Chase-escape with conversion on the complete graph

TL;DR

This work analyzes chase-escape with conversion on the complete graph, establishing that conversion shifts extinction probabilities but preserves the equal-spread-rate phase transition. By mapping the dynamics to coupled birth–death processes with terminal values $\bar{\mathcal{E}}\sim\mathrm{Exp}(1)$ and $G_{\alpha}\sim\mathrm{Gamma}(\alpha,1)$, the authors adapt Kortchemski's framework to obtain explicit asymptotics: the number of converted sites scales as $\alpha\log n$ and the expected surviving white sites at fixation converges to $2\alpha$. At the critical point $\lambda=1$, the extinction probability equals $\mathbb{P}(G_{\alpha}<\bar{\mathcal{E}})=2^{-\alpha}$, highlighting how conversion enters solely through the Gamma terminal value. Overall, the paper clarifies how conversion modifies long-run behavior while preserving the underlying phase-transition structure, offering precise, closed-form limits via a clean probabilistic coupling.

Abstract

We prove that chase-escape with conversion on the complete graph undergoes a phase transition at equal fitness and derive simple asymptotic formulas for the extinction probability, the total number of converted sites, and the expected number of surviving sites.

Figures (1)

• Figure 1: Red, white, and blue population sizes at the jump times in chase-escape with conversion on $K_{101}$ with $\lambda = 1$ and $\alpha=4$. In this simulation $W(K_{101},1,4) = 2$.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7
• proof