Copyright © 2013 Xiaoqin Wang 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. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, USA, 1974.
2. J. T. Tanner, “The stability and the intrinsic growth rates of prey and predator populations,” Ecology, vol. 56, no. 4, pp. 855–867, 1975. View at Publisher · View at Google Scholar
3. J. D. Murray, Mathematical Biology, Springer, Berlin, Germany, 1989. View at Zentralblatt MATH · View at MathSciNet
4. S. B. Hsu and T. W. Hwang, “Hopf bifurcation analysis for a predator-prey system of Holling and Leslie type,” Taiwanese Journal of Mathematics, vol. 3, no. 1, pp. 35–53, 1999. View at Zentralblatt MATH · View at MathSciNet
5. R. Peng and M. Wang, “Positive steady states of the Holling-Tanner prey-predator model with diffusion,” Proceedings of the Royal Society of Edinburgh A, vol. 135, no. 1, pp. 149–164, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
6. R. Peng and M. Wang, “Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model,” Applied Mathematics Letters, vol. 20, no. 6, pp. 664–670, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
7. M. Wang, P. Y. H. Pang, and W. Chen, “Sharp spatial patterns of the diffusive Holling-Tanner prey-predator model in heterogeneous environment,” IMA Journal of Applied Mathematics, vol. 73, no. 5, pp. 815–835, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
8. H. B. Shi, W. T. Li, and G. Lin, “Positive steady states of a diffusive predator-prey system with modified Holling-Tanner functional response,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 3711–3721, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
9. X. Li, W. Jiang, and J. Shi, “Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model,” IMA Journal of Applied Mathematics. In press.
10. P. P. Liu and Y. Xue, “Spatiotemporal dynamics of a predator-prey model,” Nonlinear Dynamics, vol. 69, no. 1-2, pp. 71–77, 2012. View at Publisher · View at Google Scholar
11. W. C. Allee, Animal Aggregations: A Study in General Sociology, University of Chicago Press, 1931.
12. F. Courchamp, T. Clutton-Brock, and B. Grenfell, “Inverse density dependence and the Allee effect,” Trends in Ecology & Evolution, vol. 14, no. 10, pp. 405–410, 1999. View at Publisher · View at Google Scholar · View at Scopus
13. P. A. Stephens, W. J. Sutherland, and R. P. Freckleton, “What is the Allee effect?” Oikos, vol. 87, no. 1, pp. 185–190, 1999. View at Scopus
14. P. A. Stephens and W. J. Sutherland, “Consequences of the Allee effect for behaviour, ecology and conservation,” Trends in Ecology & Evolution, vol. 14, no. 10, pp. 401–405, 1999. View at Publisher · View at Google Scholar · View at Scopus
15. F. Courchamp, L. Berec, and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, 2008.
16. B. Dennis, “Allee effects: population growth, critical density, and the chance of extinction,” Natural Resource Modeling, vol. 3, no. 4, pp. 481–538, 1989. View at Zentralblatt MATH · View at MathSciNet
17. S. R. Zhou, Y. F. Liu, and G. Wang, “The stability of predator-prey systems subject to the Allee effects,” Theoretical Population Biology, vol. 67, no. 1, pp. 23–31, 2005. View at Publisher · View at Google Scholar
18. I. Scheuring, “Allee effect increases the dynamical stability of populations,” Journal of Theoretical Biology, vol. 199, no. 4, pp. 407–414, 1999. View at Publisher · View at Google Scholar · View at Scopus
19. M. H. Wang and M. Kot, “Speeds of invasion in a model with strong or weak Allee effects,” Mathematical Biosciences, vol. 171, no. 1, pp. 83–97, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
20. A. Kent, C. Patrick Doncaster, and T. Sluckin, “Consequences for predators of rescue and Allee effects on prey,” Ecological Modelling, vol. 162, no. 3, pp. 233–245, 2003. View at Publisher · View at Google Scholar · View at Scopus
21. J. C. Gascoigne and R. N. Lipcius, “Allee effects driven by predation,” Journal of Applied Ecology, vol. 41, no. 5, pp. 801–810, 2004. View at Publisher · View at Google Scholar · View at Scopus
22. A. Morozov, S. Petrovskii, and B. L. Li, “Bifurcations and chaos in a predator-prey system with the Allee effect,” Proceedings of the Royal Society B, vol. 271, no. 1546, pp. 1407–1414, 2004. View at Publisher · View at Google Scholar · View at Scopus
23. D. S. Boukal, M. W. Sabelis, and L. Berec, “How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses,” Theoretical Population Biology, vol. 72, no. 1, pp. 136–147, 2007. View at Publisher · View at Google Scholar · View at Scopus
24. Y. Du and J. Shi, “Allee effect and bistability in a spatially heterogeneous predator-prey model,” Transactions of the American Mathematical Society, vol. 359, no. 9, pp. 4557–4593, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
25. J. Wang, J. Shi, and J. Wei, “Predator-prey system with strong Allee effect in prey,” Journal of Mathematical Biology, vol. 62, no. 3, pp. 291–331, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
26. J. Wang, J. Shi, and J. Wei, “Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey,” Journal of Differential Equations, vol. 251, no. 4-5, pp. 1276–1304, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
27. E. González-Olivares, J. Mena-Lorca, A. Rojas-Palma, and J. D. Flores, “Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey,” Applied Mathematical Modelling, vol. 35, no. 1, pp. 366–381, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
28. G. Wang, X. G. Liang, and F. Z. Wang, “The competitive dynamics of populations subject to an Allee effect,” Ecological Modelling, vol. 124, no. 2-3, pp. 183–192, 1999. View at Publisher · View at Google Scholar · View at Scopus
29. J. D. Murray, “Discussion: Turing's theory of morphogenesis-its influence on modelling biological pattern and form,” Bulletin of Mathematical Biology, vol. 52, no. 1, pp. 117–152, 1990. View at Publisher · View at Google Scholar
30. J. Chattopadhyay and P. K. Tapaswi, “Effect of cross-diffusion on pattern formation—a nonlinear analysis,” Acta Applicandae Mathematicae, vol. 48, no. 1, pp. 1–12, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
31. W. Chen and M. Wang, “Qualitative analysis of predator-prey models with Beddington-DeAngelis functional response and diffusion,” Mathematical and Computer Modelling, vol. 42, no. 1-2, pp. 31–44, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
32. D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer, New York, NY, USA, 1981. View at MathSciNet
33. W. Wang, Q. Liu, and Z. Jin, “Spatiotemporal complexity of a ratio-dependent predator-prey system,” Physical Review E, vol. 75, no. 5, Article ID 051913, 9 pages, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
34. M. R. Garvie, “Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB,” Bulletin of Mathematical Biology, vol. 69, no. 3, pp. 931–956, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
35. A. Munteanu and R. Sole, “Pattern formation in noisy self-replicating spots,” International Journal of Bifurcation and Chaos, vol. 16, no. 12, pp. 3679–3683, 2007.