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

Territory formation from an individual-based movement-and-interaction model. Jonathan R. Potts Centre for Mathematical B iology, University of Alberta. 3 December 2012. How do territories emerge?. How do territories emerge?. How do home ranges emerge?. Outline.

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

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  1. Territory formation from an individual-based movement-and-interaction model Jonathan R. Potts Centre for Mathematical Biology, University of Alberta. 3 December 2012

  2. How do territories emerge?

  3. How do territories emerge?

  4. How do home ranges emerge?

  5. Outline • Introduction: the modelling framework

  6. Outline • Introduction: the modelling framework • Mathematics: analysing the model

  7. Outline • Introduction: the modelling framework • Mathematics: analysing the model • Biology: Application to red foxes (Vulpesvulpes). How do animals change their behaviour when populations go into decline?

  8. Outline • Introduction: the modelling framework • Mathematics: analysing the model • Biology: Application to red foxes (Vulpesvulpes). How do animals change their behaviour when populations go into decline? • Extension 1: central place foragers and stable home ranges

  9. Outline • Introduction: the modelling framework • Mathematics: analysing the model • Biology: Application to red foxes (Vulpesvulpes). 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

  10. The “territorial random walk” model Giuggioli L, Potts JR, Harris S (2011) Animal interactions and the emergence of territoriality PLoSComputBiol 7(3)

  11. The “territorial random walk” model • Nearest-neighbour lattice random walkers Giuggioli L, Potts JR, Harris S (2011) Animal interactions and the emergence of territoriality PLoSComputBiol 7(3)

  12. 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 PLoSComputBiol 7(3)

  13. 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 PLoSComputBiol 7(3)

  14. 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 PLoSComputBiol 7(3)

  15. 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)

  16. 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 PLoSComputBiol 7(3)

  17. Dynamic territories emerge from the simulations

  18. Territory border movement • Territory border mean square displacement (MSD) at long times: • Δxb2 = K2Dt/ln(Rt)

  19. Territory border movement • Territory border mean square displacement (MSD) at long times: • Δxb2 = K2Dt/ln(Rt) xb=position of territory border

  20. Territory border movement • Territory border mean square displacement (MSD) at long times: • Δxb2 = K2Dt/ln(Rt) K2D=diffusion constant of territory border xb=position of territory border

  21. Territory border movement • Territory border mean square displacement (MSD) at long times: • Δxb2 = K2Dt/ln(Rt) R=rate to make K2D a diffusion constant K2D=diffusion constant of territory border xb=position of territory border

  22. Territory border movement • Territory border mean square displacement (MSD) at long times: • Δxb2 = K2Dt/ln(Rt) • Subdiffusion: example of a 2D exclusion process R=rate to make K2D a diffusion constant K2D=diffusion constant of territory border xb=position of territory border

  23. Territory border movement • 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 R=rate to make K2D a diffusion constant K2D=diffusion constant of territory border xb=position of territory border

  24. Territory border movement • 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 R=rate to make K2D a diffusion constant K2D=diffusion constant of territory border xb=position of territory border

  25. Territory border movement • 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 R=rate to make K2D a diffusion constant K1D=diffusion constant of territory border

  26. Territory border movement • 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) R=rate to make K2D a diffusion constant K1D=diffusion constant of territory border

  27. Territory border movement • 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 R=rate to make K2D a diffusion constant K1D=diffusion constant of territory border

  28. Territory border movement 2D 1D

  29. Territory border movement 2D 1D • TTC=1/4Dρ in 2D (TTC=1/2Dρ2 in 1D) is the territory coverage time

  30. Territory border movement 2D 1D • TTC=1/4Dρ in 2D (TTC=1/2Dρ2 in 1D) is the territory coverage time • ρis the population density • D is the animal’s diffusion constant

  31. Animal movement within dynamic territories Describe in 1D first, then extend to 2D

  32. Animal movement within dynamic territories Giuggioli L, Potts JR, Harris S (2011) Brownian walkers within subdiffusing territorial boundaries Phys Rev E 83, 061138

  33. 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

  34. 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

  35. Animal movement within dynamic territories MSD of the animal is:

  36. 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

  37. 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

  38. 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

  39. 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

  40. Comparison with simulation model • Dashed = simulations; solid = analytic model • No parameter fitting

  41. Recap • 2D simulation model:

  42. Recap • 2D simulation model: (1D simulation model) • 1D reduced analytic model:

  43. Recap • 2D simulation model: (1D simulation model) • 1D reduced analytic model: • Next: 2D analytic model

  44. 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

  45. 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

  46. 2D persistent random walk within slowly moving territories Analytic 2D expression: M2D(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

  47. Fitting the model to red fox (Vulpes vulpes) data Potts JR, Harris S, GiuggioliL (in revision) Quantifying behavioural changes in territorial animals caused by rapid population declines. Am Nat

  48. Parameters before and after an outbreak of mange

  49. Parameters before and after an outbreak of mange: active scent time • TTC=1/v2Tρis the territory coverage time

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