Attraction to Equilibria in Stage-Structured Predator Prey Models and Bio-Control Problems

  • Alfonso Ruiz-HerreraEmail author


Controlling invasive species has become an important ecological issue over the last decades. A popular management strategy consists of releasing natural enemies, generally predators. From a mathematical point of view, the study of any realistic problem in bio-control normally involves models remarkably resistant to the analysis. In this paper, we propose a new iterative method for studying the dynamical behaviour of a predator-prey model in which an invasive plant is subject to predation of an insect population. We show that the dynamics of the model depends on a suitable scalar function that determines the existence of equilibria.


Global attraction Iterative method Transcritical bifurcation Invasive plants 



Funding was provided by Spanish Goverment (Grant No. MTM2014-56953-P).


  1. 1.
    Lewis, M.A., Petrovskii, S.V., Potts, J.R.: The Mathematics Behind Biological Invasions, vol. 44. Springer, Switzerland (2016)zbMATHGoogle Scholar
  2. 2.
    Parshad, R.D., Quansah, E., Black, K., Beauregard, M.: Biological control via ecological damping: an approach that attenuates non-target effects. Math. Biosci. 273, 23–44 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Smith, J.M.D., Ward, J.P., Child, L.E., Owen, M.R.: A simulation model of rhizome networks for Fallopia Japonica (Japanese knotweed) in the United Kingdom. Ecol. Model. 200, 421–432 (2007)CrossRefGoogle Scholar
  4. 4.
    Gourley, S.A., Li, J., Zou, X.: A mathematical model for biocontrol of the invasive weed Fallopia japonica. Bull. Math. Biol. 78, 1678–1702 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Williams, F.E., et al.: The economic cost of invasive non-native species on great britain. CABI (2010)Google Scholar
  6. 6.
    Gourley, S.A., Liu, R., Wu, J.: Slowing the evolution of insecticide resistance in mosquitoes: a mathematical model. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 467, 2127–2148 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Liu, R., Gourley, S.A.: A model for the biocontrol of mosquitoes using predator fish. Discrete Contin. Dyn. Syst. Ser. B 19, 3283–3298 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gourley, S.A., Lou, Y.: A mathematical model for the spatial spread and biocontrol of the Asian longhorned beetle. SIAM J. Appl. Math. 74, 864–884 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gourley, S.A., Zou, X.: A mathematical model for the control and eradication of a wood boring beetle infestation. SIAM Rev. 53, 321–345 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Shaw, R.H., Bryner, S., Tanner, R.: The life history and host range of the Japanese knotweed psyllid, Aphalara itadori Shinji: potentially the first classical biological weed control agent for the European Union. Biol. Control 49, 105–113 (2009)CrossRefGoogle Scholar
  11. 11.
    Liz, E., Ruiz-Herrera, A.: Attractivity, multistability, and bifurcation in delayed Hopfields model with non-monotonic feedback. J. Differ. Equ. 255, 4244–4266 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Smith, H.L.: Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, vol. 41. American Mathematical Society (2008)Google Scholar
  13. 13.
    Enatsu, Y., Nakata, Y., Muroya, Y.: Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model. Nonlinear Anal. Real World Appl. 13, 2120–2133 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Liz, E., Ruiz-Herrera, A.: Global dynamics of delay equations for populations with competition among immature individuals. J. Differ. Equ. 260, 5926–5955 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    El-Morshedy, H.A., Ruiz-Herrera, A.: Geometric methods of global attraction in systems of delay differential equations. J. Differ. Equ. 263, 5968–5986 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Beverton, R.J., Holt, S.J.: On the Dynamics of Exploited Fish Populations, vol. 11. Springer, Berlin (2012)Google Scholar
  17. 17.
    El-Morshedy, H.A., Röst, G., Ruiz-Herrera, A.: Global dynamics of delay recruitment models with maximized lifespan. ZAMP 67, 1–15 (2016)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Singer, D.: Stable orbits and bifurcation of maps of the interval. SIAM J. Appl. Math. 35, 260–267 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Faria, T.: Stability and bifurcation for a delayed predatorprey model and the effect of diffusion. J. Math. Anal. Appl. 254, 433–463 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de OviedoOviedoSpain

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