DYNAMIC BEHAVIORS OF A NONAUTONOMOUS

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Ann. of Diff. Eqs. 29:4(2013), 379-389

DYNAMIC BEHAVIORS OF A NONAUTONOMOUS RATIO-DEPENDENT LESLIE SYSTEM INCORPORATING A PREY REFUGE ∗ Wanlin Chen, Fengde Chen†, Xiaojie Gong, Kun Yang (College of Math. and Computer Science, Fuzhou University, Fuzhou 350108)

Abstract A nonautonomous ratio-dependent Leslie system incorporating a prey refuge is studied in this paper. By applying the comparison theorem of differential equations and constructing a suitable Lyapunov function, a set of sufficient conditions which guarantee the persistent property and global attractivity of the system is obtained. Also, by applying the comparison theorem of differential equations and Fluctuation Lemma, a set of sufficient conditions which ensure the extinction of the prey species and the global attractivity of predator species is obtained. This result shows that for the Lotka-Volterra type predator-prey system, when the value of prey refuge increases, predator species will be driven to extinction due to the lack of food. Our study shows that the alternative food resource predator species is always permanent, which means that prey refuge has no influence on the permanence of predator species. However, refuge plays an important role in the persistent property of the prey species: large enough prey refuge could keep the persistent property of the prey species, while small enough refuge could lead to the extinction of prey species. Numerical simulations show the feasibility of the main results. Keywords Leslie system; refuge; ratio-dependent; global attractivity; extinction 2000 Mathematics Subject Classification 34D23; 92B05; 34D40

1

Introduction

As was pointed out by Ma et al [1], prey species make use of refuges to decrease predation risk, here refuge means a place or situation where predation risk is somehow reduced. Examples include spatial refuges (burrows, heavy vegetation), temporal refuges. During the last decade, many scholars have studied the dynamic behaviors of predator-prey system incorporating a prey refuge, they have found that the influence of refuge depends on the system investigated ([2-14]). Chen, Ma and Zhang [2] incorporated a constant number of refuge to the traditional Lotka-Volterra predator-prey system, to ensure that the positive equilibrium to the system is globally attractive, and found that the refuge should be restricted to a certain range. T. Kar [4] considered an autonomous Lotka-Volterra predator-prey system with Holling type II response function incorporating a prey refuge, and found that if prey refuge size is large enough, prey population would outbreak and predator species would be extinct because of the lack of food resource. Chen, Chen and Xie [5] studied the influence of refuge on an autonomous Leslie-Gower predator-prey model, they found that the system always admits a unique globally attractive positive equilibrium, which means that the refuge has no ∗ †

Manuscript received May 23, 2013 Corresponding author. E-mail: [email protected]

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influence on the persistent property of the system. Based on the work of [5], Wu [6] studied a modified Leslie-Gower predator-prey model, and obtained sufficient conditions for the existence and global stability of the positive equilibrium, also found that the prey species would be driven to extinction if prey refuge size is small enough. Considering the environment fluctuation, it is nature to study the nonautonomous predator-prey system with prey refuges ([7,10-14]). Zhu and Liu [10] studied a non-autonomous Lotka-Volterra predator-prey system incorporating a time varying prey refuge, and obtained some sufficient conditions which ensure the existence and global stability of the positive periodic solution. However, as was pointed out by Wu, Chen and Ma [11], for this system, the comparison theorem of the differential equation could not be applied directly, hence we doubt that the results of [10] is correct. By applying the fluctuation theorem, the authors [11] obtained a set of sufficient conditions which guarantees the extinction of the predator species. Stimulated by the works of [5,6], Wu, Chen and Zhang [12] considered a nonautonomous modified Leslie-Gower predator-prey model incorporating a prey refuge, and found that enough large amount of refuge could guarantee the persistent of the system. Increasing other food resources can increase the amount of the predator population, but this has a negative effect on the density of prey species. We know that all the works of [2,5,6,10-12] assumed that the predator functional response depends only on prey abundance, but this is not always appropriate. Recently, more and more biological evidences show that in many conditions, such as when the predators have to share or compete for food, predator-prey system which relies on the theory of ratio-dependence is more realistic and general. Already there have been several scholars investigating the nonautonomous case of such kind of systems ([14-17]). Yu, Xiong and Qi [14] considered a non-autonomous ratio-dependent predator-prey system with delays and refuge. They obtained sufficient conditions for the permanence and global stability of the system and found that suitable refuge size could ensure the persistent of the system; Chen, Chen and Shi [15] studied a non-autonomous predator-prey system with Beddington-DeAngelis functional response, and obtained a set of sufficient conditions which ensures that the predator is driven to extinction. Sarwardi, Haque and Mandal [16] considered the modified ratio-dependent Bazykin model with delay in predator equation, and the essential mathematical features were proposed by analysis the equilibria, local and global stability and bifurcation theory. Recently, by applying the fluctuation theorem, Yu [17] obtained a set of sufficient conditions which ensure the global asymptotic stability of the positive equilibrium to a predator-prey model with modified Leslie-Gower and Holling-Type II schemes. In this paper, based on the works of [2,5,6,11,15], we study a nonautonomous predatorprey model incorporating a prey refuge, where each of two species obeys the logistic law of growth, and the carrying capacity of the predator’s environment is proportional to the sum of prey densities and other food resource. It is assumed that the predator response function is ratio-dependence. The goal of this paper is to find out the possible answer about the influence of the prey refuge and other food resource.

2

Model Formulation In 1948, Leslie [18] introduced the following famous Leslie predator-prey model

No.4 W.L. Chen, etc., A TYPE OF RATIO-DEPENDENT LESLIE MODEL 381  ( ) dx    dt = x(t) a − bx(t) − h(x)y(t), ( dy y(t) )    = y(t) r1 − r2 , dt x(t)

(2.1)

where x(t) and y(t) denote the densities of the prey and predator at time t, respectively; a, b, ri , i = 1, 2 are all positive constants; a and r1 stand for intrinsic growth rates; a/b and x(t)/r2 stand for the carrying capacities of the environment, where the carrying capacity of the predator’s environment is proportional to the densities of prey species; h(x) is predator functional response to prey. When h(x) is of Holling type II functional response, that is, h(x) = cx/(d + x), then we have  ( ) dx cx(t)    dt = x(t) a − bx(t) − d + x(t) y(t), (2.2) (  dy y(t) )   = y(t) r1 − r2 , dt x(t) where c is the maximal predator per capita consumption rate; d is the half-saturation constant, and both of them are positive constants. If predator-prey system relies ( )on the theory of ratio-dependence, the functional response of Holling-Tanner type is h x(t) y(t) . When h(x) is a functional response of Holling type II, the ratio-dependent Leslie predator-prey system with the functional response of Holling-Tanner is as follows:  ( ) cx(t) dx   = x(t) a − bx(t) − y(t),  dt dy(t) + x(t) (2.3) (  dy y(t) )   = y(t) r1 − r2 . dt x(t) If predator’s most favorite food (x) is not available in abundance, it would choose other species as its food resource. This situation can be represented by adding a positive constant k to the denominator, where k denotes the abundance of other food resource [19]. Hence, the ratio-dependent modified Leslie predator-prey system can be written as follows:  ( ) cx(t) dx    dt = x(t) a − bx(t) − dy(t) + x(t) y(t), (2.4) (  dy y(t) )   = y(t) r1 − r2 . dt x(t) + k Considering that prey may go to the refuge to avoid predation, we further assume that there is a refuge which is proportional to the prey species and denoted by mx(t), thus the prey available for the predators is (1 − m)x(t), where m ∈ (0, 1). This leads to the following system:  ( ) c(1 − m)x(t) dx    dt = x(t) a − bx(t) − dy(t) + (1 − m)x(t) y(t), (2.5) ( )  dy y(t)   = y(t) r1 − r2 . dt (1 − m)x(t) + k Considering that environment fluctuation, the nonautonomous one is more realistic, this motivated us to study the following system:

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 ( ) ( ) c(t) 1 − m(t) x(t)  dx   ( ) y(t),  dt = x(t) a(t) − b(t)x(t) − d(t)y(t) + 1 − m(t) x(t) ( )  dy y(t)   ) ,  dt = y(t) r1 (t) − r2 (t) ( 1 − m(t) x(t) + k(t)

(2.6)

where a(t), b(t), c(t), d(t), k(t), m(t), ri (t), i = 1, 2 are all continuous and bounded above and below by positive constants on (−∞, +∞). And system (2.6) is assumed to satisfy initial conditions x1 (0) = x01 > 0, x2 (0) = x02 > 0. It is easy to see that x(t) > 0, y(t) > 0 for all t ≥ 0. We shall now investigate the dynamics of the system (2.6). For the rest of the paper, given a bounded continuous function f defined on R, let f l and f u be defined as f l = inf {f (t)}, t∈R

3 3.1

f u = sup{f (t)}, t∈R

fεl = f l − ε,

fεu = f u + ε.

Main Results Permanence of the System

As a direct corollary of Lemma 2.2 of [20], we have: dx Lemma 3.1 If a > 0, b > 0 and ≥ x(b−ax)(≤ x(b−ax)), when t ≥ 0 and x(0) > 0, dt we have b ( b) lim inf x(t) ≥ lim sup x(t) ≤ . t→+∞ a a t→+∞ Lemma 3.2 Let (x(t), y(t))T be any positive solution to system (2.6), then we have lim sup x(t) ≤ xu , t→+∞

u

lim sup y(t) ≤ y u , t→+∞

u

where x and y are defined by (3.1) and (3.2) respectively. Proof From the first equation of system (2.6) and the positivity of the solution to system (2.6), one has ( ) ( ) dx ≤ x(t) a(t) − b(t)x(t) ≤ x(t) au − bl x(t) . dt Applying Lemma 3.1 to the above differential inequality leads to au def lim sup x(t) ≤ l = xu , (3.1) b t→+∞ hence, for any ε > 0 sufficiently small, there exists an enough large T1 > 0 such that x(t) < xuε , t > T1 . From the second equation of system (2.6) and the above inequality, one has ( ) dy r2l y(t) ≤ y(t) r1u − , t > T1 , dt (1 − ml )xuε + k u so, ( ) r1u (1 − ml )xuε + k u lim sup y(t) ≤ . r2l t→+∞ Setting ε → 0, it follows ( ) r1u (1 − ml )xu + k u def u lim sup y(t) ≤ = y . (3.2) r2l t→+∞ This ends the proof of Lemma 3.2.

No.4 W.L. Chen, etc., A TYPE OF RATIO-DEPENDENT LESLIE MODEL 383 Theorem 3.1 Assume that ml > 1 −

al dl cu

(H1 )

holds, then system (2.6) is permanent. Proof Let (x(t), y(t))T be any positive solution to system (2.6). By the first equation of system (2.6), one has ( ) ( ) c(t) 1 − m(t) dx ≥ x(t) a(t) − b(t)x(t) − y(t) dt d(t)y(t) ( ) ( u c 1 − ml ) ≥ x(t) al − bu x(t) − . (3.3) dl According to Lemma 3.1 and condition (H1 ), we obtain lim inf x(t) ≥ t→+∞

al dl − cu (1 − ml ) def l = x. dl bu

(3.4)

From the second equation of system (2.6), one has ( ) ( ) r2 (t) ru dy ≥ y(t) r1 (t) − y(t) ≥ y(t) r1l − 2l y(t) , dt k(t) k so, lim inf y(t) ≥ t→+∞

r1l k l def l = y. r2u

(3.5)

(3.1), (3.2), (3.4) and (3.5) show that system (2.6) is permanent under assumption (H1 ). This ends the proof of Theorem 3.1. Remark 3.1 Theorem 3.1 shows that due to the alternative food resource, predator species is always permanent. Furthermore, from the proof of (3.3) and (3.4), we know that if al dl > cu , then prey species is also permanent, in which case, refuge has no influence on the persistent property of the system. However, if al dl < cu , one could always choose refuge which is large enough (it means m is large enough) such that inequality ml > 1 − al dl /cu holds, then the refuge could protect enough amount of the prey, this results in less prey available to the predator. The chance for the prey alive are improved and this finally leads to the permanence of the prey species.

3.2

Global Attractivity of the System

Let

( )2 ( ) { c(t) 1 − m(t) y u r2 (t) 1 − m(t) y u } λ1 = inf b(t) − ( ( ) )2 − ( ( ) )2 , t∈R+ d(t)y l + 1 − m(t) xl k(t) + 1 − m(t) xl ( )2 { r (t)k(t) + r (t)(1 − m(t))xl } c(t) 1 − m(t) xu 2 2 λ2 = inf+ −( ( ( ) )2 ( ) )2 . t∈R k(t) + 1 − m(t) xu d(t)y l + 1 − m(t) xl

Theorem 3.2 In addition to (H1 ), assume further that λ1 > 0,

λ2 > 0

hold, then the positive solution to system (2.6) is globally attractive.

(H2 )

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Proof Let (x(t), y(t))T , (x∗ (t), y ∗ (t))T be any two positive solutions to system (2.6). From condition (H2 ), there exists an ε > 0 (which is small enough and ε < min{xl , y l }/2) such that ( )2 ( ) { c(t) 1 − m(t) yεu r2 (t) 1 − m(t) yεu } λ1ε = inf b(t) − ( ( ) )2 − ( ( ) )2 > 0, t∈R+ d(t)yεl + 1 − m(t) xlε k(t) + 1 − m(t) xlε ( )2 { r (t)k(t) + r (t)(1 − m(t))xl } c(t) 1 − m(t) xuε 2 2 ε λ2ε = inf+ −( ( ( ) )2 ( ) )2 > 0. (3.6) t∈R k(t) + 1 − m(t) xuε d(t)yεl + 1 − m(t) xlε According to Theorem 3.1, for the above ε > 0, there exists T2 > T1 > 0 which is large enough, such that xlε < x(t), x∗ (t) < xuε , yεl < y(t), y ∗ (t) < yεu , t > T2 .

(3.7)

Now define Lyapunov functions as follows: V1 (t) = | ln x(t) − ln x∗ (t)|,

V2 (t) = | ln y(t) − ln y ∗ (t)|,

and V (t) = V1 (t) + V2 (t). Calculating the right upper derivative of V1 along the positive solution to system (2.6), we have ( )2 c(t) 1 − m(t) y(t)|x(t) − x∗ (t)| )( ) D+ V1 (t) ≤ −b(t)|x(t) − x∗ (t)| + ( d(t)y(t) + (1 − m(t))x(t) d(t)y ∗ (t) + (1 − m(t))x∗ (t) ( )2 c(t) 1 − m(t) x(t)|y(t) − y ∗ (t)| ( ( ) )( ( ) ) + d(t)y(t) + 1 − m(t) x(t) d(t)y ∗ (t) + 1 − m(t) x∗ (t) ( )2 c(t) 1 − m(t) yεu ∗ ∗ ≤ −b(t)|x(t) − x (t)| + ( ( ) )2 |x(t) − x (t)| d(t)yεl + 1 − m(t) xlε ( )2 c(t) 1 − m(t) xuε ∗ +( (3.8) ( ) )2 |y(t) − y (t)|, t > T2 , l l d(t)yε + 1 − m(t) xε ( ) k(t)r2 (t) + r2 (t) 1 − m(t) x∗ (t) + (( ) )(( ) ) |y(t) − y ∗ (t)| D V2 (t) ≤ − 1 − m(t) x(t) + k(t) 1 − m(t) x∗ (t) + k(t) ( ) r2 (t) 1 − m(t) y ∗ (t) ) )(( ) ) |x(t) − x∗ (t)| + (( 1 − m(t) x(t) + k(t) 1 − m(t) x∗ (t) + k(t) ( ) k(t)r2 (t) + r2 (t) 1 − m(t) xlε ∗ ≤ − (( ) )2 |y(t) − y (t)| 1 − m(t) xuε + k(t) ( ) r2 (t) 1 − m(t) yεu ∗ + (( (3.9) ) )2 |x(t) − x (t)|, t > T2 . 1 − m(t) xlε + k(t) It follows from (3.8) and (3.9) that D+ V (t) ≤ −λ1ε |x(t) − x∗ (t)| − λ2ε |y(t) − y ∗ (t)|,

t > T2 .

No.4 W.L. Chen, etc., A TYPE OF RATIO-DEPENDENT LESLIE MODEL 385 From (3.6), there exists a positive constant µ such that min{λ1ε , λ2ε } > µ > 0, thus ( ) D+ V (t) ≤ −µ |x(t) − x∗ (t)| + |y(t) − y ∗ (t)| , t > T2 . (3.10) Integrating both sides of (3.10) from T2 to t deduces ∫ t ( ) V (t) + µ |x(s) − x∗ (s)| + |y(s) − y ∗ (s)| ds < V (T2 ) < +∞,

t > T2 .

T2

Then



t

lim sup t→+∞

(

T2

) V (T2 ) |x(s) − x∗ (s)| + |y(s) − y ∗ (s)| ds < < +∞. µ

dy It follows from Theorem 3.1 and system (2.6) that dx dt and dt are uniformly bounded, ∗ ∗ therefore |x(t)−x (t)|+|y(t)−y (t)| is uniformly continuous, and by the above inequality, we get |x(t)−x∗ (t)|+|y(t)−y ∗ (t)| ∈ L1 ([T2 , +∞)). By Barbalat’s lemma [21], we conclude that

lim |x(t) − x∗ (t)| = 0,

t→+∞

lim |y(t) − y ∗ (t)| = 0.

t→+∞

This completes the proof of Theorem 3.2.

3.3

Extinction of the Prey Species

Theorem 3.1 shows that prey species is permanent if refuge is large enough to ensure that inequality (H1 ) holds. Some interesting questions are proposed: What will happen if inequality (H1 ) does not hold? Will prey species be driven to extinction in this case? We try to find out the answers in this subsection. Consider the Logistic equation ( ) dy(t) r2 (t) = y(t) r1 (t) − y(t) . (3.11) dt k(t) According to Lemma 4.1 of [15], it has exactly one solution y ∗∗ (t) bounded above and below by positive constants on R, which is given by (∫ )−1 t ( ∫ t ) r2 (s) ∗∗ y (t) = exp − r1 (ξ)dξ ds . −∞ k(s) s Theorem 3.3 Assume that

{ η = sup a(t) − def

t∈R+

( ) c(t) 1 − m(t) y l } ( ) 0 which is sufficiently small such that ( ) { c(t) 1 − m(t) yεl } ( ) < 0. (3.12) ηε = sup a(t) − d(t)yεl + 1 − m(t) xuε t∈R+ From the proof of Lemma 3.2 and Theorem 3.1, for the above ε > 0, there exists T3 > T1 > 0 which is large enough such that 0 < x(t) < xuε ,

yεl < y(t),

y ∗∗ (t) < yεu ,

t > T3 .

(3.13)

Let g(y) = c(1 − m)y/[dy + (1 − m)x], then we could easily verify that g(y) is a nondecrea-

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sing function of y, so,

( ) ( ) c(t) 1 − m(t) yεl c(t) 1 − m(t) y(t) ( ) ( ) , ≤ d(t)yεl + 1 − m(t) x(t) d(t)y(t) + 1 − m(t) x(t)

t > T3 .

According to the first equation of system (2.6), (3.13) and the above inequality, it follows ( ) ( ) c(t) 1 − m(t) y(t) dx ( ) ≤ x(t) a(t) − dt d(t)y(t) + 1 − m(t) x(t) ( ) ( c(t) 1 − m(t) yεl ) ( ) ≤ x(t) a(t) − ≤ ηε x(t), t > T3 . d(t)yεl + 1 − m(t) xuε Form (3.12), there exists a negative constant δ such that ηε < δ < 0, so, (∫ t ) ( ) x(t) ≤ x(T3 ) exp ηε (s)ds ≤ x(T3 ) exp δ(t − T3 ) → 0, t → +∞. T3

Therefore, lim x(t) = 0. t→+∞

Let ω(t) = 1/y(t), ω ∗∗ (t) = 1/y ∗∗ (t) and z(t) = ω(t) − ω ∗∗ (t), then it follows that dω r2 (t) ) , = −r1 (t)ω(t) + ( dt 1 − m(t) x(t) + k(t) hence

dω ∗∗ r2 (t) = −r1 (t)ω ∗∗ (t) + , dt k(t)

( ) r2 (t) 1 − m(t) x(t) dz (( ) ). = −r1 (t)z(t) − dt k(t) 1 − m(t) x(t) + k(t)

(3.14)

(3.15)

From the definitions of ω(t) and ω ∗∗ (t), and the boundedness of y(t) and y ∗∗ (t), ω(t) and ω ∗∗ (t) are bounded, so z(t) is a bounded differentiable function. By the Fluctuation lemma [15], there exist sequences σn → +∞, θn → +∞, such that dz dz z(σn ) → lim inf z(t), → 0, z(θn ) → lim sup z(t), → 0 (n → +∞). t→+∞ dt σn dt θn t→+∞ We shall show that lim inf z(t) = lim sup z(t) = 0. t→+∞

t→+∞

From (3.15) we have

( ) r2 (t) 1 − m(t) x(t) 1 dz (( ) ). z(t) = − − r1 (t) dt r1 (t)k(t) 1 − m(t) x(t) + k(t)

Since 0
0.495 ≈ al dl /cu , λ1 ≈ 0.78 > 0, λ2 ≈ 0.58 > 0, then conditions (H1 ) and (H2 ) hold. By Theorem 3.2, system (4.1) is globally attractive. Figure 1 shows the dynamics behaviors of system (4.1) for m = 0.8 with the initial conditions (x(0), y(0))T = (0.1, 0.5)T , (0.5, 1.8)T , (1.5, 2)T and (0.8, 1)T , respectively.

No.4 W.L. Chen, etc., A TYPE OF RATIO-DEPENDENT LESLIE MODEL 389 [5] F.D. Chen, L.J. Chen, X.D. Xie, On a Leslie-Gower predator-prey model incorporating a prey refuge, Nonlinear Analysis: Real World Applications, 10:5(2009),2905-2908. [6] H.H. Wu, On a predator-prey model with modified Leslie-Gower and prey refuge, Journal of Fuzhou University (Natural Science), 38:3(2010),342-346. (in Chinese) [7] Y.B. Zhang, W.X. Wang, Y.H. Duan, Analysis of prey-predator with Holling III functional response and prey refuge, Mathematics in Practice and Theory, 40:24(2010),149-154. (in Chinese) [8] L.J. Chen, F.D. Chen, L.J. Chen, Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a constant prey, Nonlinear Analysis: Real World Applications, 11:1(2010),246-252. [9] Y.J. Huang, F.D. Chen, Z. Li, Stability analysis of a prey-predator model with holling type III response function incorporating a prey refuge, Applied Mathematics and Computation, 182:1(2006),672-683. [10] J. Zhu, H.M. Liu, Permanence of the two interacting prey-predator with refuges, Journal of Northwest University (Natural Edition), 27:62(2006),1-3. (in Chinese) [11] Y.M. Wu, F.D. Chen, Z.Z. Ma, Extinction of predator species in a nonautonomous predatorprey system incorporating prey refuge, Applied Mathematics: A Journal of Chinese Universities, 27A:3(2012),359-366. [12] Y.M. Wu, W.L. Chen, H.Y. Zhang, On permanence and global attractivity of a nonautonomous modified Leslie-Gower predator-prey system incorporating a prey refuge, Journal of Minjiang University, 33:5(2012),17-20. (in Chinese) [13] G.M. Xu, X.H. Chen, Persistence and periodic solution for three interacting predator-prey system with refuges, Yinshan Academic Journal, 23:1(2009),14-17. (in Chinese) [14] S.P. Yu, W.T. Xiong, H. Qi, A ratio-dependent one predator-two competing prey model with delays and refuges, Mathematica Applicata, 23:1(2010),198-203. (in Chinese) [15] F.D. Chen, Y.M. Chen, J.L. Shi, Stability of the boundary solution of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response, Journal of Mathematical Analysis and Applications, 344:2(2008),1057-1067. [16] S. Sarwardi, M. Haque, P.K. Mandal, Ratio-dependent predator-prey model of interacting population with delay effect, Nonlinear Dynamics, 69(2012),817-836. [17] S.B. Yu, Global asymptotic stability of a predator-prey model with modified Leslie-Gower and Holling-Type II schemes, Discrete Dynamics in Nature and Society, Volume 2012 (2012), Article ID 208167, 8 pages. [18] P.H. Leslie, Some further notes on the use of matrices in population mathematics, Biometrika, 35(1948),213-245. [19] M.A. Aziz-Alaoui, M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Applied Mathematics Letters, 16(2003),1069-1075. [20] F.D. Chen, On a nonlinear nonautonomous predator-prey model with diffusion and distributed delay, Journal of Computational and Applied Mathematics, 180:1(2005),33-49. [21] I. Barbalat, Systemes d’quations differential d’scillations nonlinearies, Revue Roumaine de Mathematiques Pures et Appliquees, 4(1959),267-270.

(edited by Liangwei Huang)

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