Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 353 / 2

SetThe stated curl-free condition says that $C_j=({\kappa }^{-1}{)}_{ji}{a}_i$ is a gradient, $C=\nabla \Psi$. DefineThen $a=-D\nabla V$, and the Fokker-Planck probability current becomesThis is the potential condition for a Fokker--Planck equation.

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Normalization makes the derivative of the additive $1$ in $\delta F/\delta P=V+\log P+1$ vanish. Using $\dot{P}=-\nabla \cdot J$, the no-flux boundary condition, and integration by parts givesSubstituting the gradient current,Positive definiteness makes equality possible only when $\nabla (V+\log P)=0$, equivalently $J=0$. Thus $F$ is a strict Fokker--Planck free-energy functional away from stationarity.

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Let $Z={\int }_X{e}^{-V}dx$ and $P_B=e^{-V}/Z$. ThenBy nonnegativity of Kullback-Leibler divergence,with equality exactly when $P=P_B$ almost everywhere. Hence a normalizable $P_B$ is the unique minimizer and, because $\dot{F}<0$ elsewhere, the unique steady density compatible with the boundary condition.

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The independent additive noises give diagonal diffusion $\kappa =\mathrm{diag}(b_1^2,b_2^2)$. ThusThe mixed derivatives of ${\kappa }^{-1}a$ areThey agree precisely when

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Under the potential condition, integrate ${\partial }_i\Psi =a_i/b_i^2$ to obtainSince $V=-2\Psi$, the zero-current steady density on the positive quadrant isThe equivalent coefficient $r_2{c}_2/b_2^2$ may be used for the cross term.

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If $c_1,c_2<0$, the cross term in the exponent becomes positive. Along rays with both densities large it grows quartically, while the self-limiting death terms are only negative cubics. The candidate density is therefore not normalizable and the Fokker--Planck free-energy functional is not bounded below. Deterministically, mutual nutrient enhancement eventually overwhelms each species' quadratic crowding death and drives runaway growth, potentially in finite time. The stochastic model consequently has no steady probability density and sends probability toward arbitrarily large populations. This signals failure of the idealized growth law at high density; resource depletion or stronger saturation must regularize a biological model.

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