Qualitative analysis of a harvested predator-prey system with Holling type III functional response

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1 Jiang and Wang Advances in Difference Equations 201, 201:249 R E S E A R C H Open Access Qualitative analysis of a harvested predator-prey system with Holling type III functional response Qiaohong Jiang * and Jinghai Wang * Correspondence: fzdxjqh@126.com College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 50002, P.R. China Abstract A harvested predator-prey system with Holling type III functional response is considered. By applying qualitative theory of differential equations, we show the instability and global stability properties of the equilibria and the existence and uniqueness of limit cycles for the model. The possibility of existence of a bionomic equilibrium is discussed. The optimal harvesting policy is studied from the view point of control theory. Numerical simulations are carried out to illustrate the validity of our results. Keywords: limit cycles; global stability; bionomic equilibrium; optimal harvest policy; Holling type III 1 Introduction The dynamical relationship between predators and their preys is one of the dominant subjects in ecology and mathematical ecology due to its universal importance, see [1]. Harvesting generally has a strong impact on the population dynamics of a harvested species. The severity of this impact depends on the harvesting strategy implemented which in turn may range from rapid depletion to complete preservation of the population. Problems related to the exploitation of multispecies systems are interesting and difficult both theoretically and practically. The problem of inter-specific competition between two species which obey the law of logistic growth was considered by Gause [2]. But he did not study the effect of harvesting. Clark [] considered harvesting of a single species in a twospecies ecologically competing population model. Modifying Clark s model, Chaudhuri [4, 5] studied combined harvesting and considered the perspectives of bioeconomics and dynamic optimization of a two-species fishery. Matsuda [6] showed that the total yield and lowest stock levels of harvested resources increase if fishers may focus their effort on a temporally abundant stock. Matsuda [7] showed that the maximum sustainable yield from the multispecies systems does not guarantee persistence of all resources. Dai [8] analyzed the global behavior of a predator-prey system with some functional response in the presence of constant harvesting. Xiao [9] studied Bogdanov-Takens bifurcations in a predator-prey system with constant rate harvesting. 201 Jiang and Wang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Jiangand WangAdvances in Difference Equations 201, 201:249 Page 2 of 17 Kar [10] studied global dynamics and controllability of a harvested prey-predator system with Holling type III functional response: dx dτ = x x βx2 y E 1+x 2 1 x, dy dτ = 1.1 β 1x 2 y γ y E 1+x 2 2 y. The aim of this paper is to consider the instability and global stability properties of the equilibria and the optimal harvesting policy, the existence and uniqueness of limit cycles of a harvested predator-prey system with Holling type III functional response: dx dt = x x βx2 y E 1+x 2 1 x 2, dy dt = 1.2 β 1x 2 y ry E 1+x 2 2 y, where x, y denote prey and predator population respectively at any time t;, β, β 1, r, E 1 and E 2 are positive constants. Here, β, β 1 and r are the same with system 1.1, E 1 and E 2 denote the harvesting efforts for the prey and predator respectively, the term βx 2 /1 + x 2 denotes the functional response of the predator, which is known as Holling type III response function [11]. ThisworkismotivatedbythepaperbyKarandMatsuda.Theystudiedtheabovesystem 1.1 and showed the local stability of equilibria and uniqueness of limit cycle, but they did not discuss the global stability of the positive equilibrium, the possibility of existence of a bionomic equilibrium, and the optimal harvesting policy. This paper gives the complete qualitative analysis for model 1.2 and the results improve and extend the corresponding results of the above system 1.1. This paper is organized as follows. Basic properties such as the boundedness, existence, stability and instability of the equilibria of the model are given in Section 2.InSection, sufficient conditions for the global stability of the unique positive equilibrium are obtained. In Section 4, we derive the existence and uniqueness of limit cycle. In Section 5,we study a bionomic equilibrium. In Section 6, we study optimal harvest policy. In Section 7, we give numerical stimulations of system Basic properties of the model Let R + 2 = {x, y x 0, y 0}. For practical biological meaning, we simply study system 1.2inR + 2. Lemma 2.1 All the solutions xt,yt of system 1.2 with the initial values x0 > 0, y0 > 0 are uniformly bounded. Proof Let us consider the function w = x + β β 1 y. Differentiating w with respect to τ and using 1.2, we get dw dτ = dx dτ + β β 1 dy dτ = x x E 1x 2 βr β 1 y βr β 1 E 2 y.

3 Jiangand WangAdvances in Difference Equations 201, 201:249 Page of 17 Now we have dw dt + rw = x x+rx E 1x 2 βr E 2 y + rx x 2 + r2 β E 1 = L say. Applying the theory of differential inequality [9], we obtain 0<wx, y< L r 1 e rt + w x0, y0 e rt, which, upon letting t, yields0 < w <L/r.So,wehavethatallthesolutionsofsystem 1.2thatstartinR + 2 are confined to the region B,where B = {x, y R +2 : w = Lr } + ε for any ε >0. This completes the proof. Now we find all the equilibria admitted by the system and study their stability properties. The equilibria of 1.2 are the intersection points of the prey isocline on which ẋ =0and the predator isocline on which ẏ =0.Obviously,P 0 0, 0 is the trivial equilibrium and P 1, 0 is the only axial equilibrium of system 1.2. The third and the most interesting equilibrium point is P 2 x, y, where x and y are given by x r + E 2 =, y = [ x ]1 + x β 1 r E 2 βx Thus the existence condition for the positive interior equilibrium point P 2 depends upon the restrictions β 1 > r + E and namely 0<x <, 2. r + E 2 >. 2.4 β 1 r E 2 We assume that the system parameters are such that they satisfy conditions 2.2 and 2.. From the expressions for x and y,weobservethatx increases with E 2 and y decreases with E 1.ThisisnaturalbecauseanincreaseinE 2 decreases the predator population and hence enhances the survival rate of the prey; on the other hand, an increase in E 1 results in the loss of food for the predator. Lemma P 0 0, 0 is a saddle point;

4 Jiangand WangAdvances in Difference Equations 201, 201:249 Page 4 of 17 2 If E 2 > β 1 2 β rholds, P 1,0 is a stable focus or a stable node. If E 2 < rholds, then P ,0 is a saddle point; P 1,0 is a stable node point for β E 2 = 1 2 r; P x, y isastablefocusorastablenodefor < 2x ; P 2 x, y is an unstable focus or an unstable node for > 2x ; P 2 x, y is a center or a focus for = 2x. Proof The Jacobian matrix of system 1.2isgivenby Jx, y= 21+E1 x 2βxy 1+x 2 2 βx2 1+x 2 2β 1 xy β 1 x 2 r E 1+x x Since det J0, 0 = r + E 2 <0,wederivethatP 0 0, 0 is a saddle point; 2 The Jacobian matrix of system 1.2 for the equilibrium point P 1,0isgivenby J,0 = β β r E 2. If β 1x 2 β r E 1+x 2 2 < 0 holds, namely E 2 > 1 2 r,thendet J 2 + 2,0>0,p = tr J,0> 0, we can derive that P 1, 0 is a stable focus or a stable node. If β 1x 2 r E 1+x 2 2 >0holds, β namely E 2 < 1 2 r, thendet J 2 + 2,0 < 0, we can derive that P 1,0 is a saddle point. If E 2 = 1 2 r,thendet J β,0=0,p 1, 0 is a higher-order singular point. At this time, point P 1,0 and point P 2 x, y are the same point. So, we can derive that P 1, 0 is a stable node point. The Jacobian matrix of system 1.2 for the equilibrium point P 2 x, y isgivenby J x, y = 21+E1 x 2βx y βx 2 1+x x 2 2β 1 x y 0 1+x As det Jx, y = 2ββ 1x y 1+x 2, P = tr J x, y = [ 2 x 2βx y ] = + x 2 2 x, 1 + x x 2 we know that if + x 2 2 x >0,namely < 2x,wederivethatP >0, P 2 x, y is a stable focus or a stable node. If + x 2 2 x <0,namely > 2 x,wederivethatp <0,P 2 x, y is an unstable focus or an unstable node. If + x 2 2 x =0,namely = 2x, P 2 x, y is a center or a focus. The proof is completed. We make the following transformation: dt = [ 1+x 2] dτ.

5 Jiangand WangAdvances in Difference Equations 201, 201:249 Page 5 of 17 Substituting this into system 1.2, then replacing τ with t gives dx dt = x[ x]1 + x 2 βx 2 y, dy dt = y[β 1 r E 2 x 2 r E 2 ]. 2.5 From Lemma 2.2 we know that if P = +x 2 2 x = 0 holds, namely = 2x, 1+x 2 P 2 x, y is a center focus. We can make further conclusions as follows. Lemma 2. 1 If C 0 >0holds, namely = 2x > [Aβ+11]x, P 2 x, y is a stable fine focus with order one; 2 If C 0 <0holds, namely = 2x focus with order one, where C 0 = β 1 r E 2 y [ Aβx 11 x +] 2A β 2 x. < [Aβ+11]x, P 2 x, y is an unstable fine Proof First use the coordinate translation, that is, translate the origin of coordinates into the point P 2 x, y. Then we make the following substitutions for model 2.5: x = x x, y = y y, dt = βx 2 dt. Replacing x, y, t with x, y, t respectively, we have dx dt = y 2 x xy + 1 [x 1+E βx x 2 βy ]x 2 1 x 2 y + 1 [ 41+E x 2 βx 2 1 x ]x x 4, βx 2 dy dt = 2β 1 r E 2 y βx x + β 1 r E 2 y x 2 + 2β 1 r E 2 βx 2 βx xy + β 1 r E 2 x 2 y. βx We denote A = 2β1 r E 2 y βx > 0 and make the following transformations: u = x, v = 1 A y, dτ = Adt.Replacingu, v, τ with x, y, t,respectively,gives dx dt = y + Dxy Ex2 + Nx 2 y Fx + Lx 4 = y + 4 j=2 P jx, y ˆPx, y, dy dt = x Gx2 Hxy Ix 2 y = x + j=2 Q 2.7 jx, y ˆQx, y, where D = 2 x, E = 1 [x 1+E Aβx x 2 βy ], N = 1, F = 1 [ 41+ x 2 Aβx 2 E 1 x ], L =, G = β 1 r E 2 y, H = 2β 1 r E 2 Aβx 2 A 2 βx 2 Aβx, I = β 1 r E 2. Aβx 2 It is obvious that D 2 =4N, andhd =4I. Then we make use of the Poincare method to calculate the focus value. Construct a form progression Fx, y=x 2 + y 2 + k= F kx, y, where F k x, y isthekth homogeneous multinomial with x and y. Considering df dt 2.7 = F t ˆPx, y+ F ˆQx, y=0, t we can obtain that three multinomials and four multinomials of Fx, y areequaltozero separately. Noting that 2xP 2 x, y+2yq 2 x, y= H x, y, we can obtain Let H =2Ex +2G 2Dx 2 y +2Hxy 2. F x, y=a 0 x + a 1 x 2 y + a 2 xy 2 + a y,

6 Jiangand WangAdvances in Difference Equations 201, 201:249 Page 6 of 17 thenwecanobtainthefollowingform: y F x x F y = y a 0 x 2 +2a 1 xy + a 2 y 2 x a 1 x 2 +2a 2 xy +a y 2 = a 1 x +a 0 2a 2 x 2 y +2a 1 a xy 2 + a 2 y. From y F x x F y = H,wecanderivethat a 1 x +a 0 2a 2 x 2 y +2a 1 a xy 2 + a 2 y =2Ex +2G 2Dx 2 y +2Hxy 2. By the comparison method of correlates, we can obtain that a 0 = 2G 2D, a 1 = 2E, a 2 =0, a = 2H+4E, then F x, y= 2G 2D x 2Ex 2 y 2H+4E y and H 4 =2xP +2yQ + F x P 2 + F y Q 2 =2EG +2D 2G 2Fx 4 +2HG +4EG 2I 4DEx 2 y 2 + 2N +2GD +4E 2 +2EH 2D 2 x y + 2H 2 +4EH xy. Let x = r cos θ, y = r sin θ,thenwecanderivethat C 0 = 2 0 H 4 cos θ, sin θ dθ = β 1 r E 2 y [ Aβx 11 x +] 2A β 2 x. Hence the point O0, 0 of system 2.5 is an unstable fine focus with order one when C 0 >0,namely = 2x > [Aβ+11]x. But considering the time change dτ = Adt, P 2 x, y is a stable fine focus with order one. And P 2 x, y is an unstable fine focus with order one when C 0 <0,namely = 2x < [Aβ+11]x. The proof is completed. Global stability of the unique positive equilibrium Theorem.1 Suppose that < holds, thereisnocloseorbitaroundsystem2.5 in the first quadrant. And if x < holds, the positive equilibrium P 2 x, y of system 2.5 is globally asymptotically stable for x 1 + E 1 < < min{ 2x, 1 + E 1 }. Proof By Lemmas 2.1and 2.2,the solutionxt,yt of system 1.2 with the initial values x0 > 0, y0 > 0 is unanimous bounded for all t 0, and the point P 2 x, y is globally asymptotically stable under conditions of Theorem.1.We should prove that system2.5 does not have limit cycle if < 1 + E 1 holds. Define a Dulac function Bx, y=x 2 y 1,thenalongsystem2.5, we have W = BP x + BQ y = x2 [ x 2 2 x ]. y

7 Jiangand WangAdvances in Difference Equations 201, 201:249 Page 7 of 17 Let f x =x 2 2 x,thenf x =2x 6 x 2, we can derive that the roots of f x=0arex 1 =0,andx 2 =. As f 0 = <0,weknowthatf xreachesthemaximumvalue + at x Hence if < 1 + E 1 holds,w <0forallx 0, then system 2.5 does not have any close orbit. Then we can obtain that if x < holds,thepositiveequilibriump 2 x, y exists and is globally asymptotically stable for x 1 + E 1 < < min{ 2x, 1 + E 1 }. This completes the proof. 4 Existence and uniqueness of limit cycle Theorem 4.1 Suppose that > 2x, then system 2.5 has at least one limit cycle in the first quadrant. Proof We see that dx dt x= 1+E = βx2 y <0withy >0,sowehavex = 1 1+x 2 is an untangent line of the system. And the positive trajectory of system 1.2 goes through from its right side to its left side when it meets the line x =. Construct a Dulac function wx, y=y + β 1 x l, computing w = 0 along the trajectories β of system 1.2, we have dw dt = dy dt + β 1 dx β dt = r + E 2y + β 1x[ 1+E1 x ] β = β 1x[ 1+E1 x ] r + E 2 l β 1 β β x. If l > 0 is large enough, we have dw <0,where0<x < dt.so,theliney + β 1 β x = l goes through upper to lower part in the region {x, y 0<x <,0<y < l β 1 β }.Forsystem 1.2, construct a Bendixson ring ÔABC including P 2 x, y. Define OA, AB, BC as the lengths of line L 1 = y =0,L 2 = x =0,L = y + β 1 x l separately. The outer boundary β line of Bendixson ring ÔABC is J. Due to Lemma 2.1, we know that there exists a unique unstable singular point P 2 x, y inthebendixsonringôabc. By the Poincare-Bendixson theorem, system 2.5 hasatleastonelimitcycleinthefirstquadrant. Thiscompletesthe proof. Note = 2x =. From the proof of Lemma 2.2,weknowthat d dt trace J 1 P 2 = =1 1+x = r + E β 1 Hence by the Hopf bifurcation theorem [12], the system enters into a Hopf-type small amplitude periodic solution at the parametric value = near the positive interior equilibrium point P 2. Now we consider the problem of uniqueness of the limit cycle arising from Hopf bifurcation at the parametric value =. There are different techniques for studying the uniqueness of limit cycles. Kuang [1]gave the following result on the uniqueness of limit cycles for the system: dx = xρx yφx, dt x0 > 0, dy dt = y v + ϕx, y0 > 0, 4.2

8 Jiangand WangAdvances in Difference Equations 201, 201:249 Page 8 of 17 where v >0;allthefunctionsaresufficientlysmoothon[0, ] and satisfy φ0 = 0, ϕ0 = 0 and φ x>0, ϕ x>0 forx >0. 4. Theorem 4.2 Assume that 4. holds. If there exist constants ξ and η with 0<ξ < η such that φξ=v and x ηρx<0 for x η; 4.4 d xρx >0; 4.5 dx φx x=ξ d xρ x+ρx xρx φ x φx 0 for x ξ. 4.6 dx v + ϕx Then system 4.2 has exactly one limit cycle which is globally asymptotically stable [1]. Theorem 4. If conditions 2.2, 2. and > 2x 1.2 has a unique limit cycle. > [Aβ+11]x hold, then system Proof We can rewrite system 1.2assystem4.2withρx= x, φx= βx2 1+x 2 and ϕx= β 1x 2 1+x 2 = eφx.. It is clear that φxandϕxsatisfyassumption4.. Let ξ = x = r+e2 β 1 r E 2,andη = Then by 2. weseethatξ <.Assumption4.4 issatisfied.infact,wehaveϕξ= r + E 2 ; ρx>0if0<x < and ρx<0ifx >. For the sake of convenience, let and Sx=xρ x+ρx xρx φ φ Sx Tx= for x x. ϕx r E 2 Since d xρx = Sx dx φx φx, we get d dx xρx φx x=x = +x 2 2 x > 0. Hence, assumption 4.5issatisfied. βx 2 Now Sx 1 [ Tx= = + x 2 21+E ϕx r E 2 β 1 r E 2 x 2 1 x ]. r E 2 Differentiating this equality and using the fact that r + E 2 =β 1 r E 2 x 2,weobtain T x= β 1 r E 2 [ 21 + E1 x 4 61+E [β 1 r E 2 x 2 1 x 2 x 2 +2x 2 x 2x ]. r E 2 ]

9 Jiangand WangAdvances in Difference Equations 201, 201:249 Page 9 of 17 Taking > 2x into account, we have Hence 2 > E 1x =41+E x 2 1 x. 1 T x> β 1 r E 2 [ 21 + E1 xux ], [β 1 r E 2 x 2 r E 2 ] where Ux=x x 2 x +2x. Since U x=x 2 x 2, Ux has a minimum value at x = x. Therefore, T x 0. Thus assumption 4.6issatisfied. Thus we observe that whenever the nontrivial equilibrium of system 1.2 isunstable, then all the solutions of the system, initiating in the interior of the positive quadrant of the x, y plane, except at the equilibrium, approach a unique limit cycle eventually. But by Lemma 2., we know that when = 2x > [Aβ+11]x, P 2 x, y isastablefine focus with order one. So, when and only when > 2x > [Aβ+11]x,system1.2 has a unique limit cycle. The proof is completed. Remark 1 System 1.2 has no limit cycle whenever the harvesting efforts E 1 and E 2 satisfy the condition < 2x. So, local stability of the equilibrium point x, y implies global asymptotic stability. From the point of view of ecological managers, it may be desirable to have a unique positive equilibrium which is globally asymptotically stable in order to plan harvesting and maintain sustainable development of an ecosystem. Remark 2 If we wish, we can prevent the cycles in the system considered. Let Ê 1, Ê 2 be the harvesting efforts for which system 1.2 admits a limit cycle. Then these efforts must satisfy the condition > 2x > [Aβ+11]x. Nowlet x, ỹ be the required limiting value for the solutions of the system. Let E1, E 2 besuchthatx E1, y E2 = x, ỹ. Hence x, ỹ will be asymptotically stable only if the harvesting efforts E1, E 2 satisfy the condition < 2x. Therefore, by choosing the effort functions E 1 t, E 2 t as E 1 0, E 2 0 = Ê 1, Ê 2 ande 1, E 2 = E1, E 2, it is possible to prevent cycles and drive to the steady state x, ỹ. 5 Bionomic equilibrium Motivated by the paper by Kar [14], we consider the bionomic equilibrium and optimal harvest policy of system 1.2. Let q 1 be the constant prey species cost per unit effort. Let q 2 be the constant predator species cost per unit effort, let be the constant price per unit biomass of the prey species, and let be the constant price per unit biomass of the predator species. The economic rent net revenue at any time is given by N = x 2 q 1 E1 + y q 2 E 2 = N 1 + N 2, 5.1 where N 1 = x 2 q 1 E 1 and N 2 = y q 2 E 2, they denote the economic rent of the prey species and the predator species separately.

10 Jiangand WangAdvances in Difference Equations 201, 201:249 Page 10 of 17 The bionomic equilibrium of the predator-prey system implies both a biological equilibrium and an economic equilibrium. The biological equilibrium occurs when dx dt = dy dt =0, and the economic equilibrium is defined to be achieved when the economic rent is completely dissipated i.e.,whenn =0. Hence we can derive the bionomic equilibrium from the following system: x x βx2 y 1+x E 1x 2 =0, 2 β 1 x 2 y 1+x ry E 2y =0, 2 N = x 2 q 1 E1 + y q 2 E 2 = Now the following cases may arise. Case 1: If x q1 and y > q 2,thenN 1 0andN 2 > 0. In this case, we should stop harvesting the prey species, but we can harvest the predator species. Hence we can derive that E 1 =0,andfromN = x 2 q 1 E 1 + y q 2 E 2 =0,wehavey = q 2. We substitute E 1 =0andy = q 2 into x x βx2 y E 1+x 2 1 x 2 =0,thenwehave x x βq 2 =0. 5. Let f x =x x βq 2 =x 2 1x + βq 2,weknowthatf0 = <0, f = βq 2 >0.So,f x = 0 has at least one positive root in 0,. Now we discuss the roots of f x = 0 as follows. For f x =x 2 2x +1+ βq 2, we can derive that f x >0whenx, andf = βq 2 >0,thenf x = 0 does not have any roots in [,+.Hencewediscusstherootsof f x=0onlyin0,. 1 If βq 2,wehavef x 0,thenf x=0has only one positive root in 0, βq 2 2 If 2 >1+ βq 2, we can derive that x 1 = the roots of f x=0. It is obvious that 0<x 1 < x 2 < βq 2 and x 2 = + are <, f x i= 2x i x 2 i +. We can further obtain that: i If f x 1 >0and f x 2 <0,thenfx=0has three positive roots in 0,. ii If f x 1 =0or f x 2 =0,thenfx=0hastwopositiverootsin0,. iii If f x 1 <0or f x 2 >0,thenfx=0has one positive root in 0,. Let the root of f x=0bex, then we substitute x and y into β 1x 2 y ry E 1+x 2 2 y =0. WecanobtainthatE 2 = β 2 1x 1+x 2 r.fore 2 >0,thenwemusthaver < β 2 1x 1+x 2.Atlast,we have the bionomic equilibrium x, y,0,e 2. Case 2: If x > q1 and y q 2,thenN 1 >0andN 2 0. In this case, we should stop harvesting the predator species, but we can harvest the prey species. Hence we can derive that E 2 =0,andfromN = x 2 q 1 E 1 + y q 2 E 2 =0,wehavex = q1. Substituting E 2 =0andx = q1 into β 1x 2 y ry E 1+x 2 2 y =0,wehaver = β 1 +q 1 and y is any positive number. Then we substitute x and y into x x βx2 y E 1+x 2 1 x 2 = 0, we can obtain that E 1 = x 1+x 2 βx y.fore x 1+x 2 1 >0,thenwemusthavey < q1 + q 1 1 p 1 β q1.hencey is

11 Jiangand WangAdvances in Difference Equations 201, 201:249 Page 11 of 17 any positive number but it must satisfy y < q1 + q 1 1 p 1 equilibrium x, y, E 1,0. β q1.atlast,wehavethebionomic Case : If x q1 and y q 2,thenN 1 0andN 2 0. In this case, we should stop harvesting the predator species and the prey species. Hence we can derive that E 1 =0 and E 2 = 0. Then the bionomic equilibrium does not exist. Case 4: If x > q1 and y > q 2,thenN 1 >0andN 2 >0.Inthiscase,wecanharvestboththe predator species and the prey species. Hence, from N = x 2 q 1 E 1 + y q 2 E 2 =0,we have x = q1 and y = q 2. Then we substitute x and y into x x βx2 y E 1+x 2 1 x 2 =0, β 1 x 2 y ry E 1+x 2 2 y = 0. We can obtain that E 1 = x 1+x 2 βx y and E x 1+x 2 2 = β 2 1x 1+x 2 r.for E 1 >0andE 2 >0,thenwemusthaveβ < +q 1 q 1 p 2 q 2 q1 and β 1 > r+q 1 q 1.Atlast,we have the bionomic equilibrium x, y, E 1, E 2. 6 Optimal harvest policy The fundamental problem in determination of an optimal harvest policy in a commercial predator-prey system is to determine the optimal trade-off between the current and future harvests. So, for determining an optimal harvesting policy, we consider the present value J of a continuous time stream of revenues given by J = 0 Nx, y, E 1, E 2, te δt dt, where δ denotes the instantaneous annual rate of discount. N is the economic rent net revenue at any time t given by Nx, y, E 1, E 2, t= x 2 q 1 E 1 + y q 2 E 2,whereq 1 and q 2 are the cost per unit biomass of the x and y species, separately; and are the prices per unit biomass of the x and y species, respectively. We shall optimize the objective function J = 0 e δt[ x 2 q 1 E1 + y q 2 E 2 ] dt, 6.1 subject to the state Eq. 1.2 by using Pontryagin s maximal principle. Let us now construct the Hamiltonian function H = e δt[ x 2 q 1 E1 + y q 2 E 2 ] + λ 1 [ x x βx2 y 1+x 2 E 1x 2 ] [ β1 x 2 ] y + λ 2 1+x ry E 2y, where λ 1 t andλ 2 t aretheadjointvariables.e 1 and E 2 are the control variables. Now we wish to find the maximum equilibrium x δ, y δ, E 1δ, E 2δ ofthehamiltonianfunctionh. For the control variables E 1 and E 2 are the linear function of H, we have that the optimal equilibrium must occur at the extreme point, namely H E i =0, i =1,2. 6.

12 Jiangand WangAdvances in Difference Equations 201, 201:249 Page 12 of 17 By the maximal principle, there exist adjoint variables λ 1 t andλ 2 t for all t 0such that [ dλ 1 {2e dt = H x = δt E 1 x + λ 1 2x 2βxy ] 1 + x 2 2E 1x + 2β } 1λ 2 xy x 2 2 and dλ 2 dt = {e H y = δt E 2 λ 1βx 2 [ 1+x + λ β1 x 2 ]} x r E Using Eq. 6.wecanderivethat λ 1 t=e δt q 1 x and λ 2 t=e δt q 1 q y By Eqs. 6.6 and6.7 weknowthatλ i te δt i =1,2areinvariablebythechangeof time t. Using 5., Eqs. 6.6and6.7, we have that Eqs. 6.4and6.5become [ 2 x βxy 1+x 2 + 2β 1xy 1 + x 2 2 [ β1 x 2 ] 1+x r 2 q 2 y ] + q 1 δ q 1 x 2 x 2 [ 2βxy q 1 x 2 βx 2 1+x δ x 2 + 2βxy ] 2 1+x 2 =0, 6.8 q 2 y = By Eqs. 6.8 and6.9,wecanobtainthepositiverootx δ, y δ. Then substituting the values of the positive root into 5.2, we get E 1δ = x δ1 + x 2 δ βx δ y δ, 6.10 x δ 1 + x δ2 E 2δ = β 2 1x δ r x 2 δ 7 Numerical simulation Example 1 For simulation let us take =1,β =2,β 1 =,r =0.5. We verify the limit cycle, global stability and controllability of system 1.2. See Figures 1-6. To show the controllability of the system, we take examples as follows: 1 E 1, E 2 =.5,2see Figure ; 2 E 1, E 2 =4.5,2see Figure 4; E 1, E 2 =5,2see Figure 5; 4 E 1, E 2 =1,200, 2 see Figure 6. Figures -6 show the dependence of the dynamic behavior of system 1.2 on the harvesting efforts for the prey E 1.Figures-6 show that when E 1 is small, both the prey and

13 Jiangand WangAdvances in Difference Equations 201, 201:249 Page 1 of 17 Figure 1 Phase diagram of the limit cycle with different initial conditions E 1 =0.5,E 2 =1.5.Here harvesting efforts lie in the region > 2 x x 2 > [Aβ+11 ]x 1. Figure 2 Phase plane trajectories corresponding to different initial levels E 1 =2,E 2 =1.5.The figure clearly indicates that the interior equilibrium point P 2 is globally asymptotically stable. predator population converge to their equilibrium values respectively, but as E 1 is large, the predator does not have enough food and approaches extermination at last. As E 1 is large enough, both the prey and predator approach extermination at last. Which means that if we change the value of E 1, it is possible to prevent the cyclic behavior of the predator-prey system and to drive it to a required stable state.

14 Jiangand WangAdvances in Difference Equations 201, 201:249 Page 14 of 17 Figure The quantity of predator is larger than the prey when they approach globally asymptotically stable. Figure 4 The quantity of predator is less than the prey when they approach globally asymptotically stable. Example 2 For =0.4,q 1 =0.2, =0.5,q 2 =0.2,δ = 1.25, and the remaining parameter values being the same as above, then Eqs. 6.8, 6.9 can be simplified as follows: x 1.6xy 1+x xy 1 2 x x 2 + 4xy 2 1+x 2 + 6xy y 1 + x x 2 =0, 7.1

15 Jiangand WangAdvances in Difference Equations 201, 201:249 Page 15 of 17 Figure 5 The prey approaches stable state, but the prey approaches extermination at last. Figure 6 Both the predator and the prey approach extermination at last. 1.5x x x2 x = x 2 y We can solve roots of Eqs. 7.1 and7.2 by the tool of Maple. The roots of Eqs. 7.1 and 7.2 are solved as follows: {x = ,y = }; {x = I, y = I};

16 Jiangand WangAdvances in Difference Equations 201, 201:249 Page 16 of 17 {x = I, y = I}; {x = I, y = I}; {x = I, y = I}; {x = I, y = I}; {x = I, y = I}; {x = I, y = I}; {x = I, y = I}. But only {x = ,y = } satisfies q1 x δ > = 2 2, y δ > q 2 = 0.5. So, we can derive that x δ = ,y δ = Then, substituting the values of x δ and y δ into 5.2, we get E 1δ = ,E 2δ = So,wecan derive that an optimal equilibrium solution is , , , Conclusion In this paper, qualitative analysis of a harvested predator-prey system with Holling type III functional response is considered. This work presents analysis of the effect of harvesting efforts on the prey-predator system. We have proved that exactly one stable limit cycle occurs in the system under certain conditions and have proved the global stability of the positive equilibrium. It was also found that it is possible to control the system in such a way that the system approaches a required state, using the efforts E 1 and E 2 as the control. The results we have obtained may be helpful for the fishery managers wishing to maintain a globally sustainable yield. Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript. Received: 1 July 201 Accepted: 2 August 201 Published: 20 August 201 References 1. Berryman, AA: The origins and evolutions of predator-prey theory. Ecology 7, Gause, GF: La Theorie mathematique de la lutte pour la vie. Hermann, Paris 195. Clark, CW: Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd edn. Wiley, New York Chaudhuri, KS: A bioeconomic model of harvesting a multispecies fishery. Ecol. Model. 2, Chaudhuri, KS: Dynamic optimization of combined harvesting of a two species fishery. Ecol. Model. 41, Matsuda, H, Katsukawa, T: Fisheries management based on ecosystem dynamics and feedback control. Fish. Oceanogr. 116, Matsuda, H, Abrams, PA: Maximal yields from multi-species fisheries systems: rules for systems with multiple trophic. Ecol. Appl. 161, Dai, G, Tang, M: Coexistence region and global dynamics of a harvested predator-prey system. SIAM J. Appl. Math. 581, Xiao, D, Ruan, S: Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting. Fields Inst. Commun. 21, Kar, TK, Matsuda, H: Global dynamics and controllability of a harvested prey-predator system with Holling type III function response. Nonlinear Anal. Hybrid Syst. 1,

17 Jiangand WangAdvances in Difference Equations 201, 201:249 Page 17 of Chaudhuri, KS: Dynamic optimization of combined harvesting of a two species fishery. Ecol. Model. 41, Hassard, BD, Kazarinoff, ND, Wan, YH: Theory and Application of Hopf Bifurcation. Cambridge University Press, Cambridge Kuang, Y, Freedman, HI: Uniqueness of limit cycles in Gauss-type models of predator-prey systems. Math. Biosci. 88, Kar, TK, Chaudhuri, KS: On non-selective harvesting of a two competing fish species in the presence of toxicity. Ecol. Model , doi: / Cite this article as: Jiang and Wang: Qualitative analysis of a harvested predator-prey system with Holling type III functional response. Advances in Difference Equations :249.