Study On Nonliear Models Of Microbial Pestcide

In the recent ten years, the sustainable development of agriculture and the har-monious relationship between human beings and the environment have aroused people's attention. In particular, because of the negative effects brought by chemical pesticides and the rising cost of screening new pesticides, some enterprises and researchers began to develop efficient biological pesticides which are compatible with the environment and safe to human beings and animals. As a new pest bio-insccticidcs, cntomopathogenic ne-matode can be widely used in preventing and killing pests in agriculture, forestry, grass, flowers and some sanitary pests. The cntomopathogenic nematodc possesses substantial application potential in sustainable pest management and it is an important biological control agent. Though the study of entomopathogenic ncmatode has been conducted by many researchers, there is still no theoretical analysis of it from the perspective of math-ematical model. In this dissertation, two classes of mathematical models are established based on preying characteristic of Entomopathogenic Nematode. Dynamic behavior of the established models, including the existence and stability of equilibriums, Hopf bifur-cation, the existence and stability of the periodic solutions, etc. are investigated by using the qualitative theory of differential equation, bifurcation theory and impulsive differen-tial equation theory. Moreover, the parameters'influence on the dynamical behaviors is also explored here. This dissertation, therefore, sets up some theoretical foundations for explaining, predicting and controlling some phenomena in pest management. The main results of this dissertation may be summarized as follows:In Chapter 3, the mathematical models for Entomopathogenic Nematode with the Malthus growth rate are formulated and investigated. Firstly, the model of continuous release of Entomopathogenic Nematode is studied. In the case, by using qualitative analytical methods of ordinary differential equations we found that the boundedness of the solution, global asymptotic stability of feasible equilibrium points, conditions under which there is not closed trajectory, and the conditions of the existence and uniqueness of limit cycles. The theoretical results are verified by numerical simulations. Our results indicate that with the increasing release of nematodes, the amount of pests changes from infinite oscillation to cycle oscillation and tends to the positive equilibrium point. Finally the pests arc extinct. Secondly, the model of periodic impulsive release of Entomopathogenic Nematode is studied. In this case, we obtain the critical value for the global asymptotic stability of pest-free periodic solution by means of Floquet theorem and small amplitude perturbation skills. In a certain limiting case, it is shown that a nontrivial periodic solution emerges. The main results are verified by using numerical simulation. Finally, the model of state-dependent impulsive release of Entomopathogenic Nematode is discussed. When the number of pests has reached a dangerous level, the Entomopathogenic Nematodes are distributed, meanwhile the pesticides are sprayed. We researched the existence and stability of order one periodic solution by using Poincare map and Brouwer's fixed point theorem. It is shown that the system tends to a stable periodic solution, which depends on the feedback state, the control parameters and the initial densities of nematodes and pests. In particular, we prove that the system without impulsive state feedback control is global asymptotic stability of negative direction. And then, we obtain that the system with impulsive state feedback control exists singular order one periodic solution. The numerical simulations verify the theoretical results.In Chapter 4, the mathmatical models for Entomopathogenic Nematode with the Monod growth rate are formulated and investigated. Firstly, the model of continuous release of Entomopathogenic Nematode is studied. By using Poincare-Bendixson theo-rem, decision methods for the Center and Focus and the Friedrich's method of bifurcation problem, a complete qualitative analysis for the system is undertaken. It indicates that under some conditions, the system is globe asymototically stable if positive equilibrium point is stable and the system has at least one limit cycle if positive equilibrium point is unstable. The Hopf bifurcation conditions for the model are obtained by using the Friedrich's method. The stability of the periodic solutions is determined as well. Com-bined with numerical simulations, the ecological significance of these conclusions is inter-preted. Secondly, the model of periodic impulsive release of Entomopathogenic Nematode is investigated, and the critical value for the global asymptotic stability of pest-free pe-riodic solution is also obtained. We prove that a nontrivial periodic solution emerges in a certain limiting case. This result indicates the numbers of nematodes and pests are os-cillation. The numbers of pests would be under Economic threshold by adjusting control parameters. Finally, the model of state-dependent impulsive release of Entomopathogenic Nematode is discussed. By employing geometrical method, we proved the existence and stability of order one periodic solution. global asymptotic stability of negative direction of the system without pulse and the existence of singular order one periodic solution of the system with pulse. This shows that if the pest population density is below economic injury level, pests and nematodes can co-exist. That is to say, this method can not only be employed to control pests, but also be used to avoid pesticide contamination of environment.