= Chemotactic instability with logistic growth and saturating production
{title2=$\kappa\eta>(\sqrt{d_n}+\sqrt{\mu d_b})^2$}
For $b_t=d_bb_{xx}-\chi_b\partial_x[b(1+n)^{-2}n_x]+b(1-b)$ and $n_t=d_nn_{xx}+\lambda b^2/(b_0^2+b^2)-\mu bn$, the positive homogeneous <equilibrium> is $(1,n_*)$, where $n_*=\lambda/[\mu(1+b_0^2)]$. Set $\kappa=\chi_b/(1+n_*)^2$ and $\eta=\lambda(b_0^2-1)/(1+b_0^2)^2$. The <linear stability> <matrix> at squared <wavenumber> $q$ is $\left(\begin{smallmatrix}-1-d_bq&\kappa q\\\eta&-\mu-d_nq\end{smallmatrix}\right)$. Its <trace> is negative and its <determinant> is $d_bd_nq^2+(d_n+\mu d_b-\kappa\eta)q+\mu$. 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.
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