Distributed predator-prey co-evolution

An evolutionary distribution z (x, t) is a distribution of the density of organisms along the values of adaptive traits x at time t. Predator and prey evolutionary distributions with competition, assortative mating and mutations display dynamics that include cyclic waves and other esoteric phenomena. Such phenomena depend on the parameter space, on the instability of the predator-prey dynamics, on initial conditions and on the nature of both intra- and inter-distribution interactions. Traditional reaction-diffusion models treat polynomial reaction terms. Here the reaction terms are rational. Diffusion and reaction terms in the reaction-diffusion equations emerge from first principles, including selective predation, assortative mating, mutations and competition. Non-homogeneous distributions of phenotypes along trait values emerge from non-homogeneous perturbations. The non-homogeneity, albeit stable, depends on the specific non-homogeneous perturbation. The emerging patterns are internal to population processes (predator-prey). When external, non-homogeneous selection is imposed, the predator-prey dynamics exhibit high-frequency changes in evolutionary types on the boundaries of the smooth transitions of external selective pressures.