DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 393729 10.1155/2013/393729 393729 Research Article Permanence and Global Attractivity of the Discrete Predator-Prey System with Hassell-Varley-Holling III Type Functional Response Wu Runxin Li Lin De la Sen M. Department of Mathematics and Physics Fujian University of Technology Fuzhou, Fujian 350108 China fjut.edu.cn 2013 27 3 2013 2013 06 11 2012 18 02 2013 19 02 2013 2013 Copyright © 2013 Runxin Wu and Lin Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

By constructing a suitable Lyapunov function and using the comparison theorem of difference equation, sufficient conditions which ensure the permanence and global attractivity of the discrete predator-prey system with Hassell-Varley-Holling III type functional response are obtained. An example together with its numerical simulation shows that the main results are verifiable.

1. Introduction

Recently, there were many works on predator-prey system done by scholars . In particular, since Hassell-Varley  proposed a general predator-prey model with Hassell-Varley type functional response in 1969, many excellent works have been conducted for the Hassell-Varley type system [1, 713].

Liu and Huang  studied the following discrete predator-prey system with Hassell-Varley-Holling III type functional response: (1)x(k+1)=x(k)exp{a(k)-b(k)x(k)-A(k)x(k)y(k)r(k)x2(k)+y2R(k)},y(k+1)=y(k)exp{-d(k)+B(k)x2(k)r(k)x2(k)+y2R(k)},R(0,1), where x(k), y(k) denote the density of prey and predator species at the kth generation, respectively. a, b, A, r, d, B are all periodic positive sequences with common period X. Here a(k) represents the intrinsic growth rate of prey species at the kth generation, and b(k) measures the intraspecific effects of the kth generation of prey species on their own population; d(k) is the death rate of the predator; A(k) is the capturing rate; B(k) is the maximal growth rate of the predator. Liu and Huang obtained the necessary and sufficient conditions for the existences of positive periodic solutions by applying a new estimation technique of solutions and the invariance property of homotopy. As we know, the persistent property is one of the most important topics in the study of population dynamics. For more papers on permanence and extinction of population dynamics, one could refer to [25, 1417] and the references cited therein. The purpose of this paper is to investigate permanence and global attractivity of this system.

We argue that a general nonautonomous nonperiodic system is more appropriate, and thus, we assume that the coefficients of system (1) satisfy the following:

(A) a, b, A, r, d, B are nonnegative sequences bounded above and below by positive constants.

By the biological meaning, we consider (1) together with the following initial conditions as (2)x(0)>0,y(0)>0.

For the rest of the paper, we use the following notations: for any bounded sequence {h(k)}, set hu=supkN{h(k)} and hl=infkN{h(k)}.

2. Permanence

Now, let us state several lemmas which will be useful to prove our main conclusion.

Definition 1 (see [<xref ref-type="bibr" rid="B16">5</xref>]).

System (1) said to be permanent if there exist positive constants m and M, which are independent of the solution of system (1), such that for any positive solution {x(k),y(k)} of system (1) satisfies (3)mliminfk+{x(k),y(k)}limsupk+{x(k),y(k)}M.

Lemma 2 (see [<xref ref-type="bibr" rid="B9">14</xref>]).

Assume that {x(k)} satisfies x(k)>0 and (4)x(k+1)x(k)exp{a(k)-b(k)x(k)}, for kN, where a(k) and b(k) are all nonnegative sequences bounded above and below by positive constants. Then, (5)limsupk+x(k)1blexp(au-1).

Lemma 3 (see [<xref ref-type="bibr" rid="B9">14</xref>]).

Assume that {x(k)} satisfies (6)x(k+1)x(k)exp{a(k)-b(k)x(k)},kN0,limsupk+x(k)x*, and x(N0)>0, where a(k) and b(k) are all nonnegative sequences bounded above and below by positive constants and N0N. Then, (7)liminfk+x(k)min{albuexp{al-bux*},albu}.

Theorem 4.

Assume that (H1)al-AuM21-R2rl>0,(H2)Bl-duru>0 hold, then system (1) is permanent, that is, for any positive solution {x(k),y(k)} of system (1), one has (8)m1liminfk+x(k)limsupk+x(k)M1,m2liminfk+y(k)limsupk+y(k)M2, where (9)m1=al-(AuM21-R/2rl)bu×exp{al-AuM21-R2rl-buM1},m2=min{{(Bl-rudu)m12du}1/2R,{(Bl-rudu)m12du}1/2R×exp{-du+Blm12rum12+M22R}{{(Bl-rudu)m12du}1/2R,{(Bl-rudu)m12du}1/2R}},M1=1blexp(au-1),M2={BuM12dl}1/2Rexp{-dl+Burl}.

Proof.

We divided the proof into four steps.

Step 1. We show (10)limsupk+x(k)M1. From the first equation of (1), we have (11)x(k+1)x(k)exp{a(k)-b(k)x(k)}. By Lemma 2, we have (12)limsupk+x(k)1blexp(au-1)=M1. Previous inequality shows that for any ε>0, there exists a k1>0, such that (13)x(k)M1+ε,kk1.

Step 2. We prove limsupk+y(k)M2 by distinguishing two cases.

Case 1. There exists a l0k1, such that y(l0+1)y(l0).

By the second equation of system (1), we have (14)-d(l0)+B(l0)x2(l0)r(l0)x2(l0)+y2R(l0)0, which implies (15)-d(l0)+B(l0)x2(l0)y2R(l0)0. The previous inequality combined with (13) leads to y(l0){Bu(M1+ε)2/dl}1/2R. Thus, from the second equation of system (1), again we have (16)y(l0+1)=y(l0)exp{-d(l0)+B(l0)x2(l0)r(l0)x2(l0)+y2R(l0)}{Bu(M1+ε)2dl}1/2Rexp{-dl+Burl}=defM2ε. We claim that (17)y(k)M2εkl0. By a way of contradiction, assume that there exists a p0l0 such that y(p0)>M2ε. Then p0l0+2. Let p0~l0+2 be the smallest integer such that y(p0~)>M2ε. Then y(p0~)>y(p0~-1). The previous argument produces that y(p0~)M2ε, a contradiction. This proves the claim. Therefore, limsupk+y(k)M2ε. Setting ε0 in it leads to limsupk+y(k)M2.

Case 2. Suppose y(k+1)<y(k) for all kk1. Since y(k) is nonincreasing and has a lower bound 0, we know that limk+y(k) exists, denoted by y-, we claim that (18)y-{BuM12dl}1/2R. By a way of contradiction, assume that y->{BuM12/dl}1/2R.

Taking limit in the second equation in system (1) gives (19)limk+{-d(k)+B(k)x2(k)r(k)x2(k)+y2R(k)}=0, however, (20)limk+{-d(k)+B(k)x2(k)r(k)x2(k)+y2R(k)}limsupk+{-d(k)+B(k)x2(k)r(k)x2(k)+y2R(k)}-dl+BuM12y-2R<0, which is a contradiction. It implies that y-{BuM12/dl}1/2R. By the fact Bu>dlrl, we obtain that (21)y-{BuM12dl}1/2R{BuM12dl}1/2Rexp{-dl+Burl}=M2. Therefore, we have (22)limk+y(k)=y-M2. Then, (23)limsupk+y(k)M2.

Step 3. We verify (24)liminfk+x(k)m1. Conditions (H1) imply that for enough small positive constant ε, we have (25)al-Au(M2+ε)1-R2rl>0. For the previous ε, it follows from Steps 1 and 2 that there exists a k2 such that for all kk2(26)x(k)M1+ε,y(k)M2+ε. Then, for kk2, it follows from (26) and the first equation of system (1) that (27)x(k+1)x(k)exp{al-Au(M2+ε)1-R2rl-bux(k)}. According to Lemma 3, one has (28)liminfk+x(k)min{m1*,al-Au(M2+ε)1-R/2rlbu}=m1*, where (29)m1*=al-Au(M2+ε)1-R/2rlbu×exp{al-Au(M2+ε)1-R2rl-bu(M1+ε)}.

Setting ε0 in (28) leads to (30)liminfk+x(k)al-AuM21-R/2rlbu×exp{al-AuM21-R2rl-buM1}=m1. By the fact that minxR+{[exp(x-1)]/x}=1, we see that M1=exp(au-1)/blau/blal/bu(al-AuM21-R/2rl)/blm1.

This ends the proof of Step 3.

Step 4. We present two cases to prove that (31)liminfk+y(k)m2. For any small positive constant ε<m1/2, from Step 1 to Step 3, it follows that there exists a k3k2 such that for all kk3(32)x(k)m1-ε,x(k)M1+ε,y(k)M2+ε.

Case 1. There exists a n0k3 such that y(n0+1)y(n0), then (33)-d(n0)+B(n0)x2(n0)r(n0)x2(n0)+y2R(n0)0. Hence, (34)y(n0){(Bl-rudu)(m1-ε)2du}1/2R=defc1ε, and so, (35)y(n0+1){(Bl-rudu)(m1-ε)2du}1/2R×exp{-du+Bl(m1-ε)2ru(m1-ε)2+(M2+ε)2R}=defc2ε. Set (36)m2ε=min{c1ε,c2ε}. We claim that (37)y(k)m2εkn0. By a way of contradiction, assume that there exists a q0n0, such that y(q0)<m2ε. Then q0n0+2. Let q0~n0+2 be the smallest integer such that y(q0~)<m2ε. Then y(q0~)<y(q0~-1), which implies that y(q0)m2ε, a contradiction, this proves the claim. Therefore, liminfk+y(k)m2ε, setting ε0 in it leads to liminfk+y(k)m2.

Case 2. Assume that y(k+1)>y(k) for all kk3, then, limk+y(k) exists, denoted by y_, then limk+y(k)=y_. We claim that (38)y_m2. By a way of contradiction, assume that y_<m2. Taking limit in the second equation in system (1) gives (39)limk+{-d(k)+B(k)x2(k)r(k)x2(k)+y2R(k)}=0, which is a contradiction since (40)limk+{-d(k)+B(k)x2(k)r(k)x2(k)+y2R(k)}liminfk+{-d(k)+B(k)x2(k)r(k)x2(k)+y2R(k)}-du+Blm12rum12+y_2R>0. This proves the claim, then we have (41)limk+y(k)=y_m2. So, (42)liminfk+y(k)m2. Obviously, M2={BuM12/dl}1/2Rexp{-dl+Bu/rl}{(Bl-rudu)m12/du}1/2Rm2. This completes the proof of the theorem.

3. Global Attractivity Definition 5 (see [<xref ref-type="bibr" rid="B18">18</xref>]).

System (1) is said to be globally attractive if any two positive solutions (x1(k),y1(k)) and (x2(k),y2(k)) of system (1) satisfy (43)limk+|x1(k)-x2(k)|=0,limk+|y1(k)-y2(k)|=0.

Theorem 6.

Assume that (H1) and (H2) hold. Assume further that there exist positive constants α, β, and δ such that (H3)αmin{bl,2M1-bu}-αAuM21-R4m2R-αAuM24rlm12-βBuM2R2rlm1R>δ,(H4)βmin{G1,G2,G3,G4}-αAuM14m22R-αAu(M2+ε)R4rlm1(m2+ε)R-αAuR2rlm1max{(M2m2)1-R,(M2m2)R}>δ, where (44)G1=2RBlm12m22R-1(ruM12+M22R)2,G2=2RBlm12M22R-1(ruM12+M22R)2,G3=2M2-2RBuM12M22R-1(rlm12+m22R)2,  G4=2M2-2RBuM12m22R-1(rlm12+m22R)2.

Then, system (1), with initial condition (2), is globally attractive, that is, for any two positive solutions (x1(k),y1(k)) and (x2(k),y2(k)) of system (1), we have (45)limk+|x1(k)-x2(k)|=0,limk+|y1(k)-y2(k)|=0.

Proof.

From conditions (H3) and (H4), there exists an enough small positive constant ε<min{m1/2,m2/2} such that (46)αmin{bl,2M1+ε-bu}-αAu(M2+ε)1-R4(m2-ε)R-αAu(M2+ε)4rl(m1-ε)2-βBu(M2+ε)R2rl(m2-ε)R(m1-ε)>δ,βmin{G1ε,G2ε,G3ε,G4ε}-αAu(M1+ε)4(m2-ε)2R-αAu4rl(m1-ε)(M2+εm2-ε)R-αAuR2rl(m1-ε)×max{(M2+εm2-ε)1-R,(M2+εm2-ε)R}>δ, where (47)G1ε=2RBl(m1-ε)2(m2-ε)2R-1[ru(M1+ε)2+(M2+ε)2R]2,    G2ε=2RBl(m1-ε)2(M2+ε)2R-1[ru(M1+ε)2+(M2+ε)2R]2,G3ε=2M2+ε-2RBu(M1+ε)2(M2+ε)2R-1[rl(m1-ε)2+(m2-ε)2R]2,G4ε=2M2+ε-2RBu(M1+ε)2(m2-ε)2R-1[rl(m1-ε)2+(m2-ε)2R]2.

Since (H1) and (H2) hold, for any positive solutions (x1(k),y1(k)) and (x2(k),y2(k)) of system (1), it follows from Theorem 4 that (48)m1liminfk+xi(k)limsupk+xi(k)M1,m2liminfk+yi(k)limsupk+yi(k)M2,i=1,2. For the previous ε and (48), there exists a k4>0 such that for all k>k4, (49)m1-εxi(k)M1+ε,m2-εxi(k)M2+ε,i=1,2. Let (50)V1(k)=|lnx1(k)-lnx2(k)|. Then from the first equation of system (1), we have (51)V1(k+1)=|lnx1(k+1)-lnx2(k+1)||lnx1(k)-lnx2(k)-b(k)(x1(k)-x2(k))|+A(k)|x1(k)y1(k)r(k)x12(k)+y12R(k)-x2(k)y2(k)r(k)x22(k)+y22R(k)|. Using the mean value theorem, we get (52)x1(k)-x2(k)=exp(lnx1(k))-exp(lnx2(k))=ξ1(k)(lnx1(k)-lnx2(k)),y12R(k)-y22R(k)=2Rξ22R-1(k)(y1(k)-y2(k)), where ξ1(k) lies between x1(k) and x2(k), ξ2(k) lies between y1(k) and y2(k).

It follows from (51) and (52) that (53)V1(k+1)|lnx1(k)-lnx2(k)|-(1ξ1(k)-|1ξ1(k)-b(k)|)|x1(k)-x2(k)|+|A(k)r(k)x1(k)x2(k)y1(k)(r(k)x12(k)+y12R(k))(r(k)x22(k)+y22R(k))|×|x1(k)-x2(k)|+|A(k)y12R(k)y2(k)(r(k)x12(k)+y12R(k))(r(k)x22(k)+y22R(k))|×|x1(k)-x2(k)|+|A(k)r(k)x12(k)x2(k)(r(k)x12(k)+y12R(k))(r(k)x22(k)+y22R(k))|×|y1(k)-y2(k)|+|A(k)x1(k)y12R(k)(r(k)x12(k)+y12R(k))(r(k)x22(k)+y22R(k))|×|y1(k)-y2(k)|+|A(k)x1(k)y1(k)(r(k)x12(k)+y12R(k))(r(k)x22(k)+y22R(k))×2Rξ22R-1(k){A(k)x1(k)y1(k)(r(k)x12(k)+y12R(k))(r(k)x22(k)+y22R(k))}||y1(k)-y2(k)|. And so, for k>k4, (54)ΔV1-min{bl,2M1+ε-bu}|x1(k)-x2(k)|+Au(M2+ε)1-R4(m2-ε)R|x1(k)-x2(k)|+Au(M1+ε)4rl(m1-ε)2|x1(k)-x2(k)|+Au(M1+ε)4(m2-ε)2R|y1(k)-y2(k)|+Au4rl(m1-ε)(M2+εm2-ε)R|y1(k)-y2(k)|+RAu2rl(m1-ε)max{(M2+εm2-ε)1-R,(M2+εm2-ε)R}×|y1(k)-y2(k)|. Let (55)V2(k)=|lny1(k)-lny2(k)|. Then, from the second equation of system (1), we have (56)V2(k+1)=|lny1(k+1)-lny2(k+1)|=|{×(x12(k)r(k)x12(k)+y12R(k)-x22(k)r(k)x22(k)+y22R(k))}lny1(k)-lny2(k)+B(k)×(x12(k)r(k)x12(k)+y12R(k)-x22(k)r(k)x22(k)+y22R(k))||{-B(k)x22(k)(y12R(k)-y22R(k))(r(k)x12(k)+y12R(k))(r(k)x22(k)+y22R(k))}lny1(k)-lny2(k)-B(k)x22(k)(y12R(k)-y22R(k))(r(k)x12(k)+y12R(k))(r(k)x22(k)+y22R(k))|+|B(k)y22R(k)(x1(k)+x2(k))(r(k)x12(k)+y12R(k))(r(k)x22(k)+y22R(k))|×|x1(k)-x2(k)|. Using the mean value theorem, we get (57)y1(k)-y2(k)=exp(lny1(k))-exp(lny2(k))=ξ3(k)(lny1(k)-lny2(n)),y12R(k)-y22R(k)=2Rξ22R-1(k)(y1(k)-y2(k)), where ξ3(k),  ξ2(k) lies between y1(k) and y2(k), respectively. Then, it follows from (56) and (57) that for k>k4, (58)ΔV2-({-B(k)x22(k)2Rξ22R-1(k)(r(k)x12(k)+y12R(k))(r(k)x22(k)+y22R(k))|}1ξ3(k)-|{B(k)x22(k)2Rξ22R-1(k)(r(k)x12(k)+y12R(k))(r(k)x22(k)+y22R(k))}1ξ3(k)-B(k)x22(k)2Rξ22R-1(k)(r(k)x12(k)+y12R(k))(r(k)x22(k)+y22R(k))|)×|y1(k)-y2(k)|+B(k)y22R(k)(x1(k)+x2(k))(r(k)x12(k)+y12R(k))(r(k)x22(k)+y22R(k))×|x1(k)-x2(k)|-min{G1ε,G2ε,G3ε,G4ε}×|y1(k)-y2(k)|+Bu(M2+ε)R2rl(m1-ε)(m2-ε)R|x1(k)-x2(k)|.

Now, we define a Lyapunov function as follows: (59)V(k)=αV1(k)+βV2(k). Calculating the difference of V along the solution of system (1), for k>k4, it follows from (54) and (58) that (60)ΔV-[{βBu(M2+ε)R2rl(m2-ε)R(m1-ε)}αmin{bl,2M1+ε-bu}-αAu(M2+ε)1-R4(m2-ε)R-αAu(M2+ε)4rl(m1-ε)2-βBu(M2+ε)R2rl(m2-ε)R(m1-ε)]×|x1(k)-x2(k)|-[βmin{G1ε,G2ε,G3ε,G4ε}-αAu(M1+ε)4(m2-ε)2R-αAu(M1+ε)(M2+ε)R4rl(m1-ε)(m2-ε)R-αAu2R2rl(m1-ε)×max{(M2+εm2-ε)1-R,(M2+εm2-ε)R}{{{min{G1ε,G2ε,G3ε,G4ε}-αAu(M1+ε)4(m2-ε)2R}}}]×|y1(k)-y2(k)|-δ(|x1(k)-x2(k)|+|y1(k)-y2(k)|). Summating both sides of the previous inequalities from k4 to k, we have (61)p=k4k(V(p+1)-v(p))-δp=k4k(|x1(p)-x2(p)|+|y1(p)-y2(p)|), which implies (62)V(k+1)+δp=k4k(|x1(p)-x2(p)|+|y1(p)-y2(p)|)V(k4). It follows that (63)p=k4k(|x1(p)-x2(p)|+|y1(p)-y2(p)|)V(k4)δ. Using the fundamental theorem of positive series, there exists small enough positive constant ε>0 such that (64)p=k4+(|x1(p)-x2(p)|+|y1(p)-y2(p)|)V(k4)δ+ε, which implies that (65)limk+(|x1(k)-x2(k)|+|y1(k)-y2(k)|)=0, that is, (66)limk+|x1(k)-x2(k)|=0,limk+|y1(k)-y2(k)|=0. This completes the proof of Theorem 6.

4. Extinction of the Predator Species

This section is devoted to study the extinction of the predator species y.

Theorem 7.

Assume that (H5)-dl+Burl<0. Then, the species y will be driven to extinction, and the species x is permanent, that is, for any positive solution (x(k),y(k)) of system (1), (67)limk+y(k)=0,m*liminfk+x(k)limsupk+x(k)M1, where (68)m*=albuexp{al-buM1},M1=1blexp(au-1).

Proof.

For condition (H5), there exists small enough positive γ>0, such that (69)-dl+Burl<0, for all kN, from (69) and the second equation of the system (1), one can easily obtain that (70)y(k+1)=y(k)exp{-d(k)+B(k)x2(k)r(k)x2(k)+y2R(k)}<y(k)exp{-dl+Burl}<y(k)exp{-γ}. Therefore, (71)y(k+1)<y(0)exp{-kγ}, which yields (72)limk+y(k)=0. From the proof of Theorem 4, we have (73)limsupk+x(k)M1. For enough small positive constant ε>0, (74)al-Auε1-R/2rlbu>0. For the previous ε, from (72) and (73) there exists a k5>0 such that for all k>k5, (75)x(k)<M1+ε,y(k)<ε. From the first equation of (1), we have (76)x(k+1)x(k)exp{al-al-Auε1-R/2rlbu-bux(k)}. By Lemma 3, we have (77)liminfk+x(k)al-Auε1-R/2rlbu×exp{al-Auε1-R2rl-bu(M1+ε)}. Setting ε0 in (72) leads to (78)liminfk+x(k)albuexp{al-buM1}=defm*. The proof of Theorem 7 is completed.

5. Example

The following example shows the feasibility of the main results.

Example 8.

Consider the following system: (79)x(k+1)=x(k)exp{{1.7x(k)y(k)0.3x(k)2+y(k)}{1.7x(k)y(k)0.3x(k)2+y(k)}0.85+0.05cos(k)-2.4x(k)-1.7x(k)y(k)0.3x(k)2+y(k)},y(k+1)=y(k)exp{-4.1+1.6x(k)20.3x(k)2+y(k)}.

One could easily see that (H6)al-AuM21-R2rl=0.1228>0,(H7)Bl-duru=0.37>0. Clearly, conditions (H6) and (H7) are satisfied. It follows from Theorem 4 that the system is permanent. Numerical simulation from Figure 1 shows that solutions do converge and system is permanent and globally attractive.

Dynamics behaviors of system (1) with initial conditions (x(0),y(0))=(0.3,0.3),(0.4,0.2),(0.2,0.4), respectively.

6. Conclusion

In this paper, a discrete predator-prey model with Hassell-Varley-Holling III type functional response is discussed. The main topics are focused on permanence, global attractivity, and extinction of predator species. The numerical simulation shows that the main results are verifiable.

The investigation in this paper suggests the following biological implications. Theorem 4 shows that the coefficients, such as the death rate of the predator, the capturing rate, and the intraspecific effects of prey species, influence permanence. Conditions (H1) and (H2) imply that the higher the intraspecific effects of prey species are, the more favourable permanence is. Those results have further application on predator-prey population dynamics. However, the conditions for global attractivity in Theorem 4 is so complicated that its application is very difficult. A further study is required to simplify the application.

Acknowledgment

This work is supported by the Foundation of Fujian Education Bureau (JA11193).

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