Dynamical Analysis Of Models For The Continuous Culture Of Microorganisms With Noise Excitation And Flocculation Effect

Due to the widespread existence of randomness in the natural environment,the continuous culture of microorganisms and the flocculation process of microorganisms are inevitably subject to randomness.Based on the mechanism of the flocculation method,several classes of stochastic microorganism flocculation models are constructed and studied in this thesis.The global dynamics of the constructed models are investigated mainly by using the theories of stochastic differential equations and ordinary differential equations,and numerical simulations,and some control strategies with practical biological significance are obtained.The dynamical responses of stochastic models are observed which are quite different from those of the deterministic models.The specific contents of this thesis are as follows:In Chapter 3,a class of stochastic microorganism flocculation model with complementary nutrients is constructed.Firstly,the well-posedness of the stochastic model is considered.Then,by constructing appropriate stochastic Lyapunov functions,some sufficient conditions for the existence of an ergodic stationary distribution and persistence of the stochastic model are given.The results show that the microorganisms in chemostat can be collected continuously.Furthermore,based on sensitivity analysis techniques,some control strategies for the sustainable collection of microorganisms are discussed.Finally,we carry out some numerical simulations to illustrate the applications of theoretical results and give the empirical probability densities in numerical forms.In particular,the numerical simulations show that,when the stochastic perturbations of the environment is large,the growth of the microorganisms in chemostat can be transformed from the state of tending to extinction to the state of persistence.The interesting observation reveals that the stochastic perturbations may have positive biological effects and we call it the noise-induced stochastic transition phenomenon.In Chapter 4,based on Chapter 3,considering the influence of seasonal variations,a class of stochastic nonautonomous microorganism flocculation model with complementary nutrients and periodic parameters is further investigated.The dynamical behaviors of stochastic nonautonomous model are investigated by theoretical analysis and numerical simulations.The well-posedness of the stochastic nonautonomous model is considered,Further,by utilizing Khasminskii’s theory for periodic Markov processes,sufficient conditions for the existence of the stochastic nontrivial positive periodic solution are obtained.The existence of the stochastic nontrivial positive periodic solution implies periodic change of microorganisms’density.Some sufficient conditions for the global attractivity of the boundary periodic solution of the stochastic nonautonomous model are also derived.At last,numerical simulations are performed to illustrate the theoretical results.It is found numerically that the stable positive periodic solution and the stable boundary periodic solution of the corresponding deterministic nonautonomous model may coexist.For appropriate stochastic perturbations,the population of the microorganisms in certain density ranges can change from an endangered state to an oscillatory persistence state.In Chapter 5,a class of deterministic microorganism flocculation model with complementary nutrients and general functional responses along with its stochastic version are further studied.For the deterministic microorganism flocculation model,the well-posedness of the solutions,the dissipativeness and the existence of the equilibria(forward/backward bifurcation)are discussed.Sufficient conditions are given for the global asymptotic stability of the boundary equilibrium.If the basic reproduction number is greater than one,the model is uniformly persistent,and an explicit expression for the estimation of the ultimately lower bound of microorganism populations is given by using techniques different from the existing literature.For the stochastic microorganism flocculation model,the existence of an ergodic stationary distribution is analyzed by establishing two thresholds which automatically become the basic reproduction number when stochastic perturbations are not considered.Furthermore,sufficient conditions are given for the extinction of microorganism populations.The numerical simulations reveal that noise excitation may make the prediction and control of the evolutionary trend of microorganism populations more difficult and complex.In Chapter 6,a class of high-dimensional stochastic microorganism flocculation model with multiple substitutable nutrients and general functional response functions is considered.The study of asymptotic behavior of solutions of this stochastic model can provide feasible control strategies for the sustainable collection of microorganisms.The main theoretical results are sufficient conditions for the permanence and extinction of the high-dimensional stochastic microorganism flocculation model,which are also extensions of some results in the existing literature and Chapter 5.In addition,through numerical simulations,we vividly demonstrate the statistical characteristics of the stochastic model.And the stochastic model also exhibits the stochastic transition phenomenon.The main innovations of this thesis are summarized as:1.Considering the randomness that cannot be ignored in the natural environment,a class of stochastic microorganism flocculation model with complementary nutrients are constructed and studied.By constructing suitable stochastic Lyapunov functions,the threshold for the existence of stationary distribution of the stochastic model is obtained,and this threshold can degenerate to the basic reproduction number of the corresponding deterministic microorganism flocculation model.2.For a class of stochastic nonautonomous microorganism flocculation model with complementary nutrients and periodic parameters,using Khasminskii’s theory for periodic Markov processes,sufficient conditions for the existence of the stochastic nontrivial positive periodic solution and the global attractivity of the boundary periodic solution are obtained by skillfully constructing auxiliary functions.3.For a class of deterministic microorganism flocculation model with complementary nutrients and general functional response functions as well as its stochastic version,the uniform persistence of the deterministic microorganism flocculation model is obtained.And an explicit expression for the estimation of the ultimately lower bound of microorganism populations is given,which improves the results in the existing literature;the new stochastic Lyapunov functions are constructed to overcome the difficulties associated with the general functional response functions,and the threshold conditions for the existence of stationary distribution of the stochastic microorganism flocculation model are obtained,which extends the results in Chapter 3;and the conditions for the extinction of microorganism populations improve the results in Chapter 3.4.For a class of high-dimensional stochastic microorganism flocculation model with multiple substitutable nutrients and general functional response functions,through carefully analysis,the permanence and extinction results of a more generalised stochastic microorganism flocculation model are obtained.