespecific competition (Baot et al. 1999), distubance (Dale 1999), hebivoe pessue (Jetsch et al. 1999), succession (Begon et al. 1976, Connell & Slatye 1977) and zonation (Roels 2001). These ecological pocesses ae impotant factos that detemine the spatial stuctue and the oganization of a community. 45

50 46 Spatial Patten Analysis - an Application Unde the assumption that the spatial heteogeneity of abiotic factos does not pedetemine plant distibution, studies in plant ecology have shown that a tendency to egulaity may be a esult of competition fo limited esouces such as wate, light and nutients (King & Woodel 1973). By contast, a tendency to clusteing may be an indication of facilitation pocesses, such as amelioative envionmental conditions (Mulle 1953, Haase et al. 1996), heteogeneous edaphic conditions (Couteon & Kokou 1997), local seed dispesal effects (Baot et al. 1999), stess gadients in the physical envionment (Malkison et al. 2003) o envionmental heteogeneity (Klaas et al. 2000). Unde the assumption that the spatial heteogeneity of abiotic factos does not pedetemine plant distibution, studies in plant ecology have shown that a tendency to egulaity may be a esult of competition fo limited esouces such as wate, light and nutients (King & Woodel 1973). By contast, a tendency to clusteing may be an indication of facilitation pocesses, such as amelioative envionmental conditions (Mulle 1953, Haase et al. 1996), heteogeneous edaphic conditions (Couteon & Kokou 1997), local seed dispesal effects (Baot et al. 1999), stess gadients in the physical envionment (Malkison et al. 2003) o envionmental heteogeneity (Klaas et al. 2000). In summay, egula and clumped pattens can be associated with competition and facilitation pocesses espectively. Recent studies show that positive (facilitation) and negative (competition) inteactions usually occu togethe simultaneously, exeting a combined affect on the stuctue of the plant community. The elative impotance of these pocesses depends on the intensity of the envionmental stess (Betness & Callaway 1994). It should be noted that the intepetation of the spatial point patten depends on the scale of the obsevation in compaison to the scale of the study site. Hee I assumed that vaiations at lowe scales can be attibuted to plant-plant inteactions and that lage scale vaiations ae due to envionmental heteogeneity (Pelissie & Goeaud 2001) o povide evidence of an invasion pocess, by a species new to the location (Goeaud et al. 1996). In Table 3.1 I pesent a summay of some ecological pocesses and thei possible associated spatial point pattens. In ode to analyze the spatial configuation of individual tees in the sites Lagoa A and B, I applied the Ripley K-function in combination with the Aea method intoduced in Chapte 2.

51 3.1 Intoduction 47 Figue 3.1. Coastal zone of nothe Bazil whee the study sites Lagoa A and Lagoa B (black dots) ae located.

52 48 Spatial Patten Analysis - an Application Spatial Patten Clusteing (+) Undelying Pocess Seed-dispesion effect Nuse-plant effect Succession Zonation Envionmental heteogeneity Regulaity (-) Intaspecific competition Intespecific competition Table 3.1. Summay of undelying pocess and an possible associated spatial point patten. (+) Positive and (-) negative inteaction Mangove Foest Mangove foests ae defined as associations of woody tees and bushes which pospe in mangal (mangove habitat), an inhospitable habitat between land and sea. But they can also occu in othe types of habitats (Hogath 1999). They ae elatively stable ecosystems dominated by only a few species (Tomlison 1986, Ricklefs & Latham 1993, Duke et al. 1998). A wide vaiety of plant species can be found in mangove habitat, but of the 110 ecoded species only about 54 species, belonging to 20 genea in 16 families, ae consideed "tue mangoves", that is, species that occu almost exclusively in mangove habitats and aely elsewhee (Hogath 1999). Mangove habitats ae constantly changing, gowing, eestablishing and egeneating themselves. The main chaacteistics that pemit mangove foests to suvive, occupy, dominate and stabilize tidal locations ae thei notable toleance to saltwate tidal conditions and the fact that they ae highly dispesive plants with floating popagules, which fequently display vivipay (Tomlison 1986, Duke 2001). They ae complex ecosystems that can be consideed at diffeent spatial scales. At tee level scale, they ae stuctually and physiologically well adapted to espond to the conditions in the immediate envionment, in paticula to physical factos such as salinity. But at lage scales, local vaiation in physical factos influences the oveall stuctue of the mangove foest (Hogath 1999).

53 3.1 Intoduction 49 Mangoves have a highly specialized method of popagation. Thei seeds geminate into seedlings while on the paent tee and afte an initial peiod of development, the seedlings fall down fom the paent tee to the sands below. They then can eithe spout o be caied with the tide to colonize othe locations. Some popagules emain viable fo peiods of weeks o a yea o moe (Rabinowitz 1978). Mangoves help to potect coastlines fom eosion, stom damage, and wave action. They pevent shoeline eosion by acting as buffes and tapping alluvial mateials, thus stabilizing land elevations by sediment accetion that balances sediment loss (Hogath 1999). Mangove ecosystems have taditionally been utilized by local populations fo the poduction of food, medicines, tannins, fuel wood, and constuction mateials. Fo millions of indigenous coastal esidents, mangove foests povide dependable basic livelihoods and sustain thei taditional cultues (Quato 2001). Mangoves ae almost exclusively topical and can be found between the latitudes of 32 degees noth and 38 degees south, along the topical coasts of Afica, Austalia, Asia, and the Ameicas. This distibution is an indication of a limitation by tempeatue and they aely occu outside the winte position of the 20 C isothem (Hogath 1999). In this study, I ae paticulaly inteested in the mangoves species found in Notheasten Bazilian State of Paa, the egion whee the sites Lagoa A and Lagoa B ae located. The main species found in at these sites ae Lagunculaia acemosa, Avicennia geminans and Rhizophoa mangle and in the following sections I biefly descibe these thee species Avicennia geminans Black mangove o Avicennia geminans occus in peiodically immesed and fully teestial envionments (see Figue 3.2). It toleates aibone salt and a degee of wate salinity, but favos fesh wate envionments. It pefes loamy o muddy substates, but toleates sand and it is also easonably toleant of cutting back and mild fost conditions. It does not gow on pop oots; athe it possesses pneumatophoes that allow its oots to beathe even when gowing in standing wate. It occus mainly in topical Atlantic egions whee it thives on sandy and muddy shoes. Like many othe mangoves, it epoduces by vivipay. Thei seeds ae cased inside a fuit until this falls into the wate to elease the geminated seedling. Avicennia expels absobed salt mainly fom its leathey leaves. It is widely distibuted along Atlantic coasts of topical Ameica and is found in Bemuda, the Bahamas, and the West Indies, in southeasten USA as fa as nothen Floida and southeasten Texas, and fom nothen Mexico southwads along the Atlantic Coast to Bazil. It is also found on the Pacific Coast fom Mexico to Ecuado including the Galapagos Islands, and as fa as nothwesten Peu, and on coasts of westen Afica (Little 1983, Kjefve & Laceda 1993, Hogath 1999). Fo additional infomation see, Table 3.2 and Table 3.3.

54 50 Spatial Patten Analysis - an Application Figue 3.2. Black mangove o Avicennia geminans Rhizophoa mangle Red mangove o Rhizophoa mangle (see Figue 3.3) can live in wate and in peiodically immesed envionments; it also occus as a fully teestial plant in well-hydated conditions. It toleates fesh, backish o full seawate but cannot adapt to maked changes in salinity. It favos fine sand o muddy substates but can suvive on couse substates. It is the most tempeatue sensitive of thee genea listed hee. It equies wame tempeatues and is also the most sensitive to cutting back. It geneally occus in intetidal aeas which ae inundated daily by the tides. Rhizophoa has a numbe of adaptations suited to this envionment, namely popagules that allow them to beath in an anaeobic envionment (Little 1983, Hogath 1999). Fo additional infomation, see Table 3.2 and Table 3.3. Figue 3.3. Red mangove o Rhizophoa mangle.

55 3.1 Intoduction Lagunculaia acemosa White mangove o Lagunculaia acemosa (see Figue 3.4) is a fully teestial plant which toleates aibone salt, but not highly saline wate. It is vey toleant of cutting back and is modeately to vey toleant of occasional fost conditions. It can be found on both coasts of topical Ameica, fom nothen Mexico to Bazil and Ecuado, including the Galapagos Islands and as fa as nothwesten Peu, as well as in the West Indies, Bemuda, in southen and cental Floida, and in West Afica fom Senegal to Cameoon (Little 1983, Hogath 1999). Fo additional infomation, see Table 3.2 and Table 3.3. Figue 3.4. White mangove o Lagunculaia acemosa. Species Weight (g) Length (cm) Floating (d) Longevity (d) Avicennia always 110S Rhizophoa Lagunculaia F and 31S 35S Table 3.2. Table 3.2 Chaacteistics of popagules of thee mangove species (adapted fom Rabinowitz 1978). (S) Salt wate conditions and (F) Fesh wate conditions.

56 52 Spatial Patten Analysis - an Application Species Shade toleance Salinity Avicennia Intoleant 100(MS) and <40(OG) Rhizophoa Intoleant 70(MS) Lagunculaia Intoleant 90(MS) Table 3.3. Ecological Chaacteistics of vaious mangove species. Salinity in ( 0 / 00 ). MS = Maximum poe wate salinity measued in the fields at sites whee the species was gowing, OG = Salinity fo optimum gowth based on cultue studies. Adapted fom Smith III (1991) Mangove foest evolution Ealy studies used a fou stage model to descibe the development of mangove stands: mangove fist establish themselves duing a colonization phase and continue though late phases of ealy development and matuity. Finally a new cycle of colonization begins duing the senescence stage. The duation of a complete cycle was estimated to be about yeas (Jimenez & Lugo 1985, Fomad et al. 1988). Late Duke (2001) updated this appoach by consideing gap dynamics explicitly. Howeve, ecent studies in mangove foest evolution show that the ealy development stage is much longe then assumed in this model. Menezes (2006) has poposed subdividing the ealy development stage in two stages, which he denominates "ealy development and "young foest" stages (see Table 3.4). Development Colonization Ealy Young Matue Senescence Stage development Foest Stand Density low to vey high medium low high high Biomass low medium medium high high to low Self-thinning minimal high high to modeate minimal modeate to low Size nomal L-shape L-shape nomal J-shape distibution Table 3.4. A peliminay model of stand development in mangoves. The foest collapses with senescence when the cycle esumes with e-colonization. Adapted fom Duke (2001), Silvetown & Doust (1993) and Menezes et al. (2004). The colonization stage stats with the establishment of popagules in gaps and on unoccu-

57 3.1 Intoduction 53 pied and damaged tidal aeas. Recuitment and gowth is fast, self-thinning is minimal and the density of plants inceases thoughout this stage. Lagunculaia acemosa is the dominant species at this stage. This occus because Lagunculaia pesents some chaacteistics of pionee species, including a low shade toleance and a high nutient use efficiency. Fo this eason, this species is often the fist colonize of newly ceated mud banks. Unde optimal light o nutient conditions, Lagunculaia ovetops Avicennia and Rhizophoa in tems of gowth ate (Lovelock & Felle, 2003). As soon as the conditions become suboptimal, Lagunculaia looses this initial advantage. This stage lasts about 4 yeas, until canopy closue is lagely achieved (Duke 2001, Bege et al. 2006, Menezes et al. 2006). The ealy development stage is chaacteized by vey intense self-thinning and a apid decline in density while the seedling bank is fomed unde the closed canopy. Duing this stage, the height of the stand inceases moe slowly until the canopy appoaches "site maximal canopy height. and/o appea in the stand. (Duke 2001, Menezes et al. 2006). This stage lasts about 5 yeas. It should be noted that this desciption is deived fom studies undetaken on the Acaao Peninsula (Bazil), whee the study sites Lagoa A and Lagoa B ae located (Bege 2006). In the young foest stage, the numbe of individuals continues to decease due to selfthinning effects. At this point thee two possibilities fo the futue development the stand: eithe a change in dominance fom Lagunculaia acemosa to Rhizophoa mangle (Ball 1980) o a change in dominance fom Lagunculaia acemosa to Avicennia geminans (Bege et al. 2006). The matue stand stage commences when the "site maximal canopy height is achieved. At this stage the tees stat to gain biomass while tee density continues to decease due to self-thinning effects. Neighbohood competition vaies between low and modeate values. The mean age of the tees vaies between yeas (Duke 2001, Menezes et al. 2006). The senescence stage stats when lage individuals begin to die standing o they fall ove. At this stage, tee density is expected to be low and self-thinning minimal. (Duke 2001). Duing this stage, the stand is dominated by few old and lage tees. Thee ae wide gaps in the canopy and a lack of egeneation. It cannot be consideed as an inteval development of the stand, because in this case, the whole foest collapses (Duke 2001, Menezes 2006). Analysis of the shape of the tee dbh histogam could povide a good indication of the stage of development a stand. Studies have shown that, initially, plant size (in this case, I am consideing dbh) in dense populations geneally has a nomal distibution, which quickly skews to an L-shape distibution with a many small individuals and a few lage ones. As the tees gow, the motality caused by the self-thinning may totally emove the smallest individuals, poducing a moe symmetical size distibution once again. In an advanced stage of the development, the numbe of small tees deceases and the stand is dominated by old and lage tees and thei dbh distibution pesents a J-shape (Silvetown & Doust 1993).

58 54 Spatial Patten Analysis - an Application 3.2 Results Lagoa A Site Lagoa A contains a total of 812 tees (including 252 dead tees) (see Figue 3.5). All tees (n = 812) y x Figue 3.5. All tees at stand site Lagoa A. (black coss) dead tee, (blue dot) Lagunculaia acemosa and (ed dot) Avicennia geminans. The size of dot is popotional to the dbh of Avicennia and Lagunculaia (thee s no infomation about the dbh of the dead tees). (scale in metes)

59 3.2 Results 55 Type mean(dbh) va(dbh) max(dbh) n n/n All 4.41* 14.11* 18.46* Avicennia geminans Lagunculaia acemosa Dead tees Table 3.5. Shot statistical summay of the mean stem diamete in beast height (dbh) fo Lagoa A site. (-) Thee s no infomation about the dead tees s dbh. (*) Excluding the dead tees. (dbh in cm). a b c dbh dbh dbh Figue 3.6. Lagoa A - Histogams showing the size class distibution of the mean stem diamete in beast height (dbh) in cm obtained fo (a) all tees (excluding dead tees), (b) Avicennia geminans and (c) Lagunculaia acemosa.(scaleincm)

60 56 Spatial Patten Analysis - an Application All tees (n = 812) L function y L() x Figue 3.7. (left) Spatial point patten elative to all tees of the stand Lagoa A and its (ight) espective L-function (black) and 99% simulation envelope (dashed ed). The simulation envelope was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations. Avicennia (n = 309) L function y L() x Figue 3.8. (left) Spatial point patten elative to the species Avicennia geminans within stand Lagoa A. Its (ight) espective L-function (black) and 99% simulation envelope (dashed ed). The simulation envelope was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations.

61 3.2 Results 57 Lagunculaia (n = 251) L function y L() x Figue 3.9. (left) Spatial point patten elative to the species Lagunculaia acemosa within stand Lagoa A. Its (ight) espective L-function (black) and 99% simulation envelope (dashed ed). The simulation envelope was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations. Dead tees (n = 252) L function y L() x Figue (left) Spatial point patten elative to dead tees within stand Lagoa A. Its (ight) espective L-function (black) and 99% simulation envelope (dashed ed). The simulation envelope was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations.

62 58 Spatial Patten Analysis - an Application a c e y x b L1() y x d L2() y x f L12() Figue The point pattens (a) and (c) epesents espectively the big tees (n=232) and small tees (n=328). (b) and (d) epesents thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesent the lage tees (blue) and small tees (ed). (f) epesents its bivaiate L-function (black) and 99% simulation envelope fo spatial independence hypothesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations.

63 3.2 Results 59 a c e y x b L1() y x d L2() y x f L12() Figue The point pattens (a) and (c) epesents espectively the lage Avicennia (n=100) and small Avicennia (n=209). (b) and (d) epesent thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesents the big Avicennia (blue) and small Avicennia (ed). (f) epesents its bivaiate L-function (black) and 99% simulation envelope fo spatial independence hypothesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations.

64 60 Spatial Patten Analysis - an Application a c e y x b L1() y x d L2() y x f L12() Figue The point pattens (a) and (c) epesents espectively the lage Lagunculaia (n=132) and small Lagunculaia (n=132). (b) and (d) epesents thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesents the big Lagunculaia (blue) and small Lagunculaia (ed). (f) epesents its bivaiate L-function (black) and 99% simulation envelope fo spatial independence hypothesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations.

65 3.2 Results 61 a c e y x b L1() y x d L2() y x f K1() K2() Figue The point pattens (a) and (c) epesents espectively the dead tees (n=252) and living tees (n=560). (b) and (c) epesents thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesents the dead tees (blue) and living tees (ed). (f) epesents its K 1 () K 2 () (black) and 99% simulation envelope fo andom labeling hyphotesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations.

66 62 Spatial Patten Analysis - an Application a c e y x b L1() y x d L2() y x f L12() Figue The point pattens (a) and (c) epesents espectively the species Avicennia (n=309) and Lagunculaia (n=251). (b) and (c) epesents thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesents the species Avicennia (blue) and Lagunculaia (ed). (f) epesents its bivaiate L-function (black) and 99% simulation envelope fo spatial independence hypothesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations. All tees in Lagoa A pesent clusteing at scales = 2 m and = 4 m and a cetain tendency to egulaity at scale = 0.20 m. This patten is pobably a esult of a supeposition of vaious ecological pocesses that occu at diffeent scales within the same community. Because of this, an intepetation of the oveall patten is not staightfowad. Moe infomation can be obtained by analysing the pattens exhibited by diffeent species and diffeent goups, as follows. (see Figue 3.7) Avicennia geminans tees exhibit clusteing at scale = 0.5 m (see Figue 3.8). Lagunculaia acemosa tees exhibit clusteing at diffeent scales, with a maximum at scale = 4 m (see Figue 3.9).

67 3.2 Results 63 Dead tees exhibit clusteing at lowe and intemediate scales, with maxima at scales =1mand=4m(seeFigue 3.10). Lage tees geneally exhibit CSR but with a tendency to egulaity at scale = 2 m and small tees tends to be clumped at lowe and intemediate scales, in paticula at scale = 4 m(see Figue 3.11). The spatial elation between the lage and small tees shows some tendency to epulsion at lowe scales (see Figue 3.11). Lage Avicennia tees tend to be clumped at scale = 1 m and small Avicennia tees tend to be clumped at scales = 1 m and = 3 m (see Figue 3.12). The spatial elation between lage and small Avicennia tees additionally pesents a tendency to epulsion at scale = 2 m (see Figue 3.12). Lage Lagunculaia tees tend to exhibit CSR and Lagunculaia tees exhibit clumping at all scales (with a maximum at scale = 4 m) (see Figue 3.13). The spatial elation between lage and small Lagunculaia tees exhibits spatial independence (see Figue 3.13). Living tees exhibit clumping at scales between = 1 m and = 4 m (see Figue 3.14). The spatial elation between dead and living tees exhibits aggegation at lowe scales (with a maximum at = 2 m) (see Figue 3.14). The spatial elation between Lagunculaia and Avicennia tees exhibits a tendency to epulsion at lowe scales (in paticula at scale = 0.50 m) (see Figue 3.15). In Table 3.6 and Table 3.7 I pesent a summay of the esults obtained fom the spatial statistical analyses descibed above. Lagoa A - Univaiate Case Type L() All Regulaity ( 0.2 m) Clumping at LS ( 1 mand 4m) Avicennia Clumping at 1 m. Lagunculaia Clumping at all scales ( 4 m) Dead tees Clumping at LS ( 1 mand 4m) Table 3.6. Summay of the univaiate L-function analysis obtained fo the site Lagoa A. (*) tendency and (**) in paticula. LS (Lowe Scales), IS (Intemediate Scales) and HS (Highe Scales).

68 64 Spatial Patten Analysis - an Application Lagoa A - Bivaiate Case Type L 1 () L 2 () L 12 () 1 Lage tees Regulaity Clumping at LS Repulsion at LS 2Smalltees ( 2 m) andis( 4 m) ( 2 m) 1LageAvi. Clumping Clumping Repulsion at LS 2SmallAvi. ( 1 m) ( 1 mand 3 m) ( 2 m) 1LageLag. Spatial Clumping at all scales Spatial 2SmallLag. independence ( 4 m) independence 1 Dead tees Clumping at LS Clumping at LS Aggegation at LS 2 Alive tees ( 1 mand 4 m) andis( 4 m) ( 2 m) 1 Avi. Clumping Clumping at all scales Repulsion 2 Lag. ( 1 m) ( 4 m) ( 1 m) Table 3.7. Summay of the univaiate and bivaiate L-function analysis obtained fo the site Lagoa A. small = (dbh 5 cm) and lage = (dbh>5 cm).(*) tendency and (**) in paticula. LS (Lowe Scales), IS (Intemediate Scales) and HS (Highe Scales).

69 3.2 Results Lagoa B Site Lagoa B contains a total of 543 tees (including 116 dead tees) (see Figue 3.16). All tees (n = 550) y x Figue All tees of the stand Lagoa B. (black coss) dead tee, (blue dot) Lagunculaia acemosa and (ed dot) Avicennia geminans and (geen dot) Rhizophoa mangle. The size of dot is popotional to the dbh of Avicennia and Lagunculaia (thee s no infomation about the dbh of the dead tees).

70 66 Spatial Patten Analysis - an Application Type mean(dbh) va(dbh) max(dbh) n n/n All 5.18* 25.54* 24.83* N= Avicennia geminans Lagunculaia acemosa Rhizophoa mangle Dead tees Table 3.8. Shot statistical summay of the mean stem diamete in beast height (dbh) fo Lagoa B site. (-) Thee s no infomation about the dead tees s dbh. (*) Excluding the dead tees. (dbh in cm). a b c dbh dbh dbh Figue Lagoa B - Histogams showing the size class distibution of mean stem diamete in beast height (dbh) in cm obtained fo (a) all tees (excluding dead tees), (b) Avicennia geminans and (c) Lagunculaia acemosa.

71 3.2 Results 67 All tees (n = 550) L function y L() x Figue (left) Spatial point patten elative to all tees of the stand Lagoa B and its (ight) espective L-function (black) and 99% simulation envelope (dashed ed). The simulation envelope was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations. Avicennia (n = 256) L function y L() x Figue (left) Spatial point patten elative to the species Avicennia geminans within stand Lagoa B. Its (ight) espective L-function (black) and 99% simulation envelope (dashed ed). The simulation envelope was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations.

72 68 Spatial Patten Analysis - an Application Lagunculaia (n = 171) L function y L() x Figue (left) Spatial point patten elative to species Lagunculaia acemosa within stand Lagoa B. Its (ight) espective L-function (black) and 99% simulation envelope (dashed ed). The simulation envelope was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations. Dead tees (n = 116) L function y L() x Figue (left) Spatial point patten elative to dead tees within stand Lagoa A. Its (ight) espective L-function (black) and 99% simulation envelope (dashed ed). The simulation envelope was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations.

73 3.2 Results 69 a c e y x b L1() y x d L2() y x f L12() Figue The point pattens (a) and (c) epesents espectively the lage tees (n=179) and small tees (n=255). (b) and (d) epesents thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesents the big tees (blue) and small tees (ed). (f) epesents its bivaiate L-function (black) and 99% simulation envelope fo spatial independence hypothesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations.

74 70 Spatial Patten Analysis - an Application a c e y x b L1() y x d L2() y x f L12() Figue The point pattens (a) and (c) epesents espectively the lage Avicennia (n=46) and small Avicennia (n=210). (b) and (d) epesents thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesents the big Avicennia (blue) and small Avicennia (ed). (f) epesents its bivaiate L-function (black) and 99% simulation envelope fo spatial independence hypothesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations.

75 3.2 Results 71 a c e y x b L1() y x d L2() y x f L12() Figue The point pattens (a) and (c) epesents espectively the lage Lagunculaia (n=129) and small Lagunculaia (n=42). (b) and (d) epesents thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesents the big Lagunculaia (blue) and small Lagunculaia (ed). (f) epesents its bivaiate L-function (black) and 99% simulation envelope fo spatial independence hyphotesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations.

76 72 Spatial Patten Analysis - an Application a c e y x b L1() y x d L2() y x f K1() K2() Figue The point pattens (a) and (c) epesents espectively the dead tees (n=116) and living tees (n=434). (b) and (d) epesents thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesents the dead tees (blue) and living tees (ed). (f) epesents its K 1 () K 2 () (black) and 99% simulation envelope fo andom labeling hyphotesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations.

77 3.2 Results 73 a c e y x b L1() y x d L2() y x f L12() Figue The point pattens (a) and (c) epesents espectively the species Avicennia (n=256) and Lagunculaia (n=171). (b) and (d) epesents thei L-function (black) and 99% simulation envelope fo CSR hyphotesis (dashed ed) espectively. The point patten (e) epesents the species Avicennia (blue) and Lagunculaia (ed). (f) epesents its bivaiate L-function (black) and 99% simulation envelope fo spatial independence hyphotesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations.

78 74 Spatial Patten Analysis - an Application a c e y x b L1() y x d L2() y x f K1() K2() Figue The point pattens (a) and (c) epesents espectively the species Lagunculaia (n=171) and dead tees (n=171). (b) and (d) epesents thei L-function (black) and 99% simulation envelope fo CSR hypothesis (dashed ed) espectively. The point patten (e) epesents the species Lagunculaia (blue) and dead tees (ed). (f) epesents its bivaiate L-function (black) and 99% simulation envelope fo spatial independence hypothesis (dashed ed). The simulation envelopes wee calculated via Monte Calo method (Besag 1977) with simulations.

79 3.2 Results 75 All tees in Lagoa B exhibit clusteing at lowe scales (with a maximum at scale = 1 m) (see Figue 3.18). Avicennia geminans tees exhibit clusteing at lowe and intemediate scales (with a maximumatscale=4m)(seefigue 3.19). Lagunculaia acemosa tees exhibit clusteing at lowe, intemediate and highe scales (with a maximum at scale = 1 m) (see Figue 3.20). Dead tees exhibit clusteing at lowe, intemediate and highe scales (see Figue 3.21). Lage tees exhibit clusteing at scale = 1 m and small tees exhibit clusteing at intemediate scales (with a maximum at scale = 4 m) (see Figue 3.22). The spatial elation between the lage and small tees exhibits a tendency to epulsion at scale = 6 m (see Figue 3.22). Lage Avicennia tees exhibit CSR. Small Avicennia tees exhibit clusteing at intemediate scales (with a maximum at scale = 4 m) (see Figue 3.23). The spatial elation between lage and small Avicennia tees exhibits a tendency to aggegation at lowe and intemediate scales (with a maximum at = 4 m) (see Figue 3.23). The spatial elation between lage and small Lagunculaia tees exhibits aggegation at scale = 1 m (see Figue 3.24). Living tees exhibit clusteing at lowe scales (with a maximum at = 1 m) (see Figue 3.25). The spatial elation between the dead and living tees exhibits aggegation at intemediate and highe scales (see Figue 3.25). The spatial elation between Avicennia and Lagunculaia tees exhibits epulsion at scale = 1 m and a tendency to epulsion at scale = 4 m (see Figue 3.26). The spatial elation between Lagunculaia and dead tees exhibits aggegation at lowe scales (see Figue 3.27). In Table 3.9 and Table 3.10 I pesent a summay of the esults obtained fom the spatial statistical analyses descibed above.

80 76 Spatial Patten Analysis - an Application Lagoa B - Univaiate Case Type L() All Clumping at lowe scales ( 1 m) Avicennia Clumping LS and IS ( 4 m) Lagunculaia Clumping at all scales ( 1 m) Dead tees Clumping at all scales Table 3.9. Summay of the univaiate L-function analysis obtained fo the site Lagoa B. (*) tendency and (**) in paticula. LS (Lowe Scales), IS (Intemediate Scales) and HS (Highe Scales). Lagoa B - Bivaiate Case Type L 1 () L 2 () L 12 () 1Lagetees Clumping Clumping at IS Repulsion 2 Small tees ( 1 m) ( 4 m) ( 6 m) 1LageAvi. CSR Clumping at IS Aggegation at LS 2SmallAvi. patten ( 4 m) andis( 4 m) 1LageLag. Clumping at LS Clumping at LS Aggegation 2SmallLag. andis( 2 m) ( 2 m) ( 1 m) 1Deadtees Clumping at Clumping at LS Aggegation at LS 2Alivetees all scales ( 1 m) and IS 1 Avi. Clumping at LS Clumping at all scales Repulsion 2 Lag. andis( 4 m) ( 1 m) ( 1 m) 1Lag. Clumping at all scales Clumping at Aggegation at LS 2Deadtees ( 1 m) all scales ( 1 m) Table Summay of the univaiate and bivaiate L-function analysis obtained fo the site Lagoa B. small = (dbh 5 cm) and lage = (dbh>5 cm). (*) tendency and (**) in paticula. LS (Lowe Scales), IS (Intemediate Scales) and HS (Highe Scales). 3.3 Discussion The esults in Table 3.5 and Table 3.8 povide some impotant infomation about the pobable stages of the development of mangove foests at sites Lagoa A and Lagoa B. It should be noted that is vey difficult to obtain this infomation about a mangoves foest. The poblem is that mangoves show a high degee of plasticity in esponse to envionmental conditions. If conditions ae hash, the tees gow moe slowly. Thus, tees can be small because they ae young o because of poo conditions fo gowth in the stand. The same agument applies to the succession stages (Hogath 1999). Moeove I have no infomation about the inundation egimes and abiotic conditions at these sites. Fo these easons, the

81 3.3 Discussion 77 discussion pesented below can be no moe that a peliminay intepetation of pobable undelying ecological pocesses occuing at sites Lagoa A and Lagoa B. Fou points can be made: Fist, tee density at the sites Lagoa A and Lagoa B is 0.90 ind/m2 and 0.61 ind/m2 espectively (see Table 3.5 and Table 3.8). Silvetown & Dout (1993) show that the density of the tees geneally tends to decease duing the development of a foest stand,, due to self-thinning effects. Second, some individuals of Rhizophoa mangle ae pesent at Lagoa B and by contast, none wee found at Lagoa A (see Table 3.5 and Table 3.8). Ball (1980) shows that Rhizophoa mangle geneally colonizes a mangove foest only in elatively advanced stages of the development. Thid, mean(dbh) of tees at Lagoa A and Lagoa B pesent = 4.41 cm and 5.18 cm espectively (see Table 3.5 and Table 3.8). Jimenez et al. (1985) and Fomad et al. (1988) show that, duing the development of a foest stand, the mean dbh of the tees geneally inceases. Finally, the sites Lagoa A and Lagoa B contain the same popotion of Lagunculaia acemosa tees in the stands (31%). Howeve, the popotion of Avicennia geminans at sites Lagoa A and Lagoa B is 38% and 47% of espectively (see Table 3.5 and Table 3.8). Duke (2001) and Menezes (2006) show that, duing the development of a stand, Avicennia geminans is the second species to establish itself. Taken togethe, the esults and thei intepetation pesented above povide a stong indication that the mangove foest at Lagoa A is at an ealie stage of development than at site Lagoa B. Site Lagoa A contains a high popotion of Lagunculaia acemosa and no specimens of Rhizophoa mangle (see Table 3.5). The dbh histogam calculated fo all tees in this site pesents a L-shape distibution (see Figue 3.6). On this basis, I sumise that the foest at site Lagoa A could be at an ealy development stage (Ball 1980). Site Lagoa B also contains a high popotion of Lagunculaia acemosa, but in contast to the site Lagoa A, it also contains some individuals of Rhizophoa mangle, which is stating to colonize the stand (see Table 3.8). The dbh histogam calculated fo all tees in this site pesents a L-shape distibution. On this basis, I sumise that the foest at site Lagoa B is also at an ealy development stage. In summay, the sites Lagoa A and Lagoa B seem to be in the same stage of development, but Lagoa B seems to be moe advanced in elation to the site Lagoa A. Now I move on to conside the undelying ecological pocesses at the two sites, basing ou infeences on the statistical analysis of the spatial configuation of the tees in sites pesented in Chapte 2. It is impotant to note that these methods have some limitations. They assume that the spatial pattens being analyzed ae homogeneous. The hypothesis of homogeneity means that the second-ode chaacteistics of a point patten ae invaiant unde tanslation and otation. But it is well-known that heteogeneity is common in natue and it is unlikely that the sites Lagoa A and Lagoa B ae exceptions to this ule.

82 78 Spatial Patten Analysis - an Application The spatial configuation of all tees in sites Lagoa A and Lagoa B is pobably a esult of the supeposition of diffeent ecological pocesses occuing at diffeent scales within the same community (see Figue 3.7 and Figue 3.18). In ode to ty to isolate and identify these diffeent pocesses, I pefomed univaiate and bivaiate L-function analyses, sepaating the tees into diffeent species and goups. In site Lagoa A, Avicennia geminans show clusteing at lowe scales (see Figue 3.8). This could be eithe a seed dispesal (Stene et al. 1986) o a nuse-plant effect (Tielböge & Kadmon 2000). By contast, Avicennia geminans in site Lagoa B exhibits clusteing at intemediate scales (see Figue 3.19). This could be esult of envionmental heteogeneity within the stand (Klaas et al. 2000). Lagunculaia acemosa in site Lagoa A exhibits clusteing at highe scales (see Figue 3.9). On the othe hand, Lagunculaia acemosa in site Lagoa B exhibits clusteing at intemediate scales (see Figue 3.20). These esults could be the esult of spatial envionmental heteogeneity within the stands (Klaas et al. 2000). Although the L-function has no spatial esolution, visually it is possible to make out lage egions with a high density of Lagunculaia acemosa tees, both in Lagoa A (see Figue 3.9) and Lagoa B (see Figue 3.20). The indication of clumping at the scale = 4 m in Lagoa A is esult of a cluste located at the bottom ight of the site (see Figue 3.9). Dead tees in site Lagoa A don t exhibit heteogeneity (see Figue 3.10). This means that the death of tees at this site has been a homogeneous pocess. By contast, dead tees of the site Lagoa B exhibit clusteing at highe scales (see Figue 3.21). This could be a esult of envionmental heteogeneity (Klaas et al. 2000). Pobably, the death of the tees is a pocess that has occued at specific locations within this site. The egula patten of lage tees in site Lagoa A (see Figue 3.11) couldbeaesult of a competition effect (Wiegand & Moloney 2004). In contast, the small tees pesent clusteing (see Figue 3.11). The spatial elationship between the lage and small tees (see Figue 3.11) indicates pobably the existence of light gaps (Duke 2001), whee the small tees tends to occupy the space that exists between the big tees. In contast, the lage tees in site Lagoa B exhibit clusteing, and small tees in the Lagoa B exhibit the same spatial patten (see Figue 3.22). But the spatial elationship between these lage and small tees exhibits epulsion at intemediate scales (see Figue 3.22). In summay, in site Lagoa B, tees of the same goup tends to fom clustes, but thee is a tendency to epulsion between tees of diffeent goups. That could be esult of a succession pocess (Begon et al. 1976, Connel & Slatye 1977) o invasion by a species new to the location (Goeaud et al. 1996). Lage and small Avicennia in site Lagoa A exhibit clumping at lowe scales (see Figue 3.12). But in contast, the spatial elationship between the lage and small Avicennia pesents some tendency to epulsion at lowe scales (see Figue 3.12). This could be a indication of inta-specific competition (Kenkel 1988). In site Lagoa B, the lage Avicennia tees exhibit CSR (see Figue 3.23). On the othe hand, small tees exhibit clumping at intemediate scales (see Figue 3.23). The spatial elationship between lage and small Avicennia tees (see Figue 3.23) shows some tendency to aggegation, in paticula at

83 3.3 Discussion 79 an intemediate scale, poviding an indication of envionmental heteogeneity within the stand (Klaas et al. 2000). In site Lagoa A, lage Lagunculaia exhibit CSR and small Lagunculaia exhibit clumping (see Figue 3.13). This could be esult of envionmental heteogeneity in the stand (Klaas et al. 2000). Additionally, these lage and small Lagunculaia tees exhibit spatial independence at all scales (see Figue 3.13). In site Lagoa B, lage Lagunculaia exhibit clusteing at intemediate scales (see Figue 3.24). This could be indication of envionmental heteogeneity within the stand (Klaas et al. 2000). Lage and small Lagunculaia pesents aggegation at lowe scales (see Figue 3.24). This could be an indication of nuse-plant effect (Tielböge & Kadmon 2000) o a seed-dispesion effect (Stene 1986). In site Lagoa A, dead and living tees pesent clumping patten at lowe scales (see Figue 3.14). But the spatial elationship between dead and living tees shows aggegation at lowe scales (see Figue 3.14). The death of the tees has pobably occued homogeneously within this site. In site Lagoa B, dead and living tees (see Figue 3.25) show indications of envionmental heteogeneity within the stand (Klaas et al. 2000). In contast with site Lagoa A, the death of the tees has not occued homogeneously within this site. Visually, it is easy to notice that dead tees seem to occu moe fequently in aeas with a high density of Lagunculaia tees (see Figue 3.27). The spatial elation between Lagunculaia and dead tees shows aggegation at lowe scales, confiming ou hypothesis. In the sites Lagoa A and Lagoa B, the spatial elationship between the species Avicennia geminans and Lagunculaia acemosa exhibits a tendency to epulsion at lowe and intemediate scales (see Figue 3.15 and Figue 3.26 espectivelly). This could be an indication of inte-specific competition occuing within the stand (Baot et al. 1999), succession (Begon et al. 1976, Connel & Slatye 1977) o evidence of invasion by a new species (Goeaud et al. 1996). In summay, analysis of the spatial configuation of individual tees at the sites Lagoa A and Lagoa B shows few similaities between the two sites, despite the shot distance between them (about 15 m). This could be a esult of the diffeent inundation conditions occuing at these sites. The fequency of inundation is pesumably lowe at Lagoa A than at Lagoa B, due to its lowe basin (Haum 2006). Thee ae moe indications of spatial heteogeneity at site Lagoa B that at Lagoa A. This could be an indication of geate envionmental heteogeneity at Lagoa B (Klaas et al. 2000). This hypothesis is difficult to pove, because I have no infomation about the abiotic conditions at the study sites Lagoa A and Lagoa B, but the spatial configuation exhibited by the tees could be a indication of envionmental heteogeneity and so I cannot exclude this possibility. The esults povide an indication of a pobable succession pocess (Begon et al. 1976, Connel & Slatye 1977) occuing among the Avicennia and Lagunculaia tees within sites Lagoa A and Lagoa B. If I conside that mangove foests ae in a continuous pocess of gowth and constantly establishing and enewing themselves (Duke 2001), the hypothesis of succession occuing among the individuals of species Avicennia and Lagunculaia within

84 80 Spatial Patten Analysis - an Application the stands is a plausible one. The esults suggest a cetain tendency among Lagunculaia and Avicennia tees to occupy specific egions of the stand (i.e they exhibit epulsion). But the Ripley K-function cannot detect this because it has no spatial esolution. That is a limitation of the Ripley K-function. The method povides infomation about the scales which the ecological pocesses occu, but no infomation about whee these pocesses occu. Despite the heteogeneity pesented within sites Lagoa A and Lagoa B and the inability of the Ripley K-function (hypothesis of heteogeneity) to povide infomation about this, the Ripley K-function did povide impotant infomation about the undelying ecological pocesses occuing in these stands. Late in Chapte 4, I pesent a methodology that can be used in combination with the standad spatial statistical methods applied to spatial patten analysis (Ripley K-function, fo example), in ode to ovecome these limitations and povide, unde cetain conditions, scale-spatial infomation about the ecological pocesses that occu in the stand.

85 Chapte 4 Object Patten Analysis 4.1 Intoduction In plant ecology, each individual plant is mapped as a point with the Catesian coodinates (x, y) epesenting the cente of thei stems. But, depending on the dimensions of the individuals in elation to the scale that I want to analyze, this tansfomation can cause poblems. To analyze the spatial configuation of individuals (tees, coal, etc...) using taditional statistical spatial methods (quadat counts and the Ripley K-function, fo example), each individual within the study site Ω is epesented as a point (x, y) Ω R 2,fothetwo dimensional case. Poblem can aise due to the loss of infomation duing the tansfomation, which involves the following steps: The fist step T 1 epesents a thee dimensional plant individual as a two dimensional abstaction, epesenting only its stem and cown. The second step T 2 tansfoms this two dimensional epesentation into a single point (see Figue 4.1). Figue 4.1. Steps of a tansfomation of a tidimensional object (tee) tee into a point. (left) Real tees, (middle) bidimensional abstaction of a eal tee with cown (geen) and stem (bown) and (ight) point patten epesenting the bidimensional abstaction. The limitations of this pocedue whic consides a thee dimensional plant individual as a point, affect the intepetation of esults obtained via standat spatial methods, which can not coespond to what is eally happening at the study site. Fo example, this pocedue can indicate egulaity at lowe scales instead of a significant small-scale aggegation (Simbeloff 1979, Pentice & Wege 1983) (see Figue 4.2). 81

86 82 Object Patten Analysis Figue 4.2. The cicle-to-point tansfomation T 2 indicates egulaity at lowe scales, instead of small-scale aggegation. One way of minimizing the effects of this limitation, could be to utilize the definition of functional scale 4.1 adopted by Malkinson et al (2003). The lowe bounday of the functional scale can be estimated by calculating the mean plant bh of the plants in the community. Then, only the ecological pocesses that occu above this scale should be consideed. Fo example, tee inteactions would be consideed as occuing at distances of between one and ten metes (Stoyan & Penttinen 2000). Howeve this can lead to infomation being lost due to the failue to take of account of infomation about inteactions at smalle scales. Futhemoe, Wiegand et al. (2006) poposed a methodology, based on a gid-based appoach, to pefom the analysis of spatial configuation of objects of finite size and iegula shape. The method basically consists to discetize each individual in a gid and utilizes gidbased vesions of the bivaiate functions K 12 (),L 12 () and g 12 () (Wiegand & Moloney 2004) to analyze the spatial elationship of these tee individuals within a study site. Similaly, the main objective of this chapte is to povide a suitable method to pefom spatial analysis of individuals, consideing these individuals as a cicula objects, athe than points. The idea behind the method is to appoximate each individual as a cicle and to pefom the spatial analysis consideing these individuals as cicula objects. Thee exists at least two basic diffeences among the method poposed by Wiegand et al. (2006) and the method poposed in this chapte. The fist discetizes the individuals in a gid and do not pemits the ovelaping among these individuals. On the othe hand, ou method epesents each individual analytically as a cicle and pemits the ovelaping between these objects. Of couse, these method is only applicable if the shape of the analysed individuals ae appoximately cicula. The methodology was applied to the stand sites Lagoa A and Lagoa B and compaed with the esults obtained in the Chapte 2. Additionally, I want to know if the method has sufficient sensibility to detect a pobable succession pocess occuing among the tees of species Avicennia geminans and Lagunculaia acemosa within these stands The scales at which inteactions among plants occu in each study site.

87 4.2 Method Method The idea was to adapt the Ripley K-function to analyze the spatial patten of objects (cicles, in ou case). This pocedue consides the spatial distibution of cicula objects, athe than points. Hee c i ( i ) and c j ( j ) ae defined espectively as a cicle of adius i centeed at a point p i and a cicle of adius j centeed at a point p j and c i () is a seach cicle with adius centeed at a point p i. These cicles ae located inside a study egion Ω. (see Figue 4.3). Figue 4.3. Two cicula objects c i ( i ) and c j ( j ) and a seach cicle c i () inside a study egion Ω Univaiate Analysis IdefineK c (), a function adapted to pefom spatial analysis of cicula objects, as K c ()=μ 1 E[A ij ()] = μ 1 1 n n w i () n i=1 j=1 A ij (), (4.1) whee w i () is an edge effect coection facto based on aea. The paamete μ is the specific aea. It is the aea of the cicula objects pe unit of aea of the study egion unde consideation (Cessie 1991). It is defined as μ = A c A Ω, (4.2)

88 84 Object Patten Analysis whee A c is the aea of all cicles inside the study egion Ω and A Ω is the aea of the study egion Ω. ThefunctionA ij (), is the aea of the inteception between the seach cicles c i () and the cicula object c j ( j ), sepaated by a distance d, and it is defined as ( ) ( ) d A ij ()= 2 j cos j 2 d + 2 cos j 2d j 2d 2 1 ( d + j )(d + j )(d j + )(d + j + ) 2. (4.3) The idea of the method is to estimate the aea of the objects contained within a distance of an abitay cicula object inside egion Ω and compae this esult with the expected aea, consideing that μ is homogeneous inside Ω o μ(x, y) =μ, (x,y) Ω (see Figue 4.4). Figue 4.4. Estimating the expected specific aea within a distance of an abitay cicula object of the study egion Ω. The expected intecepted aea within a cicle of adius is Substituting equation (4.4) in the equation (4.1) I obtain E[A ij ()] = π 2 μ. (4.4) E[K c ()] = μ 1 E[A ij ()] = μ 1 π 2 μ = π 2. (4.5) If the estimated aea fo a fixed scale is geate than the expected aea, the cicula objects exhibit aggegation at this scale. If the aea fo a fixed is smalle than the expected aea, they exhibit epulsion. The intepetation of the K c () is show in Table 4.1.

89 4.2 Method 85 K c ()=π 2 K c ()>π 2 K c ()<π 2 CSR Clusteing Repulsion Table 4.1. Intepetation of the K c () function. Now I popose a modified function L c () to nomalize the function K c (). Itisdefinedas and the intepetation is shown in Table 4.2 below. L c ()= K c(), (4.6) π L c ()=0 L c ()>0 L c ()<0 CSR Clusteing Repulsion Table 4.2. Intepetation of the L c () function. Seveal studies show that the classical cumulative Ripley K-function can confuse effects at lage distances with effects at shot distances (Getis & Fanklin 1987, Condit et al. 2000, Revilla & Palomaes 2002). The function K c () has the same limitation. One way of avoiding this poblem is to use ings, athe than cicles (Wiegand & Moloney, 2004). Than I define a function R ε () (heeinafte object ing method) as R ε ()=μ 1 1 n n n w i ( + ε/2)a ij ( + ε/2) i=1 j=1 w i ( ε/2)a ij ( ε/2), (4.7) whee μ is the specific aea, n is the numbe of cicula objects, w i is an edge effect coection factobasedonaeaanda ij is the function defined in equation (4.3). The expected intecepted aea fo the ing in Figue 4.5 is defined as E[A ij ( + ε/2) A ij ( ε/2)] = [π( + ε/2) 2 π( ε/2) 2 ]μ = = πμ( 2 + ε + ε 2 /4 2 + ε ε 2 /4) = 2πμε, (4.8) and substituting in the equation (4.5) I obtain E[R ε ()] = μ 1 π2με =2πε. (4.9) The intepetation of the R ε () function is shown in Table 4.3 below.

90 86 Object Patten Analysis R ε ()=2πε R ε () > 2πε R ε () < 2πε CSR Clusteing Regulaity Table 4.3. Intepetation of the function R ε (). The nomalization of the function R ε () is obtained as and its intepetation is shown in Table 4.4. R ε()= R ε() 2πε, (4.10) Figue 4.5. Estimating the expected specific aea inside aing. R ε()=1 R ε() > 1 R ε() < 1 CSR Clusteing Regulaity Table 4.4. Intepetation of the function R ε(). The function R ε() is simila to the pai-coelation g() associated to the K-Ripley function K() and it can be defined also as R ε()= 1 2π dk c(). (4.11) d

91 4.2 Method Bivaiate Analysis In a simila way, I define the bivaiate vesion of the function R ε (). Fist of all, I have to define the function R ε 12 (). This function estimates the aea of the objects of type 2 contained within a distance of an abitay cicula object of type 1 inside egion Ω. It is defined as R n 1 n 2 ε ()=μ 2 w i ( + ε/2)a ij( + ε/2) n 1 i=1 j=1 w i ( ε/2)a ij ( ε/2), (4.12) whee n 1 and n 2 ae the numbe of objects of type 1 and type 2 espectively, w i is an edge coection facto based on aea, A ij is the function defined in the equation (4.3) and μ 2 is the specific aea of the objects of type 2. It is defined as μ 2 = A c 2 A Ω, (4.13) whee A c 2 and A Ω ae the aea of the cicles of type 2 and the aea of the study site espectively. Similaly, I define the function R ε 21 (). This function estimates the aea of the objects of type 1 contained within a distance of an abitay cicula object of type 2 inside egion Ω. Itisdefinedas R n 2 n 1 ε ()=μ 1 w i ( + ε/2)a ij( + ε/2) n 2 i=1 j=1 w i ( ε/2)a ij ( ε/2), (4.14) whee n 1 and n 2 ae the numbe of objects of type 1 and type 2 espectively, w i is an edge coection facto based on aea, A ij is the function defined in the equation (4.3) and μ 1 is the specific aea of the objects of type 1. It is defined as μ 1 = A c 1 A Ω, (4.15) whee A c 1 and A Ω ae the aea of the cicles of type 1 and the aea of the study site espectively. Now I define the bivaiate vesion of the R ε () function as Rˆε B = n1 R n ε 2R ε, (4.16) n 1 + n 2

92 88 Object Patten Analysis whee n 1 and n 2 ae the numbe of objects of type 1 and type 2 espectively. The intepetation of the functions RˆεB is simila to the function R ε () and its nomalization is obtained by and its intepetation is shown in Table 4.5. R ε B ()= Rˆε B () 2πε, (4.17) R ε B ()=1 R ε B () > 1 R ε B () < 1 CSR Aggegation Repulsion Table 4.5. Intepetation of the function R B ε(). It is inteesting to notice that the expected values of the functions R ε () and Rˆε B do not depend on the value of the paametes μ, μ 1 and μ Simulations Envelope In ode to detect if a tendency to clusteing/aggegation o egulaity/epulsion is statistically significant, I have to compae the obseved function value obseved with an adequate null model. Now I define the ZOI (Zone of influence) of each tee as a cicula zone suounding each tee, within which a tee influences its neighbos and is influenced by its neighbos (Bege & Hildenbandt 2000). The equation that defines the adius of the ZOI is R ZOI = a bh b, (4.18) whee bh stands fo the stem adius at beast height in metes. The paametes a and b ae scaling paametes specific fo each specie of tee. In this case, I use the same paametes, a = and b = 0.654, fo both species Avicennia geminans and Lagunculaia acemosa, following Bege & Hildenbandt (2000). The object patten model used to obtain the simulation envelope via Monte Calo method (Besag 1977) fo the univaiate case is vey simila to a soft coe model (Tomppo 1986). Fist I have to simulate a CSR point patten with the same numbe of objects as the study site and the adii of the objects having a nomal distibution N(R ZOI,va(R ZOI )),wheer ZOI and va(r ZOI ) ae espectively the mean R ZOI and the vaiance of R ZOI fo the obseved object patten (Goeaud et al. 1996). Heeinafte, I define this model as Model I. The simulation of the envelope in the bivaiate case is vey simila and it is obtained by simulating a CSR point patten with the same numbe of the objects of type 1 and type 2.

93 4.3 Results 89 The adii of type 1 objects has nomal distibution N(R ZOI1, va(r ZOI1 )) and the adii of type 2 objects has a nomal distibution N(R ZOI2, va(r ZOI2 )), wheer ZOI1 and R ZOI2 ae the means R ZOI of type 1 and type 2 objects and va(r ZOI1 ) and va(r ZOI2 ) ae the vaiances of R ZOI of type 1 and type 2 objects espectively. Heeinafte, I define this model as Model II. Infomation about the paametes used fo the calculation of the simulation envelopes can be found at the Table 4.6 and Table Results Now I apply the methodology to the datasets Lagoa A and Lagoa B, obtained fom the study sites which have aleadybeen descibed in Chapte 2. But in this case, I conside only the tees with dbh > 0. I also exclude the dead tees, because lack of infomation about the dbh fo goup. It is also impotant to note that, fo this analysis of these two study sites, I applied only the ing functions R ε() and R ε B () Lagoa A The Table 4.6 and Figue 4.6 povides a summay of the basic statistics and shows the esulting histogam fo each goup analyzed in this simulation expeiment. Goup n R ZOI va(r ZOI ) min(r ZOI ) max(r ZOI ) All Small Lage Avic Lag Small Avic Small Lag Lage Avic Lage Lag Table 4.6. Basic statistics fo Lagoa A. R ZOI, min(r ZOI) andmax(r ZOI) inm and va(r ZOI) inm 2.

94 90 Object Patten Analysis a b c Fequency R ZOI (m) R ZOI (m) d Fequency R ZOI (m) g Fequency Fequency e Fequency R ZOI (m) h Fequency Fequency R ZOI (m) f Fequency R ZOI (m) i Fequency R ZOI (m) R ZOI (m) R ZOI (m) Figue 4.6. Histogam calculated fo the R ZOI distibution elative to (a) all tees, (b) small tees (dbh<5 cm), (c) lage tees (dbh 5 cm),(d) Avicennia geminans,(e)smallavicennia (dbh<5 cm), (f) lage Avicennia (dbh 5 cm), (g) Lagunculaia acemosa,(h) smalllagunculaia (dbh<5 cm) and (i) lage Lagunculaia (dbh 5 cm).

95 4.3 Results 91 a b y(m) R ε () x(m) Figue 4.7. (a) Object patten elative to all tees within study site Lagoa A. (b) Object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model I hypothesis with 200 simulations. a b y(m) R ε () x(m) Figue 4.8. (a) Object patten elative to Avicennia geminans within study site Lagoa A. (b) Object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model I hypothesis with 200 simulations.

96 92 Object Patten Analysis a b y(m) R ε () x(m) Figue 4.9. (a) Object patten elative to Lagunculaia acemosa within study site Lagoa A. (b) Object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model I hypothesis with 200 simulations. a b y(m) R B ε() x(m) Figue (a) Object patten elative to small (dbh<5 cm) (ed) and lage (dbh 5 cm) (blue) tees within study site Lagoa A. (b) Bivaiate object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model II hypothesis with 200 simulations.

97 4.3 Results 93 a b y(m) R B ε() x(m) Figue (a) Objectpattenelative to Avicennia geminans (ed) and Lagunculaia acemosa (blue) tees within study site Lagoa A. (b) Bivaiate object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model II hypothesis with 200 simulations. a b y(m) R B ε() x(m) Figue Object patten elative to small (dbh<5 cm) (ed) and lage (dbh 5 cm) (blue) Avicennia tees within study site Lagoa A. (b) Bivaiate object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model II hypothesis with 200 simulations.

98 94 Object Patten Analysis a b y(m) R B ε() x(m) Figue Object patten elative to small (dbh<5 cm) (ed) and lage (dbh 5 cm) (blue) Lagunculaia tees within study site Lagoa A. (b) Bivaiate object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model II hypothesis with 200 simulations. The patten pesented by all tees shows a tendency to egulaity at scales 1,5and8 m. Howeve they show a tendency to clusteing at loi and intemediate scales (see Figue 4.7). The patten pesented by the goup Avicennia geminans shows egulaity at scale 2 m. This egula patten at loi scales could be associated with competition between the tees (King & Woodel 2004) (see Figue 4.8). The patten pesented by the goup Lagunculaia acemosa shows clusteing at scale 3m and at lage scales. This esult could be an indication of envionmental heteogeneity (Klass et al. 2000) (see Figue 4.9). The spatial elation between individuals of goups of small (dbh < 5 cm) and lage (dbh 5 cm) tees exhibits some tendency to clusteing at lowe scales. Howeve, the patten shows some tendency to epulsion at scales 5 and 8 m. This could be an indication of intaspecific competition (Kenkel 1998) and/o light gaps (Duke 2001) (see Figue 4.10). The spatial elation between individuals of species Avicennia geminans and Lagunculaia acemosa exhibits epulsion at scales 5 m. Again, this could be a indication of succession (Begon et al. 1976) and/o intespecific competition (Begon et al. 1976, Connel & Slattie 1977) (see Figue 4.11). The spatial elation between individuals of goups of small (dbh < 5 cm) and lage (dbh 5 cm) Avicennia geminans exhibits some tendency to epulsion at scales 2 and 4 m. This could be an indication of inta-specific competition (Kenkel 1988) (see Figue 4.12).

99 4.3 Results 95 The spatial elation between individuals of goups of small (dbh<5 cm) and lage (dbh 5 cm) Lagunculaia acemosa exhibits aggegation at intemediate scales and epulsion at lage scales. This esult could be a indication of eithe inta-specific competition (Kenkel 1988) and/o envionmental heteogeneity (Klass et al. 2000) Lagoa B The Table 4.7 and Figue 4.14 pesent a summay of the basic statistics and the histogam calculated of each analyzed goups at this expeiment espectively. Goup n R ZOI va(r ZOI ) min(r ZOI ) max(r ZOI ) All Small Lage Avic Lag Small Avic Small Lag Lage Avic Lage Lag Table 4.7. Basic statistics fo Lagoa B. R ZOI, min(r ZOI) andmax(r ZOI) inm and va(r ZOI) inm 2.

100 96 Object Patten Analysis a b c Fequency R ZOI (m) R ZOI (m) d Fequency R ZOI (m) g Fequency Fequency e Fequency R ZOI (m) h Fequency Fequency R ZOI (m) f Fequency R ZOI (m) i Fequency R ZOI (m) R ZOI (m) R ZOI (m) Figue Histogam calculated fo the R ZOI distibution elative to (a) all tees, (b) small tees (dbh<5 cm), (c) lage tees (dbh 5 cm),(d) Avicennia geminans,(e)smallavicennia (dbh<5 cm), (f) lage Avicennia (dbh 5 cm), (g) Lagunculaia acemosa,(h) smalllagunculaia (dbh<5 cm) and (i) lage Lagunculaia (dbh 5 cm).

101 4.3 Results 97 a b y(m) R ε () x(m) Figue (a) Object patten elative to all tees within study site Lagoa B. (b) Object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model I hypothesis with 200 simulations. a b y(m) R ε () x(m) Figue (a) Object patten elative to Avicennia geminans within study site Lagoa B. (b) Object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model I hypothesis with 200 simulations.

102 98 Object Patten Analysis a b y(m) R ε () x(m) Figue (a) Object patten elative to Lagunculaia acemosa within study site Lagoa B. (b) Object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model I hypothesis with 200 simulations. a b y(m) R B ε() x(m) Figue (a) Object patten elative to small (dbh<5 cm) (ed) and lage (dbh 5 cm) (blue) tees within study site Lagoa A. (b) Bivaiate object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model II hypothesis with 200 simulations.

103 4.3 Results 99 a b y(m) R B ε() x(m) Figue (a) Objectpattenelative to Avicennia geminans (ed) and Lagunculaia acemosa (blue) tees within study site Lagoa A. (b) Bivaiate object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model II hypothesis with 200 simulations. a b y(m) R B ε() x(m) Figue Object patten elative to small (dbh<5 cm) (ed) and lage (dbh 5 cm) (blue) Avicennia tees within study site Lagoa A. (b) Bivaiate object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model II hypothesis with 200 simulations.

104 100 Object Patten Analysis a b y(m) R B ε() x(m) Figue Object patten elative to small (dbh<5 cm) (ed) and lage (dbh 5 cm) (blue) Lagunculaia tees within study site Lagoa A. (b) Bivaiate object ing analysis (blue) and espective 90% simulation envelope (ed) obtained via Monte Calo Method (Besag 1977) fo the Model II hypothesis with 200 simulations. The spatial elation pesented by all tees exhibits egulaity at intemediate and lage scales. This patten could be esult of envionmental heteogeneity (Klass et al. 2000) (see Figue 4.15). The tees of goup Avicennia geminans exhibits clusteing at lowe, intemediate and highe scales. This could be an indication of eithe envionmental heteogeneity within the stand (Klass et al. 2000) (see Figue 4.16). The tees of goup Lagunculaia acemosa shows clusteing at lowe and intemediate scales. This could be an indication of envionmental heteogeneity (Klass et al. 2000) (Figue 4.17). The spatial elation between individuals of goups of small (dbh < 5 cm) and lage (dbh 5 cm) tees exhibits epulsion at small, intemediate and lage scales. This could be an indication of eithe envionmental heteogeneity (Klass et al. 2000) and/o succession (Begon et al. 1976) (Figue 4.18). The spatial elation between individuals of species Avicennia geminans and Lagunculaia acemosa shows epulsion at all ange of scales. This could be an indication of eithe envionmental heteogeneity (Klass et al. 2000) o succession (Begon et al. 1976) (Figue 4.19).

105 4.4 Discussion 101 The spatial elation between small (dbh < 5 cm) and lage (dbh 5 cm) Avicennia acemosa exhibits aggegation at lowe scales. That could be an indication of eithe seed dispesal (Baot et al. 1999) and/o nuse-plant effects (Tielböge & Kadmon 2000). Additionally, they also show clusteing at lage scales. This could be a indication of eithe succession (Begon et al. 1976) o envionmental heteogeneity (Klass et al. 2000) (Figue 4.20). The spatial elation between small (dbh < 5 cm) and lage (dbh 5 cm) Lagunculaia acemosa pesents aggegation at scale 3 m and at lage scales. This could be a esult of eithe succession (Begon et al. 1976) o envionmental heteogeneity (Klass et al. 2000) (Figue 4.21). 4.4 Discussion It is vey difficult to compae the esults descibed in this chapte with the esults in Chapte 3 and thee ae at least two easons fo this. Fist, I applied a diffeent analytical appoach. Secondly, the dataset is completely diffeent, because I excluded the tees with dbh > 0 and the dead tees fom the analysis. This pocedue significantly educed the numbe of individuals compaed to ou oiginal dataset. But despite the diffeences between the two methods (Ripley K-function and object ing function R ε()), some of the esults obtained wee equivalent in qualitative sense. The analysis was applied to diffeent goups of tees in the sites Lagoa A and Lagoa B. The main aim of ou study was to see whethe the object ing function R ε() is sufficiently sensitive to detect the spatial inteactions occuing between tees of species Lagunculaia acemosa and Avicennia geminans within these study sites. A paticula inteest was to see whethe the method could povide indiect evidence of the succession pocesses that ae pobably occuing in these stands. The spatial elation between tees of species Lagunculaia acemosa and Avicennia geminans exhibits epulsion at intemediate scale in Lagoa A and epulsion at all scales in Lagoa B (Figue 4.11 and Figue 4.19 espectively). These esults ae equivalent to the esults obseved in Figue 3.15 and Figue 3.26 of the Chapte 3 espectively. These esults shows only a tendency to epulsion between the species Lagunculaia and Avicennia. The esult in Figue 4.19, shows that the ing method displays geate sensitivity in detecting the epulsion pocess between tees of species and within the sites Lagoa A and Lagoa B. These pattens could be a esult of eithe succession (Begon et al. 1976) o envionmental heteogeneity (Klass et al. 2000). The hypothesis of envionmental heteogeneity is difficult to pove, because I have no infomation about abiotic conditions in the two studies sites Lagoa A and Lagoa B; howeve the spatial heteogeneity exhibited by the tees could be a indication of envionmental heteogeneity and so I cannot exclude this possibility. Mangoves foests ae in a continuous pocess of gowth and constantly establishing and enewing themselves (Duke 2001). Theefoe, the hypothesis of succession occuing among the individuals of species and in the stands is a plausible one.

106 102 Object Patten Analysis It s impotant to emphasize that both hypotheses could be coect and the spatial distibution of the tees could be a esult of thei combined action. The main chaacteistic of the ing function R ε() is that is consides individual tees as cicles, athe than points. This pocedue ovecomes the poblems associated with the eduction of a thee dimensional individual into a point, minimizing the loss of infomation caused by the tansfomation step T2. It is impotant to note the method can only be applied when the shape of the individuals consideed by the analysis is appoximately cicula. Othewise, the method can not to be applied. I consideed just two simple models to simulate the spatial distibution of cicula objects and to geneate the simulation envelope using the Monte Calo method (Besag 1977). Othe null models could be used to test othe hypothesis, but at this stage this is no moe than an idea fo futue wok. Finally, the object ing method, like the Ripley K-function, has the limitation that although it can povide infomation about the scale of pocesses occuing in the stands, it does not povide infomation whee the whee these pocesses ae occuing; o in othe wods, neithe method has spatial esolution. In the next chapte, I pesent a method that can povide spatial-scale infomation, (unde cetain conditions) about the pocesses that ae occuing in a study site.

107 Chapte 5 Wavelet Tansfom applied to Ecology 5.1 Intoduction Taditional spatial statistical methods, such as the Ripley K-function, povide infomation about whethe individual plants exhibit clumped, egula o andomly distibution in the plot and at which scale these pattens occu. Howeve, this infomation is insufficient when the location of cetain distibution pattens has to be taken into account, because these methods have no spatial-scale esolution. The majoity of spatial statistical methods used to analyze spatial point pattens (including the Ripley K-function) wee developed to deal with homogeneous point pattens, but spatial heteogeneity 5.1 is a common featue of natual ecosystems (Kolasa & Pickett 1991). The use of such methods to analyze a heteogeneous point patten can thus lead to misintepetation of the spatial point pocess that occus in a study aea (Pelissie & Goeaud 2001). Spatial heteogeneity in plant distibution can occu when abiotic factos (soil, climate, nutients, etc) vay significantly fom one location to anothe (Pelissie & Goeaud 2001). Additionally, natual pocesses (bith, development, epoduction, competition, pedation and senescence) can also poduce spatial heteogeneity in ecological populations (Stene et al. 1986, Kenkel 1988, Foget 1994, Blate et al. 1998, Couteon 1998, Desouhant et al. 1998). The analysis of spatial vaiations of point locations depends on the scale of obsevation in elation to the size of the study site. In this study, I conside spatial vaiations in point position occuing at highe scales as spatial heteogeneity, wheeas lowe and intemediate scale vaiations can be consideed as elements of the stuctue (Wiens 1989, Kolasa & Rollo 5.1. If a spatial point patten vaies fom location to location, it is thus called heteogeneous (Ripley 1981). 103

108 104 Wavelet Tansfom applied to Ecology 1991, Holling 1992, He et al. 1994, Goeaud 2000). Fo example, the patchy distibution of individual tees can give ise epeated stuctues in a study aea, wheeas a single patch can display heteogeneity at a fine scale (Pelissie & Goeaud 2001). In this chapte, I considee a numbe of possible appoaches to the statistical analysis of spatial heteogeneity. One possible would be to apply a multiesolution method to analyze and decompose a spatial point patten at diffeent scales. A possibility is to apply the well known Fouie Tansfom method (FT heeinafte) to obtain this multiscale epesentation of a spatial point patten. This method epesents a signal as the sum of a seie of sines and cosines. It pemits all the fequencies pesent in a time-seies to be detected, but does not povide infomation about the location of the fequencies. That is the main limitation of this method. It has only fequency esolution and no time esolution and theefoe the FT can be only used fo decomposition of stationay signals. But ecological and envionmental time seies obseved in natue ae typically apeiodic, noisy and tansient and the analysis of such time seies by means of FT can lead to poblems in the intepetation of the esults. Fo example, Figue 5.5 shows two diffeent signals (Figues 5.5a and 5.5b) and thei espective FT analysis (Figues 5.5c and 5.5d). The esults show that the FT is not able to diffeentiate the signals and only povides infomation about the existence of two fequencies that in these signals. To ovecome these limitations of the FT method, the natual choice would be to apply the Multiesolution Decomposition Analysis (heeinafte MDA) obtained via the Wavelet Tansfom method (heeinafte WT). The WT method is a mathematical tool that pemits spatial infomation to be obtained about the stuctues contained within an image at diffeent scales. The method decomposes an image into vaious maps and each map epesents a ange of scales contained in the oiginal image. The MDA is obtained via DWT (Discete Wavelet Tansfom), the discete vesion of the WT method. The main advantage of this method in compaison to othe spectal methods, such as Fouie Tansfom (heeinafte FT), is its spatial-scale esolution. Subject to cetain estictions (connected with the Heisenbeg uncetainty pinciple 5.2 ), the MDA povides infomation about the scale of the point pocesses occuing in the study site and thei espective spatial-location. This singula capability pemits allows the method to be used to analyze the scale and positions of heteogeneous point patten configuations. ButitisimpotanttomentionthattheMDA method can not be diectly applied to a spatial point patten. The fist step equied is to tansfom this spatial point patten into a density map (an image) using a Kenel Density Estimation Method (heeinafte KDE). This density map povides spatial infomation about the density of the points ove the study site (see Figue 5.1) The uncetainty pinciple states that the time-position and fequency cannot both be measued, exactly, at the same time (Wene Heisenbeg ).

109 5.1 Intoduction 105 The second and final step is to decompose the density map obtained via KDE method at diffeent scales using the MDA obtained via WT method (see Figue 5.1). This enables us to obtain spatial infomation about the density of the points within the study site at diffeent scales and positions. Additionally, I calculate the vaiogam to veify the decomposition of the density map duing each step of the MDA. The vaiogam is a geostatistical that povides infomation about the scales of the stuctues contained in an image. Figue 5.1. Repesentation of the steps of ou methodology. (left) Point Patten (middle) Density Map (ight) Multiesolution Analysis. I pefom thee applications of the methodology. The fist methodology is applied to decompose a heteogeneous study aea into smalle homogeneous egions that can subsequently be analyzed individually using classical spatial statistical methods that equie a hypothesis of homogeneity. A second application is pefomed to see if the method is sufficiently sensitive to detect the epulsive spatial patten between the tees of species Lagunculaia and Avicennia within Lagoa A and Lagoa B that was detected by the spatial statistical methods pesented in Chapte 2 and Chapte 3. The idea behind the final application is to obtain scale-position infomation about the spatial point pocesses occuing at heteogeneous study site and use this infomation to simulate spatial point pattens using an inhomogeneous Poisson pocess.

110 106 Wavelet Tansfom applied to Ecology 5.2 Methods The fist step of the methodology (see Figue 5.2) is to tansfom a spatial point patten that epesents the spatial stuctue of a foest stand into a density map using a Kenel Density Estimation (Diggle, 1985). This fist step is equied, because is not possible to apply the MDA diectly to spatial point pattens Kenel Density Estimation Method Conside (s 1,s 2,,s n ) the spatial position of n = N(A) events in a study egion Ω R d. The KDE method (Cessie 1993) is geneally defined as { λ ĥ (s)= 1 n p h (s) i=1 } κ h (s s i ), s Ω, (5.1) whee κ h ( ) is a kenel function symmetical about the oigin, h>0 is the kenel bandwidth which detemines the amount of smoothing of the density map and p h (s)= Ω κ h(s u)du is an edge coection facto (Diggle 1985). The selection of an appopiate h depends on the estimation of λ( ) (Silveman 1978). But it is impotant to notice, that the choice of an appopiated kenel bandwidth h is also diectly elated to the scales that I want to analyze and will depend on the level of decomposition I want to achieve using the MDA via WT method. To analyze stuctues with dimension scale s, the discetization of ou density map must be at least equal to s/2 (Nyquist fequency). Additionally, if ou density map is a matix with dimensions 2 d 2 d,themaximallevel of decomposition that is achievable is d. In this case, I conside that the algoithm used to pefom the MDA is dyadic (Mallat 1988). In this study, I conside the Quatic Kenel defined in two dimensions as κ h = ρ h (u 1 ) ρ h (u 2 ),whee { h ρ h (u)= 1 [1 (u/h) 2 ] 2, h u h, 0, othewise. (5.2) In Figue 5.1, I pesent an application of the method. It shows a density map obtained fom a spatial point patten via KDE method using a Quatic Kenel.

111 5.2 Methods 107 Figue 5.2. The Kenel Density Estimation method tansfoms a spatial point patten (left) into a density map (ight). The scale of the study site is povided in m and of the unity of the density map is points/m Vaiogam Analysis The vaiogam 5.3 function γ(h) calculated fo the density map and fo each step in the decomposition of the density map obtained via MDA. This function povides an indication of the spatial coelation between measuements taken at sample locations. It is a cucial paamete in geostatistics (Matheon 1963) and is commonly epesented as a function that shows the vaiance in measuements of distance between all pais in sampled locations. The classical estimato of the Vaiogam function is defined as γˆ(h) 1 2 N(h) N(h) (Z(u) Z(u + h)) 2, (5.3) whee u is the vecto of spatial coodinates (a point on the density map), Z(u) is the vaiable unde consideation as a function of spatial location (in this case, density), Z(u + h) is the lagged vesion of the vaiable unde consideation, h is the lag vecto epesenting the sepaation between two spatial locations and N(h) is the numbe of pais sepaated by lag h. The main chaacteistics of a vaiogam function ae pesented below: Sill is the semivaiance value at which the vaiogam levels off (C ). Range is the lag distance at which the semivaiogam (o semivaiogam component) eaches the sill value. Pesumably, autocoelation is essentially zeo beyond the ange. Nugget is the semivaiogam value at the oigin (C 0 ). In theoy this value at the oigin (0 lag) should be zeo. If it is significantly diffeent fom zeo fo lags vey close to zeo, then this semivaiogam value is efeed to as the nugget. The nugget 5.3. Heeinafte I define the classical semivaiogam as vaiogam.

112 108 Wavelet Tansfom applied to Ecology epesents vaiability at distances smalle than the typical sample spacing, including measuement eo. These chaacteistics ae epesented gaficaly in Figue 5.3. Figue 5.3. Chaacteistics of a Vaiogam Wavelet Tansfom The second step of the methodology (Figue 5.4) is to use the MDA obtained via WT method in ode to decompose an image (the density map geneated by the KDE method) at diffeent scales. Figue 5.4. The Multiesolution Decomposition Analysis pefomed to decompose a density map (left) in diffeent scales (ight). The unit of the study site is povided in m and of the density map is points/m 2.

113 5.2 Methods 109 Figue 5.5. The gaphic (a) epesents a time seie that contains a supeposition of a low fequency signal (sin10t) and a high fequency signal (sin20t). The gaphic (b) epesents a time seie that contains a low fequency signal (sin10t) in the fist half and a signal with high fequency signal (sin20t) in the second half. The gaphics (c) and (d) epesent the esponse of the FT to the time seies epesented at the gaphics (a) and (b) espectively. The gaphics (e) and (f) ae epesent the esponse of the WT tothetimeseies epesented at gaphics (a) and (b) espectively Continuous Wavelet Tansfom The WT method was developed to investigate and analyze the tempoal development of a non-stationay time-seies. It is able to pefom a time-fequency analysis of a time seies and to estimate the spectal chaacteistics of the signal as a function of time (Meyes 1993, Toence & Compo 1998). The main advantage of the WT in elation to othe spectal analysis methods (like FT, fo example), is its capacity to detect the fequencies that exist in the signal and thei espective tempoal-localization. In summay, WT uses a multiesolution technique to analyze diffeent fequencies at diffeent esolutions. The WT method is an integal tansfomation whose integation kenels ae called wavelets (Chi 1992, Daubechies 1992, Mallat 1998). These wavelets have the popeties of being located in time and fequency (space and wave numbe if I ae consideing the spatial case), which pemits us to analyze signals that contain non-stationay powe at diffeent fequencies (Daubechies 1990). The WT decomposes a signal f(x) using scaled

114 110 Wavelet Tansfom applied to Ecology and shifted vesions of a function called mothe wavelet and it is geneally defined as + Wf(λ,t)= f(u)ψ λ,t (u)du, (5.4) whee f(t) is the signal to be analyzed and ψ λ,t (u) is a family of functions defined as ψ λ,t (u)= 1 ψ λ ( u t λ ), (5.5) and λ is a scale paamete (elated to the fequency), t isashiftingpaamete(elatedto time position) and ψ(u) is a mothe wavelet. The choice of a wavelet function ψ(t) is not abitay and it has to satisfy two conditions: that the function ψ(t) is nomalized o ψ(t) 2 dt=1 and ψ(t)dt=0.thefacto λ maintains the vaiance of the shifted and scaled wavelet identical to those of the mothe wavelet. Changes in the values of scale paamete λ modify the wavelet width. Lage values of λ dilate the wavelet width, while small values of λ compess the wavelet width. Compessed vesions of the mothe wavelet ae used to detect high fequency components (o low scale featues) contained within the analyzed signal and dilated vesion of the mothe wavelet ae used to detect low fequency components (o high scale featues) that ae contained within the analyzed signal (see Figue 5.6). The paamete t contols the shifting of the mothe wavelet ovetheentiesignalf(x) (see Figue 5.7). In summay, it contols the position of the wavelet in the signal. Futhe featues of the WT ae that the method peseves the vaiance of the analyzed signal f(t), o + f(t) 2 dt= 1 C g + 0 W f (λ,t) 2 ψ λ,t (u)du dλ λ 2, (5.6) and that the oiginal signal f(t) can be ecoveed by the invese wavelet tansfom defined as whee and f(t)= 1 C g λ 2W f(λ,t)ψ λ,t (u)dtdλ, (5.7) + ψˆ(f) C g = f 2 df, (5.8) + ψˆ(f)= ψ(t)e i2πf t dt. (5.9)

115 5.2 Methods 111 Thee is a elation between fequency (o pseudo-fequency) and scale (Aby 1997) defined as F a = ΔF c a, (5.10) whee a is the scale of the wavelet, Δ isthesamplepeiod,f c is the cental fequency in Hz of the wavelet (that value is specific fo each wavelet family) and F a is the pseudo-fequency associated with the scale a in Hz. Figue 5.6. The gaphics (a),(b) and (c) epesent the same mothe wavelet with paamete λ =1, λ = 0.5 and λ = 0.25 espectively. Figue 5.7. The gaphics (a) and (b) epesentthe same mothe wavelet with paamete t=0 and t= 0.25 espectively. The coefficients Wf(λ, t) measue the match between the signal and the wavelet at the position t and scale λ. In summay, lage values of Wf(λ, t) indicate a high degee of similaity between the wavelet ψ with scale λ and the signal f(x) at the position t. In

116 112 Wavelet Tansfom applied to Ecology contast, low values of Wf(λ, t) indicate a low degee of similaity between the wavelet ψ with scale λ and the signal f(x) at the position t and finally, negative values of Wf(λ, t) indicates an out-of-phase association between the wavelet ψ with scale λ and the signal f(x) at the position t. The WT computed ove a continuous ange of values t and λ is called Continuous Wavelet Tansfom (heeinafte CWT) and the esult is a thee dimensional suface Wf(λ,t) called as scalogam Discete Wavelet Tansfom The WT computed ove a discete ange of paametes λ and t is called DWT. The DWT will povide us with the MDA, the multiscale decomposition of an image at diffeent scales. It is a special case of the WT and it povides a compact epesentation of a signal in time and fequency that can be computed efficiently. In this case the scale paamete is λ = λ m 0,wheem is an intege and λ 0 is a fixed scale paamete geate than 1, and the shifting paamete is assumed to be t = nt 0 λ m 0,wheet 0 > 0 and depends on ψ(t) and n is an intege. Then, I can define the CWT applied to a signal f(t) as Wf(λ,t)= f(t)ψ m,n (t)dt, (5.11) whee ψ m,n (t)=λ 0.5m 0 ψ(λ m 0 t nt 0 ). (5.12) In pactice, the DWT is pefomed consideing the paametes λ 0 =2and t 0 =1 (Othogonal Wavelet Tansfom). Then I have ψ m,n (t)=2 0.5m ψ(2 m t n). (5.13) It is possible to constuct a class of othogonal wavelets ψ m,n (t) satisfying the following condition ψ j,k (t)ψ m,n (t)dt= δ jm δ jm, (5.14) whee δ ij is the Konecke Delta function defined as { 1, if i = j δ ij = 0, othewise. (5.15) Now I can appoximate a function f(t) with a linea combination of othogonal wavelets ψ m,n (t) (Kuma & Foufola 1988) as f(t)= + + m= n= D m,n ψ m,n (t), (5.16)

117 5.2 Methods 113 whee m and n ae the scale and tempoal index espectively, with D m,n = f(t)ψ m,n (t)dt. (5.17) as Supposing now thee ae intemediate scales m 1 and m 2, I can edefine the equation 5.15 f(t)= + + m=m 2 +1 n= m m= n= D m,n ψ m,n (t)+ m 2 + D m,n ψ m,n (t)+ D m,n ψ m,n (t), m=m 1 n= (5.18) that epesents the decomposition of the signal f(t) at thee ange scales: the fist summation epesents the highe scales chaacteistics of the signal (>m 2 ), the second one epesents the intemediate scales chaacteistics (m 2 m 1 ) and the final one epesents the lowe scale chaacteistics (<m 1 ). In pactice, the MDA of a signal at diffeent scales is pefomed by successive use of lowpass and highpass filtes. In this case, the function φ(t) o scale function coesponds to the discete lowpass filte, that etains the low fequencies featues of the signal and the function ψ(t) coesponds to the discete highpass filte, that etains the high fequencies of the signal (Daubechies & Mallat 1998). In summay, the MDA is pefomed by decomposing the oiginal signal into two pats: a pat named detail that contains the high fequency featues (D) of the signal and anothe one named appoximation that contains the low fequencies featues (A) of the signal. The MDA geneally (depending on the type of the filte used at the decomposition) pemits a pefect econstuction (Mallat 1998) of the oiginal signal f(t). TheMDA at one level (see Figue 5.8) can be defined as f(t)=a 0 = A 1 + D 1, (5.19) whee A 0 is the oiginal signal f(t), A 1 and D 1 ae espectively appoximation and detail at level 1. Figue 5.8. MDA applied to a signal f(t). A 1 is the appoximation at level 1 and D 1 is the detail at level 1.

118 114 Wavelet Tansfom applied to Ecology The same decomposition pocess can be epeated with the appoximation A 1 (see Figue 5.9). Thus, I have A 1 = A 2 + D 2, (5.20) whee A 2 and D 2 ae espectively appoximation and detail at level 2. Figue 5.9. MDA applied to A 1. A 2 is the appoximation at level 2 and D 2 is the detail at level 2. Then combining Equation 5.18 and Equation 5.19, I obtain f(t)=a 0 = A 2 + D 1 + D 2, (5.21) and by induction, a decomposition pefomed at level n can be epesented as f(t)=a 0 = A n + D D n = A n + i=1 n D i, (5.22) whee A n is the appoximation at level at level n and D i is the detail at level i. Insummay, to decompose an image at level n I need to apply the highpass and lowpass filtes to the image successively n times (see Figue 5.10). Figue MDA at level n applied to a signal f(t)=a 0.

119 5.2 Methods 115 It is impotant to note that the methodology can be also applied to a 2D signal (Image) without loss of geneality. Fo moe details, see Mallat (1998). Figue 5.11 shows an example of an image decomposition pefomed by MDA at level two (highe, intemediate and lowe scales). This decomposition can be epesented as Oiginal Image = A 0 = A 2 + D 1 + D 2, (5.23) whee A 2,D 2 and D 1 contains the highe, intemediate and lowe scale chaacteistics of the oiginal image. In fact, the Equation 5.22 is equivalent to Equation Figue MDA applied to an image (density map). The oiginal image (a density map) was decomposed at its highe, intemediate and lowe scale components Inhomogeneous Poisson Pocess The classical K-function equies the assumption that the point patten is spatially homogeneous (stationay and/o isotopic). It supposes that the fist ode intensity ove the entie aea Ω is constant: that is, it is the same fo all locations, o fomally that λ(x, y) λ, (x, y) Ω. The poblem occus when the point patten does not show homogeneity and then the analysis of this patten obtained via the classical K-function can lead sometimes to a misintepetation of the spatial point pocesses that occu inside the study aea (Pelissie & Goeaud 2001). A possibility way to analyze an inhomogeneous point patten is to use the inhomogeneous K-function, a semi-paametic method that supposes that the point patten was geneated by a inhomogeneous Poisson pocess (Baddeley et al 2000). This method assumes that the fist ode intensity λ(x, y) ovetheentiestudyaeaω is a function that depends on the location (x, y). Fomally the inhomogeneous K-function is defined as K i ()= 1 n w i () n A i=1 j=1 δ ij () 1 λ i λ j, (5.24)

120 116 Wavelet Tansfom applied to Ecology whee A is the aea of the study egion Ω, w i () is an aea based edge coection facto, δ ij () is an indicato function defined as with { 1, if dij δ ij ()= 0, othewise, (5.25) d ij = (x i x j ) 2 +(y i y j ) 2, (5.26) as the Euclidian distance between the points of the point patten (x i,y i ) and (x j,y j ) and λ i and λ j ae the local densities calculated at the locations (x i,y i ) and (x j,y j ) espectively. To simulate a point patten inside a study aea Ω and subject to a fist ode intensity λ(x, y), I have fist to define the constant λ 0 = max (λ(x, y)), (5.27) (x,y) Ω and I then simulate a point (x, y) following a classical homogeneous Poisson pocess inside the aea Ω and I accept this point as the location of a tee with pobability (Diggle λ(x, y) λ , Tomppo 1986). The infomation about the local density λ(x, y) is obtained using the MDA method Density Map Geneation To geneate the density map, I applied a KDE method to the spatial point patten using a Quatic Kenel. I computed the local density with a kenel bandwidth of h =1 matthe nodes of a 0.47 m x 0.47 m systematic gid coveing the whole stand in ode to obtain a density map with 64 x 64 cells. This enables us to decompose the density map at 7 levels using the MRA method. The kenel bandwidth was chosen in ode to peseve stuctues with scales geate than o equal to 2 m (Nyquist fequency). In ode to decompose the density map geneated at diffeent scales, I applied a classical MDA method using an algoithm implemented in the softwae R, a language and envionment fo statistical computing and gaphics. The multiesolution analysis of the density maps ae based on a Daubechies D8 wavelet pefomed using the RpackageWaveslim. The multiesolution decomposition was obtained up to level 4. This coesponds to filtes out all the stuctues with scales up to 12 m. It coesponds to 40% of the dimensions of ou study sites Lagoa A and Lagoa B.

121 5.3 Results Results Heteogeneity detection Fist I apply the MDA method in ode to decompose a heteogeneous site into smalle homogeneous subplots that can be analyzed using classical statistical methods. In this case, I ae consideing heteogeneity as a deviance of the point patten fom a CSR patten at lage scales Lagunculaia acemosa -LagoaA The esults obtained fom the L-function (Figue 3.9) show that the spatial configuation of the tees of species Lagunculaia acemosa within stand Lagoa A exhibit clusteing at lage scales ( 10 m). This could be an indication of heteogeneity in the stand (Goeaud & Pelissie 2001). The fist step of the methodology was to tansfom this point patten (Figue 3.9) into adensitymap(figue 5.12) usingthekde method. The vaiogam calculated (Figue 5.12) indicates that the density map shows stuctues with scales geate than 2 metes. It is inteesting to note, that the density of the point patten inside this stand is λ p = 0.279, while the mean value obtained fo the density map is λ d = This means that the tansfomation pefomed by the KDE peseved the global density of the stand. a b c y Density γ() x Pixel value Figue (a) Density map obtained fom spatial point patten elative to Lagunculaia acemosa within stand Lagoa A. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a).

122 118 Wavelet Tansfom applied to Ecology The second step was to decompose the density map into diffeent scales using the MDA method. I applied the decomposition up to level 4 and in the following I descibe the steps of the decomposition. The appoximation A1 obtained at the fist step of the decomposition shows no visible changes in the vaiogam (Figue 5.13). The mean density calculated fo the image A1 is λ d = a b c y Fequency γ() x Pixel value Figue (a) Appoximation A1 elative to the density map at the Figue (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a). The appoximation A2 obtained at the second step of the decomposition now pesents changes in the vaiogam (Figue 5.14). This shows that the decomposition filteed all stuctues with scales smalle then o equal to 3 metes out the density map. The mean density calculated fo the image A2 is λ d = a b c y Fequency γ() x Pixel value Figue (a) Appoximation A2 elative to the density map at the Figue 5.13a. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a).

123 5.3 Results 119 The vaiogam calculated fo the appoximation A3 (Figue 5.15) shows that the decomposition filteed out all stuctues with scales smalle then o equal to 6 metes. The mean density calculated fo the image A3 is λ d = a b c y Fequency γ() x Pixel value Figue (a) Appoximation A3 elative to the density map at the Figue (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a). The vaiogam calculated fo the appoximation A4 shows that the decomposition filteed out all stuctues with scale smalle than o equal to 10 metes(figue 5.18). That is the scale of ou inteest and the decomposition stops hee. The mean density calculated fo the image A4 is λ d = Visually I can see fom the image A4 (Figue 5.16) that thee is a lage egion located at the left side of the stand with a high density of tees. The density cutoff chosen to sepaate this egion fom the est of the stand was λ cutoff = The Figue 5.17 shows the study site Lagoa A divided into two egions. a b c y Fequency γ() x Pixel value Figue (a) Appoximation A4 elative to the density map at the Figue 5.13a. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a).

124 120 Wavelet Tansfom applied to Ecology To complete the analysis, I calculated the L-function fo the points contained within the ed and yellow egions (Figue 5.17) sepaately. The esult obtained fo the L-function calculated fo the point patten within the ed egion (Figue 5.17) shows clusteing at lowe scales ( 5 m). It is inteesting to note that the point patten contained within the yellow egion exhibits a CSR patten. a b c L() L() Figue (a) Point patten elative to Lagunculaia acemosa whit-in stand Lagoa A, now divided at two egions (ed and yellow). (b) and (c) epesents the the L-function (black) and 99% simulations inteval (dashed ed) calculated fo the point patten inside the ed and yellow espectively. The simulations inteval was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations. Consideing that low and intemediate scale vaiations in the stuctues of the study site can be consideed as elements of the stuctue (Wiens 1989, Kolasa & Rollo 1991, Holling 1992, He et al. 1994, Goeaud 2000). I can affim that I divided the whole stand site with a stong indication of heteogeneity into two subsets that exhibited homogeneity Lagunculaia acemosa -LagoaB The tees of the species Lagunculaia acemosa within the stand Lagoa B also exhibit clusteing at lage scales (see Figue 3.20). This could be an indication of heteogeneity in the stand (Pelissie and Goeaud 2001). I applied the same methodology descibed above with this goup of tees, but in the following I do not descibe all the intemediate steps, just the final esult. The vaiogam in Figue 5.18 shows that the density map pesents stuctues with scales geate than 3 metes. The density calculated fo the point patten inside this stand is λ p = 0.190, while the mean value obtained fo the density map is λ d =

125 5.3 Results 121 a b c y Density γ() x Pixel value Figue (a) Density map obtained fom spatial point patten elative to Lagunculaia acemosa within stand Lagoa B. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a). The vaiogam elative to the appoximation A4 shows that the decomposition filteed out all stuctues with scale smalle then o equal to 12 metes (Figue 5.19). The mean density calculated fo the image A4 emains λ d = Visually I can see fom the image A4 (Figue 5.19) that thee ae some aeas in the stand with a high density of tees. The cutoff, chosen visually to divide the stand, was λ cutoff = and the Figue 5.20 shows the study site Lagoa B divided into two distinct egions. a b c y Fequency γ() x Pixel value Figue (a) Appoximation A4 elative to the density map at the Figue (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a). The esult obtained fo the L-function calculated fo the point patten within the ed egion (Figue 5.20) pesents clusteing at lowe scales ( 1m).Thepointpatten contained within the yellow egion (Figue 5.20) pesents clusteing at lowe scales ( 5 m). Again, I sepaated a lage egion that pesents heteogeneity at two egions that pesents some homogeneity.

126 122 Wavelet Tansfom applied to Ecology a b c L() L() Figue (a) Point patten elative to Lagunculaia acemosa whit-in stand Lagoa B, now divided at two egions (ed and yellow). (b) and (c) epesents the the L-function (black) and 99% simulations inteval (dashed ed) calculated fo the point patten inside the ed and yellow espectively. The simulations inteval was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations Dead Tees - Lagoa B The dead tees within stand Lagoa B (Figue 3.21) also exhibit clusteing at lage scales. This could be an indication of heteogeneity in the stand (Pelissie and Goeaud 2001). The vaiogam in Figue 5.21 indicates that the density map shows stuctues with scales geate than 2 metes. The density calculated fo the point patten inside this stand is λ p = 0.128, while the mean value obtained fo the density map is λ d = Again the tansfomation pefomed by the KDE peseved the global density of the stand. a b c y Density γ() x Pixel value Figue (a) Density map obtained fom spatial point patten elative to dead tees within stand Lagoa A. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a).

127 5.3 Results 123 The vaiogam elative to the appoximation A4 shows that the decomposition filteed out all stuctues with scale smalle then o equal to 12 metes (Figue 5.22). The mean density calculated fo the image A4 emains λ d = Visually I can see fom the image A4 (Figue 5.22) that thee ae some aea in the stand with a high density of dead tees. The cutoff chosen was λ cutoff = and the Figue 5.23 shows the study site Lagoa B divided into two egions. a b c y Fequency γ() x Pixel value Figue (a) Appoximation A4 elative to the density map at the Figue 5.25a. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a). The esult obtained fo the L-function calculated fo the point patten within the ed egion (Figue 5.26) exhibits a CSR patten. The point patten in the yellow egion (Figue 5.22) exhibits clusteing at lowe scales ( 1 m) and egulaity at intemediate scales ( 5 m). Again I divided a lage egion that shows heteogeneity into two egions that exhibit some homogeneity. a b c L() L() Figue (a) Point patten elative to dead tees whit-in stand Lagoa B, now divided at two egions (ed and yellow). (b) and (c) epesents the the L-function (black) and 99% simulations inteval (dashed ed) calculated fo the point patten inside the ed and yellow espectively. The simulations inteval was calculated via Monte Calo method (Besag 1977) fo the CSR hypothesis with simulations.

128 124 Wavelet Tansfom applied to Ecology Spatial-Scale Resolution Now the main objective of this expeiment is ty to povide evidence fom spatial pattens of the pobable succession pocesses occuing among the tees of species Lagunculaia acemosa and Avicennia geminans within studies sites Lagoa A and Lagoa B Spatial elationship between Lagunculaia acemosa and Avicennia geminans - Lagoa A The esults pesented at Figue 3.15 show that Avicennia tees tends to be clumped at scales 1 m. Lagunculaia tees exhibit clumping at all scales, with a maximum at scale 4 m. The spatial elation between the Avicennia and Lagunculaia tees exhibits a tendency to epulsion at lowe scales, in paticula at scale 0.50 m. The vaiogam pesented in Figue 5.24 (Avicennia geminans) indicates that the densitymappesentsstuctueswithscalesgeatethan2metes.thedensitycalculatedfothe point patten inside this stand is λ p = 0.343, while the mean value obtained fo the density map is λ d = a b c y Density γ() x dmap$v Figue (a) Density map obtained fom spatial point patten elative to Avicennia geminans within stand Lagoa A. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a). The vaiogam pesented in Figue 5.25 (Lagunculaia acemosa) indicates that this density map also shows stuctues with scales geate than 2 metes. The density calculated fo the point patten inside this stand is λ p = 0.279, while the mean value obtained fo the density map is λ d =

129 5.3 Results 125 a b c y Density γ() x dmap$v Figue (a) Density map obtained fom spatial point patten elative to Lagunculaia acemosa within stand Lagoa A. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a). The vaiogam elative to the appoximation A4 shows that the decomposition filteed out all stuctues with scales smalle then o equal to 12 metes (Figue 5.26). The mean density calculated fo the image A4 emains λ d = Visually I can see fom the image A4 (Figue 5.26) that thee ae some aeas in of the stand with a high density of tees of species Avicennia geminans. a b c y Fequency γ() 0e+00 4e 04 8e x density Figue Avicennia geminans (a) Appoximation A4 elative to the density map at the Figue (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a). The vaiogam elative to the appoximation A4 shows that the decomposition filteed out all stuctues with scales smalle then o equal to 12 metes (Figue 5.27). The mean density calculated fo the image A4 emains λ d = Visually I can see fom the image A4 (Figue 5.37) that thee ae some aea in the stand with a high density of tees of species Lagunculaia acemosa.

130 126 Wavelet Tansfom applied to Ecology a b c y Fequency γ() x density Figue Lagunculaia acemosa (a) Appoximation A4 elative to the density map at the Figue (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a). The esult pesented in Figue 5.28 shows a epulsive spatial patten at lage scales exhibited by tees of species Lagunculaia and Avicennia. Visually, it can be clealy obseved that thee is a high density of Lagunculaia tees in places with a low density of Avicennia tess and vice-vesa. Figue (a) Appoximation A4 elative to the density map obtained fo the Avicennia geminans and (b) Appoximation A4 elative to the density map obtained fo the Lagunculaia acemosa within Lagoa A Spatial elationship between Lagunculaia acemosa and Avicennia geminans - Lagoa B The esult shows that Avicennia tees exhibit clusteing at lowe and intemediate scales, with a maximum at 4 m. The Lagunculaia tees exhibit clumping at lowe and intemediate scales, in with a maximum at 1 m. The spatial elation between Avicennia and Lagunculaia tees exhibits epulsion at scale 1 m and a tendency to epulsion at scale 4 m(seefigue 3.26).

131 5.3 Results 127 The vaiogam in Figue 5.29 indicates that the density map shows stuctues with scales geate than 2 metes. The density calculated fo the point patten inside this stand is λ p = 0.284, while the mean value obtained fo the density map is λ d = Again the tansfomation pefomed by the KDE peseved the global density of the stand. a b c y Density γ() x dmap$v Figue (a) Density map obtained fom spatial point patten elative to Avicennia geminans within stand Lagoa B. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a). The vaiogam in Figue 5.30 (Lagunculaia acemosa) indicates that the density map also shows stuctues with scales geate than 2 metes. The density calculated fo the point patten inside this stand is λ p = 0.190, while the mean value obtained fo the density map is λ d = Again the tansfomation pefomed by the KDE peseved the global density of the stand. a b c y Density γ() x dmap$v Figue (a) Density map obtained fom spatial point patten elative to Lagunculaia acemosa within stand Lagoa B. (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a). The vaiogam elative to the appoximation A4 (Figue 5.31) shows that the decomposition filteed out all stuctues with scale smalle then o equal to 12 metes. The mean

132 128 Wavelet Tansfom applied to Ecology density calculated fo the image A4 emains the same (λ d = 0.236). Visually I can see fom the image A4 (Figue 5.31) that thee ae some aeas in the stand with a high density of tees of species Avicennia geminans. a b c y Fequency γ() x density Figue Avicennia geminans (a) Appoximation A4 elative to the density map at the Figue (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a). This decomposition was also pefomed up to level 4. The vaiogam elative to the appoximation A4 shows that the decomposition filteed out all stuctues with scale smalle then o equal to 12 metes (Figue 5.32). The mean density calculated fo the image A4 emains the same (λ d = 0.191). Visually I can see fom image A4 (Figue 5.32) thatthee ae some aeas of the stand with a high density of tees of species Lagunculaia acemosa. a b c y Fequency γ() x density Figue Lagunculaia acemosa (a) Appoximation A4 elative to the density map at the Figue (b) and (c) ae espectively the Histogam and the Vaiogam obtained fom this density map in (a). The esults pesented in Figue 5.33 also show a epulsive spatial patten at lage scales exhibited by the tees of species Lagunculaia and Avicennia. The patten is the same as in Lagoa A. Thee ae aeas with a high density of Lagunculaia tees located in egions with low density of Avicennia tees and vice-vesa.

133 5.3 Results 129 Figue (a) Appoximation A4 elative to the density map obtained fo the Avicennia geminans and (b) Appoximation A4 elative to the density map obtained fo the Lagunculaia acemosa within Lagoa B Simulating an Inhomogeneous Poisson Pocess Lagunculaia acemosa -LagoaB In this expeiment, I simulated an inhomogeneous Poisson pocess to epoduce the point patten pesented by the Lagunculaia tees in the site Lagoa B. The local density λ(x, y) equied fo this simulation was the appoximation A4 aleady shown in Figue L() Figue Inhomogeneous L-function (black) and 90% simulations inteval (dashed ed) calculated fo the Lagunculaia tees inside Lagoa B. The simulations inteval was calculated via Monte Calo method (Besag 1977) fo the inhomogeneous Poisson pocess hypothesis with 1000 simulations.