Cyclic dominance is a phenomenon where different states (species,
strategies...) in a prey-predator relationship dominate each other in a
cyclic fashion: A preys on B preys on C preys on A. It has been observed
in real ecological systems, evolutionary game theory, etc.

Modelling cyclic dominance through Lotka-Volterra-type models yields
heteroclinic cycles, where the states take turn in dominating almost the
whole space before being unseated by the next state. Models with a
spatial component yield similar phenomena, where similar states cluster
together and dominate local regions before being driven out.

In this work, we consider the simplest spatial model for cyclic
dominance - one dimension, 3 states, synchronous and deterministic
updates (cellular automata) - and choose different initial densities for
each state. As the states cluster together, we prove that the asymptotic
probability that each state dominates corresponds to the initial density
of its prey ("You become what you eat"). Similar phenomena had been
observed empirically but not in such a simple model, and this is the
first formal proof to our knowledge. The main tools are based on
discrete probability, in particular particle systems and random walks.

This is a joint work with Yvan le Borgne (LaBRI, Université de Bordeaux).