Kinetic Analysis Of A Class Of Food Chain Chemostat Models

It is an important method to simulate the environment of chemostat by differential equations and to research and solve biological problems by mathematical theory and method.In recent decades,the research of chemostat models has received great attention from many biological and mathematical workers.With the deepening of research,scholars have introduced factors such as time delay,inhibitors,periodic input and output into the classic chemostat models,which have greatly enriched the theoretical results of the chemostat compet.ition model.Compared with the competition model,the study of predator-prey model is more difficult,so there is less related research work.To reduce the complexity of the problem,scholars first ignored the mortality of species in predator-prey models.Combined with biological reality,this paper deals with a food chain chemostat model with species mortality.In this case,the system does not satisfy the conservation of biomass.Hence it cannot be reduced to a lower dimensional predator-prey system,which makes the study of its dynamical behavior more difficult.In this paper,the existence,uniqueness,stability,and long-time behavior of the positive solutions of the model are researched by using the comparison principle,the local and global bifurcation theory,linear operator stability theory,uniform persistence theory,and numerical simulation.The main results are as follows:In the first chapter,we introduce the background and related researches of the models and put forward main results and preliminary knowledge.In the second chapter,we investigate well-stirred food-chain chemostat model.Firstly,the existence and uniqueness of the equilibrium solutions of the model are obtained by analyzing the monotonicity of the functions.Secondly,the stability at the equilibrium points are studied with the help of the linear operator stability theory and Hurwitz discriminant.Then the long-time behavior of the solution is considered by using the uniform persistence theory.The results show that species can coexist when the dilution rate D satisfies some conditions.Perturbation theory and numerical simulation are used to study the dynamic behavior of the system solution with small mortality.Finally,the effects of parameters on the dynamic behavior of the system are studied by means of numerical simulations.In the third chapter,we study the equilibrium and long-time behavior of an unstirred food-chain chemostat model.Firstly,the boundedness and global existence of the model solution are given by using the comparison principle and the maximum principle.Secondly,the stability of equilibrium solution of the system is analyzed by using the linear operator stability theory.Then,by using the local bifurcation theory,the positive bifurcation from the semi-trivial solution is researched by taking the mortality m as the bifurcation parameter.The global bifurcation theory is applied to extend the local bifurcation to the global one.The long-time behavior of the system solution is investigated by using the theory of uniform persistence.Finally,the effects of parameters on the dynamic behavior of the system are further analyzed by means of numerical simulations.