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Bifurcation Analysis of a Delayed Predator-Prey Model with Holling Type III Functional Response and Predator Harvesting

DOI: 10.1155/2014/543041

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Abstract:

This paper tries to highlight a delayed prey-predator model with Holling type III functional response and harvesting to predator species. In this context, we have discussed local stability of the equilibria, and the occurrence of Hopf bifurcation of the system is examined by considering the harvesting effort as bifurcation parameter along with the influences of harvesting effort of the system when time delay is zero. Direction of Hopf bifurcation and the stability of bifurcating periodic solutions are also studied by applying the normal form theory and the center manifold theorem. Lastly some numerical simulations are carried out to draw for the validity of the theoretical results. 1. Introduction and Model Description Differential equation models for interactions between species are one of the classical applications of mathematics to biology, dating back to the first half of this century. The development and use of analytical techniques and the growth of computer power have progressively improved our understanding of these types of models. The study of population dynamics with harvesting is a subject of mathematical bioeconomics, which in turn related to the optimal management of renewable resources, Clark [1]. Generally the concept of optimal resource management is based on the standard cost benefit criterion which maximizes present values of net economic revenues. This criterion is relevant to both private and public management decisions, although the specification of costs and benefits are not necessarily the same in both cases. Regulation of exploitation of biological resources has become a problem of major concern nowadays in view of the dwindling resource stocks and the deteriorating environment. Exploitation reduces the biomass of the concerned species, exhibits oscillation, and even causes extinction of some other species. Rosenweig-MacAurtho model experiences oscillation under selective effort of Kar and Ghosh [2]. Legovic et al. [3] have concluded that harvesting the prey species at maximum level causes the extinction of the predator species in traditional prey-predator system. Kar and Ghosh [4] and Ghosh and Kar [5] show that harvesting the prey species at maximum sustainable yield (MSY) level never causes the extinction of the predator species in both ratio-dependent and Holling-Tanner prey-predator systems. It is shown that harvesting the prey species at MSY level may or may not drive the predator population to extinction if intraspecific competition is present among the predator species, Kar and Ghosh [4]. More recently, Ghosh and Kar [6]

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