Extinction and survival in inherited sterility

TL;DR

The paper introduces the IS(λ,p) process to model inherited sterility on $\mathbb{Z}^d$, where fertile offspring reproduce at rate $λ$ and are fertile with prob. $p$ or sterile with prob. $1-p$, with all individuals dying at rate $1$. It proves a partial phase transition: extinction whenever $λp\le λ_c(d)$ and survival for large $p$ when $λ>λ_c(d)$, using two auxiliary constructs—the Spont(λ,p) process (a contact-process-like model in a dynamic random environment) and a coupling with the standard contact process—to circumvent the non-monotonicity of IS in $p$. By exploiting graphical representations and a renormalization approach, the authors show Spont itself undergoes a phase transition in $p$ and that IS inherits survival from Spont via order-preserving couplings with the contact process. The work highlights that non-monotone interactions can still yield sharp large-scale behavior through careful domination and percolation-based arguments, contributing to the theory of interacting particle systems with nonlinear inheritance mechanisms and informing related pest-control modeling strategies.

Abstract

We introduce an interacting particle system which models the inherited sterility method. Individuals evolve on $\mathbb{Z}^d$ according to a contact process with parameter $λ>0$. With probability $p \in [0,1]$ an offspring is fertile and can give birth to other individuals at rate $λ$. With probability $1-p$, an offspring is sterile and blocks the site it sits on until it dies. The goal is to prove that at fixed $λ$, the system survives for large enough $p$ and dies out for small enough $p$. The model is not attractive, since an increase of fertile individuals potentially causes that of sterile ones. However, thanks to a comparison argument with attractive models, we are able to answer our question.

Figures (2)

• Figure 1: Graphical representation for a one dimensional $IS$ process. The blue, resp. red arrows, correspond to births of fertile, resp. sterile individuals. The black crosses, resp. red dots, correspond to deaths of fertile, resp. sterile individuals. The blue paths correspond to space-time active paths along which individuals survive. The red segments correspond to blocked sites due to the presence of a sterile individual, that is, sites where fertile individuals cannot be born, until the sterile individual blocking the site dies.
• Figure 2: At $t=0$ there is a translate $\mathcal{C}$ of $[-K,K]$ such that the contact process restricted to $\mathcal{C}$ survives with probability greater than $1-\gamma/2$. At time $T_1$, with high probability, all $-1$'s initially present have died and, up to time $T$, no new $-1$'s have appeared. From $t=0$ to $T_1$, the leftmost, resp. rightmost $1$, has not reached $-2N$, resp. $2N$ (blue dashed lines). From $t=0$ to $T$, w.h.p, the leftmost, resp. rightmost $1$ in a supercritical contact process with parameter $\lambda p$ starting from $\mathcal{C}$, has reached $I_1$, resp. $I_{-1}$ (red dashed line). At $T$, the supercritical contact process is not extinct with probability greater than $1-\gamma/2$ and, w.h.p, on $[-3N,3N]$ it is coupled to a contact process where all sites are initially filled with $1$'s.

Theorems & Definitions (35)

• Definition 1
• Theorem 1
• Definition 2
• Proposition 1
• Remark 1
• Remark 2
• Theorem 2
• Definition 3
• Theorem 3
• Proposition 2