Hart: Yield- and biomass-per-recruit analysis of rotational fisheries 



45 



simulations. Analysis of pulse fishing, although simple to 

 apply and understand, is not applicable to those situations 

 where only a portion of the available resource is removed 

 when an area is opened periodically to fishing. Per-recruit 

 analysis of a single cohort is not applicable to relatively 

 nonselective multiple age-group fisheries. 



Botsford et al. (1993) developed a mixed-age rotational 

 yield-per-recruit model for red sea urchins. They showed 

 that rotational fishing for these urchins would increase 

 egg production considerably, while slightly decreasing 

 yield-per-recruit. Recently, Myers et al. (2000) presented 

 a mixed-age per-recruit analysis of a possible rotational 

 fishery strategy for sea scallops. The emphasis of this 

 study was on the effect of putative high levels of indirect 

 (noncatch) fishing mortality on yield-per-recruit, and on 

 a proposed rotational plan that Myers et al. suggested 

 would help ameliorate this effect. 



The purpose of the present article is to present a general 

 theory for any type of periodic or rotational fishing strat- 

 egy for a mixed-age sessile or sedentary stock. This work 

 generalizes many of the above mentioned studies (in par- 

 ticular, that of Botsford et al., 1993) and does not require 

 an assumption of constant recruitment or specific spatial 

 configuration (or both). This theory is applied to the Atlan- 

 tic sea scallop fishery of Georges Bank. 



Measures of fishing mortality and overfishing defini- 

 tions are usually based on models where fishing is as- 

 sumed constant in space and time. In rotational fisheries, 

 or in cases where part of a fishing ground has been closed 

 indefinitely to fishing, these assumptions may be seriously 

 violated, especially for stocks that are relatively immobile. 

 Alternative measures of fishing effort and overfishing 

 definitions are presented here that are more appropriate 

 to fisheries of nonmobile stocks where rotational or indefi- 

 nite closures are used. 



Methods 



The object of this analysis was to compute the expected 

 yield-per-recruit and biomass-per-recruit of a cohort 

 located in an area where fishing mortality may depend on 

 the year and the variation in fishing mortality is periodic 

 with time. Rotational fishing is usually thought of as a 

 sequence of periodic closures and openings of different 

 areas. The theory described here is more general, and 

 can be applied to any situation where fishing mortality is 

 varied periodically in a given area. 



Suppose the fully recruited fishing mortality in an area 

 during year k is F^ and that fishing mortality rates vary pe- 

 riodically with period p ( where p is in years ), so that F , = 

 Ff. for all k. Let ^wi; be the mean of F^.F.^ F^^ For sim- 

 plicity, it is assumed that there is one recruitment event 

 and one new cohort per year. However, extension of the 

 theory to multiple cohorts per year is straightforward. 



There are p possible patterns of fishing mortality expe- 

 rienced by a cohort, depending on the point of the cycle 

 when it enters the fishery. The cohort that enters in the 

 first year will experience fully recruited fishing mortality 

 rates: 



F F F F F F F 



(1) 



during successive years. The next cohort will experience 

 the same fishing mortality rates, but in a different order: 



F„F„F„...,F^.F,K. 



(2) 



and so on. 



Two special cases are of particular interest: pulse rota- 

 tion and symmetric rotation. Pulse rotation means that 

 F^ = for ^=l,2,...,p-l (the area is closed for p-1 years), 

 then F >0 (the area is pulse fished for one year), and then 

 Ff^=0 for k=p-¥l,p+2,..., 2p-l (the area is closed again), etc. 

 Symmetric rotation, where p is even, means that F/, = for 

 1 < ^ < p/2, and Ff. = 2F^v^. for p/2 < k < p, i.e. the area is 

 closed for p/2 years and then fished at a constant rate for 

 the next p/2 years. 



For each of the p patterns of fishing mortality, yield- and 

 biomass-per-recruit can be calculated by using standard 

 per-recruit techniques. Here, a method similar to the "ge- 

 neric per-recruit" model described in Quinn and Deriso 

 (1999) is used (see Appendix). The only unusual aspect 

 is that the mortality terms Z and F^. in Equations 11-13 

 (see Appendix) depend explicitly on time, i.e. on the year of 

 the rotational cycle. Each of the p cohorts will have differ- 

 ent yield-per-recruit Y^,Y.^...,Y , and biomass-per-recruit 

 ByB.^.-Mp values because the ages at which they experi- 

 ence the fishing mortalities F^,F2,..-,F are different. 



Define y^VG ^"^^ ^avg ^^ be the means of the p patterns 

 of cohort yield- and biomass-per recruit, respectively. Y^yg 

 is the expected yield of a recruit chosen randomly with re- 

 spect to cohort. In other words, F^vc '^ ^^^ long-term mean 

 yield-per-recruit that can be expected from the rotational 

 fishing strategy. Similarly, Bavg '^ the expected long-term 

 mean biomass-per-recruit. Note that unlike conventional 

 per-recruit theory, yield- and biomass-per-recruit vary 

 with cohort, so that the mean yield- and biomass-per-re- 

 cruit obtained at any point in time may be different from 



^AVG and Bavg- 



Let y^',y2',...,y'P' be yield-per-recruit of the p cohorts, 

 in decreasing order, so that y" is the highest yield-per- 

 recruit out of all the p cohorts and Y'l'' the lowest, y" is 

 an upper bound on the yield-per-recruit that might be 

 obtained with a rotational strategy if, for example, the 

 closure pattern were timed to optimize yield-per-recruit 

 from a large year class. 



It is important when comparing rotational and con- 

 stant fishing strategies to compare alternatives that have 

 the same long-term survival rates, i.e the same natural 

 mortalities and mean fishing mortality rates. If this is 

 not done, then effects of rotation can be confounded with 

 those due to variations in fishing mortalities. If there are 

 initially N^ fully recruited individuals in an area that are 

 fished at a constant rate F„, then there will be 



N=N,,exp{-pM-pFj 



(3) 



of these individuals remaining alive after p years. If 

 instead, fishing mortality was varied on ap year rotation, 

 so that in each year of the cycle, fishing mortality in an 



