On the uniqueness and nonexistence of limit cycles for predator-prey systems

In this paper, a general predator-prey system is considered. By utilizing simple but crucial changes of variables, the system is reduced to a generalized Lienard system where a wealth of existing methods and results are applicable. A new uniqueness theorem of limit cycle for generalized Lienard system is obtained. Some conditions of known uniqueness theorems and nonexistence theorems are modified so that they are more easily applied. As an application, criteria for the uniqueness of limit cycles and global stability of a unique positive equilibrium of the general predator-prey system are derived, which include some results of Kuang and Freedman (1988 Math. Biosci. 88 67-84). Several examples are given to illustrate our results.