= Solution
<Chemotaxis> supplies positive feedback: a local increase in bacterial density produces more attractant, whose <gradient> draws in more bacteria. Ordinary cell <diffusion> opposes aggregation. Chemical <diffusion> spreads the signal and removes very sharp gradients, while decay and logistic growth stabilize the uniform kinetics.
The <dispersion relation> from the preceding matrix makes the competition quantitative: the positive $D_nD_ak^4$ term stabilizes sufficiently short wavelengths, whereas the negative chemotactic contribution $-\alpha\chi k^2$ can destabilize intermediate wavelengths. Increasing cell diffusivity or chemical decay raises the attraction threshold. A growing allowed spatial mode produces correlated patterns in cell density and chemical concentration; the <eigenvector> relation $(\lambda+\beta+D_ak^2)\delta a=\alpha\delta n$ puts their peaks in phase for the growing real mode.
This predicts initial pattern formation. Nonlinear saturation and stability of the eventual pattern require the nonlinear equations, beyond the linear instability test.
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