Two Kinds Of Reaction Diffusion Model Of Qualitative Analysis And Numerical Simulation

As an important mathematical form to describe the real world, reaction-diffusion equations become more and more attracted to mathematicians and researchers studying on other subjects because of high practical value. The needs and curiosity of people have prompted scientists to put forward various of reaction-diffusion mod-els. Meanwhile, these models are analyzed and studied to feedback and describe the real world.Two kinds of reaction-diffusion models are concerned in this paper. One is a chemical reaction model with zero-flux (Neumann) boundary condition The other is a general Gause-type predator-prey model with non-homogeneous Dirichlet boundary conditionFor the first model, the effect of interaction ratio is mainly considered. By integrating equations, the properties and non-existence of steady-state solutions are discussed. When the reaction region is too small or the diffusion coefficient of sub-stance u is large enough, spatially non-homogeneous structures do not exist. Treat-ing the interaction ratioγas a parameter, the existence of non-constant positive steady-states is followed from the bifurcation theory. Especially, the steady-state bifurcation from the double eigenvalue is derived. If interaction ratioγis sufficiently large, the steady-state system creates spatially non-homogeneous structures. In ad-dition, the Hopf bifurcation to both ODE and PDE systems is discussed in detail. The periodic solution generates whenγis close toγ0. Furthermore, examples of nu-merical simulations are shown to complement the analytical conclusions. And more complicated structures of steady-states are seen.For the second model, the properties of steady-state solutions are investigated. By using linear stability theory and constructing Lyapunov function, the local and global stable conditions of the positive constant solution are given. On the basis of priori estimates, non-existence of non-constant nonnegative steady-states is dis-cussed by Cauchy and Poincare inequalities. The steady-state system does not cre-ate non-homogeneously spatial structures when the diffusion coefficients are large enough. Lastly, by using degree theory and global bifurcation theorem, the exis-tence of positive steady-state solutions is researched. The bifurcation curve from the bifurcation point extends to infinity provided that the parameters satisfy some conditions.