The replicator equation in R

evolutionary ecology

evolutionary game theory

population dynamics

ecology tutorial

Simulate evolutionary game dynamics in R with the replicator equation: convergence to an ESS, and the rock-paper-scissors cycles that never settle.

Author

Tidy Ecology

Published

2026-08-05

An evolutionarily stable strategy tells you where a population can rest, but not whether it ever arrives there, nor what it does if no such rest exists. For that you need dynamics. The replicator equation is the workhorse: it grows each strategy in proportion to how much its payoff exceeds the population mean, and it turns a payoff matrix into a trajectory. This post codes it in base R, confirms it converges on an ESS, and then shows the case that has no ESS at all: rock-paper-scissors.

The equation

For strategy frequencies x and a payoff matrix A, the payoff to strategy i is (A x)_i, the mean payoff is x' A x, and each frequency changes as

\[ \dot{x}_i = x_i\left[(\mathbf{A}\mathbf{x})_i - \mathbf{x}^\top\mathbf{A}\mathbf{x}\right]. \]

A strategy above the average grows, one below it shrinks, and the frequencies always sum to one. Here it is as a plain Euler integrator.

replicator <- function(A, x0, dt = 0.005, tmax = 200) {
  x <- x0; out <- matrix(0, tmax / dt, length(x0))
  for (i in seq_len(nrow(out))) {
    fit <- as.numeric(A %*% x)
    x <- x + dt * x * (fit - sum(x * fit))
    x <- x / sum(x)                      # guard against numerical drift off the simplex
    out[i, ] <- x
  }
  out
}

Rock, paper, scissors: no ESS

The lizard Uta stansburiana runs a real rock-paper-scissors among three male types, where each beats one and loses to another (Sinervo and Lively 1996). Encode the cycle as a zero-sum payoff matrix: winning pays +1, losing -1.

A <- matrix(c( 0, -1,  1,
               1,  0, -1,
              -1,  1,  0), 3, byrow = TRUE)
out <- replicator(A, c(0.5, 0.3, 0.2))
cat(sprintf("time-averaged frequencies: %.3f %.3f %.3f (interior point is 1/3 each)\n",
            mean(out[, 1]), mean(out[, 2]), mean(out[, 3])))

time-averaged frequencies: 0.333 0.336 0.331 (interior point is 1/3 each)

cat(sprintf("strategy 1 ranges over %.3f to %.3f: it never settles\n", min(out[, 1]), max(out[, 1])))

strategy 1 ranges over 0.157 to 0.550: it never settles

The interior point where all three are at 1/3 is an equilibrium, but not an ESS: it is a neutrally stable centre, not an attractor. Started off it, the population orbits forever without converging. The time-average of each strategy is 1/3, yet at no single moment is the population actually there. Reporting “the equilibrium is one-third each” would describe a state the population never occupies.

library(ggplot2)
tt <- seq_len(nrow(out)) * 0.005
d <- rbind(data.frame(t = tt, x = out[, 1], s = "rock"),
           data.frame(t = tt, x = out[, 2], s = "paper"),
           data.frame(t = tt, x = out[, 3], s = "scissors"))
ggplot(d, aes(t, x, colour = s)) +
  geom_hline(yintercept = 1/3, colour = "#93a87f", linewidth = 0.4) +
  geom_line(linewidth = 0.7) +
  scale_colour_manual(values = c(rock = "#275139", paper = "#c9b458", scissors = "#b5534e")) +
  labs(x = "Time", y = "Frequency", colour = NULL) +
  theme_minimal(base_size = 12)

Three coloured lines oscillating out of phase over time, none settling to a constant value. — Figure 1: Rock-paper-scissors replicator dynamics. The three frequencies chase each other in sustained cycles and never reach the one-third equilibrium; the mean is 1/3 but the population is never at rest.

When it does settle

Not every game cycles. A game with a genuine ESS, like hawk-dove, sends the replicator straight to its stable mix and holds it there. The difference is structural: hawk-dove has negative frequency dependence pulling toward a point, while rock-paper-scissors has a rotational flow with nothing at the centre to pull toward. The same equation produces both, so you cannot assume an equilibrium is where a population ends up. Whether an interior point attracts, repels, or merely circles is a question about the flow around it (see basins of attraction), not about the equilibrium alone.

An honest limit

The replicator equation assumes an infinite, well-mixed population reproducing in proportion to payoff, with fixed payoffs and no mutation. Each assumption bends real cases. Finite populations add drift that can push a cycling system to the boundary and lose a strategy for good; spatial structure can stabilise cycles that would otherwise wander; and the payoff matrix is, as ever, assumed rather than measured. The equation is a clear lens on which strategies win and whether they settle, not a literal population forecast. The checking post takes up finite populations and payoff sensitivity directly.

References

Hofbauer J, Sigmund K 2003. Bulletin of the American Mathematical Society 40(4):479-519 (10.1090/S0273-0979-03-00988-1)

Nowak MA, Sigmund K 2004. Science 303(5659):793-799 (10.1126/science.1093411)

Sinervo B, Lively CM 1996. Nature 380(6571):240-243 (10.1038/380240a0)

--- title: "The replicator equation in R" description: "Simulate evolutionary game dynamics in R with the replicator equation: convergence to an ESS, and the rock-paper-scissors cycles that never settle." date: "2026-08-05 10:00" categories: [evolutionary ecology, evolutionary game theory, population dynamics, R, ecology tutorial] image: thumbnail.png image-alt: "Three strategy frequencies oscillating in sustained cycles under rock-paper-scissors replicator dynamics" --- An [evolutionarily stable strategy](../hawk-dove-and-the-ess/) tells you where a population can rest, but not whether it ever arrives there, nor what it does if no such rest exists. For that you need dynamics. The replicator equation is the workhorse: it grows each strategy in proportion to how much its payoff exceeds the population mean, and it turns a payoff matrix into a trajectory. This post codes it in base R, confirms it converges on an ESS, and then shows the case that has no ESS at all: rock-paper-scissors. ## The equation For strategy frequencies `x` and a payoff matrix `A`, the payoff to strategy `i` is `(A x)_i`, the mean payoff is `x' A x`, and each frequency changes as $$ \dot{x}_i = x_i\left[(\mathbf{A}\mathbf{x})_i - \mathbf{x}^\top\mathbf{A}\mathbf{x}\right]. $$ A strategy above the average grows, one below it shrinks, and the frequencies always sum to one. Here it is as a plain Euler integrator. ```{r} #| label: replicator replicator <- function(A, x0, dt = 0.005, tmax = 200) { x <- x0; out <- matrix(0, tmax / dt, length(x0)) for (i in seq_len(nrow(out))) { fit <- as.numeric(A %*% x) x <- x + dt * x * (fit - sum(x * fit)) x <- x / sum(x) # guard against numerical drift off the simplex out[i, ] <- x } out } ``` ## Rock, paper, scissors: no ESS The lizard *Uta stansburiana* runs a real rock-paper-scissors among three male types, where each beats one and loses to another (Sinervo and Lively 1996). Encode the cycle as a zero-sum payoff matrix: winning pays `+1`, losing `-1`. ```{r} #| label: rps A <- matrix(c( 0, -1, 1, 1, 0, -1, -1, 1, 0), 3, byrow = TRUE) out <- replicator(A, c(0.5, 0.3, 0.2)) cat(sprintf("time-averaged frequencies: %.3f %.3f %.3f (interior point is 1/3 each)\n", mean(out[, 1]), mean(out[, 2]), mean(out[, 3]))) cat(sprintf("strategy 1 ranges over %.3f to %.3f: it never settles\n", min(out[, 1]), max(out[, 1]))) ``` The interior point where all three are at `1/3` is an equilibrium, but not an ESS: it is a neutrally stable centre, not an attractor. Started off it, the population orbits forever without converging. The time-average of each strategy is `1/3`, yet at no single moment is the population actually there. Reporting "the equilibrium is one-third each" would describe a state the population never occupies. ```{r} #| label: fig-cycles #| fig-cap: "Rock-paper-scissors replicator dynamics. The three frequencies chase each other in sustained cycles and never reach the one-third equilibrium; the mean is 1/3 but the population is never at rest." #| fig-alt: "Three coloured lines oscillating out of phase over time, none settling to a constant value." library(ggplot2) tt <- seq_len(nrow(out)) * 0.005 d <- rbind(data.frame(t = tt, x = out[, 1], s = "rock"), data.frame(t = tt, x = out[, 2], s = "paper"), data.frame(t = tt, x = out[, 3], s = "scissors")) ggplot(d, aes(t, x, colour = s)) + geom_hline(yintercept = 1/3, colour = "#93a87f", linewidth = 0.4) + geom_line(linewidth = 0.7) + scale_colour_manual(values = c(rock = "#275139", paper = "#c9b458", scissors = "#b5534e")) + labs(x = "Time", y = "Frequency", colour = NULL) + theme_minimal(base_size = 12) ``` ## When it does settle Not every game cycles. A game with a genuine ESS, like [hawk-dove](../hawk-dove-and-the-ess/), sends the replicator straight to its stable mix and holds it there. The difference is structural: hawk-dove has negative frequency dependence pulling toward a point, while rock-paper-scissors has a rotational flow with nothing at the centre to pull toward. The same equation produces both, so you cannot assume an equilibrium is where a population ends up. Whether an interior point attracts, repels, or merely circles is a question about the flow around it (see [basins of attraction](../basins-of-attraction/)), not about the equilibrium alone. ## An honest limit The replicator equation assumes an infinite, well-mixed population reproducing in proportion to payoff, with fixed payoffs and no mutation. Each assumption bends real cases. Finite populations add drift that can push a cycling system to the boundary and lose a strategy for good; spatial structure can stabilise cycles that would otherwise wander; and the payoff matrix is, as ever, assumed rather than measured. The equation is a clear lens on which strategies win and whether they settle, not a literal population forecast. The [checking post](../checking-a-game-theory-model/) takes up finite populations and payoff sensitivity directly. ## Related tutorials - [The hawk-dove game and the ESS](../hawk-dove-and-the-ess/) - [Sex ratio evolution and the ESS](../sex-ratio-evolution/) - [The Moran process and fixation probability](../the-moran-process/) - [Basins of attraction](../basins-of-attraction/) ## References Hofbauer J, Sigmund K 2003. Bulletin of the American Mathematical Society 40(4):479-519 (10.1090/S0273-0979-03-00988-1) Nowak MA, Sigmund K 2004. Science 303(5659):793-799 (10.1126/science.1093411) Sinervo B, Lively CM 1996. Nature 380(6571):240-243 (10.1038/380240a0)