Deriving space use patterns from animal interaction mechanisms
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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.

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Deriving space use patterns from animal interaction mechanisms

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Deriving space use patterns from animal interaction mechanisms

Deriving space use patterns from animal interaction mechanisms

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


From mechanism to pattern

From mechanism to pattern

Movement


From mechanism to pattern1

From mechanism to pattern

Directinteractions


From mechanism to pattern2

From mechanism to pattern

Mediated interactions


From mechanism to pattern3

From mechanism to pattern

Environmental interactions


From mechanism to pattern4

From mechanism to pattern


Outline

Outline


Outline1

Outline

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


Outline2

Outline

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

  • Adding in environmental interactions: step selection functions


Outline3

Outline

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

  • Adding in environmental interactions: step selection functions

  • Including animal-animal interactions: coupled step selection functions


Outline4

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


Movement correlated random walk1

Movement: correlated random walk

Example step length distribution:


Movement correlated random walk2

Movement: correlated random walk

Example step length distribution:

Example turning angle distribution:


Mathematical formulation

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

Adding environmental interactions

A, B, C different habitats. B = worse, A = better, C = best.


The step selection function

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 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.


Example 1 amazonian bird flocks

Example 1: Amazonian bird flocks

  • is a function denoting the value of each point in the study area

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 flocks1

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).

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 flocks2

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 .

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

Hypotheses

1. Birds are more likely to move to higher canopies:


Hypotheses1

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

Maximum likelihood technique

1. Find the that maximises:

where and are, respectively, the sequence of positions and trajectories from the data, and


Maximum likelihood technique1

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: 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

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 pdes1

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 :

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

Moorcroft and Barnett (2008) Mechanistic home range models and resource selection analysis: a reconciliation and unification.Ecology 89(4), 1112–1119


Deriving space use patterns from animal interaction mechanisms

Movement data

Statistical tests, e.g. MLE

Step selection functions

Simulations

Master equations, PDEs

Mathematical analysis


C oupled step selection functions

Coupled step selection functions

One 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 memoryof 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

Amazon birds: testing hypotheses

Territorial marking (vocalisations):

if any flock is at position at time t

otherwise.


Amazon birds testing hypotheses1

Amazon birds: testing hypotheses

Territorial marking (vocalisations):

if any flock is at position at time t

otherwise.

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


Amazon birds testing hypotheses2

Amazon birds: testing hypotheses

Territorial marking (vocalisations):

if any flock is at position at time t

otherwise.

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

Amazon birds: space use patterns

between competing models


Acknowledgements

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)


Deriving space use patterns from animal interaction mechanisms

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!

Spatial patterns

Mathematical analysis


Thanks for listening

Thanks for listening!


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