The Self-diffusion And Staggered From The Proliferation Of Turing Instability

Mathematical Ecology is a science that uses mathematical models to predict and interpret what we observe. It is a fast growing, well recognized subject and is the most exciting modern application of mathematics. There are two kinds of continuous models in mathematical ecology. One is the ordinary differential equation (ODE), and the other is the partial differential equation (PDE) with diffusion. Since there existes diffusion in the PDE, a little interesting change happened in some properties of the two different models. One of the great examples is Turing instability phenomenon, which was claimed in the paper"The chemical basis of morphogenesis"by Alan. M. Turing—one of the greatest scientists in the 20th century in England.Turing suggested that, in the absence of the diffusion, the basic chemicals tend to a linearly stable uniform steady state, while, under certain conditions, the uniform steady state can become unstable, and spatial inhomogeneous patterns can evolve through bifurcations. In the other word, under certain conditions, a constant equilibrium solution can be asymptotically stable with the kinetic equation, but it is unstable with its corresponding reaction-diffusion system.Over the years, Turing's ideas have attracted much more attention and successfully developed on the many scientific backgrounds, such as chemistry, physics, biology, mathematics, the learning communicating by letter. The paper"The chemical basis of morphogenesis"is now regarded as the foundation of basic chemical theory or reaction diffusion theory of morphogenesis and Turing instability phenomenon in the experimental laboratory is realized successfully.In the first part of this dissertation, the background and history about the related work of the Turing instability are introduced. In section 2, an example about Turing instability only with self-diffusion in ( 0,π) and the proof of the example is given. Second we deals with the case which has time-delay on the bounded domainΩin Rn ( n≥1) with no-flux boundary condition. Using the linearizing method, we get the sufficient condition for the system to have Turing instability. We show that the time-delay affects the Turing instability phenomenon and if the ODE is absolute stability, then the PDE must be absolute stability. Last, we discuss some Lotka-Volterra models under what condition the general Lotka-Volterra model with the self-diffusion will have Turing instability phenomenon according to the above theory, and we give a concrete model and illustrate the above conclusions, the numerical simulations using software Maple 9.0 are also given.Recent researches show that the cross-diffusion phenomenon is universal besides the self-diffusion phenomenon. Discussing the nonlinear partially differential equation system with cross-diffusion is difficult, which needs more abstruse theory and deeper analytical capability, but with the advance of modern science and technology, more and more researchers begin to study the PDEs with cross-diffusion nowadays. The migration of group is a classical cross-diffusion phenomenon. It is found that some ecological phenomenon is induced by cross-diffusion not self-diffusion. Thus studying the system with cross-diffusion is more reasonable and meaningful. In section 3, the instability of the uniform equilibrium of a spruce budworm-aphid interaction model is discussed and we show that the cross-diffusion can induced the Turing instability phenomenon. In the first part of this section we introduce the background and history about this model to see the practical significance of the model. In the following two parts, we get the sufficient condition of Turing instability. We show that under certain conditions cross-diffusion can induce the instability of uniform equilibrium, which is stable for the kinetic system and for the self-diffusion reaction system; On the other hand, cross-diffusion can stabilize the uniform equilibrium, which is stable is stable for the kinetic system but unstable for the self-diffusion reaction system, that is to say, cross-diffusion can make the model happen Turing instability and also can make Turing instability vanish. Lastly we present some numerical simulation to illustrate our results.