A model of cell surface receptor aggregation.

In this paper we construct and analyze a model of cell receptor aggregation. Experiments have shown that receptors in an aggregated state have greatly reduced mobility. We model the effects of this reduced mobility with a density dependent diffusion and study the impact of density dependent diffusion on aggregate formation in a one-dimensional domain. Critical values of receptor diffusivity and receptor activation are found and compared with numerical simulations. We find that the role of density dependant diffusion is quite limited in the formation of aggregate structures. In the case of receptor activation, the analytical results agree very well with the numerical calculations. Finally, we consider our model in higher dimensional domains. In this case our analysis is primarily numerical.

A threshold area ratio of organic to conventional agriculture causes recurrent pathogen outbreaks in organic agriculture.

Conventional agriculture uses herbicides, pesticides, and chemical fertilizers that have the potential to pollute the surrounding land, air and water. Organic agriculture tries to avoid using these and promotes an environmentally friendly approach to agriculture. Instead of relying on herbicides, pesticides and chemical fertilizers, organic agriculture promotes a whole system approach to managing weeds, pests and nutrients, while regulating permitted amendments. In this paper, we consider the effect of increasing the total area of agricultural land under organic practices, against a background of conventional agriculture. We hypothesized that at a regional scale, organic agriculture plots benefit from existing in a background of conventional agriculture, that maintains low levels of pathogens through pesticide applications. We model pathogen dispersal with a diffusive logistic equation in which the growth/death rate is spatially heterogeneous. We find that if the ratio of the organic plots to conventional plots remains below a certain threshold l(c), the pest population is kept small. Above this threshold, the pest population in the organic plots grows rapidly. In this case, the area in organic agriculture will act as a source of pest to the surrounding region, and will always infect organic plots as they become more closely spaced. Repeated localized epidemics of pest outbreaks threaten global food security by reducing crop yields and increasing price volatility. We recommend that regional estimates of this threshold are necessary to manage the growth of organic agriculture region by region.

Stability of asymmetric spike solutions to the Gierer-Meinhardt system.

In this paper, we study the spectra of asymmetric spike solutions to the Gierer-Meinhardt system. It has previously been shown that the spectra of such solutions may be determined by finding the generalized eigenvalues of matrices, which are determined by the positions of the spikes and various parameters from the system. We will examine the spectra of asymmetric solutions near the point at which they bifurcate off of a symmetric branch. We will confirm that all such solutions are unstable in a neighborhood of the bifurcation point and we derive an explicit expression for the leading order terms of the critical eigenvalues.