Spatial Dynamics Of Population Models With Stage Structure

In population biology, stage structure is an important factor. For example, mammals exhibit distinct age stages which can influence the evolution of such populations in a significant way. Therefore, it is important to model and study the population models with stage structure. By using the theory of dynamical system and nonlinear analysis, the thesis focuses on the spatial dynamics of some classes of population models with stage structure, including the traveling wave solutions, spreading speeds and entire solutions. The main results are as follows.1. We study the traveling wave fronts of an age-structured population model in a 2D lattice strip (i.e. a spatially discrete reaction-diffusion equation with delay). Under the monostable and quasi-monotone assumptions, the uniqueness result is first proved by using the sliding method. Then, applying the squeezing technique, the asymptotic stability of the traveling wave front with non-minimal speed is established.2. We study the entire solutions for the age-structured population model in a 2D lattice strip under monostable assumptions, i.e., solutions defined in the whole space and for all time t G R. In the quasi-monotone case, we first establish the existence and asymptotic behavior of solutions of the equation without j (∈ Z) variable. Combining traveling wave fronts with different speeds above minimal wave speeds and a solution without j variable, the existence and qualitative features of entire solutions are then proved. In the non-quasi-monotone case, we introduce two auxiliary quasi-monotone equations and establish a comparison argument for the three systems. Some new entire solutions are then constructed by using the compar-ison argument, the traveling wave fronts and a solution without j variable of the auxiliary equations. Finally, some entire solutions related to the minimal wave fronts (i.e the traveling wave fronts with minimal wave speed) are constructed by establishing a new comparison theorem.3. We study the pulsating traveling fronts and entire solutions for a spatially discrete peri-odic reaction-diffusion system with a quiescent stage. Under the monostable assumption, the existence and asymptotic behavior of pulsating traveling fronts and spatially periodic solutions connecting two periodic steady states are first established. Combining the left-ward and rightward pulsating traveling fronts with different speeds and a spatially periodic solution, the existence and qualitative properties of entire solutions are then proved.4. We study the spreading speed and traveling wave solutions of a non-quasi-monotone delayed reaction-diffusion model for a single species population with separate mobile and stationary states. By using comparison arguments, Schauder’s fixed-point theorem and a limiting process, we establish the existence of the spreading speed and characterize it as the minimal wave speed for traveling wave solutions. The upward convergence of the spreading speed and traveling wave solutions are also established by applying a fluctuation method. In particular, the effects of the delay and transfer rates on the spreading speed are investigated.