# Where to learn about whether a travelling wave solution to the reaction diffusion equation is a pushed or pulled wave?

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 $$$$\label{eq:1} \frac{\partial n}{\partial t}= D\frac{\partial^2 n}{\partial x^2} + nf(n)$$$$

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?

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