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Question
An important problem in fishery science is to estimate the number of fish presently spawning in streams and use this information to predict the number of mature fish or “recruits” that will return to the rivers during the reproductive period. If S is the number of spawners and R the number of recruits, “Beverton-Holt spawner recruit function” is R(S) = `"S"/((alpha"S" + beta)` where `alpha` and `beta` are positive constants. Show that this function predicts approximately constant recruitment when the number of spawners is sufficiently large
Solution
Given R(S) = `"S"/((alpha"S" + beta)`
Where S is the number of spawners, R is the number of recruits. α, β are positive constants.
When the number of spawners is sufficiently large s → ∞
`lim_("s" -> oo) "R"("S") = lim_("s" -> oo) "S"/(alpha"S" + beta)`
= `lim_("s" -> oo) "S"/("S"(alpha + beta/"S")`
= `lim_("s" -> oo) 1/(alpha + beta/"S")`
= `1/(alpha + 0)`
= `1/alpha`
When the number of spawners is sufficiently large, the number of recruits is `1/alpha`
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