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Journal of Mathematical Biology

, Volume 54, Issue 3, pp 407–451 | Cite as

Prediction of predator–prey populations modelled by perturbed ODEs

  • Sorana FrodaEmail author
  • Sévérien Nkurunziza
Article

Abstract

In this paper we explore a stochastic model in continuous time for predator–prey interactions, which accounts for the periodical behaviour observed in many animal populations. More precisely, we consider a solution to the classical Lotka–Volterra system of equations, but we view the actual population sizes as random perturbations of the solutions to this ODE system. Namely, we assume that the perturbations follow correlated Ornstein–Uhlenbeck processes; this approach generalizes the one of Froda and Colavita [Aust N Z J Stat 2:235–254, 2005] who considered only i.i.d. errors. This type of perturbed deterministic model allows to perform parameter estimation and to predict population sizes at future times. On the other hand, the present model refines the previous one since it takes into account the variability due to external factors and the time dependence in the random component. Moreover, this more flexible model improves the predictions of population sizes at future times. In order to illustrate this last point, we analyse two data sets.

Keywords

Predator–prey system of differential equations Closed orbit Periodic solutions Correlated Ornstein–Uhlenbeck processes Weighted least squares Estimation and prediction Stochastic calculus 

Mathematics Subject Classification (2000)

Primary 62M10 Secondary 92D25 Secondary 62P10 

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Département de mathématiquesUniversité du Québec à MontréalMontrealCanada
  2. 2.Department of Mathematics and StatisticsUniversity of WindsorWindsorCanada

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