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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-3/13a/a/iii/solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 13A / a / iii / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

At late times the first sine mode dominates, with growth exponent

μ - \frac{D π ^{2}}{L ^{2}} .

(1)

Thus

μ_{c} (D, L) = \frac{D π ^{2}}{L ^{2}},

(2)

and the population grows precisely when

μ > μ_{c}

. Faster diffusion carries organisms to the lethal ends more quickly and raises the threshold; a longer channel lowers the principal diffusion eigenvalue and therefore lowers the threshold.

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