Stability Analysis Of Coupled Biological System Model With Delay

This paper will focus on discussing three kinds of coupled prey-predator systems with delay, based on the theories of delay differential equation, the theories of Hopf bifurcation and symmetric local Hopf bifurcation theorem. Though discussing the distribution of characteristic equation roots, the stability of the delay differential equations (DDE) and the existence of Hopf bifurcation are obtained, and then analyzes the nature of the whole system. Main results are as follows:In the third part, an three-dimentional coupled Lotka-Volterra ring model with delay is introduced. The existence conditions of equilibrium point is given by using the theories of delay differential equation. Bifurcations of multiple periodic solutions is given by using the theories of bifurcation, among which symmetry is an important condition of producing multiple periodic solutions. Bifurcations of phase-locked periodic solutions of model is obtained by using symmetry group theory. At last, under the help of Matlab software, the numerical simulation of the system is presented.In the fourth part, an three-dimensional infected prey-predator system with delay is considered.We obtain an infected prey-predator system model with delay by introducing time delays into the original three-dimensional infected predator-prey system. Stability analysis around the different equilibriums is studied by using the theories of delay differential equation. According to Rouche's theorem, the condition of Hopf bifurcation is obtained, and then analyzes the nature of the whole system. The existence of the local Hopf bifurcation can be obtained when the parameter passes a sequence of critical values. Some numerical simulation examples are given to illustrate the obtained results.The fifth part further explores the model of the coupled prey-predator system with delay though the same analysis method that have been introduced in the previous part. Stability analysis and the periodic solutions around the different equilibriums are studied by using the theories of delay differential equation. According to Hurwitz criterion demonstrates the sufficient conditions of stability around the equilibrium point, also demonstrates the sufficient conditions of periodic solutions generation. At last, under the help of Matlab software, the numerical simulation of the system is presented.