The evolution of periodic population systems under random environments

Abstract

In this paper we study the behavior of trajectories of the Lotka-Volterra com- petition system with periodic coefficients under t elegraph noise. We give sufficient conditions for the average permanence. Furthermore, we determine the ω-limit sets of the system. 1. Introduction. In this paper we study the behavior of trajectories of the Lotka- Volterra competition system with periodic coefficients under telegraph noise. Until now, many models have revealed the effect of environmental variability on the population dynamics in mathematical ecology (10, 14). In particular, a great effort has been made to find the possibil- ity of the coexistence of competing species under the unpredictable or rather predictable (such as seasonal) environmental fluctuations. It is well recognized that the seasonality has similar effects to stochastic variation (4, 9), but as Lo reau (11) pointed out, the theory of coexistence in a seasonal environment needs further development to reveal the variety of possibilities that seasonal fluctuations may cause. Among these, Namba and Takahashi (13) review the results on Lotka-Volterra competition systems with periodic coefficients, and show the new modes of the possibilities of stable periodic solutions even when the stable coexistence cannot be realized in the corresponding classical Lotka-Volterra system with constant coefficients. Here, we restrict the competition parameters so that there is no possibility of the multi- ple periodic solutions that (13) shows. Then we consider the situation where the interacting populations experience pseudo-stochastic environmental fluctuations with unpredictable dis- continuous change, such as seasonality in a year with 'a cycle of three cold days and four warm days'. In a separate paper (6), we analyze the Lotka-Volterra competition system with constant coefficients under telegraph noises, i.e., environmental variability causes parameter switching between two systems. Our focus of attention is on the intermediate case where environments have both pre- dictable and unpredictable aspects. This case is studied by using a combined system of two Lotka-Volterra systems with periodic coefficients. In this system, it is assumed that at ev- ery moment the population dynamics is governed by one of the two Lotka-Volterra systems with periodic coefficients. That is, the populations usually experience predictable changes of environments. However, it is also assumed that the population dynamics abruptly becomes governed by another Lotka-Volterra system. This abrupt switch between two systems occurs