1. Jonathan R. Potts, Luca Giuggioli, Steve Harris, Bristol Centre for Complexity Sciences & School of Biological Sciences, University of Bristol. 20 September 2011 Territorial dynamics

2. What is “territorial dynamics”? The moving territorial patterns that arise from animal movements and interactions.

3. Outline What is “territorial dynamics”?

4. Outline What is “territorial dynamics”? An agent-based model of territory formation in scent-marking animals

5. Outline What is “territorial dynamics”? An agent-based model of territory formation in scent-marking animals Mathematical analysis of the model

6. 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

7. 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)

8. Outcomes of the simulations

9. 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 T AS /T TC increases, where D is the animal’s diffusion constant, T TC =1/4Dρ is the territory coverage time and ρ is the population density

10. 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 T AS /T TC increases, where D is the animal’s diffusion constant, T TC =1/4Dρ is the territory coverage time and ρ is the population density 1D simulations show analogous results but the border MSD scales as Kt 1/2

11. 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

12. 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

13. 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 exactly 1 When p>1/2, MSD of one particle at long times is Δx(t) 2 = 2a 2 F(1-p) t where 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

14. Animal movement within dynamic territories Use an adiabatic approximation, assuming boundaries 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 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

15. 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

16. 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 T AS /T TC is low

17. Obtaining T AS 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

18. Obtaining T AS 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

19. Obtaining T AS 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

20. Obtaining T AS from movement data Run simulations using movement patterns from red fox Obtain a curve relating K to T AS /T TC (right)

21. Obtaining T AS from movement data Run simulations using movement patterns from red fox Obtain a curve relating K to T AS /T TC (right)

22. Obtaining T AS from movement data Run simulations using movement patterns from red fox Obtain a curve relating K to T AS /T TC (right) Long-time MSD data gives K-value

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

24. Conclusions Dynamic territorial patterns emerge from systems of moving, interacting animals

25. 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

26. 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

27. 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

28. Thanks for listening References 1.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, 061138 3.Potts JR, Harris S and Giuggioli L (in review) An anti-symmetric exclusion process for two particles on an infinite 1D lattice 4. Giuggioli L, Potts JR, Harris S (submitted) Predicting oscillatory dynamics in the movement of territorial animals Working title 1.Potts JR, Harris S and Giuggioli L (in prep) The effect of animal movement and interaction strategies on territorial patterns