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I'm trying to understand pushed and pulled waves as seen in many biology articles such as:

  • Gene Surfing in Expanding Populations by Hallatschek

  • Spatial gene drives and pushed genetic waves by Tanaka et al

  • Evolution transforms pushed waves into pulled waves by Erm et al

  • Range expansions transition from pulled to pushed waves as growth becomes more cooperative in an experimental microbial population by Ghandi et al

Specifically, I would like to analyse travelling wave solutions $ \hat{n}(\xi) = \hat{n}(x-ct) = n(x, t) $ to the reaction diffusion equation \begin{equation} \label{eq:1} \frac{\partial n}{\partial t}= D\frac{\partial^2 n}{\partial x^2} + nf(n) \end{equation}

and be able to classify a solution as pushed or pulled depending on $f(n)$ or the wave speed $c$. In biological terms, a pulled wave occurs when the Allee effect is small (source: Range expansion transitions...), meaning that at small population sizes the disadvantage to the proliferation rate $\frac{dN}{dt}=F(N)$ is weak.

My guess is that if $F'(0)<0$ this leads to a pushed wave and if $F'(0)\geq 0$ it leads to a pulled wave. However I need a source for this, and also a way to adapt this to a reaction-diffusion travelling wave solutions framework.

Are there any textbooks/papers/articles that define pushed/pulled waves mathematically?

There is Front propagation into unstable states by Wim van Saarloos which is very long and quite advanced, is there anything more beginner friendly?

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The supplemental information for "Fluctuations uncover a distinct class of traveling waves" by Gabriel Birzu is a very good overview.

Also, the definition for pulled wave is a nonlinear front whose asymptotic propagation speed equals $v^*$. A pushed wave is a nonlinear front whose asymptotic speed $v^\dagger$ is larger than $v^∗$.

For $$\frac{\partial n}{\partial t}=D\frac{\partial ^2 n}{\partial ^2 t} +r(n)n $$

we have $$v^*=2 \sqrt{Dr(0)}.$$

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