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, T
AS
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, 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)

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)

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)

Dynamic territories emerge from the simulations

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

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

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

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

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

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

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

Territory border movement
•
Territory border mean square displacement (MSD) at long times:
Δx b
2 = K
2D t/ln(Rt)
R=rate to make
K
2D a diffusion constant
• 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 K
1D
=diffusion constant of
Δx b
2 = K
1D t 1/2 R -1/2 territory border

Territory border movement
•
Territory border mean square displacement (MSD) at long times:
Δx b
2 = K
2D t/ln(Rt)
R=rate to make
K
2D a diffusion constant
• 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 K
1D
=diffusion constant of
Δx b
2 = K
1D t 1/2 R -1/2 territory border
• Single file diffusion phenomenon (1D exclusion)

Territory border movement
•
Territory border mean square displacement (MSD) at long times:
Δx b
2 = K
2D t/ln(Rt)
R=rate to make
K
2D a diffusion constant
• 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 K
1D
=diffusion constant of
Δx b
2 = K
1D t 1/2 R -1/2 territory border
• Single file diffusion phenomenon (1D exclusion)
• Henceforth just write K for K
2D or K
1D

Territory border movement
2D 1D

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

Territory border movement
2D 1D
• 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

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(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

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

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 <x 2 >=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 <x 2 >=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: 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
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
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
•
T
TC
=1/v 2 Tρ 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
• (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)

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

•
AS

•
AS
•

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)