## Stock and recruitment

Fish are a renewable resource, and underlying the system is the relationship between abundance of the spawning stock (reproductively active adults) and the number or biomass of new fish (recruits) produced. This is generally called the stock-recruitment relationship, and we encountered one version (the Ricker equation) of it in Chapter 2, in the discussion of discrete dynamical systems. Using S size of the spawning stock and R for the size of the recruited population, we have

where the parameters a and b respectively measure the maximum per capita recruitment and the strength of density dependence. Another commonly used stock-recruitment relationship is due to Beverton and Holt (1957)

where the parameters a and b have the same general interpretations as before (but note that the units of b in Eq. (6.1) and in Eq. (6.2) are different) as maximum per capita reproduction and a measure of the strength of density dependence. When S is small, both Eqs. (6.1) and (6.2) behave according to R ~ aS, but when S is large, they behave very differently (Figure 6.3).

The Ricker and Beverton-Holt stock-recruitment relationships each have a mechanistic derivation. The Ricker is somewhat easier, so we start there. Each spawning adult makes a potential number of offspring, a, so that aS offspring are potentially produced by S spawning adults. Suppose that each offspring has probability per spaw-ner p of surviving to spawning status itself. Then assuming independence, when there are S spawners the probability that a single offspring survives to spawning status is pS. The number of recruits will thus be R = aSpS. If we define b = |log(p)|, then pS = exp(—bS) and Eq. (6.1) follows directly, this is the traditional way of representing the Ricker stock-recruitment relationship (we could have left it as R = aSpS).

Figure 6.3. The Ricker and Beverton-Holt stock-recruitment relationships are similar when stock size is small but their behavior at large stock sizes differs considerably. I have also shown the 1:1 line, corresponding to R = S (and thus a steady state for a semelparous species).

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