Chemotactic instability with logistic growth and saturating production ($\kappa \eta >(\sqrt{d_n}+\sqrt{\mu d_b}{)}^2$)

For $b_t=d_b{b}_{xx}-{\chi }_b{\partial }_x[b(1+n{)}^{-2}{n}_x]+b(1-b)$ and $n_t=d_n{n}_{xx}+\lambda b^2/(b_0^2+b^2)-\mu bn$, the positive homogeneous equilibrium is $(1,n_{\ast })$, where $n_{\ast }=\lambda /[\mu (1+b_0^2)]$. Set $\kappa ={\chi }_b/(1+n_{\ast }{)}^2$ and $\eta =\lambda (b_0^2-1)/(1+b_0^2{)}^2$. The linear stability matrix at squared wavenumber $q$ is $\left(\begin{array}{cc}-1-d_{b}q& \kappa q\\ \eta & -\mu -d_{n}q\end{array}\right)$. Its trace is negative and its determinant is $d_b{d}_n{q}^2+(d_n+\mu d_b-\kappa \eta )q+\mu$. Minimizing this quadratic polynomial over $q>0$ proves the displayed continuous-wavenumber growth criterion. It requires $b_0>1$. On a bounded domain an actual permitted nonzero Fourier mode must lie strictly inside the negative-determinant interval. This is a criterion for growth of infinitesimal perturbations, not a general exclusion of finite-amplitude patterns.