Asymptotic behaviour of the one-dimensional “rock-paper-scissors” cyclic cellular automaton
Université Paris-Sud - CNRS - CentraleSupélec, Université Paris-Saclay, France
Univ. Bordeaux, Bordeaux INP, CNRS, LaBRI, UMR5800, F-33400 Talence, France
The one-dimensional three-state cyclic cellular automaton is a simple spatial model with three states in a cyclic “rock-paper-scissors” prey-predator relationship. Starting from a random configuration, similar states gather in increasingly large clusters; asymptotically, any finite region is filled with a uniform state that is, after some time, driven out by its predator, each state taking its turn in dominating the region (heteroclinic cycles).
We consider the situation where each site in the initial configuration is chosen independently at random with a different probability for each state. We prove that the asymptotic probability that a state dominates a finite region corresponds to the initial probability of its prey. The proof methods are based on discrete probability tools, mainly particle systems and random walks.
Keywords: cyclic dominance, heteroclinic cycles, cellular automata, self-organisation, random walk
Cyclic dominance is a general term for phenomena where different states (species, strategies, etc.) are in prey-predator relationships that form a cycle: A preys on B preys on C… preys on A. This phenomenon occurs in many natural or theoretical systems, among which a few examples are:
- Population ecology
- Game theory
- Infection models
May and Leonard’s  is the first effort to model the evolution of three species with cyclic dominance, using the standard Lotka-Volterra equations; it is a mean-field approximation, that is, it assumes the population is well-mixed. The system exhibits so-called heteroclinic cycles where each species in turn dominates almost the whole space before being replaced by its predator. Consequently, cyclic dominance has been proposed as a mechanism to explain the coexistence of various strategies or species  (biodiversity), the regular oscillations in population sizes of different species , and some counter-intuitive phenomena such as the “survival of the weakest” . In other contexts, heteroclinic cycles appear to coincide with important concepts: for example, social choice among three cyclically dominant choices can lead to an heteroclinic cycle along the so-called bipartisan set .
Mean-field models do not take into account spatial aspects of the evolution of populations, such as the effect of population structure, mobility, dispersal, local survival, etc. This is why spatial models have been introduced both in ecology [6, 31] and in so-called evolutionary game theory . In both cases agents have a spatial location and can only interact with their neighbours at short range. There is some variety in spatial models:
a lattice in one, two or more dimensions, or a graph with more structure;
discrete or continuous time, synchronous or asynchronous updates;
usually a predator replaces a prey by a copy of itself (replicator dynamics). The model can include empty space, different ranges, threshold effects, invasion probabilities, etc.;
infinite, periodic or fixed boundary conditions, choice of the initial configuration.
In this article, we consider arguably the simplest spatial model for cyclic dominance: the one-dimensional, 3-state cyclic cellular automaton. Each site on the lattice is initially associated a state in . At each (discrete) time step, every site is updated synchronously: if any of the two neighbouring sites contains a predator, it becomes the new state for this site. While the restriction to one dimension may not be ecologically realistic (two-dimensional models being the object of more interest ), it has two benefits. First, its simple spatial structure makes many questions mathematically tractable, while the two-dimensional models have much more complex dynamics with structured interfaces between regions . Second, its dynamics is similar to a interacting particle system with borders progressing at constant speed and annihilating on contact (ballistic annihilation - see Figure 1); this is a subject of independent interest  and many tools have been developed for it .
In all space-time diagrams of this article, the initial configuration is drawn horizontally at the bottom and time goes from bottom to top. States are represented by colours following the convention .
The seminal work of Fisch  focused on the case where each site is independently assigned a random state with uniform probability. He proved a clustering phenomenon: for 3 or 4 states, large monochromatic regions emerge and grow, but each region keeps changing state arbitrarily late (fluctuation, the spatial counterpart of heteroclinic cycle); for 5 states or more, the regions reach a limit size then stay unchanged (fixation). These behaviours, illustrated in Figure 2, are considered as a prime example of self-organisation in a relatively simple model . These results were later refined in terms of cluster growth rate, number of state changes, etc. [9, 10, 20].
The present article focuses on the asymptotic behaviour of the -state cyclic cellular automaton when the initial configuration is chosen independently at random, but with distinct probabilities for each state, breaking the symmetry. It is not hard to see that the same clustering phenomenon as in the uniform case occurs. Our main result (Theorem 6) is that the asymptotic probability for any region to be dominated by a given state corresponds to the initial probability of its prey; this completely determines the limit probability measure. A similar relationship was observed empirically between invasion rates and asymptotic probability in more complex models ; see , Section 7.7 for a detailed account. However, we could not find a conjecture for this phenomenon in such a simple model, and this is the first formal proof of a similar result to our knowledge.
Our approach is based on a correspondence between the time evolution of the borders and some well-chosen random walk, a method that was already used in the study of one-dimensional cellular automata . Compared to previous work, the random walk is not the standard symmetric walk and the probability of a step up or down depends on the current position.
1.1 Symbolic space
For a finite alphabet, define the set of finite patterns (or words) and the set of (one-dimensional) configurations, that is, the set of bi-infinite words over the alphabet . For , denote its length, and for , define the cylinder , with . Cylinders form a clopen basis of for the product topology. A word is a factor of a configuration if for some .
Define the shift function by for any . From a finite pattern define the infinite -periodic configuration by and .
A cellular automaton is a pair where is a continuous function that commutes with (i.e. ). Alternatively is defined by a finite neighbourhood and a local rule in the sense that .
In the figures, we represent the time evolution of cellular automata starting from an initial configuration by a two-dimensional space-time diagram .
The frequency of a finite word in a configuration is defined as:
1.2 Cyclic cellular automata
Definition 1 (-state cyclic cellular automaton).
is the -state cyclic cellular automaton defined on the neighbourhood by the local rule :
All operations concerning -state cyclic automata are assumed to be modulo .
As should be clear from Figure 2, the self-organisation is driven by borders between monochromatic regions behaving as particles. We call particles the factors of length (with ) in a configuration. Each particle moves “from predator to prey”, that is, left if , right if , and stays put otherwise. This motivates the following definitions:
- Positive particles
- Negative particles
- Neutral particles
and we write as a shorthand for : it means that a positive particle occurs at position . Notice that for . Figure 1 illustrate the particle dynamics for .
1.3 Probability measures on
Let be the Borel sigma-algebra of . Denote by the set of probability measures on defined on the sigma-algebra . Since the cylinders form a basis of the product topology on , a measure is entirely characterised by the values .
In this paper, we only consider -invariant probability measures, and therefore write instead of .
- Measures supported by a periodic orbit
For a word , we define the -invariant measure supported by by taking the mean of the Dirac measures along its -orbit:
When is a single letter , we obtain the measure supported on the single monochromatic configuration .
- Bernoulli measure
Let be a vector of real numbers such that for all and . Let be the discrete probability distribution on such that for all (a generalisation of the standard Bernoulli law with outcomes).
The associated Bernoulli measure on is the product measure , that is,
In other words, each cell is drawn in an i.i.d. manner according to . We denote the set of Bernoulli measures on with nonzero parameters .
- Uniform measure
In particular, if we take for all in the previous definition, we obtain the uniform (Bernoulli) measure .
The image measure of by a cellular automaton is defined as for all . This defines an action .
We endow with the weak topology: for a sequence and a measure , we have if, and only if:
This topology makes continuous and is compact.
A measure is -ergodic if, for every subset such that -almost everywhere, we have or . The set of -ergodic measures is denoted . In particular, all examples above are -ergodic and the image of a -ergodic measure under the action of a cellular automaton is -ergodic.
As an example of a non--ergodic measure, consider the average of two Dirac measures (the set is -invariant and has measure ).
We make use of the following corollary to Birkhoff’s theorem:
Let and . Then:
where means for -almost all (that is, for all in some set of measure ).
2 Known and new results
The first main result on one-dimensional cyclic cellular automata is the following. It consider the values of the sequence for an arbitrary site (here ) when iterating on a uniform random configuration.
Theorem 3 (Fisch , Theorem 1).
Draw an initial configuration according to be the uniform Bernoulli measure on , and consider the sequence . Then:
If , then ( fluctuates);
If , then ( fixates).
Since changes of values corresponds to times when a particle or crosses the column, this result can be interpreted in terms of limit measures. For , some particles (“walls”) survive asymptotically () and delimit walled areas where the remaining moving particles or cannot enter; for , and moving particles cross each column infinitely often. This result can be intuited on Figure 2.
Notice that the previous result only applies when the initial measure is uniform. The following result follows from , Corollary 1; it is weaker but applies on the much more general setting of -ergodic measures:
Let be any -ergodic measure on . Then at least two of the following are true:
For Bernoulli measures, the state of the art is summed up in the following proposition:
If is a Bernoulli measure, then and In particular, if , any limit point of is a convex combination of the measures .
If furthermore the Bernoulli uniform measure, the unique limit point of is for both cases .
In the case where is a Bernoulli measure, or more generally a measure invariant by the mirror involution , the only possible non-zero case is . Indeed, since and the mirror operation sends to and conversely, we have .
For , since , there is asymptotically no particle at all, so all limit points must be some convex combination of the measures .
If furthermore the Bernoulli uniform measure, Theorem 3 gives us in the case as well. Since this measure is invariant by the state-transposing operation and , the unique limit point is for both cases . ∎
The previous results, fluctuation in particular, can be interpreted in terms of heteroclinic cycles. For -almost every configuration , no state ever dominates the whole space in the sense that (by Corollary 1) for every state (we use the fact that the image under of a -ergodic measure is -ergodic).
However, Proposition 5 implies that, for any fixed window and -almost every , is monochromatic (in topological terms, it is close to one of the , ) except for some sequence of times of zero density. Theorem 3 further shows that does not converge to one of the as , but that the window keeps changing state (as a particle crosses the central column), less and less often, letting each state dominate the central window in turn. In this sense, the 3-state cyclic cellular automaton exhibits heteroclinic cycles in local regions.
Our main new result determines the unique limit point for non-uniform Bernoulli measures:
Theorem 6 (Main result).
Let be a Bernoulli measure on with nonzero parameters . Then:
Theorem 6 can be interpreted as follows. Draw an initial configuration according to a Bernoulli measure with nonzero parameters , and consider a fixed arbitrary window . By Proposition 5, the probability that contains at least two different states (i.e. a particle) tends to . Theorem 6 further shows that the probability that for tends to as tends to infinity.
Remarkably, the parameters of the limit measure are a simple cyclic permutation of the parameters of the initial Bernoulli measure: each state reaches asymptotically the initial frequency of its “prey” . This is illustrated on Figure 3.
3 Proof of the main result
In this section, we use the one-sided version of to simplify proofs:
Definition 7 (One-sided cyclic CA).
is the one-sided -state cyclic cellular automaton defined on the neighbourhood by the local rule:
A (computer assisted) proof by enumeration of all factors of length , shows that
Hence proving Theorem 6 on implies a similar result on .
The proof proceeds in 4 steps:
- Section 3.1
where we associate a random walk to each configuration and relate the properties of this random walk to the orbit of the configuration under ;
- Section 3.2
where we translate Theorem 6 on the random walk and establish the objects that will be relevant to the proof.
- Section 3.3
where we introduce a second random walk “embedded” in the previous one, which is symmetric (hence easier to analyse) and captures its large-scale behaviour.
- Section 3.4
where we bring back the results from the embedded walk to the initial walk and bring all tools together to conclude the proof.
3.1 Random walk associated with a configuration
In this section, we introduce tools to turn the study of the dynamics of the -state cyclic automaton, in particular of (defined above), into the study of some random walk built from the initial configuration .
To a configuration we associate a random walk on such that and made up of steps in as follows:
for all , is the value in such that ,
and for , is the value in such that .
and this encoding is an injection.
Figure 4 provides an example of this encoding (black configuration to black walk).
We denote by the positions of the walk on from time to time . Notice that we call time in the context of the random walk what corresponds to space in the configuration , which is different from the time corresponding to the iteration of cellular automaton. Context should make clear which notion of time we refer to.
The main interest of this correspondence is to deduce the state of a cell after iterations from the maximal height in the first steps of the walk associated to the initial configuration :
For , we have
We will prove that the iterations of keep for the following invariant:
When this invariant is expressed for and , we deduce the expected identity:
We prove this invariant in the case and any . The cases follow by replacing and .
We describe how to obtain from by a 3-step transformation: . Each of these steps, illustrated in Figure 4, preserves the invariant.
By definition, for , . We notice that cases where becomes are exactly the steps in the walk (factors , or in ).
For , define if and otherwise. In addition . Notice that is also a walk on made up of steps . The maximal height is preserved since any visit at maximal height in may not be followed by a step.
The only case where is when and . In this case, for , define and otherwise. The maximal height may be decreased by , but it is preserved .
We remove the last position in the walk to obtain . This preserves the maximal height: if was the first visit to the maximal height, the first step ensures that . Therefore , so can not be the first occurrence of the maximal height and can be safely removed.
3.2 Analysing the random walk
Recall that the measure on the initial configuration is the Bernoulli measure of parameters . From this and the bijection with walks on we forget its relationship with to study it for itself as a random variable, directly sampling as follows (each choice being independent):
with probability for ,
then for all , with probability , is the value in such that for ,
and for , with probability , is the value in such that for .
Similarly, we can sample the factor by assuming by convention that to ensure that . Then the only rule is with probability , independently from other choices.
In the proofs, we will need such walks starting from an arbitrary . Formally, define a random walk on of length and starting from as:
where are i.i.d. random variables in for all , and for all .
Theorem 10 (Main result of this section).
For any and any ,
We first consider the case (and ), i.e. ; the other cases will follow.
Our proof proceeds by conditioning this event to the length of the 3-tail (defined below), and describing the probability in terms of the value of other probabilities (also defined below).
Definition 11 (Record, tail).
A record occurs at time in the random walk if ; notice a walk can have multiple records sharing the same value .
The -tail of is the suffix , where is the last occurrence of a record divisible by ; the -tail for may not exist.
We make use of the -tail and the -tail in the proof. The length of the -tail is usually denoted by .
is the set of walks on steps which start from and remain on values strictly lower than .
is the probability that a random walk belongs to .
Proposition 12 (Description conditioned by -tail).
For any and any possible -tail length , we have: