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 𝜃0 is:
𝑓 𝒙 𝒚, 𝜃0 = 𝜌 𝒙 − 𝒚 𝑉 𝒙, 𝒚, 𝜃0
where 𝜌 𝑟 is the step length distribution and
𝑉 𝒙, 𝒚, 𝜃0 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 𝜃0 is:
𝑓 𝒙 𝒚, 𝜃0 ∝ 𝜌 𝒙 − 𝒚 𝑉 𝒙, 𝒚, 𝜃0 𝑊(𝒙, 𝒚, 𝐸)
•
•
•
•
𝜌 𝑟 is the step length distribution,
𝑉 𝒙, 𝒚, 𝜃0 is the turning angle distribution
𝑊(𝒙, 𝒚, 𝐸) 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.
𝑓 𝒙 𝒚, 𝜃0 ∝ 𝜌 𝒙 − 𝒚 𝑉 𝒙, 𝒚, 𝜃0 𝑊(𝒙, 𝒚, 𝐸)
Example 1: Amazonian bird flocks
• 𝐸: Ω ⟶ ℝ is a function denoting the value of
each point in the study area Ω
• 𝑊 𝒙, 𝒚, 𝐸 = 𝐸 𝒙
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
𝑓 𝒙 𝒚, 𝜃0 ∝ 𝜌 𝒙 − 𝒚 𝑉 𝒙, 𝒚, 𝜃0 𝑊(𝒙, 𝒚, 𝐸)
Example 1: Amazonian bird flocks
• 𝐸: Ω ⟶ ℝ is a function denoting the value of
each point in the study area Ω
• 𝑊 𝒙, 𝒚, 𝐸 = 𝐸 𝒙
• Notice that 𝑊 𝒙, 𝒚, 𝐸 is independent of 𝒚
(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
𝑓 𝒙 𝒚, 𝜃0 ∝ 𝜌 𝒙 − 𝒚 𝑉 𝒙, 𝒚, 𝜃0 𝑊(𝒙, 𝒚, 𝐸)
Example 1: Amazonian bird flocks
• 𝐸: Ω ⟶ ℝ is a function denoting the value of
each point in the study area Ω
• 𝑊 𝒙, 𝒚, 𝐸 = 𝐸 𝒙
• Notice that 𝑊 𝒙, 𝒚, 𝐸 is independent of 𝒚
(please ask).
• Use this to test various
hypotheses about the
nature of 𝐸: 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:
𝐸1 𝒙 = (canopy height at 𝐱)𝜶
Hypotheses
1. Birds are more likely to move to higher canopies:
𝐸1 𝒙 = (canopy height at 𝐱)𝜶
2. In addition, birds are more likely to move to lower
ground:
𝐸2 𝒙 = (canopy height at 𝐱)𝜶 (ground height at 𝐱)−𝜷
Maximum likelihood technique
1. Find the 𝛼 that maximises:
𝑁
𝑓1 𝒙𝑛 𝒙𝑛−1 , 𝜃𝑛−1 , 𝛼
𝑛=1
where 𝒙0 , … , 𝒙𝑁 and 𝜃0 , … , 𝜃𝑁 are, respectively,
the sequence of positions and trajectories from
the data, and
𝑓1 𝒙 𝒚, 𝜃0 ∝ 𝜌 𝒙 − 𝒚 𝑉 𝒙, 𝒚, 𝜃0 𝑊(𝒙, 𝒚, 𝐸1 )
Maximum likelihood technique
2. Find the 𝛽 that maximises:
𝑁
𝑓2 𝒙𝑛 𝒙𝑛−1 , 𝜃𝑛−1 , 𝛼𝑚𝑎𝑥 , 𝛽
𝑛=1
where 𝛼𝑚𝑎𝑥 is the value of 𝛼 that maximises the
likelihood function on the previous page, and
𝑓2 𝒙 𝒚, 𝜃0 ∝ 𝜌 𝒙 − 𝒚 𝑉 𝒙, 𝒚, 𝜃0 𝑊(𝒙, 𝒚, 𝐸2 )
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:
𝜋
𝑢 𝒙, 𝜃, 𝑡 + Δ𝑡 =
−𝜋
𝑑𝜃0
𝛾(𝜃,𝒙)
𝑑𝒚𝑓 𝒙 𝒚, 𝜃0 𝑢(𝒚, 𝜃0 , 𝑡)
where 𝛾 𝜃, 𝒙 is the intersection of Ω with the half-line starting
at 𝒙 and continuing on a bearing of 𝜃 + 𝜋.
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:
𝜋
𝑢 𝒙, 𝜃, 𝑡 + 𝜏 =
−𝜋
𝑑𝜃0
𝑑𝒚𝑓 𝒙 𝒚, 𝜃0 𝑢(𝒚, 𝜃0 , 𝑡)
𝛾(𝜃,𝒙)
where 𝛾 𝜃, 𝒙 is the intersection of Ω with the half-line starting
at 𝒙 and continuing on a bearing of 𝜃 + 𝜋.
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 𝑊 𝒙, 𝒚, 𝐸 = 𝐸 𝒙 :
𝜕𝑢
= −𝛻 𝑐 𝒙 𝑢 𝒙, 𝑡 + 𝛻 2 [𝑑 𝒙 𝑢 𝒙, 𝑡 ]
𝜕𝑡
𝑐 𝑥 =
𝑀2 (𝜏)𝐸𝑥 (𝑥)
lim
,
𝜏𝐸(𝑥)
𝜏→0
𝑑 𝑥 =
𝑀2 (𝜏)
lim
𝜏→0 2𝜏
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 𝐶𝑖 (𝒙, 𝒚, 𝑃𝑖,𝑡 ):
𝑓𝑖,𝑡 𝒙 𝒚, 𝜃0 ∝ 𝜌𝑖 𝒙 − 𝒚 𝑉𝑖 𝒙, 𝒚, 𝜃0 𝑊𝑖 (𝒙, 𝒚, 𝐸)𝐶𝑖 (𝒙, 𝒚, 𝑃𝑖,𝑡 )
where 𝑃𝑖,𝑡 represents both the
population positions and any
traces of their past positions
left either in the environment
or in the memory of agent 𝑖.
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):
𝑃𝑖,𝑡 𝒙 = 𝑇 if any flock 𝑗 ≠ 𝑖 is at position 𝒙 at time t
𝑃𝑖,𝑡 𝒙 = min{𝑃𝑖,𝑡−𝜏 𝒙 − 𝜏, 0} otherwise.
Amazon birds: testing hypotheses
Territorial marking (vocalisations):
𝑃𝑖,𝑡 𝒙 = 𝑇 if any flock 𝑗 ≠ 𝑖 is at position 𝒙 at time t
𝑃𝑖,𝑡 𝒙 = min{𝑃𝑖,𝑡−𝜏 𝒙 − 𝜏, 0} otherwise.
Hypothesis 1 (tendency not to go into another’s territory):
𝐶𝑖 𝒙, 𝒚, 𝑃𝑖,𝑡 = {[𝑇 − max 𝑃𝑗,𝑡 (𝒙)]/𝑇}𝛾
𝑖≠𝑗
Amazon birds: testing hypotheses
Territorial marking (vocalisations):
𝑃𝑖,𝑡 𝒙 = 𝑇 if any flock 𝑗 ≠ 𝑖 is at position 𝒙 at time t
𝑃𝑖,𝑡 𝒙 = min{𝑃𝑖,𝑡−𝜏 𝒙 − 𝜏, 0} otherwise.
Hypothesis 1 (tendency not to go into another’s territory):
𝐶𝑖 𝒙, 𝒚, 𝑃𝑖,𝑡 = {[𝑇 − max 𝑃𝑗,𝑡 (𝒙)]/𝑇}𝛾
𝑖≠𝑗
Hypothesis 2 (tendency to retreat after visiting another’s territory):
𝐶𝑖 𝒙, 𝒚, 𝑃𝑖,𝑡 = 𝑉 𝑘𝐼 max 𝑃𝑗,𝑡 𝒙 > 0 , 𝜃 − 𝜃𝑐
𝑖≠𝑗
where 𝑉(𝑘, 𝜙) is a von Mises distribution, 𝜃 is the bearing from 𝒙 to 𝒚 and
𝜃𝑐 is the bearing from 𝒚 to a central point within the territory and 𝐼 𝑋 = 1
if X is true and 0 otherwise.
Amazon birds: space use patterns
Δ𝐴𝐼𝐶 = 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!