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Dynamical analysis of a competition and cooperation system with multiple delays
Boundary Value Problems volumeÂ 2018, ArticleÂ number:Â 111 (2018)
Abstract
This paper is concerned with a competition and cooperation system with multiple constant delays relating to economic enterprise. The stability of the unique positive equilibrium is investigated and the existence of Hopf bifurcations is demonstrated by analysing the associated characteristic equation. Furthermore, the explicit formulae determining the stability and the direction of periodic solutions bifurcating from Hopf bifurcations are obtained by applying centre manifold theory and the normal form method. Finally, special attention is paid to some numerical simulations in order to support the theoretical predictions.
1 Introduction
Delay differential equations (DDEs) which involve delays and change in time [1, 2] exhibit considerably more complex dynamical behaviour than ordinary differential equations (ODEs), since the delays could cause a stable equilibrium to become unstable and fluctuate. Many complicated and largescale systems in nature and society can be modelled as DDEs due to their flexibility and generality for representing virtually any natural and manmade structure. Research of the dynamical behaviour of DDEs has received much attention in interdisciplinary subjects, including natural sciences [3â€“6], engineering [7, 8], life sciences [9] and others [10â€“16]. In recent decades, scientists have focused on the stability and bifurcation phenomena of the continuoustime autonomous predatorâ€“prey system with multiple delays (see, for example, [17â€“22]). In fact, there is a strong relationship between how species coevolve in nature and how different enterprises coexist in societal economics, leading to significant research into the delayed competition and cooperation model for business enterprises [23â€“25], which are governed by the following system of ODEs:
where \(x_{1}(t)\), \(x_{2}(t)\) denote the output of enterprise \(x_{1}\) and enterprise \(x_{2}\) at time t, respectively, \((x_{1}(t),x_{2}(t))\in\mathbb{R}^{1}\times\mathbb{R}^{1}\); \(r_{i}\) (\(i=1,2\)) represents the intrinsic growth rate for the output of the two enterprises; \(K_{i}\) (\(i=1,2\)) is a measure of the load capacity of the two enterprises in an unrestricted natural market; Î±, Î² are the coefficients of competition of enterprise \(x_{1}\) and \(x_{2}\), respectively; \(c_{i}\) (\(i=1,2\)) denotes the initial production of them. All the parameters assume strictly positive values.
Let \(a_{1}=\frac{r_{1}}{K_{1}}\), \(a_{2}=\frac{r_{2}}{K_{2}}\), \(b_{1}=\frac{\alpha r_{1}}{K_{2}}\), \(b_{2}=\frac{\beta r_{2}}{K_{1}}\), \(d_{i}=r_{i}a_{i}c_{i}\), \(\forall i=1,2\), \(u(t)=x_{1}(t)c_{1}\), \(v(t)=x_{2}(t)c_{2}\). System (1.1) becomes
where \((u(t),v(t))\in\mathbb{R}^{1}\times\mathbb{R}^{1}\).
Taking into account the influence of the prior history of the enterprises, authors have introduced a time delay, Ï„, to the feedback in model (1.2) [26], which is a more realistic approach for understanding competition and cooperation dynamics. Delays can induce oscillations and periodic solutions through bifurcations as the delay is increased. Therefore, it is interesting to investigate the following delayed model:
where \(y_{i}(t)\) (\(i=1,2\)) denotes the output of two enterprises at time t, \(a_{i}\), \(b_{i}\) (\(i=1,2\)) denote the intraspecific competition rate and interspecific effect rate between them, where \(a_{i}\), \(b_{i}\), \(c_{i}\), \(d_{i}\) (\(i=1,2\)) are positive constants. \(\tau_{1}\) denotes the interior delays of themselves, \(\tau_{i}\) (\(i=2,3\)) denotes the exterior delays between each other, and \(\tau_{i}\) (\(i=1,2,3\)) is nonnegative constant delays.
We define \(\mathbb{R}_{+}\equiv\{x\in\mathbb{R}:x\geq0\}\), \(\operatorname{int} \mathbb{R}_{+}\equiv\{x\in\mathbb{R}:x >0\}\), \(\hat{\tau}=\max\{\tau_{1},\tau_{2},\tau_{3}\}\). Denote by \(C([\hat{\tau},0],\mathbb{R}_{+})\) the infinite dimensional Banach space of continuous functions from the interval \([\hat{\tau},0]\) into \(\mathbb{R}_{+}\), equipped with the uniform norm. We assume that the initial data for model (1.3) is taken from
The variables \(y_{1}(t)\) and \(y_{2}(t)\) in model (1.3) belong to X for \(t\in[\hat{\tau},0]\).
By [1] (Theorem 2.1 and 2.3, Chap.Â 2, p.Â 41), solutions of system (1.3) with the initial value in C exist and are unique for all \(t>0\).
Liao [23] assumed \(\tau_{i}\ (i=1,2,3)=\tau\) and Li [24] considered \(\tau_{1}=0\), regarding Ï„ and \(\tau_{2}+\tau_{3}\) as the bifurcation parameters, respectively. They investigated the existence of the unique positive equilibrium and proved that the Hopf bifurcation can occur as the bifurcation parameter crosses some critical value, and studied the direction of Hopf bifurcation and stability of the periodic solutions. In [27], Liao considered \(\tau_{2}=\tau_{3}\neq\tau_{1}\), analysed the stability of the positive equilibrium and the existence of local Hopf bifurcation and provided some numerical simulations. However, they did not give the underlying description of the bifurcated periodic solution.
In the more realistic competition and cooperation model, the interior delays exist (i.e. \(\tau_{1}\neq0\)) and the exterior delays are not necessarily equal (i.e. \(\tau_{2}\neq \tau_{3}\)). Based on these observations, we studied the system of (1.3) that could better describe the real system behaviour. Compared with the models from the literature [23, 27], the dynamical behaviour of system (1.3) is more complicated than the above models.
In this paper, we have taken the delay \(\tau_{1}:=\tau_{2}+\tau_{3}\) as the bifurcation parameter and show that when \(\tau_{1}\) passes through the critical values, the positive equilibrium loses its stability and a Hopf bifurcation occurs. Furthermore, we give details of the bifurcation values that describe the direction of the Hopf bifurcation and the stability of the bifurcated periodic solution using centre manifold theory and the normal form method introduced by Hassard et al. [28]. Finally, some numerical simulations and conclusions are given to illustrate the theoretical predictions.
2 The existence and the property of the local Hopf bifurcation
In this section, we give the following results about the existence and stability of the positive equilibrium of system (1.3).
Proposition 1
For system (1.3), assume that \(a_{2}\), \(b_{1}\), \(d_{1}\), \(d_{2}\) are positive constants such that
 \((\mathrm{H}_{1})\) :

\(a_{2}^{2}d_{1}>b_{1}d_{2}^{2}\)
holds, then the system has a unique positive equilibrium \(E^{\ast }=(y_{1}^{\ast},y_{2}^{\ast})\). Furthermore, when system (1.3) has no delay, i.e. \(\tau_{i}\ (i=1,2,3)=0\), then \(E^{\ast}\) is globally asymptotically stable.
Proof
For system (1.3), assumption \((\mathrm{H}_{1})\) is the parameter condition which ensures the existence of the positive equilibrium \(E^{\ast}\). The proof for the existence of \(E^{\ast}\) is similar to that in [23], we omit it here.
We now prove the global asymptotic stability. When \(\tau_{i}\ (i=1,2,3)=0\), system (1.3) is reduced to the ODE system (1.1). Defining Dulac function as \(B(x_{1},x_{2})=\frac{1}{x_{1}x_{2}}\), and
we easily get \(D<0\) in the \(\operatorname{int} \mathbb{R}_{+}^{2}=\{ (x_{1},x_{2}):x_{1}>0,x_{2}>0\}\) space. By [29] (Theorem 4.1.2, Chap.Â 4, p.Â 72), it follows from Dulacâ€™s principle that the system has no closed path curve. So \(E^{\ast}\) is globally asymptotically stable when system (1.3) has no delay and also when the nonnegative delays are sufficiently small.â€ƒâ–¡
If a pair of complex roots with negative real parts and nonzero imaginary parts cross the imaginary axis as Ï„ increases, this potentially results in Hopf bifurcation and the positive equilibrium \(E^{\ast}\) loses stability. Now we discuss the existence of a local Hopf bifurcation occurring at \(E^{\ast}\). Let \(u_{1}(t)=y_{1}(t)y_{1}^{\ast}\), \(u_{2}(t)=y_{2}(t)y_{2}^{\ast}\), then system (1.3) becomes
the linearization of system (2.1) at \(E^{\ast}\) is
and the associated characteristic equation of (2.2) is
For the above characteristic equation, it is hard to do the complete analysis for the distribution of the roots, so we assume that
 \((\mathrm{H}_{2})\) :

\(\tau_{2}+\tau_{3}= \tau_{1}\)
holds, and \(\tau_{1}\triangleq\tau\).
Hence, the characteristic equation is equivalent to
where \(p=a_{1}(y_{1}^{\ast}+c_{1})+a_{2}(y_{2}^{\ast}+c_{1})\), \(q=4b_{1}b_{2}(y_{2}^{\ast})^{2}(y_{1}^{\ast}+c_{1})(y_{2}^{\ast}+c_{1})\), \(r=a_{1}a_{2}(y_{1}^{\ast}+c_{1})(y_{2}^{\ast}+c_{1})\).
Since the characteristic equation (2.3) has the same form as equation (2.4) in [30], so by Theorem 2.5 in [30], we can get the following result, which presents the conditions for a Hopf bifurcation to occur in system (1.3).
Proposition 2
Suppose that \((\mathrm{H}_{1})\) and \((\mathrm{H}_{2})\) hold. Then
are Hopf bifurcation values at \(E^{\ast}\), where \(i\omega _{k}\) (\(k=1,2,3,4\)) are the roots of (2.3). And \(E^{\ast}\) is locally asymptotically stable for \(\tau\in[0,\tau _{1}^{0}]\) and unstable where \(\tau>\tau_{1}^{0}\).
Remark 1
The characteristic equation (2.3) has some pairs of purely imaginary roots denoted by \(\lambda=\pm i\omega_{k}\) with \(\tau=\tau^{j}_{k}\) under the condition of \((\mathrm{H}_{1})\), \((\mathrm{H}_{2})\). Define \(\tau^{0}=\tau^{0}_{k_{0}}=\min_{1\leq k\leq4}\{\tau^{0}_{k}\} \), \(\omega_{0}=\omega_{k_{0}}\), where \(k_{0}\in\{1,2,3,4\}\). Then \(\tau^{0}\) is the first value of Ï„ such that (2.3) has purely imaginary roots. For convenience, we denote \(\tau^{j}_{k}\) by \(\tau^{j}\) (\(j=0,1,2,\ldots\)) for fixed \(k\in\{1,2,3,4\}\).
Remark 2
Let \(\lambda(\tau)=\alpha(\tau)\pm i\omega(\tau)\) be the roots of (2.3) near \(\tau=\tau^{j}\) satisfying \(\alpha(\tau^{j})=0\), \(\omega(\tau^{j})=\omega_{0}\) (\(j=0,1,2,\ldots\)). By the theory of DDEs, for \(\forall\tau^{j}_{k}\), \(\exists\varepsilon>0\) s.t. \(\lambda(\tau )\) in \(\tau\tau^{j}_{k}<\varepsilon\) about Ï„ is continuous and differentiable. The transversality condition \(\frac{{d}\operatorname{Re}\lambda(\tau)}{{d}\tau } _{\tau=\tau_{j}}>0\) is satisfied (more details are provided in [30]).
In the previous part, it was shown that system (2.1) undergoes a Hopf bifurcation under certain conditions. Here we will derive explicit formulae determining the direction of the Hopf bifurcation and the stability of the periodic solutions bifurcating from \(E^{\ast}\) at \(\tau^{j}\) (\(j=0,1,2,\ldots\)), by employing centre manifold theory and the normal form method. For convenience, denote \(\tau^{j}\) by Ï„Ìƒ and \(\tau =\widetilde{\tau}+\mu\), \(\mu\in\mathbb{R}\), then \(\mu=0\) is the Hopf bifurcation value for system (1.3), where \(\widetilde{\tau}=\widetilde{\tau_{2}}+\widetilde{\tau_{3}}\), \(\tau=\widetilde{\tau_{2}}+\widetilde{\tau_{3}}+\mu\). Without loss of generality, assume \(\widetilde{\tau_{2}}<\widetilde{\tau_{3}}\).
The discussion will be divided into five steps as follows.
Step 1. Transform system ( 2.1 ) into the abstract ODE.
System (2.1) can locally be represented as the following DDE in \(C=C([\widetilde{\tau},0],R^{2})\):
where \(u(t)= (u_{1}(t),u_{2}(t) )^{T}\), \(u_{t}(\theta)=u(t+\theta )\), \(L_{\mu}:C\rightarrow R\) is a bounded linear operator and \(F:R\times C\rightarrow R\) is continuous and differentiable with
and
where \(\phi=(\phi_{1}(\theta),\phi_{2}(\theta))\in C\).
By the Riesz representation theorem, there exists a \(2\times2\) matrix whose elements are a bounded variation function \(\eta(\theta,\mu)\) in \(\theta\in[\widetilde{\tau},0]\) such that
where \(\eta(\theta,\mu)\) can be chosen as
with
For \(\phi\in C\), let
then system (2.4) is equivalent to the following abstract operator equation:
Step 2. Calculate the eigenfunctions of \(A=A(0)\) and the adjoint operator \(A^{\ast}\) corresponding to \(i\omega_{0}\widetilde{\tau}\) and \(i\omega _{0}\widetilde{\tau}\) .
For \(\psi\in C([0,\widetilde{\tau}],\mathbb{(}{C}^{2})^{\ast})\), where \(\mathbb{(}{C}^{2})^{\ast}\) is the twodimensional complex space of row vectors, we define the adjoint operator \(A^{\ast}\) of A
and the bilinear form is given by
where \(\eta(\theta)=\eta(\theta,0)\). Then \(A=A(0)\) and \(A^{\ast}(0)\) are adjoint operators.
By [27], \(\pm i\omega_{0}\widetilde{\tau}\) are eigenvalues of \(A(0)\), so they are also eigenvalues of \(A^{\ast}(0)\). Suppose that \(q(\theta)=(1,\alpha)^{T}e^{i\omega_{0}\theta}\) is the eigenfunction of \(A(0)\) corresponding to the eigenvalue \(i\omega _{0}\widetilde{\tau}\) and \(q^{\ast}(s)=G(\beta,1)e^{i\omega_{0}s}\) is the eigenfunction of \(A^{\ast}\) corresponding to the eigenvalue \(i\omega_{0}\widetilde{\tau}\), where
which assures that \(\langle{q}^{\ast}(s),q(\theta)\rangle=1\), \(\langle{q}^{\ast }(s),\overline{q}(\theta)\rangle=0\).
Step 3. Obtain the reduced system on the centre manifold.
In this part, we will use the same notations as in [28] and compute the coordinates to describe the centre manifold \(\mathbf {C}_{0}\) at \(\mu=0\) (a local centre manifold is in general not unique, and the dimension of local centre manifold is 2). Let \(u_{t}\in C\) be the solution of system (2.5) when \(\mu=0\), and define
where z and zÌ… are local coordinates for the centre manifold \(\mathbf{C}_{0}\) in the direction of \(q^{\ast}\) andÂ \(\overline{q}^{\ast}\). On the centre manifold \(\mathbf{C}_{0}\), we have \(W(t,\theta )=W(z(t),\overline{z}(t),\theta)\), where
The existence of a centre manifold enables us to reduce (2.5) to an ODE on \(\mathbf{C}_{0}\). Note that W is real if \(u_{t}\) is real, we consider only real solutions. For solution \(u_{t}\in\mathbf{C}_{0}\) of system (2.5) at \(\mu=0\),
with
Rewriting (2.8), we obtain that the reduced system on \(\mathbf {C}_{0}\) is described by
where
We will mainly discuss equation (2.9) in the following part.
Step 4. Obtain the values of \(g_{20}\) , \(g_{11}\) , \(g_{02}\) , \(g_{21}\) in ( 2.10 ).
In this part, we calculate the coefficients \(W_{20}(\theta)\), \(W_{11}(\theta)\), \(W_{02}(\theta)\), â€¦ and substitute them in (2.8) to get the reduced system (2.9) on \(\mathbf{C}_{0}\).
It follows from (2.6) that
And we have
It follows that together with \(F(\mu,\phi)\) we get
Substituting (2.11) into (2.12), then this substitution into (2.8), and comparing the coefficients with (2.10), we obtain
Since there are \(W_{20}(\theta)\) and \(W_{11}(\theta)\) in \(g_{21}\), we still need to compute them.
where
On the other hand, near the origin, on the centre manifold \(\mathbf {C}_{0}\), according to (2.7), we obtain
Substituting (2.7) into the righthand side of (2.14), equating terms of \(\frac{z^{2}}{2}\) and zzÌ… of (2.14) with (2.15), we obtain
According to the definition of A and from (2.16), (2.17) for \(\theta\in[\widetilde{\tau},0)\), we get
Solving for \({W}_{20}(\theta)\) and \({W}_{11}(\theta)\), we obtain
where \(E_{1}=(E^{(1)}_{1},E^{(2)}_{1})^{T}\in R^{2}\) and \(E_{2}=(E^{(1)}_{2},E^{(2)}_{2})^{T}\in R^{2}\) are constant vectors.
In what follows we shall seek appropriate \(E_{1}\) and \(E_{2}\) in (2.18) and (2.19), respectively. According to the definition of A and (2.16), (2.17) for \(\theta=0\), we have
where \(\eta(\theta)=\eta(0,\theta)\) and
Substituting (2.18) into (2.20), we obtain
that is
Similarly, substituting (2.19) into (2.21), we get
that is
We have obtained the values of \(E_{1}\) and \(E_{2}\) as (2.22) and (2.23) and, ultimately, the reduced system (2.9).
Step 5. Obtain the key values \(\mu_{2}\) , \(\beta_{2}\) , \(T_{2}\) to determine the property of the Hopf bifurcation.
As with the calculation of the ODE Hopf bifurcation parameter and as in [28], according to the analysis above and the expressions of \(g_{20}\), \(g_{11}\), \(g_{02}\) and \(g_{21}\), we can compute the following values:
where \(\lambda(\tau)=\alpha(\tau)\pm i\omega(\tau)\) is the characteristic root of (2.3), which is a continuous differentiable family. \(\alpha'(\widetilde{\tau })\) and \(\omega'(\widetilde{\tau})\) can be obtained by taking the derivative of the two sides of (2.3) and taking values at Ï„Ìƒ.
These formulae give a description of the Hopf bifurcation periodic solution of system (1.3) at \(\tau=\tau^{j}\) (\(j=0,1,2,\ldots\)) on the centre manifold. Thus, we can obtain the following results according to the discussion about properties of Hopf bifurcating periodic solutions of dynamical system in [30].
Proposition 3
Assume that \((\mathrm{H}_{1})\) and \((\mathrm{H}_{2})\) hold. Then

(i)
\(\mu_{2}\) determines the direction of the Hopf bifurcation. If \(\mu_{2}>0\) (\(\mu_{2}<0\)), then the Hopf bifurcation is supercritical (subcritical);

(ii)
\(\beta_{2}\) determines the stability of the bifurcating periodic solutions. If \(\beta_{2}<0\) (\(\beta_{2}>0\)), then bifurcating periodic solution is stable (unstable);

(iii)
\(T_{2}\) determines the period of the bifurcating periodic solutions. If \(T_{2}>0\) (\(T_{2}<0\)), then periods of the periodic solutions increase (decrease).
3 Numerical simulations and conclusions
In this section, we shall give some numerical simulations to support the theoretical analysis discussed in the previous section. We also present our conclusions and limitations of the analysis.
Firstly, we study the following specific model:
with initial values \((y_{1}(t),y_{2}(t))=(0.2,0.2)\), which satisfies \((\mathrm {H}_{1})\), \((\mathrm{H}_{2})\). By computing, \(E^{\ast}=(1,1)\) and by [30], \(h(z)=z^{4}2.56z^{3}+0.3584z^{2}2.1627z0.3244=0\), which has only one positive root \(z=2.7341\), get \(\omega=1.6535\), \(\tau ^{0}=0.5847\). By PropositionÂ 2, we find that \(E^{\ast}\) is asymptotically stable when \(0\leq\tau<\tau^{0}=0.5847\), as Figs.Â 1(a)â€“(d) illustrate, and \(E^{\ast}\) is unstable when \(\tau >\tau^{0}=0.5847\), as shown in Figs.Â 2(a)â€“(d) which are generated by dde23 [31], a Matlab tool that integrates DDEs.
According to the above numerical simulations and from an economic viewpoint, we conclude that a critical duration time of the two enterprise outputs exists. When the duration time is less than the critical delay, the cooperation between the two enterprises is very effective; though the competition between them exists, they can coexist and have developed over a long time. Alternatively, when the competition between the two enterprises is much stronger than their effective cooperation, the result will ultimately force a merger or a closure by one of the enterprises. Therefore, entrepreneurs must have a shrewd understanding of market forces and economic laws in order to maintain a viable and successful enterprise.
However, here we have only considered the problem of a local Hopf bifurcation and do not give the conditions ensuring the existence of a global Hopf bifurcation for large values of the delay. Furthermore, we do not consider systems with a spatial variable, which is the diffusive model subject to a suitable boundary condition. We intend to make the comparison between the two models, and find what is the influence on the dynamical behaviour with different delays and diffusive terms [32â€“38], and then illustrate with the theoretical predictions. Lastly, our problem is only restricted to the theoretical analysis of such economical phenomena. It may be timely and necessary to make field investigations and experimental studies for realworld scenarios, and this is left for further study.
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Acknowledgements
This research receives the grant from China Scholarship Council (CSC) and was supported by the Natural Science Foundation of Anhui Province (No. 1608085QF151, No. 1608085QF145). The authors would like to express their deep thanks to the referee for valuable suggestions for the revision and improvement of the manuscript.
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Xin Zhangâ€™s research interests are delay differential equations, dynamical systems, bifurcation theory and mathematical biology. The research interests of Zizhen Zhang mainly are bifurcation theory and population dynamics. MatthewÂ J.Â Wadeâ€™s research topics are numerical analysis and mathematical modelling.
Funding
This research is supported by the Natural Science Foundation of Anhui Province (No. 1608085QF151, No. 1608085QF145).
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The main idea of this paper was proposed by XZ and she prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
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Zhang, X., Zhang, Z. & Wade, M.J. Dynamical analysis of a competition and cooperation system with multiple delays. Bound Value Probl 2018, 111 (2018). https://doi.org/10.1186/s1366101810329
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DOI: https://doi.org/10.1186/s1366101810329