Global structure of a kind of predator-prey system

Abstract

In this paper we prove a theorem, theorem 2, on nonexistence of closed trajectory for a general predator-prey system. Then, using this theorem and another theorem on existence and uniqueness of limit cycle for predator-prey system, we complete the investigation of a concrete model of predator-prey system

$$\begin{gathered} \dot x = \gamma x({{1 - x} \mathord{\left/ {\vphantom {{1 - x} K}} \right. \kern-\nulldelimiterspace} K}) - {{yx^n } \mathord{\left/ {\vphantom {{yx^n } {(a + x^n )}}} \right. \kern-\nulldelimiterspace} {(a + x^n )}} \hfill \\ \dot y = y({{\mu x^n } \mathord{\left/ {\vphantom {{\mu x^n } {(a + x^n ) - D}}} \right. \kern-\nulldelimiterspace} {(a + x^n ) - D}}) (n = 1,2) \hfill \\ \end{gathered} $$

under the conditions of all kinds of parameters.