Some Conditions for the Nonexistence of Limit Cycles in a Predator-Prey System

Abstract

In this paper, we study the problem of the existence of limit cycles for a predator-prey system with a functional response. It is assumed that the functional response is positive, increasing, concave down, and its third derivative has a unique root. A necessary condition for the nonexistence of limit cycles is presented. Some conditions are given under which the necessary condition is also the sufficient condition for the nonexistence of limit cycles.

Geometric criteria for the nonexistence of cycles in Gause-type predator-prey systems

“…Under some general conditions on the response function p(x), Moghadas [14] showed that (for the special case K = 1) system (1) has no cycle if and only if 2x…”

“…A concrete counterexample was constructed in Hofbauer and So [6]. To different degrees Cheng et al [2], Kuang [11], Moghadas [14], and Xiao and Zhang [21] contain modifications of Hsu's result. But the sufficient conditions they proposed are difficult to satisfy and hard to verify, and the geometrical interpretation of their results is not clear.…”

“…For this reason, the existence and uniqueness of positive cycles and limit cycles of system (1) has attracted much interest in recent years. For a sample of these studies, see Cheng [1], Cheng et al [2], Kuang and Freedman [12], Huang and Merrill [8], Ding [3], Kuang [11], Kooij and Zegeling [10], Sugie et al [19], Sugie [17], Moghadas [14], and Xiao and Zhang [21]. Most of the recent work has employed technical methods, such as transforming the system to an equivalent generalized Liénard system or trying directly to exploit the special structure of the limit cycle and the prey isocline.…”