  The ubiquitous existence of microbial communities marks the importance of
understanding how species interact within the community to coexist and their
spatial organization. We study a two-species mutualistic cross-feeding model
through a stochastic cellular automaton on a square lattice using kinetic Monte
Carlo simulation. Our model encapsulates the essential dynamic processes such
as cell growth, and nutrient excretion, diffusion and uptake. Focusing on the
interplay among nutrient diffusion and individual cell division, we discover
three general classes of colony morphology: co-existing sectors, co-existing
spirals, and engulfment. When the cross-feeding nutrient is widely available,
either through high excretion or fast diffusion, a stable circular colony with
alternating species sector emerges. When the consumer cells rely on being
spatially close to the producers, we observe a stable spiral. We also see one
species being engulfed by the other when species interfaces merge due to
stochastic fluctuation. By tuning the diffusion rate and the growth rate, we
are able to gain quantitative insights into the structures of the sectors and
the spirals.
