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The symbiotic contact process. (English) Zbl 1441.60077

Summary: We consider a contact process on \(\mathbb{Z}^d\) 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 \(\mu\leq 1\) for each particle at that site. We show the critical birth rate \(\lambda_c(\mu)\) for weak survival is of order \(\sqrt{\mu}\) as \(\mu \to 0\). Mean-field calculations predict that when \(\mu<1/2\) there is a discontinuous transition as \(\lambda\) 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, [M. M. de Olivera, et. al., “Symbiotic two species contact process”, Preprint, arXiv:1205.5974], 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, [C. I. N. Sampaio Filho, et al., “The symbiotic contact process: phase transitions, hysteresis cycles, and bistability”, Preprint, arXiv:1806.11167], asserts that the transition is discontinuous above the upper critical dimension, which is 4 for oriented percolation.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
92D25 Population dynamics (general)
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References:

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