‎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 Poincarchr('39'){e}-Bendixson theorem and LaSallechr('39')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.