= Solution
Write $b=1+u$, $n=n_*+v$ and define
$$
\kappa=\frac{\chi_b}{(1+n_*)^2},\qquad
\eta=\frac{\lambda(b_0^2-1)}{(1+b_0^2)^2}.
$$
The first-order perturbation equations are
$$
u_t=d_bu_{xx}-\kappa v_{xx}-u,\qquad
v_t=d_nv_{xx}+\eta u-\mu v.
$$
The homogeneous <Neumann boundary conditions> select <Fourier modes> proportional to $\cos(m\pi x)$, with squared <wavenumber> $q_m=(m\pi)^2$ for $m=0,1,\ldots$. For a mode $e^{\sigma t}(U,V)\cos(m\pi x)$, the <eigenvalue> problem is
$$
\sigma\begin{pmatrix}U\\V\end{pmatrix}
=A(q_m)\begin{pmatrix}U\\V\end{pmatrix},\qquad
A(q)=\begin{pmatrix}-1-d_bq&\kappa q\\\eta&-\mu-d_nq\end{pmatrix}.
$$
Its <trace> is $-[1+\mu+(d_b+d_n)q]<0$. Therefore a growing mode exists exactly when its <determinant> is negative. Set
$$
C=\kappa\eta-d_n-\mu d_b.
$$
The <dispersion relation> is
$$
\sigma^2+[1+\mu+(d_b+d_n)q]\sigma+P(q)=0,\qquad
P(q)=d_bd_nq^2-Cq+\mu.
$$
The constant mode has $P(0)=\mu>0$; very short <wavelengths> also decay because $P(q)\sim d_bd_nq^2$. For a negative interval at positive $q$, the upward-opening <quadratic polynomial> must have its minimum below zero at positive $q$. This gives
$$
\boxed{\kappa\eta>d_n+\mu d_b+2\sqrt{\mu d_bd_n}
=\left(\sqrt{d_n}+\sqrt{\mu d_b}\right)^2.}
$$
In particular, $\boxed{b_0>1}$ is necessary, since otherwise $\eta\le0$ and every coefficient of $P(q)$ is positive. When $b_0>1$, a locally increased population must increase net <nutrient> production, so attraction to that <nutrient> reinforces the population increase. This feedback must overcome both <diffusion> and local relaxation. The calculation is an instance of <chemotactic instability with logistic growth and saturating production>.
The displayed continuous-<wavenumber> inequality is necessary on the finite domain, but an allowed mode must actually lie in the unstable band:
$$
q_\pm=\frac{C\pm\sqrt{C^2-4\mu d_bd_n}}{2d_bd_n},\qquad
\boxed{q_-<(m\pi)^2<q_+\quad\text{for some integer }m\ge1.}
$$
Equivalently, the exact discrete threshold is
$$
\boxed{\chi_b>\frac{(1+n_*)^2}{\eta}
\min_{m\ge1}\left[d_n+\mu d_b+d_bd_n(m\pi)^2+\frac{\mu}{(m\pi)^2}\right],\qquad \eta>0.}
$$
Equality gives a marginal zero <eigenvalue> rather than positive growth. At the unrestricted onset, $q_c=\sqrt{\mu/(d_bd_n)}$, so $k_c=(\mu/(d_bd_n))^{1/4}$. On the finite interval, <Discrete Neumann mode selection> first destabilizes the mode minimizing the discrete threshold; it need not attain $k_c$ exactly.
To predict the early form of a perturbation, use the larger growth <eigenvalue>
$$
\boxed{\sigma_+(q)=-\frac{1+\mu+(d_b+d_n)q}{2}
+\frac12\sqrt{[\mu-1+(d_n-d_b)q]^2+4\kappa\eta q}.}
$$
For $\eta>0$, the square root is real. Choose $m_{\rm fast}$ maximizing $\sigma_+(q_m)$ among allowed modes with positive growth. The corresponding perturbation has <wavelength> $2/m_{\rm fast}$ in $x$, or $2L/m_{\rm fast}$ in $X$. Its <eigenvector> satisfies
$$
\frac VU=\frac{1+d_bq_m+\sigma_+(q_m)}{\kappa q_m}>0,
$$
so <bacteria> and <nutrient> peaks initially coincide. A mode with larger $m$ has finer spatial structure. The two signs of $\cos(m\pi x)$ are both allowed, exchanging peaks and troughs.
\Image[/past-exam-of-the-further-mathematical-biology-course-of-the-university-of-oxford/oxford-2023-further-mathematical-biology-pattern-selection.png]
{title=<Dispersion relations> and early <Fourier modes> as <chemotaxis> strength changes}
The illustration holds $d_b=0.002$, $d_n=0.04$, $\mu=0.5$, $b_0=2$ and $\lambda=2.5$ fixed, giving $n_*=1$ and $\eta=0.3$, and varies $\chi_b$. Crossing the discrete threshold makes a nonuniform <Fourier mode> grow; stronger <chemotaxis> widens the unstable band and can favor finer structure. Increasing either <diffusion> coefficient raises the threshold of each fixed mode. Varying $\lambda$ can have a nonmonotone effect: with $s=1+b_0^2$,
$$
\kappa\eta=\frac{\chi_b\lambda(b_0^2-1)}{s^2(1+\lambda/(\mu s))^2}
$$
is maximal at $\lambda=\mu s$, where $n_*=1$. Additional production beyond that point weakens sensitivity enough to reduce this feedback. Changes in $L$ also change the allowed dimensional <wavenumbers> $m\pi/L$ and all three scaled transport coefficients. These are predictions of <linear stability analysis>: eventual pattern amplitude, competition between growing modes and possible merging require the nonlinear system. The threshold is a necessary condition for spontaneous pattern growth from arbitrarily small perturbations, rather than a claim that every finite-amplitude nonuniform state is excluded below it.
Back to article page
