Dynamical Analysis On Some Biochemical Reaction-diffusion Models

Biochemical reaction-diffusion models are wildly studied not only to describe the reaction process and explain complex spatiotemporal phenomena,but also to predict the long time behavior of the reactant and provide some theoretical basis for production.In this thesis,three biochemical reaction-diffusion models are investigated to explore the effect of diffusion and advection on the corresponding kinetics process,respectively.Firstly,a reaction-advection-diffusion system in a homogeneous environment is considered.The global asymptotic stability of nonconstant semi-trivial states is obtained.It is also shown that there exists a stable nonconstant co-existence state under some appropriate conditions.Numerical simulations are given to illustrate the theoretical results.Furthermore,when two species drift in opposite direction,it is observed that densities of two species are monotone with respect to the spatial variable as time proceeds.In fact,for a predator-prey problem,the two species can coexist even though they drift in the same direction.Then a reaction-diffusion model is presented to describe the microbial continuous culture with diversified growth.The existence of nonnegative solutions and attractors for the system is obtained,the stability of steady states and the steady state bifurcation are studied under three growth conditions.In the case of no growth inhibition or only product inhibition,the system admits one positive constant steady state which is stable;in the case of growth inhibition only by substrate,the system can have two positive constant steady states,explicit conditions of the stability and the steady state bifurcation are also determined.In addition,numerical simulations are given to exhibit the theoretical results.It is found that larger substrate concentration in medium can induce the proposed system with substrate inhibition to stabilize to the washout state.At last,a Schnackenberg model with crucial reversible reactions is studied.Under Neumann or Dirichlet boundary conditions,the existence and uniqueness of the strong solution is obtained by semigroup theory.For the Neumann boundary conditions,the system admits four possible positive constant steady states,and the explicit conditions of the stability,Turing instability,steady states bifurcation and Hopf bifurcation are determined,respectively.Numerical simulations are given not only to show the theoretical results,but also to depict the spatiotemporal patterns of the Schnackenberg reaction process.Besides,it is observed that the proposed system with Neumann boundary has homogenous period solutions,and the system with Dirichlet boundary admits stable nonconstant steady state.