The symbiotic contact process

Abstract

We consider a contact process on Z d with two species that interact in a symbiotic manner. Each site can either be vacant or occupied by individuals of species A and/or B. Multiple occupancy by the same species at a single site is prohibited. The name symbiotic comes from the fact that if only one species is present at a site then that particle dies with rate 1 but if both species are present then the death rate is reduced to µ ≤ 1 for each particle at that site. We show the critical birth rate λc(µ) for weak survival is of order √ µ as µ → 0. Mean-field calculations predict that when µ < 1/2 there is a discontinuous transition as λ is varied. In contrast, we show that, in any dimension, the phase transition is continuous. To be fair to the physicists that introduced the model, [27], the authors say that the symbiotic contact process is in the directed percolation universality class and hence has a continuous transition. However, a 2018 paper, [30], asserts that the transition is discontinuous above the upper critical dimension, which is 4 for oriented percolation.