jonathan r. potts, luca giuggioli, steve harris, bristol centre for complexity sciences & school...
TRANSCRIPT
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