# Asymptotic behaviour of cyclic automata

Time: 14:00 -- Location: LRI, 445

summary: Cyclic dominance describes models where different states (species, strategies...) are in some
cyclic prey-predator relationship: for example, rock-paper-scissors. This occurs in many contexts such
as ecological systems, evolutionary games on graphs, etc. Many models exhibit heteroclinic cycles where
one state dominates almost the whole space before being replaced by the next state in turn. In spatial
models, this appears often as large moving clusters that dominate locally before being replaced.

We consider the simplest spatial cyclic dominance model, which is the cyclic cellular automata with 3 states
with synchronous update. When every state has a different initial density, we prove that the asymptotic
probability of each state is the intial density of its prey ("you become what you eat"). Such phenomena
and its paradoxical consequences (survival of the weakest) have been observed empirically in more complex
systems, and we give a formal proof in this simple context using particle systems and random walks.
I will also discuss ongoing work about what kind of prey-predator graphs can be tackled with our approach.

associated preprint + ongoing work with Yvan le Borgne.