Jonathan R. Potts, Luca Giuggioli, Steve Harris,Bristol Centre for Complexity Sciences & School of Biological Sciences, University of Bristol.

20 September 2011

Territorial dynamics

What is “territorial dynamics”?

The moving territorial patterns that arise from animal movements and interactions.

Outline

• What is “territorial dynamics”?

Outline

• What is “territorial dynamics”? • An agent-based model of territory formation

in scent-marking animals

Outline

• What is “territorial dynamics”? • An agent-based model of territory formation

in scent-marking animals• Mathematical analysis of the model

Outline

• What is “territorial dynamics”? • An agent-based model of territory formation

in scent-marking animals• Mathematical analysis of the model• Using data on animal movements to obtain

information about scent-mark longevity

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)

Outcomes of the simulations

Outcomes of the simulations

• Territory border MSD scales as Kt/ln(T) where T=4tF and F is the animal’s hopping rate between lattice sites• The ratio K/D decays as TAS /TTC increases, where D is the animal’s diffusion constant, TTC=1/4Dρ is the territory coverage time and ρ is the population density

Outcomes of the simulations

• Territory border MSD scales as Kt/ln(T) where T=4tF and F is the animal’s hopping rate between lattice sites• The ratio K/D decays as TAS /TTC increases, where D is the animal’s diffusion constant, TTC=1/4Dρ is the territory coverage time and ρ is the population density• 1D simulations show analogous results but the border MSD scales as Kt1/2

A reduced analytic (1D) model

• Decouple the animal and border movement (adiabatic approximation)• Animal constrained to move within its two adjacent borders• Territories are modelled as springs with equilibrium length 1/ρ• Borders and animals have an intrinsic random movement

A reduced analytic (1D) model

• In the simulations, the borders in fact consist of two territory boundaries• The boundaries may be separated at any point in time, but they are more likely to move together than separate: p>1/2

Border movement arising from the interaction of boundaries

• Two mutually exclusive particles on an infinite 1D lattice• Perform biased, nearest-neighbour random walk• System can be solved exactly1 • When p>1/2, MSD of one particle at long times is

Δx(t)2 = 2a2F(1-p)twhere a is the lattice spacing and F the hopping rate

1. Potts JR, Harris S and Giuggioli L An anti-symmetric exclusion process for two particles on an infinite 1D lattice arxiv:1107:2020

Animal movement within dynamic territories

• Use an adiabatic approximation, assuming boundaries move slower than animal:

P(L1,L2,x,t)≈Q(L1,L2,t)W(x,t|L1,L2)

• Q(L1,L2,t) is boundary 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• 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: values of K and γ measured from simulation• Adiabatic approximation works well except when TAS/TTC is low

Obtaining TAS from movement data• Radio-tracking data on the urban red fox (Vulpes vulpes)• Obtained every 5 minute with 25m square granularity• 8000 fixes over 5 years (1990-1994)• Gathered in spring and summer so no dispersing/cuckolding

Obtaining TAS from movement data• Radio-tracking data on the urban red fox (Vulpes vulpes)• Obtained every 5 minute with 25m square granularity• 8000 fixes over 5 years (1990-1994)• Gathered in spring and summer so no dispersing/cuckolding

Obtaining TAS from movement data• Radio-tracking data on the urban red fox (Vulpes vulpes)• Obtained every 5 minute with 25m square granularity• 8000 fixes over 5 years (1990-1994)• Gathered in spring and summer so no dispersing/cuckolding

Obtaining TAS from movement data• Run simulations using movement patterns from red fox• Obtain a curve relating K to TAS/TTC (right)

Obtaining TAS from movement data• Run simulations using movement patterns from red fox• Obtain a curve relating K to TAS/TTC (right)

Obtaining TAS from movement data• Run simulations using movement patterns from red fox• Obtain a curve relating K to TAS/TTC (right)• Long-time MSD data gives K-value

Obtaining TAS from movement data• Run simulations using movement patterns from red fox• Obtain a curve relating K to TAS/TTC (right)• Long-time MSD data gives K-value• Read off from simulation curve value of TAS/TTC • TTC = ρva where v is the animal speed, ρ the population density and a is distance between fixes (25m)• Hence calculate TAS ≈ 6.5 days

Conclusions• Dynamic territorial patterns emerge from

systems of moving, interacting animals

Conclusions• Dynamic territorial patterns emerge from

systems of moving, interacting animals• Reduced, analytically-tractable models help us

understand the features that emerge from the system

Conclusions• Dynamic territorial patterns emerge from

systems of moving, interacting animals• Reduced, analytically-tractable models help us

understand the features that emerge from the system

• Such models also allow us to estimate longevity of olfactory cues from animal movement patterns

Conclusions• Dynamic territorial patterns emerge from

systems of moving, interacting animals• Reduced, analytically-tractable models help us

understand the features that emerge from the system

• Such models also allow us to estimate longevity of olfactory cues from animal movement patterns

• Demonstrated with red fox (Vulpes vulpes) data

Thanks for listeningReferences1. Giuggioli L, Potts JR, Harris S (2011) Animal interactions and the emergence of

territoriality PLoS Comput Biol 7(3) (featured research)2. Giuggioli L, Potts JR, Harris S (2011) Brownian walkers within subdiffusing territorial

boundaries Phys Rev E 83, 0611383. Potts JR, Harris S and Giuggioli L (in review) An anti-symmetric exclusion process for

two particles on an infinite 1D lattice4. Giuggioli L, Potts JR, Harris S (submitted) Predicting oscillatory dynamics in the

movement of territorial animals

Working title5. Potts JR, Harris S and Giuggioli L (in prep) The effect of animal movement and

interaction strategies on territorial patterns