Skip to main content

Theory and Modern Applications

Uniformly asymptotic stability of almost periodic solutions for a delay difference system of plankton allelopathy

Abstract

In this contribution, we investigate a delayed difference almost periodic system for the growth of two species of plankton with competition and allelopathic effects on each other. By using the methods of Lyapunov function and preliminary lemmas, sufficient conditions which guarantee the existence and uniformly asymptotic stability of a unique positive almost periodic solution of the system are established. An example together with its numerical simulations is presented to verify the validity of the proposed criteria.

1 Introduction

Allelopathy is a biological phenomenon by which individuals of a population release one or more biochemicals that have an effect on the growth, survival, and reproduction of the individuals of another population. As an important factor for ecosystem functioning, allelopathic interactions have occurred in various aspects: between bacteria [1], between bacteria and phytoplankton [2, 3], between phytoplankton and zooplankton [4], and also between calanoid copepods [5]. Especially, allelopathic interactions are widespread in phytoplankton communities, which deeply attract the attention of researchers. Thus, in aquatic ecology, the study of tremendous fluctuations in abundance of many phytoplankton communities is a significant theme. Recently, many workers have been aware that the increased population of one species of phytoplankton might restrain the growth of one or several other species by the production of allelopathic toxins. For detailed literature studies, we can refer to [615] and the references cited therein.

In [15], Qin and Liu discussed the permanence and global attractivity of the following delay difference system with plankton allelopathy:

{ x 1 ( n + 1 ) = x 1 ( n ) exp { r 1 ( n ) a 11 ( n ) x 1 ( n ) a 12 ( n ) x 2 ( n ) x 1 ( n + 1 ) = b 1 ( n ) x 1 ( n ) p = 0 M k 2 ( p ) x 2 ( n p ) } , x 2 ( n + 1 ) = x 2 ( n ) exp { r 2 ( n ) a 21 ( n ) x 1 ( n ) a 22 ( n ) x 2 ( n ) x 2 ( n + 1 ) = b 2 ( n ) x 2 ( n ) p = 0 M k 1 ( p ) x 1 ( n p ) } , x i ( Φ ) 0 , Φ [ p , 0 ] Z ; x i ( 0 ) > 0 , i = 1 , 2 ,
(1.1)

where x i (n) are the population densities of species x i at the n th generation, r i (n) stand for the intrinsic growth rates of species x i at the n th generation, a i i (n) are the intra-specific effects of the n th generation of species x i on own population, and a i j (n) measure the inter-specific effects of the n th generation of species x j on species x i , b i (n) x i (n) p = 0 M k j (p) x j (np) denote the effect of toxic substances (i,j=1,2; ij), M is a positive integer.

Notice that the environment varies due to the factors such as seasonal effects and variations in weather conditions, food supplies, mating habits, harvesting etc. Thus it is reasonable to assume that the parameters in system (1.1) are periodic. However, if the various constituent components of the temporally nonuniform environment is with incommensurable periods (non-integral multiples), then we have to consider the environment to be almost periodic, which leads to the almost periodicity of the parameters of system (1.1). The main purpose is to establish sufficient conditions for the existence and uniformly asymptotic stability of a unique positive almost periodic solution of system (1.1). To do so, we assume that { r i (n)}, { a i j (n)} and { b i (n)} for i,j=1,2 are bounded nonnegative almost periodic sequences, k i (p), i=1,2, is a bounded positive sequence.

Many recent works have been done on the existence and stability of almost periodic solutions for the discrete biological models without or with time delays (see [1621]). However, to the best of our knowledge, there are few published papers concerning the above almost periodic system (1.1). For the sake of simplicity and convenience, in the following discussion, the notations below will be used

h u = sup n Z + { h ( n ) } , h l = inf n Z + { h ( n ) } ,
(1.2)

where {h(n)} is a bounded sequence defined on the set of nonnegative integers Z + . Meanwhile, we make a convention that n = a b h(n)=0 if a>b.

The rest of this paper is organized as follows. In Section 2, we introduce some notations, definitions and lemmas which are useful for our main results. Sufficient conditions for the existence and uniformly asymptotic stability of a unique positive almost periodic solution of system (1.1) are established in Section 3. In Section 4, an example and its numerical simulations are presented to illustrate the feasibility of our main results. Finally, we give some proofs of theorems in the appendices for convenience in reading.

2 Preliminaries

In this section, we give some notations, definitions and lemmas which will be useful for the later sections.

Denote by , R + , and Z + the sets of real numbers, nonnegative real numbers, integers and nonnegative integers, respectively. R 2 and R k denote the cone of a two-dimensional and k-dimensional real Euclidean space, respectively. We also set

[ c , d ] Z =[c,d]Z,c,dZ,K= [ M , + ) Z ,

where M is defined in (1.1).

Definition 2.1 (see [22])

A sequence y:Z R k is called an almost periodic sequence if the ε-translation set of y

E{ε,y}:= { τ Z : | y ( n + τ ) y ( n ) | < ε , n Z }

is a relatively dense set in for all ε>0; that is, for any given ε>0, there exists an integer l(ε)>0 such that each interval of length l(ε) contains an integer τ=τ(ε)E{ε,y} such that

| y ( n + τ ) y ( n ) | <ε,nZ.

τ is called the ε-translation number of y(n).

Definition 2.2 (see [22])

Let g:Z×D R k , where D is an open set in C:={ϕ: [ τ , 0 ] Z R k }. g(n,ϕ) is said to be almost periodic in n uniformly for ϕD if for any ε>0 and any compact set S in D, there exists a positive integer l(ε,S) such that any interval of length l(ε,S) contains an integer τ for which

| g ( n + τ , ϕ ) g ( n , ϕ ) | <ε,nZ,ϕS.

τ is called the ε-translation number of g(n,ϕ).

Lemma 2.3 (see [23])

{y(n)} is an almost periodic sequence if and only if for any sequence { p k }Z there exists a subsequence { p k }{ p k } such that y(n+ p k ) converges uniformly on nZ as k. Furthermore, the limit sequence is also an almost periodic sequence.

Consider the following almost periodic delay difference system:

y(n+1)=g(n, y n ),n Z + ,
(2.1)

where

g: Z + × C B R, C B = { ϕ C : ϕ < B } ,C= { ϕ : [ τ , 0 ] Z R } ,

with ϕ= sup s [ τ , 0 ] Z |ϕ(s)|, g(n,ϕ) is almost periodic in n uniformly for ϕ C B and is continuous in ϕ, while y n C B is defined as y n (s)=y(n+s) for all s [ τ , 0 ] Z .

The product system of (2.1) is in the form of

y(n+1)=g(n, y n ),z(n+1)=g(n, z n ).
(2.2)

A discrete Lyapunov function of (2.2) is a function V: Z + × C B × C B R + which is continuous in its second and third variables. Define the difference of V along the solution of system (2.2) by

Δ V ( 2.2 ) (n,ϕ,ψ)=V ( n + 1 , y n + 1 ( n , ϕ ) , z n + 1 ( n , ψ ) ) V(n,ϕ,ψ),

where (y(n,ϕ),z(n,ψ)) is a solution of system (2.2) through (n,(ϕ,ψ)), ϕ,ψ C B . And Zhang and Zheng [22] obtained the following lemma.

Lemma 2.4 (see [22])

Suppose that there exists a Lyapunov function V(n,ϕ,ψ) satisfying the following conditions:

  1. (1)

    a(|ϕ(0)ψ(0)|)V(n,ϕ,ψ)b(ϕψ), where a,bP with P={α:[0,)[0,)|α(0)=0andα(u)is continuous, increasing inu}.

  2. (2)

    |V(n, ϕ 1 , ψ 1 )V(n, ϕ 2 , ψ 2 )|L( ϕ 1 ϕ 2 + ψ 1 ψ 2 ), where L>0 is a constant.

  3. (3)

    Δ V ( 2.2 ) (n,ϕ,ψ)γV(n,ϕ,ψ), where 0<γ<1 is a constant.

Moreover, if there exists a solution y(n) of system (2.1) such that y n B <B for all n Z + , then there exists a unique uniformly asymptotically stable almost periodic solution q(n) of system (2.1) which satisfies |q(n)| B for all nK. In particular, if g(n,ϕ) is periodic with period ω, then system (2.1) has a unique uniformly asymptotically stable periodic solution with period ω.

Remark 2.5 (see [19])

From the proof of [[24], Theorem 6.6], it is not difficult to prove that condition (3) of Lemma 2.4 can be replaced by the following condition:

(3)′ Δ V ( 2.2 ) (n,ϕ,ψ)c(|ϕ(0)ψ(0)|), where c{β:[0,)[0,)|β is continuous,β(0)=0 and β(s)>0 for s>0}.

Definition 2.6 (see [15])

System (1.1) is said to be permanent if there exist positive constants M i and M i such that

M i lim inf n + x i (n) lim sup n + x i (n) M i ,i=1,2

for any positive solution ( x 1 (n), x 2 (n)) of system (1.1).

Lemma 2.7 (see [15])

Assume that

min { r 1 l a 12 u M 2 , r 2 l a 21 u M 1 } > 0 , min { Δ 1 M 1 , Δ 2 M 2 } > 1 .
(2.3)

Then system (1.1) is permanent. Here, Δ i = a i i u + b i u M j ( M + 1 ) k j u r i l a i j u M j .

From the proof of [[15], Lemma 2.3], we have

lim sup n + x i (n) M i = def exp ( r i u 1 ) a i i l
(2.4)

and

lim inf n + x i (n) m i = def exp [ ( r i l a i j u M j ) ( 1 Δ i M i ) ] Δ i ,
(2.5)

where i,j=1,2, ij.

3 Main result

According to (2.4) and (2.5), we denote by Ω the set of all solutions ( x 1 (n), x 2 (n)) of system (1.1) satisfying m i x i (n) M i , i=1,2, for all nK. From Lemma 2.4, we first prove that there exists a bounded solution of system (1.1) and then construct a suitable Lyapunov function for system (1.1).

Theorem 3.1 If conditions (2.3) are satisfied, then Ωϕ.

The proof of Theorem 3.1 is given in Appendix 1.

Theorem 3.2 If conditions (2.3) and

1 max ( | 1 a 11 l m 1 | , | 1 a 11 u M 1 | ) ( b 1 u k 2 u + b 2 u k 1 u ) ( M + 1 ) M 1 M 2 a 21 u M 1 > 0 , 1 max ( | 1 a 22 l m 2 | , | 1 a 22 u M 2 | ) ( b 1 u k 2 u + b 2 u k 1 u ) ( M + 1 ) M 1 M 2 a 12 u M 2 > 0
(3.1)

are satisfied, then system (1.1) possesses a unique almost periodic solution ( x 1 (n), x 2 (n)), and it is uniformly asymptotically stable within Ω.

The proof of Theorem 3.2 is given in Appendix 2.

4 Example and numerical simulations

In this section, to verify the validity of our main results, we give an example and its corresponding numerical simulations.

Example 4.1 Consider the following discrete system with a delay:

{ x 1 ( n + 1 ) = x 1 ( n ) exp { 0.85 + 0.02 sin ( 2 n π ) ( 0.80 0.01 sin ( 2 n π ) ) x 1 ( n ) x 1 ( n + 1 ) = ( 0.03 + 0.01 sin ( 2 n π ) ) x 2 ( n ) x 1 ( n + 1 ) = ( 0.02 0.01 cos ( 2 n π ) ) x 1 ( n ) [ 0.83 x 2 ( n ) + 0.83 x 2 ( n 1 ) ] } , x 2 ( n + 1 ) = x 2 ( n ) exp { 0.80 + 0.01 cos ( 2 n π ) x 2 ( n + 1 ) = ( 0.02 + 0.01 cos ( 2 n π ) ) x 1 ( n ) x 2 ( n + 1 ) = ( 0.65 + 0.02 sin ( 2 n π ) ) x 2 ( n ) x 2 ( n + 1 ) = ( 0.03 + 0.02 sin ( 2 n π ) ) x 2 ( n ) [ 0.73 x 1 ( n ) + 0.73 x 1 ( n 1 ) ] } ,
(4.1)

with the following initial conditions:

x 1 (1)=1.06, x 1 (0)=1.02, x 2 (1)=0.85, x 2 (0)=0.98.
(4.2)

By a computation, we get

M 1 1.1115 , M 2 1.3126 , m 1 0.7305 , m 2 0.8010 , Δ 1 1.1259 , Δ 2 0.9927 , r 1 l a 12 u M 2 0.7775 > 0 , r 2 l a 21 u M 1 0.7567 > 0 , Δ 1 M 1 1.2514 > 1 , Δ 2 M 2 1.3030 > 1
(4.3)

and

min { r 1 l a 12 u M 2 , r 2 l a 21 u M 1 } >0,min{ Δ 1 M 1 , Δ 2 M 2 }>1.
(4.4)

A further calculation shows that

1 max ( | 1 a 11 l m 1 | , | 1 a 11 u M 1 | ) ( b 1 u k 2 u + b 2 u k 1 u ) ( M + 1 ) M 1 M 2 a 21 u M 1 0.3646 > 0 , 1 max ( | 1 a 22 l m 2 | , | 1 a 22 u M 2 | ) ( b 1 u k 2 u + b 2 u k 1 u ) ( M + 1 ) M 1 M 2 a 12 u M 2 0.2729 > 0 .
(4.5)

Clearly, the assumptions of Theorem 3.2 are satisfied, and hence system (4.1) has a unique uniformly asymptotically stable positive almost periodic solution. From Figure 1, we can see that there exists a positive almost periodic solution ( x 1 (t), x 2 (t)), and the two-dimensional and three-dimensional phase portraits of almost periodic system (4.1) are revealed in Figure 2, respectively. Figure 3 shows that any positive solution ( x 1 (n), x 2 (n)) tends to the almost periodic solution ( x 1 (n), x 2 (n)).

Figure 1
figure 1

Positive almost periodic solution of system ( 4.1 ). (a) Time-series x 1 (n) with initial values x 1 (1)=1.06, x 1 (0)=1.02 for n[0,100]. (b) Time-series x 2 (n) with initial values x 2 (1)=0.85, x 2 (0)=0.98 for n[0,100].

Figure 2
figure 2

Phase portrait. (a) Two-dimensional phase portrait of almost periodic system (4.1). Time-series x 1 (n) and x 2 (n) with initial values x 1 (1)=1.06, x 1 (0)=1.02, x 2 (1)=0.85, x 2 (0)=0.98 for n[0,100]. (b) Three-dimensional phase portrait of almost periodic system (4.1). Time-series x 1 (n) and x 2 (n) with the above initial values for n[0,100].

Figure 3
figure 3

Uniformly asymptotic stability. (a) Time-series x 1 (n) with initial values x 1 (1)=1.06, x 1 (0)=1.02 and x 1 (n) with initial values x 1 (1)=0.87, x 1 (0)=0.92 for n[0,100], respectively. (b) Time-series x 2 (n) with initial values x 2 (1)=0.85, x 2 (0)=0.98 and x 2 (n) with initial values x 2 (1)=1.15, x 2 (0)=1.25 for n[0,100], respectively.

Appendix 1: Proof of Theorem 3.1

By the almost periodicity of { r i (n)}, { a i j (n)} and { b i (n)}, i,j=1,2, any sequence { τ k } Z + , with τ k + as k+, is such that

r i (n+ τ k ) r i (n), a i j (n+ τ k ) a i j (n), b i (n+ τ k ) b i (n),i,j=1,2,
(A.1)

as k+ for n Z + . Let ε be an arbitrary small positive number. It follows from (2.4) and (2.5) that there exists a positive integer N 0 such that

m i ε x i (n) M i +εfor all n> N 0 .
(A.2)

Let x i k (n)= x i (n+ τ k ) for n N 0 +M τ k , k=1,2, . For any positive integer q, we can see that there exists a sequence { x i k (n):kq} such that the sequence { x i k (n)} has a subsequence, denoted by { x i k (n)} again, converging on any finite interval of K as k+. So, we have a sequence { y i (n)}, i=1,2, satisfying

x i k (n) y i (n)for nK as k+,
(A.3)

which, together with (A.1) and

{ x 1 k ( n + 1 ) = x 1 k ( n ) exp { r 1 ( n + τ k ) a 11 ( n + τ k ) x 1 k ( n ) a 12 ( n + τ k ) x 2 k ( n ) x 1 k ( n + 1 ) = b 1 ( n + τ k ) x 1 k ( n ) p = 0 M k 2 ( p ) x 2 k ( n p ) } , x 2 k ( n + 1 ) = x 2 k ( n ) exp { r 2 ( n + τ k ) a 21 ( n + τ k ) x 1 k ( n ) a 22 ( n + τ k ) x 2 k ( n ) x 2 k ( n + 1 ) = b 2 ( n + τ k ) x 2 k ( n ) p = 0 M k 1 ( p ) x 1 k ( n p ) } ,
(A.4)

yields

{ y 1 ( n + 1 ) = y 1 ( n ) exp { r 1 ( n ) a 11 ( n ) y 1 ( n ) a 12 ( n ) y 2 ( n ) y 1 ( n + 1 ) = b 1 ( n ) y 1 ( n ) p = 0 M k 2 ( p ) y 2 ( n p ) } , y 2 ( n + 1 ) = y 2 ( n ) exp { r 2 ( n ) a 21 ( n ) y 1 ( n ) a 22 ( n ) y 2 ( n ) y 2 ( n + 1 ) = b 2 ( n ) y 2 ( n ) p = 0 M k 1 ( p ) y 1 ( n p ) } .
(A.5)

We can easily see that ( y 1 (n), y 2 (n)) is a solution of system (1.1) and m i ε y i (n) M i +ε for nK. Since ε is small enough, it follows that

m i y i (n) M i ,i=1,2 for nK.

This completes the proof.

Appendix 2: Proof of Theorem 3.2

We first make the change of variables

p 1 (n)=ln x 1 (n), p 2 (n)=ln x 2 (n).

It follows from system (1.1) that

{ p 1 ( n + 1 ) = p 1 ( n ) + r 1 ( n ) a 11 ( n ) e p 1 ( n ) a 12 ( n ) e p 2 ( n ) p 1 ( n + 1 ) = b 1 ( n ) e p 1 ( n ) p = 0 M k 2 ( p ) e p 2 ( n p ) , p 2 ( n + 1 ) = p 2 ( n ) + r 2 ( n ) a 21 ( n ) e p 1 ( n ) a 22 ( n ) e p 2 ( n ) p 2 ( n + 1 ) = b 2 ( n ) e p 2 ( n ) p = 0 M k 1 ( p ) e p 1 ( n p ) .
(B.1)

From Theorem 3.1, it is easy to see that system (B.1) has a bounded solution ( p 1 (n), p 2 (n)) satisfying

ln m 1 p 1 (n)ln M 1 ,ln m 2 p 2 (n)ln M 2 for all nK.
(B.2)

Thus | p 1 (n)| A 1 , | p 2 (n)| A 2 , where A i =max{|ln m i |,|ln M i |}, i=1,2. Suppose that Y n (s)=( p 1 (n+s), p 2 (n+s)), Z n (s)=( q 1 (n+s), q 2 (n+s)) (n Z + , s [ M , 0 ] Z ) are any two solutions of system (B.1) defined on S, where S={( p 1 (n), p 2 (n))|ln m i p i (n)ln M i ,i=1,2,nK}. Define the norm

Y n ( s ) = ( p 1 ( n + s ) , p 2 ( n + s ) ) = sup s [ M , 0 ] Z { | p 1 ( n + s ) | + | p 2 ( n + s ) | } ,

where ( p 1 (n+s), p 2 (n+s)) R 2 , then

Y n B, Z n B,

where B= A 1 + A 2 . Consider the associate product system of system (B.1)

{ p 1 ( n + 1 ) = p 1 ( n ) + r 1 ( n ) a 11 ( n ) e p 1 ( n ) a 12 ( n ) e p 2 ( n ) p 1 ( n + 1 ) = b 1 ( n ) e p 1 ( n ) p = 0 M k 2 ( p ) e p 2 ( n p ) , p 2 ( n + 1 ) = p 2 ( n ) + r 2 ( n ) a 21 ( n ) e p 1 ( n ) a 22 ( n ) e p 2 ( n ) p 2 ( n + 1 ) = b 2 ( n ) e p 2 ( n ) p = 0 M k 1 ( p ) e p 1 ( n p ) , q 1 ( n + 1 ) = q 1 ( n ) + r 1 ( n ) a 11 ( n ) e q 1 ( n ) a 12 ( n ) e q 2 ( n ) q 1 ( n + 1 ) = b 1 ( n ) e q 1 ( n ) p = 0 M k 2 ( p ) e q 2 ( n p ) , q 2 ( n + 1 ) = q 2 ( n ) + r 2 ( n ) a 21 ( n ) e q 1 ( n ) a 22 ( n ) e q 2 ( n ) q 2 ( n + 1 ) = b 2 ( n ) e q 2 ( n ) p = 0 M k 1 ( p ) e q 1 ( n p ) .
(B.3)

Construct a Lyapunov function V(n)=V(n, Y n , Z n ) defined on Z + ×S×S as follows:

V ( n ) = V ( n , Y n , Z n ) = | p 1 ( n ) q 1 ( n ) | + | p 2 ( n ) q 2 ( n ) | + p = 0 M m = n p n 1 b 2 u k 1 u M 1 M 2 | p 1 ( m ) q 1 ( m ) | + p = 0 M m = n p n 1 b 1 u k 2 u M 1 M 2 | p 2 ( m ) q 2 ( m ) | .
(B.4)

It is easy to see that

| Y n ( 0 ) Z n ( 0 ) | V ( n ) | p 1 ( n ) q 1 ( n ) | + | p 2 ( n ) q 2 ( n ) | + p = 0 M m = n p n 1 D { | p 1 ( m ) q 1 ( m ) | + | p 2 ( m ) q 2 ( m ) | } [ 1 + ( 1 + M ) M D 2 ] sup s [ M , 0 ] Z { | p 1 ( n + s ) q 1 ( n + s ) | + | p 2 ( n + s ) q 2 ( n + s ) | } = λ Y n Z n ,
(B.5)

where

| Y n ( 0 ) Z n ( 0 ) | = ( p 1 ( n ) q 1 ( n ) ) 2 + ( p 2 ( n ) q 2 ( n ) ) 2 | p 1 ( n ) q 1 ( n ) | + | p 2 ( n ) q 2 ( n ) | ,
(B.6)

and λ=1+ ( 1 + M ) M D 2 , D=max{ b 1 u k 2 u M 1 M 2 , b 2 u k 1 u M 1 M 2 }. Let a,bC( R + , R + ), a(x)=x, b(x)=λx, so condition (1) in Lemma 2.4 is satisfied.

For y,z, y ˜ , z ˜ R, one has

| | y z | | y ˜ z ˜ | | = { | y z | | y ˜ z ˜ | , if  | y z | | y ˜ z ˜ | , | y ˜ z ˜ | | y z | , if  | y ˜ z ˜ | > | y z | { | ( y y ˜ ) + ( z ˜ z ) | , if  | y z | | y ˜ z ˜ | , | ( y ˜ y ) + ( z z ˜ ) | , if  | y ˜ z ˜ | > | y z | { | y y ˜ | + | z ˜ z | , if  | y z | | y ˜ z ˜ | , | y ˜ y | + | z z ˜ | , if  | y ˜ z ˜ | > | y z | = | y y ˜ | + | z z ˜ | .
(B.7)

Hence, for Y n , Z n , Y ˜ n , Z ˜ n S, by (B.7) we have

| V ( n , Y n , Z n ) V ( n , Y ˜ n , Z ˜ n ) | = | | p 1 ( n ) q 1 ( n ) | + | p 2 ( n ) q 2 ( n ) | + p = 0 M m = n p n 1 b 2 u k 1 u M 1 M 2 | p 1 ( m ) q 1 ( m ) | + p = 0 M m = n p n 1 b 1 u k 2 u M 1 M 2 | p 2 ( m ) q 2 ( m ) | | p ˜ 1 ( n ) q ˜ 1 ( n ) | | p ˜ 2 ( n ) q ˜ 2 ( n ) | p = 0 M m = n p n 1 b 2 u k 1 u M 1 M 2 | p ˜ 1 ( m ) q ˜ 1 ( m ) | p = 0 M m = n p n 1 b 1 u k 2 u M 1 M 2 | p ˜ 2 ( m ) q ˜ 2 ( m ) | | | | p 1 ( n ) q 1 ( n ) | + | p 2 ( n ) q 2 ( n ) | | p ˜ 1 ( n ) q ˜ 1 ( n ) | | p ˜ 2 ( n ) q ˜ 2 ( n ) | | + p = 0 M m = n p n 1 b 2 u k 1 u M 1 M 2 | | p 1 ( m ) q 1 ( m ) | | p ˜ 1 ( m ) q ˜ 1 ( m ) | | + p = 0 M m = n p n 1 b 1 u k 2 u M 1 M 2 | | p 2 ( m ) q 2 ( m ) | | p ˜ 2 ( m ) q ˜ 2 ( m ) | | { | p 1 ( n ) p ˜ 1 ( n ) | + | p 2 ( n ) p ˜ 2 ( n ) | + | q 1 ( n ) q ˜ 1 ( n ) | + | q 2 ( n ) q ˜ 2 ( n ) | } + p = 0 M m = n p n 1 b 2 u k 1 u M 1 M 2 { | p 1 ( m ) p ˜ 1 ( m ) | + | q 1 ( m ) q ˜ 1 ( m ) | } + p = 0 M m = n p n 1 b 1 u k 2 u M 1 M 2 { | p 2 ( m ) p ˜ 2 ( m ) | + | q 2 ( m ) q ˜ 2 ( m ) | } [ 1 + ( 1 + M ) M D 2 ] sup s [ M , 0 ] Z { | p 1 ( n + s ) p ˜ 1 ( n + s ) | + | p 2 ( n + s ) p ˜ 2 ( n + s ) | + | q 1 ( n + s ) q ˜ 1 ( n + s ) | + | q 2 ( n + s ) q ˜ 2 ( n + s ) | } λ ( Y n Y ˜ n + Z n Z ˜ n ) ,
(B.8)

where λ=1+ ( 1 + M ) M D 2 , D=max{ b 1 u k 2 u M 1 M 2 , b 2 u k 1 u M 1 M 2 }. Condition (2) in Lemma 2.4 is also satisfied.

Using the mean-value theorem, we derive that

e p i ( n ) e q i ( n ) = ξ i (n) ( p i ( n ) q i ( n ) ) ,
(B.9)
e p i ( n p ) e q i ( n p ) = η i (np) ( p i ( n p ) q i ( n p ) ) ,
(B.10)

i=1,2, where ξ i (n) lies between e p i ( n ) and e q i ( n ) and η i (np) lies between e p i ( n p ) and e q i ( n p ) , respectively. So, m i ξ i (n), η i (np) M i , n Z + .

In view of system (B.3) together with (B.9) and (B.10), we have

| p 1 ( n + 1 ) q 1 ( n + 1 ) | + | p 2 ( n + 1 ) q 2 ( n + 1 ) | = | p 1 ( n ) q 1 ( n ) a 11 ( n ) ( e p 1 ( n ) e q 1 ( n ) ) a 12 ( n ) ( e p 2 ( n ) e q 2 ( n ) ) b 1 ( n ) ( e p 1 ( n ) p = 0 M k 2 ( p ) e p 2 ( n p ) e q 1 ( n ) p = 0 M k 2 ( p ) e q 2 ( n p ) ) | + | p 2 ( n ) q 2 ( n ) a 21 ( n ) ( e p 1 ( n ) e q 1 ( n ) ) a 22 ( n ) ( e p 2 ( n ) e q 2 ( n ) ) b 2 ( n ) ( e p 2 ( n ) p = 0 M k 1 ( p ) e p 1 ( n p ) e q 2 ( n ) p = 0 M k 1 ( p ) e q 1 ( n p ) ) | = | p 1 ( n ) q 1 ( n ) a 11 ( n ) ( e p 1 ( n ) e q 1 ( n ) ) a 12 ( n ) ( e p 2 ( n ) e q 2 ( n ) ) b 1 ( n ) ( e p 1 ( n ) p = 0 M k 2 ( p ) e p 2 ( n p ) e p 1 ( n ) p = 0 M k 2 ( p ) e q 2 ( n p ) + e p 1 ( n ) p = 0 M k 2 ( p ) e q 2 ( n p ) e q 1 ( n ) p = 0 M k 2 ( p ) e q 2 ( n p ) ) | + | p 2 ( n ) q 2 ( n ) a 21 ( n ) ( e p 1 ( n ) e q 1 ( n ) ) a 22 ( n ) ( e p 2 ( n ) e q 2 ( n ) ) b 2 ( n ) ( e p 2 ( n ) p = 0 M k 1 ( p ) e p 1 ( n p ) e p 2 ( n ) p = 0 M k 1 ( p ) e q 1 ( n p ) + e p 2 ( n ) p = 0 M k 1 ( p ) e q 1 ( n p ) e q 2 ( n ) p = 0 M k 1 ( p ) e q 1 ( n p ) ) | = | p 1 ( n ) q 1 ( n ) a 11 ( n ) ξ 1 ( n ) ( p 1 ( n ) q 1 ( n ) ) a 12 ( n ) ξ 2 ( n ) ( p 2 ( n ) q 2 ( n ) ) b 1 ( n ) e p 1 ( n ) p = 0 M k 2 ( p ) [ η 2 ( n p ) ( p 2 ( n p ) q 2 ( n p ) ) ] b 1 ( n ) ξ 1 ( n ) ( p 1 ( n ) q 1 ( n ) ) p = 0 M k 2 ( p ) e q 2 ( n p ) | + | p 2 ( n ) q 2 ( n ) a 21 ( n ) ξ 1 ( n ) ( p 1 ( n ) q 1 ( n ) ) a 22 ( n ) ξ 2 ( n ) ( p 2 ( n ) q 2 ( n ) ) b 2 ( n ) e p 2 ( n ) p = 0 M k 1 ( p ) [ η 1 ( n p ) ( p 1 ( n p ) q 1 ( n p ) ) ] b 2 ( n ) ξ 2 ( n ) ( p 2 ( n ) q 2 ( n ) ) p = 0 M k 1 ( p ) e q 1 ( n p ) | | 1 a 11 ( n ) ξ 1 ( n ) | | p 1 ( n ) q 1 ( n ) | + a 12 u M 2 | p 2 ( n ) q 2 ( n ) | + b 1 u k 2 u M 1 M 2 p = 0 M | p 2 ( n p ) q 2 ( n p ) | + b 1 u k 2 u M 1 M 2 ( M + 1 ) | p 1 ( n ) q 1 ( n ) | + | 1 a 22 ( n ) ξ 2 ( n ) | | p 2 ( n ) q 2 ( n ) | + a 21 u M 1 | p 1 ( n ) q 1 ( n ) | + b 2 u k 1 u M 1 M 2 p = 0 M | p 1 ( n p ) q 1 ( n p ) | + b 2 u k 1 u M 1 M 2 ( M + 1 ) | p 2 ( n ) q 2 ( n ) | { | 1 a 11 ( n ) ξ 1 ( n ) | + b 1 u k 2 u M 1 M 2 ( M + 1 ) + a 21 u M 1 } | p 1 ( n ) q 1 ( n ) | + b 2 u k 1 u M 1 M 2 p = 0 M | p 1 ( n p ) q 1 ( n p ) | + { | 1 a 22 ( n ) ξ 2 ( n ) | + b 2 u k 1 u M 1 M 2 ( M + 1 ) + a 12 u M 2 } | p 2 ( n ) q 2 ( n ) | + b 1 u k 2 u M 1 M 2 p = 0 M | p 2 ( n p ) q 2 ( n p ) | .
(B.11)

Based on (B.11), we calculate the difference of V along the solution of system (B.3)

Δ V ( B.3 ) ( n ) = V ( n + 1 ) V ( n ) = | p 1 ( n + 1 ) q 1 ( n + 1 ) | + | p 2 ( n + 1 ) q 2 ( n + 1 ) | + p = 0 M m = n + 1 p n b 2 u k 1 u M 1 M 2 | p 1 ( m ) q 1 ( m ) | + p = 0 M m = n + 1 p n b 1 u k 2 u M 1 M 2 | p 2 ( m ) q 2 ( m ) | | p 1 ( n ) q 1 ( n ) | | p 2 ( n ) q 2 ( n ) | p = 0 M m = n p n 1 b 2 u k 1 u M 1 M 2 | p 1 ( m ) q 1 ( m ) | p = 0 M m = n p n 1 b 1 u k 2 u M 1 M 2 | p 2 ( m ) q 2 ( m ) | { | 1 a 11 ( n ) ξ 1 ( n ) | + b 1 u k 2 u M 1 M 2 ( M + 1 ) + a 21 u M 1 } | p 1 ( n ) q 1 ( n ) | + p = 0 M b 2 u k 1 u M 1 M 2 | p 1 ( n p ) q 1 ( n p ) | + { | 1 a 22 ( n ) ξ 2 ( n ) | + b 2 u k 1 u M 1 M 2 ( M + 1 ) + a 12 u M 2 } | p 2 ( n ) q 2 ( n ) | + p = 0 M b 1 u k 2 u M 1 M 2 | p 2 ( n p ) q 2 ( n p ) | + p = 0 M m = n + 1 p n b 2 u k 1 u M 1 M 2 | p 1 ( m ) q 1 ( m ) | + p = 0 M m = n + 1 p n b 1 u k 2 u M 1 M 2 | p 2 ( m ) q 2 ( m ) | | p 1 ( n ) q 1 ( n ) | | p 2 ( n ) q 2 ( n ) | p = 0 M m = n p n 1 b 2 u k 1 u M 1 M 2 | p 1 ( m ) q 1 ( m ) | p = 0 M m = n p n 1 b 1 u k 2 u M 1 M 2 | p 2 ( m ) q 2 ( m ) | = { | 1 a 11 ( n ) ξ 1 ( n ) | + ( b 1 u k 2 u + b 2 u k 1 u ) M 1 M 2 ( M + 1 ) + a 21 u M 1 1 } | p 1 ( n ) q 1 ( n ) | + { | 1 a 22 ( n ) ξ 2 ( n ) | + ( b 1 u k 2 u + b 2 u k 1 u ) M 1 M 2 ( M + 1 ) + a 12 u M 2 1 } | p 2 ( n ) q 2 ( n ) | { 1 max ( | 1 a 11 l m 1 | , | 1 a 11 u M 1 | ) ( b 1 u k 2 u + b 2 u k 1 u ) ( M + 1 ) M 1 M 2 a 21 u M 1 } | p 1 ( n ) q 1 ( n ) | { 1 max ( | 1 a 22 l m 2 | , | 1 a 22 u M 2 | ) ( b 1 u k 2 u + b 2 u k 1 u ) ( M + 1 ) M 1 M 2 a 12 u M 2 } | p 2 ( n ) q 2 ( n ) | θ ( | p 1 ( n ) q 1 ( n ) | + | p 2 ( n ) q 2 ( n ) | ) θ ( p 1 ( n ) q 1 ( n ) ) 2 + ( p 2 ( n ) q 2 ( n ) ) 2 = θ ( | Y n ( 0 ) Z n ( 0 ) | ) ,
(B.12)

where θ = min{1max(|1 a 11 l m 1 |, |1 a 11 u M 1 |)( b 1 u k 2 u + b 2 u k 1 u )(M+1) M 1 M 2 a 21 u M 1 , 1max(|1 a 22 l m 2 |, |1 a 22 u M 2 |)( b 1 u k 2 u + b 2 u k 1 u )(M+1) M 1 M 2 a 12 u M 2 } is a positive constant. Let cC( R + , R + ), c(x)=θx, thus condition (3)′ in Remark 2.5 is satisfied. Based on Lemma 2.4 and Remark 2.5, there exists a unique uniformly asymptotically stable almost periodic solution ( p 1 (n), p 2 (n)) of system (B.1), that is, there is a unique uniformly asymptotically stable positive almost periodic solution ( x 1 (n), x 2 (n)) of system (1.1) which satisfies m i x i (n) M i , i=1,2, for all nK. The proof is complete.

References

  1. Chao L, Levin BR: Structured habitats and the evolution of anticompetitor toxins in bacteria. Proc. Natl. Acad. Sci. USA 1981, 78: 6324-6328. 10.1073/pnas.78.10.6324

    Article  Google Scholar 

  2. Cole JJ: Interactions between bacteria and algae in aquatic ecosystems. Ann. Rev. Ecolog. Syst. 1982, 13: 291-314. 10.1146/annurev.es.13.110182.001451

    Article  Google Scholar 

  3. Kitaguchi H, Hiragushi N, Mitsutani A, Yamaguchi M, Ishida Y: Isolation of an algicidal marine bacterium with activity against the harmful dinoflagellate Heterocapsa circulatisquama (Dinophyceae). Phycologia 2001, 40: 275-279. 10.2216/i0031-8884-40-3-275.1

    Article  Google Scholar 

  4. Turner JT, Tester PA: Toxic marine phytoplankton, zooplankton grazers, and pelagic food webs. Limnol. Oceanogr. 1997, 42: 1203-1214. 10.4319/lo.1997.42.5_part_2.1203

    Article  Google Scholar 

  5. Folt C, Goldman CR: Allelopathy between zooplankton: a mechanism for interference competition. Science 1981, 213: 1133-1135. 10.1126/science.213.4512.1133

    Article  Google Scholar 

  6. Rice EL: Allelopathy. 2nd edition. Academic Press, New York; 1984.

    Google Scholar 

  7. Liu ZJ, Wu JH, Chen YP, Haque M: Impulsive perturbations in a periodic delay differential equation model of plankton allelopathy. Nonlinear Anal., Real World Appl. 2010, 11: 432-445. 10.1016/j.nonrwa.2008.11.017

    Article  MathSciNet  MATH  Google Scholar 

  8. Liu ZJ, Hui J, Wu JH: Permanence and partial extinction in an impulsive delay competitive system with the effect of toxic substances. J. Math. Chem. 2009, 46: 1213-1231. 10.1007/s10910-008-9513-1

    Article  MathSciNet  MATH  Google Scholar 

  9. Tian CR, Lin ZG: Asymptotic behavior of solutions of a periodic diffusion system of plankton allelopathy. Nonlinear Anal., Real World Appl. 2010, 3: 1581-1588.

    Article  MathSciNet  MATH  Google Scholar 

  10. He MX, Chen FD, Li Z: Almost periodic solution of an impulsive differential equation model of plankton allelopathy. Nonlinear Anal., Real World Appl. 2010, 11: 2296-2301. 10.1016/j.nonrwa.2009.07.004

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen YP, Chen FD, Li Z: Dynamics behaviors of a general discrete nonautonomous system of plankton allelopathy with delays. Discrete Dyn. Nat. Soc. 2008., 2008: Article ID 310425

    Google Scholar 

  12. Li Z, Chen FD, He MX: Global stability of a delay differential equations model of plankton allelopathy. Appl. Math. Comput. 2012, 218: 7155-7163. 10.1016/j.amc.2011.12.083

    Article  MathSciNet  MATH  Google Scholar 

  13. Liu ZJ, Chen LS: Positive periodic solution of a general discrete non-autonomous difference system of plankton allelopathy with delays. J. Comput. Appl. Math. 2006, 197: 446-456. 10.1016/j.cam.2005.09.023

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu ZJ, Chen LS: Periodic solution of a two-species competitive system with toxicant and birth pulse. Chaos Solitons Fractals 2007, 32: 1703-1712. 10.1016/j.chaos.2005.12.004

    Article  MathSciNet  MATH  Google Scholar 

  15. Qin WJ, Liu ZJ: Asymptotic behaviors of a delay difference system of plankton allelopathy. J. Math. Chem. 2010, 48: 653-675. 10.1007/s10910-010-9698-y

    Article  MathSciNet  MATH  Google Scholar 

  16. Niu CY, Chen XX: Almost periodic sequence solutions of a discrete Lotka-Volterra competitive system with feedback control. Nonlinear Anal., Real World Appl. 2009, 10: 3152-3161. 10.1016/j.nonrwa.2008.10.027

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhang TW, Li YK, Ye Y: Persistence and almost periodic solutions for a discrete fishing model with feedback control. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 1564-1573. 10.1016/j.cnsns.2010.06.033

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang Z, Li YK: Almost periodic solutions of a discrete mutualism model with feedback controls. Discrete Dyn. Nat. Soc. 2010., 2010: Article ID 286031

    Google Scholar 

  19. Li YK, Zhang TW: Almost periodic solution for a discrete hematopoiesis model with time delay. Int. J. Biomath. 2012., 5: Article ID 1250003

    Google Scholar 

  20. Li Z, Chen FD, He MX: Almost periodic solutions of a discrete Lotka-Volterra competition system with delays. Nonlinear Anal., Real World Appl. 2011, 12: 2344-2355. 10.1016/j.nonrwa.2011.02.007

    Article  MathSciNet  MATH  Google Scholar 

  21. Muroya Y: Persistence and global stability in discrete models of Lotka-Volterra type. J. Math. Anal. Appl. 2007, 330: 24-33. 10.1016/j.jmaa.2006.07.070

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang SN, Zheng G: Almost periodic solutions of delay difference systems. Appl. Math. Comput. 2002, 131: 497-516. 10.1016/S0096-3003(01)00165-5

    Article  MathSciNet  MATH  Google Scholar 

  23. Yuan R, Hong JL: The existence of almost periodic solutions for a class of differential equations with piecewise constant argument. Nonlinear Anal., Theory Methods Appl. 1997, 28: 1439-1450. 10.1016/0362-546X(95)00225-K

    Article  MathSciNet  MATH  Google Scholar 

  24. He CY: Almost Periodic Differential Equations. Higher Education Press, Beijing; 1992. (Chinese version)

    Google Scholar 

Download references

Acknowledgements

The work is supported by the National Natural Science Foundation of China (Nos. 11261017, 61261044), the Key Project of Chinese Ministry of Education (No. 212111), the Scientific Research Foundation of the Education Department of Hubei Province of China (No. B20111906) and the Key Subject of Hubei Province (Forestry).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Zhijun Liu.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Each of the authors, QW and ZL, contributed to each part of this work equally and read and approved the final version of the manuscript.

Authors’ original submitted files for images

Below are the links to the authors’ original submitted files for images.

Authors’ original file for figure 1

Authors’ original file for figure 2

Authors’ original file for figure 3

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Wang, Q., Liu, Z. Uniformly asymptotic stability of almost periodic solutions for a delay difference system of plankton allelopathy. Adv Differ Equ 2013, 283 (2013). https://doi.org/10.1186/1687-1847-2013-283

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-1847-2013-283

Keywords