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Transcription

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2 ') ;:" / ' - Macom Fitz-Ear, Ian McMian, Lonard Butr and Dougas Robson* A GENERAL THEORY OF EGG PRODUCTION by Dpartmnt of Zooogy Univrsity of Toronto Toronto, Ontario, Canada Introduction Egg production in organisms which ay ggs continuousy throughout thir adut iftims, can gnray b subdividd into two componnts. Th first componnt is an incrasing phas to a maximum gg production rat and th scond componnt a dcrasing phas from this maximum. For typica gg production profis th simpst mod invovs a von Brtaanffy growth componnt and an xponntia dcay componnt. Th ovra profi is th rsutant of ths two componnts. Th Mod For simpicity, considr gg production as a two stag procss. Matur ggs dvop from 'primordia' gg cs and ar thn xpd from th ovary as 'dvopd' ggs. Assum that th initia numbr of primordia gg cs (A ) is fixd at tim t at which tim thy bgin to dvop at a constant instantanous rata. Thn th numbr of primordia gg cs rmaining at tim t~ t is A(t) = A Assum that matur ggs ar dpositd at a constant instantanous ratc(. *Biomtrics Unit, Corn Univrsity, Ithaca, Nw York, U.S.A.

3 - 2- If B( t) is tho n.m:-:.bcr of rr.o.turo ggs prsnt in th fma at tim t, a..'1d B ( t ) =, -1-' vnn -1 ' ) ' \ -c ::; o<b(t), whr X(t) is t~1 rat of dposition of matur ggs. Th 1 rat of chang in th numbr of matur ggs prsnt in th fma is th diffrnc btwn thir rat of dvopmnt and thir rat of dposition. I. B (t)"' i\a(t)- o<.:s(t). Thrfor, Soving this quation with th initia conditions A( t ) = A and B( t ) =., wy hav : v1hr c =,... 1 B(t) = \ c<'t I constant and A( t) = A J o<t'aa(t) dt + - ~(t - t o ) c ott.i Ronco B(t) "' ot -'At dt,.,t -('A- «)t dt + c o<t -o<t + C [:.. ; -. But :S(to) - «Xto = and sinc ::1 c ::: - ota f.. A 1\_oC f (-A.) (/\-~:!.) ' «to ).to -('A-:- «)t So that B(t) - ~t =?\A ~to { (?.- o<) ('!; - to)}' "A-OC. Sinc N( t) = ( :s( t), N(t) = or, otting and dfining }.1= o< I\ A 1\- cj. ot w cbt~in th mod:

4 -3- :;( t ) = I.1 ~ 1 v1hr I\i is th potntia maximum daiy gg production to is th initia day of gg aying s is th rat of incras in gg production d.. is th rat of dcras in gg production Drivations from th mod Th tota c;g prociuc-:ion of a fma during hr productiv iftim mccybo fo~~d by intgrating th function N(t) ovr th tim intrva (t, t 6 ), whr:o t > i::j th tim at which th fma bgins gg production and t 6 is. ac at. dath, T(t, t ) "" s t ~s i :M ~(t - t ) - O(t (1 - - ) dt = M - cc:t Ltting t ~ c.o s th bracktd trm of this xprssion approachs unity and Yi obtain potntia iftim gg production, - ~t T(t, oo) = M C(( + ~3) Aso, T(t, ~) may aso b corratd with th production T(t 1, t 2 ) during a spcific tim intrva, By diffrntiating th quation for th gg production mod, '. N ( t) = :M ( ( ) - i - ~(t- ~ + ~ +~ to)~o--~t Thcroforo, maximum gg production occurs at

5 t = t max \~+"' } og.( ~ II.. It foov,rs that th maximum gg production rachd is, r;.('s 11 ) (... --of-to(~. ) (1_ +~1) -:s +\A ~hrfor th rationship btwn th potntia tota iftim, T(t, co), and th maximum rrg production rachd is r \ 1 T(t w) =.::.. ~ o' ~ t Intrstrai:n gg production comparisons Potntia iftim gg production, T(:~, co) has -bn xprssd as = - c<t M Th constant M was dfind as M"" ~c s+ ~) s - ~t M A ()(.to Thrfor A = == T(t,(X)) o<( + t;;(1),. Sinc A is dtrmind at t, it is indpndnt of o< and ~ =-o( + ).. Thrfor ) th potntia iftim gg production T(t, ) is fixd at t and is indpndnt of th paramtrs ()( and ~ Lt th numbr of ggs actuay aid by th fma during hr iftim b T(t, t~) whor t ij s r(t~) = T(t, t ) /A ~ s production actuay raisd. HoncG r(t ) = s is th tim of dath of th fma. Thn is th proportion of th tota potntia gg 1 1 <'\ - r.( ( t ~ so..., -t ) -'A( t so -t ) 1 - rt.- o< A

6 - 5 - r.'\ E ~ r( -t) ~ "' p (.A, c<) 'JD :::: ~ E{ ' ).. ('A - <);). -( (,(- ).)(t -t ) - [:- -.n _ G s s "(;,p :::: A. ('A - o( )( t -t ) jt ('A- o<)~ A(t -t (to<- ~)(ts-to) f o ) [. J 1 E { ( 'A - t. )( t s -to). -( ( t s -t o ) C1 IJ.'hrfor p ('A, II() is an. incrasing function of' both r;j... and A sction prssur, '-'< and A woud tnd to incras indfinity. With no Howvr, nvironmnta sction woud dtrmin intrmdiat vaus of' ths paramtrs to ~aiioiz fitnss. This impis that strains of' animas of th sam spcis. rard in th sam nvironmnt vvoud hav simiar p (<X, 'A). is not a suitab basis f'or comparison of gg production btwn such strains. - Sinc T( t,oo) is indpndnt of' o< and A it thrfor bcoms th ogica basis of intrstrain gg production comparisons. Rfrncs McMian, I., M.Fitz-Ear and D.S.Robson, Quantitativ gntics J of frtiity. I Liftim gg production in Drosophia manogastr - Thortica. GE~~TICS (In prss) :r:icmia..'", I., ]:.Fi tz-ear and L.Butr, Quantitativ gntics of frtiity. II Liftim gg production in Drosophia manogastr - Exprimnta. (subrni ttd for pubication) ]!Irtz, David B., 1969 Lif History Phnomna in Incrasing and Dc~asing Popuations. Unpubishd MS 39 pp.

7 - 6 - Appndix A modification of th mod to incud gg oss bfor dposition Primordia gg cs Matur ggs kab ~d A(t) - B (t) Dpositd \~ Lost k ao Lost ~ Th gg production mod as givn arir dos not tak into considration oss ~f ggs ithr at th primordia stag or at th matur gg stag. In practic, such gntic wastag coud xist through th atrophy and rabsorption of (1) th unfrtiisd dvoping ggs or (2) th frtiisd ggs, through th ffct of th ma on zygot viabiity. Th first drivation of th mod invovd a two 'compartmnt' systm: th primordia gg cs (A(t))dvoping into matur ggs (B(t)) at an instantanous rat kab' thn bing dpositd from th ovary at an instantanous rat ~d' If w now incud th oss rat~ from ach compartmnt, kao and kbo for th primordia and matur stags rspctivy, thn with initia conditions A(t ) = A and B(t ) = as bfor, th quations to b sovd ar I A (t) I B (t) ~h soutions ar - (k + k b)(t- t) ao a o A(t) = A

8 B(t) = k a b A o -(k ao + k a b)(t- t o )} Thus, th rat of gg dposition is N(t) ; - -[(k b+k )-(kd+~ )](t-t) a ao -o -oo o Ltting k b + k a ao = thn ~ = A - o< and = M. 4tnc w obtain th mod in th arir form: N(t) - -~ (t-t ) - ( t = M( - ) ~ It is to b notd that sinc two additiona oss rats hav now bn introducd I into th mod, th k s cannot b stimatd. Sinc A, th numbr of primordia gg cs at tim t, is indpndnt of a variabs, it is th bst gg production paramtr for charactrising a strain. ar matd with a varity of mas, Hnc whn fmas of a crtain strain X it might b xpctd that A is charactristic to a fmas and thrfor th vaus of A shoud b th sam irrspctiv of th ma. In practic, this is not th cas and th discrpancis coud b accountd for by th various gg osss which hav bn outind abov and incudd in th modifid mod. Now A can b radiy obtaind from th rationship A =. tit I M /x(+ ~) = T(t 5 oo). Howvr, this dos not us th oss factors incudd in th modifid mod and is thrfor inaccurat. Unfortunaty, it ~s impossib to obtain vaus for th k 1 s and hnc A cannot b stimatd using A