A Prey-Predator Model with a Reserved Area

Transcription

1 Nonlinear Analysis: Modelling and Conrol, 2007, Vol. 12, No. 4, A Prey-Predaor Model wih a Reserved Area B. Dubey Mahemaics Group, Birla Insiue of Technology and Science Pilani , Rajashan, India bdubey@bis-pilani.ac.in Received: Revised: Published online: Absrac. In his paper, a mahemaical model is proposed and analysed o sudy he dynamics of a prey-predaor model. I is assumed ha he habia is divided ino wo zones, namely free zone and reserved zone. Predaors are no allowed o ener ino he reserved zone. Crieria for he coexisence of predaor-prey are obained. The role of reserved zone is invesigaed and i is shown ha he reserve zone has a sabilizing effec on predaor-prey ineracions. Keywords: prey-predaor, reserve zone, sabiliy. 1 Inroducion The biosphere is an imporan zone for biological aciviies ha are mainly responsible for he changes in ecology and environmen. The co-exisence of ineracing biological species has been of grea ineres in he pas few decades and has been sudied exensively using mahemaical models by several researchers [1 10]. Many biological species have been driven o exincion and many ohers are a he verge of exincion due o several exernal forces such as overexploiaion, over predaion, environmenal polluion, mismanagemen of he habia, ec. In order o proec hese species, appropriae measures such as resricion on harvesing, creaing reserved zones/refuges, ec. should be adoped ha will decrease he ineracion of hese species wih exernal forces. The role of reserve zones/refuges in predaor-prey dynamics has received considerable aenion and has also been invesigaed by several researchers [11 21]. In paricular, Collings [11] sudied he nonlinear behavior of predaor-prey model wih refuge proecing a consan proporion of prey and wi emperaure dependen parameers chosen appropriaely for a mie ineracion on frui species. He showed he exisence of a emperaure inerval in which increasing he amoun of refuge dynamically desabilizes he sysem; and on par of his inerval he ineracion is less likely o persis in ha predaor and prey minimum populaion densiies are lower han when no refuge is available. Krivan [12] proposed a mahemaical model and invesigaed he effecs of opimal anipredaor behavior of prey in predaor-prey sysem. He showed ha opimal anipredaor behavior of prey leads o persisence and reducion of oscillaions in populaion densiies. Chaopadhyay e al. [13] 479

2 B. Dubey sudied a prey-predaor model wih some cover on prey species. They observed ha global sabiliy of he sysem around posiive equilibrium does no necessarily imply he permanence of he sysem. Recenly, Kar [18] proposed a predaor-prey model incorporaing a prey refuge and independen harvesing on eiher species. He showed ha using he harvesing effors as conrol, i is possible o break he cyclic behavior of he sysem. In he above invesigaions, he dynamics of predaor living in unreserved zone ogeher wih prey has no been sudied explicily. The reserve zone plays a vial role in aquaic environmen for he proecion of fishery resources from is overexploiaion [22 26]. In paricular, Dubey e al. [22] proposed and analyzed a mahemaical model o sudy he dynamics of a fishery resource sysem in an aquaic environmen consising of wo zones, namely a free fishing zone and a reserve zone where fishing is sricly prohibied. I was suggesed ha even if fishery is exploied coninuously in he unreserved zone, fish populaions can be mainained a an appropriae equilibrium level in he habia. The model presened in his paper will be of grea use in a Naional Park where prey-predaor are living ogeher. The prey species which are o be conserved can be proeced from predaors by creaing an arificial boundary or sheler ha will divide he habia ino wo zones one reserved and oher unreserved. The enry of predaors ino reserved zone can be resriced by he arificial boundary ha may be in he form of fencing of suiable mesh size hrough which prey can pass bu predaors can no. The model sudied in Secion 4 when predaor is parially dependen on he prey) can also be used in fishery resources where fisherman can be hough of as predaor in fac, generalis predaor) and fishing is no permied in a paricular zone, called he reserved zone. Keeping his in view, we consider a habia consising of wo zones: an unreserved zone where prey and predaor can move freely and a reserved zone where prey can live bu predaors are no allowed o ener inside. We consider he wo cases: one when he predaor is wholly dependen on he prey and oher when he predaor is parially dependen on he prey in he unreserved zone. In fac, we consider he model developed in [22] by incorporaing an addiional equaion for predaor in he unreserved zone. Then we sudy he coexisence and sabiliy behavior of predaor-prey sysem in he habia. 2 Mahemaical model Consider a habia where prey and predaor species are living ogeher. I is assumed ha he habia is divided ino wo zones, namely, reserved and unreserved zones. I is assumed ha predaor species are no allowed o ener inside he reserved zone whereas he free mixing of prey species from reserved o unreserved zone and vice-versa is permissible. Le x) be he densiy of prey species in unreserved zone, y) he densiy of prey species in reserved zone and z) he densiy of he predaor species a any ime 0. Le be he migraion rae coefficien of he prey species from unreserved o reserved zone and he migraion rae coefficien of prey species from reserved o unreserved zone. I is assumed ha he prey species in boh zones are growing logisically. Keeping hese in view and following Dubey e al. [22], he dynamics of sysem may 480

3 A Prey-Predaor Model wih a Reserved Area be governed by he following sysem of ordinary differenial equaions: dx d = rx 1 x ) x + y β 1 xz, K dy d = sy 1 y ) + x y, L dz d = Qz) β 0z, x0) 0, y0) 0, z0) 0. 1) In model 1), r and s are inrinsic growh rae coefficiens of prey species in unreserved and reserved zones respecively; K and L are heir respecive carrying capaciies. β 1 is he depleion rae coefficien of he prey species due o he predaor, and β 0 is he naural deah rae coefficien of he predaor species. In model 1), he funcion Qz) represens he growh rae of predaor. The model 1) is analyzed in wo differen cases, namely, i) Qz) = β 2 xz, 2) i.e. when predaor is wholly dependen on he prey species; ) ii) Qz) = bz 1 zm0 + β 2 xz 3) i.e. when he predaor is parially dependen on he prey. In his case, he prey species of densiy x) can be hough of as an alernaive resource for he predaor. By denoing a = b β 0 > 0, M = M 0 b β 0 )/b we noe ha he hird equaion of model 1) can be wrien as dz d = az 1 z M ) + β 2 xz. 4) In model sysem 1) 4), r, s,,, β 1, β 2, β 0 and a are assumed o be posiive consans. Now we presen he analysis of model 1) in wo cases 2) and 3) by using sabiliy heory of ordinary differenial equaions [27]. 3 Case I: when predaor is wholly dependen on he prey In his case, Qz) saisfies equaion 2). 3.1 Exisence of equilibria I can be checked ha model 1), when Qz) saisfies 2), has only hree nonnegaive equilibria, namely E 0 0, 0, 0), E 1 x, ŷ, 0) and Ex, y, z). The equilibrium E 0 exiss obviously and we shall show he exisence of E 1 and E as follows: 481

4 B. Dubey Exisence of E 1 x, ŷ, 0) Here x and ŷ are he posiive soluions of he following algebraic equaions: rx sy 1 x ) x + y = 0, K 1 y L From equaion 5a), we have 5a) ) + x y = 0. 5b) y = 1 [ ] rx 2 K r )x. 6) Subsiuing he value of y from equaion 6) ino equaion 5b), a lile algebraic manipulaion yields where ax 3 + bx 2 + cx + d = 0, 7) a = sr2 Lσ2 2, b = 2rsr ) K2 KLσ2 2, c = sr ) 2 Lσ2 2 s )r K, d = r )s ). I may be noed ha equaion 7) has a unique posiive soluion x = x if he following inequaliies hold: sr ) 2 < s )r, L K 8a) r )s ) <. 8b) From he model sysem 1) we noe ha if here is no migraion of he prey species from reserved o unreserved zone i.e. = 0) and r < 0, hen dx d < 0. Similarly if here is no migraion from of he prey species from unreserved o reserved zone i.e. = 0) and s < 0, hen dy d < 0. Hence i is naural o assume ha r > and s >. 8c) Knowing he value of x, he value of ŷ can be compued from equaion 6). I may also be noed ha for ŷ o be posiive, we mus have x > K r r ). 9) 482

5 A Prey-Predaor Model wih a Reserved Area Exisence of Ex, y, z) Here x, y, z are he posiive soluions of he following algebraic equaions: rx 1 x ) x + y β 1 xz = 0, K sy 1 y ) + x y = 0, L β 2 xz β 0 z = 0. Solving he above equaions, we ge, x = β 0 β 2, y = 1 2sβ 2 [ z = β 2 β 0 β 1 10a) s ) + s ) 2 + 4s Lβ 0 β 2 ], 10b) [ y + r ) β ] 0 rβ2 0 β 2 Kβ c) For z o be posiive, we mus have y + r ) β 0 > rβ2 0 β 2 Kβ ) Equaion 11) gives a hreshold value of he carrying capaciy of he free access zone for he survival of predaors. In he following lemma, we show ha all soluions of model 1) are nonnegaive and bounded. Lemma 1. The se { Ω = x, y, z) R + 3 : 0 < w = x + y + z µ } η is a region of he aracion for all soluions iniiaing in he inerior of he posiive orhan, where η is a consan such ha 0 < η < β 0, µ = K 4r r + η)2 + L 4s s + η)2, β 1 β 2. Proof. Le w) = x) + y) + z) and η > 0 be a consan. Then dw d + ηw = r + η)x rx2 K + s + η)y sy2 L β 1 β 2 )xz β 0 η)z. 12) Since β 1 is he depleion rae coefficien of prey due o is inake by he predaor and β 2 is he growh rae coefficien of predaor due o is ineracion wih heir prey, and hence i is naural o assume ha β 1 β

6 B. Dubey Now choose η such ha 0 < η < β 0. Then equaion 12) can be wrien as dw d + ηw r + η)x rx2 K sy2 + s + η)y L {x K2r } 2 r + η) = K 4r r + η)2 r K {y L2s s + η) } 2 + L 4s s + η)2 s L K 4r r + η)2 + L 4s s + η)2 = µsay). By using he differenial inequaliy [28], we obain 0 < w x), y), z) ) µ η 1 e η ) + x0), y0), z0) ) e η. Taking limi when, we have, 0 < w) µ η, proving he lemma. 3.2 Sabiliy analysis By compuing he variaional marices corresponding o each equilibrium, we noe he following: 1. E 0 is a saddle poin wih sable manifold locally in he z-direcion. 2. If β 2 x > β 0 hen E 1 is a saddle poin wih sable manifold locally in he xy-plane and wih unsable manifold locally in he z-direcion. 3. If β 2 x < β 0 hen E 1 is locally asympoically sable. In he following heorem, we show ha he model sysem 1) does no have any closed rajecory in he inerior of he posiive quadran of he xy-plane. Theorem 1. The model sysem 1) under he assumpion 2) can no have any periodic soluion in he inerior of he posiive quadran of he xy-plane. Proof. Le Hx, y) = 1 xy. Clearly Hx, y) is posiive in he inerior of he posiive quadran of he xy-plane. Le h 1 x, y) = rx 1 x ) x + y, K h 2 x, y) = sy 1 y ) + x y. L Then x, y) = x h 1H) + y h 2H) = 1 r y K + σ ) 2y x 2 1 s x L + σ ) 1x y 2 <

7 A Prey-Predaor Model wih a Reserved Area From he above equaion, we noe ha x, y) does no change sign and is no idenically zero in he inerior of he posiive quadran of he xy-plane. By Dulac-Bendixon crieria, i follows ha here is no closed rajecory in he inerior of he posiive quadran of he xy-plane, and hence he heorem follows. In he following heorem, we show ha E is locally asympoically sable. Theorem 2. The inerior equilibrium E is locally asympoically sable. Proof. In order o prove his heorem, we firs linearize model 1) by aking he following ransformaions: x = x + X, y = y + Y, z = z + Z. Now we consider he following posiive definie funcion: V ) = 1 2 X c 1Y c 2Z 2, where c 1 and c 2 are posiive consans o be chosen suiably. Now differeniaing V wih respec o ime along he linear version of model 1), we ge dv rx d = K + σ ) 2y sy X 2 c 1 x L + σ ) 1x Y 2 y + XY + c 1 ) + XZc 2 β 2 z β 1 x). Choosing c 2 = β1x β 2z we noe ha V is negaive definie if rx + c 1 ) 2 < 4c 1 K + σ ) 2y sy x L + σ ) 1x. y The above equaion can furher be wrien as rx c 1 ) 2 + 4c 1 < 4c 1 K + σ ) 2y sy x L + σ ) 1x. y I may be noed ha if we choose c 1 = σ2 hen he above condiion is auomaically saisfied. This shows ha V is a Liapunov funcion [27], and hence he heorem follows. In he following heorem, we are able o show ha E is globally asympoically sable. Theorem 3. The inerior equilibrium E is globally asympoically sable wih respec o all soluions iniiaing in he inerior of he posiive orhan. 485

8 B. Dubey Proof. Consider he following posiive definie funcion abou E, W) = x x xln x ) + c 1 y y y ln y x y ) + c 2 z z z ln z z Differeniaing W wih respec o ime along he soluions of model 1), we ge ). dw d = r K x x)2 c 1s L y y)2 + x x)z z)c 2 β 2 β 1 ) ) ) xy xy xy xy + x x) + c 1 y y). xx yy Choosing c 1 = yσ2 x and c 2 = β1 β 2, dw d can furher be wrien as dw d = r K x x)2 ys x L y y)2 xxy xy xy)2, which is negaive definie. Hence W is a Liapunov funcion [27] wih respec o E whose domain conains he region of aracion Ω, proving he heorem. 4 Case II: when he predaor is parially dependen on he prey In his case Qz) saisfies equaion 3) and he prey can be hough of as an alernaive food for he predaor. 4.1 Exisence of equilibria When Qz) saisfies equaion 3), hen he hird equaion of model 1) can be replaced by equaion 4). Then i can be checked ha model 1) has four nonnegaive equilibria, namely, F 0 0, 0, 0), F 1 0, 0, M), F 2 x, ỹ, 0), F x, y, z ). The equilibriums F 0 and F 1 obviously exis. As in Case I, equilibrium F 2 x, ỹ, 0) exiss if he inequaliies 8a) and 8b) are saisfied. Furher, for x o be posiive, we mus have x > K r r ). 13) To see he exisence of F, we noe ha x, y, z are he posiive soluions of he following algebraic equaions: rx 1 x ) x + y β 1 xz = 0, 14a) K sy 1 y ) + x y = 0, 14b) L z = M a a + β 2x). 14c) 486

9 A Prey-Predaor Model wih a Reserved Area Solving he above sysem of algebraic equaions, we ge where Ax 3 + Bx 2 + Cx + D = 0, 15) A = s L 2 B = 2s Lσ2 2 s C = L 2 r K + β ) 1β 2 M, a r K + β ) 1β 2 M r β 1 M), a r β 1 M) 2 s D = s r β 1 M). r K + β 1β 2 M a We noe ha he equaion 15) has a real posiive roo x = x if he following condiions are saisfied: r sr β 1 M) 2 < L s ) K + β ) 1β 2 M, 16a) a r β 1 M)s ) <, 16b) r β 1 M > 0. ), 16c) Knowing he value of x, he value of z can be compued from equaion 14c) and he value of y can be compued from he equaion given below: y = 1 [ r K + β ] 1β 2 M )x 2 r β 1 M)x. 17) a For y o be posiive, we mus have r K + β ) 1β 2 M x > r β 1 M). 18) a In he following lemma, we show ha he model sysem 1) is biologically well behaved. The proof of his lemma is similar o ha of Lemma 1, and hence omied. Lemma 2. The se } Ω 1 = {x, y, z): w) = x) + y) + z), 0 < w) µ aracs all soluions iniiaing in he inerior of he posiive orhan, where µ = K 4r r + η ) 2 + L 4s s + η ) 2 + M 4a a + η ) 2, and η is a posiive consan. η 487

10 B. Dubey 4.2 Sabiliy analysis In order o sudy he local sabiliy behavior of F, we compue he variaional marices corresponding o each equilibrium. From hese marices, we noe he following: 1. F 0 is an unsable equilibrium poin. 2. F 1 is a saddle poin wih sable manifold locally in he z-direcion and wih unsable manifold locally in he xy-plane. 3. F 2 is also a saddle poin whose sable manifold is locally in he xy-plane and unsable manifold locally in he z-direcion. Remark. I may be noed ha Theorem 1 will remain valid in he case when predaor is parially dependen on he prey. In he following heorems, local and global sabiliy behavior of F have been sudied. The proof of Theorem 4 is similar o ha of Theorem 2, and he proof of Theorem 5 is similar o ha of Theorem 3. Hence we omi he proofs of hese heorems. Theorem 4. The inerior equilibrium F is locally asympoically sable. Theorem 5. The inerior equilibrium F is globally asympoically sable wih respec o all soluions iniiaing in he inerior of posiive orhan. 5 Numerical simulaion In his secion we presen numerical simulaion o illusrae he resuls obained in previous secions. We choose he following values of parameers in model 1): a = 3, r = 4, s = 3.5, K = 40, L = 50, M = 30, β 0 = 3, β 1 = 2, β 2 = 1, = 2.5, = ) Wih he above values of parameers, we noe ha condiions 8) and 9) are saisfied. This shows ha equilibrium exiss, and i is given by x = , ŷ = ) When predaor is wholly dependen on he prey, i is noed ha he posiive equilibrium Ex, y, z) exiss and i is given by x = 3, y = , z = ) Furher, when he predaor is parially dependen on he prey, i is seen ha he posiive equilibrium F x, y, z ) exiss, and i is given by x = , y = , z = ) From 20) 22), we noe he following: 488

11 x y x x A Prey-Predaor Model wih a Reserved Area 1. When he predaor is a zero equilibrium level z = 0), he oal densiy of he prey species a equilibrium level is ). 2. When he predaor is compleely dependen on he prey, hen densiy of he predaor is while he oal densiy of he prey has decreased from o Comparing 21) and 22), i is noed ha when he predaor is parially dependen on he prey, hen densiy of he predaor has increased from o , and prey densiy has also increased from o This suggess ha an alernaive food for he predaor leads an increase in he densiy of he prey as well as predaor. Figs. 1 5 correspond o model 1) when he predaor is wholly dependen on he =2.5 =4.5 = =1.5 =2.5 = Fig. 1. Case I: graph of x verses for differen value of obained using parameers: s = 3.5, K = 40, L = 50, β 0 = 3, β 1 = 2, β 2 = 1, = Fig. 2. Case I: graph of x verses for differen value of obained wih =2.5 and oher values of parameers are same as in Fig β 1 =2 β 1 =12 β 1 = =2.5 =3.5 = Fig. 3. Case I: graph of x verses for differen value of β 1 obained wih = 2.5 and oher values of parameers are same as in Fig Fig. 4. Case I: graph of y verses for differen value of obained using he same values of parameers as in Fig

12 y B. Dubey prey. Fig. 1 shows he behavior of x wih ime for differen values of. This figure shows ha iniially x increases for some ime, hen i sars decreasing and finally aains is equilibrium level. We also noe ha iniially x decreases as increases bu afer cerain ime his behavior is jus reversal and finally x seles down a is equilibrium level. Fig. 2 shows he behavior of x wih ime for differen values of. From his figure, we noe ha iniially x increases as increases, afer cerain ime x decreases wih and finally aains is equilibrium level. From Fig. 3, we noe ha behavior of x wih ime is similar o ha of Fig. 1. Fig. 4 shows he behavior of prey species in reserved area w.r.. ime. This figure shows ha iniially y increases wih ime and afer cerain period of ime, i aains is equilibrium level. We also noe ha y increases as increases. Fig. 5 shows ha y increases wih ime and y decreases as increases, and finally seles down a is equilibrium level =1.5 =2.0 = Fig. 5. Case I: graph of y verses for differen value of obained wih = 2.5 and oher values of parameers are same as in Fig. 1. Figs correspond o model 1) when predaor is parially dependen on he prey. Figs. 6 8 show he behavior of prey species in unreserved area wih respec o ime. Fig. 6 shows ha behavior of x wih ime when predaor is parially dependen on he prey. I is noed ha x exhibis periodic behavior for some ime and finally i seles down a is equilibrium level. I is also observed ha iniially x increases as increases and afer cerain ime x decreases as increases, and finally obains is equilibrium level. From Fig. 7 we noe ha x has oscillaory behavior for cerain ime, and hen i seles down a is equilibrium level. I is also noed ha iniially x increases as increases, bu afer cerain ime his behavior is jus reversed. Fig. 8 shows he behavior of x w.r.. ime for differen values of β 1. I is noed ha if β 1 is small, hen iniially x increases and hen exhibis oscillaory behavior and finally obains is equilibrium level. Bu if β 1 is larger han a hreshold values, hen iniially x decreases, hen afer a sligh increase i obains is equilibrium level. I is also observed ha x decreases as β 1 increases. Fig. 9 and Fig. 10 show he behavior of prey species in reserved area w.r.. ime. From hese figures i is noed ha y increases wih ime and finally seles down a is equilibrium level. I is also noed ha y increases as increases whereas y decreases as increases. I is observed ha he prey species in reserved zone do no exhibi periodic behavior. 490

13 y x y x x A Prey-Predaor Model wih a Reserved Area =2.5 =6.5 = =1.5 =2.0 = Fig. 6. Case II: graph of x verses for differen value of obained using parameers: a = 3, s = 3.5, K = 40, L = 50, M = 30, β 1 = 2, β 2 = 1, = Fig.7 Fig. 7. Case II: graph of x verses for differen value of obained wih =2.5 and oher values of parameers are same as in Fig β 1 =2 β 1 =12 β 1 = =2.5 =4.5 = Fig.8 Fig. 8. Case II: graph of x verses for differen value of β 1 obained wih =2.5 and oher values of parameers are same as in Fig Fig.9 Fig. 9. Case II: graph of y verses for differen value of obained using he same values of parameers as in Fig =1.5 =2.0 =2.5 0 Fig. 10. Case II: graph of y verses for differen value of obained wih = 2.5 and oher values of parameers are same as in Fig

14 B. Dubey 6 Conclusions In his paper, a mahemaical model has been proposed and analyzed o sudy he role of a reserved zone on he dynamics of predaor-prey sysem. The model has been analyzed in wo cases: firs when predaor species are wholly dependen on he prey and second when predaor species are parially dependen on he prey in he unreserved zone. In boh cases, compuer simulaions wih MATLAB have been performed o sudy he effecs of various parameers on he dynamics of he sysem. By analyical and numerical simulaions, he following observaions have been made: 1. In he absence of predaor, he densiy of prey is maximum in reserved as well as unreserved zone. 2. In he case when predaors are wholly dependen on he prey, hen cumulaive densiy of prey decreases in comparison o he case In he case when predaors are parially dependen on he prey and alernaive food is also made available o predaors in unreserved zone, hen he cumulaive densiy of he prey decreases in comparison o case 1, bu i increases in comparison o he case 2 and densiy of predaor also increases in comparison o he case 2. This shows ha an alernaive resource for he predaor is beer suied in comparison o he wholly dependen case as i leads an increase in he densiy of he prey and predaor boh ha ensures he survival of prey and predaor in a beer way. In boh cases, i has been found ha prey species has oscillaory behavior in he unreserved zone where as oscillaory behavior has no been observed for prey species in he reserved zone. By using sabiliy heory of ordinary differenial equaions, i has been shown ha he posiive equilibrium, whenever exiss, is always globally asympoically sable in boh he cases, namely predaors are wholly or parially dependen on he prey species. This shows ha reserve zone has a sabilizing effec on he predaor-prey sysem. This sudy suggess ha he role of reserved zone is an imporan inegraing concep in ecology and evoluion. By creaing reserved zones in he habia where predaor have no access or chance of seling, he prey species can grow wihou any exernal disurbances and hence he prey species can be mainained a an appropriae level. Acknowledgemen The auhor is graeful o he referee for criical review and useful suggesions ha improved he paper. References 1. F. Albrech, H. Gazke, A. Haddad, N. Wax, The dynamics of wo ineracing populaions, J. Mah. Anal. Appl., 46, pp ,

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