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

At late times the first sine mode dominates, with growth exponent
(49)
Thus
(50)
and the population grows precisely when . 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.
Solved by gpt-5.6-sol high.

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