Stability And Hopf Bifurcation For Bacteria And Bacteriophage Coexistence Systems

In recent years, people irrational use of antibiotics, and lack of researching anddevelopmenting of new types of antibiotics, causing pathogens produced drug resistanceto some antibiotics which we often use. Therefore, all of the countries have begun toexplore the treatment options, which can substitute antibiotics, and found "phagetherapy" have a good therapeutic effect to resist pathogens. In order to apply "phagetherapy" in clinical treatment better, we need to understand the dynamic properties ofthe interaction of bacteria and bacteriophage, the dynamics of bacteria andbacteriophage coexistence in the presence of bacterial debris is represented using asystem of delay differential equations. Analyzing their dynamic types by studying theirbifurcations is very important in the fields of both mathematical biology and differentialequations.This paper mainly studies a model of bacteria and bacteriophage interaction in theexistence of bacterial debris, we study the stability of the equilibrium point and Hopfbifurcation problem of the bacteria and bacteriophage model, analyse the effect of thelimitations to this model and investigate other issues such as the director of thebifurcation. The major job is stated as follows: First, discussing the stability of theequilibrium points of the system, taking delay as the bifurcation parameter, we canfind the stability switch when crosses zero and becomes larger gradually, and Hopfbifurcation when time delays cross through the critical value, we derive a prerequisitefor the conditions above. The main way is to discuss the distribution of the roots ofcharacteristic equation for the linear system at the equilibrium point. Second, we discussthe nature of the Hopf bifurcation, and find the stability condition of cycle solution ofthe system of bacteria and bacteriophage coexistence in the existence of bacterial debris.Third, simulated using Matlab, and the corresponding waveforms and phase diagramsare drawn, which state the correctness of our conclusion.