Predator-prey dynamics : a review : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University
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With the recent publication explosion in population ecology, there is an increasing need for a review of the diverse approaches towards modelling. This thesis is concerned with modelling of two-species predator-prey ecosystems using two-dimensional dynamic systems of first-order differential equations. Chapters one and two are introductory in nature, discussing the place of theoretical models in ecology, and the development of the classical Lotka-Volterra model and its subsequent fall from favour. Chapter three looks at general aspects of predator-prey modelling. Graphical and analytical approaches are outlined in detail, as is the more recent curvature approach. Further results are obtained when growth and predation factors are considered separately, viewed as components to the model equations. Recent work on the consequences of enrichment, harvesting, stocking and natural selection are also dealt with. In chapter four, more specific predator-prey models are presented. Other, more variable qualities of predator-prey ecosystems are also considered, such as age structure and predation responses in chapter four; and time delays, spatial heterogeneity and migration in chapter five. Chapter six is a mathematical digression from the main body of the review. An analytical result for dynamic systems with a centre is proven, in an attempt to support an alternative outlook on the relationship between predator-prey ecosystems and their representative models. Finally, chapter seven briefly discusses potential applications in the future, the most promising being aspects of harvesting and control theory in resource management systems.