On the Global Stability‎, ‎Existence‎ ‎and Nonexistence of Limit Cycles in a Predator-Prey System

Document Type : Original Article

Authors
Tohid Kasbi 1
Vahid Roomi 2
1 Faculty of Mathematical Science, University of Tabriz, Tabriz, Iran.
2 Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.
10.22034/maco.2.2.2
Abstract
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.

Keywords

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Mathematical Analysis and Convex Optimization
Volume 2, Issue 2
Autumn 2021
Pages 17-26