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# Permanence and global attractivity in a discrete Lotka-Volterra predator-prey model with delays

*Advances in Difference Equations*
**volume 2014**, Article number: 208 (2014)

## Abstract

In this paper, we deal with a discrete Lotka-Volterra predator-prey model with time-varying delays. For the general non-autonomous case, sufficient conditions which ensure the permanence and global stability of the system are obtained by using differential inequality theory. For the periodic case, sufficient conditions which guarantee the existence of a unique globally stable positive periodic solution are established. The paper ends with some interesting numerical simulations that illustrate our analytical predictions.

**MSC:**34K20, 34C25, 92D25.

## 1 Introduction

After the pioneering work of Berryman [1] in 1992, the dynamic relationship between predators and their preys has become one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Dynamic nature (including the local and global stability of the equilibrium, the persistence, permanence and extinction of species, the existence of periodic solutions and positive almost periodic solutions, bifurcation and chaos and so on) of predator-prey models has been investigated in a number of notable studies [2–26]. In many applications, the nature of permanence is of great interest. For example, Fan and Li [27] made a theoretical discussion on the permanence of a delayed ratio-dependent predator-prey model with Holling-type functional response. Chen [28] addressed the permanence of a discrete *n*-species delayed food-chain system. Zhao and Jiang [29] focused on the permanence and extinction for a non-autonomous Lotka-Volterra system. Chen [30] analyzed the permanence and global attractivity of a Lotka-Volterra competition system with feedback control. Zhao and Teng *et al.* [31] established the permanence criteria for delayed discrete non-autonomous-species Kolmogorov systems. For more research on the permanence behavior of predator-prey models, one can see [32–44].

In 2010, Lv *et al.* [45] investigated the existence and global attractivity of a periodic solution to the following Lotka-Volterra predator-prey system:

where {x}_{1}(t) denotes the density of prey species at time *t*, {x}_{2}(t) and {x}_{3}(t) stand for the density of predator species at time *t*, {r}_{i},{a}_{ij}\in C(R,[0,\mathrm{\infty})) and {\tau}_{ij}(t)>0 (i,j=1,2,3). Using Krasnoselskii’s fixed point theorem and constructing the Lyapunov function, Lv *et al.* obtained a set of easily verifiable sufficient conditions which guarantee the permanence and global attractivity of system (1.1).

Many authors have argued that discrete time models governed by difference equations are more appropriate to describe the dynamics relationship among populations than continuous ones when the populations have non-overlapping generations. Moreover, discrete time models can also provide efficient models of continuous ones for numerical simulations [4, 16, 46]. Thus it is reasonable and interesting to investigate discrete time systems governed by difference equations. The principal object of this article is to propose a discrete analogue system (1.1) and explore its dynamics.

Following the ideas of [4, 11], we will discretize system (1.1). Assume that the average growth rates in system (1.1) change at regular intervals of time, then we can obtain the following modified system:

where [t] denotes the integer part of *t*, t\in (0,+\mathrm{\infty}) and t\ne 0,1,2,\dots . We integrate (1.2) on any interval of the form [k,k+1), k=0,1,2,\dots , and obtain

where for k\le t<k+1, k=0,1,2,\dots .

Let t\to k+1, then (1.3) takes the following form:

which is a discrete time analogue of system (1.1), where k=0,1,2,\dots .

For the point of view of biology, we shall consider (1.4) together with the initial conditions {x}_{i}(0)\ge 0 (i=1,2,3). The principal object of this article is to explore the dynamics of system (1.4) applying the differential inequality theory to study the permanence of system (1.4). Using the method of Lyapunov function, we investigate the global asymptotic stability of system (1.4).

We assume that the coefficients of system (1.4) satisfy the following:

(H1) {r}_{i}, {a}_{ij}, {\tau}_{ij} with i,j=1,2,3 are non-negative sequences bounded above and below by positive constants.

Let \tau ={sup}_{1\le i,j\le 3,k\in Z}\{{\tau}_{ij}(k)\}. We consider (1.4) together with the following initial conditions:

It is not difficult to see that the solutions of (1.4) and (1.5) are well defined for all k\ge 0 and satisfy

The remainder of the paper is organized as follows. In Section 2, basic definitions and lemmas are given, some sufficient conditions for the permanence of system (1.4) are established. In Section 3, a series of sufficient conditions for the global stability of system (1.4) are included. The existence and stability of system (1.4) are analyzed in Section 4. In Section 5, we give an example which shows the feasibility of the main results. Conclusions are presented in Section 6.

## 2 Permanence

For convenience, in the following discussion, we always use the notations:

where f(k) is a non-negative sequence bounded above and below by positive constants. In order to obtain the main result of this paper, we shall first state the definition of permanence and several lemmas which will be useful in the proof of the main result.

**Definition 2.1** [47]

We say that system (1.4) is permanent if there are positive constants *M* and *m* such that each positive solution ({x}_{1}(k),{x}_{2}(k),{x}_{3}(k)) of system (1.4) satisfies

**Lemma 2.1** [47]

*Assume that*
\{x(k)\}
*satisfies*
x(k)>0
*and*

*for* k\in N, *where* a(k) *and* b(k) *are non*-*negative sequences bounded above and below by positive constants*. *Then*

**Lemma 2.2** [47]

*Assume that*
\{x(k)\}
*satisfies*

{lim}_{k\to +\mathrm{\infty}}supx(k)\le {x}^{\ast} *and* x({N}_{0})>0, *where* a(k) *and* b(k) *are non*-*negative sequences bounded above and below by positive constants and* {N}_{0}\in N. *Then*

Now we state our permanence result for system (1.4).

**Theorem 2.1** *Let* {M}_{1}, {M}_{2}, {M}_{3} *and* {m}_{1} *be defined by* (2.4), (2.10), (2.15) *and* (2.20), *respectively*. *In addition to condition* (H1), *assume that the following conditions*:

*and*

*hold*, *then system* (1.4) *is permanent*, *that is*, *there exist positive constants* {m}_{i}, {M}_{i} (i=1,2,3) *which are independent of the solution of system* (1.4), *such that for any positive solution* ({x}_{1}(k),{x}_{2}(k),{x}_{3}(k)) *of system* (1.4) *with the initial condition* {x}_{i}(0)\ge 0 (i=1,2,3), *one has*

*Proof* Let ({x}_{1}(k),{x}_{2}(k),{x}_{3}(k)) be any positive solution of system (1.4) with the initial condition ({x}_{1}(0),{x}_{2}(0),{x}_{3}(0)). It follows from the first equation of system (1.4) that

It follows from (2.1) that

Substituting (2.2) into the first equation of system (1.4), we get

It follows from (2.3) and Lemma 2.1 that

For any positive constant \epsilon >0, it follows from (2.4) that there exists {N}_{1}>0 such that for all k>{N}_{1},

For k\ge {N}_{1}+{\tau}^{u}, from (2.5) and the second equation of system (1.4), we have

which leads to

Substituting (2.7) into the second equation of system (1.4), we have

Thus it follows from Lemma 2.1 and (2.8) that

Setting \epsilon \to 0, we obtain

For k\ge {N}_{1}+{\tau}^{u}, from (2.5) and the third equation of system (1.4), we have

which leads to

Substituting (2.12) into the third equation of system (1.4) leads to

In view of Lemma 2.1 and (2.13), one has

Setting \epsilon \to 0, we get

For k\ge {N}_{1}+{\tau}^{u}, it follows from the first equation of system (1.4) that

which leads to

Substituting (2.17) into the first equation of system (1.4), we obtain

According to Lemma 2.2, it follows from (2.18) that

where

Setting \epsilon \to 0 in (2.19), we can get

where

For k\ge {N}_{1}+{\tau}^{u}, from the second equation of system (1.4), we have

which leads to

Substituting (2.22) into the second equation of system (1.4) leads to

By Lemma 2.2 and (2.23), we can get

where

Setting \epsilon \to 0 in the above inequality, we have that

where

For k\ge {N}_{1}+{\tau}^{u}, it follows from the third equation of system (1.4) that

Hence

Substituting (2.27) into the third equation of system (1.4), we derive

In view of Lemma 2.2 and (2.28), one has

where

Setting \epsilon \to 0 in (2.29) leads to

where

In view of (2.4), (2.10), (2.15), (2.20), (2.25) and (2.30), we can conclude that system (1.4) is permanent. The proof of Theorem 2.1 is complete. □

**Remark 2.1** Under the assumption of Theorem 2.1, the set [{m}_{1},{M}_{1}]\times [{m}_{2},{M}_{2}]\times [{m}_{3},{M}_{3}] is an invariant set of system (1.4).

## 3 Global stability

In this section, we formulate the stability property of positive solutions of system (1.4) when all the time delays are zero.

**Theorem 3.1** *Let* {\tau}_{ij}=0 (i,j=1,2,3). *In addition to* (H1)-(H3), *assume further that* (H4)

*Then*, *for any positive solution* ({x}_{1}(k),{x}_{2}(k),{x}_{3}(k)) *and* ({x}_{1}^{\ast}(k),{x}_{2}^{\ast}(k),{x}_{3}^{\ast}(k)) *of system* (1.4), *we have*

*Proof* Let

Then system (1.4) is equivalent to

Then

where {\theta}_{ij}(k)\in [0,1] (i,j=1,2,3). To complete the proof, it suffices to show that

In view of (H4), we can choose \epsilon >0 small enough such that

For above \epsilon >0, in view of Theorem 2.1 in Section 2, there exists {k}^{\ast}\in N such that

for all k\ge {k}^{\ast}.

Noticing that {\theta}_{ij}(k)\in [0,1] (i,j=1,2,3) implies that {x}_{i}^{\ast}(k)exp({\theta}_{ji}(k){u}_{i}(k)) lies between {x}_{i}^{\ast}(k) and {x}_{i}(k). From (3.3), we have

Let \rho =max\{{\delta}_{1}^{\epsilon},{\delta}_{2}^{\epsilon},{\delta}_{3}^{\epsilon}\}, then \rho <1. By (3.8)-(3.10), for k\ge {k}^{\ast}, we have

which implies

Thus (3.4) holds true and the proof is complete. □

## 4 Existence and stability of a periodic solution

In this section, we further assume that {\tau}_{ij}=0 (i,j=1,2,3) and the coefficients of system (1.4) satisfy the following condition:

(H5) There exists a positive integer *ω* such that for k\in N, 0<{r}_{i}(k+\omega )={r}_{i}(k), 0<{a}_{ij}(k+\omega )={a}_{ij}(k) (i,j=1,2,3).

**Theorem 4.1** *Assume that* (H1)-(H5) *are satisfied*, *then system* (1.4) *with all the delays* {\tau}_{ij}=0 (i,j=1,2,3) *admits at least one positive* *ω*-*periodic solution which we denote by* ({x}_{1}^{\ast}(k),{x}_{2}^{\ast}(k),{x}_{3}^{\ast}(k)).

*Proof* As pointed out in Remark 2.1 of Section 2,

is an invariant set of system (1.4). Then we define a mapping *F* on {D}^{3} by

Clearly, *F* depends continuously on ({x}_{1}(0),{x}_{2}(0),{x}_{3}(0)). Thus *F* is continuous and maps the compact set {D}^{3} into itself. Therefore, *F* has a fixed point. It is not difficult to see that the solution ({x}_{1}^{\ast}(k),{x}_{2}^{\ast}(k),{x}_{3}^{\ast}(k)) passing through this fixed point is an *ω*-periodic solution of system (1.4). The proof of Theorem 4.1 is complete. □

**Theorem 4.2** *Assume that* (H1)-(H5) *are satisfied*, *then system* (1.4) *with all the delays* {\tau}_{ij}=0 (i,j=1,2,3) *has a globally stable positive* *ω*-*periodic solution*.

*Proof* Under assumptions (H1)-(H5), it follows from Theorem 4.1 that system (1.4) with all the delays {\tau}_{ij}=0 (i,j=1,2,3) admits at least one positive *ω*-periodic solution. In addition, Theorem 3.1 ensures that the positive solution is globally stable. Hence the proof. □

## 5 Numerical example

In this section, we will give an example which shows the feasibility of the main results (Theorem 2.1) of this paper. Let us consider the following discrete system:

Here

All the coefficients {r}_{i}(k) (i=1,2,3), {a}_{ij}(k) (i,j=1,2,3), {\tau}_{ij}(k) (i,j=1,2,3) are functions with respect to *k*, and it is not difficult to obtain that

It is easy to examine that the coefficients of system (5.1) satisfy all the conditions of Theorem 2.1. Thus system (5.1) is permanent which is shown in Figures 1-3.

## 6 Conclusions

In this paper, we have investigated the dynamic behavior of a discrete Lotka-Volterra predator-prey model with time-varying delays. Sufficient conditions which ensure the permanence of the system are established. Moreover, we also analyze the global stability of the system with all the delays {\tau}_{ij}(k)=0 (i,j=1,2,3) and deal with the existence and stability of the system. We have shown that delay has important influence on the permanence of the system. Therefore, delay is an important factor to decide the permanence of the system. When all the delays are zero, we obtain some sufficient conditions which guarantee the global stability of the system. Computer simulations are carried out to explain our main theoretical results.

## References

Berryman AA: The origins and evolution of predator-prey theory.

*Ecology*1992, 73(5):1530-1535. 10.2307/1940005Dai BX, Zou JZ: Periodic solutions of a discrete-time diffusive system governed by backward difference equations.

*Adv. Differ. Equ.*2005., 2005: Article ID 586218Gyllenberg M, Yan P, Wang Y: Limit cycles for competitor-competitor-mutualist Lotka-Volterra systems.

*Physica D*2006, 221(2):135-145. 10.1016/j.physd.2006.07.016Fan M, Wang K: Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system.

*Math. Comput. Model.*2002, 35(9-10):951-961. 10.1016/S0895-7177(02)00062-6Fazly M, Hesaaraki M: Periodic solutions for a discrete time predator-prey system with monotone functional responses.

*C. R. Math. Acad. Sci. Paris*2007, 345(4):199-202. 10.1016/j.crma.2007.06.021Gaines RE, Mawhin JL:

*Coincidence Degree and Nonlinear Differential Equations*. Springer, Berlin; 1997.Kar TK, Ghorai A: Dynamic behaviour of a delayed predator-prey model with harvesting.

*Appl. Math. Comput.*2011, 217(22):9085-9104. 10.1016/j.amc.2011.03.126Sen M, Banerjee M, Morozov A: Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect.

*Ecol. Complex.*2012, 11: 12-27.Haque M, Venturino E: An ecoepidemiological model with disease in predator: the ratio-dependent case.

*Math. Methods Appl. Sci.*2007, 30(14):1791-1809. 10.1002/mma.869Braza PA: Predator-prey dynamics with square root functional responses.

*Nonlinear Anal., Real World Appl.*2012, 13(4):1837-1843. 10.1016/j.nonrwa.2011.12.014Wiener J: Differential equations with piecewise constant delays. Lecture Notes in Pure and Applied Mathematics 90. In

*Trends in Theory and Practice of Nonlinear Differential Equations*. Dekker, New York; 1984.Xu R, Chen LS, Hao FL: Periodic solution of a discrete time Lotka-Volterra type food-chain model with delays.

*Appl. Math. Comput.*2005, 171(1):91-103. 10.1016/j.amc.2005.01.027Zhang JB, Fang H: Multiple periodic solutions for a discrete time model of plankton allelopathy.

*Adv. Differ. Equ.*2006., 2006: Article ID 90479Xiong XS, Zhang ZQ: Periodic solutions of a discrete two-species competitive model with stage structure.

*Math. Comput. Model.*2008, 48(3-4):333-343. 10.1016/j.mcm.2007.10.004Zhang RY, Wang ZC, Chen YM, Wu JH: Periodic solutions of a single species discrete population model with periodic harvest/stock.

*Comput. Math. Appl.*2009, 39(1-2):77-90.Zhang WP, Zhu DM, Bi P: Multiple periodic positive solutions of a delayed discrete predator-prey system with type IV functional responses.

*Appl. Math. Lett.*2007, 20(10):1031-1038. 10.1016/j.aml.2006.11.005Zhang ZQ, Luo JB: Multiple periodic solutions of a delayed predator-prey system with stage structure for the predator.

*Nonlinear Anal., Real World Appl.*2010, 11(5):4109-4120. 10.1016/j.nonrwa.2010.03.015Li YK, Zhao KH, Ye Y: Multiple positive periodic solutions of species delay competition systems with harvesting terms.

*Nonlinear Anal., Real World Appl.*2011, 12(2):1013-1022. 10.1016/j.nonrwa.2010.08.024Sun YG, Saker SH: Positive periodic solutions of discrete three-level food-chain model of Holling type II.

*Appl. Math. Comput.*2006, 180(1):353-365. 10.1016/j.amc.2005.12.015Ding XH, Liu C: Existence of positive periodic solution for ratio-dependent

*N*-species difference system.*Appl. Math. Model.*2009, 33(6):2748-2756. 10.1016/j.apm.2008.08.008Chakraborty K, Chakraborty M, Kar TK: Bifurcation and control of a bioeconomic model of a prey-predator system with a time delay.

*Nonlinear Anal. Hybrid Syst.*2011, 5(4):613-625. 10.1016/j.nahs.2011.05.004Li ZC, Zhao QL, Ling D: Chaos in a discrete population model.

*Discrete Dyn. Nat. Soc.*2012., 2012: Article ID 482459Xiang H, Yang KM, Wang BY: Existence and global stability of periodic solution for delayed discrete high-order Hopfield-type neural networks.

*Discrete Dyn. Nat. Soc.*2005, 2005(3):281-297. 10.1155/DDNS.2005.281Gopalsamy K:

*Stability and Oscillations in Delay Differential Equations of Population Dynamics*. Kluwer Academic, Dordrecht; 1992.Kuang Y:

*Delay Differential Equations with Applications in Population Dynamics*. Academic Press, New York; 1993.Fan L, Shi ZK, Tang SY: Critical values of stability and Hopf bifurcations for a delayed population model with delay-dependent parameters.

*Nonlinear Anal., Real World Appl.*2010, 11(1):341-355. 10.1016/j.nonrwa.2008.11.016Fan YH, Li WT: Permanence for a delayed discrete ratio-dependent predator-prey model with Holling type functional response.

*J. Math. Anal. Appl.*2004, 299(2):357-374. 10.1016/j.jmaa.2004.02.061Chen FD: Permanence of a discrete

*n*-species food-chain system with time delays.*Appl. Math. Comput.*2007, 185(1):719-726. 10.1016/j.amc.2006.07.079Zhao JD, Jiang JF: Average conditions for permanence and extinction in nonautonomous Lotka-Volterra system.

*J. Math. Anal. Appl.*2004, 299(2):663-675. 10.1016/j.jmaa.2004.06.019Chen FD: The permanence and global attractivity of Lotka-Volterra competition system with feedback control.

*Nonlinear Anal., Real World Appl.*2006, 7(1):133-143. 10.1016/j.nonrwa.2005.01.006Teng ZD, Zhang Y, Gao SJ: Permanence criteria for general delayed discrete nonautonomous

*n*-species Kolmogorov systems and its applications.*Comput. Math. Appl.*2010, 59(2):812-828. 10.1016/j.camwa.2009.10.011Dhar J, Jatav KS: Mathematical analysis of a delayed stage-structured predator-prey model with impulsive diffusion between two predators territories.

*Ecol. Complex.*2013, 16: 59-67.Liu SQ, Chen LS: Necessary-sufficient conditions for permanence and extinction in Lotka-Volterra system with distributed delay.

*Appl. Math. Lett.*2003, 16(6):911-917. 10.1016/S0893-9659(03)90016-4Liao XY, Zhou SF, Chen YM: Permanence and global stability in a discrete

*n*-species competition system with feedback controls.*Nonlinear Anal., Real World Appl.*2008, 9(4):1661-1671. 10.1016/j.nonrwa.2007.05.001Hu HX, Teng ZD, Jiang HJ: On the permanence in non-autonomous Lotka-Volterra competitive system with pure-delays and feedback controls.

*Nonlinear Anal., Real World Appl.*2009, 10(3):1803-1815. 10.1016/j.nonrwa.2008.02.017Muroya Y: Permanence and global stability in a Lotka-Volterra predator-prey system with delays.

*Appl. Math. Lett.*2003, 16(8):1245-1250. 10.1016/S0893-9659(03)90124-8Kuniya T, Nakata Y: Permanence and extinction for a nonautonomous SEIRS epidemic model.

*Appl. Math. Comput.*2012, 218(18):9321-9331. 10.1016/j.amc.2012.03.011Hou ZY: On permanence of Lotka-Volterra systems with delays and variable intrinsic growth rates.

*Nonlinear Anal., Real World Appl.*2013, 14(2):960-975. 10.1016/j.nonrwa.2012.08.010Li CH, Tsai CC, Yang SY: Analysis of the permanence of an SIR epidemic model with logistic process and distributed time delay.

*Commun. Nonlinear Sci. Numer. Simul.*2012, 17(9):3696-3707. 10.1016/j.cnsns.2012.01.018Chen FD, You MS: Permanence for an integrodifferential model of mutualism.

*Appl. Math. Comput.*2007, 186(1):30-34. 10.1016/j.amc.2006.07.085Berezansky L, Baštinec J, Diblík J, Šmarda Z: On a delay population model with quadratic nonlinearity.

*Adv. Differ. Equ.*2012., 2012: Article ID 230 10.1186/1687-1847-2012-230Baštinec J, Berezansky L, Diblík J, Šmarda Z: On a delay population model with a quadratic nonlinearity without positive steady state.

*Appl. Math. Comput.*2014, 227: 622-629.Tang XH, Cao DM, Zou XF: Global attractivity of positive periodic solution to periodic Lotka-Volterra competition systems with pure delay.

*J. Differ. Equ.*2006, 228(2):580-610. 10.1016/j.jde.2006.06.007Tang XH, Zou XF: Global attractivity in a predator-prey system with pure delays.

*Proc. Edinb. Math. Soc.*2008, 51: 495-508.Lv X, Lu SP, Yan P: Existence and global attractivity of positive periodic solutions of Lotka-Volterra predator-prey systems with deviating arguments.

*Nonlinear Anal., Real World Appl.*2010, 11(5-6):574-583.Chen YM, Zhou ZF: Stable periodic of a discrete periodic Lotka-Volterra competition system.

*J. Math. Anal. Appl.*2003, 277(1):358-366. 10.1016/S0022-247X(02)00611-XChen FD: Permanence for the discrete mutualism model with time delays.

*Math. Comput. Model.*2008, 47(3-4):431-435. 10.1016/j.mcm.2007.02.023

## Acknowledgements

The first author was supported by the National Natural Science Foundation of China (No. 11261010), the Soft Science and Technology Program of Guizhou Province (No. 2011LKC2030), the Natural Science and Technology Foundation of Guizhou Province (J[2012]2100), the Governor Foundation of Guizhou Province ([2012]53) and the Doctoral Foundation of Guizhou University of Finance and Economics (2010). The second author was supported by the National Natural Science Foundation of China (No. 11101126). The third author was supported by the Natural Science Innovation Team Project of Guizhou Province ([2013]14). The authors would like to thank the referees and the editor for helpful suggestions incorporated into this paper.

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Xu, C., Wu, Y. & Lu, L. Permanence and global attractivity in a discrete Lotka-Volterra predator-prey model with delays.
*Adv Differ Equ* **2014**, 208 (2014). https://doi.org/10.1186/1687-1847-2014-208

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DOI: https://doi.org/10.1186/1687-1847-2014-208