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.
References
 1.Hale, J.: Theory of Functional Differential Equations, 2nd edn. Applied Mathematical Sciences, vol. 3 Springer, Berlin (1977) CrossRefzbMATHGoogle Scholar
 2.Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Mathematics in Science and Engineering, vol. 191. Academic Press, Boston (1993) zbMATHGoogle Scholar
 3.Rezounenko, A.V., Wu, J.: A nonlocal PDE model for population dynamics with stateselective delay: local theory and global attractors. J. Comput. Appl. Math. 190(1–2), 99–113 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
 4.Cooke, K., Van den Driessche, P., Zou, X.: Interaction of maturation delay and nonlinear birth in population and epidemic models. J. Math. Biol. 39(4), 332–352 (1999) MathSciNetCrossRefzbMATHGoogle Scholar
 5.Hill, D.C., Shafer, D.S.: Asymptotics and stability of the delayed Duffing equation. J. Differ. Equ. 265(1), 33–68 (2018) MathSciNetCrossRefzbMATHGoogle Scholar
 6.Liu, X., Zhang, T.: Bogdanov–Takens and triple zero bifurcations of coupled Van der Pol–Duffing oscillators with multiple delays. Int. J. Bifurc. Chaos 27(9), Article ID 1750133 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
 7.AlvarezVázquez, L.J., Fernández, F.J., MuñozSola, R.: Analysis of a multistate control problem related to food technology. J. Differ. Equ. 245(1), 130–153 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
 8.Antman, S., Marsden, J., Sirovich, L.: Surveys and Tutorials in the Applied Mathematical Sciences. Springer, Berlin (2007) Google Scholar
 9.Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Texts in Applied Mathematics, vol. 57. Springer, New York (2011) zbMATHGoogle Scholar
 10.Zhang, W., Li, J., Chen, M.: Global exponential stability and existence of periodic solutions for delayed reaction–diffusion BAM neural networks with Dirichlet boundary conditions. Bound. Value Probl. 2013(1), Article ID 105 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
 11.Song, Y., Zhang, T., Tadé, M.O.: Stability switches, Hopf bifurcations, and spatiotemporal patterns in a delayed neural model with bidirectional coupling. J. Nonlinear Sci. 19(6), Article ID 597 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
 12.Song, Y., Tade, M.O., Zhang, T.: Bifurcation analysis and spatiotemporal patterns of nonlinear oscillations in a delayed neural network with unidirectional coupling. Nonlinearity 22(5), 975 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
 13.Bianca, C., Guerrini, L., Riposo, J.: A delayed mathematical model for the acute in ammatory response to infection. Appl. Math. Inf. Sci. 9(6), 2775–2782 (2015) Google Scholar
 14.Bianca, C., Guerrini, L.: Existence of limit cycles in the Solow model with delayedlogistic population growth. Sci. World J. 2014, Article ID 207806 (2014) CrossRefGoogle Scholar
 15.Cai, Y., Zhang, C.: Hopf–Pitchfork bifurcation of coupled Van der Pol oscillator with delay. Nonlinear Anal., Model. Control 22(5), 598–613 (2017) MathSciNetCrossRefGoogle Scholar
 16.Ozturk, O., Akin, E.: On nonoscillatory solutions of two dimensional nonlinear delay dynamical systems. Opusc. Math. 36(5), 651–669 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
 17.Bianca, C., Pennisi, M., Motta, S., Ragusa, M.A.: Immune system network and cancer vaccine. In: Numerical Analysis and Applied Mathematics. ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics. AIP Conference Proceedings, vol. 1389, pp. 945–948. AIP, New York (2011) Google Scholar
 18.Bianca, C., Pappalardo, F., Pennisi, M., Ragusa, M.: Persistence analysis in a Kolmogorovtype model for cancerimmune system competition. In: 11th International Conference of Numerical Analysis and Applied Mathematics 2013: ICNAAM 2013. AIP Conference Proceedings, vol. 1558, pp. 1797–1800. AIP, New York (2013) Google Scholar
 19.Liu, W., Jiang, Y.: Bifurcation of a delayed Gause predator–prey model with Michaelis–Menten type harvesting. J. Theor. Biol. 438, 116–132 (2018) MathSciNetCrossRefzbMATHGoogle Scholar
 20.Huo, H.F., Li, W.T.: Periodic solution of a delayed predator–prey system with Michaelis–Menten type functional response. J. Comput. Appl. Math. 166(2), 453–463 (2004). https://doi.org/10.1016/j.cam.2003.08.042 MathSciNetCrossRefzbMATHGoogle Scholar
 21.Bairagi, N., Jana, D.: On the stability and Hopf bifurcation of a delayinduced predator–prey system with habitat complexity. Appl. Math. Model. 35(7), 3255–3267 (2011). https://doi.org/10.1016/j.apm.2011.01.025 MathSciNetCrossRefzbMATHGoogle Scholar
 22.Song, Y., Wei, J.: Local Hopf bifurcation and global periodic solutions in a delayed predator–prey system. J. Math. Anal. Appl. 301(1), 1–21 (2005). https://doi.org/10.1016/j.jmaa.2004.06.056 MathSciNetCrossRefzbMATHGoogle Scholar
 23.Liao, M., Xu, C., Tang, X.: Stability and Hopf bifurcation for a competition and cooperation model of two enterprises with delay. Commun. Nonlinear Sci. Numer. Simul. 19(10), 3845–3856 (2014). https://doi.org/10.1016/j.cnsns.2014.02.031 MathSciNetCrossRefGoogle Scholar
 24.Li, L., Zhang, C.H., Yan, X.P.: Stability and Hopf bifurcation analysis for a twoenterprise interaction model with delays. Commun. Nonlinear Sci. Numer. Simul. 30(1–3), 70–83 (2016) MathSciNetCrossRefGoogle Scholar
 25.Du, Z., Xu, D.: Traveling wave solution for a reaction–diffusion competitive–cooperative system with delays. Bound. Value Probl. 2016(1), Article ID 46 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
 26.Xu, C.: Periodic behavior of competition and corporation dynamical model of two enterprises on time scales. J. Quant. Econ. 29(2), 1–4 (2012) MathSciNetGoogle Scholar
 27.Liao, M., Xu, C., Tang, X.: Dynamical behaviors for a competition and cooperation model of enterprises with two delays. Nonlinear Dyn. 75(1–2), 257–266 (2014). https://doi.org/10.1007/s1107101310639 MathSciNetCrossRefzbMATHGoogle Scholar
 28.Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. London Mathematical Society Lecture Note Series, vol. 41. Cambridge University Press, Cambridge (1981) zbMATHGoogle Scholar
 29.Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2. Springer, New York (2003) zbMATHGoogle Scholar
 30.Hu, G.P., Li, W.T., Yan, X.P.: Hopf bifurcations in a predator–prey system with multiple delays. Chaos Solitons Fractals 42(2), 1273–1285 (2009). https://doi.org/10.1016/j.chaos.2009.03.075 MathSciNetCrossRefzbMATHGoogle Scholar
 31.Shampine, L.F., Thompson, S., Kierzenka, J.: Solving delay differential equations with dde23 (2000). http://www.runet.edu/~thompson/webddes/tutorial.pdf
 32.Song, Y., Zhang, T., Peng, Y.: Turing–Hopf bifurcation in the reaction–diffusion equations and its applications. Commun. Nonlinear Sci. Numer. Simul. 33, 229–258 (2016) MathSciNetCrossRefGoogle Scholar
 33.Cao, X., Song, Y., Zhang, T.: Hopf bifurcation and delayinduced Turing instability in a diffusive lac operon model. Int. J. Bifurc. Chaos 26(10), Article ID 1650167 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
 34.Colli, P., Gilardi, G., Sprekels, J.: A boundary control problem for the pure Cahn–Hilliard equation with dynamic boundary conditions. Adv. Nonlinear Anal. 4(4), 311–325 (2015) MathSciNetzbMATHGoogle Scholar
 35.Ghergu, M., Radulescu, V.: Singular Elliptic Problems: Bifurcation and Asymptotic Analysis. Oxford Lecture Series in Mathematics and Its Applications, vol. 37. Clarendon, Oxford (2008) zbMATHGoogle Scholar
 36.Wang, G.Q., Cheng, S.S.: Bifurcation in a nonlinear steady state system. Opusc. Math. 30, 349–360 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
 37.Ghergu, M., Radulescu, V.: Nonlinear PDEs: Mathematical Models in Biology, Chemistry and Population Genetics. Springer, Berlin (2011) zbMATHGoogle Scholar
 38.Squassina, M., Watanabe, T.: Uniqueness of limit flow for a class of quasilinear parabolic equations. Adv. Nonlinear Anal. 6(2), 243–276 (2017) MathSciNetzbMATHGoogle Scholar
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