Existence of limit cycles for predator–prey systems with a class of functional responses

In this paper we study the problem of the existence of limit cycles for a predator–prey system with a functional response. From the ecological point of view, the existence of a limit cycle shows the oscillatory behaviour of the populations. A necessary and sufficient condition for the existence of limit cycles when the derivative of the functional response is positive, decreasing, and concave upward is given. Global stability of the system can be established by our results. In addition it is shown that the local stability and global stability of the critical point of the system are equivalent. The results cover most of the models which have been proposed in the ecological literature for predator–prey systems. Some ecological models are given to illustrate the results.