Harvesting model for fishery resource with reserve area of bird predator and modified effort function

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1 Malaya Jounal of Matematik Vol 5 No Havesting model fo fishey esouce with eseve aea of bid pedato and modified effot function Y ouatassi * and J El Alami and N Elalami3 Abstact The pupose of this pape is to stu the namics of fisheies esouces with eseve aea in the pesence of bid pedatos Aquatic aea unde investigation is divided into two aeas : one fo the fee fishing and othe limited fo any type of fishing In the poject havesting system accoding to the modified E effot is consideed which depends on the effect of esouce density The local and stability citeia the oveall stability and instability ae established fo the poject model Finally the theoetical esults ae illustated by numeical simulations in the last section eywods Fichey effot Stability Havesting AMS Subject Classification 9D5 Mohammed V Univesity in Rabat Supeio School of Technology Sale ASTIMI Sale Moocco V Univesity in Rabat Supeio School of Technology Sale ASTIMI Sale Moocco 3 Mohammed V Univesity in Rabat Mohammadia School of Enginees Rabat Moocco *Coesponding autho: ylouatassi@gmailcom; alamijamila@gmailcom 3 pelalami@gmailcom Aticle Histoy: Received 07 June 06; Accepted 4 Octobe 07 Mohammed Contents Intoduction 660 The Model 66 3 Eistence of Equilibia 66 4 Stability Analysis 66 5 Numeical Simulation Conclusion 663 Acknowledgments 665 Refeences 665 Intoduction Renewable esouces ae consideed an impotant food souce fo the gowth and suvival of biological population But the continued opeation and unplanned these esouces can lead to the etinction of esouces and thus affecting the suvival of species dependent on esouces Fo this thee has been a significant inteest in modeling of enewable esouces such as fisheies and foesty Dynamic models fo commecial fishing have been widely studied in the light of economic and ecological factos [6 7] In ecent decades thee has been consideable inteest fo modeling the namics of fisheies c 0 MJM esouces systems [ ] Chaudhui [3] poposed a model fo two species of fish competitos each of which develops logistically It eamined the stability analysis and discussed the bionomic balance and optimal havesting policy It has also been shown [3] that thee is no limit cycle in the positive quadant Fan and Wang [] genealized the classical model of Clak [6 7] by consideing the time-dependent the ogistic equation with peiodic coefficients and they showed that thei model has a unique positive peiodic solution which is globally asymptotically stable fo positive solutions Dubey et al [8] poposed a model whee the fish population depends in pat of a esouce and is havested They eamined the stability analysis Dubey et al [9] discussed a model of a fishey esouce system in an aquatic envionment has been divided into two aeas : the fee fishing aea and the eseved aea They discussed the biological and bionomic balance and the optimal havesting policy Dubey et al [0] also poposed and analyzed a shoe-to-sea fishing model whee the fish population is havested in both zones Then they studied the stability analysis and optimal havest policy taking taation as a contol instument ouatassi et al [6] has studied the coastal sea-fishing model whee the fish population is havested in two aeas of Dubey et al [0] whee he popose a compehensive state egulation by output feedback

2 Havesting model fo fishey esouce with eseve aea of bid pedato and modified effot function 66/666 based on a yapunov function Shama et al [9] poposed a stu the namics of fishey esouce with eseve aea in the pesence of bid pedato The aquatic egion unde investigation is divided into two zones : one fee fo fishing and anothe esticted fo any kind of fishey The citeia of biological and bionomic equilibium of system ae established The points of local stability global stability and instability ae obtained fo the poposed model An optimal havesting policy is established using Pontyagin s maimum pinciple The pupose of this pape is to stu a mathematical model of fishey esouce with eseve aea in the pesence of bid pedato In the poposed havest model a modified effot function E is consideed that depends on the effect of the density of the biomass esouce ocal stability citeia global stability and instability ae established fo the poposed system The obtained theoetical esults ae illustated using numeical simulations in the last section fish abundance on fishing effot That is it do not addess the fact that highe the density of fishes lesse the amount of effot needed to catch unit havest In ode to ovecome this deficiency Idels et al [4] poposed a modified effot function which is a function of t as well as (espectively z) and is given by E (t ) α (t) (t) d E (t z) α (t) (t) z () whee αi 0 i 0 fo i ae continuous functions of t Incopoating () we get the following modified vesion of the model () as : d The Model By Shama et al [9] it is consideed a fishey esouce system consists of two zones : a fee fishing and a eseve aea whee fishing is pohibited Each zone is assumed to be homogeneous Thee ae also a bid pedatos feed on both that is fish eseved and esticted aeas It is assumed that the pedato population is havested in the aea without esevation We assume that the pey species migate between the two aeas in a andom way Gowth pey in each zone in the absence of pedato is assumed to be logistics Taking these in ode the model becomes d σ y q q q m z q α q q y sy σ y m yz k m z k m yz q α z q q q q (3) All the paametes ae assumed to be positive Hee we obseve that if thee is no migation of fish population fom the eseved aea to the uneseved aea (ie 0 ) and σ q α < 0 then < 0 Similaly if thee is no migation of fish population fom the uneseved aea to eseved aea (ie σ 0 ) and s < 0 then y < 0 Hence thoughout ou analysis we assume that σ y m z q E y () σ y m yz sy σ q α > 0 s > 0 q > 0 and q > 0 (4) k m z k m yz q E z Hee (t) and y(t) ae the espective biomass densities of the pey species inside the uneseved and eseved aeas espectively at a time t z(t) is the biomass density of pedato at time t; espectively; and s ae intinsic gowth ates of pey species inside the uneseved and eseved zones espectively; and ae the caying capacities of pey species in the uneseved and eseved zones espectively; σ and ae migation ates fom the uneseved aea to eseved aea and the eseved aea to the uneseved aea espectively; d is death ate of pedato; m and m ae the captuing ates and k and k ae the convesion ates of pey in uneseved and eseved zones espectively; q and q ae the catchability coefficient of pey and pedato in uneseved zone espectively; E and E ae the effots applied to havest the fish population and pedato in uneseved zone espectively In this case the fishing effot E is simply taken as a function of t ie E E(t) which do not addess the invese effect of To simplify the stu of the poposed model in (3) we assume that catch ates ae the same two eseves which aeas m m m and convesion ates of pey in uneseved and eseved zones ae the same that is k k k Thus unde these assumptions the system (3) becomes d σ y q q q mz q α q q y sy σ y myz α (z yz) q α z q q q (5) whee α km 66

3 Havesting model fo fishey esouce with eseve aea of bid pedato and modified effot function 66/666 Fist the following emma says it is a egion of attaction fo the model system (5) eists and is a solution of σ y m z q α emma All the solutions of the system (5) which initiate in R3 ae unifomly bounded sy 0 y σ y my z Poof We define a function w(t) (t) y(t) z(t) and η > 0 be a constant Then the time deivative of w along the solution of system (5) is 0 α ( z y z ) q α z 0 (3) 4 Stability Analysis d dw ηw η ηy ηz We now investigate the namical behaviou of system (5) at equilibium points The geneal vaiational mati coe (q ( q α q σ η) ) sponding to the system (5) is given by s y s (s η) q α m η z qm qα y m qα qd q V q q 4(q ( q α q σ η) 4s (s η) ) σ V my (4) V ( y z) 4(q ) ( q α q σ η) 4s (s η) αz αz V3 q q µ (6) q α whee V σ mz V s s mz and q α V3 dα(y)q q Fistly the equilibium point P ( y 0) has the chaacteistic η equation is λ a λ a 0 eηt w((0) y(0) z(0))eηt 0 < w((t) y(t) z(t)) whee µ (7) a (q ) (q ) y By the thoey of diffeentiel inequality [ 8] we have and fo t 0 0 < w µ/ η This poves the lemma a (q ) (q ) y s y σ y σ (4) 3 Eistence of Equilibia Theefoe The namical behavio of a system is studied at equilibium points and equilibium points of model (5) ae obtained by solving y z 0 This gives thee possible stea states namely P0 (0 0 0) P ( y 0) and P ( y z ) At P0 (0 0 0) the population is etinct and this equilibium point always eists Now conside the equilibium point P ( y 0) whee the pedato is not pesent Hee and y ae the positive solutions of σ y q α 0 (3) y sy σ y 0 λ λ λ λ Now assume hee that the inteio equilibium point P ( y z ) < 0 (q ) (q ) y (q ) (q ) y s y σ y σ > 0 (43) Theefoe all eigenvalues ae negative and hence P is locally asymptotically stable On the othe hand et us now suppose that system (5) has a unique positive equilibium P ( y z ) The vaiational mati of (5) at P is This system (3) is alea solved by Dubey et al [0] and local and global stability esults fo the system at P ( y 0) ae discussed thee V ( y z) 66 q y m q q σ s y σ y my αz q αz q 0

4 Havesting model fo fishey esouce with eseve aea of bid pedato and modified effot function 663/666 nipulation yields (44) The chaacteistic equation of vaiational mati of system (5) at P is given by λ 3 b λ b λ b3 0 whee b b b3 y (q ) (q ) dv (q )(q ) z αm (qs ) y z αm > 0 y 5 Numeical Simulation To stu namics of the system (5) with the numeical simulation fo this we choose the values of the following paametes (see [9]) : s 7 00 σ 07 0 q 00 q 00 α 0003 m 00 d 00 α 0 α 5 (5) In appopiate units with initial conditions ( ) Fo the values of the paametes listed below in (5) the conditions (4) ae satisfied Also accoding to the Theoem 4 the balance point P ( y z ) is both locally and globally asymptotically stable within the quadant Fo the set of values of paametes given in (5) the behavio of y and z with espect to time t is plotted in Figue In Figue fistly the biomass density of pey species in the egion inceases without esevation with espect to time and little deceases slightly and moved to its equilibium level Secondly it is clea that the biomass density of the pey population in the eseved aea inceases abuptly nea its caying capacity then moved to its equilibium level nea the caying capacity of this zone Finally the same figue shows that the density of the biomass of pedatos inceases with time in an almost linea way and ties to adjust to thei equilibium level (46) and hence P ( y z ) is locally asymptotically stable Now we will discuss the global stability of the endemic equilibium point P ( y z ) of the system (5) Theoem 4 The equilibium point P ( y z ) of system (5) is globally asymptotically stable if y (q )σ (y y ) (z z ) < 0 Poof et us conside the following yapunov function: l z z z ln l y y y ln yy z z (47) We note that the αi and i fo i ae impotant paametes that goven the namics of the system Theefoe we plotted the behavio of y and z with time t fo diffeent values of αi and i in Figues 4 whee l and l ae positive constants to be chosen late on Diffeentiating V ( y z) with espect to time t we get dv d l y y y (y y ) (z z ) < 0 Clealy dv / < 0 ifand only if y (y y ) (z z ) < 0 ( q )σ Theefoe P ( y z ) is globally asymptotically stable z ln l (y y ) (49) Accoding to Routh-Huwitz citeia the necessay and sufficient conditions fo local stability of equilibium point P ae b > 0 b3 > 0 and b b b3 > 0 It is evident that b > 0 and b3 > 0 Thus the stability of P is detemined by the sign of b b b3 By diect calculation we obtain y s b b b3 y σ y (q ) (q ) s y y σ s (q ) (q ) (q ) V ( y z) )σ )σ (45) y ( ) s (q αm q ( z y z ) αm y z αms y z αmσ (q ) k(q )(q ) (q ) αmσ αm q z (q )(q ) y z y z αm (q )(q ) ) y m (q s y σ y y σ (q )y s (q ) y s (q ) (q z z z 6 Conclusion (48) Choosing l (y / ) ( / ( q )σ ) and l (m( q ))/ (α( q )) a little algebaic ma663 In this pape we have is a mathematical model of eploitation of fisheies esouces with the eseve aea in the pesence of bid pedatos and the function of the modified stain has been poposed It was assumed that the aquatic ecosystem consists of two zones : a fee fishing aea and othe esticted aeas whee fishing is pohibited It was assumed that fish populations ae logistic gowth in both aeas Using the theoy

5 Havesting model fo fishey esouce with eseve aea of bid pedato and modified effot function 664/666 Figue Plot y and z vesus time t Figue 3 Plot of y veses time t to α 0 α 0 (ed) Figue Plot of veses time t to α 0 α 0 (ed) α 40 (blue) α 60 (geen) Figue 4 Plot of z veses time t to α 0 α 0 (ed) α 40 (blue) α 60 (geen) α 40 (blue) α 60 (geen) 664

6 Havesting model fo fishey esouce with eseve aea of bid pedato and modified effot function 665/666 of stability of odinay diffeential equation it has been poven that thee is inne balance unde cetain conditions and it is globally asymptotically stable [6] Acknowledgments [7] The authos would like to epess sincee gatitude to the eviewes fo his/he valuable suggestions Refeences [] [] [3] [4] [5] [6] [7] [8] [9] [0] [] [] [3] [4] [5] [8] G Bikoff and G C Rota Odinay Diffeential Equations Ginn 98 D Bhattachaya and S Begum Bionomic equilibium of two-species system Mathematical Biosciences vol 35 no pp S Chaudhui A bioeconomic model of havesting a multispecies fishey Ecological Modelling vol 3 no 4 pp S Chaudhui and S Saha Roy Bioeconomic eploitation of a lotka-voltea pey-pedato system Bulletin of Calcutta Mathematical Society vol 83 pp S Chaudhui and S S Ray On the combined havesting of a pey-pedato system Jounal of Biological Systems vol 4 no 3 pp C W Clak Mathematical Bioeconomics : The Optimal Management of Renewable Resouce John Wiley & Sons New Yok NY USA 976 C W Clak Mathematical Bioeconomics : The Optimal Management of Renewable Resouce John Wiley & Sons New Yok NY USA nd edition 990 B Dubey P Chanda and P Sinha A esouce dependent fishey model with optimal havesting policy Jounal of Biological Systems vol 0 no pp 3 00 B Dubey P Chanda and P Sinha A model fo an inshoe-offshoe fishey Jounal of Biological Systems vol no pp B Dubey P Chanda and P Sinha A model fo fishey esouce with eseve aea Nonlinea Analysis: Real Wold Applications vol 4 no 4 pp M Fan and Wang Optimal havesting policy fo single population with peiodic coefficients Mathematical Biosciences vol 5 no pp H I Feedman and J W H So Global stability and pesistence of simple food chain Mathematical Biosciences vol 76 no pp B Gupta and A Shama Havesting Model fo Fishey Resouce with Reseve Aea and Modified Effot Function Malaya J Mat 4() V Idels and M Wang Havesting Fisheies Management Sategies with Modified Effot Function Intenational Jounal of Modelling Identification and Contol 3() : T a S Misa and B Mukhopadhyay A bioeconomic model of a atio-dependent pedato-pey system 665 [9] [0] [] [] [3] [4] [5] [6] [7] [8] [9] [30] [3] [3] and optimal havesting Jounal of Applied Mathematics and Computing vol no pp Y ouatassi N Elalami and E H El Mazoudi Stability analysis and static output feedback design fo a model the fishey esouce with eseve aea Applied Mathematical Sciences vol 6 no 66 pp Y ouatassi and N Elalami Obseve Design fo a Mathematical Model fo the Management of a Renewable Resouce Population Intenational Jounal of Computing Science and Mathematics Vol 6 No pp 8 05 Y ouatassi YE El Mazoudi and N Elalami A new genealization of lemma Gonwall-Bellman Applied Mathematical Sciences 6(3) : A Shama and B Gupta Havesting Model fo Fishey Resouce with Reseve Aea and Bid Pedato Jounal of Maine Biology Volume 04 ID pages 04 J B Shukla and B Dubey Modelling the depletion and consevation of foesty esouces: effects of population and pollution Jounal of Mathematical Biology vol 36 no pp R P Agawal Diffeence Equations and Inequalities: Theoy Methods and Applications Macel Dekke New Yok 999 I Gyoi and G adas Oscillation Theoy of Delay Diffeential Equations with Applications Claendon Pess Ofod 99 D A Geogiu E A Gove and G adas Oscillations of neutal diffeence equations Appl Anal 33(989) J R Gaef and P W Spikes Asymptotic decay of oscillatoy solutions of foced nonlinea diffeence equations Dyn Syst Appl 3(994) 95 0 G H Ha J E ittle wood and G Polya Inequalities The pint of the 95 edition Cambidge Univesity Pess Cambidge U 988 J W Hooke and W T Patula A second ode nonlinea diffeence equation: Oscillation and asymptotic behaviou J Math Anal Appl 9(983) 9 9 V akshmikantham and D Tigiante Theoy of Diffeence Equations: Numeical methods and Applications Academic pess New Yok 988 G adas Ch G Philos and YG Cas Shap conditions fo the oscillation of delay diffeence equations J Appl Math Simulation (989) 0 B S alli and B G Zhang On eistence of positive solutions and bounded oscillations fo neutal diffeence equations J Math Anal Appl 66(99) 7 87 T i and Y V Rogovchenko Oscillation theoems fo second - ode nonlinea neutal delay diffeential equations Abst Appl Anal Vol 04 Aticle ID H J i and C C Yeh Oscillation citeia fo secondode neutal delay diffeence equations Comput Math Appl 36(998) 3 3 W T i and S S Cheng Oscillation citeia fo a

7 Havesting model fo fishey esouce with eseve aea of bid pedato and modified effot function 666/666 [33] [34] [35] [36] [37] [38] [39] [40] [4] [4] [43] [44] nonlinea diffeence equation Comput Math Appl 36(8)(998) A Muugesan and Ammamuthu Conditions fo oscillation and convegence of solutions to second ode neutal delay diffeence equations with vaiable coefficients Malaya J Mat 5()(07) S H Sake and S S Cheng Oscillation citeia fo diffeence equations with damping tems Appl Math Comput 48(004) 4 44 A Stenal and B Szmanda Asymptotic and oscillatoy behaviou of cetain diffeence equations EMATEMATICHE (996) Z Szafanski and B Szmanda A note on the oscillation of some diffeence equations Fasc Math (990) Z Szafanski and B Szmanda Oscillations of some linea diffeence equations Fasc Math 5(995) B Szmanda Note on the behaviou of solutions of a second ode nonlinea diffeence equation Atti Acad Naz incei Rend Sci Fiz Mat 69(980) 0 5 B Szmanda Chaacteization of oscillation of second ode nonlinea diffeence equations Bull Polish Acad Sci Math 34(986) 33 4 B Szmanda Oscillatoy behaviou of cetain diffeence equations Fasc Math (990) E Thandapani and Mahalingam Necessay and sufficient conditions fo oscillation of second ode neutal diffeence equations Tamkang J Math 34()(003) E Thandapani Asymptotic and oscillatoy behaviou of solutions on nonlinea second ode diffeence equations Indian J Pue Appl Math 4(993) E Thandapani Asymptotic and oscillatoy behaviou of solutions of a second ode nonlinea neutal delay diffeence equation Riv Mat Univ Pama (5)()(99) 05 3 B G Zhang and S S Cheng Oscillation citeia and compaison theoems fo delay diffeence equations Fasc Math 5(995) 3 3????????? ISSN(P): Malaya Jounal of Matematik ISSN(O):3 5666????????? 666

Stability analysis of a predator (bird) prey (fish) harvesting model in the reserved and unreserved area

Stability analysis of a predator (bird) prey (fish) harvesting model in the reserved and unreserved area Malaya Jounal of Matematik, Vol. 6, No. 3, 678-684, 28 https://doi.og/.26637/mjm63/32 Stability analysis of a pedato (bid) pey (fish) havesting model in the eseved and uneseved aea Kulbhushan Agnihoti