I am reading the classical reaction-diffusion paper by Alan Turing: The chemical basis of morphogenesis (1952). The paper is in the following:
https://www.mit.edu/~kardar/teaching/IITS/lectures/lec8/TuringPhilTransB1952.pdf
In this paper, Turing derived the condition (in terms of chemical constants $a,b,c,d,\mu,\nu$) such that a finite wavelength pattern occurs (also called "Turing pattern"). The condition is (Eq. 9.4a):
$bc<0$ and $ \frac{4\sqrt{\mu\nu}}{\mu+\nu} < \frac{d-a}{\sqrt{-bc}} < \frac{\mu+\nu}{\sqrt{\mu \nu}}$
In the paper, Turing provided this criteria with no detail proof -- perhaps it is very simple -- but I am still stuck.
I understand the reason for $bc<0$ is required, otherwise the dominant eigen-mode will be longest one (correspond to system size $N$). This can be seen from the eigenvalue with wavelength $s$ has formula $p_s = \frac{1}{2} (B_s+\sqrt{B_s^2-4\Delta_s})$, where $B_s \equiv (a+d)-(\mu+\nu)U_s$, $\Delta_s \equiv (a-\mu U_s)(d-\nu U_s)-bc$, and $U_s \sim \sin^2(\pi s/N) \geq 0$. If $bc>0$, $p_0$ will be the maximal eigenvalue and no Turing pattern will form.
I also understand the part $\frac{d-a}{\sqrt{-bc}} < \frac{\mu+\nu}{\sqrt{\mu \nu}}$. This constant $\frac{\mu+\nu}{\sqrt{\mu \nu}}$ occurs when Eq. 9.3 having $U_s > 0$ for a specific wavelength $1<s<N$.
What I do not understand is the part $ \frac{4\sqrt{\mu\nu}}{\mu+\nu} < \frac{d-a}{\sqrt{-bc}}$.
I know this question is quite technical. Yet, given Turing's paper is so influential, I hope someone had also went through this "take home problem" by Turing. Any hint or reference will be appreciated. Thank you! \
Edit: The answer is by solving $p_{max} > p_0$ directly, using Eq 9.2 and Eq 9.3.