predprey

Predator-Prey Dynamics for Rabbits, Trees, & Romance: 

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

Slide2: 

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

Rabbit Dynamics: 

Rabbit Dynamics Let R = # of rabbits dR/dt = bR - dR Birth rate Death rate = rR r > 0 growth r = 0 equilibrium r < 0 extinction r = b - d

Exponential Growth: 

Exponential Growth dR/dt = rR Solution: R = R0ert R t r > 0 r = 0 r < 0 # rabbits time

Logistic Differential Equation: 

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

Effect of Predators: 

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

Lotka-Volterra Equations: 

Lotka-Volterra Equations R = rabbits, F = foxes dR/dt = r1R(1 - R - a1F) dF/dt = r2F(1 - F - a2R) r and a can be + or -

Types of Interactions: 

Types of Interactions dR/dt = r1R(1 - R - a1F) dF/dt = r2F(1 - F - a2R) + + - - a1r1 a2r2 Competition Predator- Prey Prey- Predator Cooperation

Equilibrium Solutions: 

Equilibrium Solutions dR/dt = r1R(1 - R - a1F) = 0 dF/dt = r2F(1 - F - a2R) = 0 R = 0, F = 0 R = 0, F = 1 R = 1, F = 0 R = (1 - a1) / (1 - a1a2), F = (1 - a2) / (1 - a1a2) Equilibria: R F

Stable Focus(Predator-Prey): 

Stable Focus (Predator-Prey) r1(1 - a1) < -r2(1 - a2) F R R r1 = 1 r2 = -1 a1 = 2 a2 = 1.9 r1 = 1 r2 = -1 a1 = 2 a2 = 2.1 F

Stable Saddle-Node(Competition): 

Principle of Competitive Exclusion Stable Saddle-Node (Competition) a1 < 1, a2 < 1 F R R r1 = 1 r2 = 1 a1 = .9 a2 = .9 r1 = 1 r2 = 1 a1 = 1.1 a2 = 1.1 F Node Saddle point

Coexistence: 

Coexistence With N species, there are 2N 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: 

Reaction-Diffusion Model Let Si(x,y) be density of the ith species (rabbits, trees, seeds, …) dSi / dt = riSi(1 - Si - ΣaijSj) + Di2Si 2-D grid: 2Si = Sx-1,y + Sx,y-1 + Sx+1,y + Sx,y+1 - 4Sx,y ji where reaction diffusion

Typical Results: 

Typical Results

Alternate Spatial Lotka-Volterra Equations: 

Alternate Spatial Lotka-Volterra Equations Let Si(x,y) be density of the ith species (rabbits, trees, seeds, …) dSi / dt = riSi(1 - Si - ΣaijSj) 2-D grid: S = Sx-1,y + Sx,y-1 + Sx+1,y + Sx,y+1 + aSx,y ji where

Parameters of the Model: 

Parameters of the Model 1 r2 r3 r4 r5 r6 1 a12 a13 a14 a15 a16 a21 1 a23 a24 a25 a26 a31 a32 1 a34 a35 a36 a41 a42 a43 1 a45 a46 a51 a52 a53 a54 1 a56 a61 a62 a63 a64 a65 1 Growth rates Interaction matrix

Features of the Model: 

Features of the Model Purely deterministic (no randomness) Purely endogenous (no external effects) Purely homogeneous (every cell is equivalent) Purely egalitarian (all species obey same equation) Continuous time

Typical Results: 

Typical Results

Typical Results: 

Typical Results

Typical Results: 

Typical Results

Dominant Species: 

Dominant Species

Fluctuations in Cluster Probability: 

Fluctuations in Cluster Probability Time Cluster probability

Power Spectrumof Cluster Probability: 

Power Spectrum of Cluster Probability Frequency Power

Fluctuations in Total Biomass: 

Fluctuations in Total Biomass Time Derivative of biomass Time

Power Spectrumof Total Biomass: 

Power Spectrum of Total Biomass Frequency Power

Sensitivity to Initial Conditions: 

Sensitivity to Initial Conditions Time Error in Biomass

Results: 

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

Romance(Romeo and Juliet): 

Romance (Romeo and Juliet) Let R = Romeo’s love for Juliet Let J = Juliet’s love for Romeo Assume R and J obey Lotka-Volterra Equations Ignore spatial effects

Romantic Styles: 

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

Pairings - Stable Mutual Love: 

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

Love Triangles: 

Love Triangles There are 4-6 variables Stable co-existing love is rare Chaotic solutions are possible But…none were found in LV model Other models do show chaos

Summary: 

Summary Nature is complex Simple models may suffice but

References: 

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