Bifurcation Periodic Solutions And Turing Patterus In Some Reaction-Diffusion Systems

Reaction-diffusion systems are important models describing the real world. and the study of these systems has an important guiding for understanding the real world. Since the periodic solutions and Turing patterns are important phenomena in the real world, the study of both of them has been an important subject in dynamical systems and has been applied extensively in many fields such as physics chemistry and biology etc. Based on these facts, we mainly consider bifurcation periodic solutions and Turing patterns in reaction-diffusion systems in this thesisFirstly, we consider bifurcation periodic solutions in a predator-prey diffusive system with time delay. For the delay system, the positive constant steady-state is asymptotic stable when time delay is smaller than a critical value, And the condition under which periodic solutions can bifurcate from the positive constant steady-state is given. Furthermore, by using the normal form theory and the center manifold reduction for partial functional differential equations, we give the sufficient conditions for the bifurcated periodic solutions to be orbit ally asymptotically stable and unstable on the center manifold. We indicate that smaller delay has a more beneficial effect on the population to be stable at a positive constant steady-state. Especially, we find that the system with diffusion and time delay has a Bogdanov-Takens singularity at the positive constant steady state under Neumann boundary conditions, whereas this singularity does not occur for the corresponding system without diffusion.Secondly, we consider Turing patterns of a prey-predator system with diffusion and cross-diffusion. In particular, the effect of the cross-diffusion on the dynamical behavior of the prey-predator system is studied. Our results show that the presence of the cross-diffusion can not only lead to the occurrence of Turing instability but also the disappearance of Turing instability in the original reaction-diffusion system. By combining the maximal principle and Harnack inequality, we give a priori upper and lower bounds for the positive solutions to the corresponding elliptic system. Then based on the result, some sufficient conditions for the non-existence of non-constant positive steady states are shown by the method of energy integral. After this, we deduce the existence of non-constant positive steady states by Leray-Schauder degree theory. Our results show that the system has non-constant positive solutions when the predator disperses quickly from high-prey-density patch to a low density one.Thirdly, we discuss Hopf bifurcations and small amplitude travelling wave train solutions in a prey-predator diffusive system. We exhibit that the positive constant steady-state is more favorable to be stable when capturing rate in diffusion system is smaller. Moreover, the Hopf bifurcation phenomena is found and some examples for numerical simulations are also given. In addition, the result that travelling wave solution on unbounded domain admits periodicity is obtained. We find that it is more beneficial to for travelling wave train solutions to occur when capturing rate is larger, which supplies theoretical evidence to explain the periodic phenomena in population dynamics.Finally, we analyze spatially homogeneous and nonhomogeneous periodic solu-tions of a delayed Lotka-Volterra diffusion competition system. We give the con-dition under which the periodic solutions bifurcating from the positive constant steady-state in the delayed system and diffusive delayed system have the same sta-bility and direction. Moreover, we discuss the existence of spatially nonhomogeneous Hopf bifurcation periodic solutions. Our results show that large diffusivity has no effect on the Hopf bifurcation of the corresponding delay differential equations, while suitable diffusivity can lead to bifurcating spatially nonhomogeneous periodic solu-tions at the positive constant steady-state. These results demonstrate that the delay and diffusion rate have important effects on dynamical behavior.