Skip to main content
SpringerLink
Log in

Delay-driven instability and ecological control in a food-limited population networked system

  • Original paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The delay and network are incorporated to describe the spatiotemporal behavior of a food-limited population dynamical system. By using the standard approach of upper and lower solutions, we have shown the global existence and uniqueness of solutions to the system. By analyzing eigenvalue spectrum, we show that the delay can cause the long-term behavior of the system from stability to instability, that is, the positive equilibrium is asymptotically stable in the absence of delay, but loses its stability such that the Hopf bifurcation occurs when the time delay increases beyond a threshold. By the norm form and the center manifold theory, we study the stability and direction of the Hopf bifurcation. We propose some formulas to control the stability and period of the bifurcating periodic solutions. Moreover, numerical simulations reveal that the network structure can switch the type of spatiotemporal patterns.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Hutchinson, G.E.: Circular causal systems in ecology. Ann. N. Y. Acad. Sci. 50, 221–240 (1948)

    Article  Google Scholar 

  2. Smith, F.E.: Population dynamics in Daphnia magna. Ecology 44, 651–663 (1963)

    Article  Google Scholar 

  3. Gopalsamy, K., Kulenovic, M.R.S., Ladas, G.: Time lags in a food-limited population model. Appl. Anal. 31, 225–237 (1988)

    Article  MathSciNet  Google Scholar 

  4. Wan, A.Y., Wei, J.J.: Hopf bifurcation analysis of a foodlimited population model with delay. Nonlinear Anal. RWA 11, 1087–1095 (2010)

    Article  Google Scholar 

  5. Wu, J.: Theory and Applications of Partial Functional-differential Equations. Springer, New York (1996)

    Book  Google Scholar 

  6. Su, Y., Wei, J., Shi, J.P.: Hopf bifurcations in a reaction–diffusion population model with delay effect. J. Differ. Equ. 247, 1156–1184 (2009)

    Article  MathSciNet  Google Scholar 

  7. Gopalsamy, K., Weng, P.X.: Feedback regulation of logistic growth. Int. J. Math. Math. Sci. 16, 177–192 (1993)

    Article  MathSciNet  Google Scholar 

  8. Song, Y.L., Yuan, S.L.: Bifurcation analysis for a regulated logistic growth model. Appl. Math. Model. 31, 1729–1738 (2007)

    Article  Google Scholar 

  9. Li, Z., He, M.X.: Hopf bifurcation in a delayed food-limited model with feedback control. Nonlinear Dyn. 76, 1215–1224 (2014)

    Article  MathSciNet  Google Scholar 

  10. Gan, W.Z., Tian, C.R., Zhu, P.: Hopf bifurcation in a fractional diffusion food-limited models with feedback control. J. Math. Chem. 53, 1393–1411 (2015)

    Article  MathSciNet  Google Scholar 

  11. Aizerman, M.A., Gantmacher, F.R.: Absolute Stability of Regulator Systems. Holden Day, San Francisco (1964)

    Google Scholar 

  12. Gopalsamy, K., Kulenovic, M.R.S., Ladas, G.: Environmental periodicity and time delay in a food-limited population model. J. Math. Anal. Appl. 147, 545–555 (1990)

    Article  MathSciNet  Google Scholar 

  13. Gan, W., Zhou, P.: A revisit to the diffusive logistic model with free boundary condition. Discrete Cont. Dyn. B 21, 837–847 (2016)

    Article  MathSciNet  Google Scholar 

  14. Song, Y., Jiang, H., Liu, Q., Yuan, Y.: Spatiotemporal dynamics of the diffusive mussel-algae model near Turing-Hopf bifurcation. SIAM J. Appl. Dyn. Syst. 16, 2030–2062 (2017)

    Article  MathSciNet  Google Scholar 

  15. Yang, X., Li, X., Cao, J.: Robust finite-time stability of singular nonlinear systems with interval time-varying delay. J. Frank. I(355), 1241–1258 (2018)

    Article  MathSciNet  Google Scholar 

  16. Ma, J., Zhou, P., Ahmad, B., Ren, G., Wang, C.: Chaos and multi-scroll attractors in RCL-shunted junction coupled Jerk circuit connected by memristor. PloS ONE 13, e0191120 (2018)

    Article  Google Scholar 

  17. Liu, B., Wu, R., Chen, L.: Patterns induced by super cross-diffusion in a predator–prey system with Michaelis–Menten type harvesting. Math. Biosci. 298, 71–79 (2018)

    Article  MathSciNet  Google Scholar 

  18. Liu, B., Wu, R., Chen, L.: Turing–Hopf bifurcation analysis in a superdiffusive predator–prey model. Chaos 28, 113118 (2018)

    Article  MathSciNet  Google Scholar 

  19. Galiano, G., Velasco, J.: On a cross-diffusion system arising in image denoising. Comput. Math. Appl. 76, 984–996 (2018)

    Article  MathSciNet  Google Scholar 

  20. Zhang, J.: Spatial patterns of a fractional type cross-diffusion Holling–Tanner model. Comput. Math. Appl. 76, 957–965 (2018)

    Article  MathSciNet  Google Scholar 

  21. Zhang, X., Zhu, H.: Dynamics and pattern formation in homogeneous diffusive predator–prey systems with predator interference or foraging facilitation. Nonlinear Anal. RWA 48, 267–287 (2019)

    Article  MathSciNet  Google Scholar 

  22. Mukherjee, N., Ghorai, S., Banerjee, M.: Effects of density dependent cross-diffusion on the chaotic patterns in a ratio-dependent prey–predator model. Ecol. Complex. 36, 276–278 (2018)

    Article  Google Scholar 

  23. Banerjee, M., Ghorai, S., Mukherjee, N.: Study of cross-diffusion induced Turing patterns in a ratio-dependent prey–predator model via amplitude equations. Appl. Math. Model. 55, 383–399 (2018)

    Article  MathSciNet  Google Scholar 

  24. Liu, X., Zhang, T., Meng, X., Zhang, T.: Turing–Hopf bifurcations in a predator–prey model with herd behavior, quadratic mortality and prey-taxis. Physica A 496, 446–460 (2018)

    Article  MathSciNet  Google Scholar 

  25. Wang, Y., Cao, J., Li, M., Li, L.: Global behavior of a two-stage contact process on complex networks. J. Frank. I(356), 3571–3589 (2019)

    Article  MathSciNet  Google Scholar 

  26. Han, R., Dai, B.: Spatiotemporal pattern formation and selection induced by nonlinear cross-diffusion in a toxic-phytoplankton-zooplankton model with Allee effect. Nonlinear Anal. RWA 45, 822–853 (2019)

    Article  MathSciNet  Google Scholar 

  27. Zhang, X., Zhu, H.: Dynamics and pattern formation in homogeneous diffusive predator–prey systems with predator interference or foraging facilitation. Nonlinear Anal. RWA 48, 267–287 (2019)

    Article  MathSciNet  Google Scholar 

  28. Smith-Roberge, J., Iron, D., Kolokolnikov, T.: Pattern formation in bacterial colonies with density-dependent diffusion. Euro. J. Appl. Math. 30, 196–218 (2019)

    Article  MathSciNet  Google Scholar 

  29. Chen, H., Zou, L.: How to control the immigration of infectious individuals for a region? Nonlinear Anal. RWA 45, 491–505 (2019)

    Article  MathSciNet  Google Scholar 

  30. Lou, y, Zhao, X.Q., Zhou, P.: Global dynamics of a Lotka–Volterra competition–diffusion–advection system in heterogeneous environments. J. Math. Pure. Appl. 121, 47–82 (2019)

    Article  MathSciNet  Google Scholar 

  31. Liao, K., Lou, Y.: The effect of time delay in a two-patch model with random dispersal. Bull. Math. Biol. 76, 335–376 (2014)

    Article  MathSciNet  Google Scholar 

  32. Gourley, S.A., Ruan, S.: A delay equation model for oviposition habitat selection by mosquitoes. J. Math. Biol. 65, 1125–1148 (2012)

    Article  MathSciNet  Google Scholar 

  33. Hassard, B., kazarino, D., Wan, Y.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981)

    Google Scholar 

  34. Smith, H.L.: An Introduction to Delay Differential Equations with Sciences Applications to the Life. Springer, New York (2010)

    Google Scholar 

  35. Petit, J., Asllani, M., Fanelli, D., Lauwens, B., Carletti, T.: Pattern formation in a two- component reaction–diffusion system with delayed processes on a network. Physica A 462, 230–249 (2016)

    Article  MathSciNet  Google Scholar 

  36. Ruan, S.: Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator–prey systems with discrete delays. Quart. Appl. Math. 59, 159–173 (2001)

    Article  MathSciNet  Google Scholar 

  37. Freedman, H.I., Rao, V.S.H.: The trade-off between mutual interference and time lags in predator–prey system. Bull. Math. Biol. 45, 991–1004 (1983)

    Article  MathSciNet  Google Scholar 

  38. Smith, H.L.: An Introduction to Delay Differential Equations with Sciences Applications to the Life. Springer, New York (2010)

    Google Scholar 

Download references

Acknowledgements

Wenzhen Gan is supported by the National Natural Science Foundation of China under Grants 11801229. Zuhan liu is supported by the National Natural Science Foundation of China under Grants 11771380. Canrong Tian is supported by the National Natural Science Foundation of China under Grants 61877052, Jiangsu Province 333 Talent Project, and Jiangsu Province Qinglan Project.

Author information

Authors and Affiliations

  1. School of Mathematics and Physics, Jiangsu University of Technology, Changzhou, 213001, Jiangsu, China

    Wenzhen Gan & Peng Zhu

  2. School of Mathematics and Physics, Yancheng Institute of Technology, Yancheng, 224003, Jiangsu, China

    Zuhan Liu & Canrong Tian

Authors
  1. Wenzhen Gan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peng Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zuhan Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Canrong Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Canrong Tian.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gan, W., Zhu, P., Liu, Z. et al. Delay-driven instability and ecological control in a food-limited population networked system. Nonlinear Dyn 100, 4031–4044 (2020). https://doi.org/10.1007/s11071-020-05729-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-020-05729-w

Keywords

Search

Navigation