MODEL FOR VARIABLE RECRUITMENT (ontinue) Alterntive Prmeteriztions of the pwner-reruit Moels We n write ny moel in numerous ifferent ut equivlent forms. Uner ertin irumstnes it is onvenient to work with lterntive versions of the Riker n the Beverton n Holt spwner-reruit moels. The Riker R Moel R exp( ) ln( ) Define new prmeters α ln( ) Now we must trnsform the originl moel so we n use the new prmeters. α R exp( ) exp( ln( ) ) exp α R exp( α) expα exp α α Prmeter α is imensionless n hs imensions [No.Fish]. The quntity ( - / ) is the reltive evition from the equilirium vlue; e.g., 0% of the equilirium. Also, notie tht α ln( ) ln( ) mx / mx If α < > < mx If α > > mx < mx ' mx Reruits Reruits pwners pwners FW43/53 Copyright 2008 y Dvi B. mpson Reruitment3 - Pge 89
Another formultion for the Riker R Moel Mny reent stok ssessment moels hve opte yet nother formul for the spwner-reruit reltionship efine in terms of prmeters for the size of the unexploite spwning popultion ( ) n the so-lle steepness of the R urve, often enote y h. The steepness vlue is efine to e the frtion of the unexploite level of reruitment tht ours when the prentl spwning popultion is t 20% of its unexploite level. From this efinition n the formultion of the R moel given on the previous pge we hve the following reltionship. ( ) 0.2 R 0.2 0.2 exp α h R eq Beuse the unexploite reruitment is equl to the unexploite spwning popultion (R eq ), the eqution n e written s h 0.2 0.2 exp α > h 0.2 exp α ( 0.2) > α.25 ln( 5 h) The new lterntive formultion for the Riker R moel eomes R exp.25 ln( 5 h) The Beverton n Holt Moel R + Define new prmeters α α Agin, we must trnsform the originl moel so we n use the new prmeters. R + ( ) + ( ) A n sutrt ; multiply y (-)/(-). R ( ) α As in the lterntive form of the Riker moel, prmeter α is imensionless n hs imensions [No.Fish], n the quntity ( - / ) is the reltive evition from the equilirium vlue. Also, notie tht α R mx / R mx FW43/53 Copyright 2008 y Dvi B. mpson Reruitment3 - Pge 90
Beuse must e positive numer, it will lwys e true tht α < n < R mx. Another formultion for the Beverton n Holt R Moel In terms of prmeters for the unexploite spwning stok size n the steepness of the R urve the Beverton n Holt R moel n e written s R 5 h 4 h This follows from the efinition of steepness. R( 0.2 ) 0.2 h R eq h 0.2 α 0.2 α ( 0.2) h > α 5 h 4 h Hrvesting n tok-reruit Moels If fishing ours over short intervl just prior to spwning, s ours when fishing on spwning migrtions of Pifi slmon, then the inoming reruits inlue progeny tht spwn plus fish tht woul hve spwne h they not een ught prior to spwning. We n express this reltionship with the following equtions. Reruits New_pwners + Cth New_pwners Reruits Cth If we just the size of our th so tht we hrvest ll the reruits tht re in exess of the replement level R, then the popultion will e mintine t stey stte with onstnt stok size from one genertion to the next. The equilirium th per ohort is given y C eq R( ) We n think of these fish tht re ught (C eq ) s eing "surplus" reruits euse they re not neee to mintin the size of the spwning stok. If less thn C eq fish re ught the size of the spwning stok will inrese. If more thn C eq fish re ught the size of the spwning stok will erese. FW43/53 Copyright 2008 y Dvi B. mpson Reruitment3 - Pge 9
Equilirium Cth per Cohort for the Riker R Moel C eq R( ) 0 C eq exp( ) C eq C eq ( exp( ) ) Reruits pwners Another wy to look t this is to "streth" the spwner- reruit urve n let the line R eome the new horizontl xis. The vertil xis is R()-, the surplus reruits, whih is the sme s the equilirium th C eq. urplus Reruits 0 pwners The mximum equilirium th ours t tht vlue of for whih C eq ( R( ) ) 0 R( ) 0 R exp( ) ( ) > C eq exp( ) ( ) The mximum equilirium th ours t the vlue of tht stisfies exp( ) ( ) FW43/53 Copyright 2008 y Dvi B. mpson Reruitment3 - Pge 92
C Reruits mx( C eq ) pwners We nnot solve iretly for this vlue of, euse it ppers oth insie n outsie n exponent. Inste we must use numeril tehniques (whih is esy in Exel using olver) or we n get pproximte solutions y using the Tylor series expnsion for the exponentil funtion. exp( X) + X + X 2 2! X 3 + +.. 3! We n use the first orer pproximtion, exp(- ) -, to solve pproximtely for the vlue of tht mximizes C eq. Cll this vlue C. ( C) ( C) 2 ( C) + 2 C 2 / 2 ( C) + 2 C 2 0 This is qurti eqution in C. C + B C + A C 2 0 with solutions C [-B ± ( B 2-4 A C)]/(2 A) C 2 + 4 2 4 2 2 4 2 + 2 + 2 + or C 2 + 4 2 4 2 2 4 2 + 2 + 2 + Qurti equtions hve two possile solutions euse X 2 (-X) 2. FW43/53 Copyright 2008 y Dvi B. mpson Reruitment3 - Pge 93
We wnt the negtive root. The positive root oes not proue resonle solution. C The mximum equilirium th is otine y sustituting C into the eqution for C eq, mxc eq C ( exp( C) ) Here is n exmple using the sme n vlues s erlier: 5 0.00250 Here re the pproximtions for C n mxc eq se on the first orer Tylor series. C 22. mxc eq C ( exp( C) ) 45.0 Now we'll lulte the true vlue for C using Mth's root() funtion. The funtion nees strting guess tht it itertively improves. C root[ exp( C) ( C), C] 250.4 The pproximtion for C is out 2% too smll, (22.-250.4)/250.4. mxc eq C ( exp( C) ) 49. The pproximtion for mxc eq is only out % too smll. Equilirium Cth per Cohort for the Beverton & Holt R Moel 0 C eq R( ) C eq + C eq + Reruits C eq pwners FW43/53 Copyright 2008 y Dvi B. mpson Reruitment3 - Pge 94
Here is the lterntive view otine y sutrting the replement line R from the urve R(). urplus Reruits 0 pwners The equilirium th will e mximum t tht vlue of for whih C eq 0 ( R( ) ) C R ( + ) mx( C eq ) Reruits > C eq ( + ) 2 pwners The mximum C eq ours t the vlue of, ll it C, tht stisfies ( + C) 2 > ( + C) 2 2 + 2 C + 2 C 2 ( ) 2 0 2 + C + 2 C 2 A qurti eqution gin. 0 C + B X + A X 2 The generl solution is X [-B ± (B 2-4 A C)] / (2 A) The solution for C is C ( ) 2 + 4 2 2 4 2 2 2 2 We wnt C to e positive. In our eqution prmeters n re lwys positive, so only the positive root will yiel positive vlue for C. We get the following solution for the stok size tht proues the mximum equilirium th. FW43/53 Copyright 2008 y Dvi B. mpson Reruitment3 - Pge 95
C 2 + 4 2 2 2 + The mximum equilirium th is otine y sustituting C into the eqution for C eq, mxc eq + C C C mxc eq + + + ( + ) mxc eq + + ( ) 2 Espement Curves uppose our mngement strtegy is to hrvest some fixe proportion of the inoming new reruits. Hrvest Frtion: µ C R ( ) The numer of fish tht espe n spwn, the espement, is R' R µ. We n represent this grphilly y proportionlly reuing the reruitment urve y (-µ) s shown here. ' eq Reruits pwners For exmple, if 25% of the new reruits re hrveste from eh genertion, then the spwning stok size will e reue n the stok will pproh new n lower equilirium level. In the grph ove the new R'() urve is 75% of the originl urve. FW43/53 Copyright 2008 y Dvi B. mpson Reruitment3 - Pge 96
Here is nother wy to look t the onsequenes of fixe hrvest frtion. The espement is n for ext replement we must hve ( ) Espement R µ ( ) Espement > R µ Given the hrvest frtion µ, the replement line is R. µ This gives us nother wy to represent hrveste popultion. When there is hrvesting the replement line is higher euse only some of the inoming reruits (the frtion -µ) re llowe to spwn. ' eq Reruits pwners FW43/53 Copyright 2008 y Dvi B. mpson Reruitment3 - Pge 97