Abstract
Abstract In this work, we consider the optimal harvesting and stability problems of a prey-predator model with modified Leslie-Gower and Holling-type II functional response. The model is governed by a system of three differential equations which describe the interactions between prey, predator and harvesting effort. Boundedness and existence of solutions for this system are showed. The existence and local stability of the possible steady states are analyzed and the conditions of global stability of the interior equilibrium are established by using the Lyapunov function, we prove also the occurrence of Hopf bifurcation at this point. By using the Pontryagin’s maximal principle, we formulate and we solve the problem of the optimal harvest policy. In the end, some numerical simulations are given to support our theoretical results.
Highlights
Introduction and Mathematical modelIn the last recent, the mathematical modeling of prey-predator models since Lotka ( ) and Volterra ( ) is a very interesting area of research for many ecologists, mathematicians and economists
The existence and local stability of the possible steady states are analyzed and the conditions of global stability of the interior equilibrium are established by using the Lyapunov function, we prove the occurrence of Hopf bifurcation at this point
Many authors have red their models in this theme of Clark [1] who proved the optimal equilibrium policy for joint harvesting of two independent species and he suppose that each population follows a logistic growth law in the absence of harvesting and its harvest rate is proportional to both its stock level and harvesting e ort
Summary
The mathematical modeling of prey-predator models since Lotka ( ) and Volterra ( ) is a very interesting area of research for many ecologists, mathematicians and economists. Many authors have red their models in this theme of Clark [1] who proved the optimal equilibrium policy for joint harvesting of two independent species and he suppose that each population follows a logistic growth law in the absence of harvesting and its harvest rate is proportional to both its stock level and harvesting e ort. Zhang and chen [31] considered the stage structured predator prey model and optimal harvesting policy. In Roy et al [44], the authors studied the e ect of harvesting on prey and the a ect of time delay on a formulated Holling–Tanner prey–predator model with Beddington–DeAngelis functional response They studied the stability of the interior equilibrium and the existence of small amplitude periodic solutions and their stability via the Hopf bifurcation Theorem.
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