  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 $\mu \le 1$ 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 physicists the paper that introduced the model, 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 asserts that
the transition is discontinuous above the upper critical dimension, which is 4
for oriented percolation.
