Predator Population Dynamics Involving Exponential Integral Function When Prey Follows Gompertz Model

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1 Op Joral of Modllig ad Simlatio, 25, 3, 7-8 Pblishd Oli Jly 25 i SciRs. Prdator Poplatio Dyamics Ivolvig Expotial Itgral Fctio Wh Pry Follows Gomprtz Modl yl Tay Gosh *, Prachadra Rao Koya School of Mathmatical ad Statistical Scics, Hawassa Uivrsity, Hawassa, Ethiopia * ayl_tay@yahoo.com, drpraoccc@yahoo.co.i Rcivd 2 pril 25; accptd J 25; pblishd 5 J 25 Copyright 25 by athors ad Scitific Rsarch Pblishig Ic. This wor is licsd dr th Crativ Commos ttribtio Itratioal Lics (CC BY). bstract Th crrt stdy ivstigats th prdator-pry problm with assmptios that itractio of prdatio has a littl or o ffct o pry poplatio growth ad th pry s grow rat is tim dpdt. Th pry is assmd to follow th Gomprtz growth modl ad th rspctiv prdator growth fctio is costrctd by solvig ordiary diffrtial qatios. Th rslts show that th prdator poplatio modl is fod to b a fctio of th wll ow xpotial itgral fctio. Th soltio is also giv i Taylor s sris. Simlatio stdy shows that th prdator poplatio siz vtally covrgs ithr to a fiit positiv limit or zro or divrgs to positiv ifiity. Udr crtai coditios, th prdator poplatio covrgs to th asymptotic limit of th pry modl. Mor rslts ar icldd i th papr. Kywords Expotial Itgral Fctio, Gomprtz Modl, Poplatio Growth, Prdator, Pry. Itrodctio Th prdator-pry problm has b itrstig to may rsarchrs []-[7]. Modllig poplatio growth of itractig spcis ivolvs diffrtial qatios [] [2]. Th biological spcis itract i may ad complx ways that may affct th poplatio compositios ovr tim, d to atral or artificial or maagmt rasos. Prdatio ca icras, dcras, or hav littl ffct o th strgth, impact or importac of itrspcific comptitio [3]. Thy idicat that thr ar cass i which prdatio has vry littl ffct o comptitiv itractios. * Corrspodig athor. How to cit this papr: Gosh,.T. ad Koya, P.R. (25) Prdator Poplatio Dyamics Ivolvig Expotial Itgral Fctio Wh Pry Follows Gomprtz Modl. Op Joral of Modllig ad Simlatio, 3,

2 . T. Gosh, P. R. Koya It is discssd i [4] that th t ffcts of itrspcific spcis itractios o idividals ad poplatios vary i both sig (positiv, zro, gativ) ad magitd (strog to wa). Itractio otcoms ar cotxtdpdt wh th sig ad/or magitd chag as a fctio of th biotic or abiotic cotxt. Th prdator-pry problm with th assmptios of littl or o ffct of prdatio o th pry poplatio growth is stdid i [6] [7]. I this stdy, th pry poplatios ar assmd to grow accordig to logistic, Vo Brtalaffy ad Richards modls. Th rslts show that th prdatio affcts th prdator poplatio i sch a way that its growth ithr covrgs to a fiit positiv limit or to zro or divrgs to positiv ifiity. Thr ar svral optios to cosidr amog th gralizd growth modls [8]. Ths icld, for xampl, gralizd logistic, particlar cas of logistic, logistic, Richards, Vo Brtalaffy, Brody, Gomprtz, gralizd wibll, wibll, moomolclar, mitschrlich ad mor. Bhavior of th growth modls has b frthr stdid i [8]-[]. Th followig sctios ar prstd as follows: prdator-pry modls ar prstd i Sctio 2; Gomprtz modl i Sctio 3; soltio for th Prdator-pry qatios i Sctio 4; simlatio stdy i Sctio 5; aalysis of phas diagram ad qilibrim poits i Sctio 6; ad coclsios i Sctio Prdator-Pry Modls Th classical Lota-Voltrra prdator-pry modl is giv by: dx = ax bxy; dt () dy = cy + dxy. dt whr a, b, c ad d ar positiv costats. Th paramtr a is itrisic growth rat of th pry, b is rat of cosmptio of pry by prdator, c is mortality rat of prdator at absc of pry ad d is rprodctio rat of prdator d to cosmptio of pry. I th prst wor, w cosidr th cas wh th itractio of th pry ad prdator poplatios lads to a littl or o ffct o growth of th pry poplatio, that is b. W also assm th paramtr a is a fctio of tim. Ths th assmptios of th classical prdator-pry modl ar rlaxd. Th proposd prdator-pry modl is [6]: dx = r( t) x; dt (2) dy = vy + sxy. dt whr x is poplatio siz or dsity of pry; y is poplatio siz or dsity of prdator commitis i th systm. Hr w assm r to b a rlativ growth rat fctio which is positiv vald fctio of tim t. Th paramtr ss is rprodctio rat of prdator d to cosmptio of pry ad v is mortality rat of prdator at absc of pry. Both ar positiv costats. Th pry Eqatio i (2) is th first ordr diffrtial qatio. Th soltios of this first ordr diffrtial qatio ar stdid as growth modls i [9]. This implis that a pry modl ca b slctd from th larg family of growth fctios i [8] [9]. Giv pry s modl, w ca solv th diffrtial qatio of th rspctiv prdator poplatio i (2). Th gral approach for solvig Eqatio (2) cosists of th followig stps: ) ssm that th impact of prdator o pry poplatio growth is gligibl, 2) Prdator poplatio dclis i absc of pry, 3) Prdator poplatio grows with a rat proportioal to a fctio of both x ad y, 4) ssm that thr is prior iformatio abot th pry poplatio that x follows a ow growth fctio. Gomprtz growth modl i this cas. 5) Solv for prdator poplatio growth y. 3. Gomprtz Modl for Pry Poplatio Growth W assm that th growth of pry poplatio follows Gomprtz growth modl ad costrct th corrspodig 7

3 . T. Gosh, P. R. Koya prdator growth fctio. Th Gomprtz modl is giv i [9]-[2] as follows: whr B= log, x xp B t xt = (3) = is iitial poplatio siz, is asymptotic growth of th poplatio rprstig carryig capacity, ad is absolt growth rat paramtr of th pry. Th rspctiv rlativ growth rat is rt = log{ xt }. Th growth crv has a sigl poit of iflctio at tim a= ( ) log{ log ( )}. Dtaild discssio is fod i [8]-[2]. 4. Soltio of th Prdator-Pry Eqatios Hr, w solv th ordiary diffrtial qatios i (2), th dtrmi itrsctio poits at which th pry ad prdator poplatio attai sam vals, ad fially thr spcial cass of prdator poplatio ar cosidrd. 4.. Drivatio of th Modl Th soltio for th ordiary diffrtial qatio i (2) is drivd assmig that pry follows th Gomprtz modl i (3). ftr sbstittig (3) i (2), th corrspodig prdator poplatio growth fctio is drivd to b: Or qivaltly, whr y y x log log s ( v s) t =! y = y (4) y t x log = y log x log log v s s =! = rprsts iitial prdator poplatio siz. Th dtaild drivatios ar giv i ppdix. Eqatio (4) is also qivalt to th followig soltio (6) that ca b xprssd i trms of th wll ow xpotial itgral fctio Ei: y t s x vt Ei log Ei log = y (6) Not that th xpotial itgral fctio is a poplar fctio that is oft sfl i may applicatios. W bliv that this rspctiv prdator poplatio growth fctio y( t ) ca b sfl as wll. I fact, Eqatio (6) ca b xprssd algbraically as th Ei fctio: log Ei log Ei log s y s y x v = + t Th prdator modls i Eqatios (4-7) appar to b w fctios ad thy do ot match with ay o of th commoly ow growth modls Poits of Itrsctio Poits of itrsctio ar th poit of tim at which th pry ad prdator poplatios attai th sam sizs. Whvr it occrs, lt th poit of itrsctio b rprstd by p xt = yt i t, i.. w mst hav ( p) ( p) Eqatios (3) ad (4). I tryig to solv ths qatios, w gt th followig xprssio xprssd xplicitly as: t log N( t) t = sv log ( ) ( y ) (5) (7) 72

4 . T. Gosh, P. R. Koya whr N t Eqatio (7) ca oly b comptd mrically Spcial Cass log ( ) = =! To frthr drstad th modl i Eqatio (4), thr spcial cass ar idtifid which ar dpdt of birth ad dath paramtrs of prdator poplatio. Th cass ar cosidrd hr blow. Cas I ( s = v) : I this cas th fctio dscribig th poplatio growth of prdator i Eqatio (4) tas x log log s th form y( t) = y xp =!. It is frthr obsrvd that th prdator poplatio covrgs to lowr or ppr asymptot dpdig o th iitial val of th prdator poplatio. Th iitial poplatio siz ca b largr or smallr tha. Ths, th limitig val for th prdator poplatio is xp s log y y! = =. It is itrstig to ot that both th pry ad prdator poplatio sizs covrg to th sam asymptot providd that th paramtr s is assigd th val s = s log ( y) whr s =. lso th = log ( )! prdator poplatio siz covrgs to a asymptot abov or blow that of prdator poplatio siz dpdig o s = s or s = s rspctivly. Cas II ( v > s) : I this cas, th prdator poplatio dcays from y to zro as t whil th pry poplatio grows from to. Cas III ( v < s) : I this cas th prdator poplatio grows from y ad ltimatly divrgs to + as t whil th pry poplatio grows from to as xpctd or rstrictd. Th miimal poit at which th prdator growth crv trs or gts miimm val is fod to b: log ( ) tmi = ( ) log. Th th vals of pry ad prdator poplatios ar xt ( mi ) = vs ad log ( s v) s v v log log log ( ) s s y( tmi ) = y xp = log ( s v)!, rspctivly. This rslt is diffrt, for xampl, for th cas of logistic pry modl [7] for which th miimm poit occrs at s v v s tmi = log ad y( tmi ) = y s s s v v. v 73

5 . T. Gosh, P. R. Koya vs of prdator growth is propor- = whr c is a costat that ca b rlatd to a itrvtio factor applid o pry optimal siz, or a factor that ca b applid ithr to th prdator s birth parav cs vc s=. Sch itrvtio ca hc b ap- Th cass ca b gralizd to a statmt that ratio of daths to births tioal to asymptot ( ) of pry growth. That is, v s c mtr { = }, or o th prdator s dath paramtr { } plid to th pry or prdator paramtrs. Th drivatios providd i this papr corrspod to th cas that c =. For c diffrt from, th rspctiv drivatios ca similar b mad. 5. Simlatio Stdy Th simlatio stdy is carrid ot basd o Eqatio (4). Th stdy is dsigd by varyig th modl paramtrs:,, for pry poplatio followig Gomprtz modl ad y,, s v for prdator poplatio. Th stdy dsig is as follows: Pry modl: Gomprtz growth modl Pry modl paramtrs: =, = 2, =.. Prdator modl paramtrs: y =.5 is iitial poplatio siz; birth rat s, dath rat v. Cass: Cas I: s = v, Cas II: s < v, Cas III: s > v Cas I: s = & v = 8 ; s =. & v =. ; s =.356 & v =.356 ; s =. & v =.; s =.2 & v =.2 Cas II: s =. & v =.6 ; s =. & v =.2 ; s =. & v =.5 ; s =.2 & v =.25 Cas III: s =. & v =.5 ; s =. & v =.65 ; s =. & v =.95 ; s =.2 & v =.95 Th rslts of th stdy ar displayd i Figrs -3. Figr displays plots for th Cas I, whr th ratio of dath to birth of prdator is qal to asymptotic siz of pry. I th simlatio, th birth rat is varid from smallr to largr vals. Th rslt rvals that th prdator poplatio siz dcrass ad vtally covrgs to a positiv qatity at varios rats. Not that for a particlar val of th birth paramtr ss, th poplatio sizs of both pry ad prdator covrg to sam asymptotic val dotd by. Growth Cas I pry prd: s=-,v=-8 prd: s=.,v=. prd: s=.356,v=.356 prd: s=.,v=. prd: s=.2,v= Tim Figr. Plots of prdator poplatio dyamics wh pry follows Gomprtz growth modl for Cas I with =., =, = 2, y =.5. 74

6 . T. Gosh, P. R. Koya Growth Cas II pry prd: s=.,v=.6 prd: s=.,v=.2 prd: s=.,v=.5 prd: s=.2,v= Tim Figr 2. Plots of prdator poplatio dyamics wh pry follows Gomprtz growth modl for Cas II with =., =, = 2, y =.5. Growth Cas III pry prd: s=.,v=.5 prd: s=.,v=.65 prd: s=.,v=.95 prd: s=.2,v= Figr 3. Plots of prdator poplatio dyamics wh pry follows Gomprtz growth modl for Cas III with =., =, = 2, y =.5. Figr 2 displays plots for th Cas II, whr th ratio of dath to birth of prdator is largr tha. For th simlatio, w varid th rats with small magitds. Th rslt shows that th prdator poplatio dcrass ovr tim ad vtally covrgs to zro. Figr 3 displays simlatios for Cas III, whr th ratio dath/birth of prdator is lss tha. Plots ar mad for varios vals of th rats. Th rslt shows that th prdator poplatio dclis for som tim ad Tim 75

7 . T. Gosh, P. R. Koya th icrass ad vtally divrgs to ifiity. W hav show by simlatio stdy that th prdator poplatio ithr covrgs to a fiit limit or covrgs to zro or divrgs to ifiity o th positiv sid dpdig o th paramtr vals dr th assmptio that pry follows Gomprtz growth modl. Morovr, for a particlar val of birth paramtr ss, th poplatio sizs of both pry ad prdator covrg to sam asymptot. Ths fidigs ar similar with thos i [7] [8]. 6. alysis of Phas Diagram ad Eqilibrim Poits Th wly proposd prdator-pry modl (2) i its fll form ca b xprssd, i cas of Gomprtz growth of dx dt = xlog x ad dy dt = vy + sxy. Th two qilibrim pry poplatio, as th systm of qatios poits of this systm ar fod to b ( x, y ) = (,) ad ( x, y ) (,) 2 2 = sic at both ths poits th cssary ad sfficit coditios dx dt = ad dy dt = ar satisfid. lso th Jacobia matrix of th systm of qatios is J( xy, ) ( ( x) ) log =. W ow aalyz th atr of th qilibrim poits sy v + sx blow ad display th smmary i Tabl. x, y =, Natr of th Eqilibrim Poit Th Jacoba matrix at this qilibrim poit tas th form J( x, y ) J(,) ad th corrspodig igvals ar ( log ( ) ) ( ( ) ) log = = v = ad 2 = v. Rcall that all th paramtrs,, ad v ar positiv qatitis ad ths hr aris th followig two cass for whil 2 is always gativ. Coditio I log ( ) < : I this cas, both th igvals ad 2 ar gativ ad hc th { } qilibrim poit ( x, y ) = (,) is stabl. Coditio II { log ( ) } hc th qilibrim poit ( x, y ) (,) > : I this cas, th igvals is positiv whil 2 = is stabl. Natr of th Eqilibrim Poit ( x, y ) = (,) 2 2 Tabl. Smmary of stabilitis of th qilibrim poits. is gativ ad Eqilibrim poit Eigval Coditio Sig of igval Natr of qilibrim poit ( x, y ) = (,) λ log = = v 2 log < log > Both ad 2 ar gativ 2 is positiv ad is gativ Stabl Ustabl s = v 2 is gativ ad is zro Stabl ( x, y 2 2) = (,) = 2 = v + s s < v Both 2 ad ar gativ Stabl s > v 2 is gativ ad is positiv Ustabl 76

8 . T. Gosh, P. R. Koya 2 2 Th Jacoba matrix at this qilibrim poit tas th form J( x, y ) J(,) = = v + s ad th corrspodig igvals ar 2 = ad = v + s. Th paramtrs, v, s ad ar positiv ad ths implis that 2 is always gativ bt thr cass for. Coditio I { s v} is stabl. Coditio II { s v} (, x2 y2) (, ) = is stabl. Coditio III { s v} stabl. 7. Coclsios = : I this cas, v s λ = + is zro ad hc th qilibrim poit ( x, y ) = (,) < : I this cas, both th igvals ar gativ ad hc th qilibrim poit > : I this cas, oly 2 2 λ is positiv. Hc th qilibrim poit ( x, y ) (,) 2 2 = is Som mathmatical aspct of th wll ow prdator-pry problm is stdid by modifyig th rspctiv classical assmptios. W assm that th pry poplatio growths atrally with o itractio ffct d to prdatio ad rat of growth is o-costat. Th, th prdator-pry qatios ar solvd cosidrig pry grows as Gomprtz modl. Th soltio for th prdator poplatio is fod to ivolv th xpotial itgral fctio ad is qivaltly xprssd i trms of Taylor s sris. Th simlatio stdis ad frthr aalysis of th modls rval that th prdator poplatio grows i sch a way that ithr covrgs to a fiit limit or zro or divrgs to positiv ifiity. Thr is a sitatio at which both pry ad prdator poplatios covrg to th sam limit irrspctiv of thir iitial poplatio sizs. Thr is also a sitatio whr th prdator poplatio attais a miimal poit bfor it divrgs to ifiity. Morovr, two qilibrim poits ar idtifid which ar stabl oly dr som spcific coditios. Rfrcs [] Bars, B. ad Flford, G.R. (29) Mathmatical Modllig with Cas Stdis: Diffrtial Eqatios pproach sig Mapl ad MTLB. 2d Editio, Chapma ad Hall/CRC, Lodo. [2] Loga, J.D. (987) pplid Mathmatics Cotmporary pproach. J. Wily ad Sos, Hobo. [3] Chas, J.M., brams, P.., Grovr, J.P., Dihl, S., Chsso, P., Holt, R.D., Richards, S.., Nisbt, R.M. ad Cas, T.J. () Th Itractio btw Prdatio ad Comptitio: Rviw ad Sythsis. Ecology Lttrs, 5, [4] Chambrlai, S.., Brosti, J.L. ad Rdgrs, J.. (24) How Cotxt Dpdt ar Spcis Itractios? Ecology Lttrs, 7, [5] Brlow, E.L. (999) Strog Effcts of Wa Itractios i Ecological Commitis. Natr, 398, [6] Grvitch, J.J., Morriso,. ad Hdgs, L.V. (2) Th Itractio btw Comptitio ad Prdatio: Mta- alysis of Fild Exprimts. Th mrica Natralist, 55, [7] Dawd, M.Y., Koya, P.R. ad Gosh,.T. (24) Mathmatical Modllig of Poplatio Growth: Th Cas of Logistic ad Vo Brtalaffy Modls. Op Joral of Modllig ad Simlatio, 2, [8] Koya, P.R., Gosh,.T. ad Dawd, M.Y. (24) Modllig Prdator Poplatio ssmig That th Pry Follows Richards Growth Modl. Eropa Joral of cadmic Essays,, [9] Koya, P.R. ad Gosh,.T. (23) Gralizd Mathmatical Modl for Biological Growths. Op Joral of Modllig ad Simlatio,, [] Koya, P.R. ad Gosh,.T. (23) Soltios of Rat-Stat Eqatio Dscribig Biological Growths. mrica Joral of Mathmatics ad Statistics, 3, [] Gosh,.T. ad Koya, P.R. (23) Drivatio of Iflctio Poits of Noliar Rgrssio Crvs Implicatios to Statistics. mrica Joral of Thortical ad pplid Statistics, 2, [2] Wisor, C.P. (932) Th Gomprtz Crv as a Growth Crv. Procdigs of th Natioal cadmy of Scics of th Uitd Stats of mrica, 8,

9 . T. Gosh, P. R. Koya ppdix Drivatio of Prdator Poplatio Modl giv Pry follows Gomprtz Growth Modl ssm th pry poplatio growth ca b rprstd by th Gomprtz fctio: Th th prdator qatio ca b solvd as: Sbstittig (i) i (ii) givs B xt =, B= log dy dy = vy + sxy = [ v + sx] dt logy = vt s xdt dt y + (ii) B log d y = vt + s t W ow itrodc a w variabl for th prpos of valatig th itgral as So as to gt: = B d = dt dt = d To valat th itgral, w ow s Taylor s sris xpasio of Hc Usig (iv) i (iii), w gt: = or =! s logy = vt d (iii) 2 = ! 3! d = log + =! as: (iv) s logy = vt log + C = + (v)! To dtrmi th itgral costat C w ow impos th iitial coditios ad y. That rdcs (v) to b: To limiat C, sbtract (vi) from (v) to gt: t B Bt log = log = B Ths, (vi) ca b rarragd as: s logy = log + C = +! (vi) y s t t log = vt log + = y (vi)! x log =. t with ( t) = B = log, x log log s ( v s) t =! y t = y (vii) (i) 78

10 . T. Gosh, P. R. Koya Ths y( t ) is prdator poplatio siz at t. Usig y t ( x) ( ) log = y log ( x) ( ) log = log ( v s) x log log s =! i (vii), w hav Th rlatioship (viii) is a phas path qatio. It ca b sd to aalyz phas path diagrams. (viii) ppdix 2 Show th Soltio with Expotial Itgral Fctio ad th Taylor s Sris of th Prdator Poplatio ar Eqivalt. Expotial itgral fctio Ei( x ), for small vals of x, is giv by Maclari sris as: Ei x = + + (i) =! ( x) γ logx Hr γ =.5772 is calld Elr s costat. Usig (i), th xprssios for Ei log x ad Ei log ca b obtaid as follows: O sbtractig (iii) from (ii), w gt Bt th xprssio x log x x Ei log γ log log = + + = (ii)! log x (iii)! Ei log = γ + log log + = x x log log log x = + (iv)! log Ei log Ei log log = x log log log Hc, sig (v) i (iv) w gt ca b simplifid sig (2) as x log B log = log = log ( ) = B log x log log x Ei log Ei log t = + = (vi)! (v) 79

11 . T. Gosh, P. R. Koya Usig (v) i (i), w gt Or qivaltly y = y x log log s vt t + =! x log log s ( v s) t =! y = y (vii) Ths (vii) is th rqird prdator qatio xprssd sig xpotial itgral fctio Ei( x ). Sic (vii) is drivd from (i), it ca b drstood that both th soltios obtaid sig Taylor s sris xpasio ad Expotial itgral fctio agr with ach othr. Not that th idfiit itgral d togthr with th iitial coditio L th smi-dfiit itgral as a w dw. Th lowr limit is fixd. w = a ca b xprssd as 8