Abstract and Applied Analysis

Global Stability Analysis of a Nonautonomous Stage-Structured Competitive System with Toxic Effect and Double Maturation Delays

Chao Liu and Yuanke Li

Full-text: Open access

Abstract

We investigate a nonautonomous two-species competitive system with stage structure and double time delays due to maturation for two species, where toxic effect of toxin liberating species on nontoxic species is considered and the inhibiting effect is zero in absence of either species. Positivity and boundedness of solutions are analytically studied. By utilizing some comparison arguments, an iterative technique is proposed to discuss permanence of the species within competitive system. Furthermore, existence of positive periodic solutions is investigated based on continuation theorem of coincidence degree theory. By constructing an appropriate Lyapunov functional, sufficient conditions for global stability of the unique positive periodic solution are analyzed. Numerical simulations are carried out to show consistency with theoretical analysis.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 689573, 15 pages.

Dates
First available in Project Euclid: 27 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1425048718

Digital Object Identifier
doi:10.1155/2014/689573

Mathematical Reviews number (MathSciNet)
MR3273914

Zentralblatt MATH identifier
07022881

Citation

Liu, Chao; Li, Yuanke. Global Stability Analysis of a Nonautonomous Stage-Structured Competitive System with Toxic Effect and Double Maturation Delays. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 689573, 15 pages. doi:10.1155/2014/689573. https://projecteuclid.org/euclid.aaa/1425048718


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References

  • W. G. Aiello and H. I. Freedman, “A time-delay model of single-species growth with stage structure,” Mathematical Biosciences, vol. 101, no. 2, pp. 139–153, 1990.
  • G. Z. Zeng, L. S. Chen, L. H. Sun, and Y. Liu, “Permanence and the existence of the periodic solution of the non-autonomous two-species competitive model with stage structure,” Advances in Complex Systems, vol. 7, no. 3-4, pp. 385–393, 2004.
  • R. Xu, M. A. J. Chaplain, and F. A. Davidson, “Modelling and analysis of a competitive model with stage structure,” Mathematical and Computer Modelling, vol. 41, no. 2-3, pp. 159–175, 2005.
  • D. Xu and X. Zhao, “Dynamics in a periodic competitive model with stage structure,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 417–438, 2005.
  • S. Ahmad and A. C. Lazer, “Average growth and total permanence in a competitive Lotka-Volterra system,” Annali di Matematica Pura ed Applicata, vol. 185, pp. S47–S67, 2006.
  • Z. Hou, “On permanence of all subsystems of competitive Lotka-Volterra systems with delays,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 4285–4301, 2010.
  • Z. G. Lin, “Time delayed parabolic system in a two-species competitive model with stage structure,” Journal of Mathematical Analysis and Applications, vol. 315, no. 1, pp. 202–215, 2006.
  • F. Chen, “Almost periodic solution of the non-autonomous two-species competitive model with stage structure,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 685–693, 2006.
  • Z. Liu, M. Fan, and L. Chen, “Globally asymptotic stability in two periodic delayed competitive systems,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 271–287, 2008.
  • X. Xiong and Z. Zhang, “Periodic solutions of a discrete two-species competitive model with stage structure,” Mathematical and Computer Modelling, vol. 48, no. 3-4, pp. 333–343, 2008.
  • M. Kouche, N. Tatar, and S. Liu, “Permanence and existence of a positive periodic solution to a periodic stage-structured system with infinite delay,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 620–638, 2008.
  • B. D. Tian, Y. H. Qiu, and N. Chen, “Periodic and almost periodic solution for a non-autonomous epidemic predator-prey system with time-delay,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 779–790, 2009.
  • H. Hu, Z. Teng, and H. Jiang, “On the permanence in non-autonomous Lotka-Volterra competitive system with pure-delays and feedback controls,” Nonlinear Analysis: Real World Applications, vol. 10, no. 3, pp. 1803–1815, 2009.
  • F. Y. Wei, Y. R. Lin, L. L. Que, Y. Y. Chen, Y. P. Wu, and Y. F. Xue, “Periodic solution and global stability for a nonautonomous competitive Lotka-Volterra diffusion system,” Applied Mathematics and Computation, vol. 216, no. 10, pp. 3097–3104, 2010.
  • J. F. M. Al-Omari and S. K. Q. Al-Omari, “Global stability in a structured population competition model with distributed maturation delay and harvesting,” Nonlinear Analysis. Real World Applications, vol. 12, no. 3, pp. 1485–1499, 2011.
  • C. L. Shi, Z. Li, and F. D. Chen, “Extinction in a nonautonomous Lotka-Volterra competitive system with infinite delay and feedback controls,” Nonlinear Analysis: Real World Applications, vol. 13, no. 5, pp. 2214–2226, 2012.
  • Y. Li and Y. Ye, “Multiple positive almost periodic solutions to an impulsive non-autonomous Lotka-Volterra predator-prey system with harvesting terms,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 11, pp. 3190–3201, 2013.
  • J. D. Zhao, Z. C. Zhang, and J. Ju, “Necessary and sufficient conditions for permanence and extinction in a three dimensional competitive Lotka-Volterra system,” Applied Mathematics and Computation, vol. 230, pp. 587–596, 2014.
  • H. Zhang, Y. Li, B. Jing, and W. Zhao, “Global stability of almost periodic solution of multispecies mutualism system with time delays and impulsive effects,” Applied Mathematics and Computation, vol. 232, no. 1, pp. 1138–1150, 2014.
  • H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, USA, 2003.
  • G. P. Samanta, “A two-species competitive system under the influence of toxic substances,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 291–299, 2010.
  • J. Maynard Smith, Models in Ecology, Cambridge University Press, Cambridge, UK, 1974.
  • M. Bandyopadhyay, “Dynamical analysis of a allelopathic phytoplankton model,” Journal of Biological Systems, vol. 14, no. 2, pp. 205–217, 2006.
  • Z. E. Ma, G. R. Cui, and W. D. Wang, “Persistence and extinction of a population in a polluted environment,” Mathematical Biosciences, vol. 101, no. 1, pp. 75–97, 1990.
  • W. D. Wang and Z. E. Ma, “Permanence of populations in a polluted environment,” Mathematical Biosciences, vol. 122, no. 2, pp. 235–248, 1994.
  • J. Zhen and Z. E. Ma, “Periodic solutions for delay differential equations model of plankton allelopathy,” Computers & Mathematics with Applications, vol. 44, no. 3-4, pp. 491–500, 2002.
  • F. Wang and Z. Ma, “Persistence and periodic orbits for an SIS model in a polluted environment,” Computers & Mathematics with Applications, vol. 47, no. 4-5, pp. 779–792, 2004.
  • Z. Li and F. D. Chen, “Extinction in periodic competitive stage-structured Lotka-Volterra model with the effects of toxic substances,” Journal of Computational and Applied Mathematics, vol. 231, no. 1, pp. 143–153, 2009.
  • X. Y. Song and L. S. Chen, “Optimal harvesting and stability for a two-species competitive system with stage structure,” Mathematical Biosciences, vol. 170, no. 2, pp. 173–186, 2001.
  • W. Wang, G. Mulone, F. Salemi, and V. Salone, “Permanence and stability of a stage-structured predator-prey model,” Journal of Mathematical Analysis and Applications, vol. 262, no. 2, pp. 499–528, 2001.
  • R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer, Berlin, Germany, 1977. \endinput