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 ۲-dimensional predator-prey‎ ‎system and‎, ‎using Poincarchr('۳۹'){e}-Bendixson theorem and LaSallechr('۳۹')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 equations and 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 functional response which is 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.

Keywords:

Dynamical System‎ , ‎Predator-Prey system‎ , ‎Limit Cycle‎ , ‎Global Stability