Fixed points in the dynamics of an agricultural pest model
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This thesis describes the fixed point structure and dynamics of a simple deterministic population model for an insect pest on an agricultural crop. The crop is a mixture of stands of genetically modified plants that are toxic to the pest and "refuges" of conventional toxin-free plants. It is assumed that the pest population carries a genetic allele that confers resistance to the toxin. Refuges are used by farmers with the goal of preventing, or at least delaying, predominance of this allele. Chapter 1 is an introduction. Chapter 2 provides a complete qualitative atlas of the dynamics when the crop is isolated from any external influence. Refuges are found to be always beneficial. Numerical results illustrating the influence of migration between the crop and a toxin-free external pest habitat are presented in Chapter 3. Two new fixed point branches are shown to exist; one is an agriculturally desirable low-population stable state. The addition of refuges is seen to eliminate this desirable state and is thus harmful. Finally, in Chapter 4 asymptotic formulas for the new fixed points and their eigendata are developed. Excellent agreement is found between these asymptotic formulas and numerical results in cases of low migration. These formulas show how each of the many parameters influence these fixed points which are of potentially substantial agricultural significance. They should be a useful guide for the numerical study of more elaborate models.