Report

Deriving space use patterns from animal interaction mechanisms Jonathan Potts, Postdoctoral Fellow, University of Alberta, May 2013 From mechanism to pattern Movement From mechanism to pattern Direct interactions From mechanism to pattern Mediated interactions From mechanism to pattern Environmental interactions From mechanism to pattern Outline Outline • Modelling animal movement: the “correlated random walk” framework Outline • Modelling animal movement: the “correlated random walk” framework • Adding in environmental interactions: step selection functions Outline • Modelling animal movement: the “correlated random walk” framework • Adding in environmental interactions: step selection functions • Including animal-animal interactions: coupled step selection functions Outline • Modelling animal movement: the “correlated random walk” framework • Adding in environmental interactions: step selection functions • Including animal-animal interactions: coupled step selection functions • Throughout: how do these models help us understand space use phenomena? Movement: correlated random walk Movement: correlated random walk Example step length distribution: Movement: correlated random walk Example step length distribution: Example turning angle distribution: Mathematical formulation Probability of moving to position x given that the animal was previously at position y and arrived there on a trajectory 0 is: , 0 = − , , 0 where is the step length distribution and , , 0 the turning angle distribution. Adding environmental interactions A, B, C different habitats. B = worse, A = better, C = best. The step selection function Probability of moving to position x given that the animal was previously at position y and arrived there on a trajectory 0 is: , 0 ∝ − , , 0 (, , ) • • • • is the step length distribution, , , 0 is the turning angle distribution (, , ) is a weighting function E is information about the environment Fortin D, Beyer HL, Boyce MS, Smith DW, Duchesne T, Mao JS (2005) Wolves influence elk movements: Behavior shapes a trophic cascade in Yellowstone National Park. Ecology 86:1320-1330. , 0 ∝ − , , 0 (, , ) Example 1: Amazonian bird flocks • : Ω ⟶ ℝ is a function denoting the value of each point in the study area Ω • , , = Potts JR, Mokross K, Stouffer PC, Lewis MA (in review) Step selection techniques uncover the environmental predictors of space use patterns in flocks of Amazonian birds. Ecology , 0 ∝ − , , 0 (, , ) Example 1: Amazonian bird flocks • : Ω ⟶ ℝ is a function denoting the value of each point in the study area Ω • , , = • Notice that , , is independent of (please ask). Potts JR, Mokross K, Stouffer PC, Lewis MA (in review) Step selection techniques uncover the environmental predictors of space use patterns in flocks of Amazonian birds. Ecology , 0 ∝ − , , 0 (, , ) Example 1: Amazonian bird flocks • : Ω ⟶ ℝ is a function denoting the value of each point in the study area Ω • , , = • Notice that , , is independent of (please ask). • Use this to test various hypotheses about the nature of : Ω ⟶ ℝ. Potts JR, Mokross K, Stouffer PC, Lewis MA (in review) Step selection techniques uncover the environmental predictors of space use patterns in flocks of Amazonian birds. Ecology Hypotheses 1. Birds are more likely to move to higher canopies: 1 = (canopy height at ) Hypotheses 1. Birds are more likely to move to higher canopies: 1 = (canopy height at ) 2. In addition, birds are more likely to move to lower ground: 2 = (canopy height at ) (ground height at )− Maximum likelihood technique 1. Find the that maximises: 1 −1 , −1 , =1 where 0 , … , and 0 , … , are, respectively, the sequence of positions and trajectories from the data, and 1 , 0 ∝ − , , 0 (, , 1 ) Maximum likelihood technique 2. Find the that maximises: 2 −1 , −1 , , =1 where is the value of that maximises the likelihood function on the previous page, and 2 , 0 ∝ − , , 0 (, , 2 ) Deriving space use patterns: stochastic simulations Potts JR, Mokross K, Stouffer PC, Lewis MA (in review) Step selection techniques uncover the environmental predictors of space use patterns in flocks of Amazonian birds. Ecology Deriving space use patterns: master equations and PDEs • From the step selection function to a master equation: , , + Δ = − 0 (,) , 0 (, 0 , ) where , is the intersection of Ω with the half-line starting at and continuing on a bearing of + . Potts JR, Bastille-Rousseau G, Murray DL, Schaefer JA, Lewis MA (in prep) Predicting local and non-local effects of resources on animal space use using a mechanistic step-selection model Deriving space use patterns: master equations and PDEs • From the step selection function to a master equation: , , + = − 0 , 0 (, 0 , ) (,) where , is the intersection of Ω with the half-line starting at and continuing on a bearing of + . Potts JR, Bastille-Rousseau G, Murray DL, Schaefer JA, Lewis MA (in prep) Predicting local and non-local effects of resources on animal space use using a mechanistic step-selection model • PDE in the simple case where the turning angle distribution is uniform and , , = : = − , + 2 [ , ] = 2 () () lim , () →0 = 2 () lim →0 2 Moorcroft and Barnett (2008) Mechanistic home range models and resource selection analysis: a reconciliation and unification. Ecology 89(4), 1112–1119 Movement data Statistical tests, e.g. MLE Step selection functions Simulations Master equations, PDEs Mathematical analysis Coupled step selection functions One step selection function for each agent and include an interaction term (, , , ): , , 0 ∝ − , , 0 (, , ) (, , , ) where , represents both the population positions and any traces of their past positions left either in the environment or in the memory of agent . Potts JR, Mokross K, Stouffer PC, Lewis MA (in prep) A unifying framework for quantifying the nature of animal interactions Amazon birds: testing hypotheses Territorial marking (vocalisations): , = if any flock ≠ is at position at time t , = min{,− − , 0} otherwise. Amazon birds: testing hypotheses Territorial marking (vocalisations): , = if any flock ≠ is at position at time t , = min{,− − , 0} otherwise. Hypothesis 1 (tendency not to go into another’s territory): , , , = {[ − max , ()]/} ≠ Amazon birds: testing hypotheses Territorial marking (vocalisations): , = if any flock ≠ is at position at time t , = min{,− − , 0} otherwise. Hypothesis 1 (tendency not to go into another’s territory): , , , = {[ − max , ()]/} ≠ Hypothesis 2 (tendency to retreat after visiting another’s territory): , , , = max , > 0 , − ≠ where (, ) is a von Mises distribution, is the bearing from to and is the bearing from to a central point within the territory and = 1 if X is true and 0 otherwise. Amazon birds: space use patterns Δ = 4.07 between competing models Acknowledgements Mark Lewis (UofA) Karl Mokross (Louisiana State) Guillaume Bastille-Rousseau (Trent) Philip Stouffer (Louisiana State) Dennis Murray (Trent) James Schaefer (Trent) Members of the Lewis Lab (UofA) Conclusion Movement and interaction data Statistical tests Coupled step selection functions Simulations “The challenge is to develop a statistical mechanics for ecological systems” Simon Levin The final frontier! Mathematical analysis Spatial patterns Thanks for listening!