The Study On A Periodic Impulsive Predator-prey System With Functional Response

Population dynamics is the main object of mathematical ecology because of its importance. In the earlier studies, scholars often use differential equations to describe the evolution of the population. However, due to some interference, it is more practical to study the kind of development process by using impulsive dif-ferential equations. The impulsive differential equations are rapidly developed and widely used in the system of population dynamics, optimized management of bio-logical resources, integrated pest control and other fields. The relationship between predator and prey has become the main research object because of its importance. In the system, it is more accurate to describe the practical problem by considering the functional response function. In this paper, we select three representative kinds of this function:ratio-dependent, prey-dependent and Beddington-DeAngelis type of function. Due to the seasonal transformation, the occurrence of the cycle of the day and night, The periodic predator-prey system is studied. The system is more practical, so the periodic impulsive predator-prey system with functional response function is studied in this paper.In this paper, by using the basic theory of the population dynamics and dif-ferential equations, it studies several predator-prey systems with impulsive control strategy. It mainly focused on the stability of pest eradication periodic solution, the existence and uniqueness of periodic solutions of systems and the permanence of the systems, and also use the numerical analysis method to simulate the dynamic behavior.The main research contents and results in this thesis are as follows:(1) A periodic predator-prey system with diffusion and functional response is investigated. The system of impulsive harvest is modeled by a reaction-diffusion equation with impulses. The population was harvested at the fixed time. In this chapter, the main idea is to transform the impulsive partial differential equation into the corresponding ordinary differential equations by using the comparison prin-ciple. According to the biological significance, the positive solution of the system is studied. First of all, The positive property of the solution is obtained. Further, the comparison principle is applied to study the boundedness of the solution and the persistence of the system. Finally, the condition to ensure the existence and stability of the periodic solution is obtained.(2) A periodic impulsive food chain system of three species with functional response is investigated. The system is modeled by an impulsive differential equation with periodic coefficients. By using the Floquet theory and the comparison principle of impulsive differential system, as well as some analytical techniques, the dynamic behavior of the system is researched. We obtain the threshold conditions for the global asymptotic stability of the pest extinction periodic solution, and prove that the system is persistent when the threshold condition is not satisfied.(3) A three-species Beddington-DeAngelis type food chain with impulsive con-trol strategy is investigated. The system is modeled by an impulsive differential equation with periodic coefficients. By researching the dynamic behavior of the system, we obtain the threshold conditions for the global asymptotic stability of the lowest-level prey and mid-level predator extinction periodic solution, and also obtain the condition for the permanence of the system.