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 dR/dt = bR - dR Birth rateDeath rate = rR r > 0 growth r = 0 equilibrium r < 0 extinction r = b - d

Exponential Growth n dR/dt = rR n Solution: R = R 0 e rt R t r > 0 r = 0 r < 0 # rabbits time

Logistic Differential Equation n dR/dt = rR(1 - R) R t r > 0 # rabbits time 0 1

Effect of Predators n Let F = # of foxes n dR/dt = rR(1 - R - aF) Interspecies competition Intraspecies competition But… The foxes have their own dynamics...

Lotka-Volterra Equations n R = rabbits, F = foxes n dR/dt = r 1 R(1 - R - a 1 F) n dF/dt = r 2 F(1 - F - a 2 R) r and a can be + or -

Types of Interactions dR/dt = r 1 R (1 - R - a 1 F ) dF/dt = r 2 F (1 - F - a 2 R ) a1r1a1r1 a2r2a2r2 Competition Predator- Prey Prey- Predator Cooperation

Equilibrium Solutions n dR/dt = r 1 R(1 - R - a 1 F) = 0 n dF/dt = r 2 F(1 - F - a 2 R) = 0 R = 0, F = 0 R = 0, F = 1 R = 1, F = 0 R = (1 - a 1 ) / (1 - a 1 a 2 ), F = (1 - a 2 ) / (1 - a 1 a 2 ) Equilibria: R F

Stable Focus (Predator-Prey) r 1 (1 - a 1 ) < -r 2 (1 - a 2 ) F RR r 1 = 1 r 2 = -1 a 1 = 2 a 2 = 1.9 r 1 = 1 r 2 = -1 a 1 = 2 a 2 = 2.1 F

Principle of Competitive Exclusion Stable Saddle-Node (Competition) a 1 < 1, a 2 < 1 F RR r 1 = 1 r 2 = 1 a 1 =.9 a 2 =.9 r 1 = 1 r 2 = 1 a 1 = 1.1 a 2 = 1.1 F NodeSaddle point

Coexistence n With N species, there are 2 N equilibria, only one of which represents coexistence. n Coexistence is unlikely unless the species compete only weakly with one another. n Diversity in nature may result from having so many species from which to choose. n There may be coexisting “niches” into which organisms evolve. n Species may segregate spatially.

Reaction-Diffusion Model Let S i ( x,y ) be density of the i th species (rabbits, trees, seeds, …) dS i / dt = r i S i (1 - S i - Σ a ij S j ) + D i  2 S i 2-D grid:  2 S i = S x- 1, y + S x,y -1 + S x +1, y + S x,y S x,y jiji where reactiondiffusion

Typical Results

Alternate Spatial Lotka- Volterra Equations Let S i ( x,y ) be density of the i th species (rabbits, trees, seeds, …) dS i / dt = r i S i (1 - S i - Σ a ij S j ) 2-D grid: S = S x- 1, y + S x,y -1 + S x +1, y + S x,y +1 +  S x,y jiji where

Parameters of the Model 1r2r3r4r5r61r2r3r4r5r6 1a 12 a 13 a 14 a 15 a 16 a 21 1a 23 a 24 a 25 a 26 a 31 a 32 1 a 34 a 35 a 36 a 41 a 42 a 43 1 a 45 a 46 a 51 a 52 a 53 a 54 1 a 56 a 61 a 62 a 63 a 64 a 65 1 Growth rates Interaction matrix

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

Dominant Species

Fluctuations in Cluster Probability Time Cluster probability

Power Spectrum of Cluster Probability Frequency Power

Fluctuations in Total Biomass Time Derivative of biomass Time

Power Spectrum of Total Biomass Frequency Power

Sensitivity to Initial Conditions Time Error in Biomass

Results n Most species die out n Co-existence is possible n Densities can fluctuate chaotically n Complex spatial patterns spontaneously arise One implies the other

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 dR/dt = rR (1 - R - aJ ) a Narcissistic nerd Eager beaver Cautious lover Hermit r

Pairings - Stable Mutual Love Narcissistic Nerd Narcissistic Nerd Cautious Lover Cautious Lover Eager Beaver Eager Beaver Hermit 46%67% 39% 5% 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 n Simple models may suffice but

References n lectures/predprey/ (This talk) lectures/predprey/ n