The Hopf Bifurcation Of The Tick Population Dynamical Model

Ticks play an important role in the spread of some emerging infectious diseases such as lyme disease and ticks.They often live in mice,deer and other mammals,and when they bite the host,they can infect the host with a variety of diseases.Therefore,research on the dynamic system of tick popu lations is also a research hotspot in biology.But the models of the twelve life stages of ticks are more complex,and existing studies are being simplified,and scholars have found that simplified systems with delays can cause vibrations.Consider the exi stence of time delay is an important factor in the production of periodic vibration phenomenon,so based on time delay parameters,by studying the Hopf bifurcation of the system,and to explain the phenomenon of periodic vibration,and use specific examples to validate and give biological explanation on the results.This paper,mainly aiming at the existing tick population model are studied,this system contains three delay,respectively used to depict larvae for host,nymphal looking for host and adult looking for three stages of time.Firstly,the system is proved to be positive and bounded.Second,further analyzing the existence and stability of equilibrium point and the existence of Hopf bifurcation,and the study of the nature of the Hopf bifurcation,determine direction of the bifurcation and bifurcation of periodic solutions of a given formula;Finally,a concrete example is used to verify the conformance of the theoretical bifurcation value and the numerical simulation results,and try to give a biol ogical explanation.