Permanence and Extinction of a Generalized Gause-Type Predator-Prey System with Periodic Coefficients

We study the permanence and extinction of a generalized Gause-type predator-prey system with periodic coefficients. We provide a sufficient and necessary condition to guarantee the predator and prey species to be permanent and a sufficient condition for the existence of a periodic solution. In addition we prove that when the predator population tends to extinction, the prey population keeps oscillating above a positive population level. 1. Introduction Permanence of a dynamical system has always been a hot issue in the past few decades. The concept of permanence has been introduced and investigated by several authors, each using his own terminology: “cooperativity” in the earlier papers of Schuster et al. [1], and Hofbauer [2], “permanent coexistence” by Hutson and Vickers [3], “uniform persistence” in Butler et al. [4], and “ecological stability” by Svirezhev and Logofet [5–7] (for more detailed statements of the concept see [8]). Many important results have been found in recent years [1–37]. Some authors (see [21, 28, 31, 33]) have considered the following two species periodic Lotka-Volterra predator-prey system where and are periodic functions on with common period and for all . They have established sufficient and necessary conditions for the existence of positive -periodic solutions of the system by using different methods, respectively. Teng [31] has given sufficient and necessary conditions for the uniform persistence of the system. Cui [16] has considered the permanence of the following Lotka-Volterra predator-prey model with periodic coefficients: He provided a sufficient and necessary condition to guarantee the predator and prey species to be permanent. In Theorems 2.2 and 2.3 he set a precondition that . This restricts the application of the theorems more or less, since many researchers often neglect the logistic term in the predator equation when the population level of the predator is relatively low and the competition between predators can be ignored, and it proved to be an unnecessary precondition in our paper. However, the research methods in his work inspired me, and many proofs, especially in the first half of this paper, are analogous to [16]. In this paper we consider the permanence of the following generalized Gause-type predator-prey system, where , for all and , , are all periodic continuous functions with common period ; , are positive, and is nonnegative. We emphasize that our model includes the case when . In the absence of predators, system (1.3) becomes where is a real-valued function defined on Vance and Coddington [35] have