ON THE GLOBAL STABILITY, EXISTENCE AND NONEXISTENCE OF LIMIT CYCLES IN A PREDATOR-PREY SYSTEM

The existence and number of limit cycles is an important problem in the study of ordinary differential equations and dynamical systems. In this work, we consider 2- dimensional predator-prey system and, using Poincaré-Bendixson theorem and LaSalle’s invariance principle, present some new necessary and some new sufficient conditions for the existence and nonexistence of limit cycles of the system. These results extend and improve the previous results in this subject. Local or global stability of the rest points of a system is also an important issue in the study of the systems. In this paper, a sufficient condition about global stability of a critical point of the system will also be presented. Our results are sharp and are applicable for predator-prey systems with the functional response which is the function of prey and predator. At the end of the manuscript, some examples of well-known predator-prey systems are provided to illustrate our results.