Branching with selection and mutation II: Mutant fitness of Gumbel type

TL;DR

The paper studies branching processes with selection and mutation under unbounded Gumbel-type fitness tails, classifying tails into type I and II and deriving sharp growth rates for population size $X(t)$ and the leading mutant fitness $W_t$; type I yields $\frac{\log X(t)}{t\log^{(n)}(t)}\to 1/\alpha$ with $W_t$ scaling as $u_n(t)$, while type II yields nested-log growth patterns with corresponding $W_t$ asymptotics. To understand the empirical fitness distribution (EFD), the authors introduce deterministic (DFMM) and semi-deterministic (SFMM) variants and prove that the EFD converges to a Gaussian travelling wave with mean $v_n(t)$ and width $\mathfrak{s}_n(t)$, providing explicit forms for these quantities. They provide rigorous proofs for the main theorems in the type I and II regimes, accompany them with heuristic differential-equation guides, and corroborate the travelling-wave picture with numerical simulations, which also suggest similar behaviour in the MMM. The results illuminate the speed of adaptation and the structured distribution of fitness in evolving populations under mutation and selection, offering a framework potentially extendable to other tail classes and bounded-tail scenarios. The work advances understanding of how extremal fitness events shape macroscopic growth and the shape of fitness distributions in branching processes.

Abstract

We study a model of a branching process subject to selection, modeled by giving each family an individual fitness acting as a branching rate, and mutation, modeled by resampling the fitness of a proportion of offspring in each generation. For two large classes of fitness distributions of Gumbel type we determine the growth of the population, almost surely on survival. We then study the empirical fitness distribution in a simplified model, which is numerically indistinguishable from the original model, and show the emergence of a Gaussian travelling wave.

Figures (2)

• Figure 1: Semilogarithmic plot of $\sigma_t \psi(F,t)$ vs. $\Delta F/\sigma_t$ for various $\alpha$'s at $t=983~040$. For comparison, the normal distribution is plotted by a solid curve.
• Figure 2: Plots of $\psi(F,t)$ vs. $\Delta F$ at different generations for $\alpha=1$ (left), $\alpha=2$ (middle), and $\alpha=3$ (right) on a semi-logarithmic scale. For $\alpha=3$ ($\alpha=1$), the width of the traveling wave decreases (increases). For $\alpha=2$, the width of the traveling wave remains constant.

Theorems & Definitions (82)

• Theorem 1
• Theorem 2
• Theorem 3: Conjecture
• Remark 2.1
• Corollary 2.1
• proof
• Lemma 4.1
• Lemma 4.2
• proof
• Lemma 4.3