Study On The Dynamics Of Three Types Of Predator-Prey System

As we all know, biomathematics is an interdisciplinary between biology and mathematics. It developed rapidly in recent years, especially the study on the dynamics of predator-prey system has always been the hot topic concerned by lots of mathematicians and biologists.This paper consists of the following three parts:In the first part, the infinite delay predator-prey system with age-structure and Holling-Tanner III functional response function is studied. By using some basic methods in the qualitative and stability theory of ordinary differential equation as well as comparison principle, we construct an assistant system to obtain the sufficient conditions for permanent survival of the predator-prey system. Then we discuss the global asymptotic stability of the system with the help of an appropriate Lyapunov function and the Barbalat theorem, etc.In the second part, the pulse delay predator-prey system with age-structure and Holling-(n+1) functional response is studied. We obtain the boundedness of all solutions to the system through constructing and analyzing the Lyapunov function. By discussing the system’s subsystem and using the comparison theorem of impulsive differential equation as well as some basic theories of delay differential equations, we analyze the global attractability of species extinction periodic solutions and come up with sufficient conditions for system’s permanent survival.In the last part, the m dimension food chain system with Holling-(n+1) func-tional response is studied. Firstly, by analyzing the system and using comparison principle, we give sufficient conditions for system’s permanent survival. Secondly, we prove the existence of positive ω periodic solution in this system through construct-ing Poincare mapping and Brouwer fixed point theorem. And lastly, we construct an appropriate Lyapunov function to obtain the result of stationary oscillation in the system with the help of the Barbalat theorem.