Bifurcation Analysis Of Microbial Continuous Culture Model With Delays

On complicated natural environment, the research on multiple species interaction of mathematical models is becoming more and more urgent. In the species interaction model, the introduction of delay parameter becomes an inevitable trend in the widespread phenomenon. Compared with the no delay model, the delay model is more complex and difficult to deal with. One species with delay is only considered in most models. However, in order to reflect the objective world better, the multiple species interactions of mathematical model should be involved in multiple species with delays. So the study on multiple delays parameter model has attracted more and more attention.Qualitative analysis is one of the main methods in the species interaction model. Most scholars focus on the study of the existence, stability and direction of Hopf bifurcation solutions.This paper focuses on the qualitative analysis for the three-dimensional food chain model with delays and variable yield. The conditions of existence and stability of Hopf bifurcation solutions model are obtained from the qualitative analysis. The numerical simulation illustrating the theoretical results are given.Firstly, the theoretical results of the three-dimensional food chain model with variable rate and no delay are verified by numerical simulation.Secondly, the three-dimensional food chain model with variable rate of growth delay for predator is established. Through the qualitatively analysis, the influence of delay for the existence of Hopf bifurcation solutions is illustrated. The numerical simulation illustrating the theoretical results are given.Finally, the establishment of the three-dimensional food chain model with multi-delays. Based on the three-dimensional food chain model with variable rate of the same growth delay for predator and prey, the influence of the existence, and direction of Hopf bifurcation solutions is studied. The judgment for the key parameters of the Hopf bifurcation direction is given by the center manifold theorem. The numerical simulation illustrating the theoretical results are given. Then based on the three-dimensional food chain model with variable rate of the different growth delay for predator and prey, the influence of the existence, and direction of Hopf bifurcation solutions is studied. The numerical simulation illustrating the theoretical results are given.