Global Dynamics Of A Predator-prey Model With State-dependent Maturation-delay

The state-dependent delay population model is a very important model,which can describe the growth of the population more accurately.Similar to the constant time-delay model,the stability analysis of all equilibrium states is very important when studying the state-dependent delay model,and it has always attracted the attention of ecologists and mathematicians.In this paper,a state-dependent delay single-population model and a state-dependent delay predator-prey model have been established and analyzed successively.Where the state-dependent delays are taken to be increasing,convex,twice differentiable functions with respect to the number of adult populations and the number of adult predators,respectively.The models are quite different from many previous models with state-dependent delay in the sense that the derivative of delay on the time is included in the models and for the first time the general nonlinear functional response function is considered in the state-dependent delay predator-prey model.For the single-population model with a general nonlinear birth rate function,the system's positiveness,eventual uniform boundedness,and persistence / extinction threshold conditions were explored,and the local asymptotic stability of trivial equilibrium was discussed by analyzing the characteristic equations.Secondly,for a specific given birth rate function,sufficient conditions for the local asymptotic stability of the positive equilibrium state are explored.Finally,the Barbalat lemma and Fluctuation lemma are used to establish sufficient conditions for the global attractiveness of positive equilibrium.For the predator-prey model with general nonlinear functional response function,the system's positiveness,eventually uniform boundedness,and persistence / extinction threshold conditions are explored,and by analyzing the characteristic equation,the local asymptotic stability of the system's ordinary equilibrium state and the predator's extinction equilibrium state are discussed.Further,taking the Beddington-De Angelis(BD)type functional response function as an example,the sufficient conditions for local asymptotic stability of the coexistence equilibrium state are explored.Secondly,a single-population system is used as an auxiliary system,and a sufficient condition for global attractiveness of the coexistence equilibrium state is established by using the comparison principle and iterative method.The results show that if the system is persistent and the predator interference is large enough,then the coexistence equilibrium state is globally asymptotically stable.Finally,the numerical simulation of Matlab is used to investigate the effect of state-dependent time delay on the global dynamics of the system.The results show that the smaller the frequency of state-dependent time delay about state changes,the easier the system will reach stability.If the changes are too fast,the stability of the system will be destroyed.