Invasive Allele Spread under Preemptive Competition

We study a discrete spatial model for invasive allele spread in which two alleles compete preemptively, initially only the " residents " (weaker competitors) being present. We find that the spread of the advantageous mutation is well described by homogeneous nucleation; in particular, in large systems the time-dependent global density of the resident allele is well approximated by Avrami's law. The spatial and temporal characteristics of the spread of an advantageous mutation are fundamental questions in population dynamics. Fisher [1] and Kolmogorov et al. [2] first addressed these questions using the framework of a simple reaction-diffusion equation [3]. That work and many others focused on the velocity of the propagating front, which exists initially and separates the two spatial regions occupied separately by the two alleles. Both continuum and discrete spatial models have successfully tackled various aspects of these problems [3,4]. In this work we investigate how the advantageous allele emerges from " scratch " ; i.e., initially the region is fully dominated by the resident allele and the advantageous allele is introduced by rare mutations. While the mutant allele has an individual-level advantage over the original one, the low probability of mutations, combined with a discrete spatial dynamics, can prevent the spread of the mutant for long times. Here we consider a model where the original " resident " and the competitively superior " invasive " allele compete for a common limiting resource preemptively [5–8]. The details of our model are as follows. We consider an L×L lattice with periodic boundary conditions. Each site can be empty or occupied by a single allele (either a resident or an invader). A lattice site represents the minimal level of locally available resource required to sustain an individual organism, hence the " excluded volume " constraint. We introduce the local occupation numbers at site x, n i (x) = 0, 1, i = 1, 2, representing the number of resident and invader alleles, respectively. By virtue of the excluded volume constraint, n 1 (x)n 2 (x) = 0. New individuals arise through local clonal propagation only.