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

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1 Malaya Jounal of Matematik, Vol. 6, No. 3, , 28 Stability analysis of a pedato (bid) pey (fish) havesting model in the eseved and uneseved aea Kulbhushan Agnihoti * and Sheenu Nayye2 Abstact The exteme and unsustainable abuse of maine assets needs to pompt the advancement of a maine eseve as a fisheies management instument. In this pape, a pey-pedato fishey model in existence of bid pedato, with pey dispesal in a two-patch envionment has been poposed and examined. Holling type-i pedato functional esponse to pey density has been consideed fo this eseach wok. The havesting is applied on pey in an uneseved aea as well as on pedato due to a commecial value. The dynamics of the poposed famewok has been examined locally as well as globally. Finally, theoetical esults so acquied have been confimed with the suppot of numeical simulations though MATLAB. Keywods Pey-pedato; Global stability; Reseve-uneseved aea; Holling type-i; Lyapunov function; MPAs. AMS Subject Classification 35Q6, 44A, 44A5, 44A2, 44A3, 44A35,8V. Depatmentof Applied Sciences & Humanities, Shaheed Bhagat Singh State Technical Campus, Feozepu, Punjab, India. Schola, IKGPTU, Kaputhala, Punjab, India. *Coesponding autho: agnihoti69@gmail.com ; 2 sheenunayya85@gmail.com Aticle Histoy: Received 24 August 28; Accepted 3 Octobe 28 2 Reseach Contents Intoduction Fomulation of the model Existence of equilibia Stability analysis Global stability Numeical simulations Conclusion Acknowledgment Refeences Intoduction In the past few decades, the dynamics of inteacting biological species have been examined fom vaious angles. Numeous species have tuned out to be theatened o endangeed, and numeous othes ae on the vege of disappeaance as a consequence of diffeent easons like oveexploitation, ove-pedation, envionmental pollution, mismanagement of c 28 MJM. natual esouces etc. To save these species, maine potected and maine eseve aeas have been poposed as the most essential instument to peseve the maine life and sustain the ecosystem. Beaveton and Holt wee the fist in consideing the idea of maine eseves. Clak [3] intoduced the concept of economic and biological aspects of inexhaustible assets of multispecies fisheies. Recently, it has demonstated by Dubey [5] that the eseved aea has a stabilizing effect on the pedato-pey dynamics. It is confimed that egadless of whethe the fishey is exploited constantly in the uneseved zone, fishey population can be kept up at an appopiate equilibium level in the natual suoundings. Ka and Misa [7] have contemplated that the inteio equilibium level neve distubed in the absence of a pedato, continuous havesting and pesence of a pedato in the uneseved aea. Dubey [6] poved that in ecology and evolution, the eseved zone plays a vey significant ole. By ceation of a eseved zone in the habitat, povides the oppotunity fo the gowth of the pey species without any extenal distubances, whee the pedato has no access o chance of settling. In this way, the pey species can be peseved at a pope level. Ka and Chaudhui [] have investigated a pey-pedato fishey system, whee

2 Stability analysis of a pedato (bid) pey (fish) havesting model in the eseved and uneseved aea 679/684 only the pey species wee liable to havesting, by taking tax assessment as a contol instument. They looked fo an optimal tax policy and an inteio equilibium coesponding to given tax policy. As pe obsevational infomation fo Lake Kasumigaua in Japan, Kitabatake [5] built up a dynamic model fo fishey esouces with a pedato-pey elationship. Maine eseves secue the species inside the eseve aea along with incease fish ichness in adjoining aeas. Amit Shama and Bhanu Gupta [29] studied the dynamics of fishey esouce with eseve aea in the pesence of bid pedato. Vaious possibilities of the biological and bionomic equilibium of the system have been discussed. An optimal havesting policy has been established using Pontyagin s maximum pinciple. Yunfei et al. [28] investigated that maine eseves ensued the sustainability of the system. An appopiate equilibium level of pey population is always maintained in the pesence of pedatos as well as in the absence of pedatos in the uneseved zone. All the eseved and uneseved aea models in an aquatic habitat have been motivated by the existence of Maine National Pak, Kenya, a fully potected coal eef maine eseve compising appoximately 3% of fome fishing gound and Maine National Pak in the Ioise sea, a coastal sea west of Bittany (Fance). By the ceation of atificial bounday in the fom of fencing of appopiate mesh size, pedato s passageway can be esticted to the eseved zone. Latest eseaches found that MPAs ae an exceptionally viable instument fo impoving yield and in addition affimation of stocks and maintainability of jeopadized species. Keeping this in view, the pesent investigation is the modified model of Amit Shama et al. [29] within the sight of bid pedato in which Holling type I functional esponse is consideed. 2. Fomulation of the model Conside a habitat in a biological system with pey (fishes) dispesal in a two-patch envionment, one is assumed to be a fee fishing zone and othe is a eseved zone, whee fishing and othe additional activities ae confined. Both zones ae supposed to be homogeneous. Moeove, thee is a bid pedato in the famewok which may move in both the eseved and uneseved aeas of pey. We assume that the pey (fishes) species migate between the two zones andomly. The havesting is applied to pey in an uneseved aea aea as well as on the pedato. It is assumed that captuing ates ae same fom both the eseves fo the pedato. The logistic gowth is assumed only fo pey in uneseved aea. Keeping all the assumptions in view, a model is egulated by the following odinay diffeential equations. d = m m y q E d = s m m y (2.) dy = dy ( y y) q2 E2 y All the paametes of the system (2.) ae assumed to be positive and defined in the following Table: Vaiable/Paamete (t) (t) y(t) s m, E, E2 d q &q2 m Desciption biomass density of the pey species inside the uneseved aea biomass density of the pey species inside the eseved aea biomass density of the bid pedato the intinsic gowth ate of the pey species inside the uneseved aea the intinsic gowth ate of the pey species inside the eseved aea caying capacity of pey inside the uneseved aea migation ate fom the uneseved aea to eseved aea and eseved aea to uneseved aea espectively havesting effots applied to the pey (fishes) and the bid pedato espectively the natual death ate of a pedato (bid) the coefficient of catchability of pey in uneseved and pedato aea espectively captuing ate of pey in eseved and uneseved aea convesion ate of pey to pedato in uneseved and eseved aea In above model, it is assumed that if thee is no migation of fish population fom the eseved zone to the uneseved zone i.e. ( = ) and m q E, then dx <, In this case, fish species will be wiped out fom the uneseved aea. Coespondingly, if thee is no immigation of the fish pop679 ulation fom uneseved aea to eseved aea i.e. (m = 2 and s < ), then dx < holds consequently, fishes will extinct fom the eseved aea. To potect the pey (fishes) species fom extinction, migation of pey (fish) species fom both the patches ae essential. Theefoe, thoughout ou

3 Stability analysis of a pedato (bid) pey (fish) havesting model in the eseved and uneseved aea 68/684 and if analysis, we assume that F( ) = T 3 T2 2 T3 T4 < m q E > and s > (2.2) (3.5) then thee exists a positive value of (say ) in the inteval [, ]. Now, the sufficient condition fo to be unique is 3. Existence of equilibia F ( ) = 3T (2 )2 2T2 ( ) T3 < Following ae the two possible steady states of the dynamical system of Eq. (2.) value of and y will be given by q2 E2 d = y = q E m m (q2 E2 d) ( )2 I P (,, ), which always exist; (extinction of all species) II P (,, y ), (The inteio equilibium point) Fo the inteio equilibium point P (,, y ) On solving the 3d equation of diffeential equations of (2.) fo non-zeo point, we get q2 E2 d (3.) = On substituting the value of in diffeential equation of (2.), the value of y is given by y= m q E m (q2 E2 d) 2 (3.2) (3.6) Thus and Y will be positive iff following inequality holds q2 E2 d ( m q E ) < min, (3.7) Hence the equilibium P (,, y ) exist, povided condition (3.7) is satisfied. 4. Stability analysis The vaiational matix of the system of equations (2.) is Fo simplification let J(,, y) = q2 E2 d = E3. Now substituting the value of fom (3.) and y fom (3.2) in diffeential equation of (2.), we obtain following cubic equation in tems of T 3 T2 3 T3 T4 =, T = m q E λ m (3.3) s λ = d q2 E2 λ One of the eigenvalue is λ = (d q2 E2 ) < Othe two eigenvalues ae given by T3 = (( m q E )E3 se3 E3 ) T4 = E32, λ 2 λ ( m q E s ) assume F() = T4 > The chaacteistic equation of system (2.) at P (,, ) is given by (λ d q2 E2 ) λ 2 λ ( m q E s ) ( m q E )(s ) m = T2 = q E s E3 It is obvious that m m d q2 E2 Poof. F( ) = T 3 T2 2 T3 T4 s my y Theoem 4.. If the equilibium point P (,, ) exist then it will be always unstable. whee 2x K m q E my m y (( q E )(s ) m s) = (3.4) As ( m q E ) (s ) > (Assumptions) So, all the eigenvalues of the above chaacteistic equation ae not negative as thee is at least one change of sign. Theefoe the equilibium point P (,, ) is always unstable. Hence, P (,, ) is unstable. 68

4 Stability analysis of a pedato (bid) pey (fish) havesting model in the eseved and uneseved aea 68/684 Biological meaning: It is concluded that even if the system is exploited continuously in the uneseved zone, the pey o the pedato population pesist and ae not extinct fo sufficiently lage time. V (,, y) = log h log y h2 y y y log y ( ) d dv = ( ) d h (y y ) dy h2 y, h = m m h2 = Theoem 4.2. Fo the system (2.), if the inteio equilibium point P (,, y ) exist, is always locally asymptotically stable. Poof. The chaacteistic equation of the vaiational matix of the system (2.) at P is K2 m q E my m α y s my y m = m d q2 E2 that is x λ m y m m λ y m = λ λ 3 C λ 2 C2 λ C3 = Afte simplification we get ( )2 dv = ( )2 m(y y )( ) m (4.) whee Hence m > m C2 = m y ( ) C3 = m y C = dv < povided (y y )( ) m m y ( m ) m m 2 y > m m CC2 C3 = K2 2 m y (5.) So, V is negative definite, povided (5.) holds. Thus equilibium point of the system of equations (2.) is globally asymptotically stable fo the afoesaid condition. Moeove if m = i.e. if the numbe of pey migated fom uneseved aea to eseve aea ae same that of the numbe of pey migated fom eseved aea to uneseved aea then the system will become globally stable. > As C >,C3 > and CC2 C3 > Theefoe by Ruth-Huwitz citeion, P always locally asymptotically stable. 6. Numeical simulations In ode to investigate the dynamics of the system (2.) with help of numeical simulation, we choose a diffeent set of paametes. Let 5. Global stability In this section, we conside the global stability of the system of equations (2.) at inteio equilibium point P (,, y ) by constucting a suitable Lyapunov function mentioned as below: 68 = 3, s = 2, m = 2, =, m =.5, q =.2 q2 =., E =.5, E2 =.8, d =.5, =, =.2. (6.)

5 Stability analysis of a pedato (bid) pey (fish) havesting model in the eseved and uneseved aea 682/684 o this set of paametes, equilibium point P (2.4876, 5.524, ) exist and locally stable as it satisfies existence condition (3.7) and stability condition of Theoem 2 (see Fig. ). Futhe taking anothe set of paametes = 3, s = 2, m =, =.5, m =.5, q =. q2 =.2, E = 3, E2 = 4, d =.3, = 5, =.5. (6.2) Hee it is obseved that as migation ate of the fishes fom uneseved to eseved aea and vice vesa deceases, then population of pey in uneseved aea deceases wheeas population of pey in eseved aea and pedato inceases that is depicted by P (2.4685, 5.75, ) at m =, =.5, P (.8863, , 4.67)at m =.7, =.2 (see Fig. 2). Taking the same paametes as (6.2) except inceasing the caying capacity of pey in uneseved aea we get equied equilibium point P (2.9953, , 4.382)at = 3 and P (3.383, 4.47, 4.426) at = 5. Hee it is noticed that density of pey in uneseved aea and pedato inceases on the othe hand density of pey in eseved aea deceases (see Fig. 3). Figue (b). Time seies plot of (t), (t) and y(t) of P fo data set (6.) Figue 2. The phase diagam showing the local stability of P fo the data set (6.) [x(t): pey in uneseved aea; y(t): pey in eseved aea; z(t): pedato] Fo this set of paameteise values (6.2), the equilibium point P (2.6337, 4.893, 4.77) exists and also become stable. On futhe educing the havesting effots ate to E =. and E2 =.5. It is obseved that density of pey in uneseved aea as well as pedato inceases wheeas density of pey in eseved aea deceases (see Fig. 4). The extensive simulation is done and it is investigated that only P and P exist and P is locally as well as globally stable. Figue 3. Time seies plot of (t), (t) and y(t) of P fo data set (6.) except m =, =.5, K = 3 Figue (a). The phase diagam showing the global stability of P fo the data set (6.)[ (t) : pey in uneseved aea; (t) : pey in eseved aea; y(t) : pedato] Figue 4. The phase diagam showing the local stability of P fo the data set (6.2) except E =.5, E2 =.5 682

6 Stability analysis of a pedato (bid) pey (fish) havesting model in the eseved and uneseved aea 683/684 [8] 7. Conclusion A pey-pedato fishey model in the pesence of bid pedato, with pey dispesal in a two-patch envionment, has been poposed and investigated in the pesent pape. The havesting is applied on pey (fishes) in an uneseved aea as well as on the pedato (bid). A theshold fo existence, local along with the global stability at vaious equilibium points has been inspected. It has been obseved that global stability of inteio equilibium point P exists unde cetain conditions. Fom the numeical simulation, it has been veified that decease of E and E2 (the havesting effots of pey in the uneseved zone and pedato) cause a decease in the population of the pey species in eseved aea wheeas incease in the biomass density of pey in uneseved aea as well as pedato. Futhe by numeical simulation, it is exposed that incease of caying capacity of pey in an uneseved aea is esponsible fo the decease of the population of pey in eseved aea wheeas incease in the population of the pey species in uneseved aea and pedato. Moeove it is veified that as migation ate of the fishes fom uneseved to eseved aea and vice vesa deceases, then population density of pey in uneseved aea deceases wheeas population density of pey in eseved aea and pedato inceases. 8. Acknowledgment [3] [4] [5] [6] [7] [] [2] [3] [4] [5] [7] [8] [9] Refeences [2] [] [6] Authos ae thankful to anonymous eviewes fo thei caeful eading, useful comments and constuctive suggestions fo the impovement of the pesent eseach wok. We ae also thankful to edito fo his helpful comment. Futhe authos gatefully acknowledge the suppot povided by the IKG Punjab Technical Univesity, Punjab, India. [] [9] M. Agawal and R. 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7 Stability analysis of a pedato (bid) pey (fish) havesting model in the eseved and uneseved aea 684/684 [26] [27] [28] [29] N. Juneja and K. Agnihoti, Global Stability of Havested Pey Pedato Model with Infection in Pedato Species, In Infomation and Decision Sciences, 28, Spinge, Singapoe. N. Juneja, K. Agnihoti and H. Kau, Effect of delay on globally stable pey-pedato system, Chaos, Solitons & Factals, (28), Y. Lv, R. Yuan and Y. Pei, A pey-pedato model with havesting fo fishey esouce with eseve aea, Applied Mathematical Modelling, 37(5)(23), A. Shama and B. Gupta, Havesting model fo fishey esouce with eseve aea and bid pedato, Jounal of Maine Biology, 24.????????? ISSN(P): Malaya Jounal of Matematik ISSN(O): ????????? 684

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

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