DIFFUSION ANALYSIS OF A PREY PREDATOR FISHERY MODEL WITH HARVESTING OF PREY SPECIES

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1 VOL., NO., DECEMBE 7 ISSN APN Journal of Engineering and Applied Sciences 6-7 Asian esearch Publishing Nework (APN). All righs reserved. DIFFUSION ANALYSIS OF A PEY PEDATO FISHEY MODEL WITH HAVESTING OF PEY SPECIES M. N. Srinivas and A. V. S. N. Mury Deparmen of Mahemaics, School of Advanced Sciences, Vellore Insiue of Technology Universiy, Vellore, Tamil Nadu, India mnsrinivaselr@gmail.com ABSTACT In his research aricle, we considered an ecological prey predaor fishery model sysem wih a generalized case where boh he paches are accessible o boh prey and predaor. We suppose ha he prey migrae beween wo paches randomly. The growh of prey in each pach in he absence of predaors is assumed o be logisic. The predaor consumes he prey wih inrinsic growh raes in boh he paches. The exisence of is seady saes and heir sabiliy (local and global) are analyzed. I is also emphasized he diffusive sabiliy of he sysem along wih some numerical simulaions. Numerical simulaion has also been performed in suppor of analysis by using MATLAB. Keywords: prey predaor, local and global sabiliy, boundedness, harvesing, bionomic equilibrium, diffusion.. INTODUCTION There is a need o proec he fish populaion, major food recourse, by resricing exploiaion of he fish, creaing naural resources and creaing of proeced zones for hem. By hese measures he species will grow wihou any disurbances and hence proeced populaion can improve heir numbers. Mahemaical modeling is an imporan ool which involves he sudy of various disciplines like Geneics, pharmacokineics, epidemiology, ecology, biology ec. In recen years, his modeling is raising and spreading o branches of life sciences. Bio Mahemaics has been useful in recen years o various problems in ecology and epidemiology [, ]. Policymakers and Scieniss who are working in marine fisheries and ohers discussed marine proeced areas by considering he economic and social benefis []. Marine reserve may be inroduced as a defending measure because i is assumed ha adul or juvenile migraion will replenish depleed fishing grounds beyond he borders of he marine reserve. The benefis of creaing marine reserves can go beyond he safeguard of a specific overfished fish populaion. Marine reserve can guard he marine landscape from degradaion caused by damaging fishing pracices; provide an imporan prospec o learn abou marine ecosysems. I helps o invesigae on species dynamics and managemen ools on proecion of all componens of a marine communiy. Marine reserves have provided many benefis as a ool for conservaion and marine environmenal managemen []. Takashina e al. [5] invesigaed he prospecive influence of saring of marine shelered zones on aqua ecosysems using differen mahemaical ools and validaed ha founding of marine shelered zone can resul in a subsanial deerioraion of species. Wang and Takeuchi [6] suggesed ools o simulae movemens of species beween zones. They have revealed ha here exiss an arrangemen ha species collaborae well hrough adapaions such ha predaors become exinc in each pach in he absence of adapaions. Dubey [7] suggesed a mahemaical srucure o sudy he impac of a reserved zone on he dynamics of an aqua eco sysem. Many of he reimbursemens accompanying wih marine shelered zones have been exensively explored and he arena is an acive area of research in heoreical ecology and mahemaical biology [8]. The auhors [9, ] invesigaed he sabiliy of hree species and four species wih migraion, bionomic equilibrium and opimal harvesing policy. The presen invesigaion is a sudy of diffusion sabiliy of a preypredaor sysem wih logisic growh in a wo zone aqua environmen.[-7] inspired us o consider o do he presen invesigaion is on he analyical and numerical approach of diffusive sabiliy of an aqua eco sysem.. MATHEMATICAL EQUATIONS Consider a marine environmen (Figure-)) where prey and predaor species living ogeher wih he following assumpions: () I is a prey-predaor sysem in a wo pach environmen () Boh are accessible o boh prey and predaors. () Each pach is supposed o be homogeneous () We suppose ha he prey migrae beween he wo paches randomly (5) The growh of prey in each pach in he absence of predaors is assumed o be logisic (6) The predaor consumes he prey in he zone and grows logisically wih inrinsic growh raes and carrying capaciies proporional o he populaion size. Figure-. Marine environmen. 7

2 VOL., NO., DECEMBE 7 ISSN APN Journal of Engineering and Applied Sciences 6-7 Asian esearch Publishing Nework (APN). All righs reserved. Le x (), z (), E, K,, m, q, r, represens biomass densiy of prey species, biomass densiy of he predaor species, he effor applied o harves he fish populaion, carrying capaciy of prey species, he equilibrium raio of prey o predaor biomass, he prey moraliy rae due o predaion, cach abiliy coefficien, inrinsic growh rae of prey species, inrinsic growh raes of predaors, respecively in pach-. Le y (), w (), E, L,, m, q, s, represens biomass densiy of prey species, biomass densiy of he predaor species, he effor applied o harves he fish populaion, carrying capaciy of prey species, he equilibrium raio of prey o predaor biomass, he prey moraliy rae due o predaion, cachabiliy coefficien, inrinsic growh rae of prey species, inrinsic growh raes of predaors respecively, in pach-. Le, represens he migraion raes from pach - o pach- x x u, represens he biomass and vice-versa. Le densiy of prey in pach-, y y u, represens he biomass densiy of prey in pach-, z z u, represens he biomass densiy of predaor in pach-, w wu, represens he biomass densiy of predaor in pach-. D, D, D, D represens he diffusion coefficiens of prey and predaor species in pach- and pach- respecively. Keeping hese in view, he dynamics of he sysem may be governed by he following parial differenial equaions ( / ) x rx x K x y m xz q E x D x () ( / ) y sy y L x y m yw q E y D y () z z ( z / x) D z () w w ( w / y) D w () where x(), y(), z(), w() (5) Throughou his analysis we assume ha r q E, s qe (6). EXISTENCE OF EQUILIBIA AND STABILITY IN THE ABSENCE OF DIFFUSION Seady saes of equaions - are given by (i) P (,,,) (ii) P ( x, y,,) (iii) P ( x, y, z,) (iv) P (,,, ) x y z w. Since we are ineresing o sudy he assumed ecological sysem abou inerior seady sae, le x, y, z, w are posiive soluions of x( ), y( ), z( ) and w ( ), where m ( r / K) ( x ) y, z x, ( qe r) x w y and x is he soluion of a ( x ) b ( x ) c x d (7) where a ( s / L) m m ( r / K) b ( ) ( s / L) m m ( r / K) r q E / c ( s / L) m r qe s q E m ( r / K) d r qe s qe / Equaion has a unique posiive soluion if s r m r q E s q E m (8) L K r q E s q E < (9) For y o be posiive, we mus have ( x ) ( r qe ) / m ( r / K) () x m ( s / L) y s qe y / For x o be posiive, we mus have y s q E / ( s / L) m () Local sabiliy: Le us now suppose ha he above sysem has a unique posiive equilibrium a P (,,, ) x y z w and he dynamics of he Jacobian marix of he sysem a (,,, ) P x y z w is given by a a a a, where a a b a ab b b a a m y m x a a b ab ab am y m y m x m x b a ab a m y 7

3 VOL., NO., DECEMBE 7 ISSN APN Journal of Engineering and Applied Sciences 6-7 Asian esearch Publishing Nework (APN). All righs reserved. b m x m m x y rx y a ; sy x b K x L y By ouh-hurwiz crieria, he necessary and sufficien condiions for local sabiliy of equilibrium poin a, a, P ( x, y, z, w ) are a, a ( a a a ) a a and a ( a a a a a a ). I is eviden ha a, a and a. Clearly he las wo ouh-hurwiz condiions are same. I is easily esablish he condiions. Global sabiliy: Now we shall discuss he global sabiliy of he inerior equilibriums, P ( x, y, z, w ) of he sysem - wihou diffusion Theorem : If A x B and C y D, where r A, m K m K m K m r B m K m K m K m C x s s s m y L m L m L m, x s s s D m y L m L m L m hen P (,,, ) x y z w is globally asympoically sable. Proof: Le us consider Lyapunov funcion V( x, y, z, w) x x x ln x x ln y l y y y y l z z z ln z z ln w l w w w w ( x x ) dx ( y y ) dy ( z z ) dz V () l l x d y d z d ( w w ) dw l w d Choosing l ( y ) / ( x ), l /, l / ; r s y V( ) ( x x ) y y m x x z z k L x y m y y w w yx xy ( ) x xx y ( x x )( z z ) ( z z ) x x ( y y )( w w ) ( w w ) y y To prove V () o be negaive, we mus have A x B and C y D. Hence he heorem (.) concludes ha, in he presence of predaors if prey populaion lie in a cerain inerval, hey may be susained a an appropriae equilibrium level.. DIFFUSION ANALYSIS In his secion, we have inspeced he seadiness of he sysem - in he presence of diffusion. Ecologically, i means ha he movemen of species a any direcion for several reasons. If we assume he movemen of species only in he verical direcion, hen he populaion densiy variables x, y, z, w are funcions of space variable u and ime variable. In his segmen, we deliberaed he excepional influences of ransmission of x x u,, y y u,, z z( u, ), w w( u, ) where u is a space variable and xu (,), yu (,), zu (,), wu (,), for u,. he ideal srucure (-). The rivial flucuaion edges condiions are specified by z u u, w x u u, y u u,. Now, le us consider he ideal (.)-(.) u u, underneah rivial flucuaions edge ailmens. To analyze he role of ransmission on his ideal, we deliberae he linear ideal of he srucure - abou he inerior seady sae P (,,, ) x y z w as given by r X() x X m x Z D X K () s Y() y Y my W DY L () Z () () DZ W (5) () DW by puing x x X ; y y Y ;. z z Z ; w w W. Assume ha he soluions of equaions in he form X e cos pu, Y e cos pu, Z e cos pu and W e cos pu (6) where p is he wave numeral of perurbaion, is he frequency numeral & i, i,,, are he ampliudes. (7) J J J J 7

4 VOL., NO., DECEMBE 7 ISSN APN Journal of Engineering and Applied Sciences 6-7 Asian esearch Publishing Nework (APN). All righs reserved. where rx sy J p ( D D D D ); K L rx sy J D p ( D D D ) p K L sy D p D D p D D p L rx sy J D p D p ( D D ) p DD p K L sy D p DD p L rx sy J D p D p DD p K L The following is he immediae consequence of -H Crieria Theorem (): The poin P ( x, y, z, w ) is locally asympoically sable in he aendance of ransmission, if J, J, J, J ( J J J ) J J and J ( J J J J J J ). Theorem (): (i) The sysem in he absence of spaioemporal aribues a he inner seady sae P ( x, y, z, w ) aains seadiness, hen he corresponding uniform seady sae of he model - in he presence of spaioemporal aribues also aains seadiness. (ii) If he inner seady sae P ( x, y, z, w ) of he non-spaial heerogeneiy sysem is unsable, hen he respecive seady sae of he spaioemporal model - under iniial and boundary seings and aain seadiness by increasing or decreasing he spaioemporal aribues suiably. Proof: Le us define he funcion V x y z wdu, where,,, V ( ),,, l V x y z w is defined in Sabiliy analysis secion. Differeniaing w.r. o along he soluions of he diffusive model (.)- (.), we ge, Vl zz zu ww wu ( / ) ( / ) D V du D V du () u u ( / ) ( / ) u u D x x x du D y y y du D z z z du D w w w du From 7, 8 and 9, i can clearly be observed ha if I hen V l () is negaive. If I hen i is clearly showing if here is an increase in he spaioemporal aribues D, D, D and D adequaely huge numeral, V l () as negaive. Henceforh he succeeding porion of he heorem grasps. 5. NUMEICAL SIMULATIONS In his division, we esablished he analyical findings hrough numerical simulaions using MATLAB. Figure-(a). Numerical simulaion. Figure-(a) denoes he variaion of populace agains ime wih r,.8,.6, m., K, m.8, E.9, E., q.5, q.7,.5,.8,.5,., L.5, s.; wih iniial condiions x ;;5;5 V () V x V y V z V w du I I (8) l x y z w D where I V() dx ; x y z (9) w ID DV x D V y D V z D V w du D Using he analysis in [], we ge xx u yy u I D V x du D V y du Figure-(b). Figure-(b) denoes he variaion of populace agains ime wih parameers 7

5 VOL., NO., DECEMBE 7 ISSN APN Journal of Engineering and Applied Sciences 6-7 Asian esearch Publishing Nework (APN). All righs reserved. r,.8,.6, m.8, m., E.9, E., q.5, q.7,.5,.8,.5,., L, K,.; x ;;5;5 s and iniial condiions Figure- (e) Figure-(c) Figure- (f) Figure- (d) Figure-, denoes he seady flucuaions of he prey, predaor populaions in boh he paches agains space and ime wih r,.8,.6, m., m.8; E.9; E., q.5, q.7,.5,.8,.5,., K.5, s., L ; 75

6 VOL., NO., DECEMBE 7 ISSN APN Journal of Engineering and Applied Sciences 6-7 Asian esearch Publishing Nework (APN). All righs reserved. Figure-(g) Figure-(j) Figures above denoes he seady flucuaions of he prey, predaor populaions in boh he paches agains space and ime wih r 5,.8,, m., m.8; E.9; E.; q.5; q.7;.5.8;.5,., L, K.5, s.; Figure-(h) 6. CONCLUSIONS In his paper, a mahemaical model has been proposed and analyzed o sudy he sabiliy on he dynamics of a wo pachy predaor-prey sysem. The model has been analyzed in a marine environmen wih effec of harvesing for boh prey species. Iniially we have discussed abou he model and invesigaed he exisence of equilibrium poins, local sabiliy by employing ouh- Hurwiz crieria, global analysis by consrucing Lyapunov funcion. Laer, we discussed abou he dynamics of he diffusion model compuer simulaions wih MATLAB have been execued o sudy he effecs of various parameers on he dynamics of he sysem. The analyical resuls and numerical simulaion of deerminisic model sugges ha he deerminisic prey predaor model is sable in naure. The sabiliy of he sysem and variaions in growh rae for he populaion species for various parameers shows in figures (a), (b) and he figures (5.)-(5.8) represens he variaion of populaions agains ime and space. EFEENCES Figure-(i) [] T.K. Kar, M. Swarnakamal. 6. Influence of prey reserve in a prey-predaor fishery, Non- Linear Anal. 65: [] Wendi Wang, Yasuhiro Takeeuchi, Yasuhisa Saio, Shinji Nakaoka. 5. Prey-predaor sysem wih parenal care for predaors. Journal f Theoriical Biology. ():

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