Derivation of an Analytic Expression for the Mass of an Individual Fish Larvae in an Uncapped Rate Stochastic Situation.

Transcription

1 New Yrk Science Jurnal 01;5(1) hp:// Derivain f an Analyic Exprein fr he Ma f an Individual Fih Larvae in an Uncapped Rae Schaic Siuain. 1 ADEWOLE Olukrede.O and ALLI Sulaimn.G 1 Deparmen f Phyic, Univeriy f Ibadan, Nigeria. Deparmen f Mahemaic & Saiic, The Plyechnic, Ibadan, Nigeria. Crrepndence viz; kredeadewle@yah.cm ABSTRACT: There i need vividly examine he impac f chaic variain in divere prcee a may apply a ypical grwh mdel. A capped rae chaic prce culd be decribed a bunded by me limi and hrugh delineain f he variu facr affecing he grwh f fih larvae i highly eenial. The change in ma f fih larvae wa cnidered due phyilgical and meablic pahway and her relevan facr vividly examine he cncep f chaic prce a applicable he capped rae mdel, and delineae an analyically derived exprein fr he ma f an individual larvae baed n relevan chaic differenial frmulain in an uncapped rae chaic prce inundaing he I lemma. The uncapped rae iuain i nly a remval f a maximum r imped limi frm a capped rae chaic prce. [ADEWOLE Olukrede.O and ALLI Sulaimn.G. Derivain f an Analyic Exprein fr he Ma f an Individual Fih Larvae in an Uncapped Rae Schaic Siuain. N Y Sci J 01;5(1):86-91]. (ISSN: ). hp:// 13 Keywrd: Schaic prce, uncapped rae, I lemma. 1.0 INTRODUCTION Va number f mahemaical funcin are knwn ake divere frm and f variu uiabiliy a number f daily life applicain, ranging frm bilgical, chemical phyical prcee. A number f prcee defined by me relevan mahemaical funcin and equain can be mdeled. A funcin can be defined uch ha i value d n exceed me capped r maximum imped value. Inereingly, deerminiic and chaic prcee are w well encunered cae, invariably, capped rae prce n pracical grund huld incrprae me chaic variain parameer. The fih recruimen larvae have been cnidered in hi cae. An individual larva find ielf in a precariu iuain, larvae exi in a highly chaic and pachy envirnmen and pe nly limied lcmry and enry abiliy. Larva are mall relaive he paial cale f prey heergeneiy and he urbulen fluid flw a hee paial cale (Pichfrd and Brindley, 001). They al have nly lcal knwledge f heir immediae envirnmen, limied by a viual percepive diance f arund ne bdy lengh (Pichfrd e al, 003), and hey are ubjec maive mraliy, wih a newly hached individual prbabiliy f urvival meamrphi being O(1%) r le (Chamber and Trippel, 1997) driven by ypical mraliy f 10% per day in he larva age (Cuhing and Hrwd,1994). There are phyilgical limi n hw fa an individual can grw, imped by facr uch a gu ize and meablim. A iny minriy f larvae even under favurable cndiin urvive adulhd, hu average larva i m definiely dead pracically. M = min [prey encunered x cnverin efficiency meablic c maximum grwh rae] = min [f 1 (M) Z(M) f (M); g(m)] 1.0 where M repreen he ma f an individual, Z(M) i he rae a which prey i encunered and digeed (dependen n lcal prey cncenrain), f 1 (M) decribe he efficiency f cnvering prey in ma, f (M) repreen meablic c and g(m) i he maximal rae a which individual can grw. Grwh rae deermine he durain f he perid during which larvae are vulnerable gape-limied predar (Fgary e al, 1991). Equain (1.0) abve cnciely decribe a general mdel fr an individual change in ma, M during ime inerval,. Schaic and deerminiic prcee are frequenly encunered in va number f daily applicain including phyical, chemical and bilgical ec. Preciely, he grwh f fih larvae i he ubjec f hi dicuin. The cncep f chaic even r prce cann be aken wih leviy a i pan acr m apec f life and applicain. Variu frm f variain are encunered in ne r mre prcee, indeed in cience, finance menin a few, we cann bu menin chaic prce. A capped rae prce hugh ha a maximum r limi imped bu pracically, here are me flucuain ha wrh being cnidered. 86

2 New Yrk Science Jurnal 01;5(1) hp:// In real ene, flucuain r variain exi, which ake differen frm, uually randm appearing a nie. Fr inance, grwh f an individual i influenced by inernal and exernal facr. Inernal facr are baically, phyilgical and meablic pahway. The exernal facr wuld include facr like prey, predar, which are bilgical and phyical facr like emperaure, humidiy, ph, aliniy, ligh ineniy ec. In a grwh mdel, an individual i expeced grw cninuuly, realiically peaking, here i a limi r maximum grwh limi. If he change in ma i cnidered wih ime, iniially he grwh rae depend n he phyilgical and meablic pahway uch a rae f cnverin f fd ma and amun f fd cnverin uilized. Hwever, here i a maximum r capped grwh limi he individual can aain. An ideal l fr hi wrk i he Cuhing-Hrwd mdel f larval fih grwh (Cuhing & Hrwd, 1994). Accrding he grwh/mraliy hyphei (Cuhing and Hrwd, 1994; Rice e al, 1993), larvae which grw quickly hrugh a mraliy windw have a urvival advanage ver he ha d n (Campana, 1996). Fcung n he fih larvae recruimen in hi udy, he capped rae chaic prce ha been cnidered and he ma change cnidered in cnnance wih phyilgical facr, meablic c and ime wih her relevan facr. Even when a prce i capped, here i expedien need incrprae me chaiciy hereby unveiling a chaic paern..0 Mehdlgy The I inerpreain ha been applied in delineaing he capped rae prce in fih larvae recruimen wih cniderain f me chaic variable deermining he change in ma f an individual fih larvae. The mdel applied in hi wrk i he Cuhing-Hrwd mdel (1994) wih numerical reamen. The change in ma f he individual by hi mdel i given by: M = min [(b(m) Z(M) C(M) x G(M)]. An analyic exprein fr he ma f an individual larvae ha been derived. 3.0 Dicuin Many mahemaical mdel which aemp decribe hi prce ue cninuu apprximain pecifically, an rdinary differenial equain (ODE) i derived. There i an exigen need incrprae chaically in he ODE when uncerainy play a ignifican rle in he prce, fr example when prey are diribued pachly r he predar ha a high mraliy rik. Hwever, he chaic generalizain fen rely n infinieimally mall ime ep, n applicable bilgical yem. N including he unpredicable envirnmen nie in fiherie mdel can lead (and ha lead) errneu predicin f behavir f explied ck, and may have cnribued he deerirain f hee ck (Keyl and Wlff, 008). Deerminiic mdel f recruimen can prvide impran inigh in fih ppulain dynamic in he face f expliain (Fgary, 1993). Hwever, becaue he key naural phenmena are inherenly chaic, deerminiic mdel can be argued be inapprpriae fr qualifying recruimen. Raher, chaic mdel huld be cnruced arrive a recruimen prbabiliy (Pichfrd and Brindley, 001) and inveigae recruimen variabiliy (Frgary, 1991, 1993). There are phyilgical limi n hw fa an individual can grw, imped by facr uch a gu ize and meablim, a much a 99.9% f larvae die befre reaching meamrphi (Campana, 1996), and rapid grwh hrugh he larva age i hugh increae urvival prbabiliie due an increaed abiliy frage fr prey and avid predar (Cuhing and Hrwd, 1994). Tw naïve apprache chaic prce cmprie, he andard Weiner prce and Pin prce. Where M() i defined a a diffuin prce wih cnan drif; dm() = d + dw()..0.0 W() i a andard Weiner prce (Grimme and Sirzaker, 001). Wih inananeu mean zer, Anher apprach; dm() = dn () Ue a Pin nie prce (Feller, 1950). A in equain.0, M() al ha expecain in equain.0.1. Thee w chaic prce culd be applied in principle generalize equain (1.0) excep where capped rae mdel are cncerned, which culd generae n leading reul (Jame e al, 005). A a reul f he chaic prce incrpraed, a chaic differenial equain ha a luin, which ielf i a chaic prce. An earlier wrk relaed Brwnian min wa credied Bachelier, 1900 in Thery f peculain and laer fllwed up by Langevin I and Sranvich. 3.1 Furher Dicuin Cnidering he uncapped deerminiic cae dm =( (f 1 (M) z(m) f (M))d...0. If z(m) i n lnger deerminiic, he infinieimal rae z(m) wuld be replaced by z(m)d + z (M)dW() where z(m) i an average value and z (M) meaure he ineniy f he chaic perurbain by a Weiner prce. Hence he SDE bained i; dm = f 1 (M) z(m) d + z (M) dw() f (M)d

3 New Yrk Science Jurnal 01;5(1) hp:// Equain.0.3 can be lved analyically r numerically depending n he frm f he variu funcin. Schaic differenial equain are ued exenively fr inance in Phyic a Langevin equain, prbabiliy hery and financial mahemaic, numerical luin ec. A ypical chaic differenial equain i f he frm d = (, ) d + (,)db where B dene a Weiner prce (andard Brwnian min). Fr funcin fεn, he I inegral i F[f](ω)= T f(,ω)db (ω)..0.6 S In inegral frm, he equain i; + = (, u) du ( u, u) dbu..0.5 Fr funcin fεn, he I inegral i F[f](ω)= T f(,ω)db (ω)..0.6 S where B i 1-dimeninal Brwnian min. The Sranvich inegral, i an alernaive he I inegral. Unlike he I calculu, he chain rule f rdinary calculu applie Sranvich chaic inergral and he w can be cnvered viz he her fr cnvenience a demanded. A cmparin f I and Sranvich inegral culd be dne. The whie nie equain; dx b(, ) (, ) W d ha he luin given a: b(, x) d (, ) db.0.8 Adping he Sranvich equain a m apprpriae in me cae, fr -cninuuly differeniable prcee B (n) uch ha: B (n) (,ω) B(,ω) a n. Unifrmly (in ) in bunded inerval. Fr (n) each ω, le, (ω) be he luin f he crrepnding deerminiic differenial equain. ( n) dx db b(, ) (, ) d d (n) (ω) cnverge me funcin (ω) and (n) (ω) (ω) a n, unifrmly (in ) in bunded inerval. The luin here, i.e. f he Sranvich SDE = + b (, ) d (, ) db..10 Cincide wih he luin f he equain given by:.0.8. Thu, he luin f he mdified I equain i; = + 1 b (, ) d (, ) (, ) d (, ) db..11 where dene he derivaive f (, x) wih repec (Sranvich, 1966). Thu, he Sranvich inerpreain given by.11 huld be mre reanably adped cmpared he I inerpreain; = + b (, ) d (, ) db.1 which i he reul f he whie nie equain,.0.7. Cnverin beween he I and Sranvich inegral may be perfrmed uing he frmula; T 1 T T 1 ( ) dw ( ) d ( ) dw..1(b) Where i me prce, i a cninuuly differeniable funcin wih derivaive, 1 and he la i an I inegral. W() i ued in deriving a chaic differenial equain. The change in an a f he fih larva i cnidered in he curren udy. Each individual larva hache wih ma, M and grw accrding he equain; M = min ((f 1 (M) z(m) f (M); G(M)).13 Where f 1 (M) i he larva efficiency, a cnvering fd in bima, f (M) i he meablic c. Equain.13 i n lnger an ODE and becme a chaic differenial equain, SDE when he prey cnac rae Z i defined a a randm variable repreening a heergeneu fd upply. Wih he defining funcin f he rae f change i n mh in he minimum funcin f equain.13, Z fllw a Gauian diribuin. SDE are lved analyically and cmpuainally r numerically. The curren udy adped he numerical mehd f luin bain reul n he change in ma, M f fih larva recruimen fr w cae; capped rae and uncapped rae chaic prcee. The uncapped deerminiic cae i; dm = (f 1 (M) z(m) f (M))d..14 If Z i n lnger deerminiic; he infinieimal rae z(m) d huld be replaced by z(m) d + (M)dW () where z(m) i an average value and z (M) meaure he ineniy f he chaic perurbain; aumed be given by a Weiner prce. The SDE hu becme; dm = f 1 (M) z(m) d + (M)dW() f (M) d.14(b). Thi frm f he SDE can be lve analyical r numerically depending n he frm f he funcin. 88

4 New Yrk Science Jurnal 01;5(1) hp:// The ime mauriy in hi cae ha been exenively udied and me analyical luin have been fund, (Tarlin and Taylr, 1981, Grimme and Sirzaker, 001). The deerminiic capped rae cae i; dm = min( (f 1 (M) z(m) f (M), g(m))d..15 quie differen frm he uncapped chaic cae. Equain,.0.5 characeriic he behaviur f he cninuu ime chaic prce a he um f an rdinary Lebeque inegral and an I inegral. A plauible bu very helpful inerpreain f he chaic differenial equain i ha in a mall ime inerval f lengh d, he chaic prce change i value by an amun ha i nrmally diribued wih expecain (, ) and variance (,).The incremen f he Weiner prce are independen and nrmally diribued, hu hi i independen f he pa behaviur f he prce. The funcin i referred a he drif cefficien, i called he diffuin cefficien and he chaic prce i called he diffuin prce. 3. Significance f he udy A ypical ecnmic cenari i dynamic fen characerized by a number f variable. Sme f he variable are predicable a cniderable exen while her are nn-predicable fen characerized by chaic variain. The cncep f mahemaical mdeling kep n gaining grund in recen ime. A ypical mdel invlve cerain reanable aumpin and cniderain f variable ha culd be rarely aic bu uually dynamic ver a ime pace. The cncep f mdelling ha gained prminence in cienific ue and nw becme f exigen need in m her recen daily applicain, even ck exchange. Mahemaical funcin and aiical prbabiliy diribuin funcin are fen vial l in delineaing he cncep f capped-rae mdel in addiin her mdelling applicain. The idea i impe a maximum value n a mahemaical funcin ha purpr explain he eenial behaviur f he iuain in perue. Delineaing he uncapped rae chaic prce baed and frmulain f a mdel fr he ma f an individual fih larva cnidering he Wiener frmulain viz a chaic differenial equain becme exigen in explring he bilgical iuain. 4.0 POPULATION GROWTH MODEL The ppulain grwh mdel i dn = α N ; N given....1 d Where α = r +.W ; W whie nie, cnan. Auming r = r = cnan. By he I inerrelain, (,x) = x viz; = + b (, ) d (, ) d. Fr he chaic differenial equain; d = b (, ) + (, ); b (, x), x ϵ R, (, x) ϵ R. d.b where W i 1-dimeninal whie nie dn = rn d + db implie, dn N r B ( B 0)..3 uing he I frmula fr he equain g(,x) = lnx; x>.4 d(lnn ) = we have; 1. dn + 1 / N dn = 1 d N Hence; dn 1 d (ln N ) d N 1 N dn And hu; ln N 1 r B N r N = N exp 1 r B.7...8(b) Hwever, he ranvich inerpreain give; dn rn d N 0 db.9 Wih he luin; N = N exp (r + B )..30 The mehd f luin culd be analyical r numerical depending n he frm f he variu funcin and luin, where; M = min [f 1 (M) z(m) f (M)g (M)].31 i.e. M = min (prey cunered x cnverin efficiency meablic c x grwh rae) Remving he grwh rae, i.e. he uncapped rae chaic iuain; dm = f 1 (M) z(m)d f (M)d If z(m) i chaic hen; dm = f 1 (M) z(m) d + z (M) dw() f (M) d...3 where z(m) in.3 i an average value. z (M) meaure he ineniy f he chaic perurbain driven by Weiner prce. Frm.3; dm = f 1 (M) z(m)d + z (M)(M)dW() f (M)d.3b 89

5 New Yrk Science Jurnal 01;5(1) hp:// T lve i analyically, le W = 0; M(0) = M dm = [f 1 (M) z(m) f (M)]d + z (M)dW().33 z (M) = M, Okendal.v. 5.1 uing I inerpreain. dm = [f 1 (M) z(m) f (M)d + MdW()...34 Fr a mh grwh funcin i.e. an uncapped rae, i can be hwn uing a generalizain f he cenral limi herem ha hi frmulain i equivalen he SDE wih z (M) = z (M ), prvided he number f fd iem cnumed i large (Feller, 1950, Whi, 00). Fr inance, Binmial cnverge Pin a N. Fr large N and = z(m) a he mean arrival rae, a, G(M) like Pin: uppreed. ln M 1 r W M And hu; M = M exp x e..35 r 1 W behave viz inegrain f rdinary calculu and cmparin wih he I inerpreain frm he grwh mdel in.8. A an exenin f he prblem in vgue, he uncapped rae chaic prce ha been added having menined he change in ma f he fih larval recruimen al fr he capped rae chaic prce. The uncapped rae chaic cae differ by ju a remval f he grwh rae. The ime mauriy in hi cae ha been udied exenively and me analyical luin have been fund (Karlin and Taylr, 1981; Grimme and Sirzaker, 001). Having adaped he change in ma fr fih larva recruimen viz he ppulain grwh mdel, we deduced an exprein previuly aed in cmparin wih he I inerpreain fr he individual ma, M grw and becme M viz; M = M exp 1 r W ] viz inegrain f rdinary calculu. 4.1 Schaic Differenial Equain Equain (1.0) abve becme a chaic differenial equain (SDE) and n lnger an ODE when he prey encunered rae, Z i defined a a randm variable repreening a heergeneu fd upply. Schaic differenial equain incrprae Whie nie, which can be een a he derivaive f Brwnian min r he Weiner prce. I can al incrprae her ype f randm flucuain uch a jump prce. Firly, he uncapped deerminiic cae i; dm = (f 1 (M) Z(M) f (M))d (.18) If Z(M) i n lnger deerminiic, he infinieimal rae Z(M)d huld be replaced by Z(M)d + z (M) dw(), where Z(M) i an average value and z (M) meaure he ineniy f he chaic perurbain (aumed be driven by a Wiener prce). The SDE i; dm = f 1 (M) Z(M)d + z (M)dW() f (M)d...(.19) The SDE can be lved analyically r numerically depending n he frm f he variu funcin. Fr he capped rae, we have; dm = min (f 1 (M) Z(M) f (M); g(m)d (.0) When meablic c, f (M) = 0, he ime mauriy i imply he Nh arrival f he Pin prce, where N i calculaed frm M ma and f 1 (M). In he cae where f (M) 0, he prblem can be aken a ha f a randm walk hiing a mving barrier. Frm he cenral limi herem, hi i equivalen he SDE wih z (M) = Z(M), prvided he number f fd iem cnumed i large (Feller, 1950; Whi, 00). The change in ma i very eniive he ime inerval chen when n limiing prce i reached a CONCLUSION I i a wrhwhile ak explring a grwh mdel, preciely here fih recruimen larvae. The grwh mdel plauibly, culd be aumed be capped. Hwever, he aainmen f capped rae chaic iuain migh n be ufficien pracically due me chaic r randm flucuain in real life prcee, hu cniderain f an uncapped rae iuain becme expedien. The chaic variain in uncapped rae iuain and he Wiener driven nie are pracically ineviable. Fr a ypical grwh mdel, change in ma i cnidered wih repec he individual ma and al vial fr cniderain are phyical and phyilgical facr, epecially in an uncapped rae chaic iuain. Specifically, meablic c, mean prey arrival rae, cnverin efficiency f fd, ime inerval were cnidered here. Hwever, cnidering an I Inerpreain, an analyical exprein ha been derived fr deermining he ma f an individual fih larvae in an uncapped rae chaic prce r iuain. REFERENCES 1. Abramvich M & Segun IA. Handbk f Mahemaical funcin, 3rd edn. New Yrk: Dver

6 New Yrk Science Jurnal 01;5(1) hp:// Cuhing DN, Hrwd JW. The grwh and deah f fih larvae. J.Plankn Re. 1994:16: Chamber RC and Trippel AE. Early Life Hiry and Recruimen in Fih Ppulain: Lndn, Chapman and Hall Feller W. Prbabiliy hery and i applicain. New Yrk: Wiley Fgary, M.J. Recruimen in randmly varying envirnmen. ICES Jurnal f Marine Science. 1991: 50: Fgary MJ, Sieenwine MP and Chen EB. Recruimen variabiliy and he dynamic f explied marine ppulain. TRENDS in eclgy and evluin. 1991:6(8): Grimme GR & Sirzaker DR. Prbabiliy and randm prcee. 3 rd ediin, Oxfrd Univeriy Pre, Jhnn NL, Krz S. Cninuu univarae diribuin I. Bn, MA; Hughn Mifflin Cmpany Jame A, Baxer PD, Pichfrd JW. Mdelling predain a a capped rae chaic prce, wih applicain fih recruimen. Jurnal f The Ryal Sciey..005: Karlin S, Taylr H. A ecnd cure n chaic prcee. New Yrk: Academic pre, New Yrk. 1981: Økendal B. Schaic differenial equain. An inrducin wih applicain, 6h edn. New Yrk: Springer, Pichfrd J, Brindley, J. Prey pachne, predar urvival and fih recruimen. Bull. Mah. Bil. 011:63: (di:10/006/buin,001.30). 13. Sranvich. A new repreenain fr chaic inegral and inegrain: SIAM J.Cnrl. 1966:4: Whi W. Schaic prce limi. In Springer erie in perainal reearch. New Yrk: Springer /4/01 91