On The Dynamical Behaviors In Chemostat Model With Nutrient Recycling And Impulsive Input

Dynamical behaviors of chemostat model is an important conception with rich con-notation, which mainly includes extinction, permanence, local or global attractive, pe-riodicity and oscillation and so on. We can predict the growth rhythm and trend ofdevelopment of the chemostat through investigating the dynamical behaviors, and wecan have the optimal policy, then we can make the chemostat have the optimal growth.These will have very important theoretical and practical meaning to keep the populationdiversity and sustainable development of ecosystem. In this paper, we perform researchesof two chemostat models, which include: a chemostat model with nutrient recycling anddelays and a Monod type chemostat model with nutrient recycling and impulsive input.The main contents in this paper can be summarized as follows:1. In the first section we introduce the background, developmental process andsignificance of chemostat. Then we introduce the research results of chemostat. Finally,the organization of the paper is given.2. In the second section, a chemostat model with nutrient recycling and delays isstudied. Some conditions on the permanence and extinction of the chemostat are estab-lished and the globally asymptotically stability are obtained by constructing a Lyapunovfunction. Lastly, an example is given.3. In the third section, a Monod type chemostat model with nutrient recyclingand impulsive input is analyzed. we obtain some conditions about the extinction andpermanence of the chemostat. Then the globally asymptotically stability are obtained bycomparison theorem of impulsive di?erential equation. Lastly, an example is given.