STABILITY AND NEIMARK-SACKER BIFURCATION OF A SEMI-DISCRETE POPULATION MODEL

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1 Journal of Applied Analsis and Computation Website: Volume 4, Number 4, November 14 pp STABILITY AND NEIMARK-SACKER BIFURCATION OF A SEMI-DISCRETE POPULATION MODEL Cheng Wang 1 and Xiani Li 1, Abstract In this paper, a semi-discrete model is derived for a nonlinear simple population model, and its stabilit and bifurcation are investigated b invoking a ke lemma we present. Our results displa that a Neimark- Sacker bifurcation occurs in the positive fied point of this sstem under certain parametric conditions. B using the Center Manifold Theorem and bifurcation theor, the stabilit of invariant closed orbits bifurcated is also obtained. The numerical simulation results not onl show the correctness of our theoretical analsis, but also ehibit new and interesting dnamics of this sstem, which do not eist in its corresponding continuous version. Kewords Semi-discrete population model, stabilit, Neimark-Sacker bifurcation, Lapunov eponent, Chaos. MSC 9A11, 7F45, 7G5. 1. Introduction In order to describe the control of a single population of cells, Nazarenko [] proposed the nonlinear dela differential euation dξ dt = pξt + ξt r + ξ m, t, 1.1 t ω where p,, r, ω, +, m {1,,...} and > pr, ξt is the size of the population at time t, p is the death rate, the feedback is given b the function fz, zt zt ω = r+z m t ω, and ω is the generation time. Since then, E.1.1 has been well studied b several authors see [5, 16, 7, ]. In recent ears, the modern theories of difference euations have been widel applied in the discrete sstems of computer science, econom, neutral net, ecolog and control theor etc., especiall, in the applications of population dnamics. Man authors see [1, ] have argued that the discrete sstems governed b difference e- uations are more appropriate than the continuous counterparts, particularl, when the populations have nonoverlapping generations. Li [17] studied the dnamics of a discrete food-limited population model with time dela. Saker [6] investigated the nonlinear periodic solutions, oscillation and attractivit of discrete nonlinear dela the corresponding author. address:mathli@zu.edu.cnxiani Li 1 College of Mathematical Science, Yangzhou Universit, Yangzhou, 5, China The authors were supported b National Natural Science Foundation of China and the NSF of Yangzhou Universit.

2 4 C. Wang & X. Y. Li population model. Liz [18] considered the global stabilit for a discrete population model. Song and Peng [8] discussed the periodic solutions of a nonautonomous periodic model of population with continuous and discrete time. Li [19] studied the global stabilit and oscillation in nonlinear difference euations of population dnamics. Zhang etc. [4] studied the periodic solutions of a single species discrete population model with periodic harvest/stock. It is well known that the dnamics including stabilit, bifurcations and chaos etc. of a sstem have been a popular subject see [, 4, 6 8, 1 14,, 1, 4,, 5]. In this paper, motivated b the above work we discuss the analogue of the E Without loss of generalit, we ma assume ω = 1 in 1.1. In fact, b letting s = t ω, namel, ξt = ξsω = ηs, 1.1 is reduced to dη ds = pωηs + ωηs r + η m, s s 1 B resetting p b p ω and b ω, E.1. becomes dη ds = pηs + ηs r + η m, s s 1 This is just 1.1 with ω = 1. Suppose that the average growth rate in 1. changes at regular intervals of time, then we ma incorporate this aspect into 1. and obtain the following version of 1. 1 dηs = p + ηs ds r + η m, s 1,,,..., 1.4 [s 1] where [s 1] denotes the integer part of s 1, s [1, +. Euation of tpe 1.4 is known as differential euation with piecewise constant arguments and these euations occup a position midwa between differential and difference euations. B a solution of E. 1.4, we mean a function ηs, which is defined for s [1, +, and possesses the following properties: i ηs is continuous on [1, + ; ii the derivative dηs/ds eists at each point s [1, + with the possible eception of the point s {1,,,...}, where the left-side derivative eists; iii the E. 1.4 is satisfied on each internal [k, k + 1 with k = 1,,,... B integrating 1.4 on an interval [n, n + 1, n = 1,,,..., we can get ηs = ηn ep p + Letting s n + 1, we have ηn + 1 = ηn ep r + η m n 1 which is the discrete analog of E. 1. without dela. Let { n = ηn 1, s n. 1.5 p + r + η m, 1.6 n 1 n = ηn, 1.7

3 Stabilit and Neimark-Sacker bifurcation 41 then we arrive in a discrete sstem as follows: n + 1 = n, n + 1 = n ep p + r + m, n 1.8 where p,, r and m are defined as in E The main aim of this paper is to investigate the dnamics of the sstem 1.8 b using the Center Manifold Theorem, bifurcation theor [, 9, 15], [5, 9] and numerical simulations. It is shown that the sstem 1.8 possesses a Neimark-Sacker bifurcation and other comple dnamics under certain parametric conditions, which have not been considered in an known literature. The rest of this paper is organized as follows. The eistence and stabilit of the fied points for the sstem 1.8 are analzed in the net section. In Section, the sufficient conditions of the eistence for Neimark-Sacker bifurcation are obtained. In Section 4, numerical simulations are presented, which not onl illustrate our theoretical results, but also ehibit other comple dnamics of the sstem 1.8, and the Lapunov eponents are computed numericall to confirm some of its dnamics. A brief conclusion is given in Section 5.. Eistence and stabilit of fied point In this section, we first determine the eistence of the fied points of the sstem 1.8, then investigate their stabilit. The fied points of the sstem 1.8 satisf the following euations: =, ep p + r + m =. B some computations to the sstem.1, it is eas to obtain: i the trivial fied point E,, which alwas eists for all parameter values;.1 ii the uniue positive fied point E +, feasible because of > pr, where 1/m 1/m pr pr =, =.. p p Now investigate the local stabilit of ever fied point of the sstem 1.8. The Jacobian matri of the sstem 1.8 at a fied point E, is 1 J = ep p + m m 1 r + m r + m ep p + The characteristic euation associated with. is r + m.. λ TrJλ + DetJ =,.4

4 4 C. Wang & X. Y. Li where λ is the eigenvalue, TrJ and DetJ are the trace and determinant of. respectivel, namel, TrJ = ep p + r + m.5 and Hence the sstem 1.8 is see [14] m m 1 DetJ = ep p + r + m r + m..6 i a dissipative dnamical sstem if and onl if p ep m m 1 + r + m r + m < 1; ii a conservative dnamical sstem if and onl if p ep m m 1 + r + m r + m = 1; iii an undissipated dnamical sstem otherwise. In order to stud the local stabilit and bifurcation for a fied point of a general D sstem, the following lemma will be ver useful and even essential. Lemma.1. Let F λ = λ + Bλ + C, where B and C are two real constants. Suppose λ 1 and λ are two roots of F λ =. Then the following statements hold. i If F 1 >, then i.1 λ 1 < 1 and λ < 1 if and onl if F 1 > and C < 1; i. λ 1 = 1 and λ 1 if and onl if F 1 = and B ; i. λ 1 < 1 and λ > 1 if and onl if F 1 < ; i.4 λ 1 > 1 and λ > 1 if and onl if F 1 > and C > 1; i.5 λ 1 and λ are a pair of conjugate comple roots and λ 1 = λ = 1 if and onl if < B < and C = 1; i.6 λ 1 = λ = 1 if and onl if F 1 = and B =. ii If F 1 =, namel, 1 is one root of F λ =, then the other root λ satisfies λ = <, >1 if and onl if C = <, >1. iii If F 1 <, then F λ = has one root ling in 1,. Moreover, Proof. iii.1 the other root λ satisfies λ < = 1 if and onl if F 1 < =; iii. the other root λ satisfies 1 < λ < 1 if and onl if F 1 >. The proof for Lemma.1 is simple and omitted here. Remark.1. i When F 1 >, our results are the same as the ones in [1] ecept the cases ii and vi. Corresponding to the above i., the conclusion in [1] is stated as: λ 1 = 1 and λ 1 if and onl if F 1 = and B,. We think, B is redundant. Otherwise, λ 1 + λ =, together with λ 1 = 1, implies λ = 1, which is contrar to F 1 >. Therefore, B should be kicked out.

5 Stabilit and Neimark-Sacker bifurcation 4 So, our results correct the case iv of Lemma.. in [1] and give a new conclusion vi which is not considered in an known literature. ii The results for the cases F 1 = and F 1 < are completel new. Net, we recall the definition of topological tpes for a fied point,. Definition.1. Let E, be a fied point of the sstem 1.8 with multipliers λ 1 and λ. i A fied point E, is called sink if λ 1 < 1 and λ < 1, so sink is locall asmptoticall stable. ii A fied point E, is called source if λ 1 > 1 and λ > 1, so source is locall asmptoticall unstable. iii A fied point E, is called saddle if λ 1 < 1 and λ > 1or λ 1 > 1 and λ < 1. iv A fied point E, is called to be non-hperbolic if either λ 1 = 1 or λ = 1. Now, we discuss the local dnamics for the fied points of the sstem 1.8. The result for the stabilit of the fied point E, is as follows. Theorem.1. The fied point E, of the sstem 1.8 is a saddle. Proof. The Jacobian matri J of the sstem 1.8 at E is given b 1 JE = pr..7 ep r Obviousl, the eigenvalues of.7 are λ 1 = and λ = ep pr r with λ 1 < 1 and λ > 1 because of > pr. Thus E is a saddle. In the following we deduce the local dnamics of the fied point E +,. Theorem.. The sstem 1.8 has a uniue positive fied point E +,, where = = pr p 1/m. i When < mp 1, E + is a sink. ii When mp > 1, there eist three different topological tpes of E + for all permissible values of parameters: Proof. ii.1 E + is a sink if < mp r mp 1 = ; ii. E + is a source if > ; ii. E + is non-hperbolic if =. The Jacobian matri J of the sstem 1.8 at E + is given b 1 JE + = mp pr..8 1 The corresponding characteristic euation of.8 can be written as F λ = λ λ + mp pr =..9

6 44 C. Wang & X. Y. Li It is eas to verif that and F 1 = F 1 = + mp pr mp pr >.1 >..11 When < mp 1, mp pr < 1. B using Lemma.1 i.1, E..9 has two eigenvalues λ 1 and λ with λ 1 < 1 and λ < 1, so E + is a sink. When mp > 1, < >, = is euivalent to mp pr < >, = 1. B Lemma.1 and Ddfinition.1, it is eas to see E + is a sink for <, a source for > and non-hperbolic for =. Remark.. Theorem. shows that there eists a D locall stable manifold Wloc s in E + for < whereas a D locall unstable manifold Wloc u for >. Hence, one can see that there will be an occurrence of bifurcation at E + for =.. Neimark-Sacker bifurcation From Theorem. ii., it is eas to see that two eigenvalues of the fied point E + pr p 1/m, pr p 1/m are 1± i. Notice at this time that all the parameters locate in the following set: S E+ = {p,, r, m, + : m {1,,...}, > pr, mp > 1, = = mp r mp 1 }. The fied point E +, can pass through a Neimark-Sacker bifurcation when the parameters p,, r, m S E+ and varies in the small neighborhood of. Based on the previous analsis, we choose the parameter as a bifurcation parameter to stud the Neimark-Sacker bifurcation for the uniue positive fied point E +, of the sstem 1.8 b using the Center Manifold Theorem and bifurcation theor in [, 9, 15, 5, 9] in this section. We consider the sstem 1.8 with parameters p,, r, m S E+, which is described b, ep p + r + m..1 The first step. Giving a perturbation of parameter, we consider a perturbation of the sstem.1 as follows:, ep p + +. r + m, where 1. The second step. Let u = and v =, which transforms the fied point E +, to the origin O, and sstem. into u v, +. v v + ep p + r + u + m.

7 Stabilit and Neimark-Sacker bifurcation 45 The characteristic euation associated with the linerization of the sstem. at u, v =, is given b λ a λ + b =,.4 where a = ep p + + r + m, and b = ep p m m r + m r + m. Correspondingl, when varies in a small neighborhood of =, the roots of the characteristic euation are Hence and λ 1, = 1 d λ 1, d = [ a ± i ] 4b a..5 λ 1, = b 1/.6 = mm m + + r >..7 r + m m m 1/ In addition, it is reuired that λ i 1, 1, i = 1,,, 4 when =. Since a = = 1 and b = = 1, we have λ 1, = 1 1 ± i = e ±i π, which obviousl satisf λ 1, m 1, m = 1,,, 4..8 The third step. Stud the normal form of the sstem. when =. Epanding the sstem. as Talor series at u, v =, to the third order, we obtain where u a 1 u + a 1 v + a u + a 11 uv + a v + a u + a 1 u v + a 1 uv + a v + O u + v 4, v b 1 u + b 1 v + b u + b 11 uv + b v + b u + b 1 u v + b 1 uv + b v + O u + v 4, a 1 =, a 1 = 1, a =, a 11 =, a =,, a =, a 1 =, a 1 = a =, b 1 = 1, b 1 = 1, b 1 =, b =, b = m m 1 r + m + m m 1 r + m 4 m m 1 m 1 r + m, b 11 = m m 1 r + m, b 1 = m m r + m + m m r + m 4 m m 1 m r + m, b =, b = m m 1 m 6m m r + m 4 6m m r + m 5 m m r + m 6 m m 1m m r + m + 6m m 1 m r + m..9

8 46 C. Wang & X. Y. Li Let JE + = a1 a 1 1, namel, JE b 1 b + = B some computations we obtain the eigenvalues of the matri JE + are λ 1 = 1 + i and λ = 1 i. The fourth step. Find the normal form of.. Let matri T = 1 1, then T 1 1 =. 1 Using transformation u, v T = T X, Y T, the sstem.9 is transformed into the following form X 1 X Y + F X, Y + O X + Y 4, Y X + 1 Y + GX, Y + O X + Y 4, where F X, Y = and..1 b + 1 b 11 Y + b 11 XY + b 1 XY + b + 1 b 1 Y GX, Y =. The fifth step. Compute some coefficients. On the center manifold the sstem.9 has the above norm form.1. For convenience, for a function F 1,,..., n, denote F i, F i j, and F i j k as the first order, the second order and the third order partial derivative of F 1,,..., n, respectivel. Then, F XX, =, F XY, = b 11, F Y Y, = 4 b + b 11, F XXX, =, F XXY, =, F XY Y, = b 1, F Y Y Y, = 1 b + 6 b 1, G XX, =, G XY, =, G Y Y, =, G XXX, =, G XXY, =, G XY Y, =, G Y Y Y, =. The sith step. Compute the discriminating uantit a, which determines the stabilit of the invariant circle bifurcated from Nemark-Sacker bifurcation of the sstem.1 and can be computed via the formulae see [5] [1 a λλ ] = Re L 11 L 1 1 λ L 11 L + ReλL 1,.11

9 Stabilit and Neimark-Sacker bifurcation 47 where L = 1 8 [F XX F Y Y + G XY + ig XX G Y Y F XY ], L 11 = 1 4 [F XX + F Y Y + ig XX + G Y Y ], L = 1 8 [F XX F Y Y G XY + ig XX G Y Y + F XY ], L 1 = 1 16 [F XXX + F XY Y + G XXY + G Y Y Y + ig XXX + G XY Y F XXY F Y Y Y ]..1 Some computations produce [ 4 b + b 11 L = 1 8 L = 1 8 Hence, [ ] + ib 11, L 11 = 1 4 b + b b + b 11 + ib 11 ], L 1 = 1 16 a = 1 8 b 11b b 1 b, [ 1 b 1 i b + 6 ] b 1..1 = 16m m r + m m m 8m 7m m r + m 4 m 9m 8 m r + m + m m 1m m r + m. Clearl,.7 and.8 demonstrate that the transversal condition and the nondegenerate condition of the sstem 1.8 are satisfied. So, summarizing the above discussions, we obtain the following conclusion. Theorem.1. If a, then the sstem 1.8 undergoes a Neimark-Sacker bifurcation at the fied point E +, when the parameter varies in the small neighborhood of origin. Moreover, if a < resp., a >, then an attracting resp., repelling invariant closed curve bifurcates from the fied point for > resp., <. Two eamples, which illustrate the above Theorem.1, are given below. Eample.1. Consider the sstem 1.8 with r =.19, m =, p = 1, = =. Then, there is a uniue positive fied point E +.1,.1 with the multipliers λ = 1 + i and λ = 1 i, λ = 1, d λ 1, d = 5.5 >, and a = <. = Hence, according to Theorem.1, an attracting invariant closed curve bifurcates from the fied point for >. Eample.. Consider the sstem 1.8 with r = 9, m = 1, p = 1, = = 1. The uniue positive fied point is E + 1, 1 for the sstem 1.8, whose multipliers are λ = 1 + i and λ = 1 i with λ = 1, d λ 1, d =.11 > and =

10 48 C. Wang & X. Y. Li a =.5 >. Hence, Theorem.1 tells us that an repelling invariant closed curve bifurcates from the fied point for <. 4. Numerical simulation In this section, b using numeral simulation, we give the bifurcation diagrams, phase portraits and Lapunov eponents of the sstem 1.8 to confirm the previous theoretical analsis and show some new interesting comple dnamical behaviors eisting in the sstem 1.8. Without lose generalit, the bifurcation parameters are considered in the following two cases: Case 1. Fi the parameters p =.1, r = 1, m =, the initial value, =.1,.1 and assume that varies in the interval [1.8,.8]. Evidentl, < mp < 1. We see that the sstem 1.8 has the uniue positive 1 fied point E +.1, Theorem.1 i in Section Figures 1, and show the correctness of the a The dnamic behavior of b The dnamic behavior of Figure 1. The dnamic behavior for the sstem 1.8 which eist for p =.1, r = 1, m = and [1.8,.8]. 1 From Figure 1 we see that the fied point E + stable..1, 1.1 is asmptoticall 1.1, 1.1, the Taking = 1.8,.,. and.8 and submitting it into E + positive fied point is.884,.884,.16,.16,.4641,.4641 and 4.46, 4.46, respectivel. The phase portraits corresponding to Figure 1 are plotted in Figure which show that the fied point is asmptoticall stable. The maimum Lapunov eponents corresponding to Figure 1 and are computed and plotted in Figure in which we can easil see that the maimal Lapunov eponents are negative for the parameter [1.8,.8], that is to sa, the fied point E + 1.1, 1.1 is stable.

11 Stabilit and Neimark-Sacker bifurcation a = b = c = d =.8 Figure. Phase portrait of the sstem 1.8 versus. Maimal Lapunov eponents..4.6 Ma.Lap Figure. Maimal Lapunov eponent versus corresponding to Figure 1 and. Case. Choose the parameters p = 1, r =.19, m =, the initial values, =.1,.1 and assume that varies in the interval [1.9, 4.8]. 1 We see that mp > 1 and the uniue positive fied point is E +.1, 1.1. After calculation for the positive fied point of the sstem 1.8, we find that the Neimark-Sacker bifurcation emerges from the fied point.1,.1 at =, whose multipliers are λ 1, = 1±i with λ 1, = 1.

12 4 C. Wang & X. Y. Li a [1.9,.] b [.,.8] c [.8,.] d [.,.6] e [.6, 4.] f [4., 4.4] g [4.4, 4.8] h 4.8, 5.] Figure 4. Bifurcation diagrams of component for the sstem 1.8 versus.

13 Stabilit and Neimark-Sacker bifurcation 41 a [1.8,.] b [.,.8] c [.8,.] d [.,.6] e [.6, 4.] f [4., 4.4] g [4.4, 4.8] h 4.8, 5.] Figure 5. Bifurcation diagrams of component for the sstem 1.8 versus.

14 4 C. Wang & X. Y. Li In Figures 4 and 5, the bifurcation diagrams for the sstem 1.8 are plotted as a function of the control parameter for From Figures 4a and 5a, it is clear that the fied point is stable for <, and loses its stabilit at the Neimark-Sacker bifurcation parameter value =. An attracting invariant circle appears when the parameter eceeds. This shows the correctness of the Theorem.1. Figures 4 and 5 also displa the new and interesting dnamics as increases. The maimum Lapunov eponents corresponding to Figures 4 and 5 are computed and plotted in Figure 6, in which we can easil see that the maimal Lapunov eponents are negative for the parameter 1.9,., that is to sa that fied point is stable for <. For., 5., some Lapunov eponents are positive and some are negative, so there eist stable fied point or stable period windows in the chaotic region. In general, when the maimal Lapunov eponent is positive, this can be considered to be one of the characteristics for the eistence of chaos.. Maimal Lapunov eponents.4 Maimal Lapunov eponents Ma.Lap.. Ma.Lap a [1.8,.8] b [.8, 5.] Figure 6. Maimal Lapunov eponent versus corresponding to Figure 4 and 5. The phase portraits are considered in the following: An attractive fied point takes place for = 1.98, which means that the sstem orbit is a fied point, as shown in Figure 7a. Figure 7b shows that fied point E + is a stable attractor at = For this parameter value, the fied point E + occurs with =.995, =.995 and the associated comple conjugate eigenvalues are λ 1, =.5 ±.865i with λ 1, =.995, which means that the fied point E + is asmptoticall stable. Figure 7c demonstrates the behavior of the sstem 1.8 before the Neimark- Sacker bifurcation when = while Figure 7d demonstrates the behavior of the sstem 1.8 after the Neimark-Sacker bifurcation when =.1. From Figure 7c and Figure 7d, we deduce that the fied point E + loses its stabilit through a Neimark-Sacker bifurcation when the parameter varies from to.1.

15 Stabilit and Neimark-Sacker bifurcation a = b = c = d = e = f = g = h = i = j = k = l =.85 Figure 7. Phase portraits for the sstem 1.8 versus. Increasing the control parameter =., the sstem 1.8 has the fied point E +.195,.195, whose associated eigenvalues are λ 1, =.5 ± i.9686 with λ 1, = 1.9. So, one can conclude that the fied point becomes unstable and invariant closed curve is created around the fied point. Figure 7e and 7f confirms the above argument. Continuing to increase the value of, we observe that the dnamics of the fied point E + becomes comple from Figure 7g-l. There eist chaotic sets. 5. Conclusion In this paper, a semi-discrete model is derived for a nonlinear simple population model and its stabilit and bifurcation have been investigated. Our results displa that a Neimark-Sacker bifurcation phenomenon occurs in the positive fied point

16 44 C. Wang & X. Y. Li of this sstem under certain parametric conditions. Some other basic dnamical properties of the sstem 1.8 have been analzed b means of bifurcation diagrams, phase portraits, Lapunov eponents. Numerical simulations show that the sstem has more comple dnamical behaviors than its corresponding continuous case. References [1] R.P. Agarwal, Difference Euations and Ineualities: Theor, Methods and Applications, Marcel Dekker, NewYork,. [] H.N. Agiza, E.M. ELabbas, H. EL-Metwall and A.A. Elsadan, Chaotic dnamics of a discrete pre-predator model with Holling tpe II, Nonlinear Anal. RWA, 19, [] J. Carr, Application of Center Manifold Theorem, Springer-Verlag, New York, [4] B.S. Chen and J.J. Chen, Bifurcation and chaotic behavior of a discrete singular biological economic sstem, Appl. Math. Comput., 191, [5] H.A. El-Morshed, Global attractivit in a population model with nonlinear death rate and distributed delas, J. Math. Anal. Appl., 4114, [6] E.M. Elabbas, A.A. Elsadan and Y. Zhang, Bifurcation analsis and chaos in a discrete reduced Lorenz sstem, Appl. Math. Comput., 814, [7] D.J. Fan and J.J. Wei, Bifurcation analsis of discrete survival red blood cells model, Commun. Nonlinear Sci. Numer. Simulat., 149, [8] R.K. Ghaziani, W. Govaerts and C. Sonckb, Resonance and bifurcation in a discrete-time predator-pre sstem with Holling functional response, Nonlinear Anal. RWA, 11, [9] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dnamical Sstems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 198. [1] Z.M. He and X. Lai, Bifurcation and chaotic behavior of a discrete-time predatorcpre sstem, Nonlinear Anal. RWA, 111, [11] Z.Y. Hu, Z.D. Teng and H.J. Jiang, Stabilit analsis in a class of discrete SIRS epidemic models, Nonlinear Anal. RWA,11, 17-. [1] Z.Y. Hu, Z.D. Teng and L. Zhang, Stabilit and bifurcation analsis in a discrete SIR epidemic model, Math. Comput. Simul., 9714, 8-9. [1] Z.Y. Hu, Z.D. Teng and L. Zhang, Stabilit and bifurcation analsis of a discrete predator-pre model with nonmonotonic functional response, Nonlinear Anal. RWA, 111, [14] D. Jana, Chaotic dnamics of a discrete predator-pre sstem with pre refuge, Appl. Math. Comput., 41, [15] Y.A. Kuzenetsov, Elements of Applied Bifurcation Theor, nd Ed., Springer- Verlag, New York, [16] I. Kubiaczk and S.H. Saker, Oscillation and stabilit in nonlinear dela d- ifferential euations of population dnamics, Appl. Math. Model., 5, [17] Y.G. Li, Dnamics of a discrete food-limited population model with time dela, Appl. Math. Comput., 181,

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