Spatiotemporal pattern in a self- and cross-diffusive predation model with the allee effect

Discrete Dynamics in Nature and Society

Volumn 2013, Issue , 2013, Pages

  Rao, Feng ^a  

^a NANJING TECH UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84880885186     PISSN: 10260226     EISSN: 1607887X     Source Type: Journal    
DOI: 10.1155/2013/681641     Document Type: Article

References (62)

1. Murray J. D., Mathematical Biology 2003 3rd New York, NY, USA Springer
2. Lotka A. J., Elements of Physical Biology 1925 Baltimore, Md, USA Williams & Wilkins
3. Volterra V., Fluctuations in the abundance of a species considered mathematically. Nature 1926 118 2972 558 560 2-s2.0-0006170098
4. Holling C. S., The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. The Canadian Entomologist 1959 91 5 293 320
5. Holling C. S., Some characteristics of simple types of predation and parasitism. The Canadian Entomologist 1959 91 7 385 398
6. Aziz-Alaoui M. A., Daher Okiye M., Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes. Applied Mathematics Letters 2003 16 7 1069 1075 10.1016/S0893-9659(03)90096-6 MR2013074 ZBL1063.34044
7. Xu R., Chaplain M. A. J., Davidson F. A., Periodic solutions for a predator-prey model with Holling-type functional response and time delays. Applied Mathematics and Computation 2005 161 2 637 654 10.1016/j.amc.2003.12.054 MR2112430 ZBL1064.34053
8. Yafia R., El Adnani F., Alaoui H. T., Limit cycle and numerical 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 2008 9 5 2055 2067 10.1016/j.nonrwa.2006.12.017 MR2441766 ZBL1156.34342
9. Zhang S., Tan D., Chen L., Chaos in periodically forced Holling type II predator-prey system with impulsive perturbations. Chaos, Solitons & Fractals 2006 28 2 367 376 10.1016/j.chaos.2005.05.037 MR2178118 ZBL1083.37537 (Pubitemid 41491942)
10. Křivan V., Eisner J., The effect of the Holling type II functional response on apparent competition. Theoretical Population Biology 2006 70 4 421 430 2-s2.0-33845544689 10.1016/j.tpb.2006.07.004 (Pubitemid 44927744)
11. Ko W., Ryu K., Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge. Journal of Differential Equations 2006 231 2 534 550 10.1016/j.jde.2006.08.001 MR2287896 (Pubitemid 44542472)
12. Pei Y., Chen L., Zhang Q., Li C., Extinction and permanence of one-prey multi-predators of Holling type II function response system with impulsive biological control. Journal of Theoretical Biology 2005 235 4 495 503 10.1016/j.jtbi.2005.02.003 MR2158283 (Pubitemid 40780939)
13. Stephens P. A., Sutherland W. J., Consequences of the Allee effect for behaviour, ecology and conservation. Trends in Ecology & Evolution 1999 14 10 401 405 2-s2.0-0033215101 10.1016/S0169-5347(99)01684-5 (Pubitemid 29428465)
14. Stephens P. A., Sutherland W. J., Freckleton R. P., What is the Allee effect? Oikos 1999 87 1 185 190 2-s2.0-0032713166 (Pubitemid 29511251)
15. Courchamp F., Clutton-Brock T., Grenfell B., Inverse density dependence and the Allee effect. Trends in Ecology & Evolution 1999 14 10 405 410 2-s2.0-0033214426 10.1016/S0169-5347(99)01683-3 (Pubitemid 29428466)
16. Allee W. C., Animal Aggregations: A Study in General Sociology 1978 New York, NY, USA AMS Press
17. Dennis B., Allee effects: population growth, critical density, and the chance of extinction. Natural Resource Modeling 1989 3 4 481 538 MR1037193 ZBL0850.92062
18. Lewis M. A., Kareiva P., Allee Dynamics and the Spread of Invading Organisms. Theoretical Population Biology 1993 43 2 141 158 2-s2.0-0027524391 10.1006/tpbi.1993.1007
19. Owen M. R., Lewis M. A., How predation can slow, stop or reverse a prey invasion. Bulletin of Mathematical Biology 2001 63 4 655 684 2-s2.0-0034895545 10.1006/bulm.2001.0239 (Pubitemid 32721008)
20. Wang M.-H., Kot M., Speeds of invasion in a model with strong or weak Allee effects. Mathematical Biosciences 2001 171 1 83 97 10.1016/S0025-5564(01) 00048-7 MR1839210 ZBL0978.92033 (Pubitemid 32429044)
21. Petrovskii S. V., Morozov A. Y., Venturino E., Allee effect makes possible patchy invasion in a predator-prey system. Ecology Letters 2002 5 3 345 352 2-s2.0-0036113356 10.1046/j.1461-0248.2002.00324.x (Pubitemid 34523101)
22. Zhou S.-R., Liu Y.-F., Wang G., The stability of predator-prey systems subject to the Allee effects. Theoretical Population Biology 2005 67 1 23 31 2-s2.0-11844279070 10.1016/j.tpb.2004.06.007 (Pubitemid 40084670)
23. Shi J., Shivaji R., Persistence in reaction diffusion models with weak Allee effect. Journal of Mathematical Biology 2006 52 6 807 829 10.1007/s00285-006-0373-7 MR2235529 ZBL1110.92055 (Pubitemid 43735798)
24. Morozov A., Petrovskii S., Li B.-L., Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect. Journal of Theoretical Biology 2006 238 1 18 35 10.1016/j.jtbi.2005.05.021 MR2206584 (Pubitemid 41737794)
25. Berec L., Angulo E., Courchamp F., Multiple Allee effects and population management. Trends in Ecology & Evolution 2007 22 4 185 191 2-s2.0-33947307582 10.1016/j.tree.2006.12.002 (Pubitemid 46437275)
26. Çelik C., Duman O., Allee effect in a discrete-time predator-prey system. Chaos, Solitons & Fractals 2009 40 4 1956 1962 10.1016/j.chaos.2007. 09.077 MR2531270 ZBL1198.34084
27. Zu J., Mimura M., The impact of Allee effect on a predator-prey system with Holling type II functional response. Applied Mathematics and Computation 2010 217 7 3542 3556 10.1016/j.amc.2010.09.029 MR2733797 ZBL1202.92088
28. González-Olivares E., Meneses-Alcay H., González- Yañez B., Mena-Lorca J., Rojas-Palma A., Ramos-Jiliberto R., Multiple stability and uniqueness of the limit cycle in a Gause-type predator-prey model considering the Allee effect on prey. Nonlinear Analysis: Real World Applications 2011 12 6 2931 2942 10.1016/j.nonrwa.2011.04.003 MR2832938 ZBL1231.34053
29. Wang J., Shi J., Wei J., Predator-prey system with strong Allee effect in prey. Journal of Mathematical Biology 2011 62 3 291 331 10.1007/s00285-010- 0332-1 MR2771175 ZBL1232.92076
30. Wang J., Shi J., Wei J., Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey. Journal of Differential Equations 2011 251 4-5 1276 1304 10.1016/j.jde.2011.03.004 MR2812590 ZBL1228.35037
31. Cai Y., Wang W., Wang J., Dynamics of a diffusive predator-prey model with additive Allee effect. International Journal of Biomathematics 2012 5 2 11 1250023 10.1142/S1793524511001659 MR2904442
32. Wang W. M., Cai Y. L., Zhu Y. N., Guo Z. G., Allee-effect-induced instability in a reactiondiffusion predator-prey model. Abstract and Applied Analysis 2013 2013 10 10.1155/2013/487810 487810
33. Cai Y., Liu W., Wang Y., Wang W., Complex dynamics of a diffusive epidemic model with strong Allee effect. Nonlinear Analysis: Real World Applications 2013 14 4 1907 1920 10.1016/j.nonrwa.2013.01.002 MR3027935
34. Shukla J. B., Verma S., Effects of convective and dispersive interactions on the stability of two species. Bulletin of Mathematical Biology 1981 43 5 593 610 10.1016/S0092-8240(81)80085-7 MR661520 ZBL0473.92016
35. Biktashev V. N., Brindley J., Holden A. V., Tsyganov M. A., Pursuit-evasion predator-prey waves in two spatial dimensions. Chaos 2004 14 4 988 994 10.1063/1.1793751 MR2106513 ZBL1080.35036 (Pubitemid 40389665)
36. Kerner E. H., Further considerations on the statistical mechanics of biological associations. Bulletin of Mathematical Biology 1959 21 217 255 MR0104525
37. Shigesada N., Kawasaki K., Teramoto E., Spatial segregation of interacting species. Journal of Theoretical Biology 1979 79 1 83 99 10.1016/0022-5193(79)90258-3 MR540951
38. Chattopadhyay J., Sarkar A. K., Tapaswi P. K., Effect of cross-diffusion on a diffusive prey-predator system-a nonlinear analysis. Journal of Biological Systems 1996 4 2 159 169 2-s2.0-0000687142 (Pubitemid 126490873)
39. Dubey B., Das B., Hussain J., A predator-prey interaction model with self and cross-diffusion. Ecological Modelling 2001 141 1-3 67 76 2-s2.0-0035389383 10.1016/S0304-3800(01)00255-1 (Pubitemid 32784261)
40. Jorne J., The diffusive Lotka Volterra oscillating system. Journal of Theoretical Biology 1977 65 1 133 139 2-s2.0-0017337699 (Pubitemid 8036675)
41. Lou Y., Ni W.-M., Diffusion, self-diffusion and cross-diffusion. Journal of Differential Equations 1996 131 1 79 131 10.1006/jdeq.1996.0157 MR1415047 ZBL0867.35032 (Pubitemid 126169643)
42. Lou Y., Ni W.-M., Diffusion vs cross-diffusion: an elliptic approach. Journal of Differential Equations 1999 154 1 157 190 10.1006/jdeq.1998.3559 MR1685622 ZBL0934.35040
43. Kuto K., Stability of steady-state solutions to a prey-predator system with cross-diffusion. Journal of Differential Equations 2004 197 2 293 314 10.1016/j.jde.2003.10.016 MR2034162 ZBL1210.35122 (Pubitemid 38254145)
44. Kuto K., Yamada Y., Multiple coexistence states for a prey-predator system with cross-diffusion. Journal of Differential Equations 2004 197 2 315 348 10.1016/j.jde.2003.08.003 MR2034163 ZBL1205.35116 (Pubitemid 38254146)
45. Sun G.-Q., Jin Z., Liu Q.-X., Li L., Pattern formation induced by cross-diffusion in a predator-prey system. Chinese Physics B 2008 17 11 3936 3941 2-s2.0-62649117721 10.1088/1674-1056/17/11/003
46. Sun G.-Q., Jin Z., Zhao Y.-G., Liu Q.-X., Li L., Spatial pattern in a predator-prey system with both self- and cross-diffusion. International Journal of Modern Physics C 2009 20 1 71 84 10.1142/S0129183109013467 MR2494327 ZBL1155.35476
47. Wang W., Lin Y., Zhang L., Rao F., Tan Y., Complex patterns in a predator-prey model with self and cross-diffusion. Communications in Nonlinear Science and Numerical Simulation 2011 16 4 2006 2015 10.1016/j.cnsns.2010.08.035 MR2736066 ZBL1221.35423
48. Wang W. M., Guo Z. G., Upadhyay R. K., Lin Y. Z., Pattern formation in a cross-diffusive Holling-Tanner model. Discrete Dynamics in Nature and Society 2012 2012 12 10.1155/2012/828219 828219
49. Morozov A., Petrovskii S., Li B.-L., Bifurcations and chaos in a predator-prey system with the Allee effect. Proceedings of the Royal Society B 2004 271 1546 1407 1414 2-s2.0-3042846003 10.1098/rspb.2004.2733 (Pubitemid 38858318)
50. Okubo A., Levin S. A., Diffusion and Ecological Problems: Modern Perspectives 2001 14 2nd New York, NY, USA Springer xx+467 Interdisciplinary Applied Mathematics MR1895041
51. Murray J. D., Discussion: Turing's theory of morphogenesis-its influence on modelling biological pattern and form. Bulletin of Mathematical Biology 1990 52 1-2 119 152 2-s2.0-0001100811 10.1016/S0092-8240(05)80007-2
52. Munteanu A., Solé R. V., Pattern formation in noisy self-replicating spots. International Journal of Bifurcation and Chaos 2006 16 12 3679 3685 2-s2.0-33847244910 10.1142/S0218127406017063 (Pubitemid 46324104)
53. Garvie M. R., Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB. Bulletin of Mathematical Biology 2007 69 3 931 956 10.1007/s11538-006-9062-3 MR2295839 (Pubitemid 46466303)
54. Pearson J. E., Complex patterns in a simple system. Science 1993 261 5118 189 192 2-s2.0-0001102631
55. Tobias S. M., Knobloch E., Breakup of spiral waves into chemical turbulence. Physical Review Letters 1998 80 21 4811 4814 2-s2.0-0000003620 (Pubitemid 128622080)
56. Bär M., Or-Guil M., Alternative scenarios of spiral breakup in a reaction-diffusion model with excitable and oscillatory dynamics. Physical Review Letters 1999 82 6 1160 1163 2-s2.0-0000771172 (Pubitemid 129583702)
57. Sandstede B., Scheel A., Absolute versus convective instability of spiral waves. Physical Review E 2000 62 6, part A 7708 7714 10.1103/PhysRevE.62.7708 MR1807181 (Pubitemid 32087989)
58. Bär M., Brusch L., Breakup of spiral waves caused by radial dynamics: eckhaus and finite wavenumber instabilities. New Journal of Physics 2004 6 1 23 2-s2.0-3042593474
59. Xie F., Xie D., Weiss J. N., Inwardly rotating spiral wave breakup in oscillatory reaction-diffusion media. Physical Review E 2006 74 2 2-s2.0-33746918774 10.1103/PhysRevE.74.026107 026107 (Pubitemid 44200595)
60. Rao F., Wang W., Dynamics of a Michaelis-Menten-type predation model incorporating a prey refuge with noise and external forces. Journal of Statistical Mechanics: Theory and Experiment 2012 2012 3 2-s2.0-84859237545 10.1088/1742-5468/2012/03/P03014 P03014
61. Sherratt J. A., Eagan B. T., Lewis M. A., Oscillations and chaos behind predator-prey invasion: mathematical artifact or ecological reality? Philosophical Transactions of the Royal Society B 1997 352 1349 21 38 2-s2.0-0030814768 10.1098/rstb.1997.0003 (Pubitemid 27114978)
62. Medvinsky A. B., Petrovskii S. V., Tikhonova I. A., Malchow H., Li B.-L., Spatiotemporal complexity of plankton and fish dynamics. SIAM Review 2002 44 3 311 370 10.1137/S0036144502404442 MR1951363 ZBL1001.92050