A PREY-PREDATOR MODEL WITH AN ALTERNATIVE FOOD FOR THE PREDATOR AND OPTIMAL HARVESTING OF THE PREDATOR

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1 VOL. 7, O. 8, AUGUST ISS ARP Journal of Engineering and Applied Sciences 6- Asian Research Publishing ework (ARP). All righs reserved. A PREY-PREDATOR MODEL WITH A ALTERATIVE FOOD FOR THE PREDATOR AD OPTIMAL HARVESTIG OF THE PREDATOR K. Madhusudhan Reddy and K. Lakshmi arayan Vardhaman College of Engineering, Hyderabad, India SLC S Insiue of Engineering and Technology, Hyderabad, Piglipur, Hyderabad, India kmsreddy.vce@gmail.com ASTRACT The presen paper deals wih a prey - predaor model comprising an alernaive food for he predaor in addiion o he prey, and he predaor is harvesed under opimal condiions. The model is characerized by a pair of firs order nonlinear ordinary 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. Keywords: prey; predaor, equilibrium poins, sabiliy, bionomic equilibrium, opimal harvesing, normal seady sae.. 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 [] and by Volerra [3]. Since hen many mahemaicians and ecologiss conribued o he growh of his area of knowledge as repored in he reaises of Paul Colinvaux [4], Freedman [5], Kapur [6, 7] 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 [8, 9]. Chaudhuri [, ] formulaed an opimal conrol problem for he combined harvesing of wo compeing species. Models on he combined harvesing of a wospecies prey-predaor fishery have been discussed by Chaudhuri and Saha Ray []. iological and bionomic equilibria of a mulispecies fishery model wih opimal harvesing policy is discussed in deail by Kar and Chaudhari [3]. Recenly Archana Reddy [4] discussed he sabiliy analysis of wo ineracing species wih harvesing of boh species. Lakshmi arayan, Paabhiramacharyulu [5, 6] and Shiva Reddy [7] e al., have discussed differen prey-predaor models in deail. 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: ioeconomic model, AZIAM J.45 (4), ) moivaed he presen invesigaion.. MATHEMATICAL MODEL The model equaions for a wo species preypredaor sysem are given by he following sysem of nonlinear ordinary differenial equaions employing he following noaion: and are he populaions of he prey and predaor wih naural growh raes a and a respecively, 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 predaor, E is he harvesing effor and qe 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,,,,, q, E a q E are assumed o be non-negaive and ( ) consans. d d a () ( ) a q E + () 3. EQUILIRIUM STATES The sysem under invesigaion has four equilibrium saes defined by: a) The fully washed ou sae wih he equilibrium poin ; (3) b) The sae in which, only he predaor survives given ( a qe) by ; (4) c) The sae in which, only he prey survives given a by ; (5) d) The co-exisen sae (normal seady sae) given by: 3

2 VOL. 7, O. 8, AUGUST ISS ARP Journal of Engineering and Applied Sciences 6- Asian Research Publishing ework (ARP). All righs reserved. a ( a q E) ( a q E) + a ; + + This sae would exi only when a ( a q E) (6) > (7) 4. STAILITY OF THE EQUILIRIUM STATES To invesigae he sabiliy of he equilibrium saes we consider small perurbaions u, u in and over and respecively, so ha + u ; + u (8) y subsiuing (8) in () and () and neglecing second and higher powers of he perurbaions u, u, we ge he equaions of he perurbed sae: du AU (9) can be noed o be negaive. Hence he co-exisen equilibrium sae is sable. The soluions curves are: u u( λ + ) u λ λ u( λ + ) u λ λ e λ + u ( λ + ) + u u λ λ u( λ + ) + u λ λ e λ (3) e λ + e λ (4) where a () A ( a qe) + The characerisic equaion for he sysem is: [ λi ] de A () The equilibrium sae is sable only when he roos of he equaion () are negaive in case hey are real or have negaive real pars in case hey are complex. The equilibrium saes 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 + ( + ) λ+ [ ] () + Since he sum of he roos of () is negaive and he produc of he roos is posiive, he roos of which Where λ, λ, are he roos of he equaion (). 5. GLOAL STAILITY Theorem: The Equilibrium sae (, ) is globally asympoically sable. Le us consider he following Liapunov s funcion V(, ) ln + l ln (5) where l is posiive consan,o be chosen laer Differeniaing V w.r., we ge dv d d + l Subsiuing () and () in (6), we ge { ( ) ( )( ) } l ( )( ) ( ) { } dv + (6) Choosing l and wih some algebraic manipulaion yields dv ( ) ( ) < (7) Therefore, he equilibrium sae (, ) is globally asympoically sable. 6. IOOMIC EQUILIRIUM 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 predaor, p price per uni biomass of he predaor. The ne economic revenue for he predaor a any ime is given by: 4

3 VOL. 7, O. 8, AUGUST ISS ARP Journal of Engineering and Applied Sciences 6- Asian Research Publishing ework (ARP). All righs reserved. ( ) R p q c E (8) ( ) The biological equilibrium is ( ),( ),( E) ( ),( ),( E), where are he posiive soluions of a (9) ( ) a q E + () and( ) pq c E () From (), we have c { pq ( ) c}( E) ( ) () p q c a From (9) and (), we ge ( ) pq (3) From (), () and (3), we ge ( E) ( ) a (4) q pq I is clear ha ( E) > if ( ) > a (5) pq Thus he bionomic equilibrium (( ),( ),( E) ) exiss, if inequaliy (5) holds. 7. OPTIMAL HARVESTIG POLICY The presen value J of a coninuous ime-sream of revenues is given by: ( ) (6) J e p q c E Where denoes he insananeous annual rae of discoun. Our problem is o maximize J subjec o he sae equaions () and () and conrol consrains E ( E) max by invoking Ponryagin s maximum principle. The Hamilonian for he problem is given by: ( ) λ( ) ( a qe) H e p q c E+ a + 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 ) max. A ( E) max we have a singular conrol. y Ponryagin s maximum principle, dλ dλ ; ; E e ( pq c) λq λ e p E q (8) c dλ + dλ { λ( a ) λ( ) } ( λ λ E) (9) dλ dλ { e pqe λ( ) λ ( a qe ) } ( λ λ e pqe) + (3) From (8) and (9), we ge c Where p q Whose soluion is given by dλ λ e λ ( + ) e (3) dλ From (3) and (3), we ge λ e Where pqe ( + ) Whose soluion is given by λ e (3) ( + ) From (8) and (3), we ge a singular pah p q ( + ) Thus (33) can be wrien as: F ( ) p q ( + ) (33) There exis a unique posiive roo ( ) of F ( ) in he inerval < < k, if he 5

4 VOL. 7, O. 8, AUGUST ISS ARP Journal of Engineering and Applied Sciences 6- Asian Research Publishing ework (ARP). All righs reserved. following hold F () <, F (k ) >, F ( ) ) > for >. ( ), we ge ( ) For and ( ) ( ) a pq c E a q pq (34) (35) Hence once he opimal equilibrium, is deermined, he opimal harvesing (( ) ( ) ) effor ( E) can be deermined. From (8), (3) and (3), we found ha λ, λ do no vary wih ime in opimal equilibrium. Hence hey remain bounded as. From (33), we also noe ha Figure-. Shows he rajecory of he prey and predaor populaions beginning wih 45 and 35. Le a 8;.;.5; a ;.;.4; q.& E 5. p q ( + ) as Thus, he ne economic revenue of he predaor R. 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 [9] in he combined harvesing of wo ecologically independen populaions and by Chaudhuri [] in he combined harvesing of wo compeing species. 8. UMERICAL EXAMPLES Figure-3. Shows he variaion of he populaions agains he ime. Le a 3;.;.; a.5;.3;.4; q.& E. Figure-. Shows he variaion of he populaions agains he ime. Figure-4. Shows he rajecory of he prey and predaor populaions beginning wih 5 and 3. 6

5 VOL. 7, O. 8, AUGUST ISS ARP Journal of Engineering and Applied Sciences 6- Asian Research Publishing ework (ARP). All righs reserved. COCLUSIOS A prey - predaor model wih an alernaive food for he predaor, and he predaor is harvesed under opimal condiions. The model is characerized by a pair of firs order non-linear ordinary differenial equaions. All he four equilibrium poins of he model are idenified and he crieria for he sabiliy were discussed by using Lyponuv s mehod. The bio-economic equilibrium poin was idenified and he opimal harves policy was sudied, and some numerical examples were discussed. Finally i was observed ha he model was sabilized by harvesing. REFERECES [] Ponryagin L.S., olyanskii V.S., Gamkerlidge R.. and Mischenko E.F. 96. The mahemaical heory of opimal process, Wiley, ew York, USA. [] Loka A.J. 95. Elemens of Physical iology, Williams and Wilkins, alimore, ew York, USA. [3] Volerra V. 93. Leçons sur la héorie mahémaique de la lue pour la vie, Gauhier-Villars, Paris, France. [4] Paul Colinvaux A Ecology. John Wiley and Sons Inc., ew York, USA. [3] Kar T.K. and ChaudhuriK. S. 4. Harvesing in a wo-prey one-predaor fishery: A bioeconomic model. AZIAM J. 45: [4] ArchanaReddy R., Paabhiramacharyulu.Ch and KrishnaGandhi. 7. A sabiliy analysis of wo compeiive ineracing species wih harvesing of boh he species a a consan rae. In. J. of scienific compuing. () Jan- June. pp [5] Lakshmiarayan K. 5. A mahemaical sudy of a prey-predaor ecological model wih a parial cover for he prey and alernaive food for he predaor, Ph.D hesis, JTU. [6] Lakshmi arayan K. and Paabhiramacharyulu. Ch. 8. A Prey-Predaor Model wih an Alernaive Food for he Predaor, Harvesing of oh he Species and wih a Gesaion Period for Ineracion. In. J. Open Problems Comp. Mah. : [7] Shiva Reddy K., Lakshmi arayan K. and Paabhiramacharyulu. Ch.. A Three Species Ecosysem Consising of a Prey and Two Predaors. Inernaional J. of Mah. Sci. and Engg. Appls. 4(IV): [5] Freedman H. I. 98. Deerminisic Mahemaical Models in Populaion Ecology, Decker, ew York, USA. [6] Kapur J Mahemaical modeling in biology and Medicine. Affiliaed Eas Wes Press. [7] Kapur J Mahemaical modeling, wiley, easer. [8] Clark C.W ioeconomic modelling and fisheries managemen. Wiley, ew York, USA. [9] Clark C.W Mahemaical bioeconomics: he opimal managemen of renewable resources. Wiley, ew York, USA. [] Chaudhuri K.S A bioeconomic model of harvesing a mulispecies fishery. Ecol. Model. 3: [] Chaudhuri K.S Dynamic opimizaion of combined harvesing of a wo species fishery. Ecol. Model. 4: 7-5. [] Chaudhuri K.S. and Saha Ray S On he combined harvesing of a prey predaor sysem. J. iol. Sis. pp