Analysis Of The Dynamical Behavior Of Stochastic Chemostat Models Perturbed By Environmental Noises

In ecology,chemostat is a basic piece of laboratory apparatus used for the continuous culture of microorganisms.The model of chemostat occupies a central place in mathematical biology and has played an important role in many field-s.In 1950,Monod and Novick formulated the basic mathematical principle of the continuous culture of microorganisms.Since then,chemostat has been wide-ly investigated by scholars and has achieved fruitful results.The main research contents of mathematical model for chemostat which is described by deterministic differential equations,are the problems of survival and extinction for the microbial populations after a long period of time.However,whether in the real world or in the precise experiments,every microorganism will be affected by environmental random factors.Therefore,utilizing stochastic mathematical models to analyze the dynamical behaviors of species can reflect the reality more accurately.In this paper,we mainly study the asymptotic behaviors of stochastic chemostat models under environmental random noises.Contents are as follows:1.Dynamical behaviors of stochastic chemostat models perturbed by white noise.When chemostat is affected by linear perturbation,due to the ergodic the-ory employed by Khasminskii,we get the sufficient conditions for the existence of stationary distribution for stochastic chemostat with Monod-Haldane response function and with general growth response,respectively.For multi-species stochas-tic chemostat model with discrete delays and environmental noise,using stochastic Lyapunov analysis method,we investigate the asymptotic behaviors of the solutions for the stochastic system around the equilibria for the corresponding deterministic model.When the maximum growth rate of species is perturbed by white noise,we further consider the death rate of species in chemostat with Monod functional re-sponse.According to the theory of Markov semigroup,we obtain that the densities of the distribution of solutions can converge in~1to a ergodic invariant density.2.Periodic solutions of stochastic chemostat models with periodic dilution rate.Based on Khasminskii's theory for periodic Markov processes,we derive the existence for the non-trivial periodic solution of the stochastic chemostat with Mon-od growth function,and the globally attractive boundary periodic solution.These two conclusions just correspond to the existence conditions for the periodic solu-tions of deterministic periodic chemostat.For stochastic chemostat with general growth response,under two assumptions for response function p(S),we get the positive periodic solutions for the system.3.Threshold and ergodicity of stochastic chemostat models under regime switching.For chemostat perturbed by both white and color noise,we present the threshold for species survival,i.e.when threshold value is less than 1,the microor-ganism population will go to extinction exponentially;whereas the microorganis-m will be permanent in mean.Utilizing Khasminskii's theory on ergodicity and Markov regime switching,we obtain the ergodic property for stochastic chemostat with Monod response function and with general growth response,respectively.The study of stochastic mathematical model and deterministic model is com-plementary.The methods and results in this paper may enrich the research of asymptotic behavior in chemostat and help us better understanding the dynamics in stochastic sense.