Qualitative Analysis And Optimization Conditions Of Several Biochemical Reaction Processes

A great deal of biochemical reaction processes in nature can be described by mathematical models.Take advantage of the models established by differential equations,the phenomena of biochemical reactions can be reasonably explained and predicted,and the quantity of reactants can be effectively regulated and controlled,thereby providing effective solutions to actual problems.The development of light chemical industry and other industries is often accompanied by certain environmental problems while promoting economic prosperity.For example,industrial wastes without proper disposal will not only lead to environmental deterioration,endanger population survival and biodiversity,but also affect economic development directly or indirectly.Therefore,this thesis applies the theory of differential equations to conduct qualitative analysis and optimization conditions of several biochemical reaction processes.The main work is as follows:(1)Analysis of global solution and ultimate boundedness for a stochastic model consisting of toxin-producing phytoplankton,zooplankton,fish and environmental toxins.Firstly,the nonnegativity and boundedness of the solution of the deterministic system corresponding to the model are given,and the global stability of positive equilibrium is demonstrated by constructing Lyapunov function.Secondly,the existence and uniqueness of the global solution for the stochastic system are derived resorted to It(?) formula.Finally,the ultimate boundedness of the stochastic asymptotic property is obtained under hypothetical conditions.(2)The effects of time-delay and diffusion on kinetic properties of an arbitrary order autocatalysis model under homogeneous Neumann boundary conditions.First of all,the influence of time-delay on the stability of the constant positive steady state solution of the ordinary differential system is studied,and the conditions for the existence of Hopf bifurcation are derived.After that,the conditions to guarantee the existence of Turing instability induced by diffusion and the influence of time-delay on the stability of the constant positive steady state solution of the partial differential system are given.Meanwhile,the conditions for the existence of Hopf bifurcation in both spatially homogeneous and inhomogeneous fields are attained with time-delay as a parameter.Moreover,the direction of the Hopf bifurcation and the stability of periodic solutions are obtained by the center manifold theorem and the normal form theory.Finally,numerical simulations are presented to verify and illustrate the theoretical results.(3)Research on the optimization of a three-population reaction-diffusion model under the homogeneous Neumann boundary condition.Positive intervention is implemented in order to maximize the total density of three populations with predation and competition relations.The existence and uniqueness of the positive strong solution are derived by parabolic equation theory and strongly continuous semigroup theory.After that,the technique of minimal sequences is used to establish the existence and uniqueness of the optimal control.The first and second order necessary conditions of system optimization are also constructed by duality principle.(4)Study on the optimization of a three-population predator-prey model with an age structure in a polluted environment.Taking the influence of toxins within the populations and environmental toxins into account,as well as the age structure of populations,the existence and uniqueness of the positive solution are firstly proved via Banach fixed point theorem.Secondly,the necessary conditions of the optimization are obtained by means of tangent-normal cone technique.Thirdly,the existence and uniqueness of the optimal control are attained by applying Ekeland’s variational principle.