Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey

Journal of Differential Equations

Volumn 251, Issue 4-5, 2011, Pages 1276-1304

  Wang, Jinfeng ^a,b   Shi, Junping ^c   Wei, Junjie ^b  

^a HARBIN NORMAL UNIVERSITY   (China)

^b HARBIN INSTITUTE OF TECHNOLOGY   (China)

^c Dept of Economics and Thomas Jefferson Program in Public Policy at the College of William and Mary   (United States)

Author keywords

Bifurcation; Predator prey; Reaction diffusion system; Spatiotemporal patterns; Strong Allee effect

Indexed keywords

EID: 79958190219     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2011.03.004     Document Type: Article

References (60)

1. Alikakos N.D. An application of the invariance principle to reaction-diffusion equations. J. Differential Equations 1979, 33:201-225.
2. Allee W.C. Animal Aggregations: A Study in General Sociology 1931, University of Chicago Press.
3. Boukal S.D., Sabelis W.M., Berec L. How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses. Theoret. Popul. Biol. 2007, 72:136-147.
4. Cantrell R.S., Cosner C., Hutson V. Permanence in ecological systems with spatial heterogeneity. Proc. Roy. Soc. Edinburgh Sect. A 1993, 123:533-559.
5. Cantrell R.S., Cosner C. Spatial Ecology via Reaction-Diffusion Equations. Wiley Ser. Math. Comput. Biol. 2003, John Wiley & Sons, Ltd.
6. Chafee N., Infante E.F. A bifurcation problem for a nonlinear partial differential equation of parabolic type. Appl. Anal. 1974/1975, 4:17-37.
7. Chen X.-Y., Polácik P. Gradient-like structure and Morse decompositions for time-periodic one-dimensional parabolic equations. J. Dynam. Differential Equations 1995, 7(1):73-107.
8. Conway E., Hoff D., Smoller J. Large time behavior of solutions of systems of nonlinear reaction-diffusion equations. SIAM J. Appl. Math. 1978, 35(1):1-16.
9. Courchamp F., Berec L., Gascoigne J. Allee Effects in Ecology and Conservation 2008, Oxford University Press.
10. Crandall M.G., Rabinowitz P.H. Bifurcation from simple eigenvalues. J. Funct. Anal. 1971, 8:321-340.
11. Du Y.-H., Shi J.-P. Some recent results on diffusive predator-prey models in spatially heterogeneous environment. Fields Inst. Commun. 2006, vol. 48:95-135. Amer. Math. Soc., Providence, RI.
12. Du Y.-H., Shi J.-P. A diffusive predator-prey model with a protection zone. J. Differential Equations 2006, 229(1):63-91.
13. Du Y.-H., Shi J.-P. Allee effect and bistability in a spatially heterogeneous predator-prey model. Trans. Amer. Math. Soc. 2007, 359(9):4557-4593.
14. Hale J.K. Asymptotic Behavior of Dissipative Systems. Math. Surveys Monogr. 1988, vol. 25. Amer. Math. Soc., Providence, RI.
15. Hassard B.D., Kazarinoff N.D., Wan Y.-H. Theory and Applications of Hopf Bifurcation 1981, Cambridge University Press, Cambridge.
16. Henry D. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. 1981, vol. 840. Springer-Verlag, Berlin/New York.
17. Holling C.S. The components of predation as revealed by a study of small mammal predation of the European Pine Sawfly. Canad. Entomol. 1959, 91:293-320.
18. Hollis S.L., Martin R.H., Pierre M. Global existence and boundedness in reaction-diffusion systems. SIAM J. Math. Anal. 1987, 18(3):744-761.
19. Huang J.-H., Lu G., Ruan S.-G. Existence of traveling wave solutions in a diffusive predator-prey model. J. Math. Biol. 2003, 46(2):132-152.
20. Jiang J.-F., Liang X., Zhao X.-Q. Saddle-point behavior for monotone semiflows and reaction-diffusion models. J. Differential Equations 2004, 203(2):313-330.
21. Jiang J.-F., Shi J.-P. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete Contin. Dyn. Syst. 2008, 21(1):245-258.
22. Jiang J.-F., Shi J.-P. Bistability dynamics in some structured ecological models. Spatial Ecology 2009, Chapman & Hall/CRC. R.S. Cantrell, C. Cosner, S.-G. Ruan (Eds.).
23. J.-Y. Jin, J.-P. Shi, J.-J. Wei, F.-Q. Yi, Bifurcations of patterned solutions in diffusive Lengyel-Epstein system of CIMA chemical reaction, Rocky Mountain J. Math. (2011), in press.
24. Kazarinov N., van den Driessche P. A model predator-prey systems with functional response. Math. Biosci. 1978, 39:125-134.
25. Ko W., Ryu K. Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge. J. Differential Equations 2006, 231:534-550.
26. Lewis M.A., Karevia P. Allee dynamics and the spread of invading organisms. Theoret. Popul. Biol. 1993, 43:141-158.
27. Lotka A.J. Elements of Physical Biology 1925, Williams and Wilkins, Baltimore.
28. Lou Y., Ni W.-M. Diffusion, self-diffusion and cross-diffusion. J. Differential Equations 1996, 131:79-131.
29. Lou Y., Ni W.-M. Diffusion vs cross-diffusion: an elliptic approach. J. Differential Equations 1999, 154:157-190.
30. Medvinsky A.B., Petrovskii S.V., Tikhonova I.A., Malchow H., Li B.-L. Spatiotemporal complexity of plankton and fish dynamics. SIAM Rev. 2002, 44(3):311-370.
31. Mischaikow K., Smith H., Thieme H.R. Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions. Trans. Amer. Math. Soc. 1995, 347(5):1669-1685.
32. Morozov A., Petrovskii S., Li B.-L. Bifurcations and chaos in a predator-prey system with the Allee effect. Proc. R. Soc. Lond. Ser. B 2004, 271:1407-1414.
33. Morozov A., Petrovskii S., Li B.-L. Spatiotemporal complexity of patchy invasion in a predator prey system with the Allee effect. J. Theoret. Biol. 2006, 238:18-35.
34. Nisbet R.M., Gurney W.S.C. Modelling Fluctuating Populations 1982, John Wiley, New York.
35. Owen M.R., Lewis M.A. How predation can slow, stop or reverse a prey invasion. Bull. Math. Biol. 2001, 63:655-684.
36. Pao C.-V. Nonlinear Parabolic and Elliptic Equations 1992, Plenum Press, New York.
37. Peng R., Shi J.-P., Wang M.-X. Stationary pattern of a ratio-dependent food chain model with diffusion. SIAM J. Appl. Math. 2007, 65:1479-1503.
38. Peng R., Shi J.-P., Wang M.-X. On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law. Nonlinearity 2008, 21:1471-1488.
39. Petrovskii S.V., Morozov A.Y., Venturino E. Allee effect makes possible patchy invasion in a predator prey system. Ecol. Lett. 2002, 5:345-352.
40. Petrovskii S.V., Morozov A.Y., Li B.-L. Regimes of biological invasion in a predator prey system with the Allee effect. Bull. Math. Biol. 2005, 67:637-661.
41. Rabinowitz P.H. Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 1971, 7:487-513.
42. Rabinowitz P.H. Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Ser. Math. 1984, vol. 65.
43. Rosenzweig L.M. Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 1971, 171(3969):385-387.
44. Sherratt J.A., Eagan B.T., Lewis M.A. Oscillations and chaos behind predator-prey invasion: mathematical artifact or ecological reality?. Philos. Trans. R. Soc. Lond. Ser. B 1997, 352:21-38.
45. Shi J.-P. Solution Set of Semilinear Elliptic Equations: Global Bifurcation and Exact Multiplicity 2011, World Scientific Publishing.
46. Shi J.-P. Semilinear Neumann boundary value problems on a rectangle. Trans. Amer. Math. Soc. 2002, 354(8):3117-3154.
47. Shi J.-P., Shivaji R. Persistence in reaction diffusion models with weak Allee effect. J. Math. Biol. 2006, 52(6):807-829.
48. Shi J.-P., Wang X.-F. On the global bifurcation for quasilinear elliptic systems on bounded domains. J. Differential Equations 2009, 246(7):2788-2812.
49. Smoller J. Shock Waves and Reaction-Diffusion Equations. Grundlehren Math. Wiss. 1994, vol. 258. Springer-Verlag, New York. second ed.
50. Taylor C.M., Hastings A. Allee effects in biological invasions. Ecol. Lett. 2005, 8:895-908.
51. van Voorn G.A.K., Hemerik L., Boer M.P., Kooi B.W. Heteroclinic orbits indicate overexploitation in predator prey systems with a strong Allee effect. Math. Biosci. 2007, 209:451-469.
52. Volterra V. Fluctuations in the abundance of species, considered mathematically. Nature 1926, 118:558.
53. Wang J.-F., Shi J.-P., Wei J.-J. Predator-prey system with strong Allee effect in prey. J. Math. Biol. 2011, 3(3):291-331.
54. J.-F. Wang, J.-P. Shi, J.-J. Wei, Nonexistence of periodic orbits for predator-prey system with strong Allee effect in prey population, submitted for publication.
55. Wang M.-X. Non-constant positive steady states of the Sel'kov model. J. Differential Equations 2003, 190(2):600-620.
56. Wang M.-E., Kot M. Speeds of invasion in a model with strong or weak Allee effects. Math. Biosci. 2001, 171:83-97.
57. Smoller J., Wasserman A. Global bifurcation of steady-state solutions. J. Differential Equations 1981, 39(2):269-290.
58. Wu J.-H. Theory and Applications of Partial Functional-Differential Equations 1996, Springer, New York.
59. Ye Q.-X., Li Z.-Y. Introduction to Reaction-Diffusion Equations 1994, Science Press, China, (in Chinese).
60. Yi F.-Q., Wei J.-J., Shi J.-P. Bifurcation analysis of a diffusive predator-prey system with Holling Type-II function response. J. Differential Equations 2009, 246:1944-1977.