ANALYSIS OF A PREDATOR-PREY POPULATION MODEL

Transcription

1 1 AALYSIS OF A REDATOR-REY OULATIO MODEL SWARALI SHARMA G.. SAMATA Deprtment of Mthemtics Bengl Engineering nd Science University, Shibpur Howrh , IDIA Abstrct In this pper, we hve introduced frctionl-ordered predtor-prey popultion model which hs been successfully solved with the help of two powerful nlyticl methods, nmely, Homotopy erturbtion Method (HM) nd Vritionl Itertion Method (VIM). The frctionl derivtives re described in the Cputo sense. Using initil vlues, we hve derived the explicit solutions of predtor-prey popultions for different prticulr cses by using HM nd VIM. This pper represents n nlyticl s well s numericl comprison between these two methods for solving our frctionl-ordered predtor-prey popultion model for different frctionl orders. Our nlyticl nd numericl results show tht these two techniques require less computtionl wor nd provide the pproximte solutions with esily computble components. These methods re extremely efficient for obtining quntittively relible results. Keywords : redtor-prey popultion model; Logistic growth; Functionl response function; Frctionl derivtive; Homotopy perturbtion method; Vritionl itertion method. 1 MSC : 9D5 E-mil: swrnli.shrm87@gmil.com corresponding uthor, E-mil: g p smnt@yhoo.co.u, gpsmnt@mth.becs.c.in

2 1. Introduction Mthemticl modeling in popultion dynmics hs gined lot of ttention nd pprecition during the lst few decdes nd mong these models predtor-prey systems ply n importnt role. The most importnt element in predtor-prey models is the predtor functionl response on prey popultion 37,57,63], which describes the number (density) of prey consumed per predtor per unit time for given quntities (densities) of prey nd predtor. There re vrious forms of functionl responses mong which the most populr nd useful functionl responses re Lot-Voltter functionl response (Holling type I functionl response) 34,37,57,63] nd Holling type II functionl response 34,37,57,63]. The Lot-Voltter functionl response is of the form p(x) = ex nd the Holling type II functionl response is of the form p(x) = ex, where x is the popultion f+x density of prey, e is the mximum rte of predtion, i.e., the mximum number of prey tht cn be eten by predtor in ech time unit, f is the hlf sturtion constnt, i.e., the number of prey necessry to chieve one-hlf of the mximum rte e. Mny reserchers hve concentrted on the stbility of the predtor-prey systems with such functionl responses. The dynmicl reltionship between predtors nd their prey hs been n importnt topic in theoreticl ecology since the fmous Lot-Voltter eqution 37,44,57,63,73]. Two species models with different inds of functionl responses re extensively studied in ecologicl literture. Two species systems such s predtor-prey,4,5,41], plntherbivore 3] etc. hve plyed dominting roles in ecology for long time. In the lte 7s some interest in the mthemtics of tritrophic food chin models (composed by prey, predtor nd super predtor) 11,1,14,16,,1,38,41,47,66,68] emerged. Subsequently, Teuchi et l. 71], Freedmn 15,16,17,18], Run 66,67], Li nd Kung 41] hve derived some golden rules in the theory of predtor-prey models both of two species nd three species. There re mny other reserchers lie Gomez nd Zmor 1], Kuznetsov 39], Miti nd Smnt 46,47,48,49], l nd Smnt 6] etc. who hve investigted such predtor-prey popultion models. Most of the reserchers hve investigted such predtor-prey popultion models with only simple first order differentil equtions. But recently the frctionl ordered differentil equtions hve gined lot of ttention in mny field of pplied mthemtics including popultion dynmics due to their bility to provide n exct description of different non-liner phenomen 13,,35,43,6,64,65,69,7,7]. Ahmed et l. 1] hve solved the frctionl prey-predtor model nd frctionl rbies model numericlly. There re mny other reserchers lie He 3-33], Ds nd Gupt 7,8,9,1], Mous nd Rgb 56], Momni nd Odibt 5,53,54,55], Gnji et l. 19] etc. who hve successfully pplied frctionl-ordered differentil equtions in mny fields of pplied mthemtics, physics, biology nd engineering. They hve been successfully implemented mny mthemticl tools lie Homotopy erturbtion Method (HM) 19,36,4,56], Vritionl Itertion Method (VIM) 8-3,51,58,59], Adomin Decomposition Method (ADM) 5,53,54] etc. to construct the pproximte solutions of severl frctionl-ordered differentil equtions. They hve shown tht the stbility of those systems for frctionl-order is the sme s their integer order counterprt. The most powerful nd effective mthemticl tools for solving the frctionl-ordered

3 3 liner nd non-liner differentil equtions re HM nd VIM. The bsic difference of this HM from the other perturbtion techniques is tht it does not require smll prmeter in the equtions. As result it overcomes the limittions of trditionl perturbtion techniques. On the other hnd, VIM is n itertive technique which cn solve the frctionl ordered differentil equtions without using lineriztion, perturbtion or restrictive ssumptions. In this pper, we hve ten frctionl ordered predtor-prey popultion model which involves generlized Lot-Volterr Logistic growth eqution. We hve considered two different situtions, nmely, norml sitution nd hunger sitution by choosing different functionl responses nd numericl responses. In our model insted of only first order time derivtives, we hve considered frctionl derivtives of order α nd β ( < α 1, < β 1). A significnt outcome of these evolution equtions is the genertion of frctionl Brownin motion, which re Gussin in nture but in generl non-mrovin. ext we hve constructed pproximte solutions with the help of HM nd VIM. The comprison between numericl results of prey nd predtor popultions for different frctionl ordered time derivtives α, β nd for vrious time t re crried out for different cses. The results re depicted grphiclly with the help of MATLAB followed by discussions nd conclusions.. Bsic Definitions We give some bsic definitions nd properties of the frctionl clculus theory which hve been used further in this pper. Definition 1. A rel function f(x), x >, is sid to be in the spce C µ, µ R if there exists rel number p(> µ), such tht f(x) = x p f 1 (x), where f 1 (x) C, ), the clss of continuous functions on, ), nd it is sid to be in the spce C m µ iff f (m) C µ, m. Definition. The Riemnn-Liouville frctionl integrl opertor of order α, of function f C µ, µ 1, is defined s J α f(x) = 1 Γ(α) J f(x) = f(x). x (x t) α1 f(t)dt, α >, x >, roperties of the opertor J α cn be found in 45,5,61], we mention only the following: 1. J α J β f(x) = J α+β f(x),. J α J β f(x) = J β J α f(x), 3. J α x γ = (Γ(γ + 1)/Γ(α + γ + 1))x α+γ. The Riemnn-Liouville derivtive hs certin disdvntges when trying to model relworld phenomen with frctionl order differentil equtions 53]. Therefore, we re introducing modified frctionl differentil opertor Dt α proposed by Cputo in his wor on theory of viscoelsticity 6]. Definition 3. The frctionl derivtive of f(x) in the Cputo sense is defined s

4 4 D α t f(x) = J mα D m f(x) = for m 1 < α m, m, x >, f C m 1. 1 Γ(m α) Also, we need here two of its bsic properties. x (x t) mα1 f (m) (t)dt, Lemm 1. If m 1 < α m, m nd f C m µ, µ 1, then nd J α D α t f(x) = f(x) D α t J α f(x) = f(x), m1 = f () (+) x!, x >. 3. Mthemticl Model Let us consider the following frctionl ordered Logistic predtor-prey model: Cse I. For norml sitution (when c ): d α (t) = r ( ) } (t) (t) (t) dt α d β (t) dt β = (b d) (t) Cse II. For hunger sitution (when c ): ( ) } ] d α (t) r (t) = c (t) dt α with initil conditions d β (t) dt β = b c (t) d + 1 ( )] (t) T ln (t) c(t) (1) () () = c 1, () = c. (3) nd < α 1, < β 1. Here (t) nd (t) denote the popultion density of prey nd predtor respectively. The model prmeters re described below: r : Intrinsic growth rte of prey popultion,

5 5 : Crrying cpcity of prey popultion, : Mximum ttc cpcity of predtor popultion, b : Birth rte of predtor, c : Mximum predtion rte, d : turl deth rte of predtor popultion, T : The typicl time of response to hunger. Model Construction: Here we hve ten generlized Logistic lw s: d(t) = r ( ) } (t) (t), < 1. (4) dt If, the model is nown s Gompertz model nd if = 1 it becomes simple logistic model. ow, we use it in our frctionl-order predtor-prey model. Functionl Response Function φ(, ): The functionl response is function tht describes the number of prey consumed per predtor per unit time for given quntities (densities) of prey nd predtor 37,56,6]. The model is built on following two ssumptions ]: (i) The mximum functionl response is constnt, i.e., we ssume tht there is limit to the ttc cpcity of predtors. (ii) Since lrge predtor popultion cn not et the whole prey popultion t one time, we ssume tht only constnt proportion c of the prey popultion cn be cught per unit time by the whole predtor popultion. As predtor must shre c prey, the functionl response will be c. c is lso clled the mximum predtion rte. Combining the ssumptions (i) nd (ii), the functionl response function for our model is given by ],, ifc φ(, ) = c, ifc (5) In the first cse: the functionl response is equl to the ttc cpcity per predtor; we cll this sitution s the norml sitution. In the second cse: the functionl response is smller thn the ttct cpcity, i.e., if more prey were vilble the predtor would increse their ctch. We cll this sitution s the hunger sitution.

6 6 umericl Response Function ψ(, ): umericl response mens tht predtors become more bundnt s prey density increses 37,56,6]. For the numericl response function of our model we ssume tht the birth nd deth rtes of predtors depend on nutrition ]. This function (ψ) hs positive prt (ψ + ) nd negtive prt (ψ ) such tht ψ = ψ + ψ. ψ + is the totl birth rte of predtor popultion. In the norml sitution the birth rte is the mximum vlue b nd in the hunger sitution the birth rte decreses proportiontely to the food deficit. Thus ting b s the proportionlity fctor with (5) we get, b, ifc ψ + (, ) = b c, ifc (6) ψ is the deth ( rte ) of predtor popultion due to hunger. Using the formul ψ (, ) = 1 ln ] (dpted from Lssiter nd Hyne (1971) 4]) nd with (5) T φ(, ) we get,, ifc ψ (, ) = 1 ln ( ) (7) T c, ifc ow, combining the bove ssumptions nd discussions we hve finlly constructed our frctionl order predtor-prey model described by (1)-(3). 4. Approximte Solutions We re now going to construct the pproximte solutions of our frctionl order predtor-prey model (1)-(3) with the help of two powerful nd effective mthemticl tools Homotopy erturbtion Method (HM) nd Vritionl Itertion Method (VIM) Homotopy erturbtion Method (HM) Cse I. For norml sitution (when c ): According to the Homotopy erturbtion Method (HM), we construct the following homotopy of eqution (1) s ( ) } ] Dt α r (t) (t) = p (t) (t) (8) D β t (t) = p(b d) (t) If the embedding prmeter p 1 is considered s smll prmeter, pplying the clssicl perturbtion technique, we cn ssume tht the solution of eqution (8) cn be given s power series in p, i.e.,

7 7 (t) = (t) + p 1 (t) + p (t) + p 3 3 (t) +... (t) = (t) + p 1 (t) + p (t) + p 3 3 (t) +... When p 1, eqution (8) corresponds eqution (1), Eqution (9) becomes the pproximte solution of eqution (1). Substituting eqution (9) into eqution (8), we obtin the following set of liner differentil equtions: (9) p 1 : p : p : D α t (t) =, D β t (t) =. Dt α 1 (t) = r ( ) } (t) (t) (t), D β t 1 (t) = (b d) (t). Dt α (t) = r (t) ( ] (t)) 1 1 (t) + r ( ) ] (t) 1(t) 1 (t), (1) (11) (1) p 3 : D β t (t) = (b d) 1 (t). Dt α 3 (t) = r (t) ( (t)) 1 (t) ] ( 1) ( (t)) ( 1 (t)) + r 1(t) ( ] (t)) 1 1 (t) + r ( ) ] (t) (t) (t), (13) D β t 3 (t) = (b d) (t). nd so on. The method is bsed on pplying the opertors J α t nd J β t (the inverse opertors of the Cputo derivtives D α t nd D β t respectively) on both sides of equtions (1)-(13). (t) = c 1, (t) = c. (14)

8 8 1 (t) = c 1 1 (t) = (b d)c Γ(β + 1). (t) = c 1 ) } ] c t β ) } ] c ) } r t α+β ) ] (b d)c Γ(α + β + 1), t α Γ(α + 1), t α Γ(α + 1) (15) (16) t β (t) = (b d) c Γ(β + 1). 3 (t) = c 1 ( + 1) ) } ] c ) } r ) } r r c1 1 ) ] t 3α Γ(3α + 1) ) ] t α+β (b d)c Γ(α + β + 1) ) } r ) ] (17) t α+β Γ(α + 1) (Γ(α + 1)) Γ(3α + 1) t3α (b d) c Γ(α + β + 1), t 3β 3 (t) = (b d) 3 c Γ(3β + 1). roceeding in the similr wy the rest of the components of x m nd y m, m of the HM cn be obtined nd thus the series solutions re determined to the desired pproximtions. Finlly, we get the pproximte solutions for (t) nd (t) by the truncted series given by where (t) = lim M Φ M(t), (t) = lim M Ψ M(t) (18)

9 9 M1 Φ M (t) = m (t), Ψ M (t) = m= M1 m= m (t) The bove two series converge very rpidly in rel physicl problems. The rpid convergence mens tht only few terms re required to get the pproximte solutions. Cse II. For hunger sitution (when c ): According to the Homotopy erturbtion Method (HM), we construct the following homotopy of eqution () s ( ) } ] Dt α r (t) (t) = p c (t) (19) D β t (t) = p b c (t) (t) d + 1 ( )}] (t) T ln c(t) If the embedding prmeter p 1 is considered s smll prmeter, pplying the clssicl perturbtion technique, we cn ssume tht the solution of eqution (19) cn be given s power series in p, i.e., (t) = (t) + p 1 (t) + p (t) + p 3 3 (t) +... (t) = (t) + p 1 (t) + p (t) + p 3 3 (t) +... When p 1, eqution (19) corresponds eqution (), Eqution () becomes the pproximte solution of eqution (). Substituting eqution () into eqution (19), we obtin the following set of liner differentil equtions: () p : D α t (t) =, D β t (t) =. ( ) } ] p 1 : Dt α r (t) 1 (t) = c (t), D β t 1 (t) = b c (t) (t) d + 1 ( )] T ln (t). c (t) (1) ()

10 1 p : Dt α (t) = r (t) ( ] (t)) 1 1 (t) + r ( ) ] (t) 1(t) c 1 (t), D β t (t) = b c 1 1(t) (t) T ( 1 (t) (t) )] 1(t) (t) 1 (t) d + 1 ( )] T ln (t). c (t) (3) p 3 : Dt α 3 (t) = r (t) ( (t)) 1 (t) ] ( 1) ( (t)) ( 1 (t)) + r 1(t) ( ] (t)) 1 1 (t) + r ( ) ] (t) (t) c (t), D β t 3 (t) = b c 1 (t) (t) T (t) (t) + 1 (t) (t) 1 ( ) 1 (t) (t) ( ) }] ( 1 (t) 1 1 (t) 1 (t) (t) T (t) )] 1(t) (t) (t) d + 1 ( )] T ln (t). c (t) (4) nd so on. The method is bsed on pplying the opertors J α t nd J β t (the inverse opertors of the Cputo derivtives D α t nd D β t respectively) on both sides of equtions (1)-(4). (t) = c 1, (t) = c. (5)

11 11 1 (t) = c 1 1 (t) = ) } ] t α c c 1 Γ(α + 1), b c c c d + 1 ( )}] T ln c t β Γ(β + 1). (6) (t) = c 1 r (t) = c 1 r ) } ] r c ( + 1) ) } ] ( c b c + c ) T c 1 b c c c d + 1 ( )}] T ln c 1 T + d + 1 ( )}] T ln c t β Γ(β + 1). ) } ] c t α+β Γ(α + β + 1) t α Γ(α + 1) (7) 3 (t) = c 1 r ( + 1) ) } ] r c ( + 1) r c1 1 ) } ] t 3α c Γ(3α + 1) ) } ] Γ(α + 1) c (Γ(α + 1)) Γ(3α + 1) t3α

12 1 3 (t) = ( + 1) + ) } ] c ) } ] ( c b c c 1 + c ) T b c c c d + 1 ( )}] T ln c t α+β Γ(α + β + 1) 1 T + d + 1 ( )}] T ln c t 3β Γ(3β + 1) ) } ] ( c 1 c b c + c ) T c 1 1 T + d + 1 ( )}] T ln c t α+β Γ(α + β + 1) 1 b c T c c c d + 1 ( )}] T ln c Γ(β + 1) (Γ(β + 1)) Γ(3β + 1) t3β (8) c T + 1 T ) } ] Γ(α + 1) c (Γ(α + 1)) Γ(α + β + 1) tα+β ) } ] c b c c c d + 1 ( )}] T ln c Γ(α + β + 1) Γ(α + 1)Γ(β + 1)Γ(α + β + 1) tα+β. roceeding in the similr wy the rest of the components of x m nd y m, m of the HM cn be obtined nd thus the series solutions re determined to the desired pproximtions. Finlly, we get the pproximte solutions for (t) nd (t) by the truncted series given by where (t) = lim M Φ M(t), (t) = lim M Ψ M(t) (9) Φ M (t) = M1 m= m (t), Ψ M (t) = M1 m= m (t) The bove two series converge very rpidly in rel physicl problems. The rpid convergence mens tht only few terms re required to get the pproximte solutions.

13 Vritionl Itertion Method (VIM) Cse I. For norml sitution (when c ): According to the Vritionl Itertion Method (VIM), the itertion formuls for system (1) re given by t +1 (t) = (t) D α t (τ) r ( ) } ] (τ) (τ) + (τ) dτ (3) +1 (t) = (t) t D β t (τ) (b d) (τ)]dτ By the bove vritionl itertion formuls, begin with (t) = c 1 nd (t) = c, we cn obtin the following pproximtions: ( 1 c1 ) } ] (t) = c 1 + c c t, (31) 1 (t) = c + (b d)c t, (t) = c r ( r r c 1 + ) } ] c c t ) ) } ] t c c (b d)c r ( ( c 1 ) ) } ] + c c t ( ) } ] c 1 c c ( + ) r ) } ] c c t α Γ(3 α), (3) t β (t) = c + (b d)c t + (b d) t c (b d)c Γ(3 β), nd so on. roceeding in the similr wy nd cn be determined to the desired ccurcy. Cse II. For hunger sitution (when c ): According to the Vritionl Itertion Method (VIM), the itertion formuls for system () re given by

14 14 +1 (t) = (t) t ( r +1 (t) = (t) t D α t (τ) ( ) ) } ] (τ) c (τ) dτ D β t (τ) bc (τ) + d + 1 ( )} ] T ln (τ) (τ) dτ c (τ) By the bove vritionl itertion formuls, begin with (t) = c 1 nd (t) = c, we cn obtin the following pproximtions: ( 1 c1 ) } ] (t) = c 1 + c c 1 t, 1 bcc1 (t) = c + (t) = c 1 + ( r ) + c 1 c r r c 1 + ( + ) d + 1 ( )} ] T ln c c t, ) } ] c c 1 t r ( ( ) ) c 1 r ) } ] t c } c ( c 1 ) } ] + c 1 t ] c c 1 ) } ] t α c c 1 Γ(3 α), (33) (34)

15 15 bcc1 (t) = c + d + 1 ( )} ] T ln c c t + ( ) ( bcc1 bc r dc t + ( bcc1 d d + 1 ( )) }] T ln c t c bcc1 4T + 4T b r d + 1 ( )} ] T ln c c ) ) } c c 1 t β Γ(3 β) 1 ( )} ] d c + 1T ln ln c c c ( ( +c bcc1 ln c + d + 1 ( )) } ) T ln c c t ( bcc1 + d + 1 ( )) } T ln c c t ( ( bcc1 c + d + 1 ( )) } ) T ln c c t ( ( ( ( bcc1 1 + ln c + d + 1 ( )) } )))] T ln c c t ( 1 r ) ) } ( ( ) ) c c 1 t c 1 } ] c c 1 ( ( bcc1 4c d + 1 ( )) } ( T ln c bcc1 c c 1 d + 1 ( )) } T ln c c ( r + ( ( r + c c 1 ) ) } ( bcc1 c c 1 d + 1 ( )) } ) T ln c c t ( ( ( r ln c ln c 1 + ) ) } ( bcc1 c + c 1 d + 1 ( )) }) T ln c c ) ) } )) c c 1 t (35)

16 16 ( ( r + ) ) } ) c c 1 t ( ( bcc1 c + d + 1 ( )) } ) T ln c c t ( ( ( r ln c c 1 + ) ) } ))] c c 1 t nd so on. roceeding in the similr wy nd cn be determined to the desired ccurcy. 5. umericl Simultions nd Discussions In this section numericl simultions of prey nd predtor species for different frctionl Brownin motions α = β =.4,.6,.8 nd for the stndrd motion α = β = 1 re computed using HM nd VIM for vrious vlues of time t nd other vribles given in the Tble 1 nd Tble. These comprtive results re presented grphiclly through Fig.1-Fig.8. Fig.1-Fig.4 grphiclly represent the time evolution of prey-predtor popultion density for norml sitution (when c ) using both HM nd VIM under the initil conditions given in Tble 1. In this cse, we observe tht both popultions exist but prey popultion decreses nd predtor popultion increses with time t. On the other hnd, Fig.5-Fig.8 grphiclly represent the time evolution of prey-predtor popultion density for hunger sitution (when c ) using both HM nd VIM under the initil conditions given in Tble. In this cse, we observe tht both popultions exist nd increse with time t. But increse rte of prey popultion is greter thn tht of predtor popultion with time t. It cn be noted tht in both the cses it tes more time for meeting prey-predtor popultions (i.e., the biomss of both these popultions re equl) s the frctionl time derivtives increse, i.e., the vlues of α nd β increse. Finlly it tes the mximum time for the stndrd motion, i.e. for α = β = 1. In every figure we represent two comprtive digrms using HM nd VIM, which show tht pproximte solutions by these two methods (HM nd VIM) re in good greement, which is indeed remrble observtion. It is to be noted tht during the numericl computtions of the series solutions for HM only the four terms re considered in evluting the pproximte solutions nd for VIM the third itertions re used in evluting the pproximte solutions. So, the ccurcy of the results cn be improved by introducing more terms for HM nd higher itertions for VIM.

17 17 Tble 1 rmeter Vlues 1 b d 1 r c 1 1 c.5 6 for lph=bet=1 6 for lph=bet=1 5 5 opultion of two species ( nd ) opultion of two species ( nd ) Fig.1 Fig.1b Fig.1. Time series plot of prey () nd predtor ( ) popultions with vrious initil conditions, prmeter vlues re given in Tble 1 when α = β = 1: () HM; (b) VIM. 8 for lph=bet=.8 6 for lph=bet=.8 7 opultion of two species ( nd ) opultion of two species ( nd ) Fig. Fig.b Fig.. Time series plot of prey () nd predtor ( ) popultions with vrious initil conditions, prmeter vlues re given in Tble 1 when α = β =.8: () HM; (b) VIM.

18 18 1 for lph=bet=.6 6 for lph=bet= opultion of two species ( nd ) opultion of two species ( nd ) Fig.3 Fig.3b Fig.3. Time series plot of prey () nd predtor ( ) popultions with vrious initil conditions, prmeter vlues re given in Tble 1 when α = β =.6: () HM; (b) VIM. 16 for lph=bet=.4 6 for lph=bet=.4 14 opultion of two species ( nd ) opultion of two species ( nd ) Fig.4 Fig.4b Fig.4. Time series plot of prey () nd predtor ( ) popultions with vrious initil conditions, prmeter vlues re given in Tble 1 when α = β =.4: () HM; (b) VIM.

19 19 Tble rmeter Vlues 1 b c 1 d 1 r T 4 c 1.5 c 5 5 for lph=bet=1 for lph=bet=1 18 opultion of two species ( nd ) opultion of two species ( nd ) Fig.5 Fig.5b Fig.5. Time series plot of prey () nd predtor ( ) popultions with vrious initil conditions, prmeter vlues re given in Tble when α = β = 1: () HM; (b) VIM.

20 6 for lph=bet=.8 5 for lph=bet=.8 opultion of two species ( nd ) opultion of two species ( nd ) Fig.6 Fig.6b Fig.6. Time series plot of prey () nd predtor ( ) popultions with vrious initil conditions, prmeter vlues re given in Tble when α = β =.8: () HM; (b) VIM. 1 for lph=bet=.6 5 for lph=bet=.6 opultion of two species ( nd ) opultion of two species ( nd ) Fig.7 Fig.7b Fig.7. Time series plot of prey () nd predtor ( ) popultions with vrious initil conditions, prmeter vlues re given in Tble when α = β =.6: () HM; (b) VIM.

21 1 for lph=bet=.4 5 for lph=bet=.4 18 opultion of two species ( nd ) opultion of two species ( nd ) Fig.8 Fig.8b Fig.8. Time series plot of prey () nd predtor ( ) popultions with vrious initil conditions, prmeter vlues re given in Tble when α = β =.4: () HM; (b) VIM. 6. Conclusion In this pper, we hve introduced frctionl ordered predtor-prey model with generlized logistic growth lw. We hve considered two situtions, nmely, norml sitution nd hunger sitution for which functionl response nd numericl response becme different. In the model equtions insted of first order time derivtives, we hve used frctionl derivtives of order α nd β ( < α 1, < β 1). Then we hve constructed n pproximte solutions of our non-liner frctionl ordered differentil equtions with the help of very powerful nd effective mthemticl tools, nmely, Homotopy erturbtion Method (HM) nd Vritionl Itertion Method (VIM). ext we hve presented comprison of the solutions constructed by these two methods. The Homotopy erturbtion Method (HM) is more powerful thn ny other perturbtion techniques s it does not require smll perturbtion ssumption in the equtions so tht the limittions of the trditionl perturbtion methods cn be eliminted. The nother dvntge of this method is tht the pproximtions obtined by this method re vlid for both the smll nd lrge prmeters. On the other hnd, Vritionl Itertion Method (VIM) is n itertive technique to construct n pproximte solution without using lineriztion, perturbtion or restrictive ssumptions. The other importnt observtions bout these two techniques (HM nd VIM) re s follows: (i) Both of these two methods require less computtionl wor nd provide the pproximte solutions with esily computble components. (ii) The pproximte solutions computed with the help of HM nd VIM re in good greement with ech other nd lso with the exct solutions, which is very pprecible. (iii) These two methods re very powerful, esy to use nd very effective for obtining quntittively relible results. In this pper, we hve lso shown the comprtive numericl nlysis of our model

22 for two methods HM nd VIM with the help of MATLAB, which is the most importnt prt of our pper. In this prt the numericl results of prey nd predtor popultions for different frctionl Brownin motions, i.e., for α = β =.4,.6,.8 nd for the stndrd motion, i.e., for α = β = 1 re presented grphiclly using MATLAB for vrious vlues of time t nd other vribles. The most pprecible prt of this study is the nlysis of time requirement for the meeting of predtor-prey with the increse in frctionl time derivtives. In every figure we hve tried to show comprison of the solutions evluted by two methods HM nd VIM for different vlues of α nd β. From the figures it is cler to see the time evolution of prey-predtor popultion density nd it is lso very cler tht the numericl solutions of our frctionl predtor-prey popultion model continuously depend on the prmeters α nd β. It cn lso be noted tht the numericl results for HM nd VIM re good in greement with ech other, which is indeed remrble observtion. The process of development of models bsed on frctionl ordered differentil equtions hs recently gined populrity in the investigtion of dynmicl systems due to their bility to provide n exct description of different non-liner phenomen. A significnt outcome of these evolution equtions is the genertion of frctionl Brownin motions nd the best dvntge of these frctionl ordered differentil systems is tht they llow greter degree of freedom in the model. We hope tht our wor is smll step in this direction which will motivte the reserchers woring in both the res of ecologicl modelling nd frctionl clculus for further reserches. References 1. E. Ahmed, A.M.A. El-Syed, H.A.A. El-S, Equilibrium points, stbility nd numericl solutions of frctionl order predtor-prey nd rbies models, J. Mth. Anl. Appl. 35 (7) R. Arditi, J.M. Abillion, J. Vieir D Silv, A predtor-prey model with stition nd intrspecific competition, Ecol. Model. 5 (1978) M. Bndyopdhyy, R. Bhttchryy, B. Muhopdhyy, Dynmics of n utotroph herbivore ecosystem with nutrient recycling, Ecologicl Modelling 176 (4) M. Bndyopdhyy, C. G. Chrbrti, Deterministic nd stochstic nlysis of non-liner prey-predtor system, J. Biol. Syst. 11 (3) G. J. Butler, H. I. Freedmn,. Wltmn, Uniformly persistent systems, roc. Am. Mth. Soc. 96 (1986) M. Cputo, Liner models of dissiption whose Q is lmost frequency independent, rt II, J. Roy. Astron. Soc. 13 (1967) S. Ds,.K. Gupt, A mthemticl model on frctionl Lot-Volterr equtions, J. Theo. Bio. 77 (11) S. Ds,.K. Gupt, An pproximte nlyticl solution of the frctionl diffusion eqution with bsorbent term nd externl force by homotopy perturbtion method, Z. Fur. turforsch. Sec. 65 (1)

23 3 9. S. Ds,.K. Gupt, Rjeev, A frctionl predtor-prey model nd its solution, Int. J. onlin. Sci. umer. Simul. 1 (9) S. Ds,.K. Gupt, K. Vishl, Approximte pproch to the Ds model of frctionl logistic popultion growth, Appl. Appl. Mth. 5(1) (1) S. C. Ds, K. C. bru, Scope of bio-control of pests nd diseses in te plnttions, In: roc. Int. Conf. Te Res: Globl erspective Clcutt, 11-1 Jnury, 199, S. C. Ds, M. C. Borthr, B. Gope, eed for non-conventionl pproch in te pest mngement, In: roc. 3th Tocli Conf., 1989, K. Diethelm,. Ford, Anlysis of frctionl differentil equtions, J. Mth. Anl. Appl. 65 () L. H. Erbe, V. S. H. Ro, H. Freedmn, Three-species food chin models with mutul interference nd time delys, Mth. Biosci. 8 (1986) H. I. Freedmn, Deterministic Mthemticl Models in opultion Ecology, Mrcel Deer, ew Yor, 54, H. I. Freedmn,. Wltmn, Mthemticl nlysis of some three-species food chin models, Mth. Biosci. 33 (1977) H. I. Freedmn,. Wltmn, ersistence in model of three competitive popultions, Mth. BIosci. 73 (1985) H. I. Freedmn,. Wltmn, ersistence in model of three intercting predtorprey popultions, Mth. Biosci. 68 (1984) Z.Z. Gnji, D.D. Gnji, H. Jfri, M. Rostmin, Appliction of the homotopy perturbtion method to coupled system of prtil differentil equtions with time frctionl derivtives, Topol. Mth. nonlin. Anl. 31 (8) M.E. Gilpin, Enriched predtor-prey systems: theoreticl stbility, Science 177 (197) J. Gomez, R. Zmor, Top-down effects in tritrophoc system: prsitoids enhnce plnt fitness, Ecology 75 (1994) R. Gorenflo, Frctionl clculus: some numericl methods, in: A. Crpinteri, F. Minrdi (Eds.), Frctls nd Frctionl Clculus, ew Yor, J. H. He, Approximte nlyticl solution for seepge flow with frctionl derivtives in porous medi, Comput. Methods Appl. Mech. Eng. 167 (1998) J. H. He, Approximte solution of nonliner differentil equtions with convolution product nonlinerities, Comput. Methods Appl. Mech. Eng. 167 (1998) J. H. He, Asymptotic methods for strongly nonliner equtions, Internt. J. Modern hys. B (1) (6)

24 4 6. J. H. He, on-perturbtive methods for strongly nonliner problems, disserttion.de- Verlg im Internet GmbH, Berlin, J. H. He, Semi-inverse method of estblishing generlized principles for fluid mechnics with emphsis on turbomchinery erodynmics, Internt. J. Turbo Jet-Engines 14 (1) (1997) J. H. He, Vritionl itertion method- ind of non-liner nlyticl technique: some exmples, Internt. J. onliner Mech. 34 (1999) J. H. He, Vritionl itertion method for utonomous ordinry differentil systems, Appl. Mth. Comput. 114 () J. H. He, Vritionl itertion method for dely differentil equtions, Comm. onliner Sci. umer. Simul. (4) (1997) J. H. He, Vritionl principle for some nonliner prtil differentil equtions with vrible coefficients, Chos Solitons Frctls 19(4) (4) J. H. He, Vritionl theory for liner mgneto-electro-elsticity, Internt. J. onliner Sci. umer. Simul. (4) (1) J. H. He, X. H. Wu, Construction of solitry solution nd compction-lie solution by vritionl itertion method, Chos Solitons Frctls 9(1) (6) C. S. Holling, The functionl response of predtors to prey density nd its role in mimicry nd popultion regultion, Mem. Entomolog. Soc. Cn. 45 (1965) F. Hung, F. Liu, The time-frctionl diffusion eqution nd frctionl dvectiondispersion eqution, AZIAM J. 46 (5) M. Jfri, M. M. Hosseini, S. T. Mohyud-Din, Solution of nonliner singulr initil vlue problems by modified homotopy perturbtion method, Int. J. hy. Sci. 6(6) (11) M. Kot, Elements of Mthemticl Ecology, Cmbridge University ress, Cmbridge, R. Kumr, H. I. Freedmn, A mthemticl model of fculttive mutulism with popultions intercting in food chin, Mth. biosci. 97 (1989) Y. Kuznetsov, S. Rinldi, Remrs on food chin dynmics, Mth. Biosci. 134 (1996) R.R. Lssiter, D.W. Hyne, A finite model for simultion of dynmic processes in ecosystems. In: B.C. tten (Editor), Systems Anlysis nd Simultion in Ecology, Vol-I, Acdemic ress, ew Yor, (1971) B. Li, Y. Kung, simple food chin in chemostt with distinct removl rtes, J. Mth. Anl. Appl. 4 () Y. Liu, Z. Li, Y. Zhng, Homotopy perturbtion method to frctionl biologicl popultion eqution, Frc. Diff. Cl. 1(1) (11)

25 5 43. Y. Liu, B. Xin, umericl Solutions of frctionl predtor-prey system, Advnces in differentil equtions, A. Lot, Elements of hysicl Biology, Willims nd Wilins, Bltimore, Y. Lucho, R. Gorneflo, The initil vlue problem for some frctionl differentil equtions with the Cputo derivtive, reprint series A 8-98, Fchbreich Mthemti nd Informti, Freic Universitt Berlin, A. Miti, A. K. l, G.. Smnt, Effect of time dely on food chin model, Appl. Mth. compt. (8) A. Miti, A. K. l, G.. Smnt, Usefulness of biocontrol of pests in te: mthemticl model, Mth. Model. t. henom 3 (8) A. Miti, G.. Smnt, Complex dynmics of food chin model with mixed selection of functionl responses, Bull. Cl. Mth. Soc. 97 (5) A. Miti, G.. Smnt, Deterministic nd stochstic lysis of prey-dependent predtor-prey system, Int. J. Mth. Educ. Sci. Technol. 36 (6) K. S. Miller, B. Ross, An Introduction to the Frctionl Clculus nd Frctionl Differentil Equtions, Wiley, ew Yor, S. Momni, S. Abusd, Appliction of He s vritionl itertion method to Helmholtz eqution, Chos Solitons Frctls 7(5) (6) S. Momni, Z. Odibt, Anlyticl pproch to liner frctionl prtil differentil equtions rising in fluid mechnics, hys. Lett.A 355 (6) S. Momni, Z. Odibt, umericl pproch to differentil equtions of frctionl order, J. Comp. Appl. Mth. 7 (7) S. Momni, Z. Odibt, umericl comprison of methods for solving liner differentil equtions of frctionl order, Chos Solitons Frctls, in press. 55. S. Momni, Z. Odibt, I. Hshim, Algorithms for nonliner frctionl prtil differentil equtions: selection of numericl methods, Topol. Meth. onlin. Anl. 31 (8) M. M. Mous, S. F. Rgb, Appliction of the homotopy perturbtion method to liner nd nonliner sohrodinger equtions, Z. Fur. turforsch. Sec. 63 (8) J. D. Murry, mthemticl Biology, Springer-Verlg, new Yor, Z. Odibt, S. Momni, Applictions of vritionl itertion nd homotopy perturbtion methods to frctionl evolution equtions. Topol. Meth. onlin. Anl. 31 (8) Z. Odibt, S. Momni, Appliction of vritionl itertion method to nonliner differentil equtions of frctionl order, Internt. J. onliner Sci. umer. Simul. 1(7) (6) 15-7.

26 6 6. Z. Odibt, S. Momni, Approximte solutions for boundry vlue problems of time frctionl wve eqution, Appl. Mth. Comput., in press. 61. K. B. Oldhm, J. Spnier, The Frctionl Clculus, Acdemic ress, ew Yor, A. K. l, G.. Smnt, A single species popultion in polluted environment, Int. J. Biomth. 3 (1) E. C. ielou, Mthemticl Ecology, A wiley-interscience ubliction, John Wiley nd Sons, ew Yor, I. odlubny, Frctionl Differentil Equtions, Acdemic ress, Sn Diego, CA, I. odlubny, umericl solution of ordinry frctionl differentil equtions by the frctionl difference methods, in: S. Elydi, I. Gyori, G. Lds (Eds.), Advnces in Difference Equtions, Gordon nd Brech, Amsterdm, S. Run, A three-trophic-level model of plnton dynmics with nutrient recycling, Cn. Appl. Mth. Qurt. 1 (1993) S. Run, D. Xio, Globl nlysis in predtor-prey system with nonmonotonic functionl response, SIAM J. Appl. Mth. 61 (1) G.. Smnt, D. Mnn, A. Miti, Bioeconomic modelling of three-species fishery with switching effect, The Koren Journl of Computtionl nd Applied Mthemtics 1 (1-) (3) Shwgfeh, Anlyticl pproximte solutions for nonliner frctionl differentil equtions, Appl. Mth. Comput. 131() () Shwgfeh, D. Ky, Compring numericl methods for the solutions of systems of ordinry differentil equtions, Appl. Mth. Lett. 17 (4) Y. Teuchi, Y. Oshime, H. Mtsud, ersistence nd periodic orbits of threecompetitor model with refuges, Mth. Biosci. 18 (199) T. Torvi, R. L. Bgley, On the ppernce of the frctionl derivtive in the behvior of rel mterils, J. Appl. Mech. 51 (1996) V. Volterr, Vrizioni e fluttuzioni del numero di individui in specie nimli conviventi, Mem. Accd. Linc. (196)