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Territorial dynamics

Territorial dynamics. Jonathan R. Potts , Luca Giuggioli, Steve Harris, Bristol Centre for Complexity Sciences & School of Biological Sciences, University of Bristol. 20 September 2011. What is “territorial dynamics”?.

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Territorial dynamics

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

  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, 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 PLoSComputBiol 7(3)

  8. Outcomes of the simulations

  9. 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

  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 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

  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 exactly1 • When p>1/2, MSD of one particle at long times is • Δx(t)2= 2a2F(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(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

  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 <x2>=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 TAS/TTC is low

  17. Obtaining TAS from movement data • Radio-tracking data on the urban red fox (Vulpesvulpes) • 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 TAS from movement data • Radio-tracking data on the urban red fox (Vulpesvulpes) • 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 TAS from movement data • Radio-tracking data on the urban red fox (Vulpesvulpes) • 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 TAS from movement data • Run simulations using movement patterns from red fox • Obtain a curve relating K to TAS/TTC (right)

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

  22. 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

  23. 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

  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 (Vulpesvulpes) data

  28. Thanks for listening References • Giuggioli L, Potts JR, Harris S (2011) Animal interactions and the emergence of territoriality PLoS Comput Biol 7(3) (featured research) • Giuggioli L, Potts JR, Harris S (2011) Brownian walkers within subdiffusing territorial boundaries Phys Rev E 83, 061138 • Potts JR, Harris S and Giuggioli L (in review) An anti-symmetric exclusion process for two particles on an infinite 1D lattice • Giuggioli L, Potts JR, Harris S (submitted) Predicting oscillatory dynamics in the movement of territorial animals Working title • Potts JR, Harris S and Giuggioli L (in prep) The effect of animal movement and interaction strategies on territorial patterns

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