Population Dynamics In Patchy Landscapes: Steady States and Pattern Formation

Abstract

Many biological populations reside in increasingly fragmented landscapes, which arise from human activities and natural causes. Landscape characteristics may change abruptly in space and create sharp transitions (interfaces) in landscape quality. How patchy landscape affects ecosystem diversity and stability depends, among other things, on how individuals move through the landscape. Individuals adjust their movement behaviour to local habitat quality and show preferences for some habitat types over others. In this dissertation, we focus on how landscape composition and the movement behaviour at an interface between habitat patches of different quality affects the steady states of a single species and a predator-prey system. First, we consider a model for population dynamics in a habitat consisting of two homogeneous one-dimensional patches in a coupled ecological reaction-diffusion equation. Several recent publications by other authors explored how individual movement behaviour affects population-level dynamics in a framework of reaction-diffusion systems that are coupled through discontinuous boundary conditions. The movement between patches is incorporated into the interface conditions. While most of those works are based on linear analysis, we study positive steady states of the nonlinear equations. We establish the existence, uniqueness and global asymptotic stability of the steady state, and we classify their qualitative shape depending on movement behaviour. We clarify the role of nonrandom movement in this context, and we apply our analysis to a previous result where it was shown that a randomly diffusing population in a continuously varying habitat can exceed the carrying capacity at steady state. In particular, we apply our results to study the question of why and under which conditions the total population abundance at steady state may exceed the total carrying capacity of the landscape. Secondly, we model population dynamics with a predator-prey system in a coupled ecological reaction-diffusion equation in a heterogeneous landscape to study Turing patterns that emerge from diffusion-driven instability (DDI). We derive the DDI conditions, which consist of necessary and sufficient conditions for initiation of spatial patterns in a one-dimensional homogeneous landscape. We use a finite difference scheme method to numerically explore the general conditions using the May model, and we present numerical simulations to illustrate our results. Then we extend our studies on Turing-pattern formation by considering a predator-prey system on an infinite patchy periodic landscape. The movement between patches is incorporated into the interface conditions that link the reaction-diffusion equations between patches. We use a homogenization technique to obtain an analytically tractable approximate model and determine Turing-pattern formation conditions. We use numerical simulations to present our results from this approximation method for this model. With this tool, we then explore how differential movement and habitat preference of both species in this model (prey and predator) affect DDI.