Table of Contents
ISRN Biomathematics
Volume 2013, Article ID 424062, 12 pages
http://dx.doi.org/10.1155/2013/424062
Research Article

Finite Time Blowup in a Realistic Food-Chain Model

1Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY 13699, USA
2King Abdullah University of Science and Technology, Applied Mathematics and Computational Science, Thuwal 23955-6900, Saudi Arabia
3Department of Applied Mathematics, Indian School of Mines, Dhanbad, Jharkhand 826004, India
4School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh 175 001, India

Received 23 April 2013; Accepted 19 May 2013

Academic Editors: L. Pezard and A. A. Polezhaev

Copyright © 2013 Rana D. Parshad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. K. Upadhyay, “Observability of chaos and cycles in ecological systems: lessons from predator-prey models,” International Journal of Bifurcation and Chaos, vol. 19, no. 10, article 3169, 65 pages, 2009. View at Publisher · View at Google Scholar · View at Scopus
  2. J. D. Murray, Mathematical Biology, Springer, New York, NY, USA, 1993.
  3. A. Hastings and T. Powell, “Chaos in a three-species food chain,” Ecology, vol. 72, no. 3, pp. 896–903, 1991. View at Google Scholar · View at Scopus
  4. R. K. Upadhyay and V. Rai, “Why chaos is rarely observed in natural populations,” Chaos, Solitons and Fractals, vol. 8, no. 12, pp. 1933–1939, 1997. View at Google Scholar · View at Scopus
  5. C. Letellier and M. A. Aziz-Alaoui, “Analysis of the dynamics of a realistic ecological model,” Chaos, Solitons and Fractals, vol. 13, no. 1, pp. 95–107, 2002. View at Publisher · View at Google Scholar · View at Scopus
  6. M. A. Aziz-Alaoui, “Study of a Leslie-Gower-type tritrophic population model,” Chaos, Solitons and Fractals, vol. 14, no. 8, pp. 1275–1293, 2002. View at Publisher · View at Google Scholar · View at Scopus
  7. R. K. Upadhyay, “Chaotic dynamics in a three species aquatic population model with Holling type II functional response,” Nonlinear Analysis. Modelling and Control, vol. 13, no. 1, pp. 103–115, 2008. View at Google Scholar
  8. R. K. Upadhyay, S. R. K. Iyengar, and V. Rai, “Chaos: an ecological reality?” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 8, no. 6, pp. 1325–1333, 1998. View at Google Scholar · View at Scopus
  9. R. D. Parshad and R. K. Upadhyay, “Investigation of the long time dynamics of a diffusive three species aquatic model,” Dynamics of Partial Differential Equations, vol. 7, no. 3, pp. 217–244, 2010. View at Google Scholar · View at Scopus
  10. N. Kumari, “Pattern formation in spatially extended tritrophic food chain model systems: generalist versus specialist top predator,” ISRN Biomathematics, vol. 2013, Article ID 198185, 12 pages, 2013. View at Publisher · View at Google Scholar
  11. N. Mizoguchi, H. Ninomiya, and E. Yanagida, “Diffusion-induced blowup in a nonlinear parabolic system,” Journal of Dynamics and Differential Equations, vol. 10, no. 4, pp. 619–638, 1998. View at Google Scholar · View at Scopus
  12. J. Morgan, “On a question of blow-up for semilinear parabolic systems,” Differential and Integral Equations, vol. 3, pp. 973–978, 1990. View at Google Scholar
  13. H. F. Weinberger, “An example of blowup produced by equal diffusions,” Journal of Differential Equations, vol. 154, no. 1, pp. 225–237, 1999. View at Google Scholar · View at Scopus
  14. H. A. Levine, “Role of critical exponents in blowup theorems,” SIAM Review, vol. 32, no. 2, pp. 262–288, 1990. View at Google Scholar · View at Scopus
  15. P. Souplet, “A note on Diffusion-induced blow up,” Journal of Dynamics and Differential Equations, vol. 19, no. 3, pp. 819–823, 2007. View at Google Scholar
  16. R. Ferreira, A. De Pablo, F. Quirós, and J. D. Rossi, “The blow-up profile for a fast diffusion equation with a nonlinear boundary condition,” Rocky Mountain Journal of Mathematics, vol. 33, no. 1, pp. 123–146, 2003. View at Google Scholar · View at Scopus
  17. M. Chlebik and M. Fila, “On the blow-up rate for the heat equation with a nonlinear boundary condition,” Mathematical Methods in the Applied Sciences, vol. 23, pp. 1323–1330, 2000. View at Google Scholar
  18. Z. Lin, “Blowup estimates for a mutualistic model in ecology,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 8, no. 1, pp. 1–14, 2002. View at Google Scholar · View at Scopus
  19. G. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, New York, NY, USA, 1974.
  20. T. Hillen and K. J. Painter, “A user's guide to PDE models for chemotaxis,” Journal of Mathematical Biology, vol. 58, no. 1-2, pp. 183–217, 2009. View at Google Scholar · View at Scopus
  21. R. D. Parshad, A. Kasimov, and H. Abderrahmane, “Long time behavior and the Turing instability in a diffusive three species food chain model,” Under Review.
  22. J. R. Beddington, “Mutual interference between parasites on predators and its effects on searching efficiency,” Journal of Animal Ecology, vol. 44, pp. 331–340, 1975. View at Google Scholar
  23. L. H. Erbe and H. I. Freedman, “Modeling persistence and mutual interference among subpopulations of ecological communities,” Bulletin of Mathematical Biology, vol. 47, no. 2, pp. 295–304, 1985. View at Publisher · View at Google Scholar · View at Scopus
  24. L. H. Erbe, H. I. Freedman, and V. S. H. Rao, “Three-species food-chain models with mutual interference and time delays,” Mathematical Biosciences, vol. 80, no. 1, pp. 57–80, 1986. View at Google Scholar · View at Scopus
  25. H. I. Freedman, “Stability analysis of a predator-prey system with mutual interference and density-dependent death rates,” Bulletin of Mathematical Biology, vol. 41, no. 1, pp. 67–78, 1979. View at Publisher · View at Google Scholar · View at Scopus
  26. H. I. Freedman and V. Sree Hari Rao, “The trade-off between mutual interference and time lags in predator-prey systems,” Bulletin of Mathematical Biology, vol. 45, no. 6, pp. 991–1004, 1983. View at Publisher · View at Google Scholar · View at Scopus
  27. M. P. Hassell, “Mutual interference between searching insect parasites,” Journal of Animal Ecology, vol. 40, pp. 473–486, 1971. View at Google Scholar
  28. C. S. Holling, “The functional role of invertebrate predators to prey density,” Memoirs of the Entomological Society of Canada, vol. 45, pp. 3–60.
  29. R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, USA, 2001.
  30. L. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, Springer, New York, NY, USA, 2nd edition, 1998.
  31. C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, New York, NY, USA, 1993.
  32. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, NY, USA, 2nd edition, 1997.
  33. D. Henry, Geometric Theory of Semi-Linear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer, New-York, NY, USA, 1984.
  34. F. Takens and R. Mane, “Dynamical systems and turbulence,” in Warwick, R. Rand and L. S. Young, Eds., vol. 898 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1981. View at Google Scholar
  35. R. K. Upadhyay, S. R. K. Iyengar, and V. Rai, “Stability and complexity in ecological systems,” Chaos, Solitons and Fractals, vol. 11, no. 4, pp. 533–542, 2000. View at Publisher · View at Google Scholar · View at Scopus
  36. P. Grassberger and I. Procaccia, “Dimensions and entropies of strange attractors from a fluctuating dynamics approach,” Physica D, vol. 13, no. 1-2, pp. 34–54, 1984. View at Google Scholar · View at Scopus
  37. D. Lai and G. Chen, “Statistical analysis of lyapunov exponents from time series: a jacobian approach,” Mathematical and Computer Modelling, vol. 27, no. 7, pp. 1–9, 1998. View at Publisher · View at Google Scholar · View at Scopus
  38. V. Rai and R. K. Upadhyay, “Chaotic population dynamics and biology of the top-predator,” Chaos, Solitons and Fractals, vol. 21, no. 5, pp. 1195–1204, 2004. View at Publisher · View at Google Scholar · View at Scopus