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Dynamics for a discrete competition and cooperation model of two enterprises with multiple delays and feedback controls

  • Lin Lu , Yi Lian and Chaoling Li
Published/Copyright: March 13, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

This paper is concerned with a competition and cooperation model of two enterprises with multiple delays and feedback controls. With the aid of the difference inequality theory, we have obtained some sufficient conditions which guarantee the permanence of the model. Under a suitable condition, we prove that the system has global stable periodic solution. The paper ends with brief conclusions.

Keywords: Competition and cooperation model; Permanence; Feedback control; Delay; Enterprise
MSC 2010: 34K20; 34C25; 92D25

1 Introduction

It is known that the coexistence of species has become one of interesting subjects in mathematical ecology. In the past few decades, permanence dynamics of species have received great attention and have been investigated in a number of notable work. For example, Wang and Huang [1] analyzed permanence of a predator-prey model with harvesting predator. Mukherjee [2] addressed the permanence and global attractivity for facultative mutualism predator-prey model, Zhao and Jiang [3] considered the permanence and extinction for Lotka-Volterra model, Teng et al. [4] established the permanence criteria for a delayed discrete species systems, Liu et al. [5] studied the permanence and periodic solutions for reaction-diffusion food-chain system with impulsive effect. For more detailed research about this topic, one can see [6–23]. In real life, the co-existence and stability of enterprise clusters has become one of the most prevalent phenomena in our society. Thus it is important for us to study the permanence and global attractivity of enterprise clusters. However, there are few papers that consider this topic. We think that this study on the dynamics of enterprise clusters has wide application in economic performance and so on.

In 2006, Tian and Nie [24] investigated the following competition and cooperation model of two enterprises

dx1(t)dt=r1(t)x1(t)1−x1(t)K−α(x2(t)−c2)2K,dx2(t)dt=r2(t)x2(t)1−x2(t)K−β(x1(t)−c1)2K, (1)

where x1(t), x2(t) represent the output of enterprises A and B, r1, r2 are the intrinsic growth rate, K denotes the carrying capacity of mark under nature unlimited conditions, α, β are the competitive parameters of two enterprises, c1, c2 are the initial production of two enterprises. Letting a1=r1K,a2=r2K,b1=r1αK,b2=r2βK, then system (1) becomes

dx1(t)dt=x1(t)[r1−a1x1(t)−b1(x2(t)−c2)2],dx2(t)dt=x2(t)[r2−a2x2(t)+b2(x1(t)−c1)2]. (2)

Considering the effect of time delay, Liao et al. [25] modified system (2) as follows:

dx1(t)dt=x1(t)[r1−a1x1(t−τ1)−b1(x2(t−τ2)−c2)2],dx2(t)dt=x2(t)[r2−a2x2(t−τ1)+b2(x1(t−τ2)−c1)2], (3)

where τ1 is nonnegative constant which stands for the gestation periodic of production for two enterprises, τ2 in the first equation of the system (3) stands for the block delay of enterprise B to A, and τ2 in the second equation of the system (3) stands for the promoting delay of enterprise A to B. By regarding the two delays τ1 and τ2 as bifurcation parameters, Liao et al. [25] discussed the effect of different delays on the dynamical behavior of system (3). If τ1 = τ2 = τ, then system (3) becomes

dx1(t)dt=x1(t)[r1−a1x1(t−τ)−b1(x2(t−τ)−c2)2],dx2(t)dt=x2(t)[r2−a2x2(t−τ)+b2(x1(t−τ)−c1)2], (4)

By choosing the time delay τ as bifurcation parameter, Liao et al. [26] focused on the stability and Hopf bifurcation properties of system (4).

Li and Zhang [27] focused on (2) with nonconstant coefficients, which takes the following from:

dx1(t)dt=x1(t)[r1(t)−a1(t)x1(t)−b1(t)(x2(t)−c2(t))2],dx2(t)dt=x2(t)[r2(t)−a2(t)x1(t)+b2(t)(x1(t)−c1(t))2]. (5)

Using the continuation theorem of coincidence degree theory and differential inequality theory, Xu [28] established some sufficient criteria to guarantee the existence of periodic solutions of (5).

Considering that the change of environment, the output of enterprises A and B usually change rapidly, Xu and Shao [29] considered the existence and global attractivity of periodic solution for the following enterprise clusters model with impulse and varying coefficients

dx1(t)dt=x1(t)[r1(t)−a1(t)x1(t)−b1(t)(x2(t)−c2(t))2],t≠tk,dx2(t)dt=x2(t)[r2(t)−a2(t)x1(t)+b2(t)(x1(t)−c1(t))2],t≠tk,Δi(tk)=xi(tk+)−xi(tk−)=−γik(tk),i=1,2,k=1,2,⋯,q, (6)

where Δi(tk)=xi(tk+)−xi(tk−) are the impulses at moments tk and t1 < t2 < ⋯ is a strictly increasing sequence such that limk→+∞ tk = +∞ and q is a positive integer. Applying coincidence degree theory, Xu and Shao [29] obtained a set of sufficient criteria to ensure the existence of at least a positive periodic solution, and by constructing a Lyapunov functional, they established a sufficient condition to guarantee the uniqueness and global attractivity of the positive periodic solution for (6).

A lot of researchers [30–41] think that discrete systems are far better in depicting the dynamical behavior than continuous ones. In addition, discrete systems also play an important role in computer simulations for continuous systems. Considering the unpredictable forces, we know that coefficients of competition and cooperation systems of two enterprises change with time. Inspired by the discussion above, we modify (5) as follows

x 1 ( n + 1 ) = x 1 ( n ) exp ⁡ { r 1 ( n ) − a 1 ( n ) x 1 ( n − τ 1 ( n ) ) − b 1 ( n ) ( x 2 ( n − τ 2 ( n ) ) − c 2 ( n ) ) 2 − β 1 ( n ) U 1 ( n ) } , x 2 ( n + 1 ) = x 2 ( n ) exp ⁡ { r 2 ( n ) − a 2 ( n ) x 2 ( n − τ 1 ( n ) ) + b 2 ( n ) ( x 1 ( n − τ 2 ( n ) ) − c 1 ( n ) ) 2 − β 2 ( n ) u 2 ( n ) } , Δ u 1 ( n ) = − γ 1 ( n ) u 1 ( n ) + η 1 ( n ) x 1 ( n ) , Δ u 2 ( n ) = − γ 2 ( n ) u 2 ( n ) + η 2 ( n ) x 2 ( n ) , (7)

where x1(n) and x2(n) denote the output of enterprises A and B at the generation, respectively, and ui(n)(i = 1, 2) is the control variable. r1(n), r2(n), a1(n), a2(n), b1(n), b2(n), c1(n), c2(n), τ1(n) and τ2(n) are bounded nonnegative sequences. To the authors' knowledge, it is the first time one deals with system (7) with feedback control. For more related work, one can see [42–44].

We assume that

(H1)0<ril≤riu,0<ail≤aiu,0<bil≤biu,0<cil≤ciu,0<βil≤βiu(i=1,2).

Here, for any bounded sequence {f(n)}, fu = supn∊ N{f(n)} and fl = infn∊ N{f(n)}.

Let τ = supn∊ Z{τi}(n)}, τ~ = infn∊ Z{τi(n)}, i = 1, 2. The initial conditions of (7) are

xi(θ)=φi(θ)≥0,θ∈N[−τ,0]={−τ,−τ+1,⋯,0},φi(0)>0. (8)

It is not difficult to see that solutions of (7) and (8) are well defined for all n ≥ 0 and satisfy

xi(n)>0,forn∈Z,i=1,2.

The remainder of the article is organized as follows: in Section 2, some definitions and lemmas are presented and the permanence of (7) is considered. In Section 3, the existence and stability of a unique globally attractive positive periodic solution of the model are investigated. In Section 4, two examples are given to illustrate correctness of our obtained analytical results in Section 2 and Section 3. Brief conclusions are drawn in Section 5.

2 Permanence

In this section, we list several definitions and lemmas.

Definition 2.1

System (7) is permanent if there are positive constants M and m such that for each positive solution (x1(n), x2(n), u1(n), u2(n)) of system (7) satisfies

m≤limn→+∞infxi(n)≤limn→+∞supxi(n)≤M(i=1,2),m≤limn→+∞infui(n)≤limn→+∞supui(n)≤M(i=1,2).

Let us consider the following model:

N(n+1)=N(n)exp⁡(a(n)−b(n)N(n)), (9)

where {a(n)} and {b(n)} are strictly positive sequences of real numbers defined for n ∊ N = {0, 1, 2, } and 0 < al ≤ au, 0 < bl ≤ bu. Similarly to the proofs of Propositions 1 and 3 in [41], we can obtain the following Lemma 2.2.

Lemma 2.2

Any solution of system (9) with initial condition N(0) > 0 satisfies

m≤limn→+∞infN(n)≤limn→+∞supN(n)≤M,

where

M=1blexp⁡(au−1),m=albuexp⁡(al−buM).

Let consider the following equation

y(n+1)=Ay(n)+B,n=1,2,⋯, (10)

where A and B are positive constants. Following Theorem 6.2 of Wang and Wang [45, page 125], we have the following Lemma 2.3.

Lemma 2.3

([45]). If|A| < 1, then for any initial value y(0), there exists a unique solution y(n) of (10) which can be expressed as follows:

y(n)=An(y(0)−y∗)+y∗,

where y∗=B1−A. Thus, for any solution {y(n)} of system (10), limn→+∞y(n) = y*.

Lemma 2.4

([45]). Let n∈Nn0+={n0,n0+1,⋯,n0+l,⋯},r≥0. For any fixed n, g(n, r) is a nondecreasing function with respect to r, and for n ≥ n0, then

y(n+1)≤g(n,y(n)),u(n+1)≥g(n,u(n)).

If y(n0) ≤ u(n0), then y(n) ≤ u(n) for all n ≥ n0.

Proposition 2.5

Let ε > 0 be any constant. If(H1) holds, then

limn→+∞supxi(n)≤Mi,limn→+∞supui(n)≤Ui,i=1,2,

where

M1=1r1lexp⁡{r1u(τ+1)−1},Ui=ηiuMiγil(i=1,2),M2=1a2lexp⁡{(r2u+b2u(M1+ϵ+c1u)2)(τ+1)−1}.

Proof

Let (x1(n), x2(n), u1(n), u2(n)) be any positive solution of system (7) with the initial condition (x1(0), x2(0), u1(0), u2(0)) . It follows from the first equation of system (7) that

x1(n+1)≤x1(n)exp⁡{r1(n)}. (11)

Let x1(n) = exp{y1(n)}, then (11) is equivalent to

y1(n+1)−y1(n)≤r1(n). (12)

Summing both sides of (12) from n−τ1(n) to n−1, we have

∑j=n−τ1(n)n−1(y1(j+1)−y1(j))≤∑j=n−τ1(n)n−1r1(j)≤r1uτ,

which leads to

y1(n−τ1(n))≥y1(n)−r1uτ. (13)

Then

x1(n−τ1(n))≥x1(n)exp⁡{−r1uτ}. (14)

Substituting (14) into the first equation of system (7) gives

x1(n+1)≤x1(n)exp⁡{r1(n)−r1(n)exp⁡{−r1uτ}x1(n)}. (15)

It follows from (15) and Lemma 2.2 that

limn→+∞supx1(n)≤1r1lexp⁡{r1u(τ+1)−1}:=M1. (16)

For any positive constant ε > 0, it follows (16) that there exists a N1 > 0 such that for all n > N1 + τ

x1(n)≤M1+ε. (17)

For any positive constant ε > 0 and for all n > N1 + τ, by the second equation of system (7), we get

x2(n+1)≤x2(n)exp⁡{r2(n)+b2(n)(M1+ε+c1(n))2}. (18)

Let x2(n) = exp{y2(n)}, then (18) is equivalent to

y2(n+1)−y2(n)≤r2(n)+b2(n)(M1+ε+c1(n))2. (19)

Summing both sides of (19) from n − τ1(n) to n − 1, we have

∑j=n−τ1(n)n−1(y2(j+1)−y2(j))≤∑j=n−τ1(n)n−1[r2(n)+b2(n)(M1+ε+c1(n))2]≤[r2u+b2u(M1+ε+c1u)2]τ,

which leads to

y2(n−τ1(n))≥y2(n)−[r2u+b2u(M1+ε+c1u)2]τ. (20)

Then

x2(n−τ1(n))≥x2(n)exp⁡{−[r2u+b2u(M1+ε+c1u)2]τ}. (21)

By substituting (21) into the second equation of system (7) we have

x2(n+1)≤x2(n)exp⁡{r2(n)+b2(n)(M1+ε+c1(n))2−a2(n)exp⁡{−[r2u+b2u(M1+ε+c1u)2]τ}x2(n)}. (22)

It follows from (22) and Lemma 2.2 that

lim n → + ∞ sup x 2 ( n ) ≤ 1 a 2 l exp ⁡ { ( r 2 u + b 2 u ( M 1 + ε + c 1 u ) 2 ) ( τ + 1 ) − 1 } := M 2 . (23)

For any positive constant ε > 0, it follows (23) that there exists a N2 > N1 + τ such that for all n > N2 + τ

x2(n)≤M2+ε. (24)

In view of the third and fourth equations of the system (7), we can obtain

Δu1(n)≤−γ1(n)u1(n)+η1(n)(M1+ε), (25)

Δu2(n)≤−γ2(n)u2(n)+η2(n)(M2+ε). (26)

Then

u1(n+1)≤(1−γ1l)u1(n)+η1u(M1+ε), (27)

u2(n+1)≤(1−γ2l)u2(n)+η2u(M2+ε). (28)

Applying Lemmas 2.2 and 2.4, we immediately get

limn→+∞supu1(n)≤η1u(M1+ε)γ1l, (29)

limn→+∞supu2(n)≤η2u(M2+ε)γ2l. (30)

By setting ε → 0, we have

limn→+∞supu1(n)≤η1uM1γ1l:=U1, (31)

limn→+∞supu2(n)≤η2uM2γ2l:=U2. (32)

This completes the proof of Proposition 2.5.

Theorem 2.6

Let M1, M2, U1 and U2 be defined by (16), (23), (31) and (32), respectively. Assume that (H1) and (H2) r1l>b1u(M2+c2u)2+β1U1,r2l>β2uU2 hold, then system (7) is permanent.

Proof

By applying Proposition 2.5, we easily see that to end the proof of Theorem 2.6, it is enough to show that under the conditions of Theorem 2.6,

limn→+∞infx1(n)≥m1,limn→+∞infx2(n)≥m2,limn→+∞infu1(n)≥v1,limn→+∞infu2(n)≥v2.

In view of Proposition 2.5, for all ε > 0, there exists a N3 > 0, N3 ∈ N, for all n > N3,

x1(n)≤M1+ε,x2(n)≤M2+ε,u1(n)≤U1+ε,u2(n)≤U2+ε. (33)

It follows from the first equation of system (7) and (33) that

x1(n+1)≥x1(n)exp⁡{r1l−a1u(M1+ε)−b1u(M2+ε+c2u)2−β1u(U1+ε)},

for all n > N3 + τ.

Let x1(n) = exp{y1(n)}, then (34) is equivalent to

y1(n+1)−y1(n)≥r1l−a1u(M1+ε)−b1u(M2+ε+c2u)2−β1u(U1+ε). (34)

Summing both sides of (34) from n − τ1(n) to n − 1 leads to

∑j=n−τ1(n)n−1(y1(j+1)−y1(j))≥∑j=n−τ1(n)n−1[r1l−a1u(M1+ε)−b1u(M2+ε+c2u)2−β1u(U1+ε)]≥[r1l−a1u(M1+ε)−b1u(M2+ε+c2u)2−β1u(U1+ε)]τ~.

Then

y1(n−τ1(n))≤y1(n)−[r1l−a1u(M1+ε)−b1u(M2+ε+c2u)2−β1u(U1+ε)]τ~.

Thus

x1(n−τ1(n))≤x1(n)exp⁡{−[r1l−a1u(M1+ε)−b1u(M2+ε+c2u)2−β1u(U1+ε)]τ~}. (35)

Substituting (33) and (35) into the first equation of (7), we have

x1(n+1)≥x1(n)exp⁡{r1l−b1u(M2+ε+c2u)2−β1u(U1+ε)−a1ux1(n)×exp⁡{−[r1l−a1u(M1+ε)−b1u(M2+ε+c2u)2−β1u(U1+ε)]τ~}} (36)

for all n > N3 + τ.

By applying Lemmas 2.2 and 2.4, we immediately obtain

limn→+∞infx1(n)≥m1ε, (37)

where

m1ε=r1l−b1u(M2+ε+c2u)2−β1u(U1+ε)a1uexp⁡{−[r1u−a1u(M1+ε)−b1u(M2+ε+c2u)2−β1u(U1+ε)]τ~}×exp⁡{r1l−b1u(M2+ε+c2u)2−β1u(U1+ε)−a1uexp⁡{−[r1u−a1u(M1+ε)−b1u(M2+ε+c2u)2−β1u(U1+ε)]τ~}M1}.

Setting ε → 0 in (37), then

limn→+∞infx1(n)≥m1, (38)

where

m1=r1l−b1u(M2+c2u)2−β1uU1a1uexp⁡{−[r1u−a1uM1−b1u(M2+c2u)2−β1uU1]τ~}×exp⁡{r1l−b1u(M2+c2u)2−β1uU1−a1uexp⁡{−[r1u−a1uM1−b1u(M2+c2u)2−β1uU1]τ~}M1}.

By the second equation of system (7) and (33), we can obtain

x2(n+1)≥x2(n)exp⁡{r2(n)−a2(n)x2(n−τ1(n))−β2(n)u2(n)}≥x2(n)exp⁡{r2l−a2u(M2+ε)−β2u(U2+ε)} (39)

for all n > N3 + τ.

Let x2(n) = exp{y2(n)}, then (39) is equivalent to

y2(n+1)−y2(n)≥r2l−a2u(M2+ε)−β2u(U2+ε). (40)

Summing both sides of (40) from n − τ1(n) to n − 1 leads to

∑j=n−τ1(n)n−1(y2(n+1)−y2(n))≥∑j=n−τ1(n)n−1[r2l−a2u(M2+ε)−β2u(U2+ε)]≥[r2l−a2u(M2+ε)−β2u(U2+ε)]τ~.

Then

y2(n−τ1(n))≤y2(n)−[r2l−a2u(M2+ε)−β2u(U2+ε)]τ~.

Thus

x2(n−τ1(n))≤x2(n)exp⁡{−[r2l−a2u(M2+ε)−β2u(U2+ε)]τ~}. (41)

Substituting (33) and (41) into the second equation of (7), we have

x2(n+1)≥x2(n)exp⁡{r2l−β2u(U2+ε)−a2ux2(n)×exp⁡{−[r2l−a2u(M2+ε)−β2u(U2+ε)]τ~}} (42)

for all n > N3 + τ.

By applying Lemmas 2.2 and 2.4, we immediately get

limn→+∞infx2(n)≥m2ε, (43)

where

m2ε=r2l−β2u(U2+ε)a2uexp⁡{−[r2l−a2u(M2+ε)−β2u(U2+ε)]τ~}×exp⁡{r2l−β2u(U2+ε)−a2uexp⁡{−[r2l−a2u(M2+ε)−β2u(U2+ε)]τ~}M2}.

Setting ε → 0 in (43), then

limn→+∞infx2(n)≥m2, (44)

where

m2=r2l−β2uU2a2uexp⁡{−[r2l−a2uM2−β2uU2]τ~}×exp⁡{r2l−β2uU2−a2uexp⁡{−[r2l−a2uM2−β2uU2]τ~}M2}.

Without the loss of generality, we assume that ε<12min{m1,m2}. For any positive constant ε small enough, it follows from (38) and (44) that there exists large enough N4 >N3 + τ such that

x1(n)≥m1−ε,x2(n)≥m2−ε (45)

for any n ≥ N4.

From the third and fourth equations of system (7) and (45), we can derive that

Δu1(n)≥−γ1(n)u1(n)+η1(n)(m1−ε), (46)

Δu2(n)≥−γ2(n)u2(n)+η2(n)(m2−ε). (47)

Hence

u1(n+1)≥(1−γ1u)u1(n)+η1l(m1−ε), (48)

u2(n+1)≥(1−γ2u)u2(n)+η2l(m2−ε). (49)

By applying Lemmas 2.2 and 2.3, we immediately get

limn→+∞infu1(n)≥η1l(m1−ε)γ1u, (50)

limn→+∞infu2(n)≥η2l(m2−ε)γ2u. (51)

Setting ε → 0 in the above inequality leads to

limn→+∞infu1(n)≥η1lm1γ1u:=U1l, (52)

limn→+∞infu2(n)≥η2lm2γ2u:=U2l. (53)

This completes the proof of Theorem 2.6.

3 Existence and stability of periodic solution

In this section, we will study the stability of (7) under the assumption τi(n) = 0(i = 1,2), namely, we consider the following system

x1(n+1)=x1(n)exp⁡{r1(n)−a1(n)x1(n)−b1(n)(x2(n)−c2(n))2−β1(n)u1(n)},x2(n+1)=x2(n)exp⁡{r2(n)−a2(n)x2(n)+b2(n)(x1(n)−c1(n))2−β2(n)u2(n)},Δu1(n)=−γ1(n)u1(n)+η1(n)x1(n),Δu2(n)=−γ2(n)u2(n)+η2(n)x2(n). (54)

Throughout this section we always assume that ri(n), ai(n), bi(n),ci(n), γi(n) and η1(n) are all bounded nonnegative periodic sequences with a common period ω and satisfy

0<γi(n)<1,n∈N∩[0,ω],i=1,2. (55)

Also it is assumed that the initial conditions of (54) are of the form

xi(0)>0,ui(0)>0,i=1,2. (56)

In similar way, we can derive the permanence of (54). We still let Mi and Ui be the upper bound of {xi(n)} and {ui(n)}, and mi and Uil be the lower bound of {xi(n)} and {ui(n)}.

Theorem 3.1

In addition to (55), assume that (H1) and (H2) r1l>b1u(M2+c2u)2+β1U1,r2l>β2uU2 hold, then system (54) has a periodic ω solution denoted by {x¯1(n),x¯2(n),u¯1(n),u¯2(n)}.

Proof

Let Ω={(x1,x2,u1,u2)|mi≤xi≤Mj,Uil≤ui≤Ui,i=1,2}. It is easy to see that Ω is an invariant set of system (54). Then we can define a mapping F on Ω by

F(x1(0),x2(0),u1(0),u2(0))=(x1(ω),x2(ω),u1(ω),u2(ω)) (57)

for (x1(0), x2(0), u1(0),u2(0)) ∈ Ω. Obviously, F depends continuously on (x1(0), x2(0), u1(0), u2(0)) . Thus F is continuous and maps a compact set Ω into itself. Therefore, F has a fixed point (x¯1(n),x¯2(n),u¯1(n),u¯2(n)). So we can conclude that the solution (x¯1(n),x¯2(n),u¯1(n),u¯2(n)) passing through (x¯1,x¯2,u¯1,u¯2) is a periodic solution of system (54). The proof of Theorem 3.1 is complete.

Next, we investigate the global stability property of the periodic solution obtained in Theorem 3.1.

Theorem 3.2

In addition to the conditions of Theorem 3.1, assume that the following condition (H3) hold,

H3χ1=max{|1−a1lm1|,|1−a1uM1|}+2b1uc2u[M22+M2]+β1u<1χ2=max{|1−a2lm2|,|1−a2uM2|}+2b2uc1u[M12+M1]+β2u+<1,χ3=(1−γ1l)+η1uM1<1,χ4=(1−γ2l)+η2uM2<1,

then the ω periodic solution (x¯1(n),x¯2(n),u¯1(n),u¯2(n)) obtained in Theorem 3.1 is globally attractive.

Proof

Assume that (x1(n), x2(n), u1(n), u2(n)) is any positive solution of system (54). Let

xi(n)=x¯i(n)exp⁡{yi(n)},ui(n)=u¯i(n)+vi(n),i=1,2. (58)

To complete the proof, it suffices to show

limn→∞yi(n)=0,limn→∞vi(n)=0,i=1,2. (59)

Since

y1(n+1)=y1(n)−a1(n)x¯1(n)[exp⁡(y1(n))−1]−b1(n)[(x2(n)−c2(n))2−(x¯2(n)−c2(n))2]−β1(n)v1(n)=y1(n)−a1(n)x¯1(n)exp⁡{θ1(n)y1(n)}y1(n)−b1(n)x¯22(n)exp⁡{2θ2(n)y2(n)}2y2(n)+2b1(n)c2(n)x¯2(n)exp⁡{θ3(n)y2(n)}y2(n)−β1(n)v1(n), (60)

where θi(n) ∈ (0,1), i = 1, 2, 3. In a similar way, we get

y2(n+1)=y2(n)−a2(n)x¯2(n)exp⁡{θ4(n)y2(n)}y2(n)+b2(n)x¯12(n)exp⁡{2θ5(n)y1(n)}2y1(n)+2b2(n)c1(n)x¯1(n)exp⁡{θ6(n)y1(n)}y1(n)−β2(n)v2(n), (61)

where θi(n) ∈ (0,1), j = 4, 5, 6.

Also, one has

v1(n+1)=(1−γ1(n))v1(n)+η1(n)x¯1(n)[exp⁡{y1(n)}−1]=(1−γ1(n))v1(n)+η1(n)x¯1(n)exp⁡{θ7(n)y1(n)}y1(n) (62)

v 2 ( n + 1 ) = ( 1 − γ 2 ( n ) ) v 2 ( n ) + η 2 ( n ) x ¯ 2 ( n ) [ exp ⁡ { y 2 ( n ) } − 1 ] = ( 1 − γ 2 ( n ) ) v 2 ( n ) + η 2 ( n ) x ¯ 2 ( n ) exp ⁡ { θ 8 ( n ) y 2 ( n ) } y 2 ( n ) . (63)

By (H3), we can choose a ε > 0 such that

χ1ϵ=max{|1−a1l(m1−ϵ)|,|1−a1u(M1+ϵ)|}+2b1uc2u[(M2+ϵ)2+M2+ϵ]+β1u<1χ2ϵ=max{|1−a2l(m2−ϵ)|,|1−a2u(M2+ϵ)|}+2b2uc1u[(M1+ϵ)2+M1+ϵ]+β2u<1,χ3ϵ=(1−γ1l)+η1u(M1+ϵ)<1,χ4ϵ=(1−γ2l)+η2u(M2+ϵ)<1, (64)

In view of Proposition 2.5 and Theorem 2.6, there exists N5 > N4 such that

mi−ϵ≤xi(n),x¯i(n)≤Mi+ϵ,forn≤N5,i=1,2. (65)

It follows from (60) and (61) that

y1(n+1)≤max{|1−a1l(m1−ϵ)|,|1−a1u(M1+ϵ)|}|y1(n)|+2b1uc2u[(M2+ϵ)2+M2+ϵ]|y2(n)|+β1u|v1(n)|, (66)

y2(n+1)≤max{|1−a2l(m2−ϵ)|,|1−a2u(M2+ϵ)|}|y2(n)|+2b2uc1u[(M1+ϵ)2+M1+ϵ]|y1(n)|+β2u|v2(n)|, (67)

Also, for n > N5, one has

v1(n+1)≤(1−γ1l)|v1(n)|+η1u(M1+ϵ)|y1(n)|, (68)

v2(n+1)≤(1−γ2l)|v2(n)|+η2u(M2+ϵ)|y2(n)|. (69)

Let χ=max{χ1ϵ,χ2ϵ,χ3ϵ,χ4ϵ}, then 0 < χ < 1. It follows from (66) -(69) that

max{|y1(n+1)|,|y2(n+1)|,|v1(n+1)|,|v2(n+1)|}≤χmax{|y1(n)|,|y2(n)|,|v1(n)|,|v2(n)|} (70)

for n > N5. Then we get

max{|y1(n)|,|y2(n)|,|v1(n)|,|v2(n)|}≤χn−N5max{|y1(N5)|,|y2(N5)|,|v1(N5)|,|v2(N5)|}. (71)

Thus

limn→∞yi(n)=0,limn→∞vi(n)=0,i=1,2. (72)

This completes the proof.

Remark 3.3

Although Zhi et al. [46] have investigated the permanence and almost periodic solution for an enterprise cluster model based on ecology theory with feedback controls on time scales, they also do not consider the enterprise cluster model with time delays, moveover, they do not investigate the periodic solution of this model. In this paper, we consider the permanence and periodic solutions of two enterprises with multiple delays, which is more general than those models in [46]. Thus our results complement the previous work of [46].

4 Examples

Example 4.1

Consider the following system

x1(n+1)=x1(n)exp⁡{r1(n)−a1(n)x1(n−τ1(n))−b1(n)(x2(n−τ2(n))−c2(n))2−β1(n)u1(n)},x2(n+1)=x2(n)exp⁡{r2(n)−a2(n)x2(n−τ1(n))+b2(n)(x1(n−τ2(n))−c1(n))2−β2(n)u2(n)},Δu1(n)=−γ1(n)u1(n)+η1(n)x1(n),Δu2(n)=−γ2(n)u2(n)+η2(n)x2(n), (73)

where r1(n) = 0.4 + sin(n), r2(n) = 0.5 + cos(n), a1(n) = a2(n) = 1, b1(n) = 0.4 + cos(n), b2(n) = 0.3 + cos(n), c1(n) = 0.4 + cos(n), c2(n) = 0.4 + cos(n), β1(n) = 0.2 + cos(n), β2(n) = 0.4 + sin(n), γ1(n) = cos(n) + 0.3, γ2(n) = sin(n) + 0.4, η1(n) = 0.5 + cos(n), η2(n) = 0.7 + cos(n), τi(n) = 1. We can verify that all the assumptions in Theorem 2.6 are fulfilled. Then (73) is permanent which is shown in Figures 1-2.

Fig. 1 Times series of x1 and x2 for system (73), where the blue line stands for x1 and the red line stands for x2.
Fig. 1

Times series of x1 and x2 for system (73), where the blue line stands for x1 and the red line stands for x2.

Fig. 2 Times series of u1 and u2 for system (73), where the blue line stands for u1 and the red line stands for u2.
Fig. 2

Times series of u1 and u2 for system (73), where the blue line stands for u1 and the red line stands for u2.

Example 4.2

Consider the following system

x 1 ( n + 1 ) = x 1 ( n ) exp ⁡ { r 1 ( n ) − a 1 ( n ) x 1 ( n ) − b 1 ( n ) ( x 2 ( n ) − c 2 ( n ) ) 2 − β 1 ( n ) u 1 ( n ) } , x 2 ( n + 1 ) = x 2 ( n ) exp ⁡ { r 2 ( n ) − a 2 ( n ) x 2 ( n ) + b 2 ( n ) ( x 1 ( n ) − c 1 ( n ) ) 2 − β 2 ( n ) u 2 ( n ) } , Δ u 1 ( n ) = − γ 1 ( n ) u 1 ( n ) + η 1 ( n ) x 1 ( n ) , Δ u 2 ( n ) = − γ 2 ( n ) u 2 ( n ) + η 2 ( n ) x 2 ( n ) , (74)

where r1(n) = 0.4 + sin(n), r2(n) = 0.5 + sin(n), a1(n) = a2(n) = 1, b1(n) = 0.4 − cos(n), b2(n) = 0.3 − cos(n), c1(n) = 0.4 + sin(n), c2(n) = 0.3 + sin(n), α1(n) = 0.3 + cos(n), β2(n) = 0.2 + sin(n), γ1(n) = cos(n) + 0.2, γ2(n) = sin(n) + 0.3, η1(n) = 0.2 + cos(n), η2(n) = 0.2 + cos(n). We can verify that all the assumptions in Theorem 3.1 are fulfilled. Then we know that the periodic solution of system (74) is globally attractive which is illustrated in Figures 3-3.

Fig. 3 Times series of x1 and x2 for system (74), where the blue line stands for x1 and the red line stands for x2.
Fig. 3

Times series of x1 and x2 for system (74), where the blue line stands for x1 and the red line stands for x2.

Fig. 4 Times series of u1 and u2 for system (74), where the blue line stands for u1 and the red line stands for u2.
Fig. 4

Times series of u1 and u2 for system (74), where the blue line stands for u1 and the red line stands for u2.

5 Conclusions

In the present article, a discrete system with competition and cooperation model of two enterprises is proposed. Applying the difference inequality theory, we have established some sufficient conditions which guarantee the permanence of the system. We find that under some suitable conditions, the competition and cooperation of enterprises cluster can remain balanced. This shows that feedback control effect and time delays play a key role in deciding the survival of enterprises. The sufficient conditions which ensure the existence and stability of unique globally attractive periodic solution of the system without time delays are established. The obtained results are completely new and complement the published works of [25–29, 46].

Acknowledgements

The first and second authors were supported by the Key Research Institute of Philosophies and Social Sciences in Guangxi Universities and Colleges(16YC001, 16YC002). The third author was supported by Key Research Institute of Philosophies and Social Sciences in Guangxi Universities and Colleges(16YC001, 16YC002) and Key Project of Science and Technology Research in Guangxi Universities and Colleges(ZD2014058). The authors would like to thank the referees and the editor for helpful suggestions incorporated into this paper.

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Received: 2016-7-3
Accepted: 2017-1-26
Published Online: 2017-3-13

© 2017 Lu et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/math-2017-0023
Keywords for this article
Competition and cooperation model; Permanence; Feedback control; Delay; Enterprise
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