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Jonathan R. Potts Centre for Mathematical Biology, University of Alberta. 3 December 2012 Territory formation from an individual- based movement-and-interaction model

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How do territories emerge?

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How do home ranges emerge?

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Outline Introduction: the modelling framework

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Outline Introduction: the modelling framework Mathematics: analysing the model

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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?

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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

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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

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The “territorial random walk” model Giuggioli L, Potts JR, Harris S (2011) Animal interactions and the emergence of territoriality PLoS Comput Biol 7(3)

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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)

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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)

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The “territorial random walk” model Nearest-neighbour lattice random walkers Deposit scent at each lattice site visited Finite active scent time, T AS Giuggioli L, Potts JR, Harris S (2011) Animal interactions and the emergence of territoriality PLoS Comput Biol 7(3)

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The “territorial random walk” model Nearest-neighbour lattice random walkers Deposit scent at each lattice site visited Finite active scent time, T AS 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)

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The “territorial random walk” model Nearest-neighbour lattice random walkers Deposit scent at each lattice site visited Finite active scent time, T AS 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)

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The “territorial random walk” model Nearest-neighbour lattice random walkers Deposit scent at each lattice site visited Finite active scent time, T AS 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)

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Dynamic territories emerge from the simulations

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Territory border mean square displacement (MSD) at long times: Δx b 2 = K 2D t/ln(Rt) Territory border movement

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Territory border mean square displacement (MSD) at long times: Δx b 2 = K 2D t/ln(Rt) Territory border movement x b =position of territory border

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Territory border mean square displacement (MSD) at long times: Δx b 2 = K 2D t/ln(Rt) Territory border movement x b =position of territory border K 2D =diffusion constant of territory border

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Territory border movement x b =position of territory border K 2D =diffusion constant of territory border R=rate to make K 2D a diffusion constant Territory border mean square displacement (MSD) at long times: Δx b 2 = K 2D t/ln(Rt)

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Territory border movement x b =position of territory border K 2D =diffusion constant of territory border R=rate to make K 2D a diffusion constant Territory border mean square displacement (MSD) at long times: Δx b 2 = K 2D t/ln(Rt) Subdiffusion: example of a 2D exclusion process

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Territory border movement x b =position of territory border K 2D =diffusion constant of territory border R=rate to make K 2D a diffusion constant Territory border mean square displacement (MSD) at long times: Δx b 2 = K 2D t/ln(Rt) Subdiffusion: example of a 2D exclusion process No long-time steady state

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Territory border movement x b =position of territory border K 2D =diffusion constant of territory border R=rate to make K 2D a diffusion constant Territory border mean square displacement (MSD) at long times: Δx b 2 = K 2D t/ln(Rt) Subdiffusion: example of a 2D exclusion process No long-time steady state K 2D depends on both the population density, ρ, the active scent time, T AS, and the animal’s diffusion constant, D

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Territory border mean square displacement (MSD) at long times: Δx b 2 = K 2D t/ln(Rt) Subdiffusion: example of a 2D exclusion process No long-time steady state K 2D depends on both the population density, ρ, the active scent time, T AS, and the animal’s diffusion constant, D In 1D, the MSD at long times is Δx b 2 = K 1D t 1/2 R -1/2 Territory border movement R=rate to make K 2D a diffusion constant K 1D =diffusion constant of territory border

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Territory border mean square displacement (MSD) at long times: Δx b 2 = K 2D t/ln(Rt) Subdiffusion: example of a 2D exclusion process No long-time steady state K 2D depends on both the population density, ρ, the active scent time, T AS, and the animal’s diffusion constant, D In 1D, the MSD at long times is Δx b 2 = K 1D t 1/2 R -1/2 Single file diffusion phenomenon (1D exclusion) Territory border movement R=rate to make K 2D a diffusion constant K 1D =diffusion constant of territory border

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Territory border mean square displacement (MSD) at long times: Δx b 2 = K 2D t/ln(Rt) Subdiffusion: example of a 2D exclusion process No long-time steady state K 2D depends on both the population density, ρ, the active scent time, T AS, and the animal’s diffusion constant, D In 1D, the MSD at long times is Δx b 2 = K 1D t 1/2 R -1/2 Single file diffusion phenomenon (1D exclusion) Henceforth just write K for K 2D or K 1D Territory border movement R=rate to make K 2D a diffusion constant K 1D =diffusion constant of territory border

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Territory border movement 2D1D

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Territory border movement T TC =1/4Dρ in 2D (T TC =1/2Dρ 2 in 1D) is the territory coverage time 2D1D

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Territory border movement T TC =1/4Dρ in 2D (T TC =1/2Dρ 2 in 1D) is the territory coverage time ρ is the population density D is the animal’s diffusion constant 2D1D

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Animal movement within dynamic territories Describe in 1D first, then extend to 2D

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Animal movement within dynamic territories Giuggioli L, Potts JR, Harris S (2011) Brownian walkers within subdiffusing territorial boundaries Phys Rev E 83, 061138

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Animal movement within dynamic territories Use an adiabatic approximation, assuming borders move slower than animal: P(L 1,L 2,x,t)≈Q(L 1,L 2,t)W(x,t|L 1,L 2 ) Q(L 1,L 2,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

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Animal movement within dynamic territories Use an adiabatic approximation, assuming borders move slower than animal: P(L 1,L 2,x,t)≈Q(L 1,L 2,t)W(x,t|L 1,L 2 ) Q(L 1,L 2,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

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Animal movement within dynamic territories MSD of the animal is:

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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

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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

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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 =2Dt at short times

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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 =2Dt at short times

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Comparison with simulation model Dashed = simulations; solid = analytic model No parameter fitting

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Recap 2D simulation model:

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Recap 2D simulation model: (1D simulation model) 1D reduced analytic model:

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Recap 2D simulation model: (1D simulation model) 1D reduced analytic model: Next: 2D analytic model

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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

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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

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Giuggioli L, Potts JR, Harris S (2012) Predicting oscillatory dynamics in the movement of territorial animals J Roy Soc Interface Analytic 2D expression: M 2D (x,y,t|v,L,K,T,γ) v: speed of animal L: average territory width K: diffusion constant of territory borders T: correlation time of the animal movement γ: rate at which territories tend to return to an average area 2D persistent random walk within slowly moving territories

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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

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Parameters before and after an outbreak of mange

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Parameters before and after an outbreak of mange: active scent time T TC =1/v 2 Tρ is the territory coverage time

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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

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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)

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Extension: territorial central place foragers (TCPF) p = drift probability towards central place (CP) (p≥1/2) (m,n) = position of animal (m c,n c ) = 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)

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Stable home range formation MSD of the territory borders reaches a saturation value at long times for TCPF, contra to “vanilla” territorial random walkers

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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

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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)

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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)

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Extension: partial exclusion Giuggioli L, Potts JR, Rubenstein DI, Levin SA (submitted) Stigmergy, collective actions and animal social spacing

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Overlapping scented areas

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Overlaps and encounter rates

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Acknowledgements Luca Giuggioli, Bristol Centre for Complexity Sciences, University of Bristol Stephen Harris, School of Biological Sciences, University of Bristol Simon Levin, Department of Ecology and Evolutionary Biology, Princeton University Daniel Rubenstein, Department of Ecology and Evolutionary Biology, Princeton University

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Main conclusions A method for quantifying territorial interaction events (T AS ) and border movement (K) from animal movement data

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Main conclusions A method for quantifying territorial interaction events (T AS ) and border movement (K) from animal movement data Home ranges: stable or quasi- stable?

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Thanks for listening References 1.Giuggioli L, Potts JR, Rubenstein DI, Levin SA (submitted) Stigmergy, collective actions and animal social spacing 2.Potts JR, Harris S and Giuggioli L (in revision) Quantifying behavioural changes in territorial animals caused by rapid population declines. Am Nat 3.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 Interface 5.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, 061138 7.Giuggioli L, Potts JR, Harris S (2011) Animal interactions and the emergence of territoriality. PLoS Comput Biol 7(3)

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