jonathan r. potts centre for mathematical biology, university of alberta. 3 december 2012 territory...
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Jonathan R. PottsCentre for Mathematical Biology, University of Alberta.
3 December 2012
Territory formation from an individual-based movement-and-interaction model
Outline
• Introduction: the modelling framework• Mathematics: analysing the model• Biology: Application to red foxes (Vulpes
vulpes). How do animals change their behaviour when populations go into decline?
Outline
• Introduction: the modelling framework• Mathematics: analysing the model• Biology: Application to red foxes (Vulpes
vulpes). How do animals change their behaviour when populations go into decline?
• Extension 1: central place foragers and stable home ranges
Outline
• Introduction: the modelling framework• Mathematics: analysing the model• Biology: Application to red foxes (Vulpes
vulpes). How do animals change their behaviour when populations go into decline?
• Extension 1: central place foragers and stable home ranges
• Extension 2: partial territorial exclusion, contact rates and disease spread
The “territorial random walk” model
Giuggioli L, Potts JR, Harris S (2011) Animal interactions and the emergence of territoriality PLoS Comput Biol 7(3)
The “territorial random walk” model
• Nearest-neighbour lattice random walkers
Giuggioli L, Potts JR, Harris S (2011) Animal interactions and the emergence of territoriality PLoS Comput Biol 7(3)
The “territorial random walk” model
• Nearest-neighbour lattice random walkers• Deposit scent at each lattice site visited
Giuggioli L, Potts JR, Harris S (2011) Animal interactions and the emergence of territoriality PLoS Comput Biol 7(3)
The “territorial random walk” model
• Nearest-neighbour lattice random walkers• Deposit scent at each lattice site visited• Finite active scent time, TAS
Giuggioli L, Potts JR, Harris S (2011) Animal interactions and the emergence of territoriality PLoS Comput Biol 7(3)
The “territorial random walk” model
• Nearest-neighbour lattice random walkers• Deposit scent at each lattice site visited• Finite active scent time, TAS
• An animal’s territory is the set of sites containing its active scent
Giuggioli L, Potts JR, Harris S (2011) Animal interactions and the emergence of territoriality PLoS Comput Biol 7(3)
The “territorial random walk” model
• Nearest-neighbour lattice random walkers• Deposit scent at each lattice site visited• Finite active scent time, TAS
• An animal’s territory is the set of sites containing its active scent• Cannot go into another’s territory
Giuggioli L, Potts JR, Harris S (2011) Animal interactions and the emergence of territoriality PLoS Comput Biol 7(3)
The “territorial random walk” model
• Nearest-neighbour lattice random walkers• Deposit scent at each lattice site visited• Finite active scent time, TAS
• An animal’s territory is the set of sites containing its active scent• Cannot go into another’s territory• Periodic boundary conditions
Giuggioli L, Potts JR, Harris S (2011) Animal interactions and the emergence of territoriality PLoS Comput Biol 7(3)
Dynamic territories emerge from the simulations
• Territory border mean square displacement (MSD) at long times:
Δxb2 = K2Dt/ln(Rt)
Territory border movement
• Territory border mean square displacement (MSD) at long times:
Δxb2 = K2Dt/ln(Rt)
Territory border movement
xb=position of territory border
• Territory border mean square displacement (MSD) at long times:
Δxb2 = K2Dt/ln(Rt)
Territory border movement
xb=position of territory border
K2D=diffusion constant of
territory border
Territory border movement
xb=position of territory border
K2D=diffusion constant of
territory border
R=rate to make K2D a diffusion
constant
• Territory border mean square displacement (MSD) at long times:
Δxb2 = K2Dt/ln(Rt)
Territory border movement
xb=position of territory border
K2D=diffusion constant of
territory border
R=rate to make K2D a diffusion
constant
• Territory border mean square displacement (MSD) at long times:
Δxb2 = K2Dt/ln(Rt)
• Subdiffusion: example of a 2D exclusion process
Territory border movement
xb=position of territory border
K2D=diffusion constant of
territory border
R=rate to make K2D a diffusion
constant
• Territory border mean square displacement (MSD) at long times:
Δxb2 = K2Dt/ln(Rt)
• Subdiffusion: example of a 2D exclusion process • No long-time steady state
Territory border movement
xb=position of territory border
K2D=diffusion constant of
territory border
R=rate to make K2D a diffusion
constant
• Territory border mean square displacement (MSD) at long times:
Δxb2 = K2Dt/ln(Rt)
• Subdiffusion: example of a 2D exclusion process • No long-time steady state• K2D depends on both the population density, ρ, the active scent time, TAS, and the animal’s diffusion constant, D
• Territory border mean square displacement (MSD) at long times:
Δxb2 = K2Dt/ln(Rt)
• Subdiffusion: example of a 2D exclusion process • No long-time steady state• K2D depends on both the population density, ρ, the active scent time, TAS, and the animal’s diffusion constant, D • In 1D, the MSD at long times is
Δxb2 = K1Dt1/2R-1/2
Territory border movement
R=rate to make K2D a diffusion
constant
K1D=diffusion constant of
territory border
• Territory border mean square displacement (MSD) at long times:
Δxb2 = K2Dt/ln(Rt)
• Subdiffusion: example of a 2D exclusion process • No long-time steady state• K2D depends on both the population density, ρ, the active scent time, TAS, and the animal’s diffusion constant, D • In 1D, the MSD at long times is
Δxb2 = K1Dt1/2R-1/2
• Single file diffusion phenomenon (1D exclusion)
Territory border movement
R=rate to make K2D a diffusion
constant
K1D=diffusion constant of
territory border
• Territory border mean square displacement (MSD) at long times:
Δxb2 = K2Dt/ln(Rt)
• Subdiffusion: example of a 2D exclusion process • No long-time steady state• K2D depends on both the population density, ρ, the active scent time, TAS, and the animal’s diffusion constant, D • In 1D, the MSD at long times is
Δxb2 = K1Dt1/2R-1/2
• Single file diffusion phenomenon (1D exclusion) • Henceforth just write K for K2D or K1D
Territory border movement
R=rate to make K2D a diffusion
constant
K1D=diffusion constant of
territory border
Territory border movement
• TTC=1/4Dρ in 2D (TTC=1/2Dρ2 in 1D) is the territory coverage time • ρ is the population density• D is the animal’s diffusion constant
2D 1D
Animal movement within dynamic territories
Giuggioli L, Potts JR, Harris S (2011) Brownian walkers within subdiffusing territorial boundaries Phys Rev E 83, 061138
Animal movement within dynamic territories
• Use an adiabatic approximation, assuming borders move slower than animal:
P(L1,L2,x,t)≈Q(L1,L2,t)W(x,t|L1,L2)
• Q(L1,L2,t) is border probability distribution• W(x,t) is the animal probability distribution
Giuggioli L, Potts JR, Harris S (2011) Brownian walkers within subdiffusing territorial boundaries Phys Rev E 83, 061138
Animal movement within dynamic territories
• Use an adiabatic approximation, assuming borders move slower than animal:
P(L1,L2,x,t)≈Q(L1,L2,t)W(x,t|L1,L2)
• Q(L1,L2,t) is border probability distribution• W(x,t) is the animal probability distribution
Giuggioli L, Potts JR, Harris S (2011) Brownian walkers within subdiffusing territorial boundaries Phys Rev E 83, 061138
Animal movement within dynamic territories
MSD of the animal is:
• b(t) controls the MSD of the separation distance between the borders: saturates at long times
Animal movement within dynamic territories
MSD of the animal is:
• b(t) controls the MSD of the separation distance between the borders: saturates at long times• c(t) controls the MSD of the centroid of the territory: always increasing
Animal movement within dynamic territories
MSD of the animal is:
• b(t) controls the MSD of the separation distance between the borders: saturates at long times• c(t) controls the MSD of the centroid of the territory: always increasing• Other terms ensure <x2>=2Dt at short times
Animal movement within dynamic territories
MSD of the animal is:
• b(t) controls the MSD of the separation distance between the borders: saturates at long times• c(t) controls the MSD of the centroid of the territory: always increasing• Other terms ensure <x2>=2Dt at short times
Comparison with simulation model
• Dashed = simulations; solid = analytic model• No parameter fitting
Recap• 2D simulation model:
(1D simulation model)
• 1D reduced analytic model:
• Next: 2D analytic model
Giuggioli L, Potts JR, Harris S (2012) Predicting oscillatory dynamics in the movement of territorial animals J Roy Soc Interface
2D persistent random walk within slowly moving territories
Giuggioli L, Potts JR, Harris S (2012) Predicting oscillatory dynamics in the movement of territorial animals J Roy Soc Interface
Persistence => use telegrapher’s equation instead of diffusion
2D persistent random walk within slowly moving territories
Giuggioli L, Potts JR, Harris S (2012) Predicting oscillatory dynamics in the movement of territorial animals J Roy Soc Interface
Analytic 2D expression: M2D(x,y,t|v,L,K,T,γ)v: speed of animalL: average territory widthK: diffusion constant of territory bordersT: correlation time of the animal movementγ: rate at which territories tend to return to an average area
2D persistent random walk within slowly moving territories
Fitting the model to red fox (Vulpes vulpes) data
Potts JR, Harris S, Giuggioli L (in revision) Quantifying behavioural changes in territorial animals caused by rapid population declines. Am Nat
Parameters before and after an outbreak of mange: active scent time
• TTC=1/v2Tρ is the territory coverage time
Parameters before and after an outbreak of mange: active scent time
Potts JR, Harris S, Giuggioli L (in revision) Quantifying behavioural changes in territorial animals caused by rapid population declines. Am Nat
Extension: territorial central place foragers (TCPF)
Potts JR, Harris S, Giuggioli L (2012) Territorial dynamics and stable home range formation for central place foragers. PLoS One 7(3)
Extension: territorial central place foragers (TCPF)
• p = drift probability towards central place (CP) (p≥1/2)
• (m,n) = position of animal• (mc,nc) = position of CP
Potts JR, Harris S, Giuggioli L (2012) Territorial dynamics and stable home range formation for central place foragers. PLoS One 7(3)
Stable home range formation
• MSD of the territory borders reaches a saturation value at long times for TCPF, contra to “vanilla” territorial random walkers
Stable home range formation
• MSD of the territory borders reaches a saturation value at long times for TCPF, contra to “vanilla” territorial random walkers
• i.e. the utilisation distribution (home range) of the animal reaches a steady state
Stable home range formation
• MSD of the territory borders reaches a saturation value at long times for TCPF, contra to “vanilla” territorial random walkers
• i.e. the utilisation distribution (home range) of the animal reaches a steady state
Potts JR, Harris S, Giuggioli L (2012) Territorial dynamics and stable home range formation for central place foragers. PLoS One 7(3)
Stable home range formation
• Dashed (left)/black (right) = simulation. Others analytic approximation• κ: border movement, increases (a)-(d) and (e)-(h)• α: strength of central place attraction. α =0.8 for (e), (g) and 4 for (f), (h)
Extension: partial exclusion
Giuggioli L, Potts JR, Rubenstein DI, Levin SA (submitted) Stigmergy, collective actions and animal social spacing
Acknowledgements
Luca Giuggioli, Bristol Centre for Complexity Sciences, University of BristolStephen Harris, School of Biological Sciences, University of BristolSimon Levin, Department of Ecology and Evolutionary Biology, Princeton UniversityDaniel Rubenstein, Department of Ecology and Evolutionary Biology, Princeton University
Main conclusions• A method for quantifying
territorial interaction events (TAS) and border movement (K) from animal movement data
Main conclusions• A method for quantifying
territorial interaction events (TAS) and border movement (K) from animal movement data• Home ranges: stable or quasi-
stable?
Thanks for listeningReferences1. Giuggioli L, Potts JR, Rubenstein DI, Levin SA (submitted) Stigmergy, collective
actions and animal social spacing2. Potts JR, Harris S and Giuggioli L (in revision) Quantifying behavioural changes in
territorial animals caused by rapid population declines. Am Nat3. Potts JR, Harris S and Giuggioli L (2012) Territorial dynamics and stable home range
formation for central place foragers. PLoS One 7(3)4. Giuggioli L, Potts JR, Harris S (2012) Predicting oscillatory dynamics in the
movement of territorial animals. J Roy Soc Interface5. Potts JR, Harris S and Giuggioli L (2011) An anti-symmetric exclusion process for two
particles on an infinite 1D lattice. J Phys A, 44, 485003.6. Giuggioli L, Potts JR, Harris S (2011) Brownian walkers within subdiffusing territorial
boundaries. Phys Rev E, 83, 0611387. Giuggioli L, Potts JR, Harris S (2011) Animal interactions and the emergence of
territoriality. PLoS Comput Biol 7(3)