Dynamics Research Of Some Cannibalism Models With Stage Structure

When studying the population dynamics system,cannibalism phenomenon is often neglected,which often occurs to individuals in different stages of the same population,thus to a certain extent,cannibalism phenomenon affect the survival and extinction of the population.Therefore,it is more practical to consider the cannibalism model of populations to different stages.In order to study the effects of cannibalism on population development,this paper establishes three types of cannibalism models with stage structure.Mainly from the following three aspects:1.The dynamic behaviors of the cannibalism model with two stages(juvenile and adult).On the assumption that adult individuals will prey on juvenile individuals and consider the natural death of juvenile individuals,we establish a cannibalism model with two stages(juvenile and adult),consider the consistent persistence of the model,and completely analyse stability of the model equilibrium point.For the case without cannibalism,we prove the global stability by constructing the appropriate Lyapunov functions.It is found that,for the case with cannibalism,the saddle-node bifurcation can occur,and the global dynamic behaviors of the model are investigated by applying the Dulac function to rule out the existence of periodic solutions.The numerical analysis accurately verifies the conclusions obtained from the theoretical analysis.2.The dynamic behaviors of the insect cannibalism model with three stages.Based on the cannibalism model of two-stage structure,we assume that adults will prey on eggs and establish the three-stage insect cannibalism model.when there is no cannibalism,we found if has ?2(?1+?2)=?2b,the system will appear to the across critical branch,and the only survival equilibrium point is global asymptotic stability;when there is cannibalism,the local stability of the model equilibrium point is obtained by the Routh-Hurwitz discriminant method,and the second additive complex matrix method is used to prove that the only survival equilibrium point is globally stable.Then we found that the model may change from two equilibrium points to one equilibrium point in the positive invariant set,that is,the model will generate saddle node branches due to cannibalism,and the fifinal states of the populations depend on the initial conditions of the solutions.The correctness of the obtained analysis results is verifified by numerical simulations.3.The dynamic behaviors of the cannibalism model with time-delay and the optimal acquisition strategy.On the basis of assuming that the young population grows into an adult population with a fixed maturity time,we establish a cannibalism model of optimal acquisition strategy with time-delay.Firstly,we consider the uniform persistence of the model.Secondly,when there is no time-delay,the population survival equilibrium is stable,and the equilibrium capture function of capture effort is established to obtain the optimal harvest strategy.When there is time-delay,by analyzing the time-delay model,it is concluded that the model has some switching phenomena.