Complex dynamics and impulsive control of a chemostat model under the ratio threshold policy

In this paper, we study the periodic solution and global stability of a chemostat model under impulsive control. First, we investigate the positivity and boundedness of the solution of the controlled system. Second, we find the periodic solution of the controlled system by employing the Poincare map and Brouwer’s fixed-point theorem. Furthermore, we obtain a sufficient condition which allows the existence of orbitally stable order-k periodic solutions (k=1,2) by using the comparison method and the vector field analysis. We find that the controlled system exists a unique positive equilibrium point that is globally asymptotically stable (GAS) under some conditions. Finally, we provide two numerical examples to verify the correctness of the theoretical results.