⁽¹⁾ Walters (2011, 2020) developed the CDDM given the rate equations for number and biomass, respectively: and , where R_t, N_t, B_t, and F_t are the recruitment, number, biomass, and fishing mortality for time step t (see Tab. 1 for the parameters w_r, W_∞, M, and µ). Under equilibrium, i.e. assuming constant recruitment (R_t = R) and constant fishing mortality (F_t = F), setting and leads to N = R/(F + M) and B = (w_rR + µW_∞N)/(F + M + µ). In the latter equation, replacing N by R/(F + M) gives ; hence, , which is herein represented by Φ(F).

⁽²⁾ In equation (2a), exp[−(F + M)] measures the number of individuals surviving, from one recruit, at the end of a fishing regime. Equation (3) can therefore also be interpreted as the average number of individuals available during a fishing regime, given a recruit of one (1) individual that entered the very fishing regime. That recruit of 1 individual is implicit in equation (1).

⁽³⁾ Setting γʹ = 0 reduces to ζ + M = Fη, with ζ = [(F + M) F − M] e^−(F+M) and . An equivalent simplification of γʹ = 0 found in Munyandorero (2015) is such that F = M + λ, where . Thus, F_CYPR is such that M = Fη − ζ = F − λ. Note that, following F = M + λ, (i) the nontrivial solution for F (i.e. F > 0) is conditional on M's being > −λ, (ii) F_CYPR is less than M if λ < 0 and is > M otherwise, and (iii) F_CYPR is equal to M if λ = 0. But this last outcome is trivial because it entails (i) that F_CYPR = M = 0 (yet F_CYPR must be > 0 and M > 0; hence F_CYPR can never equal M) and (ii) that µ = 0 (yet µ > 0), w_r = W_∞ (a situation corresponding to catching all animals at the end of their lifespan), or both.

⁽⁴⁾ The CYPR is calculated using equations (2a) or (2b). Based on equation (2b), it is easy to show (i) that, in γʹ, the first derivative of equation (5) with respect to F is equation (4b) multiplied by l_r and (ii) that the reduced form of γʹ = 0 is as in footnote 3 and, hence, is independent of l_r.