$K$-Branching Random Walk with Noisy Selection: Large Population Limits and Phase Transitions

TL;DR

The paper analyzes a noisy Gibbs-branching random-walk model with fixed population size, showing that in the large-population limit the fitness wave evolves deterministically via a stochastic Hopf-Cole transform. It introduces two operators, $r$ (reproduction) and $s_{\sigma}$ (selection), acting on limiting log-profiles, and proves a deterministic recursion $g^{t+1}=s_{\sigma_t}\circ r(g^t)$ with an implicit selection threshold $\sigma_t$, yielding a well-defined traveling-wave regime for certain tail conditions. In the exponential-tail case ($\rho=1$), the authors establish a unique traveling wave $(G,\nu)$ and a phase transition between two regimes: selection of the luckiest ($\gamma<\gamma_c(\beta)$) and selection of the fittest ($\gamma\ge\gamma_c(\beta)$), with explicit speeds and front structures. The work combines a stochastic Hopf-Cole transformation, concentration tools, and a finite-dimensional reduction via piecewise-linear fronts to give a rigorous account of wave propagation in noisy evolutionary dynamics, including an example showing limits of the theory and a detailed traveling-wave analysis. The results contribute to understanding universal wave propagation in fitness landscapes and provide precise asymptotics for the rate of adaptation under varying noise and selection pressure.

Abstract

We analyze a variant of the Noisy $K$-Branching Random Walk, a population model that evolves according to the following procedure. At each time step, each individual produces a large number of offspring that inherit the fitness of their parents up to independent and identically distributed fluctuations. The next generation consists of a random sample of all the offspring so that the population size remains fixed, where the sampling is made according to a parameterized Gibbs measure of the fitness of the offspring. Our model interpolates between classical models of fitness waves and exhibits a novel phase transition in the propagation of the wave. By employing a stochastic Hopf-Cole transformation, we show that as we increase the population size, the random dynamics of the model can be described by deterministic operations acting on the limiting population densities. We then show that for fitness fluctuations with exponential tails, these operations admit a unique traveling wave solution with local stability. The traveling wave solution undergoes a phase transition when changing selection pressure, revealing a complex interaction between evolution and natural selection.

Figures (7)

• Figure 1: Two examples of $G$ (in red) where $c_- = \infty$ and $\beta = 0.3$, along with the limiting log-profiles after reproduction $r(G)$ (in black) and the limiting log-profile after selection $s \circ r (G) = G( \cdot -\nu)$ in blue. The first plot corresponds to the selection of the luckiest regime with $\gamma = 0.4$, and the second plot corresponds to the selection of the fittest regime with $\gamma = 0.6$. In both plots, $G$ consists of line segments with slopes $-(1-6\beta), \ldots, -(1-\beta)$, and a further line segment with slope $-1$ in the second plot.
• Figure 2: Plots of the speed of the traveling wave $\nu(\beta,\gamma)$ for 3 different values of $\beta$: $\beta = 0.3$ (in blue), $\beta=0.6$ (in red), and $\beta = 0.9$ (in teal).
• Figure 3: A potential outcome of applying the operator $r$ to the log-profile $g$ (in red), producing the functions $r(g)$ (in black).
• Figure 4: Applications of the operator $s_\sigma$ to $r$ (in black). The functions $s_\sigma(r)$ are portrayed for three values $\sigma_1 > \sigma^{\ast}(r) > \sigma_2$, where $s_{\sigma_1}(r)$ is in violet, $s(r) = s_{\sigma^\ast(r)}(r)$ is in blue, and $s_{\sigma_2}(r)$ is in orange.
• Figure 5: After applying the operator $\bar{s}$ to $r(x) = \pi(1-|x|^{1/4})$ (in black) with $\sigma^\ast(r) = 1$, the function $\bar{s}(r)$ (in orange) has a local maximum $\bar{s}(r)(1) = 0$, and $s(r)$ (in solid orange) is not the limiting log-profile of the process after selection.

Theorems & Definitions (72)

• Definition 1.1
• Definition 1.2: limiting log-profile
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• Remark 2.3
• proof