PredatorPrey Dynamics for Rabbits Trees Romance J C
Predator-Prey Dynamics for Rabbits, Trees, & Romance J. C. Sprott Department of Physics University of Wisconsin Madison Presented at the Swiss Federal Research Institute (WSL) in Birmendsdorf, Switzerland on April 29, 2002
Collaborators • Janine Bolliger Swiss Federal Research Institute • Warren Porter University of Wisconsin • George Rowlands University of Warwick (UK)
Rabbit Dynamics n Let R = # of rabbits n d. R/dt = b. R - d. R = r. R Birth rate Death rate • r>0 growth • r=0 equilibrium • r<0 extinction r=b-d
Exponential Growth n n d. R/dt = r. R Solution: R = R 0 # rabbits R rt e r>0 r=0 r<0 time t
Logistic Differential Equation n d. R/dt = r. R(1 - R) 1 # rabbits R 0 r>0 time t
Effect of Predators n Let F = # of foxes n d. R/dt = r. R(1 - R - a. F) Intraspecies competition Interspecies competition But… The foxes have their own dynamics. . .
Lotka-Volterra Equations n R = rabbits, F = foxes n d. R/dt = r 1 R(1 - R a 1 F) n d. F/dt = r F(1 F a R) 2 2 r and a can be + or -
Types of Interactions - d. R/dt = r 1 R(1 - R - a 1 F) d. F/dt = r 2 F(1 - F - a 2 R) + a 2 r 2 Prey. Competition Predator a 1 r 1 + Predator. Prey Cooperation -
Equilibrium Solutions n d. R/dt = r 1 R(1 - R - a 1 F) = 0 n d. F/dt = r 2 F(1 - F - a 2 R) = 0 Equilibria: • R = 0, F = 0 F • R = 0, F = 1 • R = 1, F = 0 R • R = (1 - a 1) / (1 - a 1 a 2), F = (1 - a 2) / (1 - a 1 a 2)
Stable Focus (Predator-Prey) r 1(1 - a 1) < -r 2(1 - a 2) r 1 = 1 r 2 = -1 a 1 = 2 a 2 = 1. 9 F R r 1 = 1 r 2 = -1 a 1 = 2 a 2 = 2. 1 F R
Stable Saddle-Node (Competition) a 1 < 1, a 2 < 1 Node r 1 = 1 r 2 = 1 a 1 =. 9 a 2 =. 9 Saddle point r 1 = 1 r 2 = 1 a 1 = 1. 1 a 2 = 1. 1 F F Principle of Competitive Exclusion R R
Coexistence n n n With N species, there are 2 N equilibria, only one of which represents coexistence. Coexistence is unlikely unless the species compete only weakly with one another. Diversity in nature may result from having so many species from which to choose. There may be coexisting “niches” into which organisms evolve. Species may segregate spatially.
Reaction-Diffusion Model • Let Si(x, y) be density of the ith species (rabbits, trees, seeds, …) • d. Si / dt = ri. Si(1 - Si - Σaij. Sj) + Di 2 Si reaction j i diffusion where 2 -D grid: 2 Si = Sx-1, y + Sx, y-1 + Sx+1, y + Sx, y+1 - 4 Sx, y
Typical Results
Alternate Spatial Lotka-Volterra Equations • Let Si(x, y) be density of the ith species (rabbits, trees, seeds, …) • d. Si / dt = ri. Si(1 - Si - Σaij. Sj) j i where 2 -D grid: S = Sx-1, y + Sx, y-1 + Sx+1, y + Sx, y+1 + a. Sx, y
Parameters of the Model Growth rates 1 r 2 r 3 r 4 r 5 r 6 Interaction matrix 1 a 21 a 31 a 41 a 51 a 61 a 12 1 a 32 a 42 a 52 a 62 a 13 a 23 1 a 43 a 53 a 63 a 14 a 24 a 34 1 a 54 a 64 a 15 a 25 a 35 a 45 1 a 65 a 16 a 26 a 36 a 46 a 56 1
Features of the Model n Purely deterministic (no randomness) n Purely endogenous (no external effects) n Purely homogeneous (every cell is equivalent) n Purely egalitarian (all species obey same equation) n Continuous time
Typical Results
Typical Results
Typical Results
Dominant Species
Cluster probability Fluctuations in Cluster Probability Time
Power Spectrum of Cluster Probability Frequency
Time Derivative of biomass Fluctuations in Total. Biomass Time
Power Spectrum of Total Biomass Frequency
Error in Biomass Sensitivity to Initial Conditions Time
Results n Most species die out n Co-existence is possible n Densities can fluctuate chaotically n One implies the other Complex spatial patterns spontaneously arise
Romance (Romeo and Juliet) n Let R = Romeo’s love for Juliet n Let J = Juliet’s love for Romeo n Assume R and J obey Lotka. Volterra Equations n Ignore spatial effects
Romantic Styles d. R/dt = r. R(1 - R - a. J) + Cautious lover a Narcissistic nerd - r Eager beaver Hermit - +
Pairings - Stable Mutual Love Narcissistic Nerd Eager Beaver Cautious Lover Hermit Narcissistic Nerd 46% 67% 5% 0% Eager Beaver 67% 39% 0% 0% Cautious Lover 5% 0% 0% 0% Hermit 0% 0%
Love Triangles n There are 4 -6 variables n Stable co-existing love is rare n Chaotic solutions are possible n But…none were found in LV model n Other models do show chaos
Summary n Nature is complex but n Simple models may suffice
References n n http: //sprott. physics. wisc. edu/ lectures/predprey/ (This talk) sprott@juno. physics. wisc. edu
- Slides: 33