Stability Of Population Models With Hierarchical Size-Structure

In the natural world,biological populations with hierarchical structure are common,and the factors affecting the structure are mainly the age and size of the individual,while study the size of individuals in the population is more intuitive and more difficult than the age of individuals.In this paper,we study three types of population models with size hierarchical structure,the main details are as follows:(1)A single population model with size hierarchical structure is established,and the existence of positive equilibrium solution of the model is obtained by applying non-zero fixed-point theorem and Frechet-Kolmogorov theorem.The characteristic equation of the positive equilibrium solution is derived by linearizing the model equilibrium solution,and discuss the stability conclusions of the equilibrium solutions of the model.In addition,the zero equilibrium solution of the model is simulated numerically.(2)We propose a class of nonlinear predator-prey of hierarchical size-structured population model,and the existence and uniqueness of model solution is proved by constructing a monotone sequence of the model and using Gronwall’s inequality.Using fixed-point theorem and FréchetKolmogorov theorem to prove the existence of positive equilibrium solution.Via deriving the characteristic equation of the zero equilibrium solution,the stability results of zero equilibrium solution is analyzed,the stability results of the zero equilibrium solution are verified by numerical simulations.(3)We study a competitive population model with hierarchical size-structure in polluted environment.Applying the theory of semigroup of linear operator to obtain the stability criterion of the positive equilibrium solution.The stability conditions of zero equilibrium solution is proved by deducing characteristic equation of the zero equilibrium solution and establishing the Lyapunov function.Finally,the stability conclusion of zero equilibrium solution is verified by numerical simulation.The paper base on the individuals size of biological populations,discussing the stability of different hierarchical size-structured population models,the existence and stability results of equilibrium solution of the models are proved,and obtain the stability threshold of zero equilibrium solution of the models.