Dynamic Analysis Of Cannibalism Model With Two-stage Density Dependence

Cannibalism model is very important in the population dynamics system,which has been concerned by ecologists and mathematicians.Cannibalism phenomenon usually occurs to individuals in different stages of the same population,which largely affects the development of the population,and the population is influenced by external factors such as itself and environment,but also by density constraints,time lag,chaos and other factors.The study of model stability,periodicity and bifurcation is becoming one of the most significant research topics.On the basis of the existing research results,we simulated the real situation in nature,and established three types of two-stage egg-maturity individual cannibalism models with density constraints,considering that mature individuals reduce their own mortality due to cannibalism behavior and are constrained by the number of eggs.The main work is as follows.1.Dynamics analysis of cannibalistic model with density dependence in mature stageBased on considering mature individuals with density dependence,and the death of eggs only due to cannibalism,a two-stage structure model of egg-mature individuals with cannibalism is established.The dynamic behavior of the equilibrium of the model is discussed from two aspects:when no cannibalism exists,the global stability of the equilibrium point is proved by constructing Lyapunov function;when cannibalism exists,it makes the model have saddle-node bifurcation by using the center manifold theorem.By constructing Dulac function,there is no limit cycle in the two-dimensional autonomous system,therefore,the global stability of the equilibrium point is obtained.Finally,the results of theoretical calculation are verified by numerical simulation.2.Dynamic behavior of cannibalism model with time delay and density constraintsBased on the two-stage cannibalism model of egg-mature individuals,a timedelay model with density constraints is established,taking into account the timelagged effects of mortality.Firstly,the existence of the equilibrium point is determined,and the local stability of the equilibrium point is proved by using the RouthHurwite criterion.Secondly,the existence of Hopf bifurcation and the stability of the bifurcation periodic solution are determined by using the central prevalence theory and the normal method with the time-delay as the bifurcation parameter.Finally,numerical simulations are applied to verify the stability and instability of the model when time-delay is used as the bifurcation parameter.3.A two-stage structure cannibalism model with time delay in the egg-mature stageConsidering the introduction of time delay into the growth term of mature individuals,assuming that eggs born from the past moment are at transformed into mature individuals,a class of cannibalism model with time delay is proposed,to analyze the impact of time-delay on the dynamic behavior of the model.If there is no time-delay,the positive equilibrium point has global asymptotic stability;on the contrary,in the presence of time-delay,the local asymptotic stability of the positive equilibrium point and the existence of the Hopf bifurcation are discussed with time-delay as the bifurcation parameter.Finally,MATLAB software is used to simulate the stability of the equilibrium point of the time-delay model.