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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

Deriving space use patterns from animal interaction mechanisms

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

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

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 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

slide28

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)

slide35

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

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