Pattern Dynamics Analysis Of Some Reaction Diffusion Systems With Super-diffusion

Anomalous diffusion phenomenon is ubiquitous in nature.Since the frac-tional diffusion equations can not only describe the memory process,genetic properties and spatial non-local characteristics of anomalous diffusion,but also accurately describe the breakthrough curve of medium migration.Math-ematical physics modeling methods are more advantageous than integer order equations in the modeling of some complex systems.In addition,anomalous diffusion is one of the essential research topics in theoretical physics,math-ematical biology,statistical mechanics,and also a underling physical process with a practical application background in the field of ecological mathematics,economic finance and engineering.It is shown that subdi:ffusion can inhibit the formation of patterns;super-diffusion leads to a significant increase in the velocity of the front wave.In addition,Levy flight system may cause the oscillation reaction mode of spiral waves and chemical turbulence.Hence the study of pattern dynamics for the superdiffusion is very interesting.Therefore,in this dissertation,we will explore the dynamical behavior of the pattern from the following three aspects.Firstly,the existence and uniqueness of the weak solution of the predator-prey model with anomalous diffusion are analyzed by using the Galerkin ap-proximation method and Gronwall inequality.On the basis of this,the optimal control problem of the superdiffusion system is further considered by the min-imum sequence theory.Secondly,Turing pattern with a chemical reaction(Lengyel-Epstein sys-tem)with superdiffusion is studied.Then complex dynamics of amplitude equations,such as the existence of homogeneous solutions,stripe and hexagon patterns,mixed structure patterns,their stability,interaction and transition between them,are analyzed.Numerical simulation shows the ratio of su-perdiffusion exponent of inhibitor and activator plays an important role in the pattern selection.If the ratio is not equal to 1,the spatial patterns are diverse.However,if the ratio is equal to 1,the only difference is in the quantity but there is no essential difference between fractional order and integer order.Finally,in Chapter 4,we explore the Turing pattern problem with a super-cross-diffusion system.Waves with two resonance vectors can lead to the ap-pearance of hot or cold spots,which is different from the results of Chapter 3.In Chapter 5 Turing-Hopf bifurcation of a predator-prey system with su-perdiffusion is investigated.It is implied that for Turing-Hopf bifurcation,when the fractional exponent decreases,secondary bifurcation occurs and also superdiffusion can generate and eliminate wave patterns.