Applied Mathematical Sciences, Vol. 10, 2016, no. 52, 2571 - 2574
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2016.65179
Small Delays in a Competition and
Cooperation Model of Enterprises
Luca Guerrini
Department of Management
Polytechnic University of Marche, Italy
c 2016 Luca Guerrini. This article is distributed under the Creative ComCopyright mons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Abstract
This paper deals with the dynamical behaviors for a competition and
cooperation model of two enterprises with two small delays introduced
by Liao et al. [7]. The simplicity of characteristic roots is proved and
an analysis of the model with small time delays is provided.
Mathematics Subject Classification: 34K18, 91B62
Keywords: Delay; enterprise; cooperation
1
Introduction
In [7], Liao et al. proposed the following competition and cooperation model
of two enterprises with two delays:
( .
x = x (r1 − a1 xd1 ) − b1 (yd2 − c2 )2 ,
(1)
.
y = y (r2 − a2 yd1 ) + b2 (xd2 − c1 )2 ,
where x = x1 −c1 , y = x2 −c2 ; xdj = x(t−τj ) and ydj = y(t−τj ); τj ≥ 0 denoting
time delays; aj = rj /Kj (j = 1, 2), b1 = αr1 /K2 and b2 = βr2 /K1 . Here, x1 and
x2 represent the output of enterprises; rj , Kj and cj denote the intrinsic growth,
the load capacity and the initial production for enterprises, respectively; α, β
are coefficients of competition and cooperation of two enterprises Liao et al. [7]
showed that system (1) has a unique positive equilibrium (x∗ , y∗ ) if condition
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Luca Guerrini
a22 d1 > b1 d22 holds. By choosing the diverse delay as a bifurcation parameter,
the complex Hopf bifurcation phenomenon at this equilibrium can occur as the
diverse delay crosses some critical values.
In this paper, following [1]-[6], our aim is to complete their analysis by proving the root of the characteristic polynomial of (1) to be simple. Furthermore,
we investigate the model when the time delays τ1 and τ2 are small.
2
Stability and existence of Hopf bifurcation
The characteristic equation of the linearization of (1) at its unique equilibrium
point (x∗ , y∗ ) is
λ2 − (A1 + B2 )λe−λτ1 + A1 B2 e−2λτ1 − A2 B1 e−2λτ2 = 0,
(2)
where
A1 = −a1 (x∗ + c1 ) ,
B1 = −b1 y∗ (x∗ + c1 ) ,
A2 = 2b2 x∗ (y∗ + c2 ) ,
B2 = −a2 (y∗ + c2 ) .
In absence of delays, the equilibrium point is locally asymptotically stable
3
Case τ1 = 0 and τ2 > 0
Eq. (2) takes the form
λ2 − (A1 + B2 )λ + A1 B2 − A2 B1 e−2λτ2 = 0,
(3)
Theorem 3.1. If (A1 B2 )2 − (A2 B1 )2 < 0, then the positive equilibrium
(x∗ , y∗ ) of system (1) is locally asymptotically stable for τ2 ∈ [0, τ20 ), and is
unstable for τ2 > τ20 . System (1) undergoes a Hopf bifurcation at the positive
equilibrium (x∗ , y∗ ) when τ2 = τ20 .
Proof. From Liao et al [7], we see it remains to show the root λ = iω20 (ω20 > 0)
at τ2 = τ20 of (3) to be simple. By contradiction, one assumes that it is not
simple. Taking the derivative of (3) with respect to λ, and then evaluating at
λ = iω20 , we must have
2
−2τ2 ω20
+ [1 − τ2 (A1 + B2 )] 2iω20 − (A1 + B2 ) + 2τ2 A1 B2 = 0.
Thus, 0 < 1 − τ2 (A1 + B2 ) = 0, an absurd.
Small delays in a competition and cooperation model of enterprises
4
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Case τ1 > 0, τ2 > 0 in its stable interval
We regard τ1 as a parameter and consider τ2 in its stable interval. Assuming
small delays τ1 ,τ2 , we can write e−λτ1 ∼
=
= 1 − 2λτ1 and e−2λτ2 ∼
= 1 − λτ1 , e−2λτ1 ∼
1 − 2λτ2 . Consequently. the characteristic equation (2) becomes
λ2 − (A1 + B2 )λ (1 − λτ1 ) + A1 B2 (1 − 2λτ1 ) − A2 B1 (1 − 2λτ2 ) = 0,
(4)
If λ = iω, ω > 0, is a root of (4), then we have
(
ω 2 [1 + (A1 + B2 )τ1 ]
= A1 B2 − A2 B1 ,
2A1 B2 τ1 + (A1 + B2 ) − 2A2 B1 τ2 =
0.
If 1 + (A1 + B2 )τ1 > 0 and τ2 < (A1 + B2 )/(2A2 B1 ), then
s
A1 B2 − A2 B1
−(A1 + B2 ) + 2A2 B1 τ2
ωc =
,
τ1 = τ1c =
> 0.
1 + (A1 + B2 )τ1
2A1 B2
Differentiating (4) with respect to τ1 yields
dλ
[2λ − (A1 + B2 ) (1 − λτ1 ) + 2A2 B1 τ2 + (A1 + B2 )λτ1 − 2A1 B2 τ1 ]
dτ1
= −(A1 + B2 )λ2 + 2A1 B2 λ.
By contradiction we have that λ = iωc is a simple root of (4) at τ1 = τ1c .
Otherwise, one would have (A1 + B2 )ωc2 + 2A1 B2 iωc = 0, and so ωc = 0. Next,
one derive
−1
2λ − (A1 + B2 ) + 2A2 B1 τ2 3τ1
dλ
=−
−
.
dτ1
(A1 + B2 )λ2
λ
Then
" −1 #
d Re(λ)
dλ
sign
= sign Re
dτ1
dτ1
τ1 =τ c
c
τ1 =τ1
1
(A1 + B2 )ωc2 − 2A2 B1 τ2 )
= sign
−(A1 + B2 )ωc2
= sign (A1 + B2 )ωc2 − 2A2 B1 τ2 ) .
We can state the following result.
Theorem 4.1. Let τ1 and τ2 be small, with τ2 in its stable interval and τ2 <
(A1 +B2 )/(2A2 B1 ). If sign(1+(A1 +B2 )τ1 ) > 0 and (A1 +B2 )ωc2 −2A2 B1 τ2 ) <
0, then the equilibrium point (x∗ , y∗ ) of system (1) is locally asymptotically
stable for τ1 ≥ 0. If sign(1+(A1 +B2 )τ1 ) > 0 and (A1 +B2 )ωc2 −2A2 B1 τ2 ) < 0,
then there exists τ1c > 0 such that the equilibrium point (x∗ , y∗ ) of system (1)
is locally asymptotically stable for τ1 ∈ [0, τ1c ) and unstable for τ1 > τ1c . System
(1) undergoes a Hopf bifurcation at the equilibrium (x∗ , y∗ ) if τ1 = τ1c .
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Luca Guerrini
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Received: June 12, 2016; Published: August 17, 2016