--- title: "Pulled and Pushed Waves: Where an Invading Population Comes From Interactive Lab | Sandbox Physics" description: "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." canonical_url: "https://sandboxphysics.com/labs/pulled-pushed-fronts/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/pulled-pushed-fronts/" last_modified: "2026-10-30" --- 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. > Interactive simulation: open this page in a web browser to change the parameters and run the model. 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 **Background** A 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 matters** Many 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 misconception** Any Allee effect makes the wave pushed. **Better mental model** For $1