YIELD-PER-RECRUIT (coninued The yield-per-recrui model applies o a cohor, bu we saw in he Age Disribuions lecure ha he properies of a cohor do no apply in general o a collecion of cohors, which is wha we usually encouner in pracice. Now we will explore some consequences for he yield-per-recrui model of he simplifying assumpion ha he populaion is in equilibrium (consan survival, consan recruimen, consan growh. Yield-per-Recrui from Muliple Cohors The Beveron and Hol model for yield-per-recrui is based on equaions for moraliy and growh of indivials wihin a single age-class. To derive he model we assumed ha F and N( e were boh consan and we brough hem from under he inegral. Y = e ( FN e e Z ( u e W( u ==> Y = FN( e e e Z ( u e W( u This sep is invalid unless F and N( e are boh consan hrough ime. If he fish populaion is in equilibrium (wih consan annual recruimen, consan raes of naural and fishing moraliy, and consan growh parameers, hen he yield-per-recrui model we derived is also valid for he annual yield-per-recrui from he enire populaion. There is a proof of his in Beveron and Hol (957 on pages 7-8. If he populaion is no in equilibrium, hen he yield-per-recrui model is only an approximaion. Recall he arificial age disribuions ha I consruced in a previous class. Year ==> If here are changes in annual survival (or recruimen, he age disribuion changes from year o year. Age 0 2 '95 '9 '97 52 2 '98 54 28 4 '99 5 0 5 S = % 52% 54% 5% When here are changes in he rae of fishing moraliy F or changes in he age-a-enry e, he populaion will be ou of equilibrium ring a ransiion period, and he yield-per-recrui ring he ransiion will no be he same as he equilibrium yield-per-recrui. Below is an arificial populaion ha illusraes he problem. R' y denoes he number of fish enering he fishery in year y, he recruimen. Cohors occur along each diagonal. The equaions ha describe survival for a cohor are reasonably simple because he recruimen erm is a common facor for each age class. However, he pseudo-cohors, which are he collecion of cohors presen in any given year, have complicaed survival equaions because here is no common facor for recruimen. FW4/5 Copyrigh 2008 by David B. Sampson YieldPerRec4 - Page 9
Year ==> Age 0 2 e R' 0 R' R' 2 R' e + R' 0 e M F 0 R' e M F R' 2 e M F 2 e + 2 R' 0 e 2M F 0 F R' e 2M F F 2 e + R' 0 e M F 0 F F 2...... λ R' 0 e λ M F 0 F.. F λ Fish older han λ are no subjec o fishing and do no appear in he cach. Le λ = λ - e. I akes λ- years o ge rid of hose cohors ha had a some ime been subjec o he F 0 rae of fishing moraliy. There is a similar delay for changes in he age-a-enry e o work hrough he exploied age classes. Yield-per-Recrui During a Transiion Suppose we have a populaion ha is a equilibrium. The annual recruimen, he biological parameers, he age-a-enry, and he rae of fishing moraliy have all been consan for a sufficienly long ime (a leas λ- years. In his case he yield-per-recrui for any cohor in he populaion will be idenical o he yield-per-recrui aken ring a single year from across he cohors. Now, if he fishing moraliy rae suddenly changes and remains a a new level, I will ake λ- years for he populaion o aain a new equilibrium. This is illusraed below. % Survival Age % Survival 0 2 4 5 2 2 2 2 0 5 7 4 8 9 4 5 8 8 There would be a similar ransiion period following an abrup change in he age-a-enry. The equaions ha describe yield-per-recrui ring a ransiion are quie complicaed and edious. If you are ineresed in he deails see Beveron and Hol (957 pages 8-89. To give you some idea of wha happens o Y/R ring a ransiion, some of he diagrams ha appear in Beveron and Hol (957 are reconsruced below. FW4/5 Copyrigh 2008 by David B. Sampson YieldPerRec4 - Page 70
Consider he Norh Sea plaice example from he las lecure. Suppose e is 4 years and F suddenly drops from 0.8 per year o 0.4 per year. The yield isopleh indicaes ha he equilibrium yield-per-recrui will shif from abou 0.20 kg per fish o almos 0. kg per fish. However, because here are cohors presen in he exploied phase, i will ake 0 years o reach his new equilibrium level of yield-per-recrui. Wha happens o he yield-per-recrui ring he 0 ransiion years? The iniial response o a drop in F will be a decrease in he yield-per-recrui. To see his, consider ha λ Y = F N( u W( u = F e e Yield is given by he fishing rae F imes he exploiable biomass. In he shor run, before changes in he biomass B( are complee, he yield will be direcly proporional o F. A decrease in F causes a decrease in yield in he shor run. However, for his paricular example he yield will increase in he long run because he biomass will graally increase under he reced rae of fishing. The ransiion will look somehing like he graph on he following page. In his example here is an increase in he Y/R wih a decrease in F because iniially he fish sock was suffering from "growh overfishing". There would be no gain in Y/R following a recion in F if he sock iniially had been "underfished" (F and e above and o he lef of he AA' line on he isopleh diagram. FW4/5 Copyrigh 2008 by David B. Sampson YieldPerRec4 - Page 7
Transiion in Y/R afer a Drop in F. ' In his paricular example he annual Y/R is less han he original level for he firs hree years of he ransiion. Yield/Recrui [kg/fish] 0.2 0. Year F=0.8 /yr F=0.4 /yr Here is anoher example wih Norh Sea plaice. Suppose e is 4 years and F suddenly increases from 0.4 per year o 0.8 per year. The response in his case is a shor-run increase in yield-per-recrui followed by a long-run decline o he new equilibrium yield-per-recrui. Transiion in Y/R afer a Rise in F. ' In his example he annual Y/R is more han he original level for he firs hree years of he ransiion. Yield/Recrui [kg/fish] 0.4 0.2 Year F=0.4 /yr F=0.8 /yr Here is a hird and final example, again wih Norh Sea plaice. Suppose F is 0.8 per year and he age-a-enry e is suddenly increased from 4 years o years by means of an increase in he mesh size. The yield-per-recrui isopleh indicaes ha he equilibrium yield-per-recrui will shif from 0.20 kg per fish o abou 0.0 kg per fish. The iniial response o a sudden increase in e will be a decrease in he yield-per- recrui, because suddenly here is less biomass available o cach. λ Y = F N( u W( u = F ' e is he new age-a-enry. ' e ' e FW4/5 Copyrigh 2008 by David B. Sampson YieldPerRec4 - Page 72
' e bu previously Y = F e N( u W ( u + F ' e N( u W( u This porion of B is unexploied under he new 'e. Only his porion of B is exploied under he new 'e. For his example, however, he new long-run Y/R will be higher han he iniial level because iniially he sock suffered from growh overfishing. Transiion in Y/R afer a Rise in e. ' Here he annual Y/R is less han he original level for he firs wo years of he ransiion. Yield/Recrui [kg/fish] 0.2 0. Year e =4 yr e = yr Wheher a change in F or e resuls in a long-run increase or decrease in yield-per-recrui depends on he sar and end posiions on he Y/R surface. Alernaive Models for Yield-per-Recrui The Beveron Hol model assumes ha growh is isomeric, W = a L. Suppose growh is allomeric, W = a L b. ( b W( = W inf exp K 0 Y = e FN( u W( u Y = FN( e W inf e e Z ( u e ( exp K u 0 b FW4/5 Copyrigh 2008 by David B. Sampson YieldPerRec4 - Page 7
If b is no an ineger (,2,,..., hen here is no exac analyic soluion o he inegral. Wih modern compuers and sofware, however, i is easy o evaluae he inegral for specific values of he parameers. Many programs have a buil-in numerical inegraion funcion ha can be used. Paulik and Gales (94 on he Supplemenal Reading lis describe a mehod for evaluaing yield-per-recrui when growh is allomeric. We can use he mehods applied here o yield-per-recrui o examine oher managemen objecives such as value-per-recrui. Die, Resrepo, and Hoenig (988 on he Supplemenal Reading lis describe one such applicaion. Ricker's Generalized Model for Equilibrium Yield-per-Recrui The Beveron and Hol model for yield-per-recrui is based on a raher resricive se of assumpions concerning growh and moraliy, namely ha he rae of naural moraliy M is consan over all ages and growh follows he von Beralanffy model. If hese assumpions are violaed, bu we have esimaes available for moraliy-a-age and weigh-a-age, hen we can use Ricker's generalized model for equilibrium yield-per-recrui. Noe ha if we apply his mehod across cohors, hen we are assuming ha moraliy, growh, and recruimen are consan hrough ime, even hough moraliy and growh may vary wih age. The mehod is essenially an approximaion o Y = e F( u = e F( u N( u W ( u This differs from he model ha underlies he Beveron and Hol model in ha F varies wih age and N(age and W(age may have enirely general forms. Sar by dividing he exploiable life span ino segmens so ha moraliy and growh occur a reasonably consan raes wihin each segmen. Age-a-enry Oldes age // age 2 λ T T T T T 0 2 λ T λ = and T i i Noe ha T 0 e = j = j If he rae of fishing F is consan wihin each inerval, hen he yield accumulaing ring each inerval is T i+ Y = F = F i i i i T i T i+ i T i Time-averaged biomass av(b i. FW4/5 Copyrigh 2008 by David B. Sampson YieldPerRec4 - Page 74
( Y = F i i i av B F i i i 2 B T i ( ( + B( T i+ The las erm is an approximaion o av(b i based on linear approximaion. Wih Ricker's mehod we calculae he biomass values a each ime poin T i using ( = N( T i W ( T i B T i and we have ha av( B i ( ( W ( T i ( W ( T i+ 2 N T + N T i i+ T T 2 In his mehod we approximae he ime-averaged biomass over each inerval by he simple average of he biomass a he wo end poins. B( B(T B(T 2 In he mehod given by Paulik and Bayliff (97, which is on he Supplemenal Reading lis, he average biomass is esimaed using av( B i ( ( + N( T i+ 2 N T i ( ( + W ( T i+ 2 W T i which is more or less equivalen o esimaing he average biomass from he proc of he average abundance imes he average weigh. The final sep in Ricker's mehod is o calculae he oal yield by summing up he yields from each ime inerval. λ Y oal = i = Y i Ricker (975, Chaper 0, on he Recommended Reading lis, provides examples of several differen mehods for calculaing yield-per-recrui. FW4/5 Copyrigh 2008 by David B. Sampson YieldPerRec4 - Page 75