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J. C. Sprott |
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Department of Physics |
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University of Wisconsin - Madison |
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Presented to |
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International Conference on Complex Systems |
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in Nashua, NH |
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on May 10, 2002 |
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R = rabbits, F = foxes |
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dR/dt = r1R(1 - R - a1F) |
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dF/dt = r2F(1 - F - a2R) |
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dR/dt = r1R(1 - R - a1F) =
0 |
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dF/dt = r2F(1 - F - a2R) =
0 |
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With N species, there are 2N
equilibria, only one of which represents coexistence. |
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Coexistence is unlikely unless the species
compete only weakly with one another. |
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Diversity in nature may result from having so
many species from which to choose. |
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There may be coexisting “niches” into which
organisms evolve. |
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Species may segregate spatially. |
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Purely deterministic |
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(no randomness) |
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Purely endogenous |
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(no external effects) |
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Purely homogeneous |
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(every cell is equivalent) |
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Purely egalitarian |
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(all species obey same equation) |
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Continuous time |
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Most species die out |
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Co-existence is possible |
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Densities can fluctuate chaotically |
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Complex spatial patterns spontaneously arise |
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Let R = Romeo’s love for Juliet |
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Let J = Juliet’s love for Romeo |
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Assume R and J obey Lotka-Volterra Equations |
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Ignore spatial effects |
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There are 4-6 variables |
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Stable co-existing love is rare |
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Chaotic solutions are possible |
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But…none were found in LV model |
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Other models do show chaos |
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Nature is complex |
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Simple models may suffice |
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http://sprott.physics.wisc.edu/
lectures/iccs2002/ (This talk) |
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sprott@juno.physics.wisc.edu |
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Notes