The CYPR14 model (equations without source are developed in this study).
|Function for the CSSR (1)||(1)||Walters (2011, 2020) and Munyandorero (2015)|
where R is the number of recruits (R = 1); equivalently,
|Function for the CYPR (2).||(2a)||Munyandorero (2015)|
|, where||Function for the CYPR||(2b)||Munyandorero (2015)|
|Function for the average recruit (2)||(3)|
where ; and
|First derivative of γ with respect to F (3)||(4a)
|, where lr = exp[−(r − 1)M]||Function for the CSSR involving (natural) survivorship from age rp to age r (lr; see Table 1 for the description of rp and r) (4)||(5)|
(1) Walters (2011, 2020) developed the CDDM given the rate equations for number and biomass, respectively: and , where Rt, Nt, Bt, and Ft are the recruitment, number, biomass, and fishing mortality for time step t (see Tab. 1 for the parameters wr, W∞, M, and µ). Under equilibrium, i.e. assuming constant recruitment (Rt = R) and constant fishing mortality (Ft = F), setting and leads to N = R/(F + M) and B = (wrR + 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, FCYPR 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) FCYPR is less than M if λ < 0 and is > M otherwise, and (iii) FCYPR is equal to M if λ = 0. But this last outcome is trivial because it entails (i) that FCYPR = M = 0 (yet FCYPR must be > 0 and M > 0; hence FCYPR can never equal M) and (ii) that µ = 0 (yet µ > 0), wr = 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 lr and (ii) that the reduced form of γʹ = 0 is as in footnote 3 and, hence, is independent of lr.
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