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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

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Collaborators Janine Bolliger Swiss Federal Research Institute Warren Porter University of Wisconsin George Rowlands University of Warwick (UK)

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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

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

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

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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...

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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 -

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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

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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

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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

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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

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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.

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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 +1 - 4 S x,y jiji where reactiondiffusion

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Typical Results

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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 jiji where

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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

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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

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Typical Results

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Dominant Species

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Fluctuations in Cluster Probability Time Cluster probability

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Power Spectrum of Cluster Probability Frequency Power

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Fluctuations in Total Biomass Time Derivative of biomass Time

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Power Spectrum of Total Biomass Frequency Power

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Sensitivity to Initial Conditions Time Error in Biomass

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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

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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

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

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Pairings - Stable Mutual Love Narcissistic Nerd Narcissistic Nerd Cautious Lover Cautious Lover Eager Beaver Eager Beaver Hermit 46%67% 39% 5% 0%

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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

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Summary n Nature is complex n Simple models may suffice but

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References n http://sprott.physics.wisc.edu/ lectures/predprey/ (This talk) http://sprott.physics.wisc.edu/ lectures/predprey/ n sprott@juno.physics.wisc.edu sprott@juno.physics.wisc.edu

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