Bifurcation Analysis Of Two Classes Of Eco-epidemical Models

Since the 20 th century,with the development of the society,the spread of infectious diseases in the population gradually get the attention of the society from all walks of life,through the establishment of mathematical model,the scientists analyzed the variation law under the influence of the population in the disease,the study of the spread of the disease,can obtain the principle and the transmission mechanism,to customize a better control strategy.In this paper,we analyze two types of eco-epidemiological models,in which the infectious disease spreads only in the prey,and the predator eats only the infected prey.Based on the qualitative and bifurcation theory of differential equations,the bounded solution of the system,the existence and stability of the boundary equilibrium point,the existence and bifurcation of the positive equilibrium point are studied in detail.The first chapter mainly introduces the research background and present situation,and the main content of this study;The second chapter introduces the necessary basic knowledge,research methods and mathematical tools.Chapters 3 and 4 respectively study two types of eco-epidemic models.The first eco-epidemiological model deals with cases where the disease is spread only among the prey and has vertical and saturated transmission rates.Firstly,the boundedness of the solution of the system is discussed.Secondly,the existence condition of the boundary equilibrium is discussed,the local asymptotic stability of the boundary equilibrium is analyzed,and the existence of the positive equilibrium is discussed.Then,the existence condition,direction and stability of the Hopf bifurcation at the positive equilibrium point of the system are discussed,and the image is obtained by numerical simulation.Finally,the conditions of Bogdanov-Takens bifurcation at positive equilibrium point,and the corresponding saddle-node bifurcation curve,Hopf bifurcation curve and homoclinic bifurcation curve are discussed.The second eco-epidemiological model deals with cases where the disease is transmitted only in the prey,with vertical transmission,but the incidence of disease is nonlinear.The boundedness of the system solution are discussed,and the existence of the boundary equilibrium and positive equilibrium is analyzed,and the stability of the boundary equilibrium is discussed.Then,using the knowledge of central manifold theory and bifurcation theory,the Hopf bifurcation of co-dimension 1 and the Bautin bifurcation of codimension 2 at the positive equilibrium point are discussed.