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

Table of contents
- i b
  - Solution i
- ii b
  - Solution ii
- iii b
  - Solution iii

i

Solution

The independent additive noises give diagonal diffusion

κ = diag (b_{1}^{2}, b_{2}^{2})

. Thus

\partial_{t} P = - \partial_{x_{1}} [(r_{1} x_{1} - r_{1} c_{1} x_{1} x_{2}^{2} - d_{1} x_{1}^{2}) P] + \frac{b _{1}^{2}}{2} \partial_{x_{1}}^{2} P - \partial_{x_{2}} [(r_{2} x_{2} - r_{2} c_{2} x_{2} x_{1}^{2} - d_{2} x_{2}^{2}) P] + \frac{b _{2}^{2}}{2} \partial_{x_{2}}^{2} P .

The mixed derivatives of

κ^{- 1} a

are

\partial_{x_{2}} \frac{a _{1}}{b _{1}^{2}} = - \frac{2 r _{1} c _{1} x _{1} x _{2}}{b _{1}^{2}}, \partial_{x_{1}} \frac{a _{2}}{b _{2}^{2}} = - \frac{2 r _{2} c _{2} x _{1} x _{2}}{b _{2}^{2}} .

They agree precisely when

r_{1} c_{1} b_{2}^{2} = r_{2} c_{2} b_{1}^{2} .

Solved by gpt-5.6-sol high.

ii

Solution

Under the potential condition, integrate

\partial_{i} Ψ = a_{i} / b_{i}^{2}

to obtain

Ψ = \frac{r _{1} x _{1}^{2}}{2 b _{1}^{2}} - \frac{d _{1} x _{1}^{3}}{3 b _{1}^{2}} + \frac{r _{2} x _{2}^{2}}{2 b _{2}^{2}} - \frac{d _{2} x _{2}^{3}}{3 b _{2}^{2}} - \frac{r _{1} c _{1} x _{1}^{2} x _{2}^{2}}{2 b _{1}^{2}} .

Since

V = - 2Ψ

, the zero-current steady density on the positive quadrant is

P_{ss} (x_{1}, x_{2}) = \frac{1}{Z} exp [\frac{r _{1} x _{1}^{2}}{b _{1}^{2}} - \frac{2 d _{1} x _{1}^{3}}{3 b _{1}^{2}} + \frac{r _{2} x _{2}^{2}}{b _{2}^{2}} - \frac{2 d _{2} x _{2}^{3}}{3 b _{2}^{2}} - \frac{r _{1} c _{1} x _{1}^{2} x _{2}^{2}}{b _{1}^{2}}] .

The equivalent coefficient

r_{2} c_{2} / b_{2}^{2}

may be used for the cross term.

Solved by gpt-5.6-sol high.

iii

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.

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