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Effect of population size in a predator–prey model

Author

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  • Campillo, F.
  • Lobry, C.

Abstract

We consider a hybrid version of the basic predator–prey differential equation model: a pure jump stochastic model for the prey variable x coupled with a differential equation model for the predator variable y. This hybrid model is derived from the classical birth and death process. The model contains a parameter ω which represents the number of individuals for one unit of prey: x=1 corresponds to ω individual prey. It is shown by the mean of simulations and explained by a mathematical analysis based on a result from the 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 discrete ones in dynamic population modeling even when we use stochastic differential equations in place of deterministic ones.

Suggested Citation

  • Campillo, F. & Lobry, C., 2012. "Effect of population size in a predator–prey model," Ecological Modelling, Elsevier, vol. 246(C), pages 1-10.
    • Handle: RePEc:eee:ecomod:v:246:y:2012:i:c:p:1-10
    DOI: 10.1016/j.ecolmodel.2012.07.015
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    Cited by:

    1. Fowler, A.C. & Winstanley, H.F., 2018. "Microbial dormancy and boom-and-bust population dynamics under starvation stress," Theoretical Population Biology, Elsevier, vol. 120(C), pages 114-120.
    2. Chapwanya, Michael & Dumani, Phindile, 2023. "Spatio-temporal dynamics of microbial population under nutrient-limiting conditions," Applied Mathematics and Computation, Elsevier, vol. 456(C).
    3. Fritsch, Coralie & Harmand, Jérôme & Campillo, Fabien, 2015. "A modeling approach of the chemostat," Ecological Modelling, Elsevier, vol. 299(C), pages 1-13.
    4. Wade, M.J. & Harmand, J. & Benyahia, B. & Bouchez, T. & Chaillou, S. & Cloez, B. & Godon, J.-J. & Moussa Boudjemaa, B. & Rapaport, A. & Sari, T. & Arditi, R. & Lobry, C., 2016. "Perspectives in mathematical modelling for microbial ecology," Ecological Modelling, Elsevier, vol. 321(C), pages 64-74.

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