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 Applied Mathematics, 2011, 2, 47-56 doi:10.4236/am.2011.21006 Published Online January 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Permanence and Global Stability for a Non-Autonomous Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes with Delays Lin Hu, Linfei Nie College of Mathematics and Systems Science, Xinjiang University, Urumqi, China E-mail: lfnie@163.com Received August 18, 2010; revised November 11, 2010; accepted November 14, 2010 Abstract In this paper, a nonautonomous predator-prey system based on a modified version of the Leslie-Gower scheme and Holling-type II scheme with delayed effect is investigated. The general criteria of integrable form on the permanence are established. By constructing suitable Lyapunov functionals, a set of easily veri- fiable sufficient conditions are derived for global stability of any positive solutions to the model. Keywords: Predator-Prey System, Leslie-Gower and Holling-Type-II Functional Response, Permanence, Global Stability 1. Introduction Predator-prey behavior is a form of very common bio- logical interaction in nature. There are many mathemati- cal models to model predator-prey behavior such as Lotka-Volterra system, Chemostat-type system, Kolmo- gorov system, etc (see [1-6]). In recent years there has been a growing interest in the study of mathematical models incorporates a modified version of Leslie-Gower functional response as well as that of the Holling-type II (see [7-9]). In particular, in [10] the authors consider the following model 1 2 d d. d d xt cyt xta bxt txtk yt eyt yt d txtk (1.1) This two species food chain model describes a prey population x which serves as food for predator y, a, b, c, e, k1 and k2 are positive parameters. They estab lished the sufficient conditions for the boundedness, existence of a positively invariant attracting set and global stability of coexisting interior equilibrium. In [11] the authors con- sidered the dynamical behavior of system (1.1) with de- lays, and establish the sufficient conditions for the exis- tence positive equilibrium, permanence and global sta- bility of positive equ ilibrium. The dynamical behavior of system (1.1) also has been discussed by many authors (see, for example, [7,12] and the references cited therein). However, we note that any biological or environmen- tal parameters are naturally subject to fluctuation in time. As [13] pointed out that the growth properties of every natural population vary through time. Most, and perhaps all, of this variation arises ultimately from fluctuations in the population's environment. Physical environmental conditions usually change greatly through the year and can influence organisms directly. Good weather can sti- mulate growth in body size and reproduction, and bad weather can cause death. Similarly, the biological envi- ronment can fluctuate in ways that influence population dynamics. These kinds of time variation in population dynamical events can exert profound effects on the ecology and evolution of individual species and on the composition of ecological commu nities. In this paper, we are concerned with the effects of the time-dependent of ecological and environmental para- meters and time delays due to gestation and negative feedbacks on the global dynamics of predator-prey sys- tems with Modified Leslie-Gower and Holling-Type II Schemes. Therefore, we consider the following delayed differential system: 1122 11 111 111 2224 22 132 d() d d d xt atxt xt rtbtxt txtkt xt atxt xtrt txtkt (1.2)  L. HU ET AL. Copyright © 2011 SciRes. AM 48 with initial conditions 1234 ,,0, ,0 ,1,2,max,,,, ii i x Ci (1.3) where ,0:,0:0, ,0,CCss 1 and 00, x t and 2 x t denote the densities of prey and predator population, respectively; 0 i 1, 2,3, 4i denote the time delays due to negative feedbacks of the prey and the predator population, 1,, ii bt rtat and 1, 2 i kt i are model para- meters. These parameters are defined as follows: 1 rt is the growth rate of prey 11 , x bt measures the strength of competition among individuals of species 11 , x at is the maximum value which per capita reduc- tion rate of 1 x can attain, 1 kt and 2 kt measure the extent to which environment provides protection to prey 1 x and predator 2 x , respectively; 2 rt is the growth rate of predator 2 x , and 2 at has a similar meaning to 1 at. The organization of this paper is as follows. In the next section, we present some basic assumptions for sys- tem (1.2) and two important lemmas on the nonauto- nomous single-species logistic system. In Section 3, we will state and prove the sufficient conditions of integra- ble form on the permanence of solutions for system (1.2). We also by means of suitable Lyapunov functionals, a set of easily verifiable sufficient conditions are derived for global stability of any positive solutions of system (1.2). Numerical result is presented to illustrate the valid- ity of our main results. 2. Preliminaries Let 0:0,R and :0,R . For a bounded continuous function g t on R, we use the following notations: :sup , utR g gt :infR lt g gt . For system (1.2), we introduce the following assump- tions. 1 H Function 1,, , iii bt rtat kt are conti- nuous and bounded on 0 R, 10at for all 0t, and 0 inf0 1,2 ti tik . 2 H There a constant 0 such that 12 infd 0, inflimd .l0im tt tt tt ba 3 H There is a constant 0 such that 12 lim linfd 0,infd 0.im tt tt tt rr It is well known by the fundamental theory of func- tional differential equations [5] that system (1.2) has a unique solution 12 , x txtxt satisfying initial conditions (1.3). If 0 1,2 i xt i on the interval of existence, then x t is said to be a positive solution. It is easy to verify that solutions of system (1.2) corres- ponding to initial conditions (1.3) are defined on 0, and remain positive for all 0t. We consider the following single-species nonauto- nomous logistic system with a parameter 2 22 ) d:,,. d atu uur tugtu tkt (2.1) Obviously, if Assumption 1 H holds, then ,,gtu is a continuous function defined on 00 ,, 0,tuRR , where 0 is constant. We easily prove that for any 00 0 ,tuRR and 0 0, system (2.1) has a unique solution ut satisfying initial condition 00 ut u . It is easy to see that ut is positive for all 0 tt if the initial value 00u. If Assumptions 13 HH hold, then the following statements can be prove to be true. 1 A For any constant 1,, ,gtu is bounded on 1 00 ,0,R . 2 A There are positive constants 12 1 2 ,,,kk and 12 kk such that 1 2 1 2 lim l inf,,0 d0, sup ,im,0d0. t t t t t t gk gk 3 A Partial derivative ,,0 g tu u exists for all 0 ,tuRR , and there are nonnegative continuous function qt and a constant 0 , satisfying inlim fd0 t t tq and a continuous function pu, satisfying 0pu for all uR , such that 0 ,,0 for all ,. gtu qt putuRR u 4 A Partial derivative ,,gtu exists for all 00 ,, 0,tuRR , and for any constant 0U, ,,gtu is also bounded on 00 , ,0,0,tu RU . In system (2.1), when parameter 0 we obtain the following system 2 22 d:,,0. d atu uur tugtu tkt (2.2.) Let * 0 ut be a fixed positive solution of system (2.2) defined on 0 R . We say that * 0 ut is globally uniformly attractive on 0 R , if for any constants 1 and 0 there is a constants ,0T such that  L. HU ET AL. Copyright © 2011 SciRes. AM 49 for any initial time 00 tR and any solution 0 ut of system (2.1) with 1 00 ,ut , one has * 00 utut for all 0,tt T . By Lemma 1 given in [14], we have the following res ult. Lemma 2.1 Suppose that Assumptions 13 AA hold, then a) There is a constants 1 M such that 100 inf sulimlim p tt M utut M for any positive solution 00 ut of system (2.2). b) Each fixed positive solution * 0 ut of system (2.2) is globally uniformly attractive on 0 R Let 0 uR , 00 tR and 0 0, , and further let ut and 0 ut be the solutions of systems (2.1) and (2.2) with initial value 00 ut u and 00 0 utu , respectively. By Lemma 2 given in [14], we further have the following result. Lemma 2.2 Suppose that assumptions 14 AA hold, then ut converges to 0 ut uniformly for 0,tt as 0 . 3. Main Results In this section, we proceed to discussion on the perma- nence and global stability of any positive solution of system (1.2) corresponding to initial conditions (1.3). We first give the result of the ultimate boundedness of any solution for system (1.2). Theorem 3.1 Suppose that Assumptions 13 HH hold, then any solution 12 , x txtxt of system (1.2) corresponding to initial conditions (1.3) are ulti- mately bounded. Proof: Let 12 , x txtxt be any solution of system (1.2) corresponding to initial conditions (1.3). From the first equation of system (1.2) we have 111 11 dfor all0. d u xt xtrt rxtt t (3.1) For any t and ,0s , integrating (3.1) from ts to t we obtain 1111 1 expexp . uu xt sxtrsxtr By assumptions 1 H, we further have 111 111 dexp d u xt xtrtbtxtr t for all 0t. It is proved in many articles, for example, see [15], that under Assumptions 13 HH any pos- itive solution ut of the following non-autonomous single-species logistic equation 11 1 dexp d u ut utr tbtutr t is ultimately bounded. Hence, using the comparison theorem, we can obtain that there is a constant 10M such that for any solution 12 , x txtxt of sys- tem (1.2) corresponding to initial conditions (1.3), there is a 10t such that 11 x tM for all 1 tt. From the second equation of system (1.2) we have 222 22 d. d u xt x tr trxt t (3.2) For any t and ,0s , integrating (3.2) from ts to t we obtain 22222 expexp . uu xtsxt rsxtr By assumptions 1 H, we further have 222 22 2 12 dexp d u xt atxt xtrtr tMkt for all 1 tt . The compar ison equa tion is the logistic equation 2 22 12 dexp . d u yta tyt ytr tr tMkt Similarly, by Assumptions 13 HH, we further can obtain that there is constant 20M such that for any solution 12 , x txtxt of system (1.2) cor- responding to initial conditions (1.3), there is a 21 tt such that 22 x tM for all 2 tt. Therefore, the so- lution x t is ultimately bounded. This completes the proof of this theorem . In particular, when parameter 24 0 in system (1.2), we obtain the following system 112 11 111 111 222 22 132 d d. d d xt atxt xt rtbtxt txtkt xt atxt xtrt txtkt (3.3) with initial conditions 11 112 ,,0,,0,00xCx (3.4) As a consequence of Theorem 3.1, we have the fol- lowing corollary on the ultimate boundedness of any solution for system (3.3) with the initial conditions (3.4). Corollary 3.1 Suppose that Assumptions 13 HH hold, then any solution 12 , x txtxt of system (3.3) corresponding to initial conditions (3.4) is ulti- mately bounded. Next, on the permanence of component 2 x of system (1.2) with the initial conditions (1.3), we have the fol- lowing result. Theorem 3.2 Suppose that Assumptions 13 HH hold, then the component 2 x of system (1.2) is perma- nent, in the sense there is a constant 0 such that  L. HU ET AL. Copyright © 2011 SciRes. AM 50 2 inflim txt for all solutions of system (1.2) corresponding to initial conditions (1.3). Proof: Let 12 , x txtxt be any solution of system (1.2) corresponding to initial conditions (1.3). From Theorem 3.1, there is constant 0M such that for any positive solution x t of system (1.2), there is a 0T such that 1, 2 i xt Mi for all tT. Therefore, from the second equation of system (1.2) we have 22 22 12 2 d d xt atM x trt xt tkt (3.5) for all tT , where 1222 sup . tR rtatMkt For any tT and ,0s , integrating (3.5) from ts to t we obtain 22 121 expexp .xts xts xt Further, we have 222 22 12 2 dexp d xt atxt xtrt tkt for all tT . By Assumptions 13 HH and Lemma 2.1, we can obtain that the component u of sys- tem 2 212 2 dexp d uta tut utrt tkt is permanent. Hence, using the comparison theorem, we can obtain the component 2 x of system (1.2) is perma- nent. This completes the proof of this theorem. In order to obtain permanence of component 1 x of system (1.2), we consider the following auxiliary system with a parameter 222 22 24 2 dexp . d u xt atxt xtrt r tkt (3.6) In particular, when 0 in system (3.6), we obtain the following system 222 22 24 2 dexp . d u xt atxt xtrt r tkt (3.7) By Assumptions 13 HH, we see that system (3.7) satisfies all conditions of Lemma 2.1. Hence, by Lemma 2.1, each positive of system (3.7) is globally asymptotically stable. Let * 20 x t be some fixed solu- tion of system (3.7) with initial value * 20 00x. On the permanence of component 1 x for system (1.2), we have the following result. Theorem 3.3 Suppose that Assumptions 13 HH hold and there is a constant 0 such that 1* 1202 1 linfd 0im , t t t as rsxss ks (3.8) then the component 1 x of system (1.2) is permanence. Proof: Let 12 , x txtxt be any solution of system (1.2) corresponding to initial conditions (1.3). From Theorems 3.1 and 3.2, there are constants 0M and 0m such that for any positive solution x t of system (1.2), there is a 0T such that 1 x tM and 2 mxt M for all tT. In fact, if inequality (3.8) is true, then by Assumption 3 H, we can choose enough small positive constants 01 ,, and 01 , and an enough large 0 TT such that 1* 1102021 1 0 d for all+. t t as rs bsx ss ks tT (3.9) For any 0 , let 2 x t be the solution of system (3.6) with initial value * 220 00xx . Hence, by con- clusion (b) of Lemma 2.1 and Lemma 2.2, there is a con- stant 00 such that *1 220 2 xt xt for all 0t and 0 0, . Let 01 min , , 2 x t be any positive solu- tion of the following system 222 22 24 2 dexp d u xt atxt xtrt r tkt (3.10) with initial value * 220 00xx . In the following, we will use two claims to complete the proof of Theorem 3.3. Claim 3.1 For the above constant , there always exist 1 suplim txt for any positive solution x t of system (1.2). In fact, if Claim 3.1 is not true, then there is a positive solution 12 , x txtxt of system (1.2) such that 1 suplim txt . Hence, there is a 10 TT such that 1 xt for all 1 tT. Further, using the comparison theorem and Lemma 2.1, we can obtain that there is a constant 21 TT such that * 1 22 201 2 xtxtx t (3.11) for all 2 tT . From the first equation of system (1.2) we have 2 112 1* 11 221 1 exp d t T xt xT as rsbsx ss ks  L. HU ET AL. Copyright © 2011 SciRes. AM 51 for all 2 tT . By (3.9) it follows that 1 xt as t. This is contradictory with 1 xt for all 2.tT From this contradiction we finally conclude that 1 suplim txt Therefore, Claim 3.1 is true. Claim 3.2 There is a constant 0 such that 1 inflim txt for any positive solution x t of system (1.2). If Claim 3.2 is not true, then there is a sequence of function 12 ,: ,0,1,2 nnnin Ci such that for the solution 12 ,,, nn xt xt of sys- tem (1.2) 12 inf,, 1,2,.lim n txt n n By Claim 3.1, for every n there are two time se- quences n q s and n q t, satisfying 11 22 0nn nnnn qq stst st and lim n qq s , such that 11 2 ,, , nn qn qn xs xt nn (3.12) and 1 2,for all,. nn nqq xtts t n n (3.13) From the ultimate boundedness of system (1.2) and Theorem 3.2, we can choose a positiv e constant n T for every n such that 1,n x tM and 2,n mxt M for all n tT . Further, there is an integer 10 n K such that nn q sT for all 1n qK. Let 1n qK, then for any , nn qq tst we have 11 111 1 01 d, , d ,, nn n xt atM xtrt btM tkt xt where 001111 suptrtbtMat ktM . In- tegrating the above inequality from n q s to n q t, we further h av e 11 0 ,,exp . nn nn qnqnq q xtxst s Consequentl y , by (3. 12) 0 2exp . nn qq ts n n Hence, 1 0 ln for a. ll nnn qq n ts qK (3.14) By (3.9), there is constant P such that 1* 11 221 1 d t t as rsbsx ss ks (3.15) for all 2 tT and P . Let 2 x t be the solution of system (3.10) with ini- tial value 22 , nn qqn xs xs . Since for any n, q and , nn qq tst we have 1,n xtX and 222 22 24 2 dexp , d u xt atxt xtrt r tkt by the comparison theorem, we have ()() 22 ,for all,. nn nqq xtx ttst (3.16) Since lim n nq s and Theorem 3.2, there is con- stant 21 nn K K for every n such that 2, n qn x sM for all 2n qK. By the comparison theorem and 2 x t is the globally uniformly attractive solution of system (3.10), we obtain that there is a constant 32 TT and such that 1 22 3 for all++. 2 n q xt xttTs (3.17) By (3.14), there is an integer 01 NN such that 302 for all,. nn n qq tsT PnNqK Further, by (3.11), (3.16) and (3.17) we ha ve * 2201 ,n xtx t (3.18) for all 3, nn qq ts Tt and 0 nN. H ence, when 0 nN and n qK, integrating the first equation of system (1.2) from 3 n q sT to n q t, by (3.12), (3.13), (3.15), and (3.18) we have 3 3 113 12 111 11 * 120 1 11 21 2 ,, (, ) exp(,)d (, ) () expd 8 . n q n q n q n q nn qn qn t n nn sT t sT xtxs T atxt rt btxtt xtk t at xt rtbtt pt kt n n  L. HU ET AL. Copyright © 2011 SciRes. AM 52 This leads to a contradiction with (3.12). Therefore, Claim 3.2 is true. Finally, from Claims 3.1 and 3.2 we see that Theorem 3.3 is proved and this completes the proof of this theo- rem. Remark 3.1 Nind ji n an d Azi z -Al aouiIn [11] discussed the following system 112 1111 11 222 22 12 d d. d d xt axt xt rbxt txtk xt axt xtr txtk (3.19) They stated that if 2 1 21 2112 1 , r r rak erka b (3.20) then system (3.19) is permanent. We note that, when system (1.2) degenerates into system (3.19), the condi- tion (3.20) clearly implies the conditio n (3.8) in Theo rem 3.3. So the theorem of A. F. Nindjin, M. Aziz-AlaouiIn (Theorem 5 in [11]) is a special case of Theorem 3.3. So our results are fresh and more general. A direct consequence of Theorem 3.3 is the following result on the permanence of system (3.3) and (3.4). Corollary 3.2 Suppose that Assumptions 13 HH hold and there is a constant 0 such that 1* 12 1 ilnf ,imd 0 t t t as rsx s ks where * 2 x t is be any solution of the following system 222 22 2 d, d x tatxt xtrt tkt then system (3.3) is permanent. Finally, we proceed to the discussion global stability of any positive solution of system (1.2). We first derive certain upperbou nd est imates for sol ut i on of system (1.2). Theorem 3.4 Let 12 , x txtxt denote any solutions of system (1.2) corresponding to initial condi- tions (1.3). Suppose that Assumptions 13 H,H hold, and 10 l b, 20 l a, then there is a constant 0T such that if tT, 112 2 0,0 , x tM xtM where 2 1212 1 12 12 ,. u ur uu r u ll reM k re MM ba (3.21) The proof of Theorem 3.4 is similar to that of Theo- rem 2.1 in [16], we therefore omit it here. We now formulate the global stability of any positive solutions of system (1.2). Theorem 3.5 Let *** 12 , x txtxt denote any positive solu tions of system (1.2). Suppose that Assump- tions 13 H,H hold, and 10 l b, 20 l a, assume further that 4 Hiim nf0li tBt where 1 1 1 34 3 12 11 2 1 22 211 1 1 2 11 111121 2 11 2 2322322 22 2 23 23 222 22 12 d d d [] [] [ t t t t t t atM Bt btkt atM rtbtMbs s kt at btMMMbss kt at Mat Mas s ks ktkt at atM Bt rt Mkt 4 4 4 12 2 2 22 2 12242 2 12 242 121 1 12 d d d ]t t t t t t as s kt ks atatMas s kt ktks at Mbss kt (3.22) Then the solution *** 12 , x txtxt is globally asymptotically stable. Proof: Let 12 , x txt be any solution of system (1.2) and (1.3). It follows from Theorem 3.4 that there exist positive constants T and i M (defined by (3.21), such that for all tT, * 0,0,1,2. iii i xt Mxt Mi (3.23) We define * 111 1 lnln .Vtxt xt Calculating the upper right derivative of 11 Vt along solutions of system (1.2), it follows that * 11 * 1111*1 1 * 11 * 12 212 2 *111 111 * 11111 sgn sgn xt xt DVtxtxtxt xt xt xt atxt atxt x tkt xt kt bt xtxt  L. HU ET AL. Copyright © 2011 SciRes. AM 53 ** 11 11111 1* 12222 1 * 112222 * 221111 122 * 11 11 1 * 1111 * 11 sgn sgn xtxtbt xtxt at ktxt xt t xtxtxt xt xtxt atxt xt xtbtt at xt xtxt 1 1 * 2222 12 2* 111 1 * 111 1* 2222 1 d t t kt xt xt atxt btxt xt t btxu xuu atxt xt kt (3.24) where * 1111111 txtktxtkt * 2132132 .txt ktxt kt On substituting (1.2) into (3.24), we derive that 1 11 12 2* 111 1 1* 2222 1 *** 1111111 ** 11 22 * 111 11111 1 112 2 111 1 1 d | t t DV t atxt btxt xt t atxt xt kt btruxu buxuxu auxuxu xu ku ruxu buxuxu auxuxu u xu ku a bt 22 * 11 1 1* 2222 1 tx txt xt t atxt xt kt 1 11111 112 2 1 ** 112211 1 * 1112 2 1 * 1111 1* 1111 1 1 * 2222 d. |t t btru buxu au kuxu u x uxu xuxu au buxu xuxu u xu xu au kuxu xuxu u xuxuu (3.25) Define 1 1 12111 11 1122 1 ** 112211 1 * 1112 2 1 1 ** 1111 11 1 1 tt ts Vtbsrubuxu au kuxu u xuxuxu xu au buxu xuxu u au x uxukuxu u xux * 112222 dd.uxu xuus (3.26) We obtain from (3.25) and (3.26) that 1 11 12 12 2* 111 1 1* 2222 1 11111 1* 12 21122 1 1 ** 11 11122 1 * 1111 d t t DV tV t atxt btxt xt t atxt xt kt bss rtbtxt at ktxtxt xt t at xt xtbtxtxtxt t a xt xt 1* 11 1 * 1112 22 2 . tktxt t xtxtxtxt (3.27)  L. HU ET AL. Copyright © 2011 SciRes. AM 54 We now define 1111213 ,VtVt VtVt (3.28) where 1 11 211 1311111 2 2 11 * 11 dd tl tl al Vtbsbl MMM kl xlxlsl (3.29) and 2 12 22 12 * 1422 12 121 * 122 12 d dd. t t l t tl as Vtxs xss ks al M bsxlxl sl kl (3.30) It then follows from (3.23) and (3.27)-(3.30) that for tT , 1 1 1 12 2 1 12 12 1111 21 1 11 1111 12 2 11 2121 * 11112 12 * 122 12 d d d. { [ t t t t t t DV t atM atM btrtbtMkt kt at bss btMMM kt at M bssxt xtkt at bssxt xt kt (3.31) Similarly, we define 22122 ,VtV tV t (3.32) where * 2122 ln lnVtxtxt (3.33) and 4 4 222 222 * 2132 22 * 22 * 132 * 2424 2 22 * 1313 2 dd. tt ts as auM Vt ru ks x uku auM xu xuxu ku xu xu auM x uxu us u (3.34) Calculating the upper right derivative of 2 Vt along solutions of system (1.2), we derive for tT that 4 4 4 2 222 2 12 2 2* 22 2 2 22222 22 2 22 22 * 13132 2* 2424 2 d d d. t t t t t t DVt at atM rt Mkt kt assxt xt ks atMatMas s ks kt kt atM xt xtkt assx tx t ks (3.35) We define a Lyapuno v f unct ional Vt as 1234 ,VtVt VtVt Vt (3.36) where 3 34 33 232 * 311 2 23 2 232 2* 11 2 223 d dd t t l t tl as M Vtxsxss ks al M as x lxlsl ks kl (3.37) and 4 44 22242 * 422 224 dd. tl tl asal M Vtxlxl sl ks kl (3.38) It then follows from (3.22), (3.31), and (3.35)-(3.38) that for tT ** 1112 22 .DVtBtxt xtBtxt xt (3.39) where 1 Bt and 2 Bt are defined in (3.22 ). By Assumption 4 H, there exist positive constants 1 , 2 and * TT such that if * tT 0, 1,2. ii Bt i (3.40) Integrating both sides of (3.39) on interval *,Tt , * 2** 1d. t iii T i VtB sxsx ssVT (3.41) It follows from (3.40) and (3.41) that  L. HU ET AL. Copyright © 2011 SciRes. AM 55 * 2*** 1dfor all. t iii T i Vtxsx ssVTtT Therefore, Vt is bounded on *,T and also ** d,1,2. ii Txs xssi By Theorem 3.4, * 1,2 ii xt xti are bounded on *,T . On the other hand, it is easy to see that * i x t and 1,2 i xt i are bounded for * tT. Therefore, * 1,2 ii xt xti are uniformly continuous on *,T . By Barbalat’s Lemma ([17], Lemmas 1.2.2 and 1.2.3), we conclude that * lim0,1, 2. ii txt xti This completes the proof of this theorem. Remark 3.2 If time delays 1 , 233 ,, and 4 are naturally subject to fluctuation in time in system (1.2). Similar Theorem 3.1-3.5, we can obtain the sufficient conditions on the permanence and globally asymptoti- cally stable of any positive solutio ns for system (1.2). Finally, we give some examples to illustrate the feasi- bility of our main results on the permanence of system (1.2). Example 3.1 In system (1.2), let 0.2 10.2 0.05re 0.1sin ,t 10.9cos ,bt 20.4,r 12 1 1, 2,aa k 122 0.1, 0.5, 1k . It is easy to verify that coeffi- cients of system (1.2) satisfy (3.8). By Theorem 3.2 and 3.3, system (1.2) is permanent. Example 3.2 In system (1.2), let 0.2 10.2re, 10.1b, 12 1aa, 12k, 20.4r, 10.1, 20.5 , 21k. By Theorem 3.2 we see that the com- ponent 2 x of system (1.2) is permanent. However, it is easy verify that 24 0.2 0.2 1122 12 0.20.42 0 r rarke kaee thus (3.8) does not hold for system (1.2) and we cannot get any information by Theorem 3.3. In this case, we note that 0.2 112212 0.20.4 20rarkkae and nu- merical simulation suggests that system (1.2) with a se- quence initial condition 12 , is permanent. In the example 3.2, from numerical simulation, we note that the time delays are harmless for the permanence. Therefore, as an improvement of Theorems 3.2 and 3.3, we give the following interesting conjecture. Conjecture: Suppose the assumptions of Corollary 3.2 hold, then system (1.2) is permanent. 4. Acknowledgements This work was supported by the National Natural Science Foundation of P.R. Chin a (11001235, 10961022) and the Natural Science Foundation of Xinjiang Univer- sity (BS100104, BS080105). 5. References [1] M. Fan, Q. Wang and X. F. Zou, “Dynamics of Non- autonomous Ratio-Dependent Predator-Prey System,” Proceedings of the Royal Society of Edinburgh: Section A, Vol. 133, No. 1, 2003, pp. 97-118. doi:10.1017/S0308 210500002304 [2] Y. Kuang, “Delay Differential Equations, with Applica- tions in Population Dynamics,” Academic Press, New York, 1993. [3] Z. Teng, “Persistence and Stability in General Non-au- tonomous Single-Species Kolmogorov Systems with De- lays,” Nonlinear Analysis: Real World Applications, Vol. 8, No. 1, 2007, pp. 230-248. doi:10.1016/j.nonrwa. 2005.08.003 [4] Z. 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