Persistence in perturbed contact models in continuum

TL;DR

This paper investigates the persistence of contact models under non-critical conditions, showing that local mortality peaks do not necessarily cause extinction and establishing the existence of invariant measures via Feynman-Kac formula.

Contribution

It extends previous contact process models by removing the criticality condition and demonstrates the existence of invariant measures in this broader setting.

Findings

Local peaks in mortality do not typically lead to extinction.

Invariant measures exist even without the criticality condition.

Invariant measures can be described using the Feynman-Kac formula.

Abstract

Can a local disaster lead to extinction? We answer this question in this work. In the paper \cite{PZ-PPI} we considered contact processes on locally compact metric spaces with state dependent birth and death rates and formulated sufficient conditions on the rates that ensure the existence of invariant measures. One of the crucial conditions in \cite{PZ-PPI} was the critical regime condition, which meant the existence of a balance between birth and death rates in average. In the present work, we reject the criticality condition and suppose that the balance condition is violated. This implies that the evolution of the correlation functions of the contact model under consideration is determined by a nonlocal convolution type operator perturbed by a (negative) potential. We show that local peaks in mortality do not typically lead to extinction. We prove that a family of invariant measures exists even without the criticality condition and these measures can be described using the Feynman-Kac formula.