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

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1 Wang and Pan Advances in Difference Equations 2012, 2012:96 R E S E A R C H Open Access Qualitative analysis of a harvested pretor-prey system with Holling-type III functional response incorporating a prey refuge Jinghai Wang * and Liqin Pan * Correspondence: fzwjh@16.com College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 50002, People s Republic of China Abstract A pretor-prey model with simplified Holling type III response function incorporating a prey refuge under sparse effect is considered. Through qualitative analysis of the model, at least two limit cycles exist around the positive equilibrium point with the result of focus value, the Hopf bifurcation under a prey refuge is obtained. We also show the influence of prey refuge. Numerical simulations are carried out to illustrate the feasibility of the obtained results and the dependence of the dynamic behavior on the prey refuge. Through the results of computer simulation, it is further shown that under certain conditions the model has three limit cycles surrounding the positive equilibrium point. Keywords: limit cycles; Holling type III; prey refuge; pretor-prey model 1 Introduction The dynamical relationship between pretors and their preys is one of the dominant subjects in ecology and mathematical ecology due to its universal importance, see [1. Some of the empirical and theoretical wor have investigated the effect of prey refuges, the refuges used by prey have a stabilizing effect on the considered interactions, and prey extinction can be prevented by the addition of refuges [4 8. Huang et al. [9 studied the stability analysis of a prey-pretor model with Holling type III response function incorporating a prey refuge: dx dt = ax bx2 αx2 y dy dt = cy + αx2 y β 2 +x 2. β 2 +x 2, (1.1) Motivated by the study of Huang et al. [9andJiandWu[10, we consider the following pretor-prey model with Holling type III response function incorporating a prey refuge under sparse effect: dx dt = rx2 (1 x ) (1 m)2 x 2 y dy dt = y[ d + u(1 m)2 x 2, a+(1 m) 2 x 2 a+(1 m) 2 x 2, 2012 Wang and Pan; 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 wor is properly cited. (1.2)

2 Wang and Pan Advances in Difference Equations 2012, 2012:96 Page 2 of 14 where x(t) is the population density of the prey and y(t) is the population density of the pretor at time t; r > 0 represents the intrinsic growth rate of the prey; is the carrying capacity of the prey in the absence of pretor and harvesting; m is a constant number of prey using refuges, which protects m of prey from pretion; the term x 2 /(a + x 2 )denotes the functional response of the pretor, which is nown as Holling type III response function; u > 0 is the conversion factor denoting the number of newly born pretors for each captured prey; d > 0 is the death rate of the pretor. From the point of view of human needs, the exploitation of biological resources and the harvest of population are commonly practiced in fishery, forestry, and wildlife management. Concerning the conservation for the long-term benefits of humanity, there is a wide range of interest in the use of bioeconomic modeling to gain insight into the scientific management of renewable resources. The problem of pretor-prey interactions under a prey refuge have been studied by some authors. For example, Hassel [2 showed that adding a large refuge to a model, which exhibited divergent oscillations in the absence of a refuge, replaced the oscillatory behavior with a stable equilibrium. McNair [11 obtained that a prey refuge with legitimate entryexit dynamics was quite capable of amplifying rather than mping pretor-prey oscillations. McNair [12 showed that several inds of refuges could exert a locally destabilizing effect and create stable, large-amplitude oscillations which would mp out if no refuge was present. Even now, prey refuges are widely believed to prevent prey extinction and mp pretor-prey oscillations. For example, Kar [6 considered a Lota-Volterra type pretor-prey system incorporating a constant proportion of prey using refuges m,which protects m of prey from pretion, with Holling type II response function and Holling type III response function, respectively. Our results indicate that refuge had a stabilizing effect on prey-pretor interactions and the dynamic behavior very much depends on the prey refuge parameter m, point that increasing the amount of refuge could increase prey densities and lead to population outbreas. This article is organized as follows. Basic properties such as the existence, stability, and instability of the equilibria of the model and the boundedness of the solutions of the system (1.2) with positive initial values are given in Section 2. InSection, sufficient conditions for the global stability of the unique positive equilibrium are obtained. Section 4 is devoted to deriving the existence of limit cycle. In Section 5, we study the Hopf bifurcation of system (1.2). In Section 6, we analyze the influence of prey refuge and give numerical stimulations. 2 Basic properties of the model Let R + 2 = {(x, y) x 0, y 0}. For practical biological meaning, we simply study system (1.2) in R + 2. The main aim of this article is to study the existence and non-existence of positive equilibrium of (1.2) by the effects of a prey refuge, that is to say, the existence and nonexistence of positive equilibrium of system (1.2)dependontheconstantm [0, 1). We mae the following substitution for model (1.2) dt = [ a +(1 m) 2 x 2 dτ.

3 Wang and Pan Advances in Difference Equations 2012, 2012:96 Page of 14 Denoting new argument τ with t again, it gives dx dt = x2 {r(1 x )[a +(1 m)2 x 2 (1 m) 2 y}, dy dt = y[ +(u d)(1 m)2 x 2. (2.1) Solutions of system (2.1) are discussed as follows: (1) If u d,system(2.1) possesses only two equilibrium points on the region R + 2,theyare the trivial solution O(0, 0) and the semi-trivial solution in the absence of pretor P 1 (,0); (2) If u > d, system(2.1) admits three equilibrium points on the region R + 2,theyarethe trivial solution O(0, 0), the semi-trivial solution in the absence of pretor P 1 (,0)andthe unique positive constant solution P 2 (x 1, y 1 ), where x 1 = 1 1 m u d, [ y 1 = r 1 1 (1 m) u d ua (u d)(1 m) 2. For the existence of positive constant solution P 2 (x 1, y 1 ), it is necessary to assume that 1 1 (1 m) u d > 0, then we derive that 0 m <1 1 u d. It turns out that the non-constant positive solutions of (2.1) mayexistforsomeranges of the parameter m. From the first equation of system (1.2), it is easy to derive that lim sup x(t). t + Lemma 2.1 The solution (x(t), y(t)) of system (1.2) with the initial value x(0) > 0, y(0) > 0 is positive and bounded for all t 0. Proof We see that dx dt x= = (1 m)2 2 y a+(1 m) 2 2 <0withy >0,sox = is a untangent line of system (1.2). 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 Dulac function w(x, y)=y + ux l, computing w = 0 along the trajectories of system (1.2) dw dt = dy ( dt + udx x = dy + urx2 1 x ) ( = urx 2 1 x ) d(l ux). dt If l > 0 is large enough, we have dw <0,where0<x <. Sotheliney + ux = l goes dt through from its upside to its downside in the region {(x, y) 0<x <,0<y <+ }.Forsystem (1.2), constructing a Bendixson ring ÔABC including P 2 (x 1, y 1 ). Define OA, AB, BC as the lengths of lines L 1 = y =0,L 2 = x =0,L = y + ux l, respectively. The bounry line of the Bendixson ring ÔABC s is J. So, the orbits of system (1.2) enter into the interior of the Bendixson ring when it meets the bounry line J. If the initial value p which is in thefirst quadrantis notin the ÔABC, we canconstruct acurvej p inthesameway,denotingp J p. Thus the positive trajectory L + p which is pass through p goes through into the interior of J p attheend.notethatthepointp 2 (x 1, y 1 )is the unique equilibrium point in the Bendixson ring ÔABC. Through the Limit collection theory, we now that the trajectory L + p goes through into the interior of the Bendixson ring ÔABC at the end. Therefore, all of the solutions (x(t), y(t)) of system (1.2) withthe initial value x(0) > 0, y(0) > 0 are positive and bounded. This completes the proof.

4 Wang and Pan Advances in Difference Equations 2012, 2012:96 Page 4 of 14 Lemma 2.2 (1) O(0, 0) is a stable node point; (2) if u dholds,p 1 (,0)isastablefocusorastablenode; () if u > dholds,thenp 1 (,0) isastablefocusorastablenodefor1 1 u d < m <1; P 1 (,0) is a stable node point for 0 m <1 1 u d ;P 1(,0) is a stable node point for m =1 1 u d.p 2(x 1, y 1 ) is an unstable focus or an unstable node for 0 m <1 u+2d 2d P 2 (x 1, y 1 ) isastablefocusorastablenodefor1 u+2d 2d u d < m <1 1 center or a focus for m =1 u+2d 2d u d. Proof The Jacobian matrix of system (2.1)isgivenby u d ; u d ;P 2(x 1, y 1 ) is a J(x, y) ( ) = r[2ax a x2 + 4(1 m) 2 x 5(1 m)2 x 4 2(1 m) 2 xy (1 m) 2 x 2 2(u d)(1 m) 2 xy +(u d)(1 m) 2 x 2. (1) Since det J(0, 0) =0, O(0, 0) is a higher-order singular point. We mae a transformation dτ = 1 dt. Substituting this into (2.1), then replacing τ with t gives dx dt = r d x2 + r d x r(1 m)2 x 4 + r(1 m)2 x 5 + (1 m)2 x 2 y P 2 (x, y), dy dt = y (2.2) (u d)(1 m)2 x 2 y = y + Q 2 (x, y). From y + Q 2 (x, y) =0,wecansolvethaty = ϕ(x) 0. Furthermore, we can derive that P 2 (x, ϕ(x)) = r d x2 + r d x r(1 m)2 x 4 + r(1 m)2 x 5,sowehave n = r 0,n = 2. According d to Zhang et al. [1, we can derive that O(0, 0) is a stable node point. (2) The Jacobian matrix of system (2.1) for the equilibrium point P 1 (,0)isgivenby ( ) r[ a (1 m) 2 (1 m) 2 2 J(,0)=. 0 +(u d)(1 m) 2 2 If u d, the eigenvalues of matrix r[ a (1 m) 2 and +(u d)(1 m) 2 2 are negative, hence P 1 (, 0) is a stable focus or a stable node. () If u > d, according to (2), we now that if +(u d)(1 m) 2 2 <0,namely1 1 u d < m <1,thenp = det J(,0)>0, hence P 1(, 0) is a stable focus or a stable node; if +(u d)(1 m) 2 2 >0,namely0 m <1 1 u d,thendet J(,0)<0,henceP 1(,0)is a saddle point. If m =1 1 u d,thendet J(,0)=0,thusP 1(, 0) is a higher-order singular point. In this situation P 1 (,0)andpointP 2 (x 1, y 1 )arethesamepoint.sop 1 (,0)isastable node point. As det J(x 1, y 1 )=2(u d)(1 m) 4 x 1 y 1 >0, [ P = r 2ax 1 +4(1 m) 2 x 1 a = r [ a x21 +2(1 m)2 x = rx 2 1 5(1 m)2 x2 1 1 (1 m)2 x 4 1, [ au +2 2(1 m) u d (u d) x (1 m) 2 x 1 y 1

5 Wang and Pan Advances in Difference Equations 2012, 2012:96 Page 5 of 14 if 2(1 m) u d au+2 >0,namely0 m <1 u+2d,wederivethatp <0,then (u d) 2d u d P 2 (x 1, y 1 ) is an unstable focus or an unstable node; if 2(1 m) 1 u+2d 2d u d < m <1 1 stable node; if 2(1 m) isacenterorafocus.theproofiscompleted. u d au+2 <0,namely (u d) u d,wederivethatp >0,thenP 2(x 1, y 1 )isastablefocusora u d au+2 =0,namelyP =0,thenwecanderivethatP (u d) 2 (x 1, y 1 ) From Lemma 2.2,ifP = rx 2 1 [2(1 m) u d au+2 = 0 holds, namely m =1 u+2d (u d) 2d P 2 (x 1, y 1 ) is a center focus. We can mae further conclusions: u d, Lemma 2. (1) if C 0 >0holds, P 2 (x 1, y 1 ) is a stable fine focus with order one; (2) if C 0 <0holds, P 2 (x 1, y 1 ) is an unstable fine focus with order one; () if C 0 =0and C 1 >0hold, P 2 (x 1, y 1 ) is a stable fine focus with second-order; (4) if C 0 =0and C 1 <0hold, P 2 (x 1, y 1 ) is an unstable fine focus with second-order, where C 0 = (G +F + I +2DE 5EG D HG) and 2 C 1 = 8 ( 10M 28 EDN 12E2 F 10 ED 48E D E2 HG 106 E2 HD 2 EDH2 +76E G +52DE 2 52GE 2 +6EDH 6EGH 2EFH + 26 EGH G2 F 100 EDG2 2 D2 F +4FN +2FH DL DGF + +4D 14GD ENG + GL +10DG2 6DN +10DH 2 10GH 2 EGD2 ). Proof First use the coordinate translation, that is translation the origin of coordinates into the point P 2 (x 1, y 1 ). Then we assume x = x x 1, y = y y 1, dt =(1 m) 2 x 2 1 dt. Replacing x, y, t with x, y, t, respectively, it gives dx dt = y 2 x 1 xy + 1 x 2 y + x 2 1 dy dt = 2(u d)y 1 r (1 m) 2 x 2 1 r (1 m) 2 x 2 1 x 1 x + (u d)y 1 x 2 1 [a +6x 2 1 (1 m)2 ax 1+10(1 m) 2 x 1 (1 m) 2 y 1 x 2 [4x 1 (1 m) 2 a+10(1 m)2 x 2 1 x + r [1 5x x 2 1 x4 r x 5, 1 x 2 1 x 2 + 2(u d) x 1 xy + u d x 2 y. x 2 1 (2.) We denote A = 2(u d)y1 x 1 > 0 and mae the following transformations u = x, v = 1 y, dτ = A Adt,andreplacingu, v, τ with x, y, t,respectively,wehave dx dt = y + Dxy Ex2 + Nx 2 y Fx Lx 4 + Mx 5 = y + 5 j=2 P j(x, y) ˆP(x, y), dy dt = x Gx2 Hxy Ix 2 y = x + j=2 Q j(x, y) ˆQ(x, y), (2.4)

6 Wang and Pan Advances in Difference Equations 2012, 2012:96 Page 6 of 14 where D = 2 r, E = x 1 A(1 m) 2 x 2 1 N = 1 x 2, F = 1 L = r Ax 2 1 [ 1 5x 1 r A(1 m) 2 x 2 1, M = r Ax 2 1 2(u d) H =, I = u d Ax 1 Ax 2. 1 [ a +6x 2 1 (1 m)2 ax 1 +10(1 m) 2 x 1 [ 4x 1 (1 m) 2 a +10(1 m)2 x 1, G = (u d)y 1 A 2 x 2, 1, (1 m) 2 y 1, It is obvious that D 2 =4N and HD =4I. Then we mae use of the method Poincare to calculate the focus value. Construct a form progression F(x, y)=x 2 + y 2 + = F (x, y), where F (x, y) istheth homogeneous multinomials with x and y. Considering df dt (2.4) = F t ˆP(x, y)+ F ˆQ(x, y)=0,wecanobtainthatthreemultinomials t and four multinomials of F(x, y)areequaltozeroseparately. Noting that 2xP 2 (x, y)+2yq 2 (x, y)= H (x, y) we can obtain H =2Ex +(2G 2D)x 2 y +2Hxy 2. Let F (x, y)=a 0 x + a 1 x 2 y + a 2 xy 2 + a y, then we can obtain the following form: 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 2D)x 2 y +2Hxy 2. Through 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 2F)x 4 +(2HG +4EG 2I 4DE)x 2 y 2 + ( 2N +2GD +4E 2 +2EH 2D 2) x y + ( 2H 2 +4EH ) xy.

7 Wang and Pan Advances in Difference Equations 2012, 2012:96 Page 7 of 14 Let x = r cos θ, y = r sin θ,thenwecanderivethat C 0 = 2 0 H 4 (cos θ, sin θ) dθ = (G +F + I +2DE 5EG D HG). 2 Hence, then point O(0, 0) of system (2.) is an unstable fine focus with order one when C 0 > 0. But considering the time change dτ = Adt, P 2 (x 1, y 1 )isastablefinefocuswith order one. And P 2 (x 1, y 1 ) is an unstable fine focus with order one when C 0 <0.IfC 0 =0, namely G +F + I +2DE 5EG D HG =0,wedenote F 4 (x, y)=b 0 x 4 + b 1 x y + b 2 x 2 y 2 + b xy + b 4 y 4. By substituting F 4 (x, y)=b 0 x 4 + b 1 x y + b 2 x 2 y 2 + b xy + b 4 y 4 into y F 4 x x F 4 y = H 4, b y 4 b 1 x 4 +(4b 0 2b 2 )x y +(b 1 b )x 2 y 2 +(2b 2 4b 4 )xy =(2G +2F 2EG 2D)x 4 +(2I +4DE 2HG 4EG)x 2 y 2 + ( 2D 2 2N 2GD 4E 2 2EH ) x y ( 2H 2 +4EH ) xy. Through the comparison method of correlates, we can obtain that b 0 = 1 2 D2 1 2 N 1 2 GD E2 2 EH 1 2 H2, b 1 =2EG +2D 2G 2F = 1 (2I +4DE 2HG 4EG), b 2 = H 2 2EH, b =0, b 4 =0, then we have ( 1 F 4 (x, y)= 2 D2 1 2 N 1 2 GD E2 2 EH 1 ) 2 H2 x 4 +(2EG +2D 2G 2F)x y ( H 2 +2EH ) x 2 y 2. Also F H 5 =2xP 4 +2yQ 4 + P x + Q F y + P F 4 2 x + Q F 4 2 y = ( 2L 2GF +2DF 2D 2 E +2NE +2GDE +4E +6E 2 H +2EH 2) x 5 + ( 2GN 4DN +10EF +2EI +2D 2GD 2 4E 2 D 6EHD 2H 2 D 6E 2 G 6DE +6GE +2GH 2 +2EGH 2DH +2GH +2FH ) x 4 y + ( 4EN +6EGD +6D 2 6GD 6FD +6EH 2 +4E 2 H +2H ) x y 2 + ( 2HI +4EI 2DH 2 4DEH ) x 2 y.

8 Wang and Pan Advances in Difference Equations 2012, 2012:96 Page 8 of 14 Noting F 5 (x, y) =d 0 x 5 + d 1 x 4 y + d 2 x y 2 + d x 2 y + d 4 xy 4 + d 5 y 5 into y F 5 x x F 5 y = H 5, through the comparison method of correlates, we can obtain that d 0 = 2 5 GN 4 5 DN 2EF +6EI GN E2 D + 8 HI E2 G DE 6 5 EG 2 5 GH2 2 5 EGH I 2 5 GH 2 5 FH, d 1 = 2L 2GF +2DF 6EN +2GDE +4E +6E 2 H +2EH 2, d 2 =2HI +2EI, d = 28 EN + 14 EGD +8N 2GD + 2 FD + 14 EH E2 H + 2 H 8 L 8 GF + 16 E, d 4 =0, d 5 = EN EGD N 4 5 GD FD EH E2 H H L 16 2 GF E. (2.5) Substituting (2.5)into F 5 H 6 =2xP 5 +2yQ 5 + P 2 x + Q F 5 2 y + P F 4 x + Q F 4 y + P F 4 x + Q F 4 y, we can derive that C 1 = 8 ( 10M 28 EDN 12E2 F 10 ED 48E D E2 HG 106 E2 HD 2 EDH2 +76E G +52DE 2 52GE 2 +6EDH 6EGH 2EFH + 26 EGH G2 F 100 EDG2 2 D2 F +4FN +2FH 2 10 DL DGF + EGD2 +4D 14GD ENG + GL +10DG2 6DN +10DH 2 10GH 2 ). Hence point O(0,0) of system (2.) is an unstable fine focus with second-order when C 1 > 0. But considering the time change dτ = Adt,wenowthatP 2 (x 1, y 1 )isastablefine focus with second-order. And P 2 (x 1, y 1 ) is an unstable fine focus with second-order when C 1 < 0. The proof is completed. Remar If C 1 =0holds,P 2 (x 1, y 1 ) is possible a fine focus with third-order. Global stability of the unique positive equilibrium Theorem.1 Suppose that 1 a m <1holdsandthereisnocloseorbitaroundsystem (2.1) in the first quadrant. Assume that: (H 1 ) d < u 4 d; (H 2 ) u > 4 d and u 4d, 1 a m <1 1 (H ) u =4d, 1 a < m <1 a. u d ;

9 Wang and Pan Advances in Difference Equations 2012, 2012:96 Page 9 of 14 Then the equilibrium P 1 (,0) of system (2.1) is globally asymptotically stable on the first quadrant, if (H 1 ) holds. And the positive equilibrium P 2 (x 1, y 1 ) of system (2.1) is globally asymptotically stable, if one of (H 2 )and(h )holds. Proof By Lemmas 2.1and 2.2,thesolution(x(t),y(t)) of system (1.2) with the initial values x(0) > 0, y(0) > 0 is unanimous bounded for all t 0andthepointP 2 (x 1, y 1 ) is globally asymptotically stable, we should proof that the system (2.1) is not exist limit cycle if 1 a m <1holds. Define a Dulac function B(x, y)=x 2 y 1,thenfromsystem(2.1), we have D = (BP) x = r [ y r y r y [ + (BQ) y a (1 m)2 x 2 +2(1 m) 2 x ( a (1 m)2 [ a + (1 m)2. ) 2 ( ) +2(1 m) 2 Hence if m 1 a and D <0forallx 0, system (2.1)doesnotexistanycloseorbit. Then we can obtain that if 1 1 ad u d 1 a holds, namely d < u 4 d,thepositive equilibrium P 2 (x 1, y 1 )doesnotexistandp 1 (,0)ofsystem(2.1) is globally asymptotically stable; if u > 4 d and u 4d,then1 a >1 u+2d a 2 d(u d) or if u =4d,then1 a =1 u+2d a 2 d(u d).sothepositiveequilibriump 2(x 1, y 1 )ofsystem(2.1) is globally asymptotically stable, if one of (H 2 )and(h ) holds. The proof is completed. 4 Existence of limit cycle Theorem 4.1 Suppose that 0 m <1 u+2d 2 limitcycleinthefirstquadrant. a d(u d).thensystem(2.1) existsatleastone Proof In the proof of Lemma 2.1, we can obtain the bounry line J of the Bendixson ring B = {(x, y) 0<x <,0<y < l ux}.letj be the outer bounry of system (2.1). Due to the Lemma 2.1, we now that there exists an unique unstable singular point P 2 (x 1, y 1 )inthe Bendixson ring B. By Poincare-Bendixson theorem, system (2.1) exists at least one limit cycle in the first quadrant. This completes the proof. 5 Hopf bifurcation By the study of Lou et al. [14 and the Lemma 2. we have the following theorem: Theorem 5.1 (1) If C 0 >0and 0<1 u+2d 2d 1 m 1 hold, then system (2.1)existsa u d limit cycle around the small neighborhood of P 2 (x 1, y 1 ). Further if the limit cycle is unique, then it is stable. (2) If C 0 <0and 0<m (1 u+2d 2d 1 ) 1 hold, then system (2.1) exists a limit cycle u d around the small neighborhood of P 2 (x 1, y 1 ). Further if the limit cycle is unique, then it is unstable.

10 Wang and Pan Advances in Difference Equations 2012, 2012:96 Page 10 of 14 () If C 1 <0, 0<δ 1, 0<δ 4 < δ 1 and 0<C 0 < δ, δ 4 < P <0hold, then system (2.1) exists at least two limit cycles around the small neighborhood of P 2 (x 1, y 1 ). By selecting the suitable values of the parameters, we can obtain these two limit cycle. (4) If C 1 >0, 0<δ 5 1, 0<δ 6 < δ 5 1 and δ 5 < C 0 <0, 0<P < δ 6 hold, then system (2.1) exists at least two limit cycles around the small neighborhood of P 2 (x 1, y 1 ). By selecting the suitable values of the parameters, we can obtain these two limit cycle. 6 The effect of prey refuge and harvesting efforts and examples 6.1 The influence of prey refuge on model (1.2) By the variable transformation dt = a +(1 m) 2 x 2 dτ,thepositiveequilibriump 2 (x 1, y 1 ) of system (1.2) taes the form x 1 = 1 1 m u d,andy 1 1 = r[1 (1 m) u d ua,where (u d)(1 m) 2 0<x 1 <. For x 1 <,wehave 1 1 m u d <,namely0<m <1 1 u d.thenweobtain dx 1 dm = 1 >0. (6.1) (1 m) 2 u d The above inequality shows that x 1 is a strictly increasing function with respect to the parameter m and that the increasing of the prey refuge increases the density of the prey. One could see that y 1 is also a continuous differential function of the parameter m.simple computation shows that dy 1 dm = r ua (u d)(1 m) [ 2. (6.2) (1 m) u d We discuss (6.2) in the following two cases. Case 1: Assume that the inequality 1 2 u d < m <1 1 u d holds, then dy 1 dm <0for all m >0,thusy 1 is a strictly decreasing function with respect to the parameter m. That is, increasing the amount of prey refuge can decrease the density of the pretor. In this case, y 1 reaches the maximum value r[1 1 u d ua at m =0. u d Case 2: Assume that the inequality 0 < m <1 2 u d holds, then dy 1 >0forallm >0, dm thus y 1 is a strictly increasing function of parameter m. Thatis,increasetheamountof prey refuge can increase the density the pretor. This analysis shows that increasing the amount of prey refuge can increase the density of the pretor due to the pretor still has enough food for pretion with m being small. 6.2 Example and simulations Example 1 Let r =0.05, =4,a =,d =2,u =4insystem(2.1). By simple computation, we have a 1 =1 4 =0.25, u +2d 2d u d =1 4 =0.567, 1 u d =1 2 =0.14, 4 d = 8 <4, u > 4 d.

11 Wang and Pan Advances in Difference Equations 2012, 2012:96 Page 11 of 14 Figure 1 P 2 (x 1, y 1 ) is globally asymptotically stable with m =0.49. Figure 2 P 2 (x 1, y 1 ) is globally asymptotically stable with m =0.4. We nown that 1 a < m <1 1,ifwetaem =0.49andm =0.4separately.By u d Theorem.1,system(2.1) admits a globally asymptotically stable equilibrium P 2 (x 1, y 1 )in the region R + 2,whichisshowninFigures1 and 2. If we tae m =0.14andm =0.11separately,byTheorem5.1,system(2.1) admits one and two limit cycles surrounding equilibrium P 2 (x 1, y 1 ), which is shown in Figures and 4. Example 2 Let r =0.05, =4,a = 10, d =2,u = 4 in the system (2.1). By simple computation, we have a 1 = =0.209,

12 Wang and Pan Advances in Difference Equations 2012, 2012:96 Page 12 of 14 Figure There is a stable limit cycle surrounding P 2 (x 1, y 1 )withm =0.14. Figure 4 There are two limit cycles surrounding P 2 (x 1, y 1 )withm = u d =1 12 =0.544, 1 u +2d 1 0 2d u d =1 = If we tae m =0.065,byTheorem5.1,system(2.1) admits three limit cycles surrounding equilibrium P 2 (x 1, y 1 ), which is shown in Figure 5. Figures 1, 2,, 4,and5 show the dependence of the dynamic behavior of system (1.2)on the prey refuge m.figures2,, 4,and5 show that when m is small enough, there are three or two limit cycles surrounding the unique positive equilibrium, when m is large there is a stable limit cycle surrounding the unique positive equilibrium and when m is large enough, the limit cycle is broen and both the prey and pretor population converge to

13 Wang and Pan Advances in Difference Equations 2012, 2012:96 Page 1 of 14 Figure 5 There are three limit cycles surrounding P 2 (x 1, y 1 )withm = their equilibrium values, respectively, which means that if we change the value of m, it is possible to prevent the cyclic behavior of the pretor-prey system and to drive it to a required stable state. 7 Conclusion This article considers a pretor-prey model with Holling type III response function incorporating a prey refuge under sparse effect. We give the complete qualitative analysis of the instability and global stability properties of the equilibria and the existence of limit cycles for the model. Our results, examples, and simulations indicate that dynamic behavior of the model very much depends on the prey refuge parameter m andincreasingthe amount of refuge could increase prey densities and lead to population outbreas. 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: 26 May 2012 Accepted: 21 June 2012 Published: 2 July 2012 References 1. Holling, CS: Some characteristics of simple 240 types of pretion and parasitism. Can. Entomol. 91, (1959) 2. Hassell, MP: The Dynamics of Arthropod Pretor-Prey Systems. Princeton University Press, Princeton (1978). Hoy, MA: Almonds (California). In: Helle, W, Sabelis, MW (eds.) Spider Mites: Their Biology, Natural Enemies and Control. World Crop Pests, vol. 1B. Elsevier, Amsterm (1985) 4. Sih, A: Prey refuges and pretor-prey stability. Theor. Popul. Biol. 1, 1-12 (1987) 5. Krivan, V: Effects of optimal antipretor behavior of prey on pretor-prey dynamics: the role of refuges. Theor. Popul. Biol. 5, (1998) 6. Kar, TK: Stability analysis of a prey-pretor model incorporating a prey refuge. Commun. Nonlinear Sci. Numer. Simul. 10, (2005) 7. Ko, W, Ryu, K: Qualitative analysis of a pretor-prey model with Holling type II functional response incorporating a prey refuge. J. Differ. Equ. 21, (2006) 8. Collings, JB: Bifurcation and stability analysis of a temperature-dependent mite pretor-prey interaction model incorporating a prey refuge. Bull. Math. Biol. 57, 6-76 (1995) 9. Huang, Y, Chen, F, Zhong, L: Stability analysis of a prey-pretor model with Holling type III response function incorporating a prey refuge. Appl. Math. Comput. 182, (2006) 10. Ji, L, Wu, C: Qualitative analysis of a pretor-prey model with constant-rate prey refuge. Nonlinear Anal., Real World Appl. 11, (2010)

14 Wang and Pan Advances in Difference Equations 2012, 2012:96 Page 14 of McNair, JN: Stability effects of prey refuges with entry-exit dynamics. J. Theor. Biol. 125, (1987) 12. McNair, JN: The effects of refuges on pretor-prey interactions: a reconsideration. Theor. Popul. Biol. 29,8-6 (1986) 1. Zhang, ZF, Ding, TR, Huang, WZ, Dong, ZX: Qualitative Theory of Differential Equations, 1st edn. Science Publishing Company, Beijing (1985) 14. Lou, DJ, Zhxng, X, Dong, ZX: Qualitative and Branch Theory of the Dynamic System. Science Publishing Company, Beijing (1999) doi: / Cite this article as: Wang and Pan: Qualitative analysis of a harvested pretor-prey system with Holling-type III functional response incorporating a prey refuge. Advances in Difference Equations :96.