Dynamic Analysis Of A Reaction-diffusion Chemostat Model

In 1950,Novick and Sizilard designed a chemostat in order to train the species of microorganisms in the laboratory,which is a very classical species competition model.Since the invention,the chemostat model has attracted many biological and mathematical scholars.The chemostat can also be seen as a complex microbial habitat,such as a pond or lake,in addition to its role as an experimental facility for laboratory bacteria culture.This makes the model of reaction diffusion chemostat have high research value.We study the competition of two species for a single resource in a reaction-diffusive chemostat model with Neumann boundary conditions.Moreover,the chemostat has different diffusion coefficients in this paper.Firstly,we study the existence and stability of semi-trivial steady state solutions.By using the mass conservation property,the system is reduced to a two-dimensional competitive sys-tem.Using the upper-lower solutions method and the instability of the semi-trivial solutions,we prove the existence of coexistent nontrivial positive solutions.The bifurcations at the semi-trivial solutions are proved by using the local bifurcation theorem.Besides,we study the existence of the global bifurcation solution branch by combining monotonicity and the global bifurcation theory,the maximum prin-ciple and the boundedness of the solution.