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PredatorPrey - Modeling the Predator-Prey Relationship...

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Modeling the Predator-Prey Relationship Michael Olinick Middlebury College MAA Session on Environmental Mathematics January 12, 2006
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A basic assumption of the classic Lotka- Volterra predator-prey model is that each species experiences exponential growth or decay in the absence of the other, Recent extensions of this model investigate logistic growth of one species when the other is absent, time delays in response by one species to population changes in the other, and multiple species interactions. A survey of consequences and predictions of the original and the modified models will be presented.
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“Nature offers a puzzling variety of interactions between species.” – Nino Boccara, Modeling Complex Systems Mutualism Competition Predation Predation Parasites : live a major period of their life in a single host doing nonlethal harm to the host (tapeworm) Grazers : attack large numbers of prey during their lifetimes, consuming part of the host without causing death ( mosquitos feeding on human blood, sheep grazing on grass). True Predators : attack many prey during their lifetimes, quickly killing the prey (wolves). Googling “Predator-Prey” returns 609,000 hits.
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Today: A very brief look at a very large and active area of modeling Review of the classic Lotka-Volterra: assumptions, model formulation, analysis and predictions, successes and failures. Modifications of the Lotka-Volterra Approach.
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A Real World Puzzle Umberto D’Ancona asks Vito Volterra: After World War I, the proportion of predatory fish caught in the Upper Adriatic was considerably higher than in the years before the war; the proportion of prey fish was lower. The war between Austria and Italy halted commercial fishing. How did the war benefit the predators more than the prey?
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Alfred James Lotka 1880-1949 Vito Volterra (1860-1940 Elements of Physical Biology, Baltimore: Williams and Wilkins, 1925. (Dover reprint Elements of Mathematical Biology , 1956) “Variazioni e fluttuazioni del numero d’individui in specie animali conviventi,” Rendiconti dell’Academia dei Lincei , 6 (2), 31 – 113 (1926). (“Fluctuations in the Abundance of a Species Considered Mathematically,” Nature , 118 , 558 – 560 (1926). Francesco M. Scudo and James R. Ziegler, eds., The Golden Age of Theoretical Ecology, 1923 – 1940, Berlin: Springer- Verlag, 1978.
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The Lotka-Volterra Model Assumptions A classic formulation of this situation is that of a population of rabbits, who feed on clover, and a population of foxes, who feed on rabbits. The assumptions usually made about the situation are these: 1. In isolation, the rate of change of population of one species is proportional to the population of that species. In the absence of foxes, it is assumed that the rabbits population will exhibit exponential growth. If there are no rabbits, the fox population will undergo a pure death process.
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