ENGS 41. SUSTAINABILITY and NATURAL RESOURCE MANAGEMENT. The Cohort. (Lynch book, Chapter 4) Benoit Cushman-Roisin January 2018
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1 ENGS 41 SUSTAINABILITY and NATURAL RESOURCE MANAGEMENT The Cohor (Lynch book, Chaper 4 Benoi Cushman-Roisin January 2018 Wha is a cohor? A cohor consiss o a group o individuals who ogeher progress hrough sages. One cohor goes hrough muliple sages. You, he sudens, make cohors: Class o 2021, now 1 s -year, will be sophomores, ec. Class o 2020, now sophomores, will be juniors, hen seniors Class o 2019, were sophomores, now juniors, will be seniors Class o 2018, were sophomores, hen juniors, now seniors 1
2 Fish cohor: Fish sar as eggs and hen evolve as larvae, ry, juveniles o evenually become adul ish. One can urher disinguish muliple sages in each o hese sages, like juvenile-1, juvenile-2, adul-1, adul-2, adul-3 o he degree o desired biological precision. Each year (or season, a new cohor is generaed, which co-lives wih he previous cohors. Time is he variable ha makes he individuals progress inside heir respecive cohor. There are oher biological sysems besides isheries, like crops and oress. Depending on he biological sysem and is managemen, he recruimen in each cohor can be eiher endogenous or exogenous. Endogenous recruimen (in he wild, unconrolled Example: Number o ish larvae proporional o adul populaion. Exogenous recruimen (humanly conrolled Examples: Nursery plans are uprooed and replaned somewhere else. In aquaculure, juveniles are reained in numbers needed. The ime o recruimen is sill se by he biology, bu he amoun o recruimen ino he cohor is decided exernally wihou relaion o he number o individuals a a laer sage. 2
3 Harvesing versus cach: During a ishing expediion, one akes rom various cohors a he same ime. oo young maure enough cach Basic decisions in harvesing are: 1. A which sage o begin? 2. How hard o work a i? Beveron-Hol cohor model: (Lynch book, Secion We deine he ollowing variables or a cohor: = age since recruimen ino he cohor W( = weigh o an individual a age N( = number o individuals a age B( = W( N( = biomass a age R = recruimen ino he cohor N(=0 = R moraliy (age independen = harvesing moraliy = age a which harvesing begins H( = cumulaive harves, rom up o age H oal = H( h = harves per recrui = H oal /R. Survival & Harves dn dn N ( N dh dh 0 B WN 0 3
4 Prior o harvesing: 0 < < Number o individuals dn N N(0 R N Re Corresponding biomass B ( WN ( ( RW ( e Biomass increases, reaches a maximum and hen decreases db dw dw R e RW( e R W e db dw 0 or W Deine p as he age a which biomass is greaes. Then, in he presence o harvesing: < Number o individuals dn ( N N Re e N ( Re Corresponding biomass Harves Toal harves ( ( B ( WN ( ( RWe ( e ( ( ( ( H ( B ( Re We ( ( ( H Re W( e oal Hoal ( ( h e W( e h(, R 4
5 Variaion in sar ime Variaion in Thus, he harves per recrui (harves inensiy is: Hoal ( ( h (, e W( e R Basic decisions in harvesing are: 1. A which sage o begin?? 2. How hard o work a i?? Mahemaically, he maximum is obained or = p and =. I is: Bp hideal h( p, R Cu all hese rees as soon as hey ge big and beore hey die! 5
6 A oal cach a he peak o biomass is usually no possible ( <. To maximize he harves wih inie, one needs o sar beore peak: p h (, ( ( 0 or h (, e W( e This is called he Eumeric Harves. We denoe i h max (. Foreser s Crierion: (Lynch, Secion You are a oreser and you manage several parcels o ores. When a parcel reaches a cerain age, say, you cu all is rees in one operaion and hen replan wih seedlings. Aer ime, he seedlings have grown ino bigger rees, and you cu again. You repea he procedure on his parcel over and over. (Over he years ha his paricular parcel akes o regrow, you deal wih he oher parcels in order o spread your income over ime. This is why you parcel ou he ores. A each cu, since you ake all he wood a he ime in one scoop, he amoun you ge is B( Since you repea his every ime inerval, your harves averaged over ime or his parcel is: B( H 6
7 Naurally, you seek o maximize your harvesing o maximize your revenue. B( Thus, you seek he maximum o H Since he age a which you cu is or you o choose, i is he variable wih respec o which you seek he maximum: dh 1 db ( B ( db ( B ( p Noe ha his sraegy makes you harves beore peak biomass. Yes, you harves early bu you harves more requenly! Wai a minue, hough! This was in order o maximize he biomass harvesed. Wha abou he revenue? Sequenial cohor The Fausman Roaion: (Lynch, Secion Neglecing he cos o harvesing or he momen (or simply subracing a ixed racion o he selling price, he income is: Need o wai ime beore he irs round, beore second round, ec. r 2r 3r pb( e pb( e pb( e 1 pb( r e 1 Maximum income or such ha This is he Fausman Crierion. db( 1 rb( rb( r 1 e 7
8 I reseeding comes a a ixed cos, hen he cos o he successive seedings is: C C C C C r r 2r 3r cce ce ce c e c 1 The proi is: Cos o iniial seeding; all oher reseedings immediaely ollow he sequenial harvess. C ( C ( C ( C pb( c c r e 1 Maximizaion occurs or chosen such ha db( c 1 rb( p e r 1 8
9 Example: (Lynch, Secion 4.2 Le s work a case or which we know he growh rae: Wih his rae o growh, he weigh o an individual over ime is: W ( W 1e g max dw g W W max During his ime, he number o individuals decreases gradually due o some moraliy according o: N( Re The oal biomass a age is: g B( W( N( RW 1e e The peak biomass B p occurs a p wih: g g p e Bp RWmax g g g max g Saring a ime wih eor, he harves h per recrui is ound o be (los o algebra here! 1 1 g h (, Wmax e e g A ixed eor, he harves is maximized i i begins a ime given by: (los o algebra here, oo! opimal 1 g p ln g p 9
10 Economic Harvesing: (Lynch, Secion 4.3 So ar, excep once, we only considered biological aspecs and ried o maximize he amoun harvesed. There were no consideraions o coss and revenues, and o value discouning over ime. Cos o eor: Cos c Revenue rom harves: dh Revenue p p B [Recall: In his chaper, harves H is cumulaive.] Ne: Proi = Revenue Cos p Bc c p B p 10
11 The proi c p B p is posiive only or B c B p 0 p BB 0 Growh prior o harves Decrease during harvesing Harves is inerruped when biomass drops below proiable level. Pos-harves moraliy T Harves sops once B drops o B 0 Now considering ha harvesing akes ime and ha uure income needs o be discouned. Firs, i eor is ininie (hanks o magical echnology!, harves is insananeous bu a ime in he uure; p B( B e 0 r This is maximized when is chosen such ha: d db db 0 r B B 0 r B( B 0 0 The economic harveser does no wai or B o peak. As soon as biological growh is less han inancial growh, B is harvesed (in an insan and sold. The money is pu in he bank and grows here. Increasing ineres rae r hasens he harvesing ime. 11
12 The siuaion is obviously more complicaed when he harves is spread over ime because echnology (, assumed consan allows only a cerain pace. The proi is now calculaed as an inegral over ime: ( r T 0 ( 0 r p B B e p B B e insananeous harves exended harves B0 c p The maximum o occurs when he ollowing relaion holds: T r B( e B( B e 0 r Mixed cohors: (Lynch, Secion During a ishing expediion, one akes rom various cohors a he same ime. oo young maure enough cach We assume: 1. A new cohor sars every year. 2. Each cohor lives muliple years, hus overlapping one anoher. 3. The various cohors live peaceully among one anoher. 4. There is no overcrowding. We ask he same quesions as or a single cohor: 1. A which age do we sar harvesing?? 2. How inensely do we harves?? 12
13 oo young maure enough cach Harvesing across cohors in a single year is equivalen o whole-lie harvesing o a single cohor. Accumulaing across cohors is he same as accumulaing across ime or a single cohor. We hus have a sysem ha will reurn, in a single year, he whole-lie harves o one cohor. This is a biological annuiy wihou urher inancial discouning: p Rh(, c Thus, he proi o one cach is: p Rh(, c I he age o he younges individuals harvesed is chosen o opimize he harves, he harves per recrui h(,m is he eumeric harves, h max (. In his case, p Rh ( max c Eor ha yields he maximum proi No need o make more eor; almos everyhing is already being caugh. 13
) were both constant and we brought them from under the integral.
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
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