The Dynamical Reseach Of Two Eco-epidemiological Models

The study of pathophoresis among interactive species, is a combination of epidemicdynamics and population dynamics.The qualitative and quantitative analysis and numericalsimulation of the relevant model, which predicts the trend of disease development and theinfluence on the population,so that people know some global states about the disease in theepidemic process among interactive species, and provide a theoretical basis for effectivedecision making for people.This paper uses the qualitative analysis and bifurcation theory of differential equationsto study two types of predator-prey models, the equilibrium points on the first octant andtheir local, global stability and bifurcation, etc., are discussed, the research of two kinds ofmodels is to assume that the disease spread only in the prey, and we don’t take into accountof the prey of the predator on susceptible prey.For the first kind of model, we assume that diseasea have a vertical infection,we firstlystudy the boundedness of the solution and obtain its global attractive domain, then we discussthe conditions for the existence of the positive equilibriums, analyze the local and globalstability of them, obtaining the conditions for the global asymptotic stability of the boundaryequilibrium and the positive equilibrium, and discuss the Hopf bifurcation of the positiveequilibrium, obtaining the direction of the Hopf bifurcation under specific parameter.For the second kind of model,another hypothesis is that the predator has constant.harvesting rate, similarly to the previous discussion, we discuss some properties of thesolutions of the system and the existence and stability of the equilibriums. Due to thecomplexity of the research about general parameters, we take some specific parameters forour study,firstly we study the Hopf bifurcation at the positive equilibrium, obtaining anunstable limit cycle bifurcated from the Hopf bifurcation.secondly we study the Bogdanov-Takens bifurcation of the positive equilibrium, obtaining the saddle-node bifurcation curve,the Hopf bifurcation curve, homoclinic bifurcation curve,finally we study the Fold-Hopfbifurcation and we get the cylindrical polar coordinate system which is topologicallyequivalent to the original model, the eco-epidemiological model with harvest rate has richerdynamical properties than that of no harvest rate.