Effect of population size in a Prey-Predator model

Fabien Campillo 1, 2 Claude Lobry 1
1 MODEMIC - Modelling and Optimisation of the Dynamics of Ecosystems with MICro-organisme
CRISAM - Inria Sophia Antipolis - Méditerranée , MISTEA - Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie
Abstract : We consider a stochastic version of the basic predator-prey differen- tial equation model. The model, which contains a parameter ω which represents the number of individuals for one unit of prey - If x denotes the quantity of prey in the differential equation model x = 1 means that there are ω individuals in the discontinuous one - is derived from the classical birth and death process. It is shown by the mean of sim- ulations and explained by a mathematical analysis based on results in singular perturbation theory (the so called theory of Canards) that qualitative properties of the model like persistence or extinction are dramatically sensitive to ω. For instance, in our example, if ω = 107 we have extinction and if ω = 108 we have persistence. This means that we must be very cautious when we use continuous variables in place of jump processes in dynamic population modeling even when we use stochastic differential equations in place of deterministic ones.
Type de document :
Pré-publication, Document de travail
2011
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  • HAL Id : hal-00645619, version 1
  • ARXIV : 1111.6460

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Fabien Campillo, Claude Lobry. Effect of population size in a Prey-Predator model. 2011. 〈hal-00645619〉

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