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Published byΤιτάνια Βούλγαρης Modified over 6 years ago
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Predator-Prey Dynamics for Rabbits, Trees, & Romance
11/10/2018 J. C. Sprott Department of Physics University of Wisconsin - Madison Presented to International Conference on Complex Systems in Nashua, NH on May 10, 2002
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Lotka-Volterra Equations
11/10/2018 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 -
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Types of Interactions - - dR/dt = r1R(1 - R - a1F)
dF/dt = r2F(1 - F - a2R) + a2r2 Prey- Predator Competition - + a1r1 Predator- Prey Cooperation -
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Equilibrium Solutions
11/10/2018 Equilibrium Solutions dR/dt = r1R(1 - R - a1F) = 0 dF/dt = r2F(1 - F - a2R) = 0 Equilibria: R = 0, F = 0 R = 0, F = 1 R = 1, F = 0 R = (1 - a1) / (1 - a1a2), F = (1 - a2) / (1 - a1a2) F R
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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.
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Alternate Spatial Lotka-Volterra Equations
11/10/2018 Alternate Spatial Lotka-Volterra Equations Let Si(x,y) be density of the ith species (rabbits, trees, seeds, …) dSi / dt = riSi(1 - Si - ΣaijSj) ji where S = Sx-1,y + Sx,y-1 + Sx+1,y + Sx,y+1 + aSx,y 2-D grid:
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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
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Typical Results
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Dominant Species
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Fluctuations in Cluster Probability
Time
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Power Spectrum of Cluster Probability
Frequency
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Sensitivity to Initial Conditions
Error in Biomass Time
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Results Most species die out Co-existence is possible
Densities can fluctuate chaotically Complex spatial patterns spontaneously arise One implies the other
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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
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Romantic Styles dR/dt = rR(1 - R - aJ) a - r - + + Cautious lover
Narcissistic nerd - + r Eager beaver Hermit -
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Pairings - Stable Mutual Love
Cautious Lover Narcissistic Nerd Eager Beaver Hermit Narcissistic Nerd 46% 67% 5% 0% Eager Beaver 67% 39% 0% 0% Cautious Lover 5% 0% 0% 0% Hermit 0% 0% 0% 0%
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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
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Simple models may suffice
11/10/2018 Summary Nature is complex Simple models may suffice but
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11/10/2018 References lectures/iccs2002/ (This talk)
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