Invasive Allele Spread under Preemptive Competition

Abstract

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.Comment: Computer Simulation Studies in Condensed Matter Physics XVIII, edited by D.P. Landau, S.P. Lewis, and H.-B. Schuttler, (Springer, Heidelberg, Berlin, in press

Fisher waves and front roughening in a two-species invasion model with preemptive competition

“…Thus, small clusters of an advantageous allele or superior species can randomly "nucleate" and subsequently grow. We have shown [8,9,19,20] that the time evolution of such systems can be well described within the framework of homogeneous nucleation and growth. The growing clusters, on average, have radial symmetry and reach an asymptotic velocity v * for sufficiently long times.…”

“…Consider, for example, that an advantageous allele or a competitively superior species is introduced through mutation within [9,19] or through geographic dispersal to [8,20] a resident population, respectively. Introductions occur rarely and stochastically in both space and time.…”

“…We have shown [8,9,19,20] that the time evolution of the invader and resident populations in such systems can be well described within the framework of homogeneous nucleation and growth [21]. In particular, in two dimensions, for sufficiently large systems, the typical time (lifetime) until competitive exclusion of the weaker competitor scales as τ ∼(Iv 2 ) −1/3 [8,9,19,20], where I is the stochastic nucleation rate per unit area of the successful clusters of the better competitor, and v is the asymptotic radial velocity of the growing (on average) circularly symmetric fronts. It is, thus, clear that the full understanding of the dependence of the lifetime on the local rates of the systems requires the knowledge of the velocity of the front separating the two species.…”

Fisher Waves and the Velocity of Front Propagation in a Two-Species Invasion Model with Preemptive Competition

“…Most notably, Fisher [3] and Kolmogorov et al [4] first addressed the velocity characteristics of a simple front by way of a reaction-diffusion equation [1], which served as a one-dimensional model for the spread of a favorable gene. Our study of front propagation envisions introduction of an advantageous allele or a competitively superior species through mutation within [5,6] or geographic dispersal to [7,8] a resident population, respectively. Introductions occur rarely, and stochastically in both space and time.…”

“…Introductions occur rarely, and stochastically in both space and time. We have shown [5][6][7][8] that the time evolution of such systems can be well described within the framework of homogeneous nucleation and growth. In particular, in two dimensions, for sufficiently large systems, the typical time scale (lifetime) until competitive exclusion of the weaker allele or species scales as τ ∼(Iv 2 ) −1/3 , where I is the stochastic nucleation rate per unit area of the successful clusters of the better competitor and v is the asymptotic radial velocity of the corresponding circular fronts.…”