A mathematical model of a biological arms race with a dangerous prey

In a recent paper, Brodie and Brodie provide a very detailed description of advances and counter-measures among predator-prey communities with a poisonous prey that closely parallel an arms race in modern society. In this work, we provide a mathematical model and simulations that provide a theory as to how this might work. The model is built on a two-dimensional classical predator-prey model that is then adapted to account for the genetics and random mating. The deterministic formulation for the genetics for the prey population has been developed and used in other contexts. Adapting the model to allow for genetic variation in the predator is much more complicated. The model allows for the evolution of the poisonous prey and for the evolution of the resistant predator. The biological paradigm is that of the poisonous newt and the garter snake which has been studied extensively although the models are broad enough to cover other examples.