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

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.

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References

  1. 1.

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

    Article  Google Scholar 

  2. 2.

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

    Article  Google Scholar 

  3. 3.

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

    MathSciNet  Article  Google Scholar 

  4. 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. 5.

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

    Google Scholar 

  6. 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)

    MathSciNet  Article  Google Scholar 

  7. 7.

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

    MathSciNet  Article  Google Scholar 

  8. 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. 9.

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

    MathSciNet  Article  Google Scholar 

  10. 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)

    MathSciNet  Article  Google Scholar 

  11. 11.

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

    Google Scholar 

  12. 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)

    MathSciNet  Article  Google Scholar 

  13. 13.

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

    MathSciNet  Article  Google Scholar 

  14. 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)

    MathSciNet  Article  Google Scholar 

  15. 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)

    MathSciNet  Article  Google Scholar 

  16. 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. 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)

    MathSciNet  Article  Google Scholar 

  18. 18.

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

    MathSciNet  Article  Google Scholar 

  19. 19.

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

    MathSciNet  Article  Google Scholar 

  20. 20.

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

    MathSciNet  Article  Google Scholar 

  21. 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)

    MathSciNet  Article  Google Scholar 

  22. 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. 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)

    MathSciNet  Article  Google Scholar 

  24. 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)

    MathSciNet  Article  Google Scholar 

  25. 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)

    MathSciNet  Article  Google Scholar 

  26. 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)

    MathSciNet  Article  Google Scholar 

  27. 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)

    MathSciNet  Article  Google Scholar 

  28. 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)

    MathSciNet  Article  Google Scholar 

  29. 29.

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

    MathSciNet  Article  Google Scholar 

  30. 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)

    MathSciNet  Article  Google Scholar 

  31. 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)

    MathSciNet  Article  Google Scholar 

  32. 32.

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

    MathSciNet  Article  Google Scholar 

  33. 33.

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

    Google Scholar 

  34. 34.

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

    Google Scholar 

  35. 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)

    MathSciNet  Article  Google Scholar 

  36. 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)

    MathSciNet  Article  Google Scholar 

  37. 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)

    MathSciNet  Article  Google Scholar 

  38. 38.

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

    Google Scholar 

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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.

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Correspondence to Canrong Tian.

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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

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Keywords

  • Delay-driven instability
  • Ecological control
  • Networked system
  • Hopf bifurcation
  • Spatiotemporal pattern