Past exam of the topics in applied mathematics course of the University of Oxford / 2011 / A / 3

Let $r$ be bacterial intrinsic growth rate, $K$ carrying density, $\sigma$ chemical production per bacterial density per time, and $\mu$ chemical decay rate. The bacterial flux iscombining random diffusion with chemotaxis up the chemical gradient. Chemical flux is $J_A=-D_A{A}_x$. The two local balances give the Keller--Segel model with logistic growth:All transport and rate parameters are positive. The production rate is constant per cell, so total production is proportional to $N$.

Choose any positive chemical concentration scale $A_0$ and setAfter dropping hats, substitution giveswhereFor explicit dimensional checks, write bacterial-density and chemical-concentration units as $\mathsf{N},\mathsf{A}$. Then $[D_N]=[D_A]=L^2/T$, $[r]=[\mu ]=T^{-1}$, $[\bar{\chi }]=L^2/(T\mathsf{A})$ and $[\sigma ]=\mathsf{A}/(T\mathsf{N})$. Since $[K]=\mathsf{N}$ and $[A_0]=\mathsf{A}$, each displayed ratio has unit one. The new spatial domain is $[0,1]$.

The nonzero homogeneous equilibrium is $\boxed{n_{\ast }=1,a_{\ast }=\alpha /\beta }$. Without transport its Jacobian matrix iswith negative eigenvalues $-1,-\beta$.

Zero chemical flux gives $a_x=0$ at the endpoints; zero bacterial flux then gives $n_x=0$. Linear perturbations therefore have Neumann boundary conditions and modes $\cos (k_{m}x)$, $k_m=m\pi$. For each mode, the amplitude matrix isIts trace is always negative. Its determinant isA mode grows exactly when $\Delta (k_m)<0$. In particular, any nonzero mode is destabilized by sufficiently large $\chi$. For continuously variable $k$, a nonempty unstable band exists precisely whenWriting $C=\alpha \chi -D_a-\beta D_n$, its squared-wavenumber endpoints are $[C\pm \sqrt{C^2-4\beta D_n{D}_a}]/(2D_n{D}_a)$. On the finite interval, at least one allowed $k_m^2$ must lie inside this band. The uniform $m=0$ mode stays stable. This is the chemotactic pattern-instability criterion.

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 $D_n{D}_a{k}^4$ 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 $(\lambda +\beta +D_a{k}^2)\delta a=\alpha \delta n$ 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.