Dynamical analysis of a competition and cooperation system with multiple delays
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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.
Keywords
Competition and cooperation system Time delays Stability Local Hopf bifurcationMSC
34K10 34K20 37L101 Introduction
By [1] (Theorem 2.1 and 2.3, Chap. 2, p. 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 τ 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
 \((\mathrm{H}_{1})\)

\(a_{2}^{2}d_{1}>b_{1}d_{2}^{2}\)
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.
 \((\mathrm{H}_{2})\)

\(\tau_{2}+\tau_{3}= \tau_{1}\)
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
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 τ 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 τ 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 τ̃ 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.
Step 2. Calculate the eigenfunctions of \(A=A(0)\) and the adjoint operator \(A^{\ast}\) corresponding to \(i\omega_{0}\widetilde{\tau}\) and \(i\omega _{0}\widetilde{\tau}\) .
Step 3. Obtain the reduced system on the centre manifold.
Step 4. Obtain the values of \(g_{20}\) , \(g_{11}\) , \(g_{02}\) , \(g_{21}\) in ( 2.10 ).
In this part, we calculate the coefficients \(W_{20}(\theta)\), \(W_{11}(\theta)\), \(W_{02}(\theta)\), … and substitute them in (2.8) to get the reduced system (2.9) on \(\mathbf{C}_{0}\).
Step 5. Obtain the key values \(\mu_{2}\) , \(\beta_{2}\) , \(T_{2}\) to determine the property of the Hopf bifurcation.
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
 (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.
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 [32, 33, 34, 35, 36, 37, 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.
Notes
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.
Availability of data and materials
Not applicable.
Authors’ information
Xin Zhang’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. Matthew J. Wade’s research topics are numerical analysis and mathematical modelling.
Authors’ contributions
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.
Funding
This research is supported by the Natural Science Foundation of Anhui Province (No. 1608085QF151, No. 1608085QF145).
Ethics approval and consent to participate
Not applicable.
Competing interests
All authors have declared that no conflict of interest exists in the submission of this manuscript, and the manuscript is approved by the authors for publication. We would like to declare on behalf of the authors that the work described was original research that has not been published previously, and not under consideration for publication elsewhere, in whole or in part. The authors listed have approved the manuscript that is enclosed.
Consent for publication
Not applicable.
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