deriving space use patterns from animal interaction mechanisms jonathan potts, postdoctoral fellow,...
TRANSCRIPT
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 walkExample step length distribution:
Movement: correlated random walkExample 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 is:
where is the step length distribution and 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 is:
• is the step length distribution,• is the turning angle distribution• is a weighting function• E is information about the environmentFortin 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.
Example 1: Amazonian bird flocks • is a function denoting the value of each point
in the study area
𝑓 (𝒙|𝒚 ,𝜃0 )∝ 𝜌 (|𝒙−𝒚|)𝑉 (𝒙 , 𝒚 , 𝜃0 )𝑊 (𝒙 , 𝒚 ,𝐸)
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
Example 1: Amazonian bird flocks • is a function denoting the value of each point
in the study area
• Notice that is independent of (please ask).
𝑓 (𝒙|𝒚 ,𝜃0 )∝ 𝜌 (|𝒙−𝒚|)𝑉 (𝒙 , 𝒚 , 𝜃0 )𝑊 (𝒙 , 𝒚 ,𝐸)
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
Example 1: Amazonian bird flocks • is a function denoting the value of each point
in the study area
• Notice that is independent of (please ask).• Use this to test various hypotheses about the nature of .
𝑓 (𝒙|𝒚 ,𝜃0 )∝ 𝜌 (|𝒙−𝒚|)𝑉 (𝒙 , 𝒚 , 𝜃0 )𝑊 (𝒙 , 𝒚 ,𝐸)
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:
Hypotheses
1. Birds are more likely to move to higher canopies:
2. In addition, birds are more likely to move to lower ground:
(
Maximum likelihood technique
1. Find the that maximises:
where and are, respectively, the sequence of positions and trajectories from the data, and
Maximum likelihood technique
2. Find the that maximises:
where is the value of that maximises the likelihood function on the previous page, and
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:
where is the intersection of 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:
where is the intersection of with the half-line starting at and continuing on a bearing of .
• PDE in the simple case where the turning angle distribution is uniform and :
Moorcroft and Barnett (2008) Mechanistic home range models and resource selection analysis: a reconciliation and unification. Ecology 89(4), 1112–1119
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
Movement data
Mathematical analysis
SimulationsStep
selection functions
Statistical tests, e.g. MLE
Master equations, PDEs
Coupled step selection functionsOne step selection function for each agent and include an interaction term :
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 hypothesesTerritorial marking (vocalisations): if any flock is at position at time totherwise.
Amazon birds: testing hypothesesTerritorial marking (vocalisations): if any flock is at position at time totherwise.
Hypothesis 1 (tendency not to go into another’s territory):
Amazon birds: testing hypothesesTerritorial marking (vocalisations): if any flock is at position at time totherwise.
Hypothesis 1 (tendency not to go into another’s territory):
Hypothesis 2 (tendency to retreat after visiting another’s territory):
where is a von Mises distribution, is the bearing from to and is the bearing from to a central point within the territory and if X is true and 0 otherwise.
Amazon birds: space use patterns
between competing models
AcknowledgementsMark 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)
Movement and interaction data
Mathematical analysis
SimulationsCoupled step selection functions
Statistical tests
The final frontier!
Conclusion
“The challenge is to develop a statisticalmechanics for ecological systems” Simon Levin
Spatial patterns
Thanks for listening!