Double Hopf Bifurcation Analysis Of Multiple Delayed Feedback Control System Of The Gut Microbiota Via MMS

Delay is a ubiquitous and critical phenomenon.There is also a significant delay phenomenon in the processes of the antibiotic injection therapy.Induced by the abuse of antibiotics,gut microbiota disorders will increase the probability of the occurance of many diseases,so the amount of injected antibiotics must be strictly controlled.To solve this problem,a novel model based on the Lotka-Volterra model has been established.The appropriate amount of antibiotics is used as a negative feedback adjustment.The densities of gut microbiota popu-lations are measured through metagenomic data with delays.In the latest work on it only the stability and Hopf bifurcation of the positive equilibrium of the delayed gut microbiota feedback control system have been investigated,but the more complex dynamics of the system,such as double Hopf bifurcation and chaos,is not yet considered.Under the support of the National Natural Science Foun-dation of China?Nos.11372282 and 11972327?,in this dissertation mainly the method of multiple scales is employed to analyze the double Hopf bifurcation of the system,and the detailed unfolding and classification are obtained.We firstly use DDE-BIFTOOL to find the double Hopf bifurcation points of the system,and plot the bifurcation diagram with two bifurcation parameters?i.e.,measurement delays?₁and?₂?.Then near these double Hopf bifurcation points,we employ the method of multiple scales to obtain its two amplitude equations,i.e.,its nor-mal form.By analyzing the bifurcation of the normal form,its unfolding and classification are obtained.It discloses that the system has complex dynamic be-haviors,such as the existence of stable equilibrium,stable periodic solution,and coexisting stable periodic solutions in different regions.Then,Win PP is used for numerical verification.The numerical simulation results agree very well with those of the theoretical analysis.It indicates that the method of multiple scales is simple,effective and accurate.The results show that measurement delays will induce complex dynamic changes in the system.We can achieve our desired re-sults by appropriate selection of parameters.So we should properly adjust the processes of the measurements.All the obtained complex dynamical phenomena in this dissertation provide a theoretical basis for studying the complex oscillation mechanism of the delayed feedback control system of the gut microbiota.They may also give some guidance and insights into microbiology and engineering.The features and innovations of this dissertation are the following.The fur-ther study for complex dynamics of the feedback control system of gut microbiota with two measurement delays such as double Hopf bifurcations are conducted.And they are more difficult higher co-dimensional bifurcations.Complex oscilla-tions in this system are discovered.