The Long-term Behavior And Steady-state Solution Analysis Of A Competitive Model With Internal Storage And Inhibitors

The chemostat is an experimental device for continuous microbial culture,which has important reference value for people to study the phenomenon of species com-petition in ecosystems.In this paper,the long-term behavior and steady state solutions of an unstirred chemostat model with internal storage and inhibitor are studied.Due to the internal storage,the singularity leads to that the standard tech-nique such as linearization and bifurcation can not be applied here.The main tools used in this paper are the classical maximum principle,upper and lower solutions,uniform persistence theory and degree theory in special cones.The main contents are as follows.In chapter 1,we introduce the known outcomes on chemostat models with internal storage and on the chemostat models with internal inhibitor.Combined with the research results,we propose an unstirred chemostat model with internal storage and inhibitor.In chapter 2,we present the preparatory knowledge that needed in this paper.Firstly,some related lemmas are listed,including uniform persistence theory and fixed point index theory.Secondly,some important results about the single species model are introduced.The main result shows that the long-term behavior of the sin-gle species model can be determined by the diffusion coefficient.When the diffusion coefficient is smaller than the critical diffusion coefficient,the species survives;when the diffusion coefficient is greater than or equal to the critical diffusion coefficient,the species goes to extinct.In chapter 3,we study the long-term behavior of the competition model.Firstly,the global existence of the classical solution is obtained by comparison principle,the upper and lower solutions.The corresponding nonlinear eigenvalue problems are introduced,and the stability of the trivial and the semi-trivial solutions can be judged according to the signs of the principal eigenvalues.When the trivial and the semi-trivial solutions are unstable,sufficient conditions of the coexistence solution are obtained by the uniform persistence theory.In chapter 4,we study the steady state problem corresponding to the com-petition model.In order to overcome the mathematical difficulties caused by the singularity in the ratio terms,the sharp priori estimates for positive steady state solutions of the system are established,which ensure that all positive steady state solutions of the system belong to a special cone.By means of the fixed point index theorem and homotopy invariance,the existence of positive steady state solutions is obtained.