Modelling And Investigation On The Dynamics Of Stochastic Models For Algae Growth And Control In Eutrophic Water

Eutrophication of water bodies is a global environmental problem,and is one of the most important environmental problems in our country.Algal blooms that induced by the eutrophication have drawn people's extensive attention.Due to the fact that the occurrence of algal blooms is inevitably affected by various external random perturbations,the studies of algae growth and control by use of stochastic dynamic models have become very important and meaningful work.Taking the stochastic dynamic systems,birth and death process,impulsive dynamic systems as background,the goal of this thesis is to establish some stochastic models for algae growth and control.We study the effects of stochastic noise,delays,chemical and biological control on the dynamics of algae growth,which set up some theoretical foundations for explaining and predicting some phenomena in algae growth.1.Considering the random influence on the mass transfer efficiency rate,we establish a stochastic model for algal bloom with nutrient recycling.The existence and uniqueness of the global positive solution of the model is proved.By constructing Lyapunov functions,sufficient conditions for the stochastic stability of its algae-free equilibrium and the persistence of the algae population are obtained,respectively.Then by using the theory of integral Markov semigroups,we show that the model has a stationary distribution.Furthermore,the extinction of the algae population is analyzed.Our results indicate that compared to the dynamics of the corresponding deterministic model,the algae-free equilibrium is still stable provided the noise intensity is small enough;Though there is not positive equilibrium for the stochastic model,the algae population is still persistent when the noise intensity is small enough,while the algae population will extinct exponentially with probability one when the noise intensity is large,that is very different from the dynamics of the deterministic model.2.Considering the stochastic factors,a stochastic algae growth model based on continuous time Markov chains is studied.First,using the cumulant generating function,the moment equations which the digital features satisfy are obtained,then the moment closure equations are derived by using of moment closure techniques based on the lognormal approximation,and according to the Euler-Maruyama method,the corresponding It? stochastic differential equations are given.To illustrate the rationality of the moment closure,numerical simulation is given to compare the deterministic model with the stochastic model and moment closure equations.3.Considering the effects of the stochastic noise and the nutrient recycling delay,a stochastic model for algal bloom with distributed delay is investigated.The existence and uniqueness of the global positive solution of the model is proved.Also,sufficient conditions for the stochastic stability of its algae-free equilibrium are obtained.Then by Fourier transform method,the spectral densities of the nutrient and the algae population are estimated.Furthermore,we analyze the effect of the large noise on the algae population.Our results indicate that when the noise intensity and the delay are small enough,the algae-free equilibrium is stochastic stable;The fluctuation intensities of the nutrient and the algae population are more sensitive to the noise intensity than to the delay,and larger noise intensity can make the algae population extinct exponentially with probability one;The numerical simulation reveals that a large delay in nutrient recycling can make the algae grow slowly.4.Taking into account random perturbations and time delays(including substrate recycling delay and bacterial reproduction delay),that may exist in the process of wastewater treatment,the dynamics of a delayed stochastic model simulating wastewater treatment process are studied.By constructing Lyapunov functionals,sufficient conditions for the stochastic stability of the stochastic model without and with delays are obtained,respectively.The combined effects of the stochastic fluctuations and delays are displayed.Our results indicate that the positive equilibrium is stochastically stable provided that the intensities of noises and the delays are small,which preserves the dynamics of its corresponding deterministic counterpart;We perform some approximate sensitivity analysis of the stochastic stability of the positive equilibrium with respect to the parameters,and find that the stochastic stability of the positive equilibrium is greatly affected by the noise intensity and the bacterial reproduction delay and less affected by the substrate recycling delay.5.We investigate the dynamics of a delayed algae growth model with both impulsive and stochastic perturbations.The impulse is introduced at fixed moments,the stochastic perturbation is of white noise type which is assumed to be proportional to the population density,and the delay is of continuous type.We start with the existence and uniqueness of the positive solution of the model,then establish sufficient conditions ensuring its global attractivity.By using the theory of integral Markov semigroups,we further derive the existence of the stationary distribution of the system.Finally,we perform the extinction analysis of the model.