RT - Journal Article T1 - On the Global Stability‎, ‎Existence‎ ‎and Nonexistence of Limit Cycles in a Predator-Prey System JF - lu-maco YR - 2021 JO - lu-maco VO - 2 IS - 2 UR - http://maco.lu.ac.ir/article-1-83-en.html SP - 17 EP - 26 K1 - Dynamical System‎ K1 - ‎Predator-Prey system‎ K1 - ‎Limit Cycle‎ K1 - ‎Global Stability AB - ‎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. LA eng UL http://maco.lu.ac.ir/article-1-83-en.html M3 10.52547/maco.2.2.2 ER -