| Summary: | 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.
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