Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 13B (Mathematical Biology)

The reaction–diffusion system for amoebae contains local reproduction $\alpha n_0^{2}n$ and density-dependent loss $-\alpha n^3$, giving carrying density $n_0$. The term ${\nabla }^{2}n$ describes diffusion with coefficient one. The chemotaxis flux is $J_{\mathrm{chem}}=\chi (n)n\nabla a$, so its contribution to the conservation equation is $-\nabla \cdot J_{\mathrm{chem}}$. Attraction towards greater chemical concentration requires $\chi (n)>0$ at the relevant positive densities; negative values would mean repulsion, and zero removes the chemotactic response.

In the chemical equation, $\beta n$ is secretion proportional to amoeba density, $-\gamma a$ is first-order chemical degradation, and $D{\nabla }^{2}a$ is chemical diffusion. The cell reaction term permits births and deaths, so the total number of amoebae is not conserved.

For a spatially homogeneous positive steady state, $\alpha n(n_0^2-n^2)=0$ and $\beta n-\gamma a=0$, givingWrite $n=n_0+\eta$, $a=a_0+\xi$, and define $A=2\alpha n_0^2>0$. Homogeneous linearization givesThe eigenvalues are $-A,-\gamma$, both negative. Hence this steady state is stable to homogeneous disturbances, including the repeated-eigenvalue case, where a possible $t{e}^{-At}$ factor still decays.

The background has $\nabla a_0=0$, so the linear chemotaxis term is $-{\chi }_0{n}_0{\nabla }^2\xi$; derivatives of $\chi$ multiply products of disturbances and do not enter at first order. For a normal mode with ${\nabla }^2=-k^2$ and time dependence $e^{pt}$, put $q=k^2$. Thenand the characteristic polynomial isIts linear coefficient is positive for every $q\ge 0$. Thus instability occurs exactly when its constant term $Q(q)$ is negative: the roots are then real with opposite signs. The positive quadratic $Q$ is negative somewhere on $q>0$ precisely whenThereforeAt equality the neutral wavenumber is $q_c=\sqrt{A\gamma /D}$ and $p=0$. Above threshold the unstable band is $q_{-}<k^2<q_{+}$, where

Only the Laplacian eigenfunctions whose eigenvalues lie in the intersection of the unstable band $(q_{-},q_{+})$ with the allowed spectrum can grow. A lower bound $k^2>k_0^2$ removes all unstable modes if $k_0^2\ge q_{+}$, even when the unrestricted threshold in (iii) is exceeded.

If all real wavenumbers above the cutoff are possible, an equivalent strict threshold isIt is unchanged when $k_0^2\le \sqrt{A\gamma /D}$, and becomes $AD+\gamma +D{k}_0^2+A\gamma /k_0^2$ when $k_0^2>\sqrt{A\gamma /D}$. A bounded geometry generally gives a discrete spectrum, so the stronger requirement is that an actual allowed eigenvalue falls in the band.