Study On Several Biological Dynamic Models With Impulsive Effects

Biomathematics is an interdisciplinary subject of mathematics and life sciences,which is the science of studying the quantitative nature and spatial pattern of life and life systems. Population dynamics is one of the important branches of biomathematics. In the classical population dynamics study, the system state is continuous. However, many population ecology phenomena are not continuous processes, and their developments are often affected by short-term disturbance. For such phenomena, the traditional continuous system is no longer applicable. And more complex impulsive differential equations axe needed to describe these phenomena.The pulse differential equations describe the rapid change or jump of certain motion states at fixed or unfixed moments, and give a natural description of the momentary factor.They combine characteristics of both discrete and continuous systems, and beyond. In recent years, although a lot of achievements have been made on pulse differential system for population dynamics, there are also many problems to be solved.This paper studies the dynamic nature of several types of population models with impulsive effects, especially the pulse effects. The paper is divided into four chapters.Chapter 1 (Introduction) briefly introduces the background and significance of the pulse differential equation in bio-dynamics, and the basic concepts of pulse differential equation.In the second chapter, a predator-prey system with a fixed-time impulse effect of Holling ? is established. The new system can be applied to the cases such as periodic artificial loading, harvesting or theorem spraying of pesticides, which cannot be handled by continuous models. The existence of periodic solutions of the system is proved by Mawhin coincidence degree theory, and verified by computer numerical simulation.In the third chapter,a non-autonomous Holling ? predator-prey model with pulsed and strong-Allee effect is established. Using the similar methods in Chapter 2. the suf-ficient conditions for the existence of periodic solutions are obtained. From theoretical and numerical simulations aspects, we prove that the system can achieve some ecological balance in the case of periodic harvest (delivery).In the fourth chapter,the fixed time pulse in the model studied in the second chapter is changed to the state feedback pulse, which makes the system more suitable to some practical situations. Using the geometry theory of semi-continuous dynamical system, we study the existence, uniqueness and stability of the periodic solution of the pulse state feedback system.Finally, we summarize the whole paper and propose subjects for the future study.