Study On The Predator-Prey Models Based On Special Constraints And Holling ⅠFunction Response

The predator-prey model, as one of the interaction between biological populations, is one of the main topics of the study of ecology and biomathematics. Due to the extensive application of the predator- prey model in ecological balance, protection of animals and plants, management and development of ecological environment, et al, the study on the predator-prey model has received much attention from mathematicians and biologists.Firstly, two types of predator-prey model based on special constraints and Holling I functional response are studied. In the first scenario, continuously harvesting of a phytoplankton-zooplankton system is considered. A predator-prey model based on continuous harvesting strategy is estabilished. The existence and local stability of equilibrium are analyzed by taking the harvest strength as a control parameter. The global asymptotic stability of the equilibrium is obtained by applying Dulac function criterion and constructed Lyapunov function. Meanwhile, to capture the profit as the goal, the condition for the existence of economic equilibrium is obtained and the corresponding optimal fishing intensity is deduced. in the second scenario, the effect of prey refuse is considered. A predator-prey model based on prey refuge effect is estabilished. By using the Lyapunov function and Utkin equivalent control method, the existence and stability of normal, virtual, pseudo equilibrium points and the tangent point are given. To verfy the correctness of the theoretical results, numerical simulations are carried out by using MATLAB software.Secondly, two predator prey models with impulsive state feedback control are studied. In the first scenario, the state-dependent harvesting strategy in fisheries management is considered, and a predator-prey model with state-dependent feedback impulsive control is estabilished. The existence and asymptotic stability of the semi-trivial periodic solution and order-one periodic solution are discussed. In the second scenario, from the aspect of pest management, a pest control predator-prey model with state-dependent feedback control is estabilished, where the pesticide strength and yield of release of predator are dependent on the monitoring state. The qualitative analysis of pest management model is carried out by using the method of the successor function and the Bendixson theorem. The existence of order-one periodic solution is proved, and the stability of order-one periodic solution is verified by the analogous Poincare criterion. In addition, an optimization problem is formulated and the optimum pest control level with a minimum control cost is obtained. The numerical simulations are carried out by using MATLAB software, and the corresponding conclusions are given.