Deterministic and stochastic models of development of resistance to genetically modified pesticidal crops
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This thesis describe the bifurcation structure and dynamics of some simple deterministic population models for an insect pest on a crop divided into two parts, one consisting of genetically modified plants that are toxic to the pest, and the other a "refuge" of conventional toxin-free plants. Refuges are used with the goal of suppressing, or at least delaying, the development of a pest population that is resistant to the toxin, but we demonstrate how they can also have the opposite effect. We have also explored a fully stochastic version of the model and the phenomena described in deterministic are also observed in the stochastic version. We develop the population model using a discrete probability distribution which involves Stirling numbers of the second kind. We computed the moments, asymptotic mean, asymptotic variance and the coefficient of variations of the distribution. In order to run stochastic simulations using this probability model, we must be able efficiently to draw deviates from the distribution which is computationally expensive to do directly. We found conditions under which the probability distribution can be approximated by Gaussian distribution with the same mean and variance which makes the simulation much faster. Chapter 1 is an introduction. In Chapter 2 we give the detailed description and derivation of the cannibalism model of population regulation. In Chapter 3 we provide the details of the calculation of the Stirling distribution. In Chapter 4 deterministic version of a simple model with single larva stage is presented and its bifurcation structure and dynamics are described in detail. In Chapter 5 we describe the stochastic version of the model presented in Chapter 4 and we presented the results. Deterministic version and stochastic version of a model with two larvae stages is discussed in Chapter 6. In Chapter 7 we summarize our results and address the apparent conflict, and agreement, with other results in the literature. In the Appendix we provide samples of the Maple code of the deterministic version of the models and samples of C++ code of the stochastic version of the models used in the computations.