Bifurcation Analysis Of Two Classes Of Ecological-Epidemic Models

Through the qualitative analysis and numerical calculation of the ecological-epidemic models,the mechanism and development trend of disease in the population,as well as the changes in population size and internal structure can be revealed.Analysis of these models can provide reference and guidance for the protection of the ecological environment,the decision-making of economic benefits,and the development and management of resources.In this paper,we mainly consider two types of eco-epidemic models with diseases which only spread among predators.The boundedness of the solution,the stability and existence of the equilibria,and the bifurcation of positive equilibria of these two types of models are discussed in detail by using the qualitative analysis and the bifurcation theory of differential equations.The first type of eco-epidemic model discusses the case where the disease is transmitted only in the predator population and the incidence of disease is bilinear.Firstly,the boundedness of the solution is discussed and the global attraction region is given.Secondly,the conditions for the existence of boundary equilibria and positive equilibria are discussed.The local asymptotic stability of boundary equilibria is analyzed,and the global stability of a boundary equilibrium point is proved by Lyapunov function and limit system theory.Finally,we discuss the conditions for the Bogdanov-Takens bifurcation near a positive equilibrium point and obtain a saddle-node bifurcation curve,a Hopf bifurcation curve and a homoclinic bifurcation curve.The second type of eco-epidemic model discusses the case where the disease is transmitted only in the predator population,but the disease incidence is the standard incidence.The boundedness of the solution,the existence and stability of the equilibria are discussed.Using the center manifold theory and bifurcation theory,the first Lyapunov coefficient is calculated,and the direction of the Hopf bifurcation and the stability of the limit cycle near a positive equilibrium point are analyzed.Conditions for the existence of the Bautin bifurcation of the codimension 2 is given.Then,we calculate the second Lyapunov coefficient.For several bifurcation behaviors of the Hopf bifurcation of the codimension 1 and the Bautin bifurcation of the codimension 2 at the positive equilibrium point of the system,a set of specific parameter values are set for numerical simulation.