%0 Journal Article
%T Bifurcation and Feedback Control of an Exploited Prey-Predator System
%A Uttam Das
%J Journal of Chaos
%D 2014
%R 10.1155/2014/418389
%X This paper makes an attempt to highlight a differential algebraic model in order to investigate the dynamical behavior of a prey-predator system due to the variation of economic interest of harvesting. In this regard, it is observed that the model exhibits a singularity induced bifurcation when economic profit is zero. For the purpose of stabilizing the proposed model at the positive equilibrium, a state feedback controller is therefore designed. Finally, some numerical simulations are carried out to show the consistency with theoretical analysis and to illustrate the effectiveness of the proposed controller. 1. Introduction and Model Description Biological resources in the prey-predator ecosystem are commercially harvested and sold with the aim of achieving economic interest. For this reason, harvesting plays an important role in the study of biological resources. Furthermore, the harvest effort is usually influenced by the variation of economic interest of harvesting. To formulate a biological economic system from an economic point of view and to investigate the dynamical behavior of the model, many scientists use differential-algebraic equations. The differential equations investigate the dynamics of prey and predators and the algebraic equation studies the harvest effort on prey from an economic perspective. Differential-algebraic system has been applied widely in power system, aerospace engineering, chemical process, social economic systems, biological systems, network analysis, and so on. With the help of the differential algebraic model for power systems and bifurcation theory, the complex dynamical behavior of power systems, specially the bifurcation phenomena which can reveal instability mechanisms of power systems, has been extensively studied by Marszalek and Trzaska [1], Ayasun et al. [2], Yue and Schlueter [3], and others. Again, the application of differential algebraic model has an immense impact on the analysis of biological system. I am aware that harvesting has a strong impact on the dynamics of populations. Depending on the applied harvesting strategy, the long run stationary density of a population may be significantly smaller than the long run stationary density of a population without harvesting. Harvesting can lead to the incorporation of a positive extinction probability, even if in the absence of harvesting, a population can be free from extinction risk. If a population is subjected to a positive extinction rate, then harvesting can drive the population density to a dangerously low level at which extinction becomes sure no
%U http://www.hindawi.com/journals/jcha/2014/418389/