Evolution dynamics of biological systems with spatial and temporal heterogeneities

Biological invasion is an important phenomenon in ecology. Mathematical studies of biological invasion involve reaction-diffusion equations which consider continuous reproduction and random movements of species, and integro-differential/difference equations which describe population dispersal via various types of dispersal kernels. The purpose of this thesis is to investigate the spatial dynamics of some reaction-diffusion and integro-differential/difference population models with spatial and temporal heterogeneities.;In Chapter 2, we present some terminologies and theorems which are based on the theories of global attractors, uniform persistence, monotone dynamical systems, asymptotic speeds of spread and traveling waves.;Chapter 3 is devoted to the study of spatial dynamics of a class of periodic integro-differential equations which describe the population dispersal process via a dispersal kernel. By appealing to the theory of asymptotic speeds of spread and traveling waves for monotone periodic semiflows, we establish the existence of the spreading speed c* and the nonexistence of time-periodic traveling wave solutions with the wave speed c < c*. In the autonomous case, we further use the method of upper and lower solutions to prove the existence of monotone traveling waves with the wave speed c > c*, which implies that the spreading speed coincides with the minimal wave speed for monotone traveling waves.;In Chapter 4, we investigate a non-local periodic reaction-diffusion population model with stage-structure. In the case of unbounded spatial domain, we establish the existence of the asymptotic speed of spread and show that it coincides with the minimal wave speed for monotone time-periodic traveling waves. In the case of bounded spatial domain, we obtain a threshold result on the global attractivity of either zero or a positive periodic solution.;Introduction and overview of mathematical investigation of biological invasions are presented in Chapter 1.;In Chapter 5, we consider a class of discrete-time population models in a periodic lattice habitat. When the recruitment function is monotone, we show that the spreading speeds coincide with the minimal wave speeds for spatially periodic traveling waves in the positive and negative directions, by appealing to the theory of spreading speeds and spatially periodic traveling waves for monotone systems in periodic environments. When the recruitment function is not monotone, we also obtain the existence and formula of the spreading speeds via the comparison method. Moreover, we prove the existence of spatially periodic traveling waves by using the Schauder fixed point theorem.;In Chapter 6, we consider a class of cooperative reaction-diffusion systems, in which one population (or subpopulation) diffuses while the other is sedentary. We use the shooting method to prove the existence of the bistable traveling wave, and then obtain its global attractivity with phase shift and uniqueness (up to translation) via the dynamical system approach. The results are applied to some specific examples of reaction-diffusion population models.;A brief summary of this thesis and some future work are presented in Chapter 7.