Journal of Dynamics and Differential Equations

, Volume 31, Issue 1, pp 435–450

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

• Alfonso Ruiz-Herrera
Article

## Abstract

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.

## Keywords

Global attraction Iterative method Transcritical bifurcation Invasive plants

## Notes

### Funding

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

## References

1. 1.
Lewis, M.A., Petrovskii, S.V., Potts, J.R.: The Mathematics Behind Biological Invasions, vol. 44. Springer, Switzerland (2016)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
18. 18.
Singer, D.: Stable orbits and bifurcation of maps of the interval. SIAM J. Appl. Math. 35, 260–267 (1978)
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)