Scaling limit of an adaptive contact process

Abstract

We introduce and study an interacting particle system evolving on the d-dimensional torus ${(\mathbb{Z}/\mathit{N}\mathbb{Z})}^{\mathit{d}}$. Each vertex of the torus can be either empty or occupied by an individual of type $\mathit{\lambda }\in (0,\infty )$. An individual of type λ dies with rate one and gives birth at each neighboring empty position with rate λ; moreover, when the birth takes place, the newborn individual is likely to have the same type as the parent but has a small probability of being a mutant. A mutant child of an individual of type λ has type chosen according to a probability kernel. We consider the asymptotic behavior of this process when $\mathit{N}\to \infty$ and, simultaneously, the mutation probability tends to zero fast enough that mutations are sufficiently separated in time so that the amount of time spent on configurations with more than one type becomes negligible. We show that, after a suitable time scaling and deletion of the periods of time spent on configurations with more than one type, the process converges to a Markov jump process on $(0,\infty )$, whose rates we characterize.