r- and K-Selection in a Completely Chaotic Population Model

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Abstract
A model of discrete-generations population dynamics originally due to Williamson was analyzed. In this model, the negative effect of population density acts suddenly and completely as the population size passes a threshold. The model almost always yields chaotic fluctuation of population size. When it does, this chaos ensues whatever the initial population size. The average population cycle length and the distribution of populations sizes over time were found. The r- and K-selection are easily analyzed in this model. Natural selection will favor cycles with more time spent above the threshold population size, and will favor increase of the threshold. When r- and K-selection oppose each other, protected polymorphism is possible in an asexual or haploid population, but only if the genotype which would be lost if there were no K-selection has the greater population fluctuation when present alone. Conditions for protected polymorphism in a 2 allele diploid population were given.

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