Stability Analysis of Three Species Model in Series Mutualism with Bionomic and Optimal Harvesting of Two Terminal Species

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1 Inrnaional Journal of Sinifi an Innovaiv Mahmaial Rsarh (IJSIMR) Volum, Issu, Dmbr, PP -5 ISS 7-7X (Prin) & ISS 7- (Onlin) Sabiliy nalysis of Thr Sis Mol in Sris Muualism wih Bionomi an Oimal arvsing of Two Trminal Sis R. Srilaha Darmn of Mahmais Sr Chaianya Ins. of Th. & Si Thimmaur, Karimnagar, Inia B. Ravinra Ry Darmn of Mahmais, JTU Collg of nginring Jagiial Karimnagar Disri, Inia. bsra: This ar als wih a hr sis Syn-o-sysm onsising of hr sis (S ), (S ) an (S ) ar in sris h ologial inraion: Muualism in airs whil h rminal sis S an S ar harvs. Th ossibiliy of xisn of bio onomi quilibrium is bing isuss an an oimal harvsing oliy is givn using Ponryagin s maximum rinil. Furhr som numrial xamls ar omu using Malab. Kywors: Sris Muualism, Oimal harvsing, Bionomi harvsing, Trminal sis. ITRODUCTIO Thr is an xnsiv suy on svral kins of ry- raor inraions afr i was iniia by Loka [] an Volrra []. Bionomis of naural rsours has lay a signifian rol in all hs inraions. Thr is a srong ima of harvsing on h ynami voluion of a oulaion. In fishry, forsry, agriulur an wil lif managmn, h xloiaion of biologial rsours an harvsing of oulaion sis an b sn. Th roblms of raorry sysms in h rsn of harvsing wr isuss by many auhors an anion on onomi oliis from harvsing hav also bn analyz. ail isussion on h issus an hniqus assoia wih h bionomi xloiaion of naural rsours was givn by Clark [, ]. suy on a lass of raor-ry mols unr onsan ra of harvsing of boh sis simulanously was ma by Braur an Souak [5, ]. Muli-sis harvsing mols ar also sui in ail by Chauhuri [7, ]. Mols on h ombin harvsing of a wo sis ry raor fishry hav bn isuss by Ragozin an Brown [9], Chauhuri an Saha Ray []. K. Shiva Ry.al [] an B. Ravinra Ry [,, 5] roos h mahmaial mols for wo an hr sis osysm wih bionomi an oimal harvsing. Thy also invsiga h sabiliy ons using various mahmaial hniqus. In his onnion, a hr sis mahmaial mol in sris muualism bas on h sysm of nonlinar quaions has bn onsru. Biologial an Bionomial quilibria of h sysm ar riv.. MTMTICL MODL Th mol quaions in his roblm ar as follows (i) quaion for h growh ra of (S ): q (ii) quaion for h growh ra of (S ): (.) (.) RC Pag

2 R. Srilaha & B. Ravinra Ry (iii) quaion for h growh ra of (S ): q oaion o: i () : Poulaion nsiy of h sis S i a im, i=,,. a i : aural growh ra of S i, i =,, a : Dras ra of S i u o is own insuffiin rsours i =,,. ii a : Inras ra of h firs sis (S ) u o inhibiion by h son sis (S ), a : Inras ra of h son sis (S ) u o (S ) a : Inras ra of h son sis (S ) u o (S ), a : Inras ra of h hir sis (S ) u o (S ) K i = a i / ii : Carrying aaiis of S i, i =,,. q i : Cah abiliy offiin of h sis (S i ), i=, : ffor ali o harvs h firs sis (S i ), i=, i q, q ar h ah-ra funions bas on h ah-r-uni-ffor hyohsis. Th variabls, an ar non-ngaiv an h mol aramrs ai, Ki, aii, a, a, a, a ar assum o b non-ngaiv onsans. Furhr a q, a q (.). QUILIBRIUM STTS Th sysm unr invsigaion has igh quilibrium sas fin by i, i,, hs ar givn hrunr. I. Fully wash ou sa: :,, II. Sas in whih only on sis survivs whil h ohr wo ar wash ou a q a :,, ; :,, a a III. a q :,, a Sas in whih wo sis surviv an h hir wash ou aa a( a q ) aa a( a q ) 5 :,, a a a a a a a a : This sa xiss only whn aa aa a q,, a a a q aa ( a q ) a aa ( a q ) a 7 :,, a a a a a a a a This sa xiss only whn aa aa an IV. Th o-xisn sa (or) ormal say sa : ( a q ) a a a aa a a ( a q ) a a a a a a a a a a a a, Inrnaional Journal of Sinifi an Innovaiv Mahmaial Rsarh (IJSIMR) Pag

3 Sabiliy nalysis of Thr Sis Mol in Sris Muualism wih Bionomi an Oimal arvsing of Two Trminal Sis aaa aa( a q ) aa ( a q) a a a a a a a a a ( a q ) a a a a a a a a a ( a q ) a a a a a a a a a This sa xiss only whn ( a aaa ) aaa.. BIO COOMIC SPCT T ITRIOR QUILIBRIUM POIT Th on of bionomi quilibrium is a union of hos of biologial quilibrium as wll as onomi quilibrium. Biologial quilibrium is givn by i, i=,,. By finiion, h bionomi quilibrium is sai o b ahiv whn h slling ri of h harvs biomass quals o h oal os ri uiliz in harvsing i. L i b h harvsing os r uni ffor for Si, i, an i b h ri r uni biomass of Si, i,. Th n rvnu or onomi rn a any im insan is hn givn by R R R, whr R ( q ), R ( ) r R rrsn n rvnu for S, i,. Th bionomi quilibrium i ( ),( ),( ),, saisfis h following quaions. a a a q (.) a a a a (.) a a a q (.) Th rvnu rurns (R) on firs an hir sis akn oghr will b R q q (.) Th ass woul aris whil rmining h bionomi quilibrium. Cas (i): if hn h os is grar han rvnu for hir sis hn is harvsing woul om o a hal ( =). Only h harvsing of firs sis rmains oraional. q q a a a a a a a a a a q (.) a a a a a a a a a a q (.7) a a q q Th oniion, for his o b osiiv fini, is ha a a q (.9) Cas (ii): If q hn h os is grar han rvnu for firs sis hn is harvsing woul om o a hal ( =). Only h harvsing of hir sis rmains oraional ( q ) Inrnaional Journal of Sinifi an Innovaiv Mahmaial Rsarh (IJSIMR) Pag 5 i, (.5) (.)

4 R. Srilaha & B. Ravinra Ry q (.) a a a a a a a a a q (.) a a a a a a a a a q (.) ow subsiuing,, in quaions (), (5) an () w g a a q q ow, whr h following oniion a a ` (.) q (.) Cas (iii): if q, hn h os is grar han h rvnu for h boh sis an sis-harvsing will om o a oal losr. Cas (iv): if q, h os is lss han rvnu rurn on h harvsing of h boh h sis, h sysm boms oraional o yil rofi. Th bionomi quilibrium ( ),( ),( ),, is h osiiv soluion of h sysm (..) Solving hs quaions, w g q, q, a a a a q q (.5) a a q q (.) a a q q (.7), a a, q a a (.) q (.9) Thus h bionomi quilibrium ( ),( ),( ),, xis if h oniions (.) an (.9) hol. 5. OPTIML RVSTIG POLICY Th rsn arg is o sl h harvsing oliy ha maximizs h rsn valu J of oninuous im sram of rvnus givn by J ( q ) ( ) ( ) ( ) (5.) whr nos h insananous annual ra of isoun. Innionally w hav o maximiz (5.) subj o h sa quaions (.) (.) by aoing Ponryagin s maximum rinil. Th onrol variabl i () is subj o h onsrains i ( ) ( ) max. Th amilonian for h roblm is givn by Inrnaional Journal of Sinifi an Innovaiv Mahmaial Rsarh (IJSIMR) Pag

5 Sabiliy nalysis of Thr Sis Mol in Sris Muualism wih Bionomi an Oimal arvsing of Two Trminal Sis ( q ) ( q ) q q whr, an ar h ajoin variabls. Consir h swihing funions ( ) ( q ) q. (5.) an ( ) ( q ) q (5.) Th oimal onrol will b a ombinaion of xrm onrols an h singular onrol. Th oimal onrol funion () an () ha maximizs mus saisfy h following oniions. ( ) max, whn ( ) i. () (5.5) whn ( ) i. () (5.) an ( ) max, whn ( ) i. () (5.7) whn ( ) i. () (5.) Thus h oimal harvsing oliy is ( ) ; ( ) max ( ) ; ( ) * ; ( ) an ( ) ; ( ) max ( ) ; ( ) * ; ( ) By Ponryagin s maximum rinil, ; ; ; Inrnaional Journal of Sinifi an Innovaiv Mahmaial Rsarh (IJSIMR) Pag 7 an q q q q q a a a q a a a a a q a q (5.) (5.9) (5.) (5.) (5.) (5.) (5.) (5.5) (5.)

6 R. Srilaha & B. Ravinra Ry fr simlifiaion w g q a q From (5.), (5.) an (5.) a q q i., whr q q Th soluion of whih an b obain as From (5.) an (5.7), q a i., Th soluion of whih an b obain as (5.7) (5.) (5.9) (5.) (5.) (5.) (5.) whr q From (5.) an (5.9), q i., Th soluion of whih an b obain as (5.) (5.5) whr q From (5.) an (5.), w g a singular ah, whih w obain From (5.) an (5.5), w g a singular ah, from (5.) Inrnaional Journal of Sinifi an Innovaiv Mahmaial Rsarh (IJSIMR) Pag

7 Sabiliy nalysis of Thr Sis Mol in Sris Muualism wih Bionomi an Oimal arvsing of Two Trminal Sis From whih w obain Thus from (5.) an (5.7), w wri as, F ( ) G ( ) (5.7) (5.) (5.9) Thr xiss a uniqu osiiv roo ( ) of F ( ) in h inrval K if h following inqualiis hol: F(), F( K ), ' F ( ) for. Similarly hr xiss a uniqu osiiv roo ( ) if G ( ) in h inrval K If h following inqualiis hol: G(), G( K ), For,, w g a a a a q q a a q q a a q q ' G( ) for (5.) (5.) (5.) n on h oimal quilibrium,, is rmin, h oimal harvsing ffor an an b rmin. From (5.), (5.) an (5.5) w obsrv ha i ( ) ( i,,) is innn of im is an oimum quilibrium. n hy saisfy h ransvrsaliy oniion a. Tha is hy rmain boun as. From (5.) an (5.7) w hav as an as Thus h oal onomi rvnu ( ),( ),( ),, ( ),( ),( ),, This imlis ha an infini isoun ra las o h oal onomi rvnu ning o zro, an hn h sysm woul rmains los.. UMRICL SIMULTIOS () L a =,α =.5,α =.5,q =.5, =,a =,α =.,α =., α =.,a =.5, α =.5, α =, q =., =, =5, = an = Inrnaional Journal of Sinifi an Innovaiv Mahmaial Rsarh (IJSIMR) Pag 9

8 oulaion Sis(S)oulaion oulaion R. Srilaha & B. Ravinra Ry Sis Sis Sis 5 im Fig.. Poulaion growh ra Variaions vrss im. 5 Sis(S)oulaion 5 Sis(S)oulaion Fig.. Phas-sa rajoris orrsoning o h sabiliis of h oulaion () L a =,α =,α =.5,q =., =,a =,α =.,α =., α =.,a =., α =., α =., q =.9, =, =, = an = 9 Sis Sis Sis im Fig.. Poulaion growh ra Variaions vrss im. Inrnaional Journal of Sinifi an Innovaiv Mahmaial Rsarh (IJSIMR) Pag 5

9 Sis(S)oulaion Sabiliy nalysis of Thr Sis Mol in Sris Muualism wih Bionomi an Oimal arvsing of Two Trminal Sis Sis(S)oulaion 5 Sis(S)oulaion 7. COCLUDIG RMRKS Fig.. Phas-sa rajoris Th bionomi quilibrium has bn analyz follow by h rminaion of oimal harvsing oliy by mloying Ponryagin s Maximum Prinil []. h say sa, h harvsing os r uni ffor is qual o h marginal rofi of h ffor. I is foun ha vn unr oninuous harvsing of h rminal sis, h oulaion may b mainain a an aroria quilibrium lvl. Som numrial xamls ar also omu using Malab. RFRCS []. Loka,.J.,lmns of Physial biology, Williams an Wilkins,Balimor,95. []. Volrra,V.,Lonssn la hori mahmaiqu la li ou lavi,gauhir-villars, Paris, 9. []. C. W. Clark, Mahmaial Bioonomis: Th Oimal Managmn of Rnwabl Rsours, Wily, w York 97. []. C.W. Clark, Bioonomi Moling an Fishris Managmn,Wily, wyork 95. [5]. F. Braur,.C. Souak, Sabiliy rgions an ransiion hnomna for harvs raor ry sysms, J. Mah. Biol. 7 (979) 9 7. []. F. Braur,.C. Souak, Sabiliy rgions in raor ry sysms wih onsan ra ry harvsing, J. Mah. Biol. (979) [7]. K.S. Chauhuri, bioonomi mol of harvsing a mulisis fishry, ol. Mol, (9), []. K.S. Chauhuri, Dynami oimizaion of ombin harvsing of a wo-sis fishry, ol. Mol. (9),.7 5. [9]. D. L. Ragozin an G. Brown, arvs oliis an nonmark valuaion in a raor ry sysm, J. nvirn.on. Manag., 95,.55. []. K.S. Chauhuri an S.S. Ray, On h ombin harvsing of a ry-raor sysm, J. Biol. Sys.. 99,.7-9 []. Ponryagin L.S., Bolyanskii V.S.,Gamkrlig R.., an Mishnko.F.,Th mahmaial hory of oimal ross(wily, w York,9). []. K. Shiva Ry an. Ch. Paabhiramaharyulu, hr sis osysm omrising of wo raors oming for a ry, vans in li sin rsarh, (), -. []. B. Ravinra Ry, Sabiliy analysis of wo omiiv inraing sis wih oimal an bionomi harvsing of h son sis, In. J. of Mahmais Trns an Thnology(), Vol., Issu, -. Inrnaional Journal of Sinifi an Innovaiv Mahmaial Rsarh (IJSIMR) Pag 5

10 R. Srilaha & B. Ravinra Ry []. B. Ravinra Ry, Sabiliy analysis of wo omiiv inraing sis wih oimal an bionomi harvsing of h firs sis, In. J. of Mahmaial rhiv-(),, -5. [5]. B. Ravinra Ry, mol of wo muually inraing sis wih limi rsours for boh h sis - a numrial aroah, In. J. of Mahmais an Comur liaions Rsarh, Vol., Issu, Mar, 5-9. Inrnaional Journal of Sinifi an Innovaiv Mahmaial Rsarh (IJSIMR) Pag 5