Abstract and Applied Analysis

Pattern Formation in a Predator-Prey Model with Both Cross Diffusion and Time Delay

Boli Xie, Zhijun Wang, and Yakui Xue

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Abstract

A predator-prey model with both cross diffusion and time delay is considered. We give the conditions for emerging Turing instability in detail. Furthermore, we illustrate the spatial patterns via numerical simulations, which show that the model dynamics exhibits a delay and diffusion controlled formation growth not only of spots and stripe-like patterns, but also of the two coexist. The obtained results show that this system has rich dynamics; these patterns show that it is useful for the diffusive predation model with a delay effect to reveal the spatial dynamics in the real model.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 392435, 7 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412279773

Digital Object Identifier
doi:10.1155/2014/392435

Mathematical Reviews number (MathSciNet)
MR3212421

Zentralblatt MATH identifier
07022296

Citation

Xie, Boli; Wang, Zhijun; Xue, Yakui. Pattern Formation in a Predator-Prey Model with Both Cross Diffusion and Time Delay. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 392435, 7 pages. doi:10.1155/2014/392435. https://projecteuclid.org/euclid.aaa/1412279773


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References

  • J. T. Tanner, “The stability and the intrinsic growth rates of prey and predator populations,” Ecology, vol. 56, pp. 855–867, 1975.
  • D. J. Wollkind, J. B. Collings, and J. A. Logan, “Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees,” Bulletin of Mathematical Biology, vol. 50, no. 4, pp. 379–409, 1988.
  • E. Saez and E. Gonzalez-Olivares, “Dynamics of a predatorprey model,” SIAM Journal on Applied Mathematics, vol. 59, pp. 1867–1878, 1999.
  • S. B. Hsu and T. W. Huang, “Global stability for a class of predator-prey systems,” SIAM Journal on Applied Mathematics, vol. 55, no. 3, pp. 763–783, 1995.
  • A. F. Nindjin, M. A. Aziz-Alaoui, and M. Cadivel, “Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay,” Nonlinear Analysis. Real World Applications, vol. 7, no. 5, pp. 1104–1118, 2006.
  • R. Xu and L. Chen, “Persistence and stability for a two-spe-cies ratio-dependent predator-prey system with time delay in a two-patch environment,” Computers & Mathematics with Applications, vol. 40, no. 4-5, pp. 577–588, 2000.
  • R. Yafia, F. El Adnani, and H. T. Alaoui, “Limit cycle and num-erical similations for small and large delays in a predator-prey model with modified Leslie-Gower and Holling-type II schemes,” Nonlinear Analysis. Real World Applications, vol. 9, no. 5, pp. 2055–2067, 2008.
  • S. Ruan, “On nonlinear dynamics of predator-prey models with discrete delay,” Mathematical Modelling of Natural Phenomena, vol. 4, no. 2, pp. 140–188, 2009.
  • S. Sen, P. Ghosh, S. Riaz, and D. Ray, “Time-delay-induced inst-abilities in reactiondiffusion systems,” Physical Review E, vol. 80, Article ID 046212, 2009.
  • P. Ghosh, “Control of the Hopf-Turing transition by time-delayed global feedback in a reaction-diffusion system,” Physical Review E, vol. 84, no. 1, Article ID 016222, 2011. \endinput