Abstract and Applied Analysis

Stability Analysis of a Population Model with Maturation Delay and Ricker Birth Function

Chongwu Zheng, Fengqin Zhang, and Jianquan Li

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A single species population model is investigated, where the discrete maturation delay and the Ricker birth function are incorporated. The threshold determining the global stability of the trivial equilibrium and the existence of the positive equilibrium is obtained. The necessary and sufficient conditions ensuring the local asymptotical stability of the positive equilibrium are given by applying the Pontryagin's method. The effect of all the parameter values on the local stability of the positive equilibrium is analyzed. The obtained results show the existence of stability switch and provide a method of computing maturation times at which the stability switch occurs. Numerical simulations illustrate that chaos may occur for the model, and the associated parameter bifurcation diagrams are given for certain values of the parameters.

Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 136707, 8 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412276917

Digital Object Identifier
doi:10.1155/2014/136707

Mathematical Reviews number (MathSciNet)
MR3198146

Zentralblatt MATH identifier
07021782

Citation

Zheng, Chongwu; Zhang, Fengqin; Li, Jianquan. Stability Analysis of a Population Model with Maturation Delay and Ricker Birth Function. Abstr. Appl. Anal. 2014 (2014), Article ID 136707, 8 pages. doi:10.1155/2014/136707. https://projecteuclid.org/euclid.aaa/1412276917


Export citation

References

  • K. Cooke, P. van den Driessche, and X. Zou, “Interaction of maturation delay and nonlinear birth in population and epidemic models,” Journal of Mathematical Biology, vol. 39, no. 4, pp. 332–352, 1999.
  • X.-Q. Zhao and X. Zou, “Threshold dynamics in a delayed SIS epidemic model,” Journal of Mathematical Analysis and Applications, vol. 257, no. 2, pp. 282–291, 2001.
  • G. Fan, J. Liu, P. van den Driessche, J. Wu, and H. Zhu, “The impact of maturation delay of mosquitoes on the transmission of West Nile virus,” Mathematical Biosciences, vol. 228, no. 2, pp. 119–126, 2010.
  • K. L. Cooke, R. H. Elderkin, and W. Huang, “Predator-prey interactions with delays due to juvenile maturation,” SIAM Journal on Applied Mathematics, vol. 66, no. 3, pp. 1050–1079, 2006.
  • S. Ruan, “Delay differential equations in single species dynamics,” in Delay Differential Equations and Applications, vol. 205 of NATO Sci. Ser. II Math. Phys. Chem., pp. 477–517, Springer, Dordrecht, The Netherlands, 2006.
  • Z. Jiang and W. Zhang, “Bifurcation analysis in single-species population model with delay,” Science China. Mathematics, vol. 53, no. 6, pp. 1475–1481, 2010.
  • E. Beretta and Y. Kuang, “Geometric stability switch criteria in delay differential systems with delay dependent parameters,” SIAM Journal on Mathematical Analysis, vol. 33, no. 5, pp. 1144–1165, 2002.
  • J. Wei and X. Zou, “Bifurcation analysis of a population model and the resulting SIS epidemic model with delay,” Journal of Computational and Applied Mathematics, vol. 197, no. 1, pp. 169–187, 2006.
  • J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, Pa, USA, 1976.
  • J. Hale, Theory of Functional Differential Equations, vol. 3 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1993. \endinput