We model the growth, dispersal and mutation of two phenotypes of a species using reaction-diffusion equations, focusing on the biologically realistic case of small mutation rates. After verifying that the addition of a small linear mutation rate to a Lotka-Volterra system limits it to only two steady states in the case of weak competition, an unstable extinction state and a stable coexistence state, we prove that under some biologically reasonable condition on parameters the spreading speed of the system is linearly determinate. Using this result we show that the spreading speed is a non-increasing function of the mutation rate and hence that greater mixing between phenotypes leads to slower propagation. Finally, we determine the ratio at which the phenotypes occur at the leading edge in the limit of vanishing mutation.

No abstract

Grass-feeding termites, Gnathamitermes sp., can be considered beneficial in pastureland, where they aerate topsoil, recycle nutrients from grasses and forbs, and excavate and redistribute soil. The primary objective of this report is to illustrate and determine the average physical dimensions of Gnathamitermes tubiformans soldiers by measuring key body parts, including left mandible length, as well as discuss observed foraging behavior. G. tubiformans soldiers have unique mandibles that can be used for accurate identification. Workers are morphologically similar compared with some other arid-land termite species, thus soldier characteristics are used for identification. The secondary objective is to describe observations of G. tubiformans foraging behavior. This report also provides detailed line drawings and a watercolor illustration for aid in identification. Termites were collected near Temple, Cotton County, Oklahoma, and measured with high precision microscopy. To ensure accurate identification, fourteen different soldier body measurements were determined. The ability to identify termites that are potentially beneficial or detrimental to forage grass biomass production may aid efforts to enhance livestock production on pastures.