Research On Dynamical Problems Of Two Delayed Biological Population Systems

Mathematical biology,as an intersecting subject with biology and mathematics,hasproduced many important branches,such as population dynamics that discusses interactions between populations or the populations interacting with environment;dynamics of infectious disease that describes the process and rule of diseases outbreaking;cell kinetics that studies the interactions between cells;dynamics of gene regulatory network that researches gene interacting with regulator factors.In recent years,domestic research has focused on two areas,that is population dynamics and dynamics of infectious disease.Delay differential equation describes the law of development of the system,which not only depends on the current state,but also is determined by the state of system on one point(or on some point)in the past.Therefore,the delay differential equation is objective descriptions of the practical problem.At the same time,the dynamic properties of the system are becoming more complicated and valuable.In this paper,two kinds of biological population models with delays are studied.Specific contents are as follows:Firstly,we study a class of discrete Logistic model with nonlinear death rate.Using stability theory of difference equation,we analyze the distribution of the root of characteristic equation in the unit circle and we get the necessary conditions that the positive equilibrium is asymptotic stability and existence of the Neimark-Sacker bifurcation.Several numerical simulations are given to verify the conclusions.Then,we discuss the stability of the positive equilibrium and Hopf bifurcation in a class of three dimensions delayed coupling Lotka-Volterra ring model.Analyzing the distribution of linearized system’s characteristic roots in complex plane,we find the necessary conditions that the positive equilibrium is stability and Hopf bifurcation occurs.According to the center manifold theorem and the normal form method,we discuss the direction and the stability of the Hopf bifurcation.Several examples are given to explain the biological significance.