## Age structure and yield per recruit

The models that we have discussed thus far are called production models because they focus on removing the ''excess production" associated with biological growth. But that production has thus far been treated in an exceedingly simple manner. We will now change that. Models that incorporate individual growth play a crucial role in modern fishery management, so we shall spend a bit of time showing that connection. Let us return to Eq. (2.13) and explicitly write a, for age, instead of t so that L(a) represents length at age a and W(a) represents weight at age a, still assumed to be given allometrically. Imagine that we follow a single cohort of fish, with initial numbers N(0) — R. In the absence of fishing mortality, the number of individuals at any other age is given by N(a) — Re-Ma.

When following a population with overlapping generations, we introduce N(a, t) as the number of individuals of age a at time t, and F(a) as the fishing mortality of individuals of age a. The dynamics of all age classes except the youngest are

since next year's 10 year olds, for example, must come from this year's 9 year olds. We assume that pm(a) is the probability that an individual of age a is mature and reproductively active, and an allo-metric relationship between length at age L(a) and egg production (—cL(a)b, with c and b constants). The total number of eggs produced in a particular year is

a and we append the dynamics of the youngest age class N(0, t + 1) — N0(E(t)), where N0(E(t)) is the relationship between the number of eggs produced by spawning adults and the number of individuals in the youngest age class. For example, in analogy to the Beverton-Holt recruitment function for we have No(0, t + 1) — aE(t)/[1 + flE(t)] and in analogy to the Ricker recruitment N0(0, t + 1)— aE(t)e-flE(t); in both cases the parameters a and fl require new interpretations from the ones that we have given previously. For example, the parameter a is now a measure of egg to juvenile survival when population size is low and the parameter fl is still a measure of the effects of density dependence.

In light of Eq. (6.23), the number of fish of age a that died in year t is N(a, t)(1 — e—(M+F(a))), and if we assume that the natural and anthropogenic components are in proportion to the contribution of total mortality m + F(a) owing to each, we conclude that a fraction M/ [M + F (a)] of the fish are lost owing to natural mortality and a fraction F (a)/[M + F (a)] of the fish are taken by the fishery. Thus, the yield of fish of age a in year t is

Y(a, t)=M + F)(a)N(a, t)(1 — e—(M+F(a»)W(a) (6.25)

where W(a) is the weight of fish of age a; the total yield in year t is Y (t) = a^o Y (a, t), where amax is the maximum age to which fish live (for most of this chapter, I will not write the upper limit).

Very often, we assume ''knife-edge'' fishing mortality, so that F(a) = 0 if a is less than the age ar at which fish are recruited to the fishery and F(a) = F, a constant, for ages greater than or equal to the age of recruitment to the fishery. Note, too, that there are now two kinds of recruitment - to the population (at age 0) and to the fishery (at age ar).

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