FR24 · Evolution and ecology

Pulled and Pushed Waves: Where an Invading Population Comes From

Does a spreading population descend from the few pioneers at its tip, or from the crowd behind them? It depends on whether growth rewards crowding. With ordinary logistic growth the wave is pulled: the sparse tip sets the speed and almost all future ancestors come from it, so diversity is lost fast. When individuals grow better in groups, the wave becomes pushed, runs faster than the Fisher speed, and its ancestry spreads into the crowd behind the tip.

Make growth cooperative and watch a stepping-stone front turn from pulled to pushed: it outruns the Fisher speed, its ancestors move inside and diversity lasts longer.

Measured front speed $\hat v=0.0456\ \text{demes/gen}$

Speed ratio and wave class $v/v_{\mathrm F}=1\ \text{pulled}$

Diversity left in the front $H=0.304$

Tagged region: expected and observed wins $\pi=0.998$

Make growth cooperative

Cooperativity of growth

0

B

Carrying capacity per deme

N=1000

\log_{10}N

Tagged region starts at

6

\zeta_{\mathrm{tag}}

Drag the gold marker under the demes to choose where the tagged region starts, then raise the cooperativity.

Sampling and playback

Generations per frame

12

\dot g

Model, method and sources

$p_i=\frac{\tfrac m2 n_i(x-1)+(1-m)n_i(x)+\tfrac m2 n_i(x+1)}{N\,[1-r(\tilde n)]},\quad r(n)=g_0\Bigl(1-\frac nN\Bigr)\Bigl(1+B\frac nN\Bigr)$

The update is the multinomial model of Birzu, Hallatschek and Korolev (2018, Methods Eq. 11) with their growth law (Eq. 3), a migration probability of 0.25 and a low-density growth rate of 0.01. The box of 120 demes moves with the front. The expected win probability of the tagged region uses the mean-field profile and the ancestry weight of the SI (Eq. S28), cut off where a deme would hold fewer than one individual. For pulled waves that cutoff is a choice, and the paper notes fluctuating fronts need a different one.

Birzu, Hallatschek & Korolev (2018), arXiv:1709.01601v2 · van Saarloos, Front propagation into unstable states (2003), arXiv:cond-mat/0308540v2

Derive the pulled and pushed speeds

Model assumptions

A row of demes, each with a fixed carrying capacity, follows the discrete model of Birzu, Hallatschek and Korolev (2018): a quarter of every deme moves to its neighbours each generation, the population grows logistically with a cooperative boost at intermediate density, and the next generation is drawn at random. Two neutral alleles track diversity and a heritable tag tracks ancestry. The pulled lattice speed, the continuum speeds and the mean-field ancestry weight are computed from the same equations; the Brunet–Derrida speed for a finite population is an asymptotic formula. Everything is simulated or computed; no field data are shown.

Front speed against the pulled and pushed predictions

Genetic diversity left in the front

Where future ancestors live in the front

More numbers and how they are computed

Pulled speed on this lattice: $0$
Speed with no number noise: $0$
Continuum and Fisher speeds: $0$
Brunet–Derrida finite-population speed: $0$
Predicted diversity loss rate: $0$
Tagged share of the front now: $0$
Individuals per deme at capacity: $0$
Generations simulated: $0$

Speeds are fitted to the total population over a sliding window. The ancestry weight uses the averaged noise-free profile and is cut off where a deme holds fewer than one individual; for pulled waves that cutoff decides the answer. The exported file lists the seed, the settings, the profile and every tag round.

Physics tutorial

Pulled by the tip or pushed by the crowd

BackgroundA species spreading into empty habitat forms a travelling wave of population density. Fisher and Kolmogorov showed that with logistic growth the wave moves at a speed set only by dispersal and the growth rate of the sparsest individuals at its tip.

Why it mattersMany organisms grow better in groups, an Allee effect. Then the crowd behind the tip can drive the wave instead. Which region drives it decides how fast the species spreads, where its future ancestors live and how quickly it loses genetic diversity.

Start with the essentials

Focus question: Does a spreading population descend from the few pioneers at its tip, or from the crowd behind them?
One-sentence intuition: Below $B=2$ the wave is pulled: it moves at $v_{\mathrm F}=2\sqrt{Dg_0}$ and its ancestors sit at the tip. Above, it is pushed and faster; past $v/v_{\mathrm F}=3/(2\sqrt2)$ the ancestry and the loss of diversity are spread over the whole front.

Core mathematical model

Growth with cooperativity

r(n)=g_0\Bigl(1-\frac nN\Bigr)\Bigl(1+B\frac nN\Bigr)

For $B\le1$ the per-capita growth is largest when the population is sparse. Larger $B$ rewards density, a weak Allee effect.

Pulled and pushed speeds

v=\begin{cases}2\sqrt{Dg_0}, & B\le2\\[2pt] \sqrt{\dfrac{g_0DB}{2}}\Bigl(1+\dfrac2B\Bigr), & B>2\end{cases}

The continuum result for this growth law, with $D=m/2$ . The lattice front in the Lab runs slightly faster than the pushed formula because the demes are discrete.

Where ancestors live

S(\zeta)\propto n^2(\zeta)\,e^{v\zeta/D}

The chance that an individual at $\zeta$ becomes the ancestor of the future front. For pulled waves it stays flat all the way to the tip, so the cutoff at the tip decides everything; for pushed waves it peaks inside the front.

Common difficulties

Faster is not the same as pushed

Typical misconceptionAny Allee effect makes the wave pushed.

Better mental modelFor $1<B\le2$ growth already favours density, yet the wave is still pulled and moves at the Fisher speed. The switch happens at $B=2$ .

Run the experiment

01
Tag the tip of a pulled wave
Keep the pulled preset and watch the gold tag that marks everyone ahead of the marker.
What to observe: The tagged tip usually takes over the whole front, even though it starts as a small minority.
02
Push the wave
Choose Fully pushed and compare the speed chart with the dashed lines.
What to observe: The front runs well above the pulled speed, and the tagged tip now loses most rounds.
03
Shrink the population
Choose Few individuals slow the front.
What to observe: The pulled front moves clearly slower than the lattice speed, close to the Brunet–Derrida estimate.

Pulled and Pushed Waves: Where an Invading Population Comes From

Pulled by the tip or pushed by the crowd

Core mathematical model

Growth with cooperativity

Pulled and pushed speeds

Where ancestors live

Common difficulties

Faster is not the same as pushed

Run the experiment

Tag the tip of a pulled wave

Push the wave

Shrink the population

Related experiments and articles

From random sampling to pulled and pushed waves