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 the patchiness of landscapes affects ecosystem diversity and stability depends, among other things, on how individuals move through the landscape. Individuals adjust their movement behavior to local habitat quality and show preferences for some habitat types over others. In this thesis, we focus on how landscape composition and the movement behaviour of individuals at an interface between patches of different quality affect the steady state of a single species.
We consider a model of reaction-diffusion equations for the temporal evolution of the density of the population in space. Individual movement is described by a diffusion process, e.g., an uncorrelated random walk. Population net growth is encapsulated in the growth function that considers birth and death of individuals, including nonlinear effects that arise from competition and/or facilitation within the species. We consider the simplest case of two adjacent one-dimensional patches, e.g., two intervals on the real line that share one boundary point. Conditions are homogeneous within a patch but differ between patches. The movement behaviour of individuals between the two patches is incorporated into matching conditions of population flux and density at the interface between patches, i.e., the boundary point that the intervals share. These matching conditions turn out to be continuous in the flux but discontinuous in the density.
Several authors have studied similar models recently. Most of these studies consider monostable dynamics on both patches, i.e., logistic growth. Under logistic growth, the net population growth rate is a strictly decreasing function of population density. Logistic population dynamics are very simple: the population extinction state is unstable and a positive steady state is globally asymptotically stable. In this work, we also include bistable dynamics, i.e., an Allee effect. Biologically, an Allee effect occurs when individuals cooperate at some level so that the net population growth rate is increasing with population density for at least some low or intermediate densities. Models with Allee growth typically exhibit bistability: there are two locally stable steady states, one at low density (possibly zero) and one at high density. This bistability makes mathematical analysis more challenging, but leads to more interesting results in return.
Mathematically, most existing work on related models is based on linear stability analysis of the extinction state. We focus on the nonlinear models and specifically on positive steady states. We establish the existence, uniqueness and - in some cases - global asymptotic stability of a positive steady state. We classify the shape of these states depending on movement behaviour. We clarify the role of movement in this context. In particular, we investigate the following prior observation: a randomly diffusing population at steady state in a continuously varying habitat can exceed its carrying capacity. Our results clarify when and under which conditions this effect can arise in our two-patch landscape.
The analysis of the model with an Allee effect on one of the two patches yields a rich and interesting structure of steady states. Under certain parameter conditions, some of these states are amenable to explicit stability calculations. These yield insights into the possible bifurcations that can occur in our system. Finally, we indicate how the model and analysis here can be extended to systems of reaction-diffusion equations on graphs that represent natural habitats with different geometries, for example watersheds.
| Identifer | oai:union.ndltd.org:uottawa.ca/oai:ruor.uottawa.ca:10393/44540 |
| Date | 18 January 2023 |
| Creators | Ketchemen Tchouaga, Laurence |
| Contributors | Lutscher, Frithjof |
| Publisher | Université d'Ottawa / University of Ottawa |
| Source Sets | Université d’Ottawa |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
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