Limit cycles in a generalized Gause-type predator–prey model

We study the problem of the existence of limit cycles for a generalized Gause-type predator–prey model with functional and numerical responses that satisfy some general assumptions. These assumptions describe the effect of prey density on the consumption and reproduction rates of predator. The model is analyzed for the situation in which the conversion efficiency of prey into new predators increases as prey abundance increases. A necessary and sufficient condition for the existence of limit cycles is given. It is shown that the existence of a limit cycle is equivalent to the instability of the unique positive critical point of the model. The results can be applied to the analysis of many models appearing in the ecological literature for predator–prey systems. Some ecological models are given to illustrate the results.