Spatially Explicit Population-genetics Modeling in a Discontinuous Habitat and in the Presence of an Allee Effect
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We introduce a population-genetics model with an Allee effect in a discontinuous habitat. The habitat consists of untreated wild area (the refuge) and pesticide-treated crop field (the toxic area). Both implicit and explicit spatial structure are implemented in our models. For the spatially explicit model, we numerically obtain a number of fixed points, among which the stable ones are tested for the convergence with respect to spatial resolution. We analyze these stable/unstable fixed points and obtain asymptotic approximations for them. During the analysis, we observe and discuss cases of homogeneous uniformly distributed population, and exponential population profile with different or the same decay exponents. Last we explore how the fixed points are affected by changes in the important parameters the resistance of the heterozygote in the toxic crop area and dispersal standard deviation. In the process of tracking fixed point loci, saddle-node bifurcations are detected. In our model, we find the Allee effect plays an important role and the high-dose refuge strategy might be harmful to the crops under certain conditions. We also understand how the fixed points interact as the important parameters vary.