Beside the point? Biodiversity research and spatial point process modelling Göttingen, November 2008
threats to biodiversity biodiversity and ecosystem functioning increasing anthropogenic impact on ecosystems recent decades have seen an increasing threat to (species) biodiversity worldwide one of the key agreements at the 1992 Earth Summit in Rio de Janeiro is the convention on biological diversity aim achieve by 2010 a significant reduction of the current rate of biodiversity loss
threats to biodiversity biodiversity and ecosystem dynamics worldwide, this has been resulting in an increased effort to maintain diversity or stop decline ecologists seek to understand community dynamics how can large numbers of species co-exist? do depauperate systems function differently from communities with higher diversity? loss of biodiversity = loss of functionality?
individual-based approaches ecological modelling in the past : mean field approach : population dynamics described based on the average abundance of different species across space BUT : only valid for highly motile species in homogeneous environments in plants, interactions take place in a spatial context interact mainly with immediate neighbours due to limited mobility today : individual-based, spatially explicit approaches
space and biodiversity individual based models modelling from the individuals perspective has become popular in (plant) ecology : spatial birth and death processes dynamic models mechanistic models... But how about spatial point process modelling? many motivating examples for spatial point processes have been derived from ecology but : however spatial point processes have rarely been used to directly answer topical ecological questions
space and biodiversity biodiversity and spatial point processes aim use existing spatial point process methods in the context of biodiversity research develop methodology that is suitable for this purpose
space and biodiversity It is essential to be able to measure biodiversity assess current measures of biodiversity develop new measures of biodiversity understand biodiversity processes that organise ecosystems and mechanisms that sustain biodiversity preserve biodiversity by understanding individual species ideal habitat preserve the habitat
space and biodiversity measuring biodiversity define suitable measures of changes in biodiversity (Studeny, Buckland, Magurran, Illian, 2008) develop measures of spatial biodiversity based on graph theory (Rajala, Illian, Penttinen (in preparation)) understanding biodiversity modelling of biodiverse plant community in arid environment (Illian, Møller, Waagepetersen, 2007) assessing predictions from biodiversity theory in rainforests (Gimona, Burslem, Illian, 2008, Illian, 2008a) modelling individuals growth in a spatial context (Schneider, Law, Illian, 2006, Berthelsen, Law, Illian (in preparation)) preserving biodiversity conservation of herbivore marsupial mammals in Australia (Illian, Beale, in preparation) conservation of muskoxen in Greenland (Illian, 2008b)
space and biodiversity measuring biodiversity understanding biodiversity preserving biodiversity conservation of herbivore marsupial mammals in Australia (Illian, Beale, in preparation) conservation of muskoxen in Greenland (Illian, 2008b)
replicated spatial point patterns preserving biodiversity Zackenberg research station, east Greenland spatial locations of muskoxen herds recorded at different time points throughout different years habitat choice : does the spatial distribution depend on habitat quality? behavioural ecology : do male and female herds interact differently with other herds?
replicated spatial point patterns preserving biodiversity Zackenberg research station, east Greenland spatial locations of muskoxen herds recorded at different time points throughout different years habitat choice : does the spatial distribution depend on habitat quality? behavioural ecology : do male and female herds interact differently with other herds?
replicated spatial point patterns animals move in and out of the observation area habitat changes over time (e.g. snow melts) group interactions change over time (e.g. in mating season) treat time as random factor and habitat and group type as fixed factors in a spatial point process model
replicated spatial point patterns animals move in and out of the observation area habitat changes over time (e.g. snow melts) group interactions change over time (e.g. in mating season) treat time as random factor and habitat and group type as fixed factors in a spatial point process model
mixed models mixed models for spatial point patterns Superposition x of the T point patterns with x = T t=1 x t. At time t consider a pairwise interaction process for x t λ θ (u t ; x t ) = b θ (u t ) n(x t) i=j,x i,t u h θ (u t, x i ), where n(x t ) number of points in point pattern x t. Consider the loglinear case where b θ (u t ) = exp(θ T B 1 (u t ) + φ T t B 2 (u t )) h θ (u t, v t ) = exp(θ T H 1 (u t, v t ) + φ T t H 2 (u t, v t )). θ is a parameter vector of fixed effects and φ T is a parameter vector of random effects, where φ T N(0, σ 2 I )
mixed models mixed models for spatial point patterns conditional intensity for the superposition x over all time points ( T λ θ,φ (u; v) = exp θ T B 1 (u) + φ T t B 2 (u t )+ θ T n(x) i=1,x i u t=1 H 1 (u, x i ) + T t=1 φ T t n(x t) H 2 (u t, x i,t ). i=1,x i,t u t
mixed models mixed models for spatial point patterns consider the pseudolikelihood with this conditional intensity over a subset A W PL A (θ, φ, x) = n(x) b θ,φ (x i ) (h θ,φ (x i, x j )) i j i=1 exp W n(x) b θ,φ (u) h θ,φ (u, x i )du. i=1
mixed models parameter estimation apply Berman-Turner device ; approximate the integral in the pseudolikelihood by a finite sum log pseudolikelihood is formally equivalent to the log likelihood of independent weighted Poisson variables since the conditional intensity is expressed as the sum of fixed and random factors, employ a generalised linear mixed model with log link and Poisson outcome estimates based on software for generalised linear mixed models using penalised quasi-likelihood
Strauss process example construct the Strauss process with mixed effects with b θ (u t ) = β + β t and h θ (u t, v t ) = γ + γ t, where β (γ) is a constant describing intensity (interaction strength) common to all time points. β t ( γ t ) is a sample from a random variable β t N(0, σ 2 β) ( γ t N(0, σ2 γ ) ) reflecting variation in intensity (interaction strength) among different time points. fit the model with a call such as lmer(y v + (1+v random.eff), family=poisson, weights=weights).
Strauss process simulation study Replicated realisations of Strauss processes with r = 0.1 in the unit square were generated (10 replicates, 100 simulation runs) with β + β t, where β = 50 and β t N(0, 20) and γ + γ t, where γ = 0.5 and γ t N(0, 0.2) comparison between analysis of individual patterns and analysis of all patterns with random effect for replicate main result : variance of estimators much smaller for mixed model than when each replicate is analysed separately
more general model more suitable model the simple Strauss process is too simplistic for the muskoxen data assumption of constant interaction within a fixed radius is not realistic ; use function with decreasing interaction strength pattern highly inhomogeneous ; use altitude and vegetation index as covariate, i.e. as a fixed effect Interaction function : { ( 1 (r/r) 2 ) 2 if 0 < r R h θ (r) = 0 else for r 0, R 0
more general model results for the year 2005 glmmpql(y v + altitude + vegetationindex, random = 1 time/v, family=poisson, weights=w). results indicate that intensity and interaction vary strongly with time both altitude and vegetation index (and their interaction) highly significant
preserving biodiversity aim prevention of decline in abundance of herbivore marsupial mammals in Australia need to determine properties of the optimal or suitable habitat for a given species
to determine properties that constitute an optimal habitat analyse the properties of the trees the animals feed on understand plant-herbivore interactions trees : defence mechanisms mammals : adaptation mechanisms and preferences determine properties of the environment that influence these interactions this study : koalas feeding on eucalyptus trees
chemical defence in eucalyptus tree foliage chemically complex defence mechanisms in foliage studies have shown that FPC (formylated phloroglycenol compounds) determine frequency of feeding by folivores toxic to most species cause slight nausea in koalas local soil fertility influences FPC concentration as well as nutritional values in the individual trees
koalas (phascolarctos cinereus) I arboreal marsupial herbivores native to Australia have a very low metabolic rate rest motionless for about 18 to 20 hours a day, sleeping most of that time live almost entirely on eucalyptus leaves selective feeding : high in nutrients and low in toxins (FPC)
koalas (Phascolarctos cinereus) II extreme example of evolutionary adaptation liver deactivates the toxic components gut is greatly enlarged to extract the maximum amount of nutrients specialists : exposed to limited range of toxins adaptation to large concentration of one or a few toxins necessary understanding koala - eucalyptus interaction is crucial for conservation
data set study conducted at the Koala Conservation Centre on Phillip Island, near Melbourne, Australia run from 1993 to 2004 20 koalas present in the reserve at all times throughout study reserve enclosed by a koala-proof fence koalas in the reserve forage and breed naturally
data set koala tree visitation several studies have concluded that tree visitation is a reasonable measure of koala foraging preferences here : diurnal tree use by individual koalas collected at monthly intervals between 1993 and March 2004 entire reserve searched for koalas identities of all koalas found and of the trees occupied were recorded. spatial autocorrelation : koala move very little and are more likely to remain in vicinity
data set foliage collection and analysis all 915 trees in woodland individually numbered and mapped circumference at a height of 130 cm recorded foliage sampled from the canopy of each eucalyptus tree samples from each tree analysed for FPC content (= toxins) nitrogen concentration (= nutrients) based on established methods and Near Infrared Reflectance Spectroscopy (NIRS) calculation of palatability : combination of toxins and nutrients based on previous studies spatial autocorrelation : nutrient levels and toxins influenced by other trees in vicinity (competition for resources)
exploratory data analysis estimated intensity palatability 5738900 5739000 5739100 5739200 5739300 348100 348200 348300 348400 348500 0.01 0.015 0.02 0.025
exploratory data analysis estimated intensity koala use 5738900 5739000 5739100 5739200 5739300 348100 348200 348300 348400 348500 0.01 0.015 0.02 0.025
exploratory data analysis estimated intensity palatability 5739100 5739120 5739140 5739160 5739180 5739200 348160 348180 348200 348220 348240 348260 0.02 0.025 0.03 0.035 0.04
exploratory data analysis estimated intensity koala use 5739100 5739120 5739140 5739160 5739180 5739200 348160 348180 348200 348220 348240 348260 0.02 0.025 0.03 0.035 0.04
exploratory data analysis mark correlation functions does the distance between trees influence the respective values of the marks? are the marks of neighbouring marks smaller (or larger) than average? where µ is the mean mark. k mm = E or (m(o) m(r)) µ 2 for r > 0 note the conditional nature of k mm note also distinct difference to (second order) summary characteristics in geostatistics ; functions are e.g. not necessarily positive definite
exploratory data analysis mark correlation functions k mm = E or (m(o) m(r)) µ 2 for r > 0 other functions than the product of the marks may be considered, leading to other characteristics estimation based on non-parametric (kernel) estimators issues of bandwidth choice...
exploratory data analysis mark correlation functions palatability of leaves koala use m(r) 0.5 0.7 0.9 1.1 m(r) 0.2 0.6 1.0 1.4 0 5 10 15 20 r 0 5 10 15 20 r
exploratory data analysis mark dependence koala use of tree 0 2 4 6 8 0 1000 2000 3000 4000 5000 6000 palatability of leaves of tree
modelling approach exploratory analysis and previous studies suggest : tree locations depend on environmental properties foliage chemistry (i.e. palatability) depends on environmental properties and local spatial structure koala visitation depends on environmental properties, local spatial structure and foliage chemistry approach : hierarchical intensity marked Cox process model two types of marks : 1. palatability of leaves 2. koala use of trees (depends on palatability)
modelling approach The tree locations are modelled using a log Gaussian Cox process, i.e. a Cox process with random intensity Λ(s) = exp{z(s)}, where {Z(s)} is a (stationary and isotropic) Gaussian random field, s R 2. conditional on the (unknown) environmental conditions Poisson process summary characteristics (intensity, pair correlation function) known analytically for details see Møller et al. (1998)
modelling approach The leave marks m L are modelled using an intensity marked log Gaussian Cox process, where the marks are modelled as : m L (x i ) Λ(x i ) N(c + d Z(x i ) + U(x i ), σ) where x i N, N unmarked point process, {Z(s)} is as above and {U(s)} another Gaussian random field (with zero mean). conditional on intensity, marks independent summary characteristics (e.g. mark correlation function) known analytically for details see Ho, Stoyan (2008), Myllymäki, Penttinen (2008)
modelling approach modelling the koala marks The koala marks m K are modelled using a hierarchically and intensity marked Cox process, where the marks are modelled as : m K (x i ) Λ(x i ) Poisson(e + f Z(x i ) + g m L (x i ) + V (x i )), where x i, {Z(s)} as above and {V (s)} another Gaussian random field (with zero mean). summary characteristics (e.g. mark correlation function) can be determined analytically for details, see Illian, Beale (in preparation)
modelling approach parameter estimation 1 minimum contrast method problems with identifiability 2 Bayesian approach imposing (uninformative) prior distributions on the parameters estimation based on MCMC ; birth and death algorithm algorithm : to update delete a point (or generate a new point) with both marks currently convergence very slow except for greatly simplified models ; problems with parallel (independent) up-dating of location and marks?
modelling approach current results (simplified) models indicate dependence of palatability on location may be confirmed but small dependence of koala use on palatability strong need to improve convergence to fit more suitable models and hence to obtain more reliable results
modelling approach general discussion koalas in ecological context data consisting of spatial locations and several dependent marks likely to be relevant in the future increasing number of parameters increasing problems with identifiability and convergence muskoxen spatial point process models with fixed and random effects may be constructed and applied to replicated patterns of muskoxen herds models become increasingly complex but standard software may be used for parameter estimation spatial point process methods may well be used to find answers to topical questions in ecology