A Prey-Predator Model with an Alternative Food for the Predator and Optimal Harvesting of the Prey

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1 Available online a Advances in Applied Science Research, 0, (4): A Prey-Predaor Model wih an Alernaive Food for he Predaor and Opimal Harvesing of he Prey K. Madhusudhan Reddy and * K. Lakshmi arayan Vardhaman College of Engineering, Hyderabad, India *SLC S Insiue of Engineering & Technology, Piglipur, Hyderabad, India _ ABSTRACT The presen paper deals wih a prey - predaor model incorporaing i) he predaor is provided wih an alernaive food in addiion o he prey, ii) he prey is harvesed under opimal condiions. The model is characerized by a pair of firs order non-linear differenial equaions. All he four equilibrium poins of he model are idenified and he crieria for he sabiliy are discussed.the possibiliy of exisence of bioeconomic equilibrium is discussed. The opimal harves policy is sudied wih he help of Ponryagin s maximum principle. Finally, some numerical examples are discussed. Key words: Prey, Predaor, equilibrium poins, sabiliy, bionomic equilibrium poin, opimal harvesing, hreshold resuls, normal seady sae, cach-per-uni-effor. _ ITRODUCTIO Ecology relaes o he sudy of living beings in relaion o heir living syles. Research in he area of heoreical ecology was iniiaed by Loka [6] and by Volerra [7].Since hen many mahemaicians and ecologiss conribued o he growh of his area of knowledge as repored in he reaises of Paul Colinvaux [3], Freedman [4], Kapur [, 3] ec. Harvesing of mulispecies fisheries is an imporan area of sudy in fishery modeling. The issues and echniques relaed o his field of sudy and he problem of combined harvesing of wo ecologically independen populaions obeying he logisic law of growh are discussed in deail by Clark [,].. Chaudhuri [9, 0] formulaed an opimal conrol problem for he combined harvesing of wo compeing species. Models on he combined harvesing of a wo-species prey-predaor fishery have been discussed by Chaudhuri and Saha Ray [8] Biological and bionomic equilibria of a mulispecies fishery model wih opimal harvesing policy is discussed in deail by Kar and Chaudhari [5]. Recenly Archana Reddy [] discussed he sabiliy analysis of wo ineracing 45

2 K. Lakshmi arayan e al Adv. Appl. Sci. Res., 0, (4): species wih harvesing of boh species. Lakshmi arayan and Paabhiramacharyulu [4, 5] and Shiva Reddy e al. [6,7] have discussed differen prey-predaor models in deail. Srilaha e al [9,0] discussed a four species model wih differen combinaion of ineracionsbeween he hem. Mos of he mahemaical models on he harvesing of a mulispecies fishery have so far assumed ha he species are affeced by harvesing only. A populaion model proposed by Kar and Chaudhuri, (c.f. Harvesing in a wo-prey one-predaor fishery: Bioeconomic model, AZIAM J.45 (004), ) and his model moivaed he presen invesigaion. In he presen invesigaion, we discussed a prey-predaor model by aking an alernaive food for he predaor and harvesing of he prey under opimal condiions. The model is characerized by a pair of firs order non-linear differenial equaions. The exisence of he possible seady saes along wih heir local sabiliy is discussed. We derive he condiions for global sabiliy of he sysem using a Liapunov funcion.the possibiliy of exisence of bioeconomic equilibrium is discussed. The opimal harves policy is sudied and he soluion is derived in he equilibrium case by using Ponryagin s maximum principle [8]. Finally, some numerical examples are discussed.. Mahemaical Model. The model equaions for a wo species prey-predaor sysem are given by he following sysem of non-linear ordinary differenial equaions employing he following noaion: and are he populaions of he prey and predaor wih naural growh raes a and arespecively, is rae of decrease of he prey due o insufficien food, is rae of decrease of he prey due o inhibiion by he predaor, is rae of increase of he predaor due o successful aacks on he prey, is rae of decrease of he predaor due o insufficien food oher han he prey; q is he cach abiliy co-efficien of he prey, E is he harvesing effor and q E is he cach-rae funcion based on he CPUE (cach-per-uni-effor) hypohesis]. Furher boh he variables and are non-negaive and he model parameers a, a,,,,, d d q, E and ( a q E) are assumed o be non-negaive consans. = a q E (.) = a + (.) 3. Equilibrium Saes. The sysem under invesigaion has four equilibrium saes defined by I. The fully washed ou sae wih he equilibrium poin = 0; = 0 (3.) a II. The sae in which, only he predaor survives given by = 0 ; = (3.) III. The sae in which, only he prey survives given by ( a q E) = ; 0 = (3.3) 45

3 K. Lakshmi arayan e al Adv. Appl. Sci. Res., 0, (4): IV. The co-exisen sae (normal seady sae) given by ( a qe ) a = + a + ( a q E) ; = + (3.4) This sae would exi only when a q E > a (3.5) 4. Sabiliy of he Equilibrium Saes To invesigae he sabiliy of he equilibrium saes we consider small perurbaions u, u in and over and respecively, so ha = + u ; = + u (4.) By subsiuing (4.) in (.) & (.) and neglecing second and higher powers of he perurbaions u, u we ge he equaions of he perurbed sae du = AU (4.) where ( a qe ) A = a + (4.3) The characerisic equaion for he sysem is [ λ ] 0 de A I = (4.4) The equilibrium sae is sable only when he roos of he equaion (4.4) are negaive in case hey are real or have negaive real pars in case hey are complex. The equilibrium poins I, II, and III are found o unsable, so we resriced our sudy o he normal seady sae only. 4. Sabiliy of he normal seady sae: In his case he characerisic equaion is ( + ) λ + [ + ] = 0 (4.5) λ + Since he sum of he roos of (4.5) is negaive and he produc of he roos is posiive, he roos of which can be noed o be negaive.hence he co-exisen equilibrium sae is sable. The soluions curves are: u0 ( λ + ) u0 u = e λ u0 ( λ + ) u0 + λ λ λ λ e λ (4.6) 453

4 K. Lakshmi arayan e al Adv. Appl. Sci. Res., 0, (4): u = u0( λ + ) + u0 e λ u0( λ + ) + u0 + e λ (4.7) λ λ λ λ Where λ, λ are he roos of he equaion (4.5) 5. Global sabiliy Le us consider he following Liapunov s funcion V (, ) = ln l + ln where l is posiive consan,o be chosen laer Differeniaing V w.r.o we ge dv d d = + l (5.) (5.) Subsiuing (.) and (.) in (5.), we ge dv = ( ) ( )( ) + l Choosing l = and wih some algebraic manipulaion yields dv = ( ) ( ) < 0. (5.3) { } { ( )( ) ( ) } Therefore, he equilibrium poin (, ) is globally asympoically sable. 6. Bionomic equilibrium The erm bionomic equilibrium is an amalgamaion of he conceps of biological equilibrium as well as economic equilibrium. The economic equilibrium is said o be achieved when he oal revenue obained by selling he harvesed biomass equals he oal cos for he effor devoed o harvesing. Le c = fishing cos per uni effor of he prey, p = price per uni biomass of he prey. The ne economic revenue for he prey a any ime is given by R = p q c E (6.) The biological equilibrium is ( ),( ),( E ) soluions of 0 0, where ( ),( ),( E) are he posiive a q E = (6.) a + = (6.3) p q c E = (6.4) and 0 454

5 K. Lakshmi arayan e al Adv. Appl. Sci. Res., 0, (4): From (6.4), we have { p q ( ) c }( E) = 0 ( ) c = (6.5) p q From (6.3) and (6.5), we ge c = a + (6.6) pq c E = a (6.7) q pq c a > (6.8) pq,, E exiss, if inequaliy (6.8) holds. From (6.), (6.5) & (6.6), we ge I is clear ha ( E) > 0 if ( ) Thus he bionomic equilibrium 7. Opimal harvesing policy The presen value J of a coninuous ime-sream of revenues is given by J = e ( pq c ) E (7.) 0 Where δ denoes he insananeous annual rae of discoun. Our problem is o maximize J subjec 0 E E by invoking o he sae equaions (.) & (.) and conrol consrains max Ponryagin s maximum principle. The Hamilonian for he problem is given by ( ) λ ( ) ( a ) H = e p q c E + a q E + λ + Where λ, λ are he adjoin variables. (7.) Le us assume ha he conrol consrains are no binding i.e. he opimal soluion does no occur a ( E we have a singular conrol. ) max E. A max By Ponryagin s maximum principle, dλ dλ = 0 ; = ; = E δ δ c = 0 e ( pq c ) λq = 0 λ = e p E q (7.3) dλ = = { e pq E + λ ( a qe ) + λ ( )} dλ = ( λ λ e pq E) (7.4) 455

6 K. Lakshmi arayan e al Adv. Appl. Sci. Res., 0, (4): dλ = = { λ ( ) + λ ( a + )} dλ = ( λ + λ ) (7.5) dλ From (7.3) & (7.5), we ge λ = A e c Where A = p q A Whose soluion is given by λ = e (7.6) + δ From (7.4) & (7.6), we ge dλ λ = Ae A Where A = pq E ( + δ ) Whose soluion is given by λ = ( ) A ( + δ ) From (7.3) & (7.7), we ge a singular pah c A p = q + δ Thus (7.8) can be wrien as c A F ( ) = p q ( + δ ) There exis a unique posiive roo e = following hold F (0)<0, F (k ) > 0, F ( ) ) > 0 for > 0. For δ = c δ = a + pq c E = a δ δ q pq we ge and (7.7) (7.8) δ of F ( ) = 0 in he inerval 0 < < k, if he (7.9) (7.0) Hence once he opimal equilibrium,( ) is deermined, he opimal harvesing effor δ δ ( E) δ can be deermined. From (7.3), (7.6) and (7.7), we found ha λ, λ do no vary wih ime in opimal equilibrium. Hence hey remain bounded as. 456

7 K. Lakshmi arayan e al Adv. Appl. Sci. Res., 0, (4): From (7.8), we also noe ha c A p = 0asδ q + δ Thus, he ne economic revenue of he prey R = 0. This implies ha if he discoun rae increases, hen he ne economic revenue decreases and even may end o zero if he discoun rae end o infiniy. Thus i has been concluded ha high ineres rae will cause high inflaion rae. This conclusion was also drawn by Clark [] in he combined harvesing of wo ecologically independen populaions and by Chaudhuri [8] in he combined harvesing of wo compeing species. 9. umerical Examples Le a = 3; = 0.; = 0.; a =.5; = 0.03; = 0.4; q = 0.0 & E = 0. Fig. 4 Fig

8 K. Lakshmi arayan e al Adv. Appl. Sci. Res., 0, (4): (i) Fig.4 shows he variaion of he populaions agains he ime and (ii) Fig.5 shows he rajecory corresponding o he prey and predaor populaions beginning wih =30 and =0. () Le a = 8; = 0.0; = 0.3; a =.5; = 0.; = 0.4; q = 0.04 & E = 5. Fig. 6 Fig. 7 (i) Fig.6 shows he variaion of he populaions agains he ime and (ii) Fig.7 shows he rajecory corresponding o he prey and predaor populaions beginning wih =0 and =

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