Invasion speed determinacy for wave propagations in partial differential equation models arising from population biology

Abstract

This thesis aims at developing the study of invasion speed determinacy for wave propagation in partial differential equations arising from population biology. Along this direction, we first investigate a reaction-diffusion-advection equation in a cylindrical domain with a Fisher-KPP type nonlinearity. Using the upper and/or lower solutions method, we obtain sufficient conditions under which the linear or nonlinear selection is realized when the model is prescribed with Neumann boundary conditions and Dirichlet boundary conditions, respectively. To study the invasion speed determinacy of a system, we investigate a reaction-diffusion-advection population model arising in stream ecology. We concentrate on how the spreading speed (the minimal wave speed) is impacted by the Allee effect in the model. Linear and nonlinear selection mechanisms for the spreading speed are first defined, and the determinacy is further established by way of the upper and lower solution method. It is found that the nonlinear determinacy is realized if there exists a lower solution with a faster decay. For a multiple species population system having diffusion, individual species possibly invade into the far end with different spreading speeds. Predicting or determining them (the fast and slow-spreading speeds) becomes challenging. Hence, we first analyze a cooperative Lotka-Volterra system, which admits a single or multiple spreading speeds (co-speed or fast-slow speeds). We successfully derive a necessary and sufficient condition for this particular model to determine whether the system has a single spreading speed or multiple spreading speeds. We define the linear and nonlinear speed selection mechanism for each case and derive new conditions to classify the speed selection. After studying the former three particular models arising from population biology, we further, in the last part, present the speed selection mechanism for an abstract time-periodic monotone semi ow. At the end of this thesis, we present our future work.