Permanence and global attractivity of a nonautonomous modified Leslie-Gower predator-prey model with Holling-type II schemes and a prey refuge

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1 Xie et al. Advances in Difference Equations :184 DOI /s R E S E A R C H Open Access Permanence and global attractivity of a nonautonomous modified Leslie-Gower predator-prey model with Holling-type II schemes and a prey refuge Xiangdong Xie 1* YalongXue 1 Jinhuang Chen and Tingting Li 3 * Correspondence: latexfzu@16.com 1 Department of Mathematics Ningde Normal University Ningde Fujian P.R. China Full list of author information is available at the end of the article Abstract A nonautonomous modified Leslie-Gower predator-prey model with Holling-type II schemes and a prey refuge is studied in this paper. Persistent property and stability property of the system are investigated. Some findings about the influence of prey refuge are obtained. MSC: 34D3; 9D5; 34D0; 34D40 Keywords: predator; prey; permanence; global stability 1 Introduction Throughout this paper for a bounded continuous function g ined on R letg L and g M be ined as g L = inf t R gt gm = sup gt. t R During the last two decades the study of dynamic behaviors of predator-prey system incorporating a prey refuge become one of the most important research topic see [1 45] and the references cited therein. In [1] Yue proposed and studied the following modified Leslie-Gower predator-prey model with Holling-type II schemes and a prey refuge: ẋt=x r 1 b 1 x ẏt=y r a y a 11 my 1 mx + k 1 1 mx + k where xtandyt denote the densities of the predator and prey species at time trespectively and all the coefficients are all positive constants 0 m < 1. Such a topic as the global stability property of the positive equilibrium was investigated. Many authors [ ] argued that with the biological and environmental change of the circumstance it is reasonable to propose and study the nonautonomous system the Xie et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use distribution and reproduction in any medium provided you give appropriate credit to the original authors and the source provide a link to the Creative Commons license and indicate if changes were made.

2 Xieetal. Advances in Difference Equations :184 Page of 11 success of [ ] motivates us to propose the corresponding nonautonomous case of system 1.1 i.e. the following system: ẋt=x r 1 t b 1 tx ẏt=y r t a 1t1 mty 1 mtx + k 1 t a ty 1 mtx + k t where xtandyt denote the densities of the predator and prey species at time trespectively; One could refer to [1] for the biological meaning of the coefficients. Throughout this paper we assume that H 1 r i t k i t a i t i =1 mt b 1 t are continuous and strictly positive functions which satisfy 1. min { r L i kl i al i ml b L 1 } >0 max { r M i k M i a M i m M b M 1 } <+. We consider 1. together with the following initial conditions: x0 > 0 y0 > It is not difficult to see that solutions of are well ined for all t 0 and satisfy xt>0 yt > 0 for all t The paper is arranged as follows: In Section we obtain sufficient conditions which guaranteethepermanenceofthe system 1.. In Section 3 we obtain sufficient conditions which ensure the global attractivity of the system 1.. In Section 4 an exampletogether with its numeric simulations illustrates the feasibility of the main results. We end this paper by a brief discussion. Permanence Lemma.1 Let xt yt T be any solution of system 1. with the initial conditions 1.3 then lim sup xt rm 1 b L 1 = M 1 lim sup yt rm 1 ml M 1 + k M a L = M..1 Proof Let xt yt T be any solution of system 1. with the initial conditions 1.3. From the first equation of system 1. it followsthat ẋt =x r 1 t b 1 tx a 1t1 mty 1 mtx + k 1 t x r M 1 b L 1 x..

3 Xieetal. Advances in Difference Equations :184 Page 3 of 11 Applying Lemma.3 in [46] to. it immediately follows that lim sup xt rm 1 b L 1 = M 1..3 For any positive constant ε > 0 small enough it follows from.3 that there exists a T 1 >0 such that xt<m 1 + ε for all t T 1..4 For t > T 1.4 together with the second equation of system 1.leadsto ẏt =y r t a ty 1 mtx + k t y r M a L 1 m L M 1 + ε+k M Applying Lemma.3 in [46] to.5 it immediately follows that lim sup Setting ε 0 then y..5 yt rm 1 ml M 1 + ε+k M a L..6 lim sup yt rm 1 ml M 1 + k M a L = M..7 Lemma. Let xt yt T be any solution of system 1. with the initial conditions 1.3 assume that r L 1 > am 1 1 ml M k L 1.8 holds then lim inf xt rl 1 kl 1 am 1 1 ml M k1 L = m 1.9 bm 1 lim inf yt rl 1 mm m 1 + k L a M = m..10 Proof Condition.8 implies that one could choose ε > 0 small enough such that r L 1 > am 1 1 ml M + ε k L 1.11 holds. For this ε it follows from.7 that there exists a T > T 1 such that yt<m + ε for all t T..1

4 Xieetal. Advances in Difference Equations :184 Page 4 of 11 Let xt yt T be any solution of system 1. with the initial conditions 1.3. For t > T.1 together with the first equation of system 1.leadsto ẋt =x r 1 t b 1 tx a 1t1 mty 1 mtx + k 1 t x r1 L am 1 1 ml M ε k1 L b M 1. x.13 Applying Lemma.3 in [46] to.13 it immediately follows that rl lim inf xt 1 am 1 1 ml M ε k1 L b M Setting ε 0 then lim inf xt rl 1 kl 1 am 1 1 ml M k1 L = m bm 1 Let ε > 0 be any positive constant small enough such that ε < 1 m 1. It then follows from.15 that there exists a T 3 > T suchthat xt>m 1 ε for all t T From the second equation of system 1. it follows that ẏt =y r t [ yt r L a ty 1 mtx + k t a M yt 1 m M m 1 ε+k L ]..17 Applying Lemma.3 in [46] to.17 it immediately follows that lim inf yt rl 1 mm m 1 ε+k L..18 a M Setting ε 0 then lim inf yt rl 1 mm m 1 + k L a M = m..19 As a direct corollary of Lemma.1 and. we have the following. Theorem.1 Assume that.8 holds then system is permanent.

5 Xieetal. Advances in Difference Equations :184 Page 5 of 11 3 Global attractivity Before we state the main result of this section we introduce some notations. Set 1 m 1 = 1 mt m 1 + k 1 t m 1 = 1 mt m 1 + k t 3.1 M 1 = 1 mt M 1 + k t and Ɣ 1 t = b 1 t a 1t1 mt M 1 m 1 Ɣ t = a tk t + a t1 mtm 1 M 1 M 1 a 1t1 mtk 1 t 1 m 1 a t1 mtm m 1 a 1t1 mt M 1. 1 m 1 3. Theorem 3.1 Assume that all the conditions of Theorem.1 hold assumefurther that { lim inf Ɣ1 t Ɣ t } > then for any positive solutions xt yt T and x 1 t y 1 t T of system 1. one has lim xt x1 t + yt y1 t =0. Proof Condition 3.3 impliesthatthereexistsasmallenoughpositiveconstant ε without loss of generality we may assume that ε < 1 {m 1 m }suchthat Ɣ 1 ε t=b 1 t a 1t1 mt M + ε 1 m ε 1 a t1 mtm + ε m ε 1 ε Ɣ ε t= a tk t M1 ε + a t1 mtm 1 ε M1 ε a 1t1 mtk 1 t 1 m ε 1 a 1t1 mt M 1 + ε 1 m ε 1 ε 3.4 where 1 m ε 1 = 1 mt m1 ε+k 1 t m ε 1 = 1 mt m1 ε+k t 3.5 M ε 1 = 1 mt M1 + ε+k t. For two arbitrary positive solutions xt yt T and x 1 t y 1 t T of system 1. for the above ε > 0 it then follows from.1.15 and.19 that there exists a T > T 3 such

6 Xieetal. Advances in Difference Equations :184 Page 6 of 11 that for all t T xt x 1 t<m 1 + ε yt y 1 t<m + ε xt x 1 t>m 1 ε yt y 1 t>m ε. 3.6 Set 1 xt x1 t = 1 mt x 1 t+k 1 t 1 mt xt+k 1 t xt x1 t = 1 mt x 1 t+k t 1 mt xt+k t. 3.7 Now we let V 1 t= ln xt ln x 1 t 3.8 V t= ln yt ln y 1 t. 3.9 Then for t > Twehave D + V 1 t sgn xt x 1 t b 1 txt a 1t1 mtyt 1 mtxt+k 1 t + b 1 tx 1 t+ a 1t1 mty 1 t 1 mtx 1 t+k 1 t b 1 t xt x 1 t + a 1 t 1 mt y 1 t 1 mtx 1 t+k 1 t yt 1 mtxt+k 1 t = b 1 t xt x 1 t + a 1t1 mt 1 xt x 1 t y 1 t 1 mt xt+k 1 t yt 1 mt x 1 t+k 1 t = b 1 t xt x 1 t + a 1t1 mtk 1 t yt y 1 t 1 xt x 1 t + a 1t1 mt y 1 txt ytx 1 t 1 xt x 1 t = b 1 t xt x1 t a 1 t1 mtk 1 t + yt y1 t 1 xt x 1 t + a 1t1 mt y 1 txt y 1 tx 1 t+y 1 tx 1 t ytx 1 t 1 xt x 1 t = b 1 t xt x1 t a 1 t1 mtk 1 t + yt y1 t 1 xt x 1 t + a 1t1 mt y 1 t xt x1 t 1 xt x 1 t + a 1t1 mt x 1 t y1 t yt 1 xt x 1 t b 1 t xt x1 t a 1 t1 mtk 1 t + yt y1 1 m ε 1 t

7 Xieetal. Advances in Difference Equations :184 Page 7 of 11 and + a 1t1 mt M + ε 1 m ε 1 xt x 1 t + a 1t1 mt M 1 + ε y1 1 m ε 1 t yt D + V t = sgn yt y 1 t a tyt 1 mtxt+k t + a ty 1 t 1 mtx 1 t+k t Now let us set = sgnyt y 1t a tyt 1 mt x 1 t+k t xt x 1 t + a ty 1 t 1 mt xt+k t = sgnyt y 1t a tk t yt y 1 t xt x 1 t + sgnyt y 1t a t 1 mt ytx 1 t y 1 txt xt x 1 t a tk t yt y1 M1 ε t sgnyt y 1 t + xt x 1 t a t 1 mt ytx 1 t y 1 tx 1 t+y 1 tx 1 t y 1 txt a tk t M1 ε yt y 1 t a t1 mtm 1 ε M1 ε yt y 1 t + a t1 mtm + ε m ε 1 xt x 1 t. Vt=V 1 t+v t. Then D + Vt Ɣ 1 ε t xt x1 t Ɣ ε t yt y1 t Integrating both sides of 3.10 onthe interval[t t Vt VT t It follows from 3.4that Vt+ε t T T [ Ɣ1 ε s xs x1 s Ɣ ε s ys y1 s ] ds for t T [ xs x 1 s + ys y 1 s ] ds VT fort T. 3.1 Therefore Vt is bounded on [T+ andalso t T [ xs x1 s + ys y1 s ] ds <

8 Xieetal. Advances in Difference Equations :184 Page 8 of 11 By Lemma.1. andtheorem.1 xt x 1 t yt y 1 t are bounded on [T+. On the other hand it is easy to see that ẋt ẏt ẋ 1 t and ẏ 1 t areboundedfort T. Therefore xt x 1 t yt y 1 t are uniformly continuous on [T+. By the Barbălat lemma [47] one can conclude that [ lim xt x 1 t + yt y 1 t ] =0. This ends the proof of Theorem Numeric example Now let us consider the following example. Example 4.1 ẋt=x 11 + cos t 6x ẏt=y y x sin t1 0.1y 1 0.1x Corresponding to system 1. onehas r 1 t=11+cos t b 1 t=6 a 1 t = sin t mt=0.1 k 1 t=1 r t= a t= k t=. And so M 1 = rm 1 b L 1 = M = rm 1 ml M 1 + k M a L = = Consequently r L 1 am 1 1 ml M k L 1 = =.98 > Equation 4.3showsthat.8 holds thus it follows from Theorem.1 that system 4.1 is permanent and numeric simulations Figures1 also support these findings. Figure 1 Dynamic behavior of the first component xt ofthesolutionxt yt ofsystem 4.1 with the initial condition x0 y0 = and 4 respectively.

9 Xieetal. Advances in Difference Equations :184 Page 9 of 11 Figure Dynamic behavior of the second component yt ofthesolutionxt yt ofsystem 4.1 with the initial condition x0 y0 = and 4 respectively. 5 Discussion In this paper we propose and study a nonautonomous modified Leslie-Gower predatorprey model with Holling-type II schemes and a prey refuge. We first obtain a set of sufficient conditions which ensure the permanence of the system after that by constructing a suitable Lyapunov function we also investigate the stability property of the system. Condition.8 implies that the prey refuge has a positive effect on the persistence property of the system indeed with the increasing of prey refuge the predators have difficulty in catching the prey species this directly increasing the survival possibility of the prey species. We mention here that in condition H 1 we make the assumption that the intrinsic growth rate of the prey species is a positive function however in some cases the growth ratemaybenegativesee[48 50] and the references therein we leave this for future investigation. Competing interests The authors declare that there is no conflict of interests. Authors contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details 1 Department of Mathematics Ningde Normal University Ningde Fujian P.R. China. College of Mathematics and Computer Science Fuzhou University Fuzhou Fujian P.R. China. 3 College of Mathematics and Computer Science Fuzhou University Fuzhou Fujian P.R. China. Acknowledgements The authors are grateful to anonymous referees for their excellent suggestions which greatly improved the presentation of the paper. The research was supported by the Natural Science Foundation of Fujian Province 015J J J05006 and the Scientific Research Foundation of Fuzhou University XRC Received: 5 May 016 Accepted: 8 June 016 References 1. Yue Q: Dynamics of a modified Leslie-Gower predator-prey model with Holling-type II schemes and a prey refuge. SpringerPlus 5 ArticleID Wu Y Chen F Chen W Lin Y: Dynamic behaviors of a nonautonomous discrete predator-prey system incorporating a prey refuge and Holling type II functional response. Discrete Dyn. Nat. Soc. 01 Article ID Chen F Wu Y Ma Z: Stability property for the predator-free equilibrium point of predator-prey systems with a class of functional response and prey refuges. Discrete Dyn. Nat. Soc. 01 Article ID Chen W Gong X Zhao L Zhang H: Dynamics of a non-autonomous discrete Leslie-Gower predator-prey system withpreyrefuge.j.fuzhouuniv Wu Y Chen WL Zhang HY: On permanence and global attractivity of a nonautonomous modified Leslie-Gower predator-prey system incorporating a prey refuge. J. Minjiang Univ Ma ZZ Chen FD Wu CQ Chen WL: Dynamic behaviors of a Lotka-Volterra predator-prey model incorporating a prey refuge and predator mutual interference. Appl. Math. Comput Chen FD Ma ZZ Zhang HY: Global asymptotical stability of the positive equilibrium of the Lotka-Volterra prey-predator model incorporating a constant number of prey refuges. Nonlinear Anal. Real World Appl Yang R Zhang C: The effect of prey refuge and time delay on a diffusive predator-prey system with hyperbolic mortality. Complexity

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