Investigating seed dispersal and natural. moment methods

Transcription

1 Investigating seed dispersal and natural enemy attack with wavelet ltvariances and moment methods Helene C. Muller Landau and Matteo Detto Smithsonian Tropical Research Institute

2 Heterogeneous spatial patterns are ubiquitous in ecology Rinorea silvatica Due to multiple processes, including local dispersal, neighborhood competition, and habitat heterogeneity.

3 Spatial processes are integral to population Connell 1971: and community dynamics The mechanism I suggest is that each tree species has host-specific enemies which attack it and any of its offspring which are close to the parent. The healthy parent tree supports a large population of these enemies without itself being killed, but the seedlings, whose growth is suppressed in the heavy shade, succumb to the attack of insects and other enemies which come from the parent tree itself or the soil below it. Janzen 1970:

4 Talk outline Wavelet variances introduced Eti Estimating seed dispersal and density dependence d d parameters from observed patterns Deriving expected wavelet variances from models using moment methods Estimating model parameters from spatial patterns Investigating how seed dispersal and natural enemy g g p y parameters influence population dynamics

5 Measuring aggregation: Pair density correlation 1 C( ) N( x) N( x) ( ) dx A 2 nd moment Focal Individual C( ) 1 A N tot i1 Ni 2d ity corre elation d 2 P air dens 1 0

6 From pair density correlation to wavelet variance Apply Fourier transforms: Apply wavelet filters: ˆ ( ) ( ) i x N N x e dx 2 C ( ) N ( ) N H ( ) ˆ To obtain the normalized wavelet variance: 2 1 ν ( ) ˆ C ( ) ( ) d 1 N H Wavelet variance Fourier scale

7 Wavelet variances vs. pair correlation densities Homogenous vs. inhomogenous habitat Poisson process (global dispersal) Detto & Muller-Landau, in press, Am Nat Differences reflecting habitat emerge at the (large) scales of habitat variation

8 Wavelet variances vs. pair correlation densities Homogenous vs. inhomogenous habitat Local dispersal Detto & Muller-Landau, in press, Am Nat Small-scale structure related to dispersal is the same in both cases

9 Wavelet variances vs. pair correlation densities Detto & Muller-Landau, in press, Am Nat

10 Deriving expected wavelet variances from individual based based, spatially explicit models using moment methods Detto & Muller-Landau, in press, Am Nat

11 Model 1: An individual-based, model with local dispersal Model formulation dn x, t dt f reproduction mortality D x xn x, tdx mn x, t N( x, t) f Dx ( ) m Population density in space and time Reproductive rate Dispersal kernel Density independent mortality rate

12 Model 1: an individual-based, model with local dispersal Steady-state solution for the wavelet variance dn x, t dt f reproduction mortality D x xn x, tdx mn x, t The steady-state solution: f=m and C ( ) ND ( ) 1 D ( ) Pair correlation density D 1 D 2 ν ( ) 1 (, ) I Wavelet variance Detto & Muller-Landau, in press, Am Nat

13 Model 1: An individual-based, model with local dispersal Relationship of dispersal parameter to wavelet variance ν ( ) 1 Dˆ (, ) 2 1 D Detto & Muller-Landau, in press, Am Nat

14 Model 1: An individual-based, model with local dispersal Differences among dispersal kernels ν ( ) Dˆ (, ) D Detto & Muller-Landau, in press, Am Nat

15 Model 2: Local dispersal and conspecific inhibition Model formulation dn( x, t) dt reproduction Density-dependent establishment Densityindepenent mortality m f Dx ( x) Nx (, t) 1 Kx ( x) Nx (, tdx ) dx mnxt (, ) f m K( x) Density-dependent establishment rate Establishment kernel

16 Model 2: Local dispersal and conspecific inhibition: Steady state wavelet variance n 2 II 1 ( ) 1 Dˆ (, ) Kˆ (, ) 1 = (f-m)/m density dependent effect D 2 = D / K scaling of dispersal to density dependence 1 K Approximate solution from closing the third moment. wavele et variance < 1 1 = 1 wavele et variance 1 > < 1 2 = > 1 2 = 1 1 = Fourier scale Fourier scale Detto & Muller-Landau, in press, Am Nat

17 Estimating process parameters from spatial patterns using moment methods and wavelets

18 Statistical properties of the wavelet variances 1. Expectations under complete spatial randomness et variance wavel expectation 1000 simulations predicted 95% CI Fourier scale (m) A predic di cted t d predicted 1.05 = 2 1 B = 4 1 C = 8 = D observed quantiles 0.8 E observed quantiles This provides a basis for statistical tests of the null hypothesis of complete spatial randomness. Detto & Muller-Landau, in press, Am Nat

19 Statistical properties of the wavelet variances 2. Expectations under nonrandom processes avelet varia ance w 10 2 For a Gaussian seed dispersal kernel analytical solution with D =1 m 10 1 Fourier scale pr redicted quan ntiles simulations predicted 95% CI = This provides a basis for estimating model parameters using maximum likelihood. Detto & Muller-Landau, in press, Am Nat

20 Model fitting results example 1 Detto & Muller-Landau Landau, in press, Am Nat scale (m)

21 Model fitting results example 2 Detto & Muller-Landau Landau, in press, Am Nat scale (m)

22 Advantages of wavelet ltvariances require only static data separate processes operating at different scales analytically tractable, thus can be linked to models Ongoing work Characterize interspecific variation Analyze spatiotemporal p patterns Models for heterogeneous environments Multi species models Detto and Muller-Landau

23 Talk outline Wavelet variances introduced Eti Estimating seed dispersal and density dependence d d parameters from observed patterns Deriving expected wavelet variances from models Estimating model parameters from spatial patterns Investigating how seed dispersal and natural enemy parameters influence population dynamics

24 When do specialized natural enemies contribute most strongly to stabilization? Janzen emphasized local distance and densitydependence, and subsequent studies have focused on quantifying this, especially the extreme cases of overcompensation and overdispersion. i l h k l ll h h But are natural enemies that attack locally the ones that contribute most strongly to stabilization?

25 How is the strength of stabilization affected by the spatial ilscales of seed dispersal and enemy attack? Kernel Den nsity (Disp persal or Competitio on) Seed dispersal kernel Enemy attack scale ~ dispersal scale Shorter enemy attack Longer enemy attack Distance from Parent/Adult

26 Will stabilization be stronger at shorter or longer scales of enemy attack? n)y nsity (Dispersa l or Competition Kernel De 0.0 Distance from Parent/Adult? Focal species frequency Per capita population growth rate

27 dn x, t dt Methods a spatial logistic model: individual-based, spatially explicit, continuous f D reproduction densityindependent mortality density dependent mortality x xn x, tdx mn x, t mn x, t Kx x N x, t N ( x, t ) Populationdensity in space and time f D ( x ) m m K ( x ) Reproductive rate Dispersal kernel Density independent mortality rate Density dependent mortality rate Conspecific competition kernel representing enemy attack dx

28 Moment methods for the spatial logistic model [Unpublished material omitted from posted pdf]

29 Results: Effects on the spatial structure [Unpublished results omitted from posted pdf]

30 Results: Effects on strength of stabilization [Unpublished results omitted from posted pdf]

31 Results: Effects on invasion growth rate [Unpublished results omitted from posted pdf]

32 These results are consistent with previous simulation results for spatially explicit community models In a discrete space simulation model Enemy dispersal scale = 10 m Enemy dispersal scale = 20 m Enemy dispersal scale = 40 m Enemy dispersal scale = 80 m Muller-Landau & Adler 2007 Adler & Muller-Landau 2005 Ecology Letters found parallel results for species diversity in a continuous space simulation model.

33 What explains these results? The shape of the curve relating per capita success to focal species abundance. Long-distance attack results in a steady decline, providing stronger stabilization... and higher diversity maintained. rate Per cap pita populat tion growth Focal species frequency Local dispersal and attack results in an initial fast decline and then a slow decline, providing less protection from extinction... and lower diversity maintained. Adler & Muller-Landau 2005 Ecology Letters

34 Summary Wavelet variances provide a key tool for investigating spatial patterns, one that separates influences operating at different spatial scales and is analytically tractable. Moment methods can provide analytical solutions or approximations of wavelet variances expected under different spatially explicit, individual based ecological process models. We provide a statistical framework for testing the null hypothesis of complete spatial randomness and for fitting ecological process models from spatial patterns using wavelet variances. Matteo Detto Detto & Muller-Landau, in press, Am Nat Specialized natural enemies that attack over long distances from adult plants (relative to seed dispersal distances) contribute more strongly to stabilizing plant populations and promoting species coexistence than do those that disperse over short distances. Ongoing/future work examines spatiotemporal patterns, habitat heterogeneity, and interspecific interactions.