Jonathan R. PottsCentre for Mathematical Biology, University of Alberta.

3 December 2012

Territory formation from an individual-based movement-and-interaction model

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 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

2D 1D

Territory border movement

• TTC=1/4Dρ in 2D (TTC=1/2Dρ2 in 1D) is the territory coverage time

2D 1D

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

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

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

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 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)