Stability Of Diffusive Models With Crowley-Martin Type Functional Response

The main purpose of this thesis is to explore the stability of diffusion models with Crowley-Martin type functional response.In real life,people are suffering from diseases.For the sake of studying the transmission mechanism of infectious diseases,many scholars have established infectious disease models to study them.Their conclusions are helpful for us to control infectious diseases.Based on the infectious disease models,Crowley-Martin type functional response is introduced to characterize the nonlinear interaction between the susceptible and the infected in infectious diseases.This thesis is divided into two parts:Firstly,Crowley-Martin type functional response is introduced into the basic epidemic models,and the asymptotic stability of the solution is studied by using Lyapunov function,La Salle invariant set and other methods.It is found that the Crowley-Martin type functional response affects the calculation of the basic regeneration number R₀,but has little effect on the existence and stability of the solution.The second is to study the diffusion epidemic model with Crowley-Martin type functional response and given constraints.Firstly,we classify the parameters of the model,and then study the existence and stability of the model solution by using the relationship between the eigenvalue and the stability of the equilibrium point for different parameter ranges,making a priori estimation of the solution,degree theory,implicit function theorem and other methods.The results show that the model converges to a endemic equilibrium with the increase of time when the patients have strong awareness of protecting others,the susceptible people have a strong sense of self-protection,or the infection rate of the disease is not high.When the awareness of protecting others is moderate,the awareness of self-protection is moderate or the infection rate of the disease is moderate,there are multiple endemic equilibrium points in the model.There is no positive steady state solution for the model,when the patients have weak awareness of protecting others,the susceptible people have weak awareness of self-protection,or the infection rate of the disease is very high.