Bifurcation In Several Kinds Of Reaction-Diffusion Equations

Bifurcation theory which has rich actual background, has become one of the central issues in nonlinear analysis, meanwhile, it is also an important issue in the study of partial differential equations. The reason why bifurcation theory is important is not only its value in pure mathematics, but also its importance in applications. The researches about bifurcation especially receive more and more attentions. Many scholars study bifurcation, and have some beautiful results.In section 2, we study the effects of time delay and diffusion on the stability of the positive equilibrium to a biological system which models the evolution of a biotic species feeding on an abiotic resource. In the spatially uniform system, we can observe that the increasing time delay destabilizes the positive equilibrium and the stability switches from stability to instability to stability occur. We can also observe oscillation phenomena in this system. In the spatially nonhomogeneous system, we investigate the effects of the delay kernels as well as the diffusion on the dynamics of the system.In section 3, we study a prey-predator system with cross diffusion, the bifurcation of the nonnegative stationary solutions and its asymptotic stability are discussed by spectral analysis and the simple eigenvalues bifurcation theorem. Moreover, the ranges of parameters are found for which there exists nontrivial and stable steady-state solutions. First, we regard a as a parameter, study the stability of the trivial solution and the weak semi-trivial solution; Then we fix a, and regard b as a parameter, discuss the existence and the stability of the strong semi-trivial solution; At last, we fix a and b, regard c as a parameter, discuss the existence and the stability of the nontrivial solution.