The role of noise in a predator-prey model with Allee effect

Journal of Biological Physics

Volumn 35, Issue 2, 2009, Pages 185-196

  Sun, Gui Quan ^a   Jin, Zhen ^a   Li, Li ^a   Liu, Quan Xing ^a  

^a NORTH UNIVERSITY OF CHINA   (China)

Author keywords

Noise; Patchy invasion; Predator prey; Spatial pattern

Indexed keywords

ARTICLE; BEHAVIOR; ECOSYSTEM; NOISE; PARAMETER; PREDATOR PREY INTERACTION; PRIORITY JOURNAL; SIMULATION;

EID: 64749087718     PISSN: 00920606     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10867-009-9139-y     Document Type: Article

References (62)

1. AM Turing 1952 The chemical basis of morphogenesis Philos. Trans. R. Soc. Lond. B 237 37 72
2. Haken, H.: Synergetics. An Introduction, Springer Series in Synergetics, vol. 1. Springer, Berlin (1977)
3. Nicolis, G., Prigogine, I.: Self-Organization in Nonequilibrium Systems. Wiley, New York (1977)
4. Weidlich, W., Haag, G.: Concepts and Models of a Quantitative Sociology. The Dynamics of Interacting Populations, Springer Series in Synergetics. Springer, Berlin (1983)
5. AB Medvinsky SV Petrovskii IA Tikhonova H Malchow B-L Li 2002 Spatiotemporal complexity of plankton and fish dynamics SIAM Rev. 44 3 311 370
6. M Baurmann T Gross U Feudel 2007 Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations J. Theor. Biol. 245 220 229
7. GQ Sun Z Jin QX Liu L Li 2008 Pattern formation induced by cross-diffusion in a predator-prey system Chin. Phys. B 17 3936 3941
8. GQ Sun Z Jin QX Liu L Li 2008 Dynamical complexity of a spatial predator-prey model with migration Ecol. Model. 219 248 255
9. Murray, J.D.: Mathematical Biology. Springer, Berlin (1989)
10. CS Holling 1973 Resilience and stability of ecological systems Ann. Rev. Ecolog. Syst. 4 1 23
11. C Folke SR Carpenter B Walker M Scheffer T Elmqvist LH Gunderson C Holling 2004 Regime shifts, resilience, and biodiversity in ecosystem management Annu. Rev. Ecol. Evol. Syst. 35 557 581
12. M Scheffer S Rinaldi YA Kuznetsov EH van Nes 1997 Seasonal dynamics of Daphnia and algae explained as a periodically forced predator-prey system Oikos 80 519 532
13. M Scheffer S Rinaldi 2000 Minimal models of top-down control of phytoplankton Freshw. Biol. 45 265 283
14. V Guttal C Jayaprakash 2007 Impact of noise on bistable ecological systems Ecol. Model. 201 420 428
15. JG Vilar RV Solé 1998 Effects of noise in symmetric two-species competition Phys. Rev. Lett. 80 18 4099 4102
16. R Mankin T Laas A Sauga A Ainsaar 2006 Colored-noise-induced Hopf bifurcations in predator-prey communities Phys. Rev. E 74 021101
17. R Mankin T Laas E Soika A Ainsaar 2007 Noise-controlled slow-fast oscillations in predator-prey models with the Beddington functional response Eur. Phys. J. B 59 1434 6036
18. R Rai H Singh 2000 Stochastic resonance without an external periodic drive in a simple prey-predator model Phys. Rev. E 62 8804 8807
19. Y-C Lai 2005 Beneficial role of noise in promoting species diversity through stochastic resonance Phys. Rev. E 72 042901
20. AL Barbera B Spagnolo 2002 Spatio-temporal patterns in population dynamics Phys. A 314 120 124
21. B Spagnolo M Cirone AL Barbera F de Pasquale 2002 Noise-induced effects in population dynamics J. Phys. Condens. Matter 14 2247 2255
22. JMG Vilar RV Solé JM Rubí 2003 On the origin of plankton patchiness Phys. A 317 239 246
23. O Kortlüke VN Kuzovkov W von Niessen 2002 Internal spatiotemporal stochastic resonance in the presence of weak noise Phys. Rev. E 66 036139
24. Sun, G.Q., Zhang, G., Jin, Z., Li, L.: Predator cannibalism can give rise to regular spatial pattern in a predator-prey system. Nonlinear Dyn. (2009, in press)
25. 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. Int. J. Mod. Phys. C (2009, in press)
26. Allee, W.C.: The Social Life of Animals. Norton, New York (1938)
27. B Dennis 1989 Allee effects: population growth, critical density, and the chance of extinction Nat. Res. Model. 3 481 538
28. P Amarasekare 1998 Interactions between local dynamics and dispersal: insights from single species models Theor. Popul. Biol. 53 44 59
29. M Gyllenberg J Hemminki T Tammaru 1999 Allee effects can both conserve and create spatial heterogeneity in population densities Theor. Popul. Biol. 56 231 242
30. SV Petrovskii ME Vinogradov AY Morozov 2002 Spatio-temporal horizontal plankton patterns caused by biological invasion in a two-species model of plankton dynamics allowing for the Allee effect Oceanology 42 384 393
31. Nisbet, R.M., Gurney, W.S.C.: Modelling Fluctuating Populations. Wiley, Chichester (1982)
32. EE Holmes MA Lewis JE Banks RR Veit 1994 Partial differential equations in ecology: spatial interactions and population dynamics Ecology 75 17 29
33. JA Sherratt 2001 Periodic travelling waves in cyclic predator-prey systems Ecol. Lett. 4 30 37
34. LA Segel JL Jackson 1972 Dissipative structure: an explanation and an ecological example J. Theor. Biol. 37 545 559
35. A Morozov S Petrovskii B-L Li 2004 Bifurcations and chaos in a predator-prey system with the Allee effect Proc. R. Soc. Lond. B 271 1407 1414
36. MA Lewis P Kareiva 1993 Allee dynamics and the spread of invading organisms Theor. Popul. Biol. 43 141 158
37. SV Petrovskii AY Morozov E Venturino 2002 Allee effect makes possible patchy invasion in a predator-prey system Ecol. Lett. 5 345 352
38. G Sun Z Jin Q-X Liu L Li 2007 Pattern formation in a spatial s-i model with non-linear incidence rates J. Stat. Mech. 11 P11011
39. T Reichenbach M Mobilia E Frey 2007 Noise and correlations in a spatial population model with cyclic competition Phys. Rev. Lett. 99 238105
40. T Reichenbach M Mobilia E Frey 2007 Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games Nature 448 1046
41. Liu, Q., Li, B., Jin, Z.: Resonance and frequency-locking phenomena in a spatially extended phytoplankton-zooplankton system with additive noise and periodic forces. J. Stat. Mech. P05011 (2008)
42. B Blasius A Huppert L Stone 1999 Complex dynamics and phase synchronization in spatially extended ecological systems Nature 399 354
43. R Mankin 2002 Trichotomous-noise-induced catastrophic shifts in symbiotic ecosystems Phys. Rev. E 65 051108
44. R Mankin 2004 Colored-noise-induced discontinuous transitions in symbiotic ecosystems Phys. Rev. E 69 061106
45. DJ Higham 2001 An algorithmic introduction to numerical simulation of stochastic differential equations SIAM Rev. 43 525
46. M Pascual 1993 Diffusion-induced chaos in a spatial predator- prey system Proc. R. Soc. B 251 1 7
47. JA Sherratt MA Lewis AC Fowler 1995 Ecological chaos in the wake of invasion Proc. Natl. Acad. Sci. U. S. A. 92 2524 2528
48. S Petrovskii B-L Li H Malchow 2004 Transition to spatiotemporal chaos can resolve the paradox of enrichment Ecol. Complex. 1 37
49. WG Wilson 1998 Resolving discrepancies between deterministic population models and individual-based simulations Am. Nat. 151 116
50. A Morozov S Petrovskii BL Li 2006 Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect J. Theor. Biol. 238 18 35
51. SV Petrovskii AY Morozov B-L Li 2005 Regimes of biological invasion in a predator-prey system with the Allee effect Bull. Math. Biol. 67 637 661
52. JA Sherratt X Lambin TN Sherratt 2003 The effects of the size and shape of landscape features on the formation of traveling waves in cyclic populations Am. Nat. 162 503 513
53. O Ovaskainen K Sato J Bascompte I Hanski 2002 Metapopulation models for extinction threshold in spatially correlated landscapes J. Theor. Biol. 215 95 108
54. J Bascompte MA Rodríguez 2000 Self-disturbance as a source of spatiotemporal heterogeneity: the case of the Tallgrass Prairie J. Theor. Biol. 204 153 164
55. J Bascompte RV Solé 1998 Effects of habitat destruction in a prey-predator metapopulation model J. Theor. Biol. 195 383 393
56. Z Neufeld 2002 Excitable media in open and closed chaotic flows Phys. Rev. E 66 066208
57. C Zhou 2003 Noise-sustained coherent oscillation of excitable media in a chaotic flow Phys. Rev. Lett. 91 150601
58. E Hernandez-Garcia C Lopez 2004 Sustained plankton blooms under open chaotic flows Ecol. Complex 1 253
59. Z Neufeld 2002 Ocean fertilization experiments may initiate a large scale phytoplankton bloom Geophys. Res. Lett. 29 1534
60. E Hernandez-Garcia C Lopez Z Neufeld 2002 Small-scale structure of nonlinearly interacting species advected by chaotic flows Chaos 12 470
61. A Tzella PH Haynes 2007 Small-scale spatial structure in plankton distributions Biogeosciences 4 173
62. R Reigada 2003 Plankton blooms induced by turbulent flows Proc. R. Soc. B 270 875