Permanence for a discrete time Lotka–Volterra type food-chain model with delays

Applied Mathematics and Computation 186 (2007) 279–285 www.elsevier.com/locate/amc

Permanence for a discrete time Lotka–Volterra type food-chain model with delays Xinyuan Liao

a,b,*,1

, Shengfan Zhou

a,1

, Yuming Chen

c,2

a

b

Department of Mathematics, Shanghai University, Shanghai 200444, PR China School of Mathematics and Physics, Nanhua University, Hengyang, Hunan 421001, PR China c Department of Mathematics, Wilfrid Laurier University, Waterloo, Ont., Canada N2L 3C5

Abstract Considered is a delayed discrete time Lotka–Volterra type food-chain model. Sufficient conditions are established for the permanence and the feasibility of the obtained results are illustrated with an example. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Permanence; Delay; Discrete Lotka–Volterra system; Food-chain; Coexistence

1. Introduction One of the most important questions in population ecology is to find the coexistence conditions for the species. In order to consider this question, several mathematical concepts of coexistence of species are developed. Permanence is one of such concepts. Permanence of Lotka–Volterra systems described by differential equations has been extensively investigated. For some recent results, see [1–7] and the references therein. However, many authors have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Discrete time models can also provide efficient computational models of continuous models for numerical simulations. Therefore, lots have been done on discrete Lotka–Volterra systems (see, [8–19]). Most of the above-mentioned references focus on the existence of periodic solutions and their stability [8,10–12,16–18]. Though some authors considered permanence for discrete Lotka–Volterra systems [9,13–15,18,19], the systems are either competitive or of predator–prey type.

*

Corresponding author. Address: School of Mathematics and Physics, Nanhua University, Hengyang, Hunan 421001, PR China. E-mail address: [email protected] (X. Liao). 1 Research was supported by the National Natural Science Foundation of China (No. 10471086) and Educational Committee Foundation of Hunan Province (No. 05C494). 2 Research was partially supported by the Natural Science and Engineering Research Council of Canada (NSERC) and the Early Research Award Program (ERA) of Ontario. 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.07.096

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X. Liao et al. / Applied Mathematics and Computation 186 (2007) 279–285

Recently, Xu et al. [17] considered the following discrete time Lotka–Volterra type food-chain model with delays: ( ) m m X X x1 ðn þ 1Þ ¼ x1 ðnÞ exp r1 ðnÞ  a11l ðnÞx1 ðn  lÞ  a12l ðnÞx2 ðn  lÞ ; (

l¼0

x2 ðn þ 1Þ ¼ x2 ðnÞ exp r2 ðnÞ þ ( x3 ðn þ 1Þ ¼ x3 ðnÞ exp r3 ðnÞ þ

m X

l¼0

a21l ðnÞx1 ðn  lÞ 

m X

l¼0

l¼0

m X

m X

a32l ðnÞx2 ðn  lÞ 

l¼0

a22l ðnÞx2 ðn  lÞ  )

m X

) a23l ðnÞx3 ðn  lÞ ;

l¼0

a33l ðnÞx3 ðn  lÞ ;

l¼0

ð1:1Þ where x1(n), x2(n) and x3(n) denote the densities of prey, predator and top predator population at the nth generation, respectively; r1(n) is the intrinsic growth rate of the prey population at the nth generation, a11l(n) measures the influence of the (n  l)th generation of the prey on the density of the prey population, and a12l(n) represents the influence at the (n  l)th generation of the predator on the prey population; r2(n) is the death rate of the predator population at the nth generation, a21l(n) stands for the influence at the (n  l)th generation of the prey on the conversion rate of the predator population, a22l(n) measures the influence of the (n  l)th generation of the predator on the density of the predator population, and a23l(n) represents the influence at the (n  l)th generation of the top predator on the predator population; r3(n) is the death rate of the top predator population at the nth generation, a32l(n) stands for the influence at the (n  l)th generation of the predator on the conversion rate of the top predator population, a33l(n) measures the influence of the (n  l)th generation of the top predator on the density of the top predator population. Under the assumption that ri, aij : Z ! Rþ are positive x-periodic, the authors derived sufficient conditions on the existence of positive periodic solutions by using the continuation theorem of coincidence degree theory, where Z is the set of all integers and Rþ ¼ ½0; 1Þ. To the best of our knowledge, no work on permanence of discrete Lotka–Volterra type food-chain has been done yet. Thus, the purpose of this paper is to fill this gap. Precisely, we study the permanence for system (1.1). For permanence of food-chain systems described by differential equations, we refer to Chen and Cohen [20] and Li and Teng [21] and the references therein. Throughout this paper, we always assume {ri(n)} and {aijl(n)} are bounded nonnegative sequences such that 0 < ri 6 ri , 0 < aijl 6  aijl , i, j = 1, 2, 3, l = 0, 1, . . . , m. Here, for any bounded sequence {a(n)}, a ¼ supn2N aðnÞ and a ¼ inf n2N aðnÞ, where N is the set of all nonnegative integers. Also, for biological reasons, we only consider solutions of (1.1) with initial conditions xi ðkÞ P 0; xi ð0Þ > 0;

k ¼ 1; 2; . . . ; m;

ð1:2Þ

i ¼ 1; 2; 3:

This paper is organized as follows. In Section 2, we establish sufficient conditions on permanence for system (1.1). Then, in Section 3, the main result is applied to a discrete time analogue of a differential system studied by Xu and Chen [5]. 2. Main result In order to present our main result, Theorem 2.5, we need some preparations. Definition 2.1. System (1.1) is said to be permanent if there exist positive constants mi and Mi, i = 1, 2, 3, such that mi 6 lim inf xi ðnÞ 6 lim sup xi ðnÞ 6 M i ; n!1

i ¼ 1; 2; 3;

n!1

for any solution x(n) = (x1(n), x2(n), x3(n)) of system (1.1) with initial condition (1.2).

X. Liao et al. / Applied Mathematics and Computation 186 (2007) 279–285

281

Lemma 2.1 (Lemma 1 [22]). Assume that {x(n)} satisfies x(n) > 0 and xðn þ 1Þ 6 xðnÞ expfrðnÞð1  axðnÞÞg for n 2 [n1, 1), where a is a positive constant and n1 2 N. Then lim sup xðnÞ 6 n!1

1 expðr  1Þ: ar

Lemma 2.2 (Lemma 2 [22]). Assume that {x(n)} satisfies xðn þ 1Þ P xðnÞ expfrðnÞð1  axðnÞÞg;

n P N 0;

limsupn!1x(n) 6 K and x(N0) > 0, where a is a constant such that aK > 1 and N 0 2 N. Then 1 lim inf xðnÞ P expfrð1  aKÞg: n!1 a Qn Before proceeding, we make a convention that i¼m F ðiÞ ¼ 1 if m > n. The main result, Theorem 2.5, will follow directly from the following two propositions. Proposition 2.3. Let x(n) = (x1(n), x2(n), x3(n)) be any solution of system (1.1) with the initial condition (1.2). Then lim sup xi ðnÞ 6 K i ;

i ¼ 1; 2; 3;

n!1

where expðr1  1Þ ; r1 lÞ l¼0 a11l expð Pm expðK 1 l¼0  a21l  r2  1Þ Pm K 2 ¼ Pm ; a expðr l a21k Þ 2  K 1l l¼0 22l k¼0  Pm expðK 2 l¼0  a32l  r3  1Þ Pm K 3 ¼ Pm : a32k Þ l¼0 a33l expðr3 l  K 2 l k¼0  K 1 ¼ Pm

Proof. First, we prove limsupn!1x1(n) 6 K1. From the first equation of system (1.1), we have x1 ðn þ 1Þ 6 x1 ðnÞ expfr1 ðnÞg: It follows that, for l = 0, 1, . . . , m and n P l, n1 Y

x1 ði þ 1Þ 6

i¼nl

n1 Y

x1 ðiÞ expfr1 ðiÞg;

i¼nl

or

(

n1 X

x1 ðnÞ 6 x1 ðn  lÞ exp

) r1 ðiÞ :

i¼nl

In other words,

(

x1 ðn  lÞ P x1 ðnÞ exp 

n1 X

) r1 ðiÞ

i¼nl

and hence

(

x1 ðn þ 1Þ 6 x1 ðnÞ exp r1 ðnÞ  ( 6 x1 ðnÞ exp r1 

m X l¼0

m X l¼0

( a11l ðnÞx1 ðnÞ exp  )

a11l expfr1 lgx1 ðnÞ :

n1 X i¼nl

)) r1 ðiÞ

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X. Liao et al. / Applied Mathematics and Computation 186 (2007) 279–285

It follows from Lemma 2.1 that lim sup x1 ðnÞ 6 K 1 : n!1

Second, we prove that limsupn!1x2(n) 6 K2. From the second equation of system (1.1), for sufficient large n, we have ( ) m X x2 ðn þ 1Þ 6 x2 ðnÞ exp r2 ðnÞ þ K 1 a21l ðnÞ : l¼0

Similar argument as above produces ( !) n1 m X X x2 ðn  lÞ P x2 ðnÞ exp r2 ðiÞ  K 1 a21k ðiÞ i¼nl

and

k¼0

( x2 ðn þ 1Þ 6 x2 ðnÞ exp r2 ðnÞ þ K 1

a21l ðnÞ 

l¼0

( 6 x2 ðnÞ exp

m X

K1

m X

 a21l  r2

l¼0

m X

( a22l ðnÞ exp

l¼0

! 

m X

(

n1 X

a22l exp l r2  K 1

l¼0

r2 ðiÞ  K 1

i¼nl m X

!) a21k

m X

!) a21k ðiÞ

) x2 ðnÞ

k¼0

)

x2 ðnÞ :

k¼0

Again, it follows from Lemma 2.1 that lim sup x2 ðnÞ 6 K 2 : n!1

Finally, from the third equation of system (1.1), we have ( ) m X x3 ðn þ 1Þ 6 x3 ðnÞ exp r3 ðnÞ þ K 2 a32l ðnÞ : l¼0

Now, it is not difficult to see that lim sup x3 ðnÞ 6 K 3 n!1

can be proved in the same manner as that for limsupn!1x2(n) 6 K2. This completes the proof. Let K1, K2, and K3 be the positive constants defined in Proposition 2.3. We introduce Pm Pm Pm a11l exp½ðK 1 k¼0  a11k þ K 2 k¼0  a12k  r1 Þlg l¼0 f Pm ; D1 ¼ r1  K 2 l¼0  a ( ) ! 12l m X 1 r1  K 2 exp a12l ð1  K 1 D1 Þ ; m1 ¼ D1 l¼0 Pm Pm Pm Pm a22l exp½ðr2  k¼0 a21k m1 þ K 2 k¼0 a22k þ K 3 k¼0 a23k Þlg l¼0 f Pm Pm D2 ¼ ; r2  K 3 l¼0 a23l l¼0 a21l m1   ( ) ! m m X X 1  exp r2 þ a23l K 3 ð1  K 2 D2 Þ ; a21l m1  m2 ¼ D2 l¼0 l¼0 Pm Pm Pm a33l exp½ðr3  k¼0 a32k m2 þ K 3 k¼0 a33k Þlg l¼0 f Pm D3 ¼ ; r l¼0 a32l m2   ( ) ! 3 m X 1  m3 ¼ a32l m2 ð1  K 3 D3 Þ : exp r3 þ D3 l¼0 Now, we make the following assumption: (H) min16i63KiDi > 1.

h

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283

Proposition 2.4. Suppose assumption (H) holds. Then, for any solution x(n) = (x1(n), x2(n), x3(n)) to system (1.1) with the initial condition (1.2), lim inf xi ðnÞ P mi ;

i ¼ 1; 2; 3:

n!1

Proof. We first prove liminfn!1x1(n) P m1. It follows from the first equation of system (1.1) that, for sufficient large n, ( ) m m X X a12l ðnÞ  a11l ðnÞx1 ðn  lÞ : x1 ðn þ 1Þ P x1 ðnÞ exp r1 ðnÞ  K 2 l¼0

Then, for l = 0, 1, . . . , m and n P l, n1 Y

x1 ði þ 1Þ P

i¼nl

n1 Y

l¼0

(

x1 ðiÞ exp r1 ðiÞ  K 2

m X

i¼nl

or, equivalently,

(

x1 ðn  lÞ 6 x1 ðnÞ exp

a12k ðiÞ 

k¼0

n1 X

K1

m X

i¼nl

a11k ðiÞ þ K 2

k¼0

m X

)! a11k ðiÞx1 ði  lÞ

;

k¼0

m X

!) a12k ðiÞ  r1 ðiÞ

:

k¼0

This, combined with the first equation of system (1.1), gives us ( x1 ðn þ 1Þ P x1 ðnÞ exp r1 ðnÞ  K 2

a12l ðnÞ 

l¼0

( P x1 ðnÞ exp

m X

r1 ðnÞ  K 2

m X

( a11l ðnÞx1 ðnÞ  exp

l¼0

m X

!

)

n1 X i¼nl

K1

m X

a11k ðiÞ þ K 2

k¼0

m X

!)) a12k ðiÞ  r1 ðiÞ

k¼0

a12l ðnÞ ð1  D1 x1 ðnÞÞ :

l¼0

Applying Lemma 2.2, we get lim inf x1 ðnÞ P m1 : n!1

Second, we prove liminfn!1x2(n) P m2. From the second equation of system (1.1), we have ( ) m m m X X X x2 ðn þ 1Þ P x2 ðnÞ exp r2 ðnÞ þ a21l ðnÞm1  a22l ðnÞx2 ðn  lÞ  a23l ðnÞK 3 : l¼0

l¼0

l¼0

The remaining proof is similar to that for liminfn!1x1(n) P m1 and hence is omitted. Finally, from the third equation of system (1.1), we have ( ) m m X X x3 ðn þ 1Þ P x3 ðnÞ exp r3 ðnÞ þ a32l ðnÞm2  a33l ðnÞx3 ðn  lÞ : l¼0

l¼0

Again, similar argument as above will yield lim inf x3 ðnÞ P m3 : n!1

Therefore, the proof is complete.

h

Combing Propositions 2.3 and 2.4, we have proved the main result of this paper, which is stated below. Theorem 2.5. Assume that (H) holds. Then system (1.1) with the initial condition (1.2) is permanent. 3. An application In this section, we apply Theorem 2.5 to the following delayed discrete three-species predator–prey system without dominating instantaneous negative feedback,

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X. Liao et al. / Applied Mathematics and Computation 186 (2007) 279–285

x1 ðn þ 1Þ ¼ x1 ðnÞ expfr1 ðnÞ  a11 ðnÞx1 ðn  bs11 cÞ  a12 ðnÞx2 ðn  bs12 cÞg; x2 ðn þ 1Þ ¼ x2 ðnÞ expfr2 ðnÞ þ a21 ðnÞx1 ðn  bs21 cÞ  a22 ðnÞx2 ðn  bs22 cÞ  a23 ðnÞx3 ðn  bs23 cÞg; x3 ðn þ 1Þ ¼ x3 ðnÞ expfr3 ðnÞ þ a32 ðnÞx2 ðn  bs32 cÞ  a33 ðnÞx3 ðn  bs33 cÞg;

ð3:1Þ

where bÆc is the floor function, {ri(n)} and {aij(n)} are bounded nonnegative sequences such that 0 < ri 6 ri and 0 < aij 6  aij , sij P 0. Again, for biological reasons, we also only consider solutions of system (3.1) with initial conditions xi ðkÞ P 0; k ¼ 1; 2; . . . ; maxfsij ; i; j ¼ 1; 2; 3g; xi ð0Þ > 0; i ¼ 1; 2; 3:

ð3:2Þ

System (3.1) can be regarded as a discrete time analog of the delayed nonautonomous three-species predator–prey Lotka–Volterra system without dominating instantaneous negative feedback, x01 ðtÞ ¼ x1 ðtÞ½r1 ðtÞ  a11 ðtÞx1 ðt  s11 Þ  a12 ðtÞx2 ðt  s12 Þ; x02 ðtÞ ¼ x2 ðtÞ½r2 ðtÞ þ a21 ðtÞx1 ðt  s21 Þ  a22 ðtÞx2 ðt  s22 Þ  a23 ðtÞx3 ðt  s23 Þ; x03 ðtÞ ¼ x3 ðtÞ½r3 ðtÞ þ a32 ðtÞx2 ðt  s32 Þ  a33 ðtÞx3 ðt  s33 Þ:

ð3:3Þ

In [5], Xu and Chen studied the permanence and the global stability of system (3.3). Denote 1 K 01 ¼ expfr1 ðbs11 c þ 1Þ  1g; a11 1 K 02 ¼ expfðK 01  a21  r2 Þðbs22 c þ 1Þ  1g; a22 1 expfðK 02  K 03 ¼ a32  r3 Þðbs33 c þ 1Þ  1g; a33  a11 expfðK 01  a11 þ K 02  a12  r1 Þbs11 cg ; D01 ¼ r1  K 02  a12 1 m01 ¼ 0 expfðr1  K 02 a21 Þð1  K 01 D01 Þg; D1  a22 K 02 þ  a23 K 03 Þbs22 cg a22 expfðr2  a21 m01 þ  0 ; D2 ¼ 0 0 a21 m1  r2   a23 K 3 1 a21 m01  a23 K 03 Þð1  K 02 D02 Þg; m02 ¼ 0 expfðr2 þ  D2  a33 K 03 Þbs33 cg a33 expfðr3  a32 m02 þ  ; D03 ¼ a32 m02  r3 1 m03 ¼ 0 expfðr3 þ  a32 m02 Þð1  K 03 D03 Þg: D3 We introduce the following assumption: (H 0 ) min16i63 K 0i D0i > 1. Then, as an application of Theorem 2.5, we immediately obtain the following result. Theorem 3.1. Assume assumption (H 0 ) holds. Then system (3.1) with the initial condition (3.2) is permanent. We illustrate Theorem 3.1 with an example to conclude this paper. Example. Consider the delayed system   1 x1 ðn þ 1Þ ¼ x1 ðnÞ exp 1  x1 ðnÞ  ð3 þ sin nÞx2 ðn  l1 Þ ; 60   2 1 x2 ðn þ 1Þ ¼ x2 ðnÞ exp 1 þ ð8 þ cos nÞx1 ðn  l2 Þ  x2 ðnÞ  ð3 þ sin nÞx3 ðn  l3 Þ ; 9 80   2 x3 ðn þ 1Þ ¼ x3 ðnÞ exp ð2 þ cos nÞ þ ð8 þ cos nÞx2 ðn  l4 Þ  x3 ðnÞ ; 9

ð3:4Þ

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