Beverton-Holt Stock-Recruitment Curve

The Beverton-Holt curve is the second classic way to turn a spawning stock $S$ into recruits $R$, the young fish that join the population. At low stock the denominator $1 + bS$ is close to one, so $R$ is close to $aS$ and each spawner produces about $a$ recruits. As the stock grows, competition for food and space among the young slows the gain, and recruitment levels off toward a ceiling of $a/b$ without ever falling. Half the ceiling is reached at $S = 1/b$, so $1/b$ is the stock that gives half of the most the curve can deliver.

It is the usual choice for cod, flatfish and other fish whose recruitment holds steady at high stock. Beverton and Holt derived it in 1957 from a model of crowded young fish, in their book On the Dynamics of Exploited Fish Populations. Where the Ricker curve drops again when spawners crowd the nests, this one only flattens, so more spawners never lower recruitment.

With $a = 5$ and $b = 0.002$, 300 spawners give $5\times300/(1 + 0.6) = 937.5$ recruits, the page's example. The ceiling is $a/b = 2500$, and at $S = 1/b = 500$ recruitment is 1250, half of it. A million spawners give $5\times10^{6}/2001 = 2498.75$, just under 2500 and never reaching it. Run backwards, $S = R/(a - bR)$ gives $937.5/(5 - 1.875) = 300$ spawners, which makes sense only while $R$ is below the ceiling $a/b$.

It is also the Beverton-Holt population model written the fisheries way: with a growth factor $R_0$ per generation and $M$ the stock at which crowding halves it, $a = R_0$ and $b = 1/M$. With $R_0 = 2$ and $M = 1000$, so $a = 2$ and $b = 0.001$, the carrying capacity is $K = (R_0 - 1)M = 1000$, and a population at $K$ replaces itself: $2\times1000/(1 + 1) = 1000$.