Pulled and pushed waves differ in who leads an expanding population

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.

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.

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.

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.

Source: W. van Saarloos · G. Birzu