Periodic Solutions And Wavefront Solutions For Some Kinds Of Reaction Diffusive Equations With Delays

In this paper, we have discussed the existence and asymptotic behaviors of periodic solutions and steady solutions of some kinds of reaction diffusion equations with delays. We also have investigated the wavefront solution of a kind of second order lattice differential equation with delays.In chapter 1, we gave some fundamental definitions and briefly introduced the main methods of investigating reaction diffusion equations and some recent fundamental results.In chapter 2 and chapter 3, we were devoted to research the existence and uniqueness of competitive and cooperative systems of two populations. We put emphasis upon the application of monotone iteration method in chapter 2 and the uniqueness of periodic solution in chapter 3.In chapter 4, we investigated the existence and asymptotic behaviors of periodic solutions and steady solutions of a type of reaction diffusion equation under different conditions (with delays and without delays) respectively.In chapter 5, we investigated Hopf bifurcation properties of Daphnia Magna equation. Based on center manifold and normal form, we get the formulas for determining the directions of Hopf bifurcation and the stability of bifurcational periodic solutions.In chapter 6, we discussed the wavefront solution of a kind of lattice differential equation which derives from diverse fields. We obtained the wavefront solution by using sup-subsolution iteration when the coupled functions are quasimonotone, and by applying Schauder fixed point theorem, we also obtained the wavefront solution when the coupled functions have a different quasimonotone property.