Deriving space use patterns from animal interaction mechanisms Jonathan Potts, Postdoctoral Fellow, University of Alberta, May 2013.

Deriving space use patterns from animal interaction mechanisms

Jonathan Potts, Postdoctoral Fellow, University of Alberta, May 2013

From mechanism to pattern

Movement


Direct interactions


Mediated interactions


Environmental interactions


Outline

Outline

• Modelling animal movement: the “correlated random walk” framework

Outline


• Adding in environmental interactions: step selection functions

Outline



• Including animal-animal interactions: coupled step selection functions

Outline



• 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


in the study area

• Notice that is independent of (please ask).




in the study area

• Notice that is independent of (please ask).• Use this to test various hypotheses about the nature of .



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


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.


Hypothesis 1 (tendency not to go into another’s territory):


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!

Deriving space use patterns from animal interaction mechanisms Jonathan Potts, Postdoctoral Fellow, University of Alberta, May 2013.

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