Research On Dynamics Of Several Kinds Of Ecological Mathematical Systems With Delay

Population dynamics is one of the important directions of biomathematics research.Recent years,scholars have established numerous ecological mathematical models to describe the quantitative evolution relationship between populations,and have carried out a lot of theoretical research on the established dynamic models,and achieved abundant results.In this paper,based on stability theory of ordinary differential equation,norm form methods and center manifold theorem,the dynamic behaviors of a delay predator-prey system with diffusion and a delay commensalism system with linear harvesting effect are studied including the stability of positive solutions,the existence of Hopf bifurcation,the direction of Hopf bifurcation and the stability of periodic solutions,the Turing instability and the Turing bifurcation.The paper is divided into five chapters.In Chapter 1,the background,significance and current status of research on two types of ecological mathematical models are introduced.In Chapter 2,some basic knowledge and theorems of differential equation and population dynamics are described.In Chapter 3,under the Neumann boundary condition,the dynamic behaviors of a delayed and diffusion predator-prey system with Holling IV and linear harvesting effect are studied.Firstly,the existence and stability of the positive equilibrium point are analyzed.Then,the influence of diffusion on the system stability without delay is studied.The conditions of the Turing instability and the expression of Turing bifurcation carve are derived.The corresponding number examples and get the spot,the coexistence of spot and stripe patterns are also gived.Next,with delay as bifurcation parameter,the conditions for the existence of the local Hopf bifurcation are analyzed.The expression for describing the Hopf bifurcation and periodic solutions properties are obtained.Finally,some numerical simulations to verify the correctness of the theoretical analysis findings are carried out.In Chapter 4,the dynamic behaviors of a commensalism system with delay and linear harvesting effect is considered.Firstly,based on the zero point distribution theorem,the stability of the only positive equilibrium of the system and the existence of the local Hopf bifurcation are researched.The local Hopf bifurcation value of the system is obtained.Secondly,the calculation formulas for determining the Hopf bifurcation direction and bifurcation periodic solutions behaviors,namely the period and stability,are determined by the normal form methods and center manifold theorem.Finally,some numerical simulations are given to verify the theory results.In Chapter 5,the summary of the paper and the prospect of the future work are given.