**Asymptotic behaviour of the 3-state cyclic cellular automaton**
Benjamin Hellouin de Menibus

*11 April 2019, 13:30*
Salle/Bat : 465/PCRI-N

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**Résumé :**
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).

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