Stability Analysis Of Age-Distributed Population Models

Since the biological data are often presented in discrete forms, and caculations with matrixs are not very complicated, matrix models are very useful for biologists. At the beginning, researchers considered the simple parameters, and studied the discrete-time population dynamics concerning with autonomous, linear models. As far as the age factor is concered, the life process of an individual contains several stages, such as eggs, juveniles, adults. Such investigations have been recently extended to age and stage structured models. A detailed analysis is required here to establish the existence and stability of the equilibria. It is simple to discuss linear models. However, density dependence is a ubiquitous theme in population ecology. Many researchers tend to consider some parameters like birth rate or mortality rate which is density-dependent. All of these considerations make the models more realistic and difficult to deal with, as well.This dissertation is focused on the dynamical systems with both stage and age structure. To investigate the dynamical properties, we concern two kinds of equilibria: trivial steady state and nontrivial steady state. The theory of matrix is applied to such discrete-time models. Based on the theories of Matrix disturbance and eigenvalue estimation, we discuss the stability of an equilibrium of a nonlinear dynamic system. By caculating the Jacobian, the conditions for stablity are obtained, which provide a solid ground for the practical use of models.The investigation of the dissertation consists of two parts: chapters 2 and 3.In chapter 2, we consider the age-structure in each stage. By evaluating the value of Jacobian matrix, we discuss the stability of the trivial steady state and the positive steaty state. It is shown that if the net reproductive number is smaller than one, then the trivival equilibrium is globally asymptotically stable. Under certain conditions, it exists a unique asymptotically stable positive equilibrium provided the net reproductive number is lager than one.Chapter 3 is devoted to the study on the global dynamics of nonlinear two-stage age-dependent populations. A basic logistic matrix model for two-stage population dynamics with age-dependence is constructed. This model discretizes a continuous two-stage and age-dependent model. It is an extension of the logistic model to the case of stage structure and age-dependence. We investigate the global behavior for a population whose members are divided into two classes: juveniles and adults with mortalities age- and density-dependent. We prove the existence of positive equilibrium, and derive some conditions for the global stability of trivial equilibria.