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.

Accounting for competition effects is an essential step in building any stand growth simulator. However, accurate modelling of competition effects depends upon a clear understanding of quantitative relationships of various aspects of stand dynamics, including distributional parameters and spatial statistics. This study addressed four aspects of competition effects: 1) competition effects on distributional parameter dynamics of tree size variables; 2) inter-specific (loblolly pine vs. hardwood) and intra-specific competition effects on basal area growth, 3) dynamics of spatial statistical characteristics of DBH and total height, and their relevance to intertree competition, 4) and spatial properties of competition measures by available stand simulators for loblolly pine (<i>Pinus taeda</i> L.) plantations. Competition was found to affect the distribution parameters such as coefficient of variation, skewness, and the general shapes of distributions for diameter measurements, total height, crown width and crown height. Competition expedites size differentiation and thereby increases distribution variability for all variables except crown height. Intertree competition also drives skewness of these variables negative, although the distributions of crown heights tends to be more symmetric. Normality assumption generally holds for diameter measurement, but distributions of total height and crown width deviate from, and those of crown height approach, normality with intensified competition. A set of competition driven equations was developed for the distribution parameters and was validated. The differences of distribution parameters among the variables studied could be attributed to their biological meanings. / Ph. D.