Lotka-Volterra Systems in Environments with Randomly Disordered Temporal Periodicity

摘要

A generalized Lotka-Volterra model for a pair of interacting populations of predators and prey is studied. The model accounts for the preyu27s interspecies competition and therefore is asymptotically stable, whereas its oscillatory behavior is induced by temporal variations in environmental conditions simulated by those in the preyu27s reproduction rate. Two models of the variations are considered, each of them combining randomness with u22hiddenu22 periodicity. The stationary joint probability density function (PDF) of the number of predators and prey is calculated numerically by the path integration (PI) method based on the use of characteristic functions and the fast Fourier transform. The numerical results match those for the asymptotic case of white-noise variations for which an analytical solution is available. Several examples are studied, with calculations of important characteristics of oscillations, for example the expected rate of up-crossings given the level of the predator number. The calculated PDFs may be of predominantly random (unimodal) or predominantly periodic nature (bimodal). Thus, the PI method has been demonstrated to be a powerful tool for studies of the dynamics of predator-prey pairs. The method captures the random oscillations as observed in nature, taking into account potential periodicity in the environmental conditions.