Research Article Dynamic Behaviors of a Discrete Periodic Predator-Prey-Mutualist System

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1 Discrete Dynamics in Nature and Society Voume 015, Artice ID 4769, 11 pages Research Artice Dynamic Behaviors of a Discrete Periodic Predator-Prey-Mutuaist System Liya Yang, 1 Xiangdong Xie, and Fengde Chen 1 Coege of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian , China Department of Mathematics, Ningde Norma University, Ningde, Fujian 35300, China Correspondence shoud be addressed to Xiangdong Xie; ndsyxxd@163.com and Fengde Chen; fdchen@63.net Received 3 June 015; Accepted 1 September 015 Academic Editor: Pave Rehak Copyright 015 Liya Yang et a. This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited. A nonautonomous discrete predator-prey-mutuaist system is proposed and studied in this paper. Sufficient conditions which ensure the permanence and existence of a unique gobay stabe periodic soution are obtained. We aso investigate the extinction property of the predator species; our resuts indicate that if the cooperative effect between the prey and mutuaist species is arge enough, then the predator species wi be driven to extinction due to the ack of enough food. Two exampes together with numerica simuations show the feasibiity of the main resuts. 1. Introduction As was pointed out by Berryman [1], the dynamic reationship between predator and prey has ong been and wi continue to be one of the dominant themes in both ecoogy and mathematica ecoogy due to its universa existence and importance. Recenty, predator-prey modes have been studied widey [ 7]. It brings to our attention that a the works of [ 7] are deaing with the reationship between two species, whie, in the rea word, the reationship among species is very compicated and it needs to consider the three-species modes. Many schoars [8 13] studied the dynamic behaviors of the three-species modes. Moreover, mutuaism is one of the most important reationships in the theory of ecoogy. Mutuaism is a symbiotic association between any two species and the interaction between the two species is beneficia to both of the species [14]. Aready, many schoars [15 1] studied the dynamic behaviors of cooperative modes. It brings to our attention that athough predator-prey and mutuaism can be recognized as major issues in both appied mathematics and theoretica ecoogy, few schoars have considered predator-prey system with cooperation in three species. But this phenomenon reay exists in nature. For exampe, whie aphids are preyed by natura enemies, they are protected by some natura friends ike ants; there ants eat the honeydew that aphids excrete and hep to overcome the resource scarcity of offspring [, 3]. In 009, Rai and Krawcewicz [4] proposed the foowing predator-prey-mutuaist system: dx dt =αx(1 x K ) βxz 1+my, dy dt =γy(1 y ), x + L 0 dz dt =z(s+ cβx 1+my ), where x(t), y(t), and z(t) denote the densities of prey, mutuaist, and predator popuation at any time t, respectivey; they appied the equivariant degree method to study Hopf bifurcations phenomenon of the system. Recenty, Yang et a. [5] argued that, due to seasona effects of weather, temperature, food suppy, mating habits, (1)

2 Discrete Dynamics in Nature and Society and so forth, a more appropriate system shoud be a nonautonomous one, and they proposed and studied the foowing system: c x=x(a 1 (t) b 1 (t) x 1 (t) z d 1 (t) +d (t) y ), y y=y(a (t) d 3 (t) +d 4 (t) x ), z=z(a 3 (t) + k 1 (t) c 1 (t) x d 1 (t) +d (t) y b (t) z). By using the Brouwer fixed pointed theorem and constructing a suitabe Lyapunov function, the authors obtained a set of sufficient conditions for the existence of a gobay asymptoticay stabe periodic soution in system (). It is we known that the discrete time modes are more appropriate than the continuous ones when the size of the popuation is rarey sma or the popuation has nonoverapping generations. It has been found that the dynamic behaviors of the discrete system are rather compex and contain more rich dynamics than the continuous ones. To the best of the authors knowedge, sti no schoar proposes and studies the discrete predator-prey-mutuaist system; this motivated us to study the foowing system: x 1 (n+1) =x 1 (n) exp {a 1 (n) b 1 (n) x 1 (n) c 1 (n) x 3 (n) d 1 (n) +d (n) x (n) }, x (n+1) =x (n) exp {a (n) x (n) d 3 (n) +d 4 (n) x 1 (n) }, x 3 (n+1) =x 3 (n) exp {a 3 (n) + k 1 (n) c 1 (n) x 1 (n) d 1 (n) +d (n) x (n) b (n) x 3 (n)}, where x 1 (n), x (n), andx 3 (n) are the popuation sizes of the prey, mutuaist, and predator at nth generation, respectivey, a 1 (n) and a (n) are the intrinsic growth rate of prey and mutuaist at nth generation, a 3 (n) is the death rate of the predator at nth generation, k 1 (n) is caed the conversion rate at nth generation, which denotes the fraction of the prey biomass being converted to predator biomass, and c 1 (n) is the capture rate of the prey at nth generation. The sequences of d 4 (n), d (n) arethemutuaismsequences.wemention here that, in system (3), we consider the density restriction term of predator species (b (n)z); such a consideration is needed since the density of any species is restricted by the environment [10]. Here, we assume that a i (n) (i = 1,, 3), b j (n), c j (n) (j = 1, ) k 1 (n), andd r (n) (r = 1,, 3, 4) are a () (3) bounded nonnegative sequences. a i (n) (i = 1,, 3), b j (n) (j = 1, ) are stricty positive sequences. Note that n1 x 1 (n) =x 1 (0) exp k=1 c 1 (k) x 3 (k) d 1 (k) +d (k) x (k) ], n1 x (n) =x (0) exp k=1 n1 x 3 (n) =x 3 (0) exp b (k) x 3 (k)]. k=1 [a 1 (k) b 1 (k) x 1 (k) [a (k) [a 3 (k) + x (k) d 3 (k) +d 4 (k) x 1 (k) ], k 1 (k) c 1 (k) x 1 (k) d 1 (k) +d (k) x (k) From the point of view of bioogy, in the sequence, we assume that x 1 (0) > 0, x (0) > 0, x 3 (0) > 0, andthenfrom (4), we know that the soutions of system (3) are positive. We use the foowing notations for any bounded sequence x(n): (4) x u = sup x (n), n N (5) x = inf x (n). n N Wearrangetherestofthepaperasfoows.InSection, we estabish a permanence resut for (3). In Section 3, the sufficient conditions about the uniqueness and goba attractivity of the periodic soution of (3) are obtained. In Section 4, the sufficient conditions about the extinction of predator species and the stabiity of prey-mutuaist species are obtained. Finay, two suitabe exampes are given to iustrate that the conditions of the main theorem are feasibe. We end this paper by a brief discussion.. Permanence Theorem 1. Assume the inequaities k u 1 cu 1 B 1/d 1 a 3 > 0, and every positive soution (x 1 (n), x (n), x 3 (n)) of system (3) satisfies where im sup x 1 (n) B 1, n im sup x (n) B, n im sup x 3 (n) B 3, n B 1 = exp {au 1 1}, b1 B =(d u 3 +du 4 B 1) exp {a u 1}, B 3 = 1 b exp {a 3 + ku 1 cu 1 B 1 d 1 1}. (6) (7)

3 Discrete Dynamics in Nature and Society 3 Proof. Since x 1 (0) > 0, x (0) > 0,andx 3 (0) > 0,thenx 1 (n) > 0, x (n) > 0,andx 3 (n) > 0,forn 0.Weonyneedtoprove that im sup x 1 (n) B 1. (8) n Sincesimiarresutcanbeshownforx (n) and x 3 (n), then (6) foows obviousy. We first assume that there exists 0 N such that x 1 ( 0 +1) x 1 ( 0 ).Then c a 1 ( 0 )b 1 ( 0 )x 1 ( 0 ) 1 ( 0 )x 3 ( 0 ) 0. (9) d 1 ( 0 )+d ( 0 )x ( 0 ) Hence, x 1 ( 0 ) a 1 ( 0 ) b 1 ( 0 ) au 1. b1 (10) It foows that x 1 ( 0 +1)=x 1 ( 0 ) exp {a 1 ( 0 )b 1 ( 0 )x 1 ( 0 ) here we used c 1 ( 0 )x 3 ( 0 ) d 1 ( 0 )+d ( 0 )x ( 0 ) } x 1 ( 0 ) exp {a 1 ( 0 ) b 1 ( 0 )x 1 ( 0 )} x 1 ( 0 ) exp {a u 1 b 1 x 1 ( 0 )} max x R exp {au 1 1} b 1 =B 1 ; (11) exp (a1) x exp (abx) =, for a, b > 0. (1) b We caim that x 1 (n) B 1, n 0. (13) By way of contradiction, assume that there exists p 0 > 0 such that x 1 (p 0 )>B 1.Thenp Let p 0 0 +be the smaest integer such that x 1 ( p 0 )>B 1.Thenx 1 ( p 0 )>B 1 x 1 ( p 0 1), which impies x 1 ( p 0 )>x 1 ( p 0 1).Theaboveargument produces that x 1 ( p 0 ) B 1, a contradiction. This proves the caim. Now, we assume that x(n + 1) < x(n) for a n N.In particuar, im n x(n) exists, denoted by x 1.Wecaimthat x 1 a u 1 /b 1. By way of contradiction, assume that x 1 >a u 1 /b 1. Taking imit in the first equation in system (3) gives im (a c n 1 (n) b 1 (n) x 1 (n) 1 (n) x 3 (n) d 1 (n) +d (n) x (n) ) (14) =0, which is a contradiction since a 1 (n) b 1 (n) x 1 (n) c 1 (n) x 3 (n) d 1 (n) +d (n) x (n) a 1 (n) b 1 (n) x 1 (n) a u 1 b 1 x 1 <0 for n N. (15) This proves the caim. Note that exp (x1) >x (x >0). exp (a u 1 1) b 1 = au 1 exp (a u 1 1) b1 a1 u > au 1. b1 (16) It foows that (8) hods. This competes the proof of the main resut. Theorem. Assume the inequaities k 1 c 1 D 1 d1 u +du B a u 3 >0, a 1 c u 1 B 3 d1 +d D >0, where B and B 3 are the same as in Theorem 1. Then where im inf x n 1 (n) D 1, im inf x n (n) D, im inf x n 3 (n) D 3, D 1 = a 1 (d 1 +d D )c u 1 B 3 b u 1 (d 1 +d D ) exp {a 1 bu 1 B c u 1 1 B 3 d1 +d D }, D =a d 3 exp {a B }, d3 D 3 = k 1 c 1 D 1 a u 3 (du 1 +du B ) (d u 1 +du B )b u Proof. We first show that exp {a u 3 + k 1 c 1 D 1 d1 u +du B b u B 3}. (H 1 ) (17) (18) im inf n x (n) D. (19) For any ε>0, there exists n Nsuch that x 1 (n) B 1 +ε, x (n) B +ε, x 3 (n) B 3 +ε, for n n. (0)

4 4 Discrete Dynamics in Nature and Society First, we assume that there exists 0 n such that x ( 0 +1) x ( 0 ).Notethat,forn 0, x (n+1) =x (n) exp {a (n) x (n) exp {a (n) x (n) d 3 (n) } x (n) exp {a x (n) }. d3 In particuar, with n= 0,weget a x ( 0 ) d 3 which impies that x ( 0 ) a d 3.Then Let x ( 0 +1) =x ( 0 ) exp {a ( 0 ) x ( 0 ) exp {a B +ε } d3 a d 3 exp {a B +ε }. d3 We caim that x ε def x (n) d 3 (n) +d 4 (n) x 1 (n) } (1) 0, () x ( 0 ) d 3 ( 0 )+d 4 ( 0 )x 1 ( 0 ) } (3) =a d 3 exp {a B +ε }. (4) d 3 x (n) x ε for n 0. (5) By way of contradiction, assume that there exists p 0 > 0 such that x (p 0 )<x ε.thenp Let p 0 0 +be the smaest integer such that x ( p 0 )<x ε.thenx ( p 0 1) x ε > x ( p 0 ),andcearyx ( p 0 1) > x ( p 0 ).Theabove argument produces that x ( p 0 ) x ε for a arge n. Then im n x (n) exists, denoted by x.wecaimthatx a d 3. By way of contradiction, assume that x <a d 3. Taking imit inthesecondequationinsystem(3)gives im (a x n (n) (n) )=0, (6) d 3 (n) +d 4 (n) x 1 (n) which is a contradiction since im (a x n (n) (n) d 3 (n) +d 4 (n) x 1 (n) ) a x d3 >0. (7) This proves the caim. Note that B d3 = du 3 exp {au 1}+du 4 exp {au 1 +au }/b 1 d 3 exp {a u 1}+du 4 exp {au 1 +au } b 1 d 3 a u + du 4 exp {au 1 +au }. b1 d 3 (8) Ceary, a B /d 3 <0,soD <a d 3.Wecaneasiyseethat (19) hods. The proof of the other two inequaities is simiar to the above anaysis and we omit the detai here. This competes the proof of the main resut. As a direct coroary of Theorems 1 and, from the definition of permanence, we have the foowing. Theorem 3. Assume that (H 1 ) hods. Then system (3) is permanent. It shoud be noticed that, from the inequaity k 1 c 1 D 1/(d u 1 + d u B )a u 3 > 0 and from the proofs of Theorems 1 and, one knows that where (H 1 ) hods, the set [D 1,B 1 ] [D,B ] [D 3,B 3 ] is an invariant set of system (3). 3. Existence and Stabiity of a Periodic Soution Duetoseasonaeffectsofweather,temperature,foodsuppy, mating habits, contact with predators, and other resources or physica environmenta quantities, we can assume tempora to be cycic or periodic [6 8]. In this section, we consider system (3) with a i (n) (i = 1,, 3), b j (n), c j (n) (j = 1, ), k 1 (n),andd r (n) (r = 1,, 3, 4) being periodic with a common period. More precisey, we assume that there exists a positive integer w such that, for n N, 0<a i (n+w) =a i (n), 0<b j (n+w) =b j (n), 0<c j (n+w) =c j (n), 0<d k (n+w) =d k (n), 0<k 1 (n+w) =k 1 (n). (9) Let B i, D i, i = 1,, 3,bethesameasinTheorems1and.Our first resut concerns the existence of a periodic soution. Theorem 4. Assume that (H 1 ) hods; then system (3) has wperiodic soution, denoted by (x 1 (n), x (n), x 3 (n)). Proof. As noted at the end of the ast section that [D 1,B 1 ] [D,B ] [D 3,B 3 ] is an invariant set of system (3), thus we can define a mapping F on [D 1,B 1 ] [D,B ] [D 3,B 3 ] by F(x 1 (0), x (0), x 3 (0)) =(x 1 (w), x (w), x 3 (w)) (30) for(x 1 (0), x (0), x 3 (0)) [D 1,B 1 ] [D,B ] [D 3,B 3 ].

5 Discrete Dynamics in Nature and Society 5 Obviousy, F depends continuousy on (x 1 (0), x (0), x 3 (0)). Thus, F is continuous and maps the compact set [D 1,B 1 ] [D,B ] [D 3,B 3 ] into itsef. Therefore, F has a fixed point (x 1 (n), x (n), x 3 (n)). It is easy to see that the soution (x 1 (n), x (n), x 3 (n)) is w-periodic soution of system (3). This competes the proof. Now, under some additiona conditions, we study the goba stabiity of the periodic soution obtained in Theorem 4. Theorem 5. Assume that (9) and (H 1 ) hod, and V (n+1) = V (n) + x (n) [1 exp {V (n)}] d 3 (n) +d 4 (n) x 1 (n) x (n) d 4 (n) x 1 (n) [1 exp {u (n)}] (d 3 (n) +d 4 (n) x 1 (n))(d 3 (n) +d 4 (n) x 1 (n)), z (n+1) =z(n) k 1 (n) c 1 (n) x 1 (n) [1 exp {u (n)}] d 1 (n) +d (n) x (n) +b (n) x 3 (n) [1 exp {z (n)}] λ 1 = max { 1bu 1 B 1, 1b 1 D 1 }+ cu 1 B 3d u B (d 1 +d D ) c u 1 + B 3 d1 +d D <1, λ = max { 1 + B d3 +d 4 D 1 d u 4 B 1B (d 3 +d 4 D 1) <1,, 1 D d3 u +du 4 B 1 λ 3 = max { 1bu B 3, 1b D 3 }+ cu 1 ku 1 B 1 d 1 +d D } (31) + k 1 (n) c 1 (n) x 1 (n) x (n) d (n) [1 exp {V (n)}]. (d 1 (n) +d (n) x (n))(d 1 (n) +d x (n)) (34) By using the mean-vaue theorem, it foows that u (n+1) =u(n) [1 b 1 (n) x 1 (n) exp {θ 1 (n) u (n)}] c 1 (n) x 3 (n) z (n) exp {θ 3 (n) z (n)} d 1 (n) +d (n) x (n) + c 1 (n) d (n) x (n) x 3 (n) exp {θ (n) V (n)} V (n) (d 1 (n) +d (n) x (n))(d 1 (n) +d (n) x (n)) + cu 1 ku 1 B 1B d u (d 1 +d D ) <1. Then for every soution (x 1 (n), x (n), x 3 (n)) of system (3), one has im (x n 1 (n) x 1 (n)) =0, im (x n (n) x (n)) =0, im (x n 3 (n) x 3 (n)) =0, (3) where (x 1 (n), x (n), x 3 (n)) is w-periodic soution obtained in Theorem 4. Proof. Let x 1 (n) = x 1 (n) exp (u (n)), x (n) = x (n) exp (V (n)), x 3 (n) = x 31 (n) exp (z (n)). Then system (3) is equivaent to u (n+1) =u(n) +b 1 (n) x 1 (n) [1 exp {u (n)}] +c 1 (n) x 3 (n) [1 exp {z (n)}] d 1 (n) +d (n) x (n) c 1 (n) x 3 (n) x (n) d (n) [1 exp {V (n)}] (d 1 (n) +d (n) x (n))(d 1 (n) +d (n) x (n)), (33) V (n+1) = V (n) [1 x (n) exp {θ (n) V (n)} d 3 (n) +d 4 (n) x 1 (n) ] d + 4 (n) x 1 (n) x (n) u (n) exp {θ 1 (n) u (n)} (d 3 (n) +d 4 (n) x 1 (n))(d 3 (n) +d 4 (n) x 1 (n)), z (n+1) =z(n) [1 b (n) x 3 (n) exp {θ 3 (n) z (n)}] + c 1 (n) k 1 (n) x 1 (n) u (n) exp {θ 1 (n) u (n)} d 1 (n) +d (n) x (n) c 1 (n) k 1 (n) d (n) x 1 (n) x (n) exp {θ (n) V (n)} V (n), (d 1 (n) +d (n) x (n))(d 1 (n) +d (n) x (n)) (35) where (θ 1 (n), θ (n), θ 3 (n)) [0, 1]. Tocompetetheproof,it suffices to show that im u (n) = im V (n) = im z (n) =0. n n n (36) In view of (31), we can choose ε>0sma enough such that λ ε 1 = max { 1bu 1 (B 1 +ε), 1b 1 (D 1 ε) } + cu 1 (B 3 +ε)d u (B +ε) c u 1 (d1 +d (D ε)) + (B 3 +ε) d1 +d (D ε) <1 λ ε = max { (B +ε) 1 d3 +d 4 (D, 1 ε)

6 6 Discrete Dynamics in Nature and Society 1 (D ε) d3 u +du 4 (B }+ du 4 (B 1 +ε)(b +ε) 1 +ε) (d3 +d 4 (D 1 ε)) <1 λ ε 3 = max { 1bu (B 3 +ε), 1b (D 3 ε) } + cu 1 ku 1 (B 1 +ε) d 1 +d (D ε) + cu 1 ku 1 (B 1 +ε)(b +ε)d u (d 1 +d (D ε)) for n n 0.Letλ=max{λ ε 1,λε,λε 3 }.Thenλ<1.Inviewof (39), we get This impies max { u (n+1), V (n+1), z (n+1) } λmax { u (n), V (n), z (n) }, n n 0. (40) <1. (37) max { u (n), V (n), z (n) } λ nn 0 max { u(n 0), V (n 0), z(n 0) }, n n 0. (41) According to Theorems 1 and, there exists n 0 Nsuch that D 1 ε x 1 (n) B 1 +ε, D ε x (n) B +ε, D 3 ε x 3 (n) B 3 +ε. D 1 ε x 1 (n) B 1 +ε, D ε x (n) B +ε, D 3 ε x 3 (n) B 3 +ε (38) for n n 0. Notice that θ 1 (n) [0, 1] impies that x 1 (n) exp{θ 1 (n)u(n)} ies between x 1 (n) and x 1 (n). Simiary,x (n) exp{θ (n)v(n)} ies between x (n) and x (n), and x 3 (n)exp{θ 3 (n)z(n)} ies between x 3 (n) and x 3 (n).from(35),weget u (n+1) u (n) max { 1bu 1 (B 1 +ε), 1b 1 (D 1 ε) }+ V (n) cu 1 (B 3 +ε)d u (B +ε) (d 1 +d (D ε)) + z (n) c u 1 (B 3 +ε) d 1 +d (D ε), V (n+1) V (n) max { 1 1 D ε d3 u +du 4 (B } 1 +ε) + u (n) du 4 (B 1 +ε)(b +ε) (d 3 +d 4 (D 1 ε)), B +ε d3 +d 4 (D, 1 ε) z (n+1) z (n) max { 1b (D 3 ε), 1bu (B c 3 +ε) }+ u(n) + V (n) cu 1 ku 1 du (B 1 +ε)(b +ε) (d 1 +d (D ε)) u 1 ku 1 (B 1 +ε) d1 +d (D ε) (39) Therefore (36) hods and the proof is compete. 4. Extinction of Predator Species and Stabiity of Prey-Mutuaist Species In this section, we aso consider system (3) with a i (n) (i = 1,, 3), b j (n), c j (n) (j = 1, ) k 1 (n),andd r (n) (r = 1,, 3, 4) being periodic with a common period w>0.bydeveoping theanaysistechniqueof[9],weshowthat,undersome suitabe assumption, the predator wi be driven to extinction whie prey-mutuaist wi be gobay attractive to a certain soution of a ogistic equation. We consider a discrete ogistic equation x (n+1) =x(n) exp (a 1 (n) b 1 (n) x (n)), n N. (4) Theorem 6. For any positive soution x of (4), one has m 1 im inf n x im sup x B 1, (43) n where m 1 = (a 1 /bu 1 ) exp{a 1 bu 1 B 1} and B 1 is defined by Theorem 1. Furthermore, there exists w-periodic soution for (4). The proof of the above caim foows that of Theorems 1 and with sight modification and we omit the detai here. Theorem 7. Assume that the inequaity b u 1 exp (au 1 1) < (H ) b 1 hods. Let x (n) be a periodic soution of (4). Then, for every positive soution x(n) of (4), one has Proof. Let Then system (4) is equivaent to im n (x (n) x (n)) =0. (44) x (n) =x (n) exp {p (n)}. (45) p (n+1) =p(n) +b 1 (n) (x (n) x(n)) =p(n) b 1 (n) x (n) [exp p (n) 1]. (46)

7 Discrete Dynamics in Nature and Society 7 By using the mean-vaue theorem, it foows that p (n+1) =p(n) [1b 1 (n) x (n) exp {θ 4 (n) p (n)}], (47) where θ 4 (n) [0, 1]. To compete the proof, it suffices to show that we first assume that im n p (n) =0; (48) λ = max { 1bu 1 B 1, 1b 1 m 1 } <1; (49) then we can choose positive constant ε>0sma enough such that λ ε = max { 1bu 1 (B 1 +ε), 1b 1 (m 1 ε) }<1. (50) According to Theorem 6, there exists n Nsuch that m 1 ε x(n), x (n) B 1 +ε, n n. (51) Notice that θ 4 (n) [0, 1] impies that x (n) exp{θ 4 (n)p(n)} ies between x (n) and x(n). From (47), we get p (n+1) p (n) max { 1bu 1 (B 1 +ε), 1b 1 (m 1 ε) } = p (n) λ ε. This impies that (5) p (n) (λ ε )nn p(n ) n n. (53) Since λ ε < 1 and ε is arbitrariy sma, we obtain im n p(n) = 0, and it means that (48) hods when λ <1. Note that thus, λ <1is equivaent to or 1b u 1 B 1 1b 1 m 1 <1; (54) 1b u 1 B 1 >1, (55) b u 1 B 1 = bu 1 exp {a u b1 1 1}<. (56) Now, we can concude that (48) is satisfied as (H ) hods, and so im n (x (n) x (n)) =0. (57) Theorem 8. Assume that the inequaity k u 1 cu 1 B 1 d1 +d D a 3 <0 (H 3) hods, where D and B 1 are defined by Theorems 1 and. Let (x 1 (n), x (n), x 3 (n)) be any positive soution of system (3); then x 3 (n) 0 as n +. Proof. From (H 3 ) we can choose positive constant ε>0sma enough such that inequaity k u 1 cu 1 (B 1 +ε) d1 +d (D ε) a 3 <0 (58) hods. Thus, there exists δ >0, k u 1 cu 1 (B 1 +ε) d1 +d (D ε) a 3 <δ <0. (59) Let (x 1 (n), x (n), x 3 (n)) be any positive soution of system (3). For any q N, according to the equation of system (3), we obtain n x 3 (q + 1) =a x 3 (q) 3 (q) + k 1 (q) c 1 (q) x 1 (q) d 1 (q) + d (q) x (q) b (q) x 3 (q) a 3 (q) + k 1 (q) c 1 (q) x 1 (q) d 1 (q) + d (q) x (q) a 3 + ku 1 cu 1 (B 1 +ε) d 1 +d (D ε) δ <0. (60) Summating both sides of the above inequations from 0 to n1, we obtain n x 3 (n) x 3 (0) <δ n, (61) and then x 3 (n) <x 3 (0) exp (δ n). (6) The above inequaity shows that x 3 (n) 0 exponentiay as n +.ThiscompetestheproofofTheorem8. Theorem 9. Assume (H ), (H 3 ),andb /(d 3 +d 4 D 1)<hod; aso a 1 c u 1 B 3 d1 +d D >0, k u 1 cu 1 B (63) 1 a d1 3 >0. Then for any positive soution (x 1 (n), x (n), x 3 (n)) of system (3), one has im (x n 1 (n) x 1 (n)) =0, (64) im (x n (n) x (n)) =0. x 1 (n) is any positive soution of system (4) and x (n) is any positive soution of the second equation of system (3).

8 8 Discrete Dynamics in Nature and Society Proof. Since (H 3 ) hods, it foows from Theorem 8 that To prove im n (x 1 (n) x 1 (n)) = 0,et im n x 3 (n) =0. (65) x 1 (n) =x 1 (n) exp (u (n)) ; (66) then from the first equation of system (3) and (66), u (n+1) =u(n) b 1 (n) x 1 (n) (exp (u (n) 1)) c 1 (n) x 3 (n) d 1 (n) +d (n) x (n). (67) Using the mean-vaue theorem, one has exp (u (n) 1) = exp (ζ 1 (n) u (n))u(n), ζ 1 (n) (0, 1). Then the first equation of system (3) is equivaent to u (n+1) =u(n) (1 b 1 (n) x 1 (n) exp (ζ 1 (n) u (n))) (68) c 1 (n) x 3 (n) d 1 (n) +d (n) x (n). (69) To compete the proof, it suffices to show that We first assume that im n u (n) =0. (70) λ=max { 1bu 1 B 1, 1b 1 D 1 } <1, (71) andthenwecanchoosepositiveconstantε>0sma enough such that λ ε = max { 1bu 1 (B 1 +ε), 1b 1 (D 1 ε) } <1. (7) For the above ε, according to Theorems 1,, and 8, there exists an integer n 1 Nsuch that D 1 ε x 1 (n) B 1 +ε, m 1 ε x 1 (n) B 1 +ε, x 3 (n) ε, Noting that m 1 D 1,then D 1 ε x 1 (n), x 1 (n) B 1 +ε, x 3 (n) ε, n n 1. n n 1. (73) (74) It foows from (74) that c 1 (n) d 1 (n) +d (n) x (n) c u 1 d1 +d (D ε) M ε, n n 1. (75) Noting that ζ 1 (n) (0, 1),itimpiesthatx 1 (n) exp(ζ 1(n)u(n)) ies between x 1 (n) and x 1(n). From (69), (7) (75), we get u (n+1) u (n) max { 1bu 1 (B 1 +ε), 1b 1 (D 1 ε) } + This impies that c u 1 ε d 1 +d (D ε) = u (n) λ ε +M ε ε, u (n) λ nn 1 ε n n 1. (76) u(n 1) + 1λnn 1 ε M 1λ ε ε, n n 1. (77) ε Since λ ε < 1 and ε is arbitrariy sma, we obtain im n u(n) = 0, and it means that (70) hods when λ<1. Note that 1b u 1 B 1 1b 1 D 1 <1; (78) thus, λ<1is equivaent to 1b u 1 B 1 >1, (79) or b u 1 B 1 = bu 1 exp {a u b1 1 1}<. (80) Now,wecanconcudethat(70)issatisfiedas(H ) hods, and so im (x n 1 (n) x 1 (n)) =0. (81) Next, we prove im (x n (n) x (n)) =0. (8) Let x (n) =x (n) exp (V (n)). (83) If k u 1 cu 1 B 1/d 1 a 3 >0and a 1 cu 1 B 3/(d 1 +d D )>0 hod, from Theorems 1 and, we know that x 1 (n), x (n) are boundedeventuay.fromthesecondinequaityof(39), V (n+1) V (n) max { 1 1 D ε d3 u +du 4 (B } 1 +ε) + u (n) du 4 (B 1 +ε)(b +ε) (d 3 +d 4 (D 1 ε)) B +ε d3 +d 4 (D, 1 ε) n>n 0. (84)

9 Discrete Dynamics in Nature and Society 9 We first assume that ρ=max { 1 B d3 +d 4 D 1, 1 D d3 u +du 4 B 1 } <1. (85) It foows from (70) that im n u(n) = 0. For any positive constant ε>0, there exists integer n max(n 0,n 1 ) such that ρ ε = max { 1 1 (B +ε) d3 +d 4 (D, 1 ε) (D ε) d3 u +du 4 (B }<1, n n 1 +ε) u (n) <ε, n n. (86) x 1,x, and x Time n Let d u 4 (B 1 +ε)(b +ε) (d 3 +d 4 (D 1 ε)) A ε. (87) From (84) (87) we can concude that This impies that V (n) ρ nn ε V (n+1) ρ ε V (n) +εa ε, n n. (88) V (n ) +εa 1ρ nn ε ε, n n 1ρ. (89) ε Since ρ ε < 1 and ε is arbitrariy sma, we obtain im n u(n) = 0.Notethat B D 1 d3 +d 4 D 1 1 d3 u +du 4 B <1; (90) 1 thus ρ ε <1is equivaent to or B 1 d3 +d 4 D >1, (91) 1 B d3 +d 4 D <. (9) 1 Now, we can concude that im n V(n) = 0.Andso We can concude that im n (x (n) x (n)) =0. (93) im (x n 1 (n) x 1 (n)) =0, im (x n (n) x (n)) =0, n n This competes the proof of Theorem 9. (94) x 1 x x 3 Figure 1: Dynamic behavior of the soution (x 1 (n), x (n), x 3 (n)) to system (95) with the initia conditions (x 1 (0), x (0), x 3 (0)) = (0.05, 0.1, 0.), (0.5, 0.8, 0.3) and (0.4, 0., 0.6),respectivey. 5. Exampes and Numeric Simuations In this section, we wi give two exampes to show the feasibiity of our resuts. Exampe 1. Consider the foowing system: x 1 (n+1) =x 1 (n) exp {( cos (n)) x 1 (n) 0.0x 3 (n) 1+x (n) }, x (n+1) =x (n) exp {0.3 x 3 (n+1) =x 3 (n) x (n) 1+0.1x 1 (n) }, exp { x 1 (n) 1+x (n) x 3 (n)}. (95) One coud easiy see that k 1 c 1 D 1/(d u 1 +du B ) a u > 0 and a 1 cu 1 B 3/(d 1 +d D ) > 0, and then condition (H 1 ) is satisfied. According to Theorem 1, system (3) is permanent. Numerica simuation (see Figure 1) indicates the permanence of system (95). Figure 1 shows the dynamic behaviors of system (95), which strongy supports our resuts. Exampe. Consider the foowing system: x 1 (n+1) =x 1 (n) exp {( cos (n)) x 1 (n) 0.0x 3 (n) 1+x (n) },

10 10 Discrete Dynamics in Nature and Society x 1,x, and x Time n x 1 x x 3 Figure : Dynamic behavior of the soution (x 1 (n), x (n), x 3 (n)) to system (96) with the initia conditions (x 1 (0), x (0), x 3 (0)) = (0.05, 0.8, 0.1), (0.5, 1., 0.) and (0.7, 1.4, 0.3),respectivey. x (n+1) =x (n) exp {( sin (n)) x 3 (n+1) =x 3 (n) x (n) +x 1 (n) }, exp { x 1 (n) 1+x (n) 50x 3 (n)}. (96) We coud easiy see that (b u 1 /b 1 ) exp(au 1 1) = 1 <, k u 1 cu 1 B 1/(d 1 +d D )a < 0, B /(d 3 + d 4 D 1) <, a 1 cu 1 B 3/(d 1 +d D ) > 0, and k u 1 cu 1 B 1/d 1 a 3 = > 0. Ceary, conditions of Theorem 9 are satisfied, and so im (x 1 (n) x 1 (n)) = 0, im n (x (n) x (n)) = 0,andim n x 3 (n) = 0. Figure shows the dynamic behaviors of system (96), which strongy supports our resuts. 6. Discussion It is we known that prey-mutuaist system can decrease predation risk; mutuaism pays an important roe in the dynamic behaviors of predator-prey popuations. For system (3), we showed that the predator-prey-mutuaist system wi be coexistent in a gobay stabe state under some suitabe conditions. We argued that it is an important topic to study the extinction of the species [9 3], since, with the deveopment of modern society, more and more species are driven to extinction; this motivated us to study the extinction of the predator species. In Section 4, our resuts indicate that if the death rate of the predator species x 3 is big enough or the cooperate effect between species x 1 and x is very strong, the predator species wi be driven to extinction due to the fewer chances of meeting prey species. This can aso be seen from (H 3 ); x 3 wi be driven to extinction when d becomes bigger. Confict of Interests The authors decare that there is no confict of interests regarding the pubication of this paper. Acknowedgments The authors are gratefu to anonymous referees for their exceent suggestions, which greaty improve the presentation of the paper. Aso, the research was supported by the Natura Science Foundation of Fujian Province (015J0101, 015J01019) and the Foundation of Fujian Education Bureau. References [1] A. A. Berryman, The origins and evoution of predator-prey theory, Ecoogy,vo.73,no.5,pp ,199. [] L.J.Chen,F.D.Chen,andL.J.Chen, Quaitativeanaysisof a predator-prey mode with Hoing type II functiona response incorporating a constant prey refuge, Noninear Anaysis: Rea Word Appications,vo.11,no.1,pp.46 5,010. [3]Z.Z.Ma,F.D.Chen,C.Q.Wu,andW.L.Chen, Dynamic behaviors of a Lotka-Voterra predator-prey mode incorporating a prey refuge and predator mutua interference, Appied Mathematics and Computation, vo.19,no.15,pp , 013. [4] L. J. Chen, F. D. Chen, and Y. Q. Wang, Infuence of predator mutua interference and prey refuge on Lotka-Voterra predator-prey dynamics, Communications in Noninear Science and Numerica Simuation,vo.18,no.11,pp ,013. [5] L.-L. Wang and W.-T. Li, Periodic soutions and permanence for a deayed nonautonomous ratio-dependent predator-prey mode with Hoing type functiona response, Computationa and Appied Mathematics,vo.16,no.,pp , 004. [6]F.D.Chen,Z.Z.Ma,andH.Y.Zhang, Gobaasymptotica stabiity of the positive equiibrium of the Lotka-Voterra preypredator mode incorporating a constant number of prey refuges, Noninear Anaysis: Rea Word Appications,vo.13,no. 6, pp , 01. [7] Y. M. Chen and Z. Zhou, Stabe periodic soution of a discrete periodic Lotka-Voterra competition system, Mathematica Anaysis and Appications,vo.77,no.1,pp , 003. [8] N. Krikorian, The Voterra mode for three species predatorprey systems: boundedness and stabiity, Mathematica Bioogy,vo.7,no.,pp ,1979. [9]S.W.ZhangandD.J.Tan, Studyforthreespeciesmode with Hoing II founctiona response and priodic coefficients, Biomathematics,vo.15,pp ,000(Chinese). [10] P. Y. H. Pang and M. X. Wang, Strategy and stationary pattern in a three-species predator-prey mode, Differentia Equations,vo.00,no.,pp.45 73,004. [11] M. Liao, X. Tang, and C. Xu, Bifurcation anaysis for a threespecies predator-prey system with two deays, Communications

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