Study On The Dynamical Behavior Of Stochastic Biochemical Reaction Models And Chemostat Models

Biochemical reaction is a chemical reaction carried out in the organism.By establishing biochemical reaction models and studying them,we can further understand the internal mechanism of the reaction process and realize the prediction,guidance and control of the process.Chemostat is a laboratory device used for the continuous culture of microorganisms.It has a wide range of applications in microbiology and ecology.Therefore,the research on biochemical reaction model and chemostat model has attracted the attention of many scholars and achieved some meaningful results.Based on the research status of these two types of models.This thesis mainly uses the theory of stochastic differential equations and related methods to deeply study the dynamic behavior of the biochemical reaction model and chemostat model disturbed by environmental noise,and uses Matlab programming experiments to verify the validity of the theoretical results.The main contents are as follows:1.The dynamical behavior of the Oregonator model and the multi-molecule biochemical reaction model perturbed by noise.For the Oregonator model with high-order perturbation,through constructing suitable Lyapunov functions,sufficient conditions for the existence of a unique ergodic stationary distribution are established.For the multi-molecule biochemical reaction model disturbed by both white noise and Lévy noise.Under appropriate assumptions,the conditions for the end and the continued progress of the reaction are derived,and the existence of positive recurrence is studied.2.The dynamical behavior of the chemostat model of competition between plasmid-bearing and plasmid-free organisms,perturbed by white noise.Firstly,we prove the system has a unique global positive solution for any initial value.Then based on Hasminskii’s theory on ergodicity,we prove the existence of a unique ergodic stationary distribution.Furthermore,utilizing the comparison theorem of stochastic differential equation and strong law of large numbers,we obtain the conditions for the extinction of plasmid-bearing organisms and the survival of plasmid-free organisms.3.The dynamical behavior of the food chain chemostat model perturbed by white noise.For the food chain chemostat model with Monod response function and the maximum growth rate of microorganisms disturbed by white noise,through studying the limit system of dimensionality reduction,we establish sufficient conditions for the existence of a unique stationary distribution and discuss the extinction of the system in two cases.Then,we extend it to the case with general response functions.By constructing a suitable and novel Lyapunov function,we prove the existence of stationary distribution under appropriate assumptions about two response functions.Finally,for the food chain chemostat model with systematic linear disturbances,the existence of stationary distribution and the extinction of microorganisms are analyzed.4.The dynamical behavior of the dual nutrient chemostat model,which is disturbed by noise.For the chemostat model with two perfectly substitutable resources and single-species,in which the dilution rate is disturbed by telegraph noise,we obtain the threshold between persistence in the mean and extinction of the microorganism.Moreover,in the case of persistence,we establish sufficient conditions for the existence of positive recurrence by constructing a suitable Lyapunov function with regime switching.On this basis,based on the Markov switching theory,we further study the existence of ergodic stationary distribution of the dual nutrient chemostat model,disturbed by both white noise and telegraph noise.