Abstract
In this paper, we investigated the dynamics of a diffusive delayed predator-prey system with Holling type II functional response and nozero constant prey harvesting on no-flux boundary condition. At first, we obtain the existence and the stability of the equilibria by analyzing the distribution of the roots of associated characteristic equation. Using the time delay as the bifurcation parameter and the harvesting term as the control parameter, we get the existence and the stability of Hopf bifurcation at the positive constant steady state. Applying the normal form theory and the center manifold argument for partial functional differential equations, we derive an explicit formula for determining the direction and the stability of Hopf bifurcation. Finally, an optimal control problem has been considered.
Highlights
Predator-prey systems model some biological phenomenons and relationship between predator and prey in the real world, which play a crucial role in mathematics and have been extensively considered in many ways by many researchers
Concerning the conservation for the long-term benefits of humanity, there is a wide-range of interest in the use of bioeconomic modeling to gain insight in the scientific management of renewable resources like fisheries and forestry [13]. Both harvesting in prey and in predator could lead some dangers in real-life harvesting such as no equilibrium exists and
The study of predator-prey models with harvesting have attracted the attention of many researchers which have rich bifurcating phenomenons
Summary
Predator-prey systems model some biological phenomenons and relationship between predator and prey in the real world, which play a crucial role in mathematics and have been extensively considered in many ways by many researchers (see e.g. [1,2,3,4,5,6] and [7,8,9,10,11,12]). Concerning the conservation for the long-term benefits of humanity, there is a wide-range of interest in the use of bioeconomic modeling to gain insight in the scientific management of renewable resources like fisheries and forestry [13] In biological, both harvesting in prey and in predator could lead some dangers in real-life harvesting such as no equilibrium exists and. When the spatial dispersal is considered in ratio-dependent predator-prey model, system has been investigated by Zhang [19] They derived the conditions for Hopf and Turing bifurcation on the spatial domain. − h2, where h1 > 0 and h2 > 0 are harvesting or removal rate for the prey and the predator, respectively They observed very rich and interesting dynamical behaviors such as the existence of multiple equilibria, homoclinic loop, Hopf bifurcation etc.
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