Past exam of the mathematical methods I course of the University of Oxford / 2014 / 11

Reading the coefficients in the ordered concentration vector gives the linear system of ordinary differential equationsIts columns sum to zero, so $d(A+B+C)/dt=0$. This three-state cyclic conversion model conserves total concentration.

Put $p=ϵ+\delta +\eta$ and $q=ϵ\delta +\delta \eta +\eta ϵ$. Direct expansion gives the characteristic polynomialThe quadratic formula therefore givesSince $p,q>0$, both nonzero eigenvalues have negative real part. They need not be real; equal rates give a complex-conjugate pair.

The remaining eigenvalue is $\boxed{{\lambda }_3=0}$. At a steady state, the three cyclic fluxes agree: $ϵA=\eta B=\delta C$. A corresponding nonzero eigenvector is thereforeor its scalar multiple $(\delta \eta ,ϵ\delta ,ϵ\eta {)}^T$. Its positive components express a possible equilibrium concentration ratio.

If $K^2\ne W$, the three eigenvalues are distinct, so a basis of eigenvectors givesThe transformed coordinates then have the displayed exponential form. If $K^2<W$, this is a complex eigenbasis; real concentrations are recovered by conjugate-paired coefficients. A real basis instead gives a decaying two-dimensional rotation block with decay rate $-K$ and angular frequency $\sqrt{W-K^2}$.

The proposed diagonalization is not valid for every positive choice of rates. For example $ϵ=\delta =1$, $\eta =4$ gives $K=-3$, $W=9$, with repeated eigenvalue $-3$ and only one corresponding eigenvector. More generally the defective repeated eigenvalue in a cyclic conversion model occurs when $K^2=W$. A Jordan basis then givesThe extra factor $t$ cannot be represented by two independent pure exponential coordinates in a diagonal basis.

All nonzero eigenmodes decay, including the $t{e}^{Kt}$ term in the repeated-root case. Total concentration is conserved and initially equals $n$. Combining $A_{\infty }+B_{\infty }+C_{\infty }=n$ with the equal-flux ratios givesThese are the unique steady state concentrations with this total. The formula remains valid when the transient eigenvalues are complex or repeated.