Two Types Of Reaction Diffusion Model Of The Nature Of The Solution

A lot of mathematical models in physics, chemistry, biology and other natural disciplines can be summarized as reaction-diffusion equations. People can explain and predict natural phenomena scientifically by studying reaction-diffusion equa-tions, so as to provide reasonable solutions to the problem, thus promote the devel-opment of natural science; In addition, the studies of reaction-diffusion equations also put forward a number of challenging problems to mathematics, which is causing the attention of more and more mathematicians, physicists, chemists, biologists and engineers.Two kinds of reaction-diffusion models are concerned in this paper, one is a predator-prey model with functional responsethe other is an SIQS epidemic model with diffusionBy means of perturbation theory and bifurcation theory, a predator-prey model between two species with functional response subject to homogeneous Neumann boundary conditions is studied in the first chapter. The local bifurcation at positive constant steady-state solution (u*,v*) is obtained by treating diffusion coefficient as the bifurcation parameter. Moreover, the structure of the solution near bifurcation points is given and the local branch can be extended to global branch. An SIQS epidemic model with constant recruitment and standard incidence is investigated in the second chapter. Quarantine is taken into consideration on the basis of SIS model. The behavior of positive solutions to a reaction-diffusion system with homogeneous Neumann boundary conditions is investigated. Sufficient conditions for the local and global asymptotical stability are given by linearization and the method of upper and lower solutions and its associated monotone iterations. Our result shows that the disease-free equilibrium is globally asymptotically stable if the contact rate is small.