The effect of the fitness gradient on fixation probability

Evolutionary biology examines how the genetic and phenotypic composition of populations changes over time. An important goal is to determine the fixation probability of a single advantageous mutant that arises in a homogeneous population of N residents. Many real populations experience environmental gradients that cause mutations to be beneficial in some spatial regions but harmful in others. Here, we study the fixation probability of a mutant placed on a simple one-dimensional spatial structure that experiences such a gradient. The mutant's fitness varies linearly from 1 - s to 1 + s, whereas the resident fitness is constant and equal to 1. The existing literature suggests that such heterogeneity in the mutant's fitness should lead to a decrease in its fixation probability. However, in this work, we find that small, non-negligible gradients ( $s<1/\sqrt{N}$ ) substantially increase the fixation probability, while larger gradients ( $s>\left(\log N\right)/\sqrt{N}$ ) substantially decrease it. Moreover, we quantify the strength of this phenomenon analytically and we precisely delimit the range of the gradients for which it occurs. Our computer simulations closely match those findings. Altogether, our results indicate that subjecting a simple population structure to natural environmental conditions can produce strong counterintuitive effects.