Functional Response Function With Time Delay Predator - Prey System Of Qualitative Analysis

Mathematical model has widely applications in theory of population control early. With the development of dynamics, differential-delay model is playing a significant role in ecology, physiology, biomechanics and even neural network, thus differential equation is led into dynamic behavior of these system. The famous of exponential law has been studied extensively in individual growth of animals and certain bacteria, unconstrained growth of groups of animals and plants, Maltacom SARS population law. With the recent development, delay differential equations exhibit much more complicated dynamics than ordinary differential equations.This article is mainly divided into the following three parts. In Chapter 1, we first introduce the background and the current situation of the delayed dynamic system. In Chapter 2, a predator-prey system with delay Holling-II functional response is studied, but the predator's numerical response has Leslie form. The permanence of system is analyzed by qualitative method; The impact of the time delay on the stability of the model is considered by choosing the delay timeτas a bifurcation parameter, the existence of Hopf bifurcation is also found by bifurcation theory;sufficient conditions are derived for the global asymptotic stability of the positive equilibrium point by constructing a suitable Lyapunov functional; using central manifold argument and normal form theory, we also establish the direction and the stability of Hopf bifurcation. Time delay could cause a stable equilibrium to become unstable and cause to fluctuate. In Chapter 3, A predator-prey system with delay Holling-III functional response is studied , but the predator's numerical response has Leslie form. The sufficient condition of the global stability of positive equilibrium point is obtained by ordinary differential qualitative and stability method whenτ= 0, we also study the impact of the time delay on the stability of the model and by choosing the delay timeτas a bifurcation parameter, we show that Hopf bifurcation can occur as the delay timeτpasses some critical values.