NEW INSIGHTS ON BIFURCATION IN A FRACTIONAL-ORDER DELAYED COMPETITION AND COOPERATION MODEL OF TWO ENTERPRISES ∗

Over the past decades, many authors establish various kinds of competition and cooperation models of two enterprises to analyze the dynamic interaction. However, they are only concerned with integer-order differential equation models, while the reports on fractional-order ones are very rare. In this article, based on the earlier studies, we propose a new fractional-order delayed competition and cooperation model of two enterprises. Letting the delay be bifurcation parameter and analyzing the corresponding characteristic equation of involved model, we establish some new sufficient conditions to guarantee the stability and the existence of Hopf bifurcation of fractional-order delayed competition and cooperation model of two enterprises. The research indicates that different delays have different effect on the stability and Hopf bifurcation of involved model. The impact of the fractional order on the stability and Hopf bifurcation of involved model is displayed. To check the correctness of theoretical analysis, we implement some computer simulations.

role in mastering the management and decision-making of enterprises. In order to grasp the internal operation mechanism among different enterprises, a lot of scholars are striving to establish the the competition and cooperation model among enterprises and investigate their dynamical behavior. Many excellent results have been reported. In 2012, Xu [50] studied the following continuous competition and cooperation model of two enterprises: ] , y 2 (t) = α 2 y 2 (t) , where y 1 (t) and y 2 (t) denote for the output of enterprise A and enterprise B, respectively; α i (i = 1, 2) stands for the intrinsic growth rate of enterprise A and enterprise B, respectively; a i (i = 1, 2) represents the natural market carrying capacity of enterprise A and enterprise B; r 1 represents the consumption coefficient of the enterprise with the output y 2 (t) to the one with the output y 1 (t) and r 2 represents the transformation coefficient of the enterprise with the output y 1 (t) to the one with the output y 2 (t); β 1 and β 2 are the initial outputs of enterprise A and enterprise B, respectively. With the help of the coincidence degree theory, Xu [50] discussed the existence of periodic solutions of model (1.1). Liao et al. [27] thought that the duration of output for enterprises has important effect on the dynamics of competition and cooperation model of two enterprises and then he established the following delayed competition and cooperation model of two enterprises: ] , y 2 (t) = α 2 y 2 (t) ] , y 1 (0) = ϕ 1 (t), y 2 (0) = ϕ 2 (t), t ∈ [−ε, 0], (1.2) where ε denotes the time delay of interior of two enterprises and ϕ 1 (t), ϕ 2 (t) ∈ C([−ε, 0], , R). Applying the normal theory and center manifold theorem, Liao et al. obtained the sufficient conditions to assure the stability and the existence of Hopf bifurcation of (1.2) and the explicit algorithms to determine the direction, period and stability of Hopf bifurcation for model (1.2). Considering the effect of different delays on the dynamics of (1.2), in 2014, Liao et al. [26] modified the model (1.2) as follows: , , Applying the Hopf bifurcation theory of delayed differential equations, Liao et al. [26] discussed the effect of two different delays on the bifurcation behavior of system (1.3). In 2016, Li et al. [22] proposed the following delayed competition and cooperation model of two enterprises with four delays: , , (1.4) By assuming ε 1 = ε 3 = 0 and ε 2 +ε 4 = ε, Li et al. [22] discussed the Hopf bifurcation issue of system (1.4). For more relative topic, one can see [3,22,25,26,30,53,61].
As is known to us, the fractional calculus is a generalization of ordinary differentiation and integration [10,12,19,21,24,36,38,40,42,43,60,[62][63][64]. However, the fractional calculus attracts little attention due to its complexity and the lack of practical background. Up to the last few decades, the fractional calculus has gained extensive applications in numerous areas of science and engineering such as mechanics, chemistry, viscoelasticity, biology, physics, finance and so on [4,59]. Many authors argued that in many cases, it is more reasonable to model the natural world by fractional-order differential equations than integer-order ones since fractional-order differential equations give a better description of the memory and hereditary nature of various materials and processes. Recently, many researchers pay much attention to the dynamical behavior of fractional-order differential systems and outstanding achievements have been made. We refer the readers to [2,18,32,41]. In particular, some fruits about Hopf bifurcation phenomenon of fractional-order differential models have also been reported. For example, Rakkiyappan et al. [35] investigated the Hopf bifurcation of fractional-order complex-valued neural networks, Abdelouahab et al. [1] analyzed the Hopf bifurcation and chaos for a fractional-order hybrid optical model, Huang and Cao [14] discussed the bifurcation behavior of fractional order neural networks with leakage delays, Deshpande et al. [8] focused on the Hopf bifurcation in a fractional order systems. In details, we refer the readers to [9, 13, 15-17, 33, 34, 44-48, 52].
Here it is worth pointing out that all the above publications about Hopf bifurcation of fractional-order differential models are mainly concerned with single or the sum of different delays. So far, there are no papers that focus on the impact of different delays on the Hopf bifurcation of involved systems. Not to speak of dealing with the impact of different delays on the Hopf bifurcation of fractional-order competition and cooperation model of two enterprises with different delays. In order to further reveal the effect of different delays on Hopf bifurcation of fractional-order competition and cooperation model of two enterprises and master the operation rules of enterprises effectively, we think that it is necessary to investigate the role of time delay in fractional-order competition and cooperation model.
Motivated by the analysis above, we modify (1.3) as a fractional-order delayed competition and cooperation model of two enterprises as follows: where q ∈ (0, 1]. All other variables and coefficients have the same meaning as those in (1.3). The primary object of this manuscript is listed as follows: (a) seeking the sufficient conditions to ensure the stability and existence of Hopf bifurcation of model (1.5); (b) revealing the impact of two different delays on Hopf bifurcation of model (1.5); (c) illustrating the effect of fractional order on the stability and Hopf bifurcation of model (1.5).
The highlights of this manuscript consist of the following aspects: • A new fractional-order competition and cooperation model of two enterprises with two delays, which can better depict the memory and hereditary nature of competition and cooperation of two enterprises, is established.
• A new set of sufficient criteria to ensure the stability and the existence of Hopf bifurcation of fractional-order competition and cooperation model of two enterprises with two delays are obtained. The influence of two different delays and fractionalorder on the stability and Hopf bifurcation of (1.5) are revealed.
• So far, no author focuses on the Hopf bifurcation of fractional-order competition and cooperation model of two enterprises with two delays.
• The research ideas of this manuscript will provide an excellent reference to handle a lot of delayed fractional-order differential systems. The rest of this manuscript is planned as follows. In Segment 2, some basic knowledge on fractional calculus is prepared. In Segment 3, the sufficient conditions to assure the stability and the existence of Hopf bifurcation of involved model are derived. The impact of two different delays on the stability and Hopf bifurcation of the involved competition and cooperation model of two enterprises is displayed. In Segment 4, computer simulations are carried out to verify the main findings. Segment 5 ends the manuscript. Remark 1.1. In (1.3), we replace the first-order derivatives by Caputo fractional derivatives of order q ∈ (0, 1]. According to the idea of [23,45] where γ is a fractional time constant.
For the detailed derivation process, one also can see [23,45].

Basic knowledge
In this section, some related definitions and lemmas about fractional calculus are displayed.

Definition 2.1 ( [31]
). The fractional integral of order q for a function g(ξ) is defined as follows: Define the Caputo fractionalorder derivative of order q of g(ξ) as follows: where ξ ≥ ξ 0 and l denotes a positive integer which satisfies l − 1 ≤ q < 1. Furthermore, if 0 < q < 1, then

Lemma 2.2 ( [7]
). For the given fractional-order system: Then the characteristic equation of the system is det |s q I − G 1 − G 2 e −sϱ | = 0. If all the roots of the characteristic equation of the system have negative real roots, then the zero solution of the system is asymptotically stable.

Impact of two different delays on Hopf bifurcation for model (1.5)
In this section, we will discuss the impact of two different delays on Hopf bifurcation for model (1.5).
. Now the assumption is given as follows: and x 20 satisfies the following condition: Proof. In view of Lemma 2.1 of Liao et al. [27], we know that system (3.2) has a unique equilibrium point (x 10 , x 20 ). Next we prove that the equilibrium point (x 10 , x 20 ) is locally asymptotically stable.
Proof. By (3.7), one has In view of (A2), we get This ends the proof of Lemma 3.2. According to analysis above, one gets the following theorem.
Here we suppose that (3.20) has eight positive real roots φ l (l = 1, 2, 3, 4, 5, 6, 7, 8). By (3.18), one has , (3.23) where k = 0, 1, 2, · · · , l = 1, 2, 3, 4, 5, 6, 7, 8. Set 2,3,4,5,6,7,8 The following hypothesis is given: . Lemma 3.4. Assume that s(ε 1 ) = ϱ(ε 1 )+iϕ(ε 1 ) is the root of (3.7) around ε 1 = ε 10 which satisfies ϱ(ε 10 ) = 0, φ(ε 10 ) = φ 0 , then Re In view of (A3), we get Re This ends the proof of Lemma 3.4. According to the analysis above, one gets the following theorem. Theorem 3.2. For system (1.5), assume that ε 2 ∈ [0, ε 20 ) and (A1)and (A3) hold true, then the equilibrium point (x 10 , x 20 ) is locally asymptotically stable for ε 1 ∈ [0, ε 10 ) and a Hopf bifurcation appears around the equilibrium point (x 10 , x 20 ) when ε 1 = ε 10 . Remark 3.1. In [5,6,11,20,37,49,51,[54][55][56][57][58], the authors considered the Hopf bifurcation of integer-order delayed models. In this paper, we investigate the stability and Hopf bifurcation of fractional-order competition and cooperation model of two enterprises with two different delays. All the obtained results and analysis methods [5,6,11,20,37,49,51,[54][55][56][57][58] can not be applied to (1.5) to obtain the stability and the existence of Hopf bifurcation for (1.5). In [22,[26][27][28]50], the authors investigated the Hopf bifurcation of competition and cooperation model of two enterprises with delay, but they did not involve the fractional-order case. Based on these viewpoints, the fruit of this paper about the stability and the existence of Hopf bifurcation for (1.5) are completely new and an important supplement to some earlier works. Remark 3.2. In [1, 8, 13-17, 33, 35, 39, 47, 48], the authors dealt with the Hopf bifurcation of fractional-order models. They did not consider the effect of different delays on the stability and Hopf bifurcation of involved models. Up to now, there are no results on the effect of different delays on Hopf bifurcation of involved fractionalorder systems. From the analysis above, the obtained results of this paper is new. Remark 3.3. Compared with the integer-order competition and cooperation model of two enterprises, the fractional-order competition and cooperation model of two enterprises can characterize memory property, history state, nonlocal effects of the output of two enterprises, which implies that the fractional-order competition and cooperation model of two enterprises have more advantage than the integer-order one. Remark 3.4. In Theorem 3.1, we assume that ε 1 = 0 which implies the single case. In a similar way, we can deal with ε 2 = 0. Considering the practical meaning of competition and cooperation of two enterprises, we focus on the effect of double delays on Hopf bifurcation of fractional-order competition and cooperation model of two enterprises.

Conclusions
The competition and cooperation among different enterprises is an important aspect in production and management of enterprises. In the manuscript, based on earlier publications, we propose a new fractional-order competition and cooperation model of two enterprises with two different delays. By regarding two different delays as bifurcation parameter, we establish two sets of sufficient conditions to assure the the stability and the existence of Hopf bifurcation for involved competition and cooperation model of two enterprises. The investigation manifests that the two different delays have different effect on the stability and Hopf bifurcation of involved model. Also the relation of fractional-order and bifurcation point are displayed. The derived results have important theoretical significance and practical value in managing the production of enterprises. Besides, we point out that how to control the duration time of output for enterprises is an interesting issue. It involves the bifurcation control issue. We will study this aspect in the near future. In addition, we will try to investigate the effect of multiple delays on Hopf bifurcation for fractional-order delayed competition and cooperation model. Now we still can not deal with the direction and stability of Hopf bifurcation of fractional order differential systems due to the lack of theory.