## Electronic Journal of Probability

### The symbiotic contact process

#### Abstract

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 $\mu \le 1$ 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, [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.

#### Article information

Source
Electron. J. Probab., Volume 25 (2020), paper no. 4, 21 pp.

Dates
Accepted: 8 December 2019
First available in Project Euclid: 16 January 2020

https://projecteuclid.org/euclid.ejp/1579143695

Digital Object Identifier
doi:10.1214/19-EJP402

#### Citation

Durrett, Rick; Yao, Dong. The symbiotic contact process. Electron. J. Probab. 25 (2020), paper no. 4, 21 pp. doi:10.1214/19-EJP402. https://projecteuclid.org/euclid.ejp/1579143695

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