Niche Occupation in Biological Species Competition
Thesis (MSc (Logistics))--University of Stellenbosch, 2008.
The primary question considered in this study is whether a small population of a biological species introduced into a resource-heterogeneous environment, where it competes for these resources with an already established native species, will be able to invade successfully. A two-component autonomous system of reaction-diffusion equations with spatially inhomogeneous Lotka-Volterra competitive reaction terms and diffusion coefficients is derived as the governing equations of the competitive scenario. The model parameters for which the introduced species is able to invade describe the realized niche of that species. A linear stability analysis is performed for the model in the case where the resource heterogeneity is represented by, and the diffusion coefficients are, two-toned functions. In the case where the native species is not directly affected by the resource heterogeneity, necessary and sufficient conditions for successful invasion are derived. In the case where the native species is directly affected by the resource heterogeneity only sufficient conditions for successful invasion are derived. The reaction-diffusion equations employed in the model are deterministic. However, in reality biological species are subject to stochastic population perturbations. It is argued that the ability of the invading species to recover from a population perturbation is correlated with the persistence of the species in the niche that it occupies. Hence, invasion time is used as a relative measure to quantify the rate at which a species’ population distribution recovers from perturbation. Moreover, finite difference and spectral difference methods are employed to solve the model scenarios numerically and to corroborate the results of the linear stability analysis. Finally, a case study is performed. The model is instantiated with parameters that represent two different cultivars of barley in a hypothetical environment characterized by spatially varying water availability and the sufficient conditions for successful invasion are verified for this hypothetical scenario.