Analysis of a Prey-Predator Fishery Model. with Prey Reserve

Transcription

1 Applied Mathematical Sciences, Vol., 007, no. 50, Analysis of a Prey-Predator Fishery Model with Prey Reserve Rui Zhang, Junfang Sun and Haixia Yang School of Mathematics, Physics & Software Engineering Lanzhou Jiaotong University, Lanzhou, Gansu , China Abstract In this paper, we consider a prey-predator fishery model with prey dispersal in a two-patch environment, one is assumed to be a free fishing zone and the other is a reserved zone where fishing and other extractive activities are prohibited. The existence of biological and bionomic equilibrium of the system is discussed. The local and global stability analysis has been carried out. An optimal harvesting policy is given using Pontryagin s maximum principle. Keywords: Prey-predator; Global stability; Optimal harvesting Introduction Biological resources are renewable resources. Economic and biological aspects of renewable resources management have been considered by Clark []. In recent years, the optimal management of renewable resources, which has a direct relationship to sustainable development, has been studied extensively by many authors [-7]. The reason is that mankind is facing the problems about shortage of resource at present. Extensive and unregulated harvesting of marine fishes can even lead to the depletion of several fish species. One potential solution to these problems is the creation of marine reserves where fishing and other extractive activities are prohibited. Marine reserve not only protect species inside the reserve area but they can also increase fish abundance Corresponding author. sunjf8@63.com

2 48 Rui Zhang, Junfang Sun and Haixia Yang in adjacent areas. Mathematical model of ecological system, reflecting these problems, has been given in Kar and Swarnakamal [6]. The paper is mainly concerned with the following prey-predator system dx = r x ( x K dt ) αx y σ x + σ x q E x, dx = r x ( x K dt )+σ x σ x, (.) dy = dy + kαx y q E y. dt Here, x (t) and y(t) are biomass densities of prey species and predator species inside the unreserved area which is an open-access fishing zone, respectively, at time t. x (t) is the biomass density of prey species inside the reserved area where no fishing is permitted at time t. All the parameters are assumed to be positive. r and r are the intrinsic growth rates of prey species inside the unreserved and reserved areas, respectively. d, α and k are the death rate, capturing rate and conversion rate of predators, respectively. and are the carrying capacities of prey species in the unreserved and reserved areas, respectively. σ and σ are migration rates from the unreserved area to the reserved area and the reserved area to the unreserved area, respectively. E and E are the effects applied to harvest the prey species and predator species in the unreserved area. q and q are the catchability coefficients. Considering the biological background, we only care about the dynamics of system (.) in the closed first quadrantr+ 3. Here we observe that, if there is no migration of fish population from the reserved area to the unreserved area (i.e., σ = 0) and r σ q E < 0, then ẋ < 0. Similarly, if there is no migration of fish population from the unreserved area to the reserved area (i.e., σ = 0) and r σ < 0, then. x < 0. Hence, throughout out analysis, we assume that r σ q E > 0, r σ > 0. (.) Existence of equilibria Equating the derivatives on the left hand sides to zero and solving the resulting algebraic equations we can find three possible equilibrium R (0, 0, 0),R ( x, x, 0) and R 3 (,x,y ).

3 Prey-predator fishery model 483 Here, x = σ [(σ + q E r ) x + r x ] and x is the positive solution of the following equation where c x 3 + c x + c 3x + c 4 =0, (.) c = r r σ > 0, c = r r (r σ q E ) σ c 3 = r (r σ q E ) σ < 0, r (r σ ) σ, c 4 = (r σ ) σ (r σ q E ) σ. Eq. (.) has a unique positive solution x = x if the following inequalities hold: For x to be positive, we must have r (r σ q E ) < r (r σ ) σ (r σ )(r σ q E ) >σ σ. (.) x > (r σ q E ) r. (.3) Again, = d+q E kα,x = {r σ +[(r σ ) +4r σ /] / }/r, y = (r σ q E ) r x K +σ x. α For y to be positive, we must have 3 Stability analysis (r σ q E ) + σ x > r x. (.4) In the absence of predator species, the model (.) becomes dx = r x ( x ) σ x + σ x q E x, dt dx dt = r x ( x )+σ x σ x, (3.)

4 484 Rui Zhang, Junfang Sun and Haixia Yang which has two equilibria, R(0, 0) and a positive equilibrium R( x, x ) if (.) and (.3) hold. The Jacobian matrix of the system (3.) is ( ) r r x σ q E σ σ r r (3.) x σ The characteristic equation of the Jacobian matrix of (3.) at R is λ (r σ q E + r σ )λ +(r σ q E )(r σ ) σ σ =0. (3.3) Since λ + λ = r σ q E + r σ > 0 and λ λ =(r σ q E )(r σ ) σ σ > 0. Hence R(0, 0) is unstable. Similarly, the characteristic equation of the Jacobian matrix of (3.) at R is λ +( r x + r x + σ x + σ x )λ + r x ( r x + σ x )+ r σ x. (3.4) x x x x Since λ + λ = ( r x + r x + σ x x + σ x x ) < 0. and λ λ = r x ( r x + σ x x )+ r σ x x > 0. Thus R( x, x ) is locally asymptotically stable. Let us now suppose that system (.) has a unique positive equilibrium R 3 (,x,y ). The Jacobian matrix of (.) at R 3 is r r αy σ q E σ α σ r r x σ 0 kαy 0 d + kα q E The characteristic equation of the Jacobian matrix of (.) at R 3 is where a = r + r x + σ x (3.5) λ 3 + a λ + a λ + a 3 =0, (3.6) + σ, x a =( r K + σ x )( r x K + σ )+σ σ + kα y, a 3 =kα y ( r x + σ ). x x

5 Prey-predator fishery model 485 According to Routh-Hurwitz criteria, the necessary and sufficient conditions for local stability of equilibrium point R 3 are a > 0, a 3 > 0 and a a a 3 > 0. It is evident that a > 0, a 3 > 0. Thus, the stability of R 3 is determined by the sign of a a a 3. By direct calculations, we obtain a a a 3 =( r + r x + σ x + σ )[( r x + kα y ( r K + σ x ) > 0. K + σ x )( r x K + σ )+σ σ ] x Hence R 3 (,x,y ) is locally asymptotically stable. Theorem. The equilibrium point R is globally asymptotically stable. Proof. Let us consider the following Lyapunov function: V (x,x )=ω (x x x ln x x )+ω (x x x x x ), where ω,ω are positive constants, to be chosen later on. Differentiating V with respect to time t, we get dv dt = ω (x x ) dx x dt + ω (x x ) dx x dt. Choosing ω ω = x x σ σ, a little algebraic manipulation yields dv dt = r ω σ x σ x (x x ) r ω (x x ) ω σ x x x ( x x x x ) < 0. Therefore, R( x, x ) is globally asymptotically stable. Theorem. R 3 (,x,y ) is globally asymptotically stable. Proof. Let us choose the Lyapunov function V (x,x,y)=ω (x x ln x )

6 486 Rui Zhang, Junfang Sun and Haixia Yang +ω (x x x ln x )+ω x 3 (y y y ln y y ), where ω,ω and ω 3 are positive constants, to be chosen later on. Differentiating V with respect to time t, we get dv dt = ω (x ) dx x dt + ω (x x ) dx x dt + ω (y y ) dy 3 y dt. Choosing ω ω 3 = k, ω ω = x σ x σ, a little algebraic manipulation yields dv dt = r ω (x K ) r ω σ x σ (x x ) ω σ (x x x ) x x < 0. Therefore, R 3 (,x,y ) is globally asymptotically stable. 4 Bionomic equilibrium Let c =fishing cost per unit effort for prey species, c =fishing cost per unit effort for predator species, p =price per unit biomass of the prey, p =price per unit biomass of the predator. Therefore, the economic rent(net revenue)at any time is given by Π=(p q x c )E +(p q y c )E =Π +Π where Π =(p q x c )E, Π =(p q y c )E i.e., Π and Π represent the net revenues for the prey and predator species, respectively. The bionomic equilibrium (x,x,y,e,e ) is given by the following simultaneous equations r x ( x ) αx y σ x + σ x q E x =0, (4.) r x ( x )+σ x σ x =0, (4.) d + kαx q E =0, (4.3) Π=(p q x c )E +(p q y c )E =0. (4.4)

7 Prey-predator fishery model 487 In order to determine the bionomic equilibrium, we now consider the following cases: Case : If c > p q y, i.e.the cost is greater than the revenue for the predator, then the predator fishing will be stopped (E = 0). Only the prey fishing remains operational (i.e.c <p q x ). We then have x = c p q,x = r {r σ +[(r σ ) r +4σ c p q ] / }. Since c <p q x <p q. Hence c p q > 0. (y,e ) will be any point on the line σ + αy + q E = r ( c p q )+ σ p q r c {r σ +[(r σ ) +4σ r c p q ] / } in the first quadrant of the ye plane. Case : If c >p q x, i.e.the cost is greater than the revenue in the prey fishing, then prey fishing will be closed (E =0). Only predator fishing remains operational (i.e.c <p q y). We then have y = c p q. Substituting y into (4.) we get x = σ [ αc p q x + σ x r x ( x K )], x is the positive solution of the following equation where c x 3 + c x + c 3x + c 4 =0, (4.5) c = r r > 0, Kσ c = r r (r σ αc p q ), σ c 3 = r (r σ αc p q ) r (r σ ), σ σ c 4 = (r σ ) (r σ αc ) σ. σ p q Now, if () r σ αc p q < 0, r (r σ σ or () r σ αc p q > 0, r (r σ σ σ σ. αc ) p q αc ) p q < r (r σ ) σ, < r (r σ ) σ, (r σ )(r σ αc p q ) >

8 488 Rui Zhang, Junfang Sun and Haixia Yang then Eq (4.5) has a unique positive solution x = x. For x to be positive, we must have Substituting x into (4.3) we get x > σ r αc p q r. E = kαx d. q E > 0, provided kαx >d. Case 3: If c >p q x,c >p q y, then the cost is greater than revenues for both the species and the whole fishery will be closed. Case 4: If c <p q x,c <p q y, then the revenues for both the species being positive, then the whole fishing will be in operation. In this case, x = c /p q and y = c /p q. Now substituting x and y into (4.), (4.) and (4.3), we get and Now, x = {r σ +[(r σ ) r c +4σ ] / }, r p q (4.6) E = r ( c )+ p σ x σ αc, q p q c q q p q (4.7) E = kαc p q q d q. (4.8) E > 0, if r q ( c p q )+ p σ x c > σ q + αc q p q, (4.9) E > 0, if kαc >d. (4.0) p q Thus the nontrivial bionomic equilibrium point (x,x,e E ) exists if conditions (4.9) and (4.0) hold. 5 Optimal harvesting policy In this section, our objective is to maximize the present value J of a continuous time stream of revenues given by J = 0 e δt {(p q x c )E (t)+(p q y c )E (t)}dt, (5.)

9 Prey-predator fishery model 489 where δ denotes the instantaneous annual rate of discount. We intend to maximize (5.) subject to the state equations (.) by invoking Pontryagin s maximal principle (Clark []). The control variable E i (t)(i =, ) are subjected to the constraints 0 E i (t) (E i ) max. The Hamiltonian for the problem is given by H =e δt [(p q x c )E +(p q y c )E ]+λ [r x ( x / ) αx y σ x + σ x q E x ]+λ [r x ( r / )+σ x σ x ] + λ 3 ( dy + kαx y q E y), where λ,λ and λ 3 are the adjoint variables (Clark []). The control variables E and E appear linearly in the Hamiltonian function H. Assuming that the control constraints are not binding i.e., the optimal solution does not occur at (E i ) max or (E i ) max, we have singular control (Clark []). According to Pontryagin s maximum principle H H dλ =0; =0; E E dt = H dλ ; x dt = H dλ 3 ; x dt = H y. Substitution and simplification yields dh =0 λ = e δt (p c ), (5.) de q x dh =0 λ 3 = e δt (p c ), (5.3) de q y dλ dt = [e δt p q E + λ (r r x αy σ q E )+λ σ + λ 3 kαy], (5.4) dλ dt = [λ σ + λ (r r x σ )], (5.5) dλ 3 dt = [e δt p q E λ αx + λ 3 ( d + kαx q E )]. (5.6) Now, Substituting λ and λ 3 into (5.6) and using equilibrium equations we get y = δc q q p q (δ + d kα)+(p q c )αq. (5.7)

10 490 Rui Zhang, Junfang Sun and Haixia Yang From (5.5), we get dλ dt A λ = A e δt, whose solution is given by λ (t) = A A + δ e δt, (5.8) where A = r x + σ,a x =(p c q )σ x. From (5.4), we get dλ A dt 3 λ = A 4 e δt, whose solution is given by λ (t) = A 4 A 3 + δ e δt, (5.9) where A 3 = r + σ x, A x 4 = p q E + A +(p A +δ c q )kαy. y From (5.) and (5.9), we get the singular path (p c )= A 4 q A 3 + δ. (5.0) Using x = r {r σ +[(r σ ) + 4r σ ] / } and y δc = q, A q p q (δ+d kα )+αq (p q c ),A 3 and A 4 can be written as A = (r σ )+ [(r σ ) +4r σ / ] / + σ r /{r σ +[(r σ ) +4r σ /] / }, A 3 = r K + σ {r r x σ +[(r σ ) +4r σ /] / }, A 4 =p q E +(p c )σ /{ q x (r σ )+ [(r σ ) +4r σ /] / + σ r /r σ +[(r σ ) +4r σ / ] / + δ} + δkαq c p /[q p q (δ + d kα )+(p q c )αq ] kαc q. Thus (5.0) can be written as F ()=(p c ) A 4 q A 3 + δ =0. There exists a unique positive root = x δ of F () = 0 in the interval 0 < <, if the following inequalities hold: F (0) < 0,F( ) > 0,F () > 0 for > 0.

11 Prey-predator fishery model 49 For = x δ, we get y = y δ from (5.7) We then have and x δ = {r σ +[(r σ ) +4r σ x δ / ] / }, r E δ = {r ( x δ ) σ + σ x δ αy δ }, q x δ E δ = q ( d + kαx δ ). Hence once the optimal equilibrium (x δ,x δ,y δ ) is determined, the optimal harvesting effort E δ and E δ can be determined. From (5.3), (5.8) and (5.9), we observe that λ i (t)(i =,, 3) do not vary with time in optimal equilibrium. Hence they remain bounded as t. 6 Concluding remarks In this paper, we have analyzed a prey-predator fishery model with prey dispersal in a two-patch environment, one is assumed to be a free fishing zone and the other is a reserved zone where fishing and other extractive activities are prohibited. we have discussed the local and global stability of the system. It has been observed that, whether in the absence or in the presence of predators, the fishing populations may be sustained at an appropriate equilibrium level. The optimal harvesting policy has been discussed using Pontryagin s maximum principle. References [] C. W. Clark, Mathematical Bioeconomics:The Optimal Management of Renewable Resources, Wiley, New York, (976). [] K. S. Chaudhuri, A bioeconomic model of harvesting of a multispecies fishery, Ecol Model, 3(986), [3] K. S. Chaudhuri, Dynamic optimization of combined harvesting of twospecies fishery, Ecol Model, 4(988), 7-5.

12 49 Rui Zhang, Junfang Sun and Haixia Yang [4] G. Sunita, S. Brahampal, The dynamics of a food web consisting of two prey and a harvesting predator, Chaos, Solitons and Fractals, in press. [5] B. Dubey, P. Chandra and P. Sinha, A resource dependent fishery model with optimal harvesting policy, J.Biol.Syst, 0(00), -3. [6] T. K. Kar, M. Swarnakamal, Influence of prey reserve in a prey-predator fishery, Nonlinear Analysis, 65(006), [7] W. Wang, L. Chen, Optimal harvesting policy for single population with periodic coefficients, Math.Biosci, 5(998). [8] Y. Takeuchi, Global dynamical properties of Lotka-Volterra system, Word Scientific Publishing Co.Pte.Ltd (996). Received: April 4, 007