The model

Published (the stochastic logistic model; Beddington and May, 1977; Reed, 1979; Alvarez and Shepp, 1998): a stock X, as a share of its unfished size, grows as dX = rX(1 − X)dt + σX dW, less the catch. The maximum sustainable yield, MSY, is r/4, taken at X = 1/2. Under a fixed escapement b the catch is everything above b; the barrier that maximises the discounted catch solves φ″(b) = 0, where φ(x) = x^θ M(θ, 2θ + 2r/σ², 2rx/σ²) is the increasing solution of the resolvent equation, M is Kummer's function and ½σ²θ(θ − 1) + rθ = δ.

Ours: the illustrative numbers, r = 0.4 a year and a discount rate δ = 5% (no stock was fitted); "collapsed" meaning below 5% of the unfished stock; Euler steps of 0.02 years. Every number on this page is computed here; the post checks the page against Python.