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 gives
$$
\dot F=\int(V+\log P+1)(-\nabla\cdot J)\,dx
=\int\nabla(V+\log P)\cdot J\,dx.
$$
Substituting the gradient current,
$$
\boxed{\dot F=-\int P\,
\nabla(V+\log P)^TD\nabla(V+\log P)\,dx\leq0}.
$$
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.

Solved by gpt-5.6-sol high.

Solution

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

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.

Solved by gpt-5.6-sol high.

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

For positive-definite $D$ and no-flux boundaries,
$$
\mathcal F[P]=\int(VP+P\log P)\,dx
$$
decreases as $\dot{\mathcal F}=-\int P\nabla(V+\log P)^TD\nabla(V+\log P)\,dx$. Its unique normalized minimizer is proportional to $e^{-V}$.


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 Incoming links: Fokker--Planck free-energy functional