Application Of Theory Of Impulsive Dynamics System In Biological Dynamics System

The theories and methods of impulsive differential equations have achieved great development and become a whole system in recent thirty years. It is well-known that many real world phenomena and human activities do have impulsive effect. In this thesis, based on impulsive differential equations, we establish and investigate two kind of predator-prey chain model, mathematically, we use a combined approach of continuous dynamics and impulsive dynamics to investigate various dynamical behavior of the systems we consider. Firstly we propose two mathematical models concerning continuous harvesting and, respectively, impulsive harvesting control strategies. In the case in which a continuous control is used, using qualitative analysis methods, we studies the stability of the equilibria under the condition of harvesting and get the optimal harvesting policy. In the case in which an impulsive control is used, we give out the condition for stability of periodic solution. Furthermore, the existence of positive periodic solution is obtained by the bifurcation theory. Secondly, based on the background of pest management in agriculture, we establish a three species two-prey one-predator chain model with two type function response periodic releasing natural enemies and spraying pesticide at fixed time. By using Floquet theory, Comparison theorem of impulsive differential equation and analytic method, we investigate the dynamics of the system in detail. We obtain the conditions for asymptotical stability of the prey-eradication periodic solution and permanence of the system.