Permanence for nonautonomous predator–prey Kolmogorov systems with impulses and its applications

Applied Mathematics and Computation 223 (2013) 54–75

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Permanence for nonautonomous predator–prey Kolmogorov systems with impulses and its applications Hongxiao Hu College of Science, University of Shanghai for Science and Technology, Shanghai 200093, PR China

a r t i c l e

i n f o

Keywords: Impulsive Kolmogorov system Permanence Predator–prey Functional response system

a b s t r a c t In this paper, the general impulsive non-autonomous predator–prey Kolmogorov system is studied. Some new criteria on the permanence and ultimate boundedness are established. As applications of these results, some special models are studied, such as a class of impulsive non-autonomous Lotka–Volterra systems, impulsive Holling I-type functional response systems, impulsive Holling ðm; nÞ-type functional response systems, impulsive Beddington–DeAngelis functional response systems, Leslie–Gower functional response systems and chemostat-type systems. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction In this paper, we consider the following two-species non-autonomous predator–prey Kolmogorov system with impulse

8 > > > > < > > > > :

dx1 ðtÞ ¼ x1 ðtÞf1 ðt; x1 ðtÞ; x2 ðtÞÞ; dt dx2 ðtÞ ¼ x2 ðtÞf2 ðt; x1 ðtÞ; x2 ðtÞÞ; dt x1 ðt þk Þ ¼ h1k x1 ðt k Þ; x2 ðt þk Þ ¼ h2k x2 ðt k Þ;

t – tk ; ð1:1Þ k ¼ 1; 2; . . . ;

where we assume that 0 6 t 1 < t 2 <    < tk <    is impulsive time sequence and limk!1 t k ¼ 1, hik are positive constant for each i ¼ 1; 2 and k ¼ 1; 2; . . . , functions fi ðt; x1 ; x2 Þ (i ¼ 1; 2) are continuous for all t 2 Rþ0 ¼ ½0; 1Þ, x1 > 0 and x2  0. But, when x1 ¼ 0, fi ðt; x1 ; x2 Þ (i ¼ 1; 2) may not have any definition for any t 2 Rþ0 and x2  0. System (1.1) include many well-known impulsive non-autonomous two-species predator–prey systems as its specific cases, for example (1) Lotka–Volterra type system with impulse

8 > < > :

dx1 ðtÞ ¼ x1 ðtÞðb1 ðtÞ  a11 ðtÞx1 ðtÞ  a12 ðtÞx2 ðtÞÞ; dt dx2 ðtÞ ¼ x2 ðtÞðb2 ðtÞ þ a21 ðtÞx1 ðtÞ  a22 ðtÞx2 ðtÞÞ; dt xi ðtþk Þ ¼ hik xi ðtk Þ; i ¼ 1; 2; k ¼ 1; 2; . . . :

t – tk ;

ð1:2Þ

(2) Holling I-type functional response system with impulse

8 > < > :

dx1 ðtÞ ¼ x1 ðtÞðb1 ðtÞ  a11 ðtÞx1 ðtÞÞ  x2 ðtÞ/1 ðt; x1 ðtÞÞ; dt dx2 ðtÞ ¼ x2 ðtÞðb2 ðtÞ þ /2 ðt; x1 ðtÞÞ  a22 ðtÞx2 ðtÞÞ; dt xi ðtþk Þ ¼ hik xi ðtk Þ; i ¼ 1; 2; k ¼ 1; 2; . . . ;

E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.07.093

t – tk ;

ð1:3Þ

H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

55

where the function

 /i ðt; x1 Þ ¼

ai ðtÞx1 ; 0 6 x1 6 x10 ; ai ðtÞx10 ; x1 > x10 ; i ¼ 1; 2;

and x10 > 0 is a constant. (3) Holling ðm; nÞ-type functional response system with impulse

8 > < > :

dx1 ðtÞ ¼ x1 ðtÞðb1 ðtÞ  a11 ðtÞx1 ðtÞÞ  x2 /1 ðt; x1 ðtÞÞ; dt dx2 ðtÞ ¼ x2 ðtÞðb2 ðtÞ þ /2 ðt; x1 ðtÞÞ  a22 ðtÞx2 ðtÞÞ; dt xi ðtþk Þ ¼ hik xi ðtk Þ; i ¼ 1; 2; k ¼ 1; 2; . . . ;

t – tk ;

ð1:4Þ

where the function

/i ðt; x1 Þ ¼

ai ðtÞxm1 ; xn1 þ bi ðtÞ

i ¼ 1; 2:

(4) Beddington–DeAngelis functional response system with impulse

8 > < > :

dx1 ðtÞ ¼ x1 ðtÞðb1 ðtÞ  a11 ðtÞx1 ðtÞÞ  x2 ðtÞ/1 ðt; x1 ðtÞ; x2 ðtÞÞ; dt dx2 ðtÞ ¼ x2 ðtÞðb2 ðtÞ þ /2 ðt; x1 ðtÞ; x2 ðtÞÞÞ; dt þ xi ðtk Þ ¼ hik xi ðtk Þ; i ¼ 1; 2; k ¼ 1; 2; . . . ;

t – tk ;

ð1:5Þ

where the function

/i ðt; x1 ; x2 Þ ¼

ai ðtÞxm1 ; 1 þ ci ðtÞxn1 þ xi ðtÞx2

i ¼ 1; 2:

(5) Leslie–Gower system with functional response with impulse

8 > > < > > :

dx1 ðtÞ ¼ x1 ðtÞðb1 ðtÞ  a11 ðtÞx1 ðtÞÞ  x2 ðtÞ/1 ðt; x1 ðtÞÞ; dt dx2 ðtÞ ¼ x2 ðtÞIðt; xx21 ðtÞ Þ; ðtÞ dt þ xi ðtk Þ ¼ hik xi ðtk Þ; i ¼ 1; 2; k ¼ 1; 2; . . . ;

t – tk ;

ð1:6Þ

where the function

/1 ðt; x1 Þ ¼

a1 ðtÞx21 x21

þ c1 ðtÞx1 þ b1 ðtÞ

and

   m x2 x2 ¼ b2 ðtÞ  a2 ðtÞ I t; : x1 x1 (6) Impulsive Chemostat-type system

8 > < > :

dx1 ðtÞ ¼ a11 ðtÞ  b1 ðtÞx1 ðtÞ  /1 ðt; x1 ðtÞÞx2 ðtÞ; dt dx2 ðtÞ ¼ x2 ðtÞðb2 ðtÞ þ /2 ðt; x1 ðtÞÞ  a22 ðtÞx2 ðtÞÞ; dt xi ðtþk Þ ¼ hik xi ðtk Þ; i ¼ 1; 2; k ¼ 1; 2; . . . ;

t – tk ;

ð1:7Þ

where the function

/i ðt; x1 Þ ¼

ai ðtÞxm1 ; xn1 þ bi ðtÞ

i ¼ 1; 2:

As we well know, in the theory of mathematical, traditional predator–prey system are very important mathematical models which describe multi-species population dynamics in a non-autonomous environment. In order to make the model more accurate, there are many well-known two-species non-autonomous predator–prey systems, such as Lotka–Volterra type systems (see [1–3]), Holling I-type functional response system (see [1,3,4]), Holling ðm; nÞ-type functional response system (see [1,3–5]), Beddington–DeAngelis functional response system (see [3,6–8]), Leslie–Gower system with functional response (see [1,3,9,10]), Chemostat-type system (see [3,11–14]). Many important and interesting results on the dynamical behaviors for such systems, such as the permanence, global asymptotic behavior and the existence and uniqueness of coexistence states (for example, positive periodic solution, positive almost periodic solution, etc.) can be found in above references and references there in.

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H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

However, biological species may undergo discrete changes of relatively short duration at a fixed time, such as fire, drought, flooding, crop-dusting, deforestation, hunting, harvesting, etc., the intrinsic discipline of biological species or ecological environment. For having a more accurate description of such system, we need to consider the impulsive differential equations. In recent years, population models with impulsive perturbations have been intensively researched, such as the Lotka–Volterra model [15–18], Holling-type [19–24] and Beddington-type [25–28]. However, to our best knowledge, for general impulsive non-autonomous Kolmogorov predator–prey system (1.1), up until now, there is not any study work for the permanence of positive solutions. In addition, we also find that for the impulsive non-autonomous predator–prey Holling functional response systems, Beddington–DeAngelis functional response systems, Leslie–Gower functional response systems and impulsive non-autonomous chemostat-type systems, etc., there is also not any study work for the permanence of positive solutions. In this paper, motivated by the above works, we study the permanence of positive solutions for general impulsive nonautonomous Kolmogorov predator–prey system (1.1) and establish a general criterion which is described by integrable form. The organization of this paper is as follows. In the next section, the impulsive non-autonomous single-species Kolmogorov system is considered and several useful lemmas are introduced. In Section 3, a general theorem for the permanence of system (1.1) is stated and proved. Finally in Section 4, as applications of above theorems, we will study the permanence of positive solutions for special cases. 2. Preliminaries Let Rþ0 ¼ ½0; 1Þ and Rþ ¼ ð0; 1Þ. Let ftk g be a time sequence, satisfying 0 6 t1 < t2 <    < t k <    and t k ! 1 as k ! 1. In this section, as a preliminary we consider the following impulsive Komogorov system with a parameter

(

duðtÞ ¼ uðtÞgðt; uðtÞ; dt uðt þk Þ ¼ hk uðt k Þ;

aÞ; t – tk ;

ð2:1Þ

k ¼ 1; 2; . . . ;

where gðt; u; aÞ is a continuous function defined on ðt; u; aÞ 2 Rþ0  Rþ  ½0; a0 , a0 > 0 is a constant and hk is positive constant for k ¼ 1; 2; . . .. We assume that for any ðt0 ; u0 Þ 2 Rþ0  Rþ and a 2 ½0; a0  system (2.1) has a unique solution ua ðtÞ satisfying ua ðt0 Þ ¼ u0 . If ua ðtÞ > 0 on the interval of existence, the ua ðtÞ is said to be a positive solution. It is easy to see that ua ðtÞ is positive solution if the initial value u0 > 0. For system (2.1) we introduce the following assumption: (A1) For any r > 1, gðt; u; aÞ is bounded on Rþ0  ½r1 ; r  ½0; a0 . (A2) There are positive constants k1 ; k2 ; x1 ; x2 and k2 > k1 such that

Z

lim inf t!1

gðs; k1 ; 0Þds þ t

t!1

! ln hk

> 0;

t6t k
Z

lim sup

X

tþx1

tþx2

gðs; k2 ; 0Þds þ

t

X

! ln hk

< 0;

t6t k
and function

hðt; v Þ ¼

X

ln hk

t6tk
is bounded on t 2 Rþ and v 2 ½0; x1 Þ. (A3) Partial derivative @gðt; u; aÞ=@u exists for all ðt; u; aÞ 2 Rþ0  Rþ  ½0; a0  and there is a nonnegative continuous function qðtÞ and a constant x > 0, satisfying

lim inf t!1

Z

tþx

qðsÞds > 0;

t

and a continuous function pðuÞ, satisfying pðuÞ > 0 for all u 2 Rþ , such that

@gðt; u; aÞ 6 qðtÞpðuÞ for all ðt; u; aÞ 2 Rþ0  Rþ  ½0; a0 : @u (A4) Partial derivative @gðt; u; aÞ=@ a exist for all ðt; u; aÞ 2 Rþ0  Rþ  ½0; a0 , and for any constant U > 0, @gðt; u; aÞ=@ a is also bounded on ðt; u; aÞ 2 Rþ0  ð0; U  ½0; a0 . In system (2.1), when parameter a ¼ 0 we obtain the following system

(

duðtÞ ¼ uðtÞgðt; uðtÞ; 0Þ; dt uðt þk Þ ¼ hk uðt k Þ;

t – tk ; k ¼ 1; 2; . . . :

Let u0 ðtÞ be a fixed positive solution of system (2.2) defined on Rþ0 .

ð2:2Þ

57

H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

Definition 2.1. System (2.2) is said to be permanent, if there are positive constants m and M such that

m 6 lim inf u0 ðtÞ 6 lim sup u0 ðtÞ 6 M; t!1

t!1

for any positive solution u0 ðtÞ of system (2.2). Definition 2.2. u0 ðtÞ is globally uniformly attractive on Rþ0 , if for any constants g > 1 and e > 0 there is a constant Tðg; eÞ > 0 such that for any initial time t 0 2 Rþ0 and any solution u0 ðtÞ of system (2.2) with u0 ðt 0 Þ 2 ½g1 ; g, one has

ju0 ðtÞ  u0 ðtÞj < e for all t P t 0 þ Tðg; eÞ: We first have the following result by using a similar argument as Lemma 1 in [3]. Lemma 2.1. Suppose that (A1)–(A3) hold, then (a) System (2.2) is permanent. (b) Each fixed positive solution u0 ðtÞ of system (2.2) is globally uniformly attractive on Rþ0 . Proof. By assumption (A2), there are positive constants d and T 0 such that for all t P T 0 we have

Z

X

tþx1

gðs; k1 ; 0Þds þ

t

ln hk > d

ð2:3Þ

ln hk < d:

ð2:4Þ

t6t k
and

Z

X

tþx2

gðs; k2 ; 0Þds þ

t

t6t k
P Since function hðt; v Þ ¼ t6tk
v 2 ½0; x1 Þ, there is a positive constant H such that for any

    X   jhðt; v Þj ¼  ln hk  < H:  t6t
ð2:5Þ

k

Let uðtÞ be any positive solution of Eq. (2.2). We first prove that there is a

s P T 0 such that

k1 expða1 x1  HÞ 6 uðtÞ 6 k2 expða2 x2 þ HÞ for all t P s; where ai ¼ supfjgðt; ki ; 0Þj : t 2 Rþ0 g. If uðtÞ P k2 , for all t P T 0 , then from (A3) we have

uðtÞ ¼ uðT 0 Þ exp

Z

gðs; uðsÞ; 0Þds þ T0

6 uðT 0 Þ exp

Z

!

X

t

ln hk

T 0 6tk
X

t

gðs; k2 ; 0Þds þ

T0

!

ln hk :

T 0 6t k
From (2.4), we easily obtain uðtÞ ! 0 as t ! 1, which is a contradiction. Hence, uðs1 Þ < k2 for some s1 P T 0 . Further, if there are s2 > s1 P s1 such that uðs2 Þ > k2 expða2 x2 þ HÞ, uðs1 Þ 6 k2 , uðsþ 1 Þ P k2 and uðtÞ P k2 for all t 2 ðs1 ; s2 Þ, then we can choose an integer p  0 such that s2 2 ½s1 þ px2 ; s1 þ ðp þ 1Þx2 Þ and obtain

k2 expða2 x2 þ HÞ < uðs2 Þ ¼ uðs1 Þ exp

Z

gðt; uðtÞ; 0Þdt þ

s1

6 k2 exp

Z

s1 þpx2

þ

6 k2 exp

s2

 gðt; k2 ; 0Þdt þ

s1 þpx2

s2

gðt; k2 ; 0Þdt þ

s1 þpx2

X s1 þpx2 6tk
uðtÞ 6 k2 expða2 x2 þ HÞ for all t 2 ½s1 ; 1Þ:

s2 P T 0 such that

uðtÞ P k1 expða1 x1  HÞ for all t 2 ½s2 ; 1Þ:

X s1 6t k
which also is a contradiction. Therefore, we have

Similarly, by (2.3), we can prove that there is a

ln hk

s1 6t k
Z

s1

Z

!

X

s2

!

ln hk

þ

X s1 þpx2 6t k
6 k2 expða2 x2 þ HÞ;

!

! ln hk

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H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

Choose s ¼ maxfs1 ; s2 g, m ¼ k1 expða1 x1  HÞ and M ¼ k2 expða2 x2 þ HÞ, then we obtain conclusion (a). Here, we prove conclusion (b). For solution u0 ðtÞ, by conclusion (a) there is a constant M 1 > 1 such that  M1 1 6 u0 ðtÞ 6 M 1

for all t 2 Rþ0 :

ð2:6Þ

For any constant g > 1 and t 0 2 Rþ0 , let u0 ðtÞ be the solution of system (2.1) with initial value u0 ðt0 Þ 2 ½g ; g. We can choose Lyapunov function as follows 1

VðtÞ ¼ j ln u0 ðtÞ  ln u0 ðtÞj: For any k ¼ 1; 2; . . ., we have

Vðtþk Þ ¼ j lnðhk u0 ðt k ÞÞ  lnðhk u0 ðtk ÞÞj ¼ Vðt k Þ: Calculating the Dini derivative, by (A3) we can obtain

@gðt; nðtÞ; 0Þ ju0 ðtÞ  u0 ðtÞj @u 6 qðtÞpðnðtÞÞju0 ðtÞ  u0 ðtÞj for all t – tk ; k ¼ 1; 2; . . . ;

Dþ VðtÞ ¼ sgnðu0 ðtÞ  u0 ðtÞÞðgðt; u0 ðtÞ; 0Þ  gðt; u0 ðtÞ; 0ÞÞ ¼

where nðtÞ is situated between u0 ðtÞ and

j ln u0 ðtÞj 6 Hence, g

1

1

g

M 2 1

j ln u0 ðtÞj

Hence, VðtÞ 6 Vðt 0 Þ for all t P t0 . Consequently, by (2.6) we have

M 21 Þ

for all t P t 0 :

þ Vðt 0 Þ 6 lnðg

M 21

6 u0 ðtÞ 6 g

M 2 1 VðtÞ

u0 ðtÞ.

6 ju0 ðtÞ 

ð2:7Þ

for all t P t0 . Further by (2.6), we obtain

u0 ðtÞj

6 gM 21 VðtÞ for all t P t 0 :

Consequently, by (2.7) it follows that

Dþ VðtÞ 6 qðtÞM 0 VðtÞ for all t – t k ; k ¼ 1; 2; . . . ; where M 0 ¼ g

1

lim inf t!1

Z

M 21

minfpðuÞ : g

1

M 2 1

6u6g

M 21 g.

ð2:8Þ

Since

tþx

qðsÞds > 0;

t

we can choose positive constants d and T 0 such that

Z

tþx

qðsÞds P d for all t P T 0 :

t

Let T 00 ¼ t 0 þ T 0 , for any t P T 00 , there is an integer nt P 0 such that t 2 ½T 00 þ nt x; T 00 þ ðnt þ 1ÞxÞ. Integrating (2.8) from T 00 to t, we have

VðtÞ 6

VðT 00 Þ exp

Z

t

T 00

ðM0 qðsÞÞds ¼

VðT 00 Þ exp

Z

T 00 þx

T 00

þ þ

Z

T 00 þnt x

T 00 þðnt 1Þx

þ

Z

!

t T 00 þnt x

ðM 0 qðsÞÞds 6 VðT 00 Þ expðM 0 dnt Þ:

Since VðT 00 Þ 6 Vðt 0 Þ 6 lnðgM 1 Þ, we further have

    VðtÞ 6 lnðgM1 Þ exp M0 dx1 ðt  T 00  xÞ ¼ M2 ðgÞ exp M0 dx1 ðt  t 0 Þ ;

where M 2 ðgÞ ¼ lnðgM 1 Þ exp ðM0 dð1 þ T 0 =xÞÞ. Hence, for any positive constant Tðg; eÞ P T 0 such that

ð2:9Þ

e, from (2.9), there is a large enough

VðtÞ < g1 M 2 for all t P t 0 þ Tðg; eÞ: 1 e Therefore, ju0 ðtÞ  u0 ðtÞj < e for all t P t0 þ Tðg; eÞ. This shows that solution u0 ðtÞ is globally uniformly attractive on Rþ0 . This completes the proof. h Lemma 2.2. Suppose that (A1)–(A4) hold. Then ua ðtÞ converges to u0 ðtÞ uniformly for t 2 ½t0 ; þ1Þ as a ! 0. Proof. For any ðt; u; aÞ 2 Rþ0  Rþ  ½0; a0 , since

gðt; u; aÞ ¼ gðt; u; 0Þ þ

@gðt; u; nÞ a; @a

where n 2 ð0; aÞ, by (A2) and (A4) there are constants T 0 > 0; c0 2 ð0; a0  and d0 > 0 such that

Z t

tþx1

gðs; k1 ; aÞds þ

X t6t k
ln hk > d0

59

H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

and

Z

X

tþx2

gðs; k2 ; aÞds þ

t

ln hk < d0

t6t k
P for all t P T 0 and a 2 ½0; c0 . Since function hðt; v Þ ¼ t6tk
v 2 ½0; x1 Þ, there is a positive

    X   jhðt; v Þj ¼  ln hk  < H:  t6t
Let ua ðtÞ be any positive solution of Eq. (2.1). From this and by (A1) and (A3), using a similar argument as in Lemma 2.1, we can prove that there is a s P T 0 such that

k1 expða1 x1  HÞ 6 ua ðtÞ 6 k2 expða2 x2 þ HÞ for all t P s; where ai ¼ supfjgðt; ki ; aÞj : t 2 Rþ0 ; a 2 ½0; a0 g for i ¼ 1; 2. Therefore, there is a constant M > 1, and M is independent of any a, such that

M1 6 ua ðtÞ 6 M

for all t P t 0 :

Let VðtÞ ¼ j ln ua ðtÞ  ln u0 ðtÞj. For any k ¼ 1; 2; . . ., we have

Vðt þk Þ ¼ j lnðhk ua ðt k ÞÞ  lnðhk u0 ðtk ÞÞj ¼ Vðt k Þ: Hence, VðtÞ is continuous for all t 2 Rþ . Then similar argument as in Lemma 2 in [3], we can prove the conclusion of Lemma 2.2 is hold. h A special case of system (2.2) is the following logistic system

(

duðtÞ ¼ uðtÞðaðtÞ  dt uðtþk Þ ¼ hk uðt k Þ;

bðtÞuðtÞÞ; t – t k ; k ¼ 1; 2; . . . ;

ð2:10Þ

where aðtÞ and bðtÞ are bounded and continuous function defined on Rþ0 . From Lemma 2.1 we obtain following result. Lemma 2.3. Suppose bðtÞ  0 and there are positive constants x1 and x2 such that

Z

lim inf t!1

lim inf t!1

aðsÞds þ

t

Z

X

tþx1

!

> 0;

ln hk

t6t k
tþx2

bðsÞds > 0

t

and function

X

hðt; v Þ ¼

ln hk

t6t k
is bounded on t 2 Rþ0 and

v 2 ½0; x1 Þ. Then the conclusions of Lemma 2.1 for system (2.10) hold.

Remark 2.1. System (2.10) has been studied by Hou and his collaborators in [18]. They obtained the same result with Lemma 2.3. Therefore, we improve and extent the results in Lemma 2.1 in [18], and our result is more general. Another special case of system (2.2) is the following linear impulsive system

(

duðtÞ ¼ aðtÞ  bðtÞu; dt uðtþk Þ ¼ hk uðt k Þ;

t – tk ;

ð2:11Þ

k ¼ 1; 2; . . . ;

where aðtÞ and bðtÞ are bounded and continuous functions defined on Rþ0 . From Lemma 2.1 we obtain Lemma 2.4. Lemma 2.4. Assume that aðtÞ  0 and there are positive constants x1 and x2 such that

Z

lim inf t!1

lim inf t!1

t

Z t

X

tþx1

bðsÞds 

t6t k
tþx2

aðsÞds > 0

!

ln hk

> 0;

60

H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

and function

X

hðt; v Þ ¼

ln hk

t6tk
is bounded on t 2 Rþ and

v 2 ½0; x1 Þ. Then the conclusions of Lemma 2.1 for system (2.11) hold.

3. Main result Let R2þ ¼ fðx1 ; x2 Þ : x1 > 0; x2 > 0g. For any point ðt0 ; x0 Þ 2 Rþ0  R2þ , let xðt; x0 Þ ¼ ðx1 ðt; x0 Þ; x2 ðt; x0 ÞÞ be the solution of system (1.1) with initial condition xðt0 ; x0 Þ ¼ x0 . We easily prove that xi ðt; x0 Þ > 0 (i ¼ 1; 2) on the interval of existence if the initial value x0 2 R2þ . Further, let a0 > 0 be a constant. For system (1.1), we introduce the following assumptions. (B1) Function f1 ðt; x1 ; x2 Þ satisfies the following conditions. (1) Partial derivative @f1 ðt; x1 ; x2 Þ=@x2 exists and is non-positive for all ðt; x1 ; x2 Þ 2 Rþ0  R2þ . (2) Partial derivative @f1 ðt; x1 ; x2 Þ=@x1 exist on ðt; x1 ; x2 Þ 2 Rþ0  Rþ  ½0; a0  and there is a nonnegative continuous function R tþx qðtÞ and a constant x > 0 satisfying lim inf t qðsÞds > 0, and a continuous function pðx1 Þ, satisfying pðx1 Þ > 0 for all x1 > 0, t!1 such that

@f1 ðt; x1 ; x2 Þ 6 qðtÞpðx1 Þ for all ðt; x1 ; x2 Þ 2 Rþ0  Rþ  ½0; a0 : @x1 (3) For any constant that

jf1 ðt; x1 ; x2 Þj 6 G1

r > 1, f1 ðt; x1 ; x2 Þ is bounded on Rþ0  ½r1 ; r  ½0; a0  and there is a constant G1 ¼ G1 ðrÞ > 0 such for all t 2 Rþ0 and 0 < xi 6 r ði ¼ 1; 2Þ:

(4) There are positive constants x1 ; x2 ; k1 and neighborhood U of origin ð0; 0Þ such that

lim inf

Z

t!1

X

tþx1

f1 ðu; x1 ; x2 Þdu þ t

! ln h1k

>0

t6t k
for any point ðx1 ; x2 Þ 2 U with xi > 0 (i ¼ 1; 2) and

lim sup

Z

t!1

X

tþx2

f1 ðu; k1 ; 0Þdu þ

t

! ln h1k

<0

t6t k
and function

X

h1 ðt; v Þ ¼

ln h1k

t6t k
is bounded on t 2 Rþ and v 2 ½0; x1 Þ. (B2) Function f2 ðt; x1 ; x2 Þ satisfying the following conditions. (1) Partial derivative @f2 ðt; x1 ; x2 Þ=@x2 exists and is non-positive for all ðt; x1 ; x2 Þ 2 Rþ0  R2þ and for any constant K > 0; @f2 ðt; x1 ; x2 Þ=@x2 is bounded on ðt; x1 ; x2 Þ 2 Rþ0  ð0; K  ½0; a0 . (2) There is a constant b0 > 0 such that partial derivative @f2 ðt; x1 ; x2 Þ=@x1 exists and is nonnegative for all ðt; x1 ; x2 Þ 2 Rþ0  ð0; b0   Rþ and for any constant K > b0 ,

supfjf2 ðt; x1 ; x2 Þj : t 2 Rþ0 ; b0 6 x1 6 K; 0 6 x2 6 Kg < 1: (3) There is a constant x3 > 0 and neighborhood U of origin ð0; 0Þ such that

lim sup t!1

Z

tþx3

f2 ðu; x1 ; x2 Þdu þ

t

X

! ln h2k

<0

t6tk
P for any point ðx1 ; x2 Þ 2 U with xi > 0 (i ¼ 1; 2), and function h2 ðt; v Þ ¼ t6tk 0 such that lim sup x2 ðtÞ < k2 for any positive solution ðx1 ðtÞ; x2 ðtÞÞ of system (1.1). t!1 We consider the following impulsive single-species non-autonomous Kolmogorov system

(

dx1 ðtÞ ¼ x1 ðtÞf1 ðt; x1 ðtÞ; 0Þ; dt x1 ðtþk Þ ¼ h1k x1 ðt k Þ;

t – tk ; k ¼ 1; 2; . . . :

ð3:1Þ

By conditions (2), (3) and (4) of (B1), we see that system (3.1) satisfies all conditions of Lemma 2.1. Hence, by Lemma 2.1, each positive solution of system (3.1) is globally asymptotically stable. Let x10 ðtÞ be some fixed positive solution of system (3.1). On the permanence of system (1.1) we have the following result.

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H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

Theorem 3.1. Suppose (B1)–(B3) hold. If there is a constant x0 > 0 such that

lim inf t!1

Z

f2 ðu; x10 ðuÞ; 0Þdu þ

t

!

X

tþx0

ln h2k

> 0;

t6tk
then system (1.1) is permanent. Proof. Let ðx1 ðtÞ; x2 ðtÞÞ be any positive solution of system (1.1). From condition (1) of (B1) we have

dx1 ðtÞ 6 x1 ðtÞf1 ðt; x1 ðtÞ; 0Þ for all t  0 and t – t k : dt By the comparison theorem of impulsive system and since x10 ðtÞ is the globally asymptotically stable positive solution of system (3.1), we obtain that for any constant e > 0 there is a constant T ¼ TðeÞ > 0 such that

x1 ðtÞ 6 x10 ðtÞ þ e for all t P T:

ð3:2Þ

From this and by (B3), it follows that all positive solutions of system (1.1) are defined on Rþ and ultimately bounded with boundary L ¼ maxfL1 ; k2 ; a0 ; b0 g, where constant L1 > maxt2R x10 ðtÞ. Let x ¼ maxfx0 ; x1 ; x2 ; x3 g. By the boundedness of function hi ðt; v Þ (i ¼ 1; 2) on ðt; v Þ 2 Rþ  ½0; xÞ, we have there is a positive constant H such that

    X   jhi ðt; v Þj ¼  ln hik  6 H  t6t
for all t 2 Rþ0 ;

v 2 ½0; xÞ:

ð3:3Þ

k

For any s1 ; s2 and s2 P s1  0, integrating directly system (1.1) we have

x1 ðs2 Þ ¼ x1 ðs1 Þ exp

Z

f1 ðt; x1 ðtÞ; x2 ðtÞÞdt þ

s1

!

X

s2

ln h1k

and

x2 ðs2 Þ ¼ x2 ðs1 Þ exp

Z

ð3:4Þ

s1 6tk
f2 ðt; x1 ðtÞ; x2 ðtÞÞdt þ

s1

!

X

s2

ln h2k :

ð3:5Þ

s1 6tk
In the following, we will use four claims to complete the proof of Theorem 3.1. Claim 1. There is a constant g > 0 such that lim sup x1 ðtÞ > g for any positive solution ðx1 ðtÞ; x2 ðtÞÞ of system (1.1). t!1

In fact, from conditions (4) of (B1) and (2), (3) of (B2), there are positive constants T 0 ; e0 ; e1 and e0 expða1 x3 þ HÞ < a0 and e1 < b0 , where a1 ¼ maxt2Rþ0 jf2 ðt; b0 ; 0Þj < 1, such that for all t P T 0

Z

tþx1

X

f1 ðu; e1 ; e0 expða1 x3 þ HÞÞdu þ

t

l, satisfying

ln h1k > l

ð3:6Þ

t6t k
and

Z

tþx3

f2 ðu; e1 ; e0 Þdu þ

t

X

ln h2k < l:

ð3:7Þ

t6t k
If Claim 1 is not true, then there is a positive solution ðx1 ðtÞ; x2 ðtÞÞ of system (1.1) such that lim sup x1 ðtÞ < e1 . Hence, there is t!1 a T 1 P T 0 such that x1 ðtÞ < e1 for all t P T 1 . Firstly, we prove that there exist a s1 P T 1 such that

x2 ðtÞ 6 e0 expða1 x3 þ HÞ for all t P s1 :

ð3:8Þ

In fact, we only need to consider the following three cases about x2 ðtÞ. Case I: there is a s1 P T 1 such that x2 ðtÞ P e0 for all t P s1 . Case II: there is a s1 P T 1 such that x2 ðtÞ 6 e0 for all t P s1 . Case III: x2 ðtÞ is oscillatory about e0 for all t P T 1 . We first consider Case I. Since x2 ðtÞ P e0 for all t P s1 , then by conditions (1) and (2) of (B2) and (3.5) we have

x2 ðtÞ 6 x2 ðs1 Þ exp

Z

t

f2 ðu; e1 ; e0 Þdu þ

s1

X

!

ln h2k

for all t P s1 :

s1 6tk
From this and by (3.7) it follows limt!1 x2 ðtÞ ¼ 0 which leads to a contradiction. Next, we consider Case III. From the oscillation of x2 ðtÞ about e0 , we can choose two sequences fqn g and fqn g satisfying

T 1 < q1 < q1 <    < qn < qn <   

62

H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

and

lim qn ¼ lim qn ¼ 1

n!1

n!1

such that

x2 ðqn Þ 6 e0 ; x2 ðtÞ P e0

x2 ðqþn Þ P e0 ;

x2 ðqn Þ P e0 ;

for all t 2 ðqn ; qn Þ

x2 ðqþ n Þ 6 e0 ; for all t 2 ðqn ; qnþ1 Þ:

x2 ðtÞ 6 e0

and

For any t P T 1 , if t 2 ðqn ; qn  for some integer n, then we can choose integer l  0 and constant 0 6 l1 < x3 such that t ¼ qn þ lx3 þ l1 . Then by condition (1) and (2) of (B2), (3.5) and (3.7) it follows that

Z

x2 ðtÞ ¼ x2 ðqn Þ exp 0

Z

¼ e0 exp @ 0 6 e0 exp @

f2 ðs; x1 ; x2 Þds þ qn

ln h2k

qn 6tk
qn þlx3

þ

Z

qn

Z

!

X

t

!

t

6 e0 exp

t

f2 ðt; e1 ; e0 Þdt þ qn

0

X

f2 ðt; e1 ; e0 Þdt þ @

qn þlx3

Z

X

t

f2 ðt; e1 ; e0 Þdt þ

qn þlx3

X

þ

qn 6tk
1

1

X

! ln h2k

qn 6tk
1

A ln h2k A

qn þlx3 6tk
0 Z ln h2k A 6 e0 exp @

qn þlx3 6tk
t

qn þlx3

f2 ðt; b0 ; 0Þdt þ

X

1 ln h2k A

qn þlx3 6tk
6 e0 expða1 x3 þ HÞ: If there is an integer n such that t 2 ðqn ; qnþ1 , then we obviously have

x2 ðtÞ 6 e0 < e0 expða1 x3 þ HÞ: Therefore, let s1 ¼ q1 , for Case III we always have

x2 ðtÞ 6 e0 expða1 x3 þ HÞ for all t P s1 : Lastly, if Case II holds, then we directly have

x2 ðtÞ 6 e0 expða1 x3 þ HÞ for all t P s1 : Therefore, (3.8) is true. Finally, by conditions (1) and (2) of (B1), (3.4) and (3.8) we have

Z

x1 ðtÞ P x1 ðs1 Þ exp

t

f1 ðu; e1 ; e0 expða1 x3 þ HÞÞdu þ s1

X

! ln h1k

s1 6tk
for all t P s1 . From this and by (3.6) it follows limt!1 x1 ðtÞ ¼ 1 which leads to a contradiction. Therefore, Claim 1 is true. Claim 2. There is a constant c > 0 such that lim inf x1 ðtÞ > c for any positive solution ðx1 ðtÞ; x2 ðtÞÞ of system (1.1). t!1

In fact, from (3.6) and (3.7) there is a constant P > 0 such that

Z

tþa

f1 ðu; e1 ; e0 expða1 x3 þ HÞÞdu þ

t

X

ln h1k > e1

ð3:9Þ

t6t k 6tþa

and

L exp

Z

tþa

f2 ðu; e1 ; e0 Þdu þ

t

X

! ln h2k

< e0

ð3:10Þ

t6tk
for all t P T 0 and a P P, where constant L is given in the above. If Claim 2 is not true, then there is a sequence of initial value fxn g  R2þ such that for the solution ðx1 ðt; xn Þ; x2 ðt; xn ÞÞ of system (1.1)

lim inf x1 ðt; xn Þ < t!1

g n2

;

n ¼ 1; 2; . . . ;

where constant g is given in Claim 1. From (3.3) we have that

eH 6 h1k 6 eH

for all k ¼ 1; 2; . . . :

Hence, we can choose an integer K > eH , for any positive solution xðtÞ of system (1.1), if

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H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

x1 ðtk Þ P

g

for some k ¼ 1; 2; . . . ;

n

then we have

x1 ðtþk Þ ¼ h1k x1 ðtk Þ P eH

g n

P

g n2

for all n P K;

and if

x1 ðtk Þ 6

g

for some k ¼ 1; 2; . . . ;

n2

then we have

x1 ðtþk Þ ¼ h1k x1 ðtk Þ 6 eH

g n2

6

g

for all n P K:

n

ðnÞ

By Claim 1 and above inequality, we obtain that there exist two time sequence fsq g and ftðnÞ q g such that for each n ¼ K; K þ 1; . . ., ðnÞ

ðnÞ

ðnÞ

ðnÞ

0 < s1 < t 1 < s2 < t2 <    < sqðnÞ < t ðnÞ q < ; ðnÞ sðnÞ as q ! 1; q ! 1; t q ! 1

x1 ðsðnÞ q ; xn Þ P

g n2

g n2

g n

g

;

6 x1 ðt; xn Þ 6

g n

g n

< x1 ðsqðnÞþ ; xn Þ 6

g

;

ð3:12Þ

;

ð3:13Þ

ðnÞ for all t 2 ðsðnÞ q ; t q Þ:

ð3:14Þ

n2

6 x1 ðt qðnÞ ; xn Þ <

ð3:11Þ

;

x1 ðtðnÞþ ; xn Þ 6 q

n

g n2

From the ultimate boundedness of system (1.1), there is a T ðnÞ P T 0 such that xi ðt; xn Þ 6 L (i ¼ 1; 2) for all t P T ðnÞ . Further, ðnÞ ðnÞ ðnÞ there is an integer N 1 > 0 such that sq > T ðnÞ for all q P N 1 . From condition (3) of (B1), there is a constant G1 ¼ G1 ðLÞ > 0 such that

jf1 ðt; x1 ; x2 Þj 6 G1 for all t 2 Rþ0 , 0 < xi 6 L (i ¼ 1; 2). Hence, from (3.4) we have 0 B x1 ðt ðnÞþ ; xn Þ ¼ x1 ðsðnÞ q q ; xn Þ exp @

Z

ðnÞ

tq ðnÞ

f1 ðt;x1 ðt;xn Þ; x2 ðt; xn ÞÞdt þ

sq

1

X ðnÞ ðnÞ sq 6t k 6tq

C ln h1k A ¼ x1 ðsðnÞ q ;xn Þ

0 Z tðnÞ Z tðnÞ Z tqðnÞ q B q  exp @ f1 ðt; x1 ðt; xn Þ; x2 ðt; xn ÞÞdt  f1 ðt; e1 ; e0 expða1 x3 þ HÞÞdt þ f1 ðt; e1 ; e0 expða1 x3 þ HÞÞdt þ ðnÞ ðnÞ ðnÞ sq

sq

ðnÞ

sq

ðnÞ

ðnÞ

ðnÞ

1

X ðnÞ

ðnÞ

C ln h1k A:

sq 6t k 6tq

ðnÞ

We can choose a positive integer lq such that tðnÞ q ¼ sq þ lq x1 þ v q , where v q 2 ½0; x1 Þ. Then from (3.6) and above equality, we can obtain

0

g n2

B ðnÞ P x1 ðt qðnÞþ ; xn Þ P x1 ðsqðnÞ ; xn Þ exp @2G1 ðtðnÞ q  sq Þ þ P

g n

Z

ðnÞ

tq ðnÞ

ðnÞ

f1 ðt; e1 ; e0 expða1 x3 þ HÞÞdt þ

sq þlq x1

X ðnÞ ðnÞ ðnÞ sq þlq x1 6t k 6t q

1 C ln h1k A

ðnÞ expð2G1 ðtðnÞ q  sq Þ  G1 x1  2HÞ:

Consequently, we have ðnÞ tðnÞ q  sq P

ln n  G1 x1  2H 2G1

ðnÞ

for all q P N1 ; n ¼ K; K þ 1; . . . :

Thus, there is an integer N 0 > K such that g=N 0 < e1 and ðnÞ tðnÞ q  sq > 2P

ðnÞ

for all n P N 0 ; q P N1 : ðnÞ

ðnÞ

ðnÞ

For any n > N 0 and q P N 1 , if x2 ðt; xn Þ P e0 for all t 2 ½sq ; sq þ P, then by condition (1) and (2) of (B2), (3.5), (3.10) and (3.14) we obtain

64

H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

e0

0 Z sðnÞ þP B q ðnÞ 6 x2 ðsðnÞ f2 ðt; x1 ðt; xn Þ; x2 ðt; xn ÞÞdt þ q þ P; xn Þ ¼ x2 ðsq ; xn Þ exp @ ðnÞ sq

0 B 6 L exp @

Z

ðnÞ

sq þP ðnÞ

f2 ðt; e1 ; e0 Þdt þ

sq

ðnÞ ðnÞ sq 6tk
ðnÞ

ðnÞ

C ln h2k A

sq 6t k
1

X

1

X

C ln h2k A < e0 : ðnÞ

ðnÞ

This leads to a contradiction. Hence, there is a s1 2 ½sq ; sq þ P such that x2 ðs1 ; xn Þ < e0 . We now prove that

x2 ðt; xn Þ 6 e0 expða1 x3 þ HÞ for all t 2 ½s1 ; tðnÞ q : ½s1 ; tðnÞ q 

In fact, if there is a s2 2

such that x2 ðs2 ; xn Þ > e0 expða1 x3 þ HÞ, then there is a s3 2 ½s1 ; s2 Þ such that

x2 ðsþ3 ; xn Þ

x2 ðs3 ; xn Þ 6 e0 ;

ð3:15Þ

P e0

and x2 ðt; xn Þ P e0

for all t 2 ðs3 ; s2 :

Choose an integer p  0 such that s2 2 ½s3 þ px3 ; s3 þ ðp þ 1Þx3 Þ, then from conditions (1) and (2) of (B2), (3.5), (3.7) and (3.14) we have

Z

x2 ðs2 ; xn Þ ¼ x2 ðs3 ; xn Þ exp

s2

f2 ðt; x1 ðt; xn Þ; x2 ðt; xn ÞÞdt þ

s3

6 e0 exp

Z

s3 þpx3

þ

Z

 f2 ðt; e1 ; e0 Þdt þ

s2

s2

ln h2k

s3 6t k
s3 þpx3

s3

6 e0 exp

Z

!

X

X

þ

s3 6t k
X

f2 ðt; e1 ; e0 Þdt þ

s3 þpx3

!

6 e0 exp

ln h2k

!

X

! ln h2k

s3 þpx3 6tk
Z

s2

s3 þpx3

s3 þpx3 6t k
 f2 ðt; b0 ; 0Þdt þ H 6 e0 expða1 x3 þ HÞ:

This leads to a contradiction. Therefore, (3.15) is true. Finally, since

x2 ðt; xn Þ 6 e0 expða1 x3 þ HÞ for all t 2 ½sqðnÞ þ P; t qðnÞ ; by conditions (1) and (2) of (B1), (3.4), (3.9) and (3.14) we obtain

0

g n2

B P x1 ðt qðnÞþ ; xn Þ ¼ x1 ðsqðnÞ þ P; xn Þ exp @ 0 P

g n2

B exp @

Z

Z

ðnÞ

ðnÞ

f1 ðt; x1 ðt; xn Þ; x2 ðt; xn ÞÞdt þ

sq þP

ðnÞ

tq

ðnÞ sq þP

X

tq

X

f1 ðt; e1 ; e0 expða1 x3 þ HÞÞdt þ ðnÞ

1

ðnÞ ðnÞ sq þP6t k 6t q

1 C ln h1k A

g C ln h1k A > 2 : n ðnÞ

sq þP6tk 6t q

This leads to a contradiction. Therefore, Claim 2 is true. Claim 3. There is a constant a > 0 such that lim sup x2 ðtÞ > a for any positive solution ðx1 ðtÞ; x2 ðtÞÞ of system (1.1). t!1

In fact, by

lim inf t!1

Z

X

tþx0

f2 ðs; x10 ðsÞ; 0Þds þ t

ln h2k

> 0;

t6t k
there are positive constants T 0 ,

juðtÞ  x10 ðtÞj < e0

!

e0 < a0 and d0 such that for any continuous function uðtÞ defined on Rþ0 , satisfying

for all t P T 0 ;

one has

Z t

tþx0

X

f2 ðs; uðsÞ; e0 Þds þ

ln h2k P d0

for all t P T 0 :

ð3:16Þ

t6tk
Consider the following system

(

dx1 ðtÞ ¼ x1 ðtÞf1 ðt; x1 ðtÞ; dt x1 ðtþk Þ ¼ h1k x1 ðt k Þ;

aÞ; t – tk ; k ¼ 1; 2; . . . ;

ð3:17Þ

H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

65

where a 2 ð0; a0  is a parameter. Let x1a ðtÞ be the solution of system (3.17) with initial value x1a ð0Þ ¼ x10 ð0Þ. Since f1 ðt; x1 ; aÞ satisfies (A1)–(A4), by Lemma 2.1, x1a ðtÞ is globally asymptotically stable. Further, by Lemma 2.2, we obtain that x1a ðtÞ uniformly for t 2 Rþ converges to x10 ðtÞ as a ! 0. Hence, there is a a > 0 and a < e0 such that

x1a ðtÞ > x10 ðtÞ 

e0

for all t 2 Rþ0 :

2

ð3:18Þ

If Claim 3 is not true, then there is a positive solution ðx1 ðtÞ; x2 ðtÞÞ of system (1.1) such that lim sup x2 ðtÞ < a. Hence, there is a t!1 T 1 > T 0 such that x2 ðtÞ < a for all t P T 1 . Since

dx1 ðtÞ P x1 ðtÞf1 ðt; x1 ðtÞ; aÞ for all t P T 1 and t – t k ; dt by the comparison theorem of impulsive differential system and global asymptotic stability of solution x1a ðtÞ, we obtain that there is a T 2 P T 1 such that

x1 ðtÞ P x1a ðtÞ 

e0

for all t P T 2 :

2

ð3:19Þ

On the other hand, by (3.2) there is a T 3 P T 2 such that

x1 ðtÞ 6 x10 ðtÞ þ e0

for all t P T 3 :

Hence, from (3.18) and (3.19) it follows that

jx1 ðtÞ  x10 ðtÞj < e0

for all t P T 3 :

ð3:20Þ

By (3.5) and condition (1) of (B2) we obtain

Z

x2 ðtÞ P x2 ðT 3 Þ exp

t

f2 ðs; x1 ðsÞ; e0 Þds þ

T3

X

! ln h2k

T 3 6t k
Thus, from (3.16) and (3.20) we finally obtain limt!1 x2 ðtÞ ¼ 1 which leads to a contradiction. Therefore, Claim 3 is true. Claim 4. There is a constant b > 0 such that lim inf x2 ðtÞ > b for any positive solution ðx1 ðtÞ; x2 ðtÞÞ of system (1.1). t!1

In fact, if Claim 4 is not true, then there is a sequence of initial value fxn g  R2þ such that, for the solution ðx1 ðt; xn Þ; x2 ðt; xn ÞÞ of system (1.1),

lim inf x2 ðt; xn Þ < t!1

a n

n ¼ 1; 2; . . . ;

;

where constant a is given in Claim 3. From (3.3) we have that

eH 6 h2k 6 eH

for all k ¼ 1; 2; . . . :

Hence, we can choose an integer K > eH , for any positive solution xðtÞ of system (1.1), if

x2 ðtk Þ P a for some k ¼ 1; 2; . . . ; then we have

x2 ðtþk Þ ¼ h2k x2 ðtk Þ P eH a >

a n

for all n P K;

and if

x2 ðtk Þ 6

a n

for some k ¼ 1; 2; . . . ;

then we have

x2 ðtþk Þ ¼ h2k x2 ðtk Þ 6 eH

a n

< a for all n P K: ðnÞ

By Claim 3, for every n ¼ K; K þ 1; . . . there are two time sequence fsq g and ft ðnÞ q g, satisfying ðnÞ

ðnÞ

ðnÞ

ðnÞ

0 < s1 < t 1 < s2 < t2 <    < sqðnÞ < t ðnÞ q <  ðnÞ

and limq!1 sq ¼ 1, such that

x2 ðsðnÞ q ; xn Þ P a;

a n

a n

6 x2 ðtðnÞ q ; xn Þ < a;

< x2 ðsðnÞþ ; xn Þ 6 a ; q x2 ðtðnÞþ ; xn Þ 6 q

a n

;

ð3:21Þ ð3:22Þ

66

H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

a n

6 x2 ðt; xn Þ 6 a for all t 2 ðsqðnÞ ; t ðnÞ q Þ:

ð3:23Þ

From (3.2), Claim 2 and the ultimate boundedness of system (1.1), we obtain that for every n there is a T ðnÞ > T 0 such that

c 6 x1 ðt; xn Þ 6 x10 ðtÞ þ e0 ; c 6 x10 ðtÞ

ð3:24Þ

ðnÞ

and xi ðt; xn Þ 6 L (i ¼ 1; 2) for all t P T , where constants e0 and T 0 is given in (3.16) and constant c is given in Claim 2. FurðnÞ ðnÞ ðnÞ ther, for every n there is an integer N 1 > 0 such that sq > T ðnÞ for all q P N 1 . By condition (2) of (B2) there is a constant G2 > 0 such that

jf2 ðt; x1 ; x2 Þj 6 G2 for all t 2 Rþ0 , c 6 x1 6 L and 0 6 x2 6 L. Hence, from (3.5), (3.16) and (3.24) we obtain

0

B x2 ðtðnÞþ ; xn Þ ¼ x2 ðsðnÞ q q ; xn Þ exp @ 0 B  exp @

Z

Z

ðnÞ

X

tq ðnÞ

f2 ðt; x1 ðt; xn Þ; x2 ðt; xn ÞÞdt þ

sq

ðnÞ ðnÞ sq 6t k 6tq

1

C ln h2k A ¼ x2 ðsðnÞ q ; xn Þ

ðnÞ

tq ðnÞ

ðf2 ðt; x1 ðt; xn Þ; x2 ðt; xn ÞÞ  f2 ðt; x10 ðtÞ; e0 ÞÞdt þ

sq

Z

ðnÞ

tq ðnÞ

f2 ðt; x10 ðtÞ; e0 Þdt þ

sq

ðnÞ

ðnÞ

ðnÞ

ðnÞ

1

X ðnÞ

ðnÞ

C ln h2k A:

sq 6t k 6t q

ðnÞ

We can choose an integer lq such that t ðnÞ q ¼ sq þ lq x0 þ v q , where v q 2 ½0; x0 Þ. Then from (3.16) and above equality, we can obtain

0

a n

B ðnÞ ðnÞ P x2 ðtðnÞþ ; xn Þ P x2 ðsðnÞ q q ; xn Þ exp @2G2 ðt q  sq Þ þ Pa

expð2G2 ðt ðnÞ q



sðnÞ q Þ

Z

ðnÞ

tq ðnÞ

ðnÞ

f2 ðt; x10 ðtÞ; e0 Þdt þ

sq þlq x0

1

X ðnÞ

ðnÞ

ðnÞ

C ln h2k A

sq þlq x0 6tk 6t q

 G2 x0  2HÞ:

Consequently, we have ðnÞ tðnÞ q  sq P

ln n  G2 x0  2H 2G2

ðnÞ

for all q P N1 ; n ¼ K; K þ 1; . . . :

By (3.16) there are positive constants P and r such that

Z

tþa

f2 ðs; uðsÞ; e0 Þds þ

t

X

ln h2k P r;

ð3:25Þ

t6t k 6tþa

for all t 2 Rþ0 and a P P. ðnÞ ðnÞ Let  x1a ðtÞ be the solution of system (3.17) with the initial condition  x1a ðsq Þ ¼ x1 ðsq ; xn Þ. Since for any n; q and ðnÞ ðnÞ t 2 ½sq ; tq  we have x2 ðt; xn Þ 6 a and

dx1 ðt; xn Þ P x1 ðt; xn Þf1 ðt; x1 ðt; xn Þ; aÞ; dt by the comparison theorem of impulsive system it follows that ðnÞ x1 ðt; xn Þ P x1a ðtÞ for all t 2 ½sðnÞ q ; t q :

ð3:26Þ

By Lemma 2.1, the solution x1a ðtÞ is globally uniformly asymptotically stable. From (3.24) we obtain ðnÞ c 6 x1 ðsðnÞ for all q P N1 : q ; xn Þ 6 L ðnÞ

Hence, there is a constant T 2 P P and T 2 is independent of any n and q P N 1 , such that

x1a ðtÞ > x1a ðtÞ 

e0 2

for all t P sðnÞ q þ T2:

Choose an integer K 0 P K such that n P K 0 and q P

ð3:27Þ ðnÞ N1 ,

ðnÞ tðnÞ q  sq > 2T 2 :

Further, from (3.18), (3.24), (3.26) and (3.27) we obtain

jx1 ðt; xn Þ  x10 ðtÞj < e0 for all t 2

ðnÞ ½sq

þ

T 2 ; t ðnÞ q .

Hence, when n P K 0 and q P

ð3:28Þ ðnÞ N1 ,

by (3.5), (3.23) and condition (1) of (B2) it follows

67

H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

0 B x2 ðtðnÞþ ; xn Þ P x2 ðsðnÞ q q þ T 2 ; xn Þ exp @

Z

ðnÞ

tq

f2 ðt; x1 ðt; xn Þ; e0 Þdt þ

ðnÞ

sq þT 2

1

X ðnÞ

ðnÞ

C ln h2k A:

sq þT 2 6t k 6t q

Finally, from (3.21), (3.23), (3.25) and (3.28) we have

0

a n

P

a n

B exp @

Z

ðnÞ

tq

f2 ðt; x1 ðt; xn Þ; e0 Þdt þ

ðnÞ

sq þT 2

1

X ðnÞ ðnÞ sq þT 2 6tk 6t q

C a ln h2k A > ; n

which leads to contradiction. Therefore, Claim 4 is true. Finally, from Claims 1–4 we see that Theorem 3.1 is proved and this completes the proof of Theorem 3.1. h Remark 3.1. If system (1.1) without the impulsive effective, that is hik ¼ 1 for all i ¼ 1; 2 and k ¼ 1; 2; . . ., then Theorem 3.1 is the same with the Theorem 1 in [3]. Therefore, our result extents the corresponding results on the permanence for the general non-autonomous predator–prey Kolmogorov systems in [3]. In Theorem 3.1, we assume that component x2 ðtÞ of all positive solutions ðx1 ðtÞ; x2 ðtÞÞ of system (1.1) is ultimately bounded, i.e. (B3). In the following we will establish a criterion which assures the ultimate boundedness of positive solutions of system (1.1). We first introduce the following assumptions: (B4) There is a constant x4 > 0 such that for any positive constant M > b0 there is a large s ¼ sðMÞ > 0 such that

lim sup

Z

t!1

tþx4

max f2 ðu; x1 ; sÞdu þ

b0 6x1 6M

t

!

X

ln h2k

< 0:

t6t k
(B5) The function f1 ðt; x1 ; x2 Þ is continuous for all ðt; x1 ; x2 Þ 2 Rþ0  R2þ0 and there is a constant x5 > 0 such that for any positive constant M; e and M > e there is a large s ¼ sðM; eÞ > 0 such that

lim sup

Z

t!1

tþx5

max f1 ðu; x1 ; sÞdu þ

e6x1 6M

t

!

X

ln h1k

< 0:

t6t k
Furthermore, for any constant G > 0

supfjf2 ðt; x1 ; 0Þj : t 2 Rþ ; 0 6 x1 6 Gg < 1: On the ultimate boundedness of system (1.1) we have the following result. Theorem 3.2. Suppose that (B1) and (B2) hold. If (B4) or (B5) holds, then system (1.1) is ultimately bounded. Proof. Choose a constant L1 such that L1 > maxfb0 ; maxt2R x10 ðtÞg, where x10 ðtÞ is some fixed positive solution of system (3.1) and constant b0 is given in condition (2) of (B2). For any positive solution ðx1 ðtÞ; x2 ðtÞÞ of system (1.1), by (3.2) there is a large T 0 > 0 such that

x1 ðtÞ 6 L1

for all t P T 0 :

ð3:29Þ

Let x ¼ maxfxi : i ¼ 0; . . . ; 5g, by the boundedness of function hi ðt; v Þ (i ¼ 1; 2) on ðt; v Þ 2 Rþ  ½0; xÞ, we have there is positive constant H such that (3.3) hold. Now, we consider component x2 ðtÞ. Firstly, we let (B4) hold. From (B4), for constant L1 , there are s ¼ sðL1 Þ > 0, T 1 P T 0 and d > 0 such that

Z t

tþx4

max f2 ðu; x1 ; sÞdu þ

b0 6x1 6L1

X

ln h2k < d for all t P T 1 :

ð3:30Þ

t6tk
Firstly, we prove that there exist a s1 P T 1 such that

x2 ðtÞ 6 s expða2 x4 þ HÞ for all t P s1 ;

ð3:31Þ

where a2 ¼ maxff2 ðt; x1 ; sÞ : t 2 Rþ0 ; b0 6 x1 6 L1 g. In fact, we only need to consider the following three cases for x2 ðtÞ, Case I: there is a s1 P T 1 such that x2 ðtÞ P s for all t P s1 . Case II: there is a s1 P T 1 such that x2 ðtÞ 6 s for all t P s1 . Case III: x2 ðtÞ is oscillatory about s for all t P T 1 . We first consider Case I. Since x2 ðtÞ P s for all t P s1 , then by condition (1) and (2) of (B2) and (3.5) we can obtain

x2 ðtÞ 6 x2 ðs1 Þ exp

Z

t

s1

f2 ðu; x1 ðuÞ; sÞdu þ

X s1 6t k
!

ln h2k

6 x2 ðs1 Þ exp

Z

t

max f2 ðu; x1 ; sÞdu þ

s1 b0 6x1 6L1

X s1 6tk
!

ln h2k

68

H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

for all t P s1 . From this and (3.30) it follows that limt!1 x2 ðtÞ ¼ 0 which leads a contradiction. Next, we consider Case III. From the oscillation of x2 ðtÞ about s, we can choose two sequences fqn g and fqn g satisfying T 1 < q1 < q1 <    < qn < qn <    and limn!1 qn ¼ limn!1 qn ¼ 1 such that

x2 ðqn Þ 6 s; x2 ðqþn Þ P s;

x2 ðqn Þ P s;

x2 ðtÞ P s for all t 2 ðqn ; qn Þ

x2 ðqþ n Þ 6 s; x2 ðtÞ 6 s for all t 2 ðqn ; qnþ1 Þ:

and

For any t P T 1 , if t 2 ðqn ; qn  for some integer n, then we can choose integer l  0 and constant 0 6 l1 < x4 such that t ¼ qn þ lx4 þ l1 . Then by condition (1) and (2) of (B2), (3.5) and (3.30) we have

Z

x2 ðtÞ 6 s exp

qn þx4

þþ qn

0

6 s exp @

Z

þ

Z

þ  þ

!

t

max f2 ðt; x1 ; sÞdt

max f2 ðt; x1 ; sÞdt þ

1

A ln h2k A

qn þlx4 6tk
X

t

1

X

þ

qn þðl1Þx4 6tk
qn þlx4 b0 6x1 6L1

b0 6x1 6L1

qn þlx4

X

qn 6tk
0

qn þlx4

qn þðl1Þx4

X

þ@

Z

1

ln h2k A 6 s expða2 x4 þ HÞ:

qn þlx4 6tk
If there is an integer n such that t 2 ðqn ; qnþ1 , then we obviously have

x2 ðtÞ 6 s < s expða2 x4 þ HÞ: Therefore, let s1 ¼ q1 , for Case III we always have

x2 ðtÞ 6 s expða2 x4 þ HÞ for all t P s1 : Lastly, if Case II holds, then we directly have

x2 ðtÞ 6 s expða2 x4 þ HÞ for all t P s1 : Therefore, (3.31) is true. Next, we let (B5) hold. Let a3 ¼ maxfjf1 ðt; x1 ; 0Þj : t 2 Rþ0 ; 0 6 x1 6 L1 g. By condition (3) of (B2) there are positive constant T 1 ; d0 ; e0 ; e1 , e0 expða3 x5 þ HÞ < b0 and e1 < L1 such that

Z

tþx3

X

f2 ðu; e0 expða3 x5 þ HÞ; e1 Þdu þ

t

ln h2k < d0

for all t P T 1 :

ð3:32Þ

t6t k
Further, by (B5) there are constant s > L1 , T 2 P T 1 and d1 such that

Z t

X

tþx5

max f1 ðu; x1 ; sÞdu þ

e0 6x1 6L1

ln h1k < d1

for all t P T 2 :

ð3:33Þ

t6tk
We first prove lim inf x2 ðtÞ 6 s for any positive solution ðx1 ðtÞ; x2 ðtÞÞ of system (1.1). In fact, if lim inf x2 ðtÞ > s, then there is a t!1 t!1 T 3 P maxfT 0 ; T 2 g such that x2 ðtÞ P s for all t P T 3 . If x1 ðtÞ P e0 for all t P T 3 , then by condition (1) of (B1), (3.4) and (3.29) we have

x1 ðtÞ 6 x1 ðT 3 Þ exp

Z

t

f1 ðu; x1 ðuÞ; sÞdu þ

T3

!

X

ln h1k

6 x1 ðT 3 Þ exp

Z

t

T3

T 3 6t k
max f1 ðu; x1 ; sÞdu þ

e0 6x1 6L1

X

! ln h1k

T 3 6tk
for all t P T 3 . Consequently, by (3.33) it follows limt!1 x1 ðtÞ ¼ 0, which leads to a contradiction. Hence, there is a s1 P T 3 such that x1 ðs1 Þ < e0 . If there is a s2 > s1 such that x1 ðs2 Þ > e0 expða3 x5 þ HÞ, then there is a s3 2 ðs1 ; s2 Þ such that

x1 ðs3 Þ 6 e0 ;

x1 ðsþ3 Þ P e0

and x1 ðtÞ P e0

for all t 2 ðs3 ; s2 :

Choose an integer p  0 such that s2 2 ½s3 þ px5 ; s3 þ ðp þ 1Þx5 Þ, then by condition (1) of (B1), (3.5) and (3.33) we have

x1 ðs2 Þ ¼ x1 ðs3 Þ exp

Z

f1 ðt; x1 ðtÞ; x2 ðtÞÞdt þ s3

6 e0 exp

Z

X s3 6t k
ln h1k

s3 6tk
þ þ

s3

þ

!

X

s2

þ  þ

Z

s3 þpx5

þ

Z

s3 þðp1Þx5

X s3 þðp1Þx5 6t k
!

s2

max f1 ðt; x1 ; sÞdt

s3 þpx5

þ

e0 6x1 6L1

X s3 þpx5 6t k
!

! ln h1k

6 e0 expða3 x5 þ HÞ;

69

H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

which is contradictory. Hence, we have x1 ðtÞ 6 e0 expða3 x5 þ HÞ for all t P s1 . From this, by conditions (1) and (2) of (B2) and (3.5) we have

Z

x2 ðtÞ 6 x2 ðs1 Þ exp

t

f2 ðu; e0 expða3 x5 þ HÞ; sÞdu þ

s1

X

! for all t P s1 :

ln h2k

s1 6t k
Therefore, by (3.32) it follows limt!1 x2 ðtÞ ¼ 0 which leads a contradiction. Further, we prove that there is a constant L2 > 0 such that for any positive solution ðx1 ðtÞ; x2 ðtÞÞ of system (1.1),

lim sup x2 ðtÞ 6 L2 :

ð3:34Þ

t!1

In fact, if (3.34) is not true, then there is a sequence of initial value fxn g  R2þ such that for the solution ðx1 ðt; xn Þ; x2 ðt; xn ÞÞ of system (1.1)

lim sup x2 ðt; xn Þ > ð2s þ 1Þn;

n ¼ 1; 2; . . . :

t!1

From (3.3) we have that

eH 6 h2k 6 eH

for all k ¼ 1; 2; . . . :

Hence, we can choose an integer K > eH , for any positive solution xðtÞ of system (1.1), if

x2 ðtk Þ P ð2s þ 1Þn for some k ¼ 1; 2; . . . ; then we have

x2 ðtþk Þ ¼ h1k x1 ðtk Þ P eH ð2s þ 1Þn > 2s for all n P K; and if

x2 ðtk Þ 6 2s for some k ¼ 1; 2; . . . ; then we have

x2 ðtþk Þ ¼ h2k x2 ðtk Þ 6 eH 2s < ð2s þ 1Þn for all n P K: ðnÞ

Hence, for every n there are two time sequence fsq g and ft ðnÞ q g, such that for each n ¼ K; K þ 1; . . . ðnÞ

ðnÞ

ðnÞ

ðnÞ

0 < s1 < t 1 < s2 < t2 <    < sqðnÞ < t ðnÞ q < ; sðnÞ q ! 1;

tðnÞ as q ! 1; q ! 1

x2 ðsðnÞ q ; xn Þ 6 2s;

ð2s þ 1Þn > x2 ðsðnÞþ ; xn Þ P 2s; q

2s < x2 ðt ðnÞ q ; xn Þ 6 ð2s þ 1Þn;

ð3:35Þ

x2 ðt qðnÞþ ; xn Þ P ð2s þ 1Þn;

ð3:36Þ

ðnÞ 2s 6 x2 ðt; xn Þ 6 ð2s þ 1Þn for all t 2 ðsðnÞ q ; t q Þ:

For every n we can choose T

x1 ðt; xn Þ 6 L1

ðnÞ

ð3:37Þ

> T 0 such that

for all t P T ðnÞ : ðnÞ

ðnÞ

ðnÞ

Further, there is an integer N 1 > 0 such that sq > T ðnÞ for all q P N 1 . From condition (2) of (B2), there is a constant G2 ¼ G2 ðL1 Þ > 0 such that

jf2 ðt; x1 ; x2 Þj 6 G2

and jf2 ðt; e0 expða3 x3 þ HÞ; e1 Þj 6 G2

for all t 2 Rþ0 , b0 6 x1 6 L1 , 0 6 x2 6 L1 . By conditions (1) and (2) of (B2) and (3.5) we have 0 B x2 ðt ðnÞþ ; xn Þ ¼ x2 ðsðnÞ q q ; xn Þ exp @

Z

f2 ðt;x1 ðt;xn Þ; x2 ðt; xn ÞÞdt þ

ðnÞ

sq

0 B 6 x2 ðsðnÞ q ; xn Þexp @ ðnÞ

ðnÞ

tq

Z

X ðnÞ ðnÞ sq 6t k 6tq

1 C ln h2k A

ðnÞ

tq ðnÞ

ðf2 ðu; x1 ðt; xn Þ; 0Þ  f2 ðu; e0 expða3 x5 þ HÞ; e1 ÞÞdu þ

sq

Z

ðnÞ

tq ðnÞ

sq

ðnÞ

ðnÞ

ðnÞ

We can choose an integer lq such that tðnÞ q ¼ sq þ lq x3 þ v q , where

f2 ðu; e0 expða3 x5 þ HÞ; e1 Þdu þ

X ðnÞ ðnÞ sq 6t k 6tq

1 C ln h2k A:

v ðnÞ q 2 ½0; x3 Þ. Then from (3.32), (3.35) and (3.36)

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H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

0 B ðnÞ ð2s þ 1Þn 6 x2 ðtðnÞþ ; xn Þ 6 2s exp @2G2 ðt ðnÞ q q  sq Þ þ

Z

ðnÞ

tq ðnÞ

f2 ðt; e0 expða3 x3 þ HÞ; e1 Þdu þ

ðnÞ

sq þlq x3

1

X ðnÞ

ðnÞ

ðnÞ

C ln h2k A

sq þlq x3 6t k 6t q

ðnÞ 6 2s expð2G2 ðt ðnÞ q  sq Þ þ G2 x3 þ 2HÞ:

Consequently, ðnÞ tðnÞ q  sq P

ln n  G2 x3  2H 2G2

ðnÞ

for all q P N1 :

By condition (1) of (B1), (3.32) and (3.33) there is a constant P > 0 such that

Z

tþa

f2 ðu; e0 expða3 x5 þ HÞ; sÞdu þ

t

X

ln h2k <  ln H

ð3:38Þ

t6tk
and

Z

X

tþa

max f1 ðu; x1 ; sÞdu þ

t

e0 6x1 6L1

ln h1k < lnð

t6t k 6tþa

e0 2L1

Þ

ð3:39Þ

for all t 2 Rþ0 and a P P. Choose an integer N 0 > K such that ðnÞ tðnÞ q  sq > 2P

ðnÞ

for all n P N0 ; q P N1 : ðnÞ

ðnÞ

ðnÞ

For any n P N 0 and q P N 1 , if x1 ðt; xn Þ P e0 for all t 2 ½sq ; sq þ P, then by condition (1) of (B1), (3.4), (3.37) and (3.39) we have

0 B x1 ðsðnÞ q þ P; xn Þ 6 L1 exp @ 0 B 6 L1 exp @

Z

ðnÞ

f1 ðt; x1 ðt; xn Þ; 2sÞdt þ

ðnÞ

sq

Z

X

sq þP ðnÞ

C ln h1k A

sq 6t k
sq þP ðnÞ

sq

ðnÞ

1

max f1 ðt; x1 ; sÞdt þ

e0 6x1 6L1

X ðnÞ ðnÞ sq 6tk
ðnÞ

1 C ln h1k A < e0 ;

ðnÞ

which is a contradiction. Hence, there is a s1 2 ½sq ; sq þ P such that x1 ðs1 ; xn Þ < e0 . Further, a similar argument as in the above we obtain that

x1 ðt; xn Þ 6 e0 expða3 x5 þ HÞ for all t 2 ½s1 ; tðnÞ q : Therefore, from (3.5), (3.37), (3.38) and conditions (1) and (2) of (B2)

0 Z tðnÞ B q ðnÞþ x2 ðtq ; xn Þ 6 ð2s þ 1Þn exp @ f2 ðt; e0 expða3 x5 þ HÞ; sÞdt þ ðnÞ sq þP

1

X ðnÞ

ðnÞ

C ln h2k A < ð2s þ 1Þn:

sq þP6t k 6t q

This leads to a contradiction with (3.36). Therefore, (3.34) is true. So system (1.1) is ultimately bounded. This completes the proof. h Remark 3.2. In (B5), we require that the function f1 ðt; x1 ; x2 Þ has definition and is continuous for x1 ¼ 0. For the case f1 ðt; x1 ; x2 Þ is discontinuous, Teng has obtained a similar result with (B5) for the predator–prey Kolmogorov system without impulse in [3]. Therefore, there is an interesting open problem is whether system (1.1) has a similar result with (B5) for the case f1 ðt; x1 ; x2 Þ is discontinuous. 4. Application In this section, we will apply the above theorems to discuss the permanence of positive solutions for systems (1.2)–(1.7). We first assume in the systems (1.2)–(1.7) that 0 6 t 1 < t2 <    < t k <    is impulsive time sequence and limk!1 tk ¼ 1 and hik for each i ¼ 1; 2 and k ¼ 1; 2; . . . are positive constants, bi ðtÞ, aij ðtÞ, ai ðtÞ, bi ðtÞ, ci ðtÞ and xi ðtÞ (i; j ¼ 1; 2) are bounded and continuous functions on Rþ0 , aij ðtÞ  0, ai ðtÞ  0, bi ðtÞ > 0, ci ðtÞ  0 and xi ðtÞ  0 for all t 2 Rþ0 , i; j ¼ 1; 2, there is a positive constant x2 such that

lim inf t!1

Z t

tþx2

a11 ðsÞds > 0:

H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

71

In addition, we assume that (H1) There is a positive constant x1 such that

Z

lim inf t!1

tþx1

b1 ðsÞds þ

t

X

!

>0

ln h1k

t6t k
and function

X

h1 ðt; v Þ ¼

ln h1k

t6t k
is bounded on t 2 Rþ0 and v 2 ½0; x1 Þ. (H2) There is a positive constant x1 such that

Z

lim inf t!1

tþx1

b1 ðsÞds 

t

X

!

ln h1k

>0

t6t k
and function

X

h1 ðt; v Þ ¼

ln h1k

t6t k
is bounded on t 2 Rþ0 and v 2 ½0; x1 Þ. Firstly, we consider the following single-species non-autonomous logistic impulsive system and linear impulsive system

(

dx1 ðtÞ ¼ x1 ðtÞðb1 ðtÞ dt x1 ðtþk Þ ¼ h1k xðt k Þ;

 a11 ðtÞx1 ðtÞÞ;

t – tk ;

ð4:1Þ

k ¼ 1; 2; . . .

and

(

dx1 ðtÞ ¼ a11 ðtÞ  b1 ðtÞx1 ðtÞ; t – tk ; dt x1 ðtþk Þ ¼ h1k xðt k Þ; k ¼ 1; 2; . . . :

ð4:2Þ

Let x10 ðtÞ be some fixed positive solution of system (4.1) or (4.2). If the assumption (H1) holds, then x10 ðtÞ is globally uniformly asymptotically stable for system (4.1). For system (4.2), If the assumption (H2) holds, by Lemma 2.4, then we have x10 ðtÞ is globally uniformly asymptotically stable. We first consider the two species non-autonomous predator–prey Lotka–Volterra system (1.2). Directly applying Theorems 3.1 and 3.2 we have the following result. Theorem 4.1. Assume that assumption (H1) holds and there is a constant x3 > 0 such that

lim sup  t!1

Z

tþx3

b2 ðsÞds þ

t

< 0;

ln h2k

t6t k
X

h2 ðt; v Þ ¼

!

X

ln h2k

t6t k
is bounded on t 2 Rþ andv 2 ½0; x3 Þ and there is a constant x4 > 0 such that

lim inf t!1

Z

tþx4

a22 ðsÞds > 0 or lim inf t!1

t

Z

tþx4

a12 ðsÞds > 0:

t

If there is a constant x0 > 0 such that

lim inf t!1

Z

X

tþx0

ðb2 ðsÞ þ a21 ðsÞx10 ðsÞÞds þ

t

! ln h2k

> 0;

t6t k
then system (1.2) is permanent. We next consider two species non-autonomous predator–prey Holling-type functional response systems (1.3) and (1.4). Applying Theorems 3.1 and 3.2 we have the following result. Theorem 4.2. Assume that assumption (H1) holds, inf t0 a11 ðtÞ > 0 and there is a constant x3 > 0 such that

lim sup  t!1

Z t

tþx3

b2 ðsÞds þ

X t6t k
!

ln h2k

< 0;

72

H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

X

h2 ðt; v Þ ¼

ln h2k

t6t k
is bounded on t 2 Rþ0 andv 2 ½0; x3 Þ, and there is a x4 > 0 such that

lim inf t!1

Z

tþx4

Z

a1 ðsÞds > 0 or lim inf t!1

t

tþx4

a22 ðsÞds > 0:

t

If there is a constant x0 > 0 such that

lim inf

Z

t!1

X

tþx0

ðb2 ðsÞ þ /2 ðs; x10 ðsÞÞÞds þ t

! ln h2k

> 0;

t6t k
then system (1.3) and (1.4) are permanent. Proof. In fact, for the two species non-autonomous predator–prey Holling I-type functional response impulsive system (1.3). Comparing with system (1.1), we have

( f1 ðt; x1 ; x2 Þ ¼

b1 ðtÞ  a11 ðtÞx1  a1 ðtÞx2 ;

0 6 x1 6 x10 ;

10 x2 ; b1 ðtÞ  a11 ðtÞx1  a1 ðtÞx x1

x1 > x10 ;

and

f2 ðt; x1 ; x2 Þ ¼



b2 ðtÞ þ a2 ðtÞx1  a22 ðtÞx2 ;

0 6 x1 6 x10 ;

b2 ðtÞ þ a2 ðtÞx10  a22 ðtÞx2 ;

x1 > x10 :

Calculating the partial derivative of fi ðt; x1 ; x2 Þ (i ¼ 1; 2) with respect to x1 and x2 we have

@f1 ðt; x1 ; x2 Þ ¼ @x1 @f1 ðt; x1 ; x2 Þ ¼ @x2 @f2 ðt; x1 ; x2 Þ ¼ @x1

(

(

a11 ðtÞ;

0 6 x1 < x10 ;

a11 ðtÞ þ

a1 ðtÞx10

a1 ðtÞ;

0 6 x1 < x10 ;

x21

10 ;  a1 ðtÞx x1



x2 ;

x1 > x10 ;

x1 > x10 ;

a2 ðtÞ; 0 6 x1 < x10 ; 0;

ð4:3Þ

ð4:4Þ

x1 > x10 ;

and

@f2 ðt; x1 ; x2 Þ ¼ a22 ðtÞ: @x2 From (4.3) we see that, when 0 6 x1 < x10 , obviously

@f1 ðt; x1 ; x2 Þ 6 a11 ðtÞ for all t 2 Rþ0 ; @x1 and when x1 > x10 , since the function

gðt; x1 Þ ¼

a1 ðtÞx10 x1

is bounded for all t 2 R10 and inf t0 a11 ðtÞ > 0, there is constant a0 > 0 such that

@f1 ðt; x1 ; x2 Þ 1 6  a11 ðtÞ @x1 2 for all t 2 Rþ0 and 0 6 x2 6 a0 . Therefore, (B1), (B2) and (B4) or (B5) hold. Thus, as a consequence of Theorems 3.1 and 3.2, we obtain that system (1.3) is permanent. Similar argument as in Theorem 4 in [3], we can obtain that system (1.4) is permanent. This complete the proof. h Further, we consider the two species non-autonomous predator–prey Beddington–DeAngelis functional response impulsive system (1.5) and Leslie–Gower functional response system (1.6). Applying Theorems 3.1 and 3.2 we have the following results. Theorem 4.3. Assume that assumption (H1) holds, inf t0 a11 ðtÞ > 0 and there is a constant x3 > 0 such that

lim sup  t!1

Z t

tþx3

b2 ðsÞds þ

X t6t k
!

ln h2k

<0

73

H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

and

X

h2 ðt; v Þ ¼

ln h2k

t6t k
is bounded on t 2 Rþ0 andv 2 ½0; x3 Þ. If there is a constant x0 > 0 such that

lim inf

Z

t!1

X

tþx0

ðb2 ðsÞ þ /2 ðs; x10 ðsÞÞÞds þ

t

ln h2k > 0;

t6tk
then system (1.5) is permanent. Proof. Comparing with system (1.1), we have

f1 ðt; x1 ; x2 Þ ¼ b1 ðtÞ  a11 ðtÞx1 

a1 ðtÞxm1 1 x2 1 þ c1 ðtÞxn1 þ x1 ðtÞx2

and

f2 ðt; x1 ; x2 Þ ¼ b2 ðtÞ þ

a2 ðtÞxm1 : 1 þ c2 ðtÞxn1 þ x2 ðtÞx2

Calculating the partial derivative of fi ðt; x1 ; x2 Þ (i ¼ 1; 2) with respect to x1 and x2 we have

@f1 ðt; x1 ; x2 Þ ½ðm  1Þð1 þ x1 ðtÞx2 Þ þ ðm  1  nÞc1 ðtÞxn1 a1 ðtÞxm2 x2 1 ¼ a11 ðtÞ  ; 2 n @x1 ½1 þ c1 ðtÞx1 þ x1 ðtÞx2 

ð4:5Þ

@f1 ðt; x1 ; x2 Þ a1 ðtÞxm1 ð1 þ c1 ðtÞxn1 Þ 1 ¼ ; @x2 ½1 þ c1 ðtÞxn1 þ x1 ðtÞx2 2 @f2 ðt; x1 ; x2 Þ a2 ðtÞxm1 ðm½1 þ x2 ðtÞx2  þ ðm  nÞc2 ðtÞxn1 Þ 1 ¼ @x1 ½1 þ c2 ðtÞxn1 þ x2 ðtÞx2 2

ð4:6Þ

and

@f2 ðt; x1 ; x2 Þ a2 ðtÞx2 ðtÞxm1 ¼ : @x2 ½1 þ c2 ðtÞxn1 þ x2 ðtÞx2 2 From (4.5) we see that, when m P n þ 1, obviously

@f1 ðt; x1 ; x2 Þ 6 a11 ðtÞ @x1 for all t 2 Rþ0 , x1 2 Rþ and x2 2 Rþ0 , and when m 6 n, since the function

gðt; x1 ; x2 Þ ¼

½ðm  1Þð1 þ x1 ðtÞx2 Þ þ ðm  1  nÞc1 ðtÞxn1 a1 ðtÞxm2 1 ½1 þ c1 ðtÞxn1 þ x1 ðtÞx2 2

is bounded for all t 2 Rþ0 and x1 ; x2 2 Rþ , there is a small enough constant a0 > 0 such that

@f1 ðt; x1 ; x2 Þ 1 6  a11 ðtÞ @x1 2 for all t 2 Rþ0 , x1 2 Rþ and 0 6 x2 6 a0 . Further, from (4.6) it is obvious that there is a constant b0 > 0 such that

f2 ðt; x1 ; x2 Þ 0 @x1 for all t 2 Rþ0 , 0 < x1 6 b0 and x2 2 Rþ0 . Therefore, (B1), (B2) and (B4) hold. Thus, as a consequence of Theorems 3.1 and 3.2, we obtain that system (1.5) is permanent. h Theorem 4.4. Assume that inf t0 a11 ðtÞ > 0 and there is a constant x3 > 0 such that lim inf t!1 stant x0 > 0 such that

lim sup t!1

Z t

tþx0

b2 ðsÞds þ

X

R tþx3 t

a2 ðsÞds > 0. If there is a con-

! ln h2k

< 0;

t6t k
then system (1.6) is permanent. We lastly consider the two-species non-autonomous predator chemostat type impulsive system (1.7).

74

H. Hu / Applied Mathematics and Computation 223 (2013) 54–75

Theorem 4.5. Assume that assumption (H2) holds and (a) inf t0 b1 ðtÞ > 0. (b) There is a constant x3 > 0 such that

Z

lim inf t!1

tþx3

b2 ðsÞds  t

!

X

ln h2k

>0

t6t k
and

X

h2 ðt; v Þ ¼

ln h2k

t6t k
is bounded on t 2 Rþ0 , v 2 ½0; x3 Þ. (c) There is a constant x4 > 0 such that

lim inf t!1

Z

X

tþx4

a22 ðsÞds 

t

ln h2k > 0:

t6t k
If there is a constant x0 > 0 such that

lim inf t!1

Z

tþx0

ðb2 ðsÞ þ /2 ðs; x10 ðsÞÞÞds þ t

X

! ln h2k

> 0;

t6t k
then system (1.7) is permanent. Remark 4.1. If system (1.2)–(1.7) without the impulsive effective, that is hik ¼ 1 for all i ¼ 1; 2 and k ¼ 1; 2; . . ., Theorems 4.1,4.2,4.3,4.4,4.5 are the same with Theorems 3–7 in [3]. Therefore, our result extents the corresponding results for the permanence for systems (1.2)–(1.7) without impulse in [3]. Remark 4.2. By our best knowledge, on the permanence for some special impulsive non-autonomous predator–prey such as systems (1.2)–(1.7) have not been studied. Therefore, our results in Theorems 4.1,4.2,4.3,4.4,4.5 is new. Acknowledgements We are very grateful to the referees for their very helpful comments and suggestions and careful reading of the manuscript. Supported by Science Foundation for The Excellent Youth Scholars of Shanghai Municipal Education Commission (No. 5113341105). References [1] H.I. Freedman, Deterministic Mathematical Models in Population Ecology, Monographs and Textbooks in Pure and Applied Mathematics, vol. 57, Marcel Dekker, New York, 1980. [2] Z. Teng, Uniform persistence of the periodic predator–prey Lotka–Volterra systems, Appl. Anal. 72 (1999) 339–352. [3] Z. Teng, Z. Li, H. Jiang, Permanence criteria in non-autonomous predator–prey Kolmogorov systems and its applications, Dyn. Syst. 19 (2004) 171–194. [4] C.S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can. 45 (1965) 3–60. [5] S. Ruan, D. Xiao, Global analysis in a predator–prey system with nonmonotonic functional response, SIAM J. Appl. Math. 61 (2001) 1445–1472. [6] J.R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol. 44 (1975) 331–340. [7] D.L. DeAngelis, R.A. Goldstein, R.V. ONeill, A model for trophic interaction, Ecology 56 (1975) 881–892. [8] H.I. Freedman, R.M. Mathsen, Persistence in predator–prey systems with ratio-dependence predation influence, Bull. Math. Biol. 55 (1993) 817–827. [9] S.B. Hsu, T.W. Huang, Global stability for a class of predator–prey systems, SIAM J. Appl. Math. 55 (1995) 763–783. [10] P.H. Leslie, J.C. Gower, The properties of a stochastic model for the predator–prey type of interaction between two species, Biometrika 47 (1960) 219– 234. [11] J.K. Hale, A.S. Somolinos, Competition for fluctuating nutrient, J. Math. Biol. 18 (1983) 255–280. [12] H.L. Smith, P. Waltman, The Theory of the Chemostat, Cambridge University Press, Cambridge, 1995. [13] S.F. Ellermeyer, S. Pilyugin, R. Redheffer, Persistence criteria for a chemostat with variable nutrient input, J. Differ. Equ. 171 (2001) 132–147. [14] J. Wu, X.-Q. Zhao, X. He, Global asymptotic behaviour in almost periodic Kolmogorov equations and chemostat models, Nonlinear World 3 (1996) 589– 611. [15] B. Liu, Y. Zhang, L. Chen, The dynamical behaviors of a Lotka–Volterra predator–prey model concerning integrated pest management, Nonlinear Anal. Real World Appl. 6 (2005) 227–243. [16] B. Liu, Y. Zhang, L. Chen, Dynamic complexities in a Lotka–Volterra predator–prey model concerning impulsive control strategy, Int. J. Bifurcation Chaos 15 (2005) 517–531. [17] B. Liu, Z. Teng, W. Liu, Dynamic behaviors of the periodic Lotka–Volterra competing system with impulsive perturbations, Chaos Solitons Fractals 31 (2007) 356–370. [18] J. Hou, Z. Teng, S. Gao, Permanence and global stability for nonautonomous N-species Lotka–Volterra competitive system with impulses, Nonlinear Anal. Real World Appl. 11 (2010) 1882–1896. [19] B. Liu, Z. Teng, L. Chen, Analysis of a predator–prey model with Holling II functional response concerning impulsive control strategy, J. Comput. Appl. Math. 193 (2006) 347–362.

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