---
title: "Pulled and pushed waves differ in who leads an expanding population"
description: "Derive the Fisher speed, see how cooperative growth makes a front pushed, find where the ancestors sit, and test speeds against lattice and continuum solutions."
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language: "en"
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last_modified: "2026-10-30"
---

[Evolution as statistical physics](https://sandboxphysics.com/stories/evolution-statistical-physics/)

L3 · Chapter 6

# Pulled and pushed waves differ in who leads an expanding population

Sandbox Physics · 2026-10-30

A species spreading into empty land advances like a wave. If lone pioneers at the tip grow fastest, the tip pulls the wave along and the future population descends from a handful of pioneers. If individuals need company to grow well, the tip lags and the crowd behind pushes the wave. The difference decides how fast the wave moves and how much genetic diversity it keeps.

## Before you begin

Chapter 4: why drift is strong at an expanding edge. Partial derivatives and the diffusion equation at the level of a first course. Exponential solutions of a linear differential equation with constant coefficients.

- Derive the Fisher speed from the linearised equation at the tip, and state the exact speed of a pushed wave for cooperative growth.
- Classify a wave as pulled, semi-pushed or fully pushed from the ratio of its speed to the Fisher speed, and test the speeds against lattice and continuum solutions.

## A wave of growth and dispersal

Let $n(x,t)$ be the population density along a line. Individuals disperse with diffusion constant $D$ and grow at the per-capita rate $r(n)$. The growth law on the right has carrying capacity $N$, growth rate $g_0$ at low density, and a cooperativity $B$. For $B=0$ growth is logistic and fastest when the population is sparse. For large $B$ individuals grow faster in company, which ecologists call an Allee effect.

The Lab runs the discrete version of this model used by Birzu, Hallatschek and Korolev: a row of demes, each holding up to about $N$ individuals. Every generation a quarter of each deme migrates to its two neighbours, the local density grows by the rule above, and the next generation is resampled at random. The random sampling is the drift of chapters 1 to 3, now acting deme by deme.

$$
\frac{\partial n}{\partial t}=D\frac{\partial^2 n}{\partial x^2}+r(n)\,n,\qquad r(n)=g_0\Bigl(1-\frac{n}{N}\Bigr)\Bigl(1+B\frac{n}{N}\Bigr)
$$

The reaction-diffusion equation and the cooperative growth law.

Source: G. Birzu

## The tip sets a minimum speed

Far ahead of the front the density is tiny, so $r(n)\approx g_0$ and the equation becomes linear. In the frame $\zeta=x-vt$ moving with the front, try $\rho\sim e^{-k\zeta}$. Substituting gives $Dk^2-vk+g_0=0$, which has a real decay rate only if $v\ge2\sqrt{Dg_0}$. This minimum is the Fisher speed $v_{\mathrm F}$.

For $B\le2$ the front moves at exactly $v_{\mathrm F}$: it is pulled by its tip, and the bulk does not matter. For $B>2$ growth at intermediate density is strong enough to push the front faster than the tip alone could. The exact speed for this growth law is given by the second formula.

$$
v_{\mathrm F}=2\sqrt{Dg_0}\quad(B\le2),\qquad v=\sqrt{\frac{g_0DB}{2}}\Bigl(1+\frac{2}{B}\Bigr)\quad(B>2)
$$

Front speed for the cooperative growth law: pulled for $B\le2$, pushed for larger $B$.

Source: G. Birzu · W. van Saarloos

## Three classes, not two

Birzu and co-workers found that pushed waves come in two kinds. Their perturbative formulas for drift and front wandering contain integrals over the front that converge only if the density at the tip falls fast enough. Using the decay rate $k$ of the tip, the integrals converge when $v/v_{\mathrm F}$ exceeds $3/(2\sqrt2)$. That point separates fully pushed waves, whose fluctuations scale as $1/N$, from semi-pushed waves, where they fall more slowly. For this growth law the boundary is $B=4$.

The figure checks the speeds. The noise-free lattice model follows the continuum curve to within $0.6$ per cent at every $B$, and a finite-difference solution of the continuum equation, extrapolated to zero grid spacing, matches the exact pushed speeds to $2\times10^{-5}$ relative error.

$$
\rho\sim e^{-k\zeta},\qquad k=\frac{v}{2D}\Bigl(1+\sqrt{1-v_{\mathrm F}^2/v^2}\Bigr),\qquad \text{fully pushed}\iff \frac{v}{v_{\mathrm F}}\ge\frac{3}{2\sqrt2}
$$

Decay of the front tip and the condition for a fully pushed wave.

Front speed divided by the Fisher speed as cooperation grows. Line: the continuum formula. Dots: noise-free lattice fronts from the Lab model, each divided by the lattice $v^*$. The front is pulled up to $B=2$, semi-pushed up to $B=4$, where the ratio reaches $3/(2\sqrt2)$, and fully pushed beyond.

Source: G. Birzu

## Worked example: a wave just past the Fisher speed

Problem. The Lab uses $D=0.125$ demes squared per generation and $g_0=0.01$ per generation. Find the speed for $B=3$, its class, and compare with the simulations.

Speeds. $v_{\mathrm F}=2\sqrt{0.125\times0.01}=0.0707$. For $B=3$, $v=\sqrt{0.01\times0.125\times1.5}\times(1+2/3)=0.0722$. The ratio is $1.021$, above $1$ and below $1.061$, so the wave is semi-pushed.

Tip. $v_{\mathrm F}^2/v^2=0.96$, so $k=0.289\times1.2=0.346$ per deme. The integrand in the drift formula grows towards the tip as $e^{(2v/D-3k)\zeta}$, and $2v/D-3k=1.155-1.039>0$, so the integral diverges. For $B=4$ the same exponent is exactly zero.

Check. The noise-free lattice front moves at $0.07242$ demes per generation, $0.34$ per cent faster than the continuum value; the lattice is not exactly the continuum equation. The continuum solver gives $0.0721684$ against the exact $0.0721688$.

Answer. $v=0.0722$, two per cent above the Fisher speed, and semi-pushed.

Source: G. Birzu

## Where the ancestors sit

In a pulled wave, the individuals at the far tip grow fastest and run into empty space first, so almost all future individuals descend from them. In a pushed wave, the tip grows slowly and ancestry spreads over the bulk of the front. The mean-field theory weighs each position by $\rho^2e^{v\zeta/D}$.

The Lab tests this by tagging every individual from a chosen position to the tip and counting how often the tag takes over, in $80$ seeded rounds for each case with $N=1000$. In the fully pushed wave at $B=10$, a tag starting at the front position won $69\pm5$ per cent of rounds against $71$ per cent expected. In the other three cases the tag won less often than the weight predicts: $29\pm5$ against $40$ per cent from two demes further forward, $49\pm6$ against $69$ in the semi-pushed wave at $B=3$ from four demes ahead, and $91\pm3$ against $99.8$ in the pulled wave from six demes ahead. The weight ignores fluctuations at the tip, which matter most in exactly these cases.

Source: G. Birzu

## Few individuals slow a pulled front

A pulled front depends on a few individuals at its tip, so it feels the discreteness of individuals strongly. Two results from van Saarloos’s review apply. A deterministic pulled front approaches its asymptotic speed $v^*$ slowly, as the first formula shows. A noisy front with $N$ individuals per deme is slower by an amount that shrinks only as one over the square of $\ln N$, the Brunet–Derrida correction.

For the Lab lattice $v^*=0.07095$. The noise-free front fitted over its second half gives $0.07061$; adding back the slow approach gives $0.07097$. With noise, $N=1000$ gives $0.0614\pm0.0012$ against $0.0636$ from Brunet–Derrida, and $N=10^4$ gives $0.0655\pm0.0005$ against $0.0668$. The formula is the leading term of an expansion in $1/\ln N$, and the simulations sit a few per cent below it. A fully pushed front at $B=10$ with $N=1000$ runs at $0.09532\pm0.00019$, about a tenth of a per cent below its noise-free speed.

$$
v(t)\approx v^*-\frac{3}{2\lambda^*t},\qquad v_N\approx v^*-\frac{\pi^2\lambda^*D^*}{\ln^2N}
$$

Slow approach of a deterministic pulled front, and the speed of a noisy pulled front with $N$ individuals per deme.

Source: W. van Saarloos · G. Birzu

## Try it in the Lab

### [Find who leads the front](https://sandboxphysics.com/labs/pulled-pushed-fronts/)

1. Choose Pulled: the tip wins. Predict whether a tagged region that starts behind the front position can take over, then move the tagged region back and watch several rounds.
2. Raise the cooperativity from zero to ten in steps. Predict where the speed ratio first rises above one and where the wave becomes fully pushed, then read the wave class.
3. Choose Few individuals slow the front and lower the carrying capacity. Predict whether a pulled or a fully pushed front slows more, then compare the measured speed for both.

## Exercises with explained answers

1. Show that $Dk^2-vk+g_0=0$ has a real root only for $v\ge2\sqrt{Dg_0}$, and find $k$ at that speed.

   Show the answer

   The discriminant $v^2-4Dg_0$ must be non-negative, so $v\ge2\sqrt{Dg_0}$. At the minimum the double root is $k=v/(2D)=\sqrt{g_0/D}$, which is $0.283$ per deme for the Lab parameters.

2. Compute the speed ratio $v/v_{\mathrm F}=\sqrt{B/8}\,(1+2/B)$ for $B=4$ and $B=10$, and classify each wave.

   Show the answer

   For $B=4$, $\sqrt{0.5}\times1.5=1.061=3/(2\sqrt2)$, exactly the boundary, so fully pushed. For $B=10$, $\sqrt{1.25}\times1.2=1.342$, fully pushed.

3. By how much does the Brunet–Derrida correction $\pi^2\lambda^*D^*/\ln^2N$ shrink when $N$ goes from $10^3$ to $10^6$? Compare with a correction proportional to $1/N$.

   Show the answer

   $\ln^2N$ grows from $47.7$ to $190.9$, so the correction falls by a factor of four. A $1/N$ correction would fall by a factor of a thousand. That slow decay is why pulled fronts stay sensitive to noise even in huge populations.

[Previous: Does a big sector mean selection? Opening angles and luck in colonies](https://sandboxphysics.com/stories/evolution-statistical-physics/does-a-big-sector-mean-selection/) [Next: Reading “Neither pulled nor pushed” and checking its drift claim](https://sandboxphysics.com/stories/evolution-statistical-physics/reading-neither-pulled-nor-pushed/)

## Sources and what we recomputed

1. [G. Birzu, O. Hallatschek and K. S. Korolev, Neither pulled nor pushed: genetic drift and front wandering uncover a new class of reaction-diffusion waves, arXiv:1709.01601v2 (2018)](https://arxiv.org/abs/1709.01601v2)

   Version 2, full text. Eq. (2): the Fisher speed. Eq. (3): the cooperative growth law. Eq. (5): perturbative rate of diversity loss. Eqs. (6)-(7): the leading-edge decay and the speed ratio $3/(2\sqrt2)$ that separates semi-pushed from fully pushed waves. Fig. 5: diversity loss in the three classes. Methods, Eq. (11): the simulation model. SI Eq. (S28): ancestry weight; Eq. (S122): closed form of the loss rate; section Duration of simulations and data collection, Eqs. (S149)-(S150).

2. [W. van Saarloos, Front propagation into unstable states, arXiv:cond-mat/0308540v2 (2003), Physics Reports](https://arxiv.org/abs/cond-mat/0308540v2)

   Version 2, full text. Section 2.2, Eqs. (24)-(27): the linear spreading speed $v^*$ and steepness $\lambda^*$ from the dispersion relation. Section 2.9.1, Eq. (78): the universal slow approach $v(t)=v^*-3/(2\lambda^*t)$ of a pulled front. Section 7.1, Eqs. (241)-(246): the cutoff correction to the pulled speed in particle models, proportional to $1/\ln^2N$; Eq. (248): the front diffusion constant falling as $1/\ln^3N$.