Report

Territory formation from an individualbased movement-and-interaction model Jonathan R. Potts Centre for Mathematical Biology, University of Alberta. 3 December 2012 How do territories emerge? How do territories emerge? How do home ranges emerge? Outline • Introduction: the modelling framework Outline • Introduction: the modelling framework • Mathematics: analysing the 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 movement • 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) xb=position of territory border Territory border movement • Territory border mean square displacement (MSD) at long times: Δxb2 = K2Dt/ln(Rt) xb=position of territory border K2D=diffusion constant of territory border Territory border movement • Territory border mean square displacement (MSD) at long times: Δxb2 = K2Dt/ln(Rt) xb=position of territory border R=rate to make K2D a diffusion constant K2D=diffusion constant of territory border Territory border movement • Territory border mean square displacement (MSD) at long times: Δxb2 = K2Dt/ln(Rt) R=rate to make K2D a diffusion constant • Subdiffusion: example of a 2D exclusion process xb=position of territory border K2D=diffusion constant of territory border Territory border movement • Territory border mean square displacement (MSD) at long times: Δxb2 = K2Dt/ln(Rt) R=rate to make K2D a diffusion constant • Subdiffusion: example of a 2D exclusion process • No long-time steady state xb=position of territory border K2D=diffusion constant of territory border Territory border movement • Territory border mean square displacement (MSD) at long times: Δxb2 = K2Dt/ln(Rt) R=rate to make K2D a diffusion constant • 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 xb=position of territory border K2D=diffusion constant of territory border Territory border movement • Territory border mean square displacement (MSD) at long times: Δxb2 = K2Dt/ln(Rt) R=rate to make K2D a diffusion constant • 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 K1D=diffusion constant of Δxb2 = K1Dt1/2R-1/2 territory border Territory border movement • Territory border mean square displacement (MSD) at long times: Δxb2 = K2Dt/ln(Rt) R=rate to make K2D a diffusion constant • 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 K1D=diffusion constant of Δxb2 = K1Dt1/2R-1/2 territory border • Single file diffusion phenomenon (1D exclusion) Territory border movement • Territory border mean square displacement (MSD) at long times: Δxb2 = K2Dt/ln(Rt) R=rate to make K2D a diffusion constant • 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 K1D=diffusion constant of Δxb2 = K1Dt1/2R-1/2 territory border • Single file diffusion phenomenon (1D exclusion) • Henceforth just write K for K2D or K1D Territory border movement 2D 1D Territory border movement 2D 1D • TTC=1/4Dρ in 2D (TTC=1/2Dρ2 in 1D) is the territory coverage time Territory border movement 2D 1D • 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 Animal movement within dynamic territories Describe in 1D first, then extend to 2D 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: 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: Recap • 2D simulation model: (1D simulation model) • 1D reduced analytic model: Recap • 2D simulation model: (1D simulation model) • 1D reduced analytic model: • Next: 2D analytic model 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 2D persistent random walk within slowly moving territories Persistence => use telegrapher’s equation instead of diffusion 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 Analytic 2D expression: M2D(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 Giuggioli L, Potts JR, Harris S (2012) Predicting oscillatory dynamics in the movement of territorial animals J Roy Soc Interface 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 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 Overlapping scented areas Overlaps and encounter rates 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 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 quasistable? 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)