Persistence in Perturbed Contact Models in Continuum

Can a local disaster lead to extinction? We answer this question in this work. In the paper [19] we considered contact processes on locally compact metric spaces with state dependent birth and death rates and formulated suf- ficient conditions on the rates that ensure the existence of invariant measures. One of the crucial conditions in [19] 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 bal- ance 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.