Adaptive Dynamics

Adaptive dynamics asks where a continuous trait — a pathogen’s transmission rate, a host’s defense investment, a body size — will settle under natural selection. The idea is to test, for every possible resident population, whether a slightly different rare mutant can invade. Following the sequence of successful invasions traces the trait to its long-term evolutionary endpoints and reveals whether those endpoints are stable or split a population in two.

A pairwise invasibility plot: the plane of resident against mutant virulence shaded where the mutant invades, with the singular strategy marked where the invasion boundary crosses the diagonal.

Invasion fitness

Fix a resident population with trait $x$ at its ecological equilibrium. Introduce a rare mutant with trait $y$ and ask whether it grows. Its long-term per-capita growth rate in the environment set by the resident is the invasion fitness $s_x(y)$ . A mutant spreads when $s_x(y) > 0$ and dies out when $s_x(y) < 0$ , and by construction a resident cannot invade itself, so $s_x(x) = 0$ always. Evolution proceeds as a succession of invasions by nearby mutants, each replacing the resident and shifting the trait a small step.

The selection gradient and singular strategies

Near the resident the direction of change is set by the slope of invasion fitness in the mutant trait, the selection gradient

$D(x) = \left.\frac{\partial s_x(y)}{\partial y}\right|_{y = x}.$

Where $D(x) > 0$ selection pushes the trait up; where $D(x) < 0$ it pushes down. A trait value $x^*$ where the gradient vanishes,

$D(x^*) = 0,$

is an evolutionarily singular strategy — a candidate endpoint where directional selection stops. What happens at $x^*$ depends on two independent second-order properties.

Evolutionary versus convergence stability

Two distinct questions must both be asked at a singular point.

Evolutionary stability (ESS). Can any nearby mutant invade once the population sits at $x^*$ ? If invasion fitness has a maximum in $y$ at $y = x^*$ ,

$\left.\frac{\partial^2 s_x(y)}{\partial y^2}\right|_{y = x = x^*} < 0,$

then no neighbor invades and $x^*$ is uninvadable.

Convergence stability. Does the succession of invasions actually lead toward $x^*$ from nearby residents? This holds when the selection gradient decreases through the singular point,

$\left.\frac{\mathrm{d} D(x)}{\mathrm{d} x}\right|_{x = x^*} < 0.$

The two properties are logically separate. A singular strategy that is both convergence stable and evolutionarily stable is a continuously stable strategy (CSS): selection carries the trait to $x^*$ and holds it there. A point that is convergence stable but not evolutionarily stable is an evolutionary branching point — the population is drawn to $x^*$ , but once there it experiences disruptive selection, splits, and diversifies into two coexisting strains.

The pairwise invasibility plot

The pairwise invasibility plot (PIP) shows the sign of $s_x(y)$ over the plane of resident $x$ against mutant $y$ . The diagonal $y = x$ is always neutral. A singular strategy sits where the boundary between the invasion and exclusion regions crosses the diagonal, and the local pattern of the shaded regions around that crossing reads off its stability: whether the vertical strip above $x^*$ lies inside or outside the invasion region distinguishes a CSS from a branching point.

A worked example

Take the transmission-virulence trade-off from the evolution of virulence. An infection with virulence $\alpha$ transmits at $\beta(\alpha) = a\sqrt{\alpha}$ and clears at total rate $\gamma + \alpha + \mu$ , so its reproduction number is

$R_0(\alpha) = \frac{a\sqrt{\alpha}}{\gamma + \alpha + \mu}.$

A rare mutant strain invades the resident’s disease-free host environment exactly when its own $R_0$ exceeds the resident’s, so invasion fitness has the sign of $R_0(\alpha') - R_0(\alpha)$ and the singular strategy maximizes $R_0$ . Setting $\mathrm{d}R_0/\mathrm{d}\alpha = 0$ with $d = \gamma + \mu$ gives $\alpha + d = 2\alpha$ , so

$\alpha^* = \gamma + \mu.$

Because $R_0(\alpha)$ has a single interior maximum, its second derivative at $\alpha^*$ is negative, so $\alpha^*$ is both evolutionarily stable and convergence stable — a CSS at $\alpha^* = \gamma + \mu$ , with no branching. With $\gamma = 0.5$ and $\mu = 0.1$ the optimum is $\alpha^* = 0.6$ .

In code

R

gamma <- 0.5; mu <- 0.1; a <- 3
R0 <- function(al) a * sqrt(al) / (gamma + al + mu)

grad <- function(x, h = 1e-6) (R0(x + h) - R0(x - h)) / (2 * h)
x_star <- uniroot(grad, c(0.05, 5))$root   # selection gradient = 0
$
d2 <- function(x, h = 1e-4) (R0(x + h) - 2 * R0(x) + R0(x - h)) / h^2
c(x_star = x_star, ess = d2(x_star) < 0)   # ~0.6, TRUE (uninvadable)

Python

import numpy as np
from scipy.optimize import brentq

gamma, mu, a = 0.5, 0.1, 3.0


def R0(al):
    return a * np.sqrt(al) / (gamma + al + mu)


def grad(x, h=1e-6):
    """Selection gradient: slope of invasion fitness at y = x."""
    return (R0(x + h) - R0(x - h)) / (2 * h)


x_star = brentq(grad, 0.05, 5.0)   # singular strategy where gradient vanishes


def curv(x, h=1e-4):
    return (R0(x + h) - 2 * R0(x) + R0(x - h)) / h**2


def dgrad(x, h=1e-3):
    return (grad(x + h) - grad(x - h)) / (2 * h)


ess = curv(x_star) < 0             # fitness maximum: uninvadable
conv = dgrad(x_star) < 0           # gradient decreasing: attracting
kind = "CSS" if ess and conv else "branching point" if conv else "repellor"

print(f"singular strategy alpha* = {x_star:.4f}")
print(f"analytic gamma + mu      = {gamma + mu:.4f}")
print(f"evolutionarily stable    = {ess}")
print(f"convergence stable       = {conv}")
print(f"classification           = {kind}")

singular strategy alpha* = 0.6000
analytic gamma + mu      = 0.6000
evolutionarily stable    = True
convergence stable       = True
classification           = CSS

Julia

using Roots

γ, μ, a = 0.5, 0.1, 3.0
R0(α) = a * sqrt(α) / (γ + α + μ)

grad(x; h = 1e-6) = (R0(x + h) - R0(x - h)) / (2h)
x_star = find_zero(grad, (0.05, 5.0))            # singular strategy

curv(x; h = 1e-4) = (R0(x + h) - 2R0(x) + R0(x - h)) / h^2
ess = curv(x_star) < 0                           # uninvadable
println("alpha* = ", round(x_star, digits = 4), "  ESS = ", ess)

Why it matters

Adaptive dynamics gives infectious-disease evolution a common language for continuous traits: virulence, drug resistance, host range, and antigenic escape are all traits under invasion selection. Its central warning is that being attracting and being uninvadable are separate properties, so a trait can evolve toward a value where selection then turns disruptive and the pathogen population diversifies — the branching-point route to coexisting strains. The same invasion-fitness logic connects to evolutionary game theory, the evolution of cooperation, and the Price equation, and it frames the treatment problem taken up in resistance evolution.