TOTAL EQUILIBRIUM YIELD

Transcription

1 TOTAL EQUILIBIUM YIELD ecall that the yield-per-recruit model enabled us to examine the problem of growth overfishing but, because we assumed that recruitment was constant, the model did not tell us whether a given level of fishing might result in recruitment overfishing. Now we will combine Beverton and Holt's model for yield-per-recruit with their model for the stock-recruit relationship and build a model for long-term equilibrium yield, which will let us examine the problem of recruitment overfishing. Although this model was originally developed and described in Beverton and Holt (1957), it has received relatively little attention from fisheries scientists. Combining Yield-per-ecruit with a Stock-ecruit Model For many fish species it is at least approximately true that the number of eggs laid per mature female is directly proportional to the weight of the female. ( No. Eggs Laid Per Female ) k ( Female Body Weight ) If this relationship is valid, then the total number of eggs laid ring the lifetime of a cohort of fish is given by E k N fem ( u) W fem ( u) t where E is the total number of eggs laid; k t N fem W fem is the number of eggs laid per female body weight; is the age at first reproction; is the number of females at age; is the female weight at age. If the amount of eggs as a proportion of bocy mass (k) is not constant with age then parameter k cannot be brought from under the integral sign. The formulation above assumes thagg proction is a continuous process, but for many fish species spawning is highly seasonal. We will ignore this complication. Also, we will assume that the sex ratio is constant and that both sexes have the same weight at age so that we do not need separate integrals for males versus females. Given these assumptions it follows that E k' N( u) W ( u) k' B( u) This equation uses k' rather than k t t because both sexes are included. The cohort's egg proction is proportional to its spawning stock biomass (). Our simplified problem is to model the spawning stock biomass and then relate that to the subsequent proction of offspring and recruits to the alt population. For simplicity let's assume that the age at recruitment t r is less than the age at first spawning t, and that the maximum exploitable age t λ is infinite. We must consider two cases: FW431/531 Copyright 2008 by David B. Sampson EquilYield1 - Page 107

2 M M+F (a) t Age, t t r t M M+F (b) t Age, t t r t For t, the spawning stock biomass is N( t ) t u t For t, the spawning stock biomass is t e N( t ) t + N( ) W ( u) u t u t t W ( u)... W ( u) u t W ( u) + t t r t e t u t u W( u)... W( u) To simplify the notation, let Ι(θ,a,b) denote the following definite integral, Ι θ, a, b b e θ ( u a) W( u) with W( u) W inf 1 e K t t 0 a 3 W inf Ι θ, a, b n where Ω 0 1, Ω 1-3, Ω 2 3, and Ω 3-1. Ω n e nk a t 0 θ + nk 1 e ( θ+ nk) ( b a) FW431/531 Copyright 2008 by David B. Sampson EquilYield1 - Page 108

3 The equilibrium yield-per-recruit (as t λ approaches infinity) is given by Y F Ι Z,, The equilibrium spawning stock biomass per recruit is given by (a) for t (b) for t + t ( t t r ) Ι ( M, t, ) ( ) (,, ) Ι Z Ι Z, t,... Notice that if t (case b), then there will always be some spawning biomass (the first term of the sum) even if fishing mortality is infinite. This is not the case for t however. Consider the graph of / as a function of fishing mortality (below). It begins at some positive value for F 0 and then decays exponentially to a horizontal asymptote. When t the graph has a horizontal asymptote of zero. When t, however, the horizontal asymptote is greater than zero, implying thaven infinite fishing mortality would not cause recruitment to fail. / Fishing Mortality, F The spawning stock biomass is represented by the area under the curve of B(t) starting at t t and extending to the right. If t, then there will be a reserve of unexploited spawning stock biomass (from ages < t ) even when the rate of fishing mortality is infinite. FW431/531 Copyright 2008 by David B. Sampson EquilYield1 - Page 109

4 t Biomass age, t Now we will take the Beverton and Holt stock-recruit relationship and manipulate it algebraically to get an equation for the equilibrium spawning stock biomass. k This is the alternate form of the k 1 α1 Beverton and Holt S model. k is a conversion factor to change biomass to numbers of fish, we previously denoted by S eq, α is the ratio of to the asymptotic number of recruits. Note that the variables and parameters are not the same as in the original Beverton and Holt stock-recruit model. The recruits here are not the fish that survive to spawn. ather, they are the fish that survive to age t r, the age of recruitment. For the stock to remain aquilibrium the number of recruits proced by a given spawning stock biomass must in turn proce the same spawning stock biomass ring their reproctive lifespan. Hence, aquilibrium the following condition must hold, 1 The left side is from the stock-recruit relationship and the right side is one over the spawning stock biomass per recruit. The following graph illustrates the idea. The curve shows how many recruits are proced by a given amount of spawning stock biomass. The diagonal lines play the same role as the 45 replacement line in a standard spawner-recruit model. Each has slope equal to one over the spawning stock biomass per recruit, but for differing rates of fishing mortality. As F increases, / decreases and the replacement line becomes steeper. FW431/531 Copyright 2008 by David B. Sampson EquilYield1 - Page 110

5 ecruits Spawning Stock Biomass In the graph above the equilibrium spawning stock biomass eq and equilibrium recruitment eq for a given rate of fishing mortality would be represented by the intersection with the stock-recruit curve of the diagonal line that corresponds to that rate of fishing mortality. In the normal stock-recruit model the replacement line intersects the curve when /k. This is noquivalent to eq because here the fish can spawn more than once. The curve () represents the number of recruits originating from a single spawning from a spawning biomass of. To derive eq algebraically, take the stock-recruiquation and divide by. k 1 > 1 α 1 k 1 α1 k k α k k 1 + α eq αk k 1 + α When a cohort's spawning stock biomass is equal to eq, the number of recruits proced by the cohort over its lifespan will exactly equal the number of fish in the cohort when they attained the age of recruitment, thus maintaining the stock aquilibrium. Notice that if k (/) is less than (1-α), then eq will be negative. In this case we would have to add fish to the population to maintain it aquilibrium; the population is not self-sustaining. We can calculate the equilibrium number of recruits from the ratio of eq over /. eq eq αk k 1 + α 1 αk k 1 α FW431/531 Copyright 2008 by David B. Sampson EquilYield1 - Page 111

6 We can calculate the equilibrium yield from the proct of eq and Y/. Y Y eq eq αk k 1 α F Ι Z,, The equilibrium yield, like the yield-per-recruit, is a function of both the rate of fishing mortality F and the age-at-entry. However, unlike the equation for yield-per-recruit, the equation for yield can proce values less than zero. These represent fishing the population to extinction. At such levels of F and we would have to add fish to the population to maintain it aquilibrium. With this model for equilibrium yield we can examine the problem of recruitment overfishing in a manner analogous to how we explored growth overfishing. In the yield-per-recruit model we defined a curved line AA' that showed combinations of F and that proced a maximum yield-per-recruit for a fixed. When F was to the right of this curve the yield-per-recruit (at equilibrium) could be increased by a rection in fishing. A completely analogous curve exists for the total yield surface. Below on the left is the equilibrium yield surface for a hypothetical stock of North Sea plaice. On the right is the yield-per-recruit surface. Equilibrium Total Yield Equilibrium Yield-per-ecruit The "horizontal" axis is F, the axis going into the page is. The two diagrams have a generally similar appearance but the vertical scales are quite different because the left-hand graph shows the Total Yield proced by thousands of recruits whereas the right-hand graph shows the yield from a single recruit. Also, the yield surface Y(F, ) above has a region in the lower right portion, corresponding to large values for F and small values for, where the yield drops to zero. FW431/531 Copyright 2008 by David B. Sampson EquilYield1 - Page 112

7 Here are the corresponding contour diagrams. Equilibrium Total Yield Equilibrium Yield-per-ecruit Like the yield-per-recruit surface, the yield surface does not have a distinct peak at any finite rate of fishing mortality. As a consequence, a yield contour will never entirely close off a region of the surface. Compared to yield-per-recruit models, equilibrium yield models as described above are not widely used. This is probably e to their greater mathematical complexity and to the difficulties that researchers have had in establishing an appropriate stock-recruit model and estimating the model's parameters. Here are a few references, all on the Supplemental eading list: Goodyear (1993), Shepherd (1982), Sissenwine and Shepherd (1987), and Walters (1969). Of special interest is Walters's study of a simulated population of brook trout. He found that by harvesting the fish periodically he could obtain harvests that were as much as 50% larger than the best yield rates obtained from continuous harvesting. One final reference is the original work on this topic by Beverton and Holt (1957), section , pages Beverton and Holt were skeptical about the usefulness of this type of equilibrium yield model. They wrote "... it is clear that any self-regenerating model in which all the parameters other than recruitment are constant can predict such extreme changes in density that compensating changes in growth and possibly natural mortality would be certain to occur in practice." To a limited degree they explored yield models in which growth and natural mortality varied with stock density. We will noxamine these more complex models in this course however. FW431/531 Copyright 2008 by David B. Sampson EquilYield1 - Page 113