The impact of Allee effect on a predator-prey system with Holling type II functional response

Applied Mathematics and Computation

Volumn 217, Issue 7, 2010, Pages 3542-3556

  Zu, Jian ^a   Mimura, Masayasu ^b  

^a MEIJI UNIVERSITY   (Japan)

^b MEIJI UNIVERSITY   (Japan)

Author keywords

Extinction risk; Heteroclinic cycle; Limit cycle; Positive interactions; Stability switches

Indexed keywords

EXTINCTION RISKS; HETEROCLINIC CYCLES; LIMIT CYCLE; POSITIVE INTERACTION; STABILITY SWITCHES;

ECOLOGY; HOPF BIFURCATION; TIME SWITCHES;

MAMMALS;

EID: 78049281430     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2010.09.029     Document Type: Article

References (33)

1. R.M. Callaway, and L.R. Walker Competition and facilitation: a synthetic approach to interactions in plant communities Ecology 78 1997 1958 1965
2. S.D. Hacker, and S.D. Gaines Some implications of direct positive interactions for species diversity Ecology 78 1997 1990 2003
3. P.A. Stephens, and W.J. Sutherland Vertebrate mating systems, Allee effects and conservation M. Apollonio, M. Festa-Bianchet, D. Mainardi, Vertebrate Mating Systems 2000 World Scientific Publishing London
4. W.C. Allee Animal Aggregations, A Study in General Sociology 1931 University of Chicago Press Chicago, USA
5. M.D. Bertness, and G.H. Leonard The role of positive interactions in communities: lessons from intertidal habitats Ecology 78 1997 1976 1989
6. M. Liermann, and R. Hilborn Depensation: evidence, models and implications Fish and Fisheries 2 2001 33 58
7. F. Courchamp, T. Clutton-Brock, and B. Grenfell Inverse density dependence and the Allee effect Trends in Ecology and Evolution 14 1999 405 410
8. M.J. Groom Allee effects limit population viability of an annual plant The American Naturalist 151 1998 487 496
9. M. Kuussaari, I. Saccheri, M. Camara, and I. Hanski Allee effect and population dynamics in the Glanville fritillary butterfly Oikos 82 1998 384 392
10. F. Courchamp, B. Grenfell, and T. Clutton-Brock Population dynamics of obligate cooperators Proceedings of the Royal Society of London Series B - Biological Sciences 266 1999 557 563
11. P.A. Stephens, and W.J. Sutherland Consequences of the Allee effect for behaviour, ecology and conservation Trends in Ecology and Evolution 14 1999 401 404
12. T. Van Kooten, A.M. de Roos, and L. Persson Bistability and an Allee effect as emergent consequences of stage-specific predation Journal of Theoretical Biology 203 2005 67 74
13. M.A. McCarthy The Allee effect, finding mates and theoretical models Ecological Modelling 103 1997 99 102
14. C. Franck, C.B. Tim, and G. Bryan Inverse density dependence and the Allee effect Perspectives 14 1999 405 410
15. M.A. Burgman, S. Ferson, and H.R. Akcakaya Risk Assessment in Conservation Biology 1993 Chapman and Hall London
16. A.J. Lotka Elements of Physical Biology 1926 Williams & Wilkins Baltimore
17. V. Volterra Lecons sur la théorie mathématique de la lutte pour la vie 1931 Gauthier-Villars Paris
18. G.F. Gause The Struggle for Existence 1934 Williams & Wilkins Baltimore
19. W.D. Wang, G. Mulone, F. Salemi, and V. Salone Permanence and stability of a stage-structured predator-prey model Journal of Mathematical Analysis and Applications 262 2001 499 528
20. M.G. Neubert, P. Klepac, and P. Driessche Stabilizing dispersal delays in predator-prey metapopulation models Theoretical Population Biology 61 2002 339 347
21. S.R. Zhou, Y.F. Liu, and G. Wang The stability of predator-prey systems subject to the Allee effects Theoretical Population Biology 67 2005 23 31
22. A. Kent, C.P. Doncaster, and T. Sluckin Consequences for predators of rescue and Allee effects on prey Ecological Modelling 162 2003 233 245
23. J.B. Ferdy, and J. Molofsky Allee effect, spatial structure and species coexistence Journal of Theoretical Biology 217 2002 413 424
24. G.A.K. Van Voorn, L. Hemerik, M.P. Boer, and B.W. Kooi Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect Mathematical Biosciences 209 2007 451 469
25. J. Guckenheimer, and P.J. Holmes Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields 1996 Springer New York
26. A. Dhooge, W. Govaerts, and Y.A. Kuznetsov Numerical continuation of branch points of limit cycles in MATCONT Lecture Notes in Computer Science 3037 2004 42 49
27. W.D. Wang, and Z.E. Ma Asymptotic behavior of a predator-prey system with diffusion and delays Journal of Mathematical Analysis and Applications 206 1997 191 204
28. J. Zu, W.D. Wang, Y. Takeuchi, B. Zu, and K.F. Wang On evolution under symmetric and asymmetric competitions Journal of Theoretical Biology 254 2008 239 251
29. F. Salemi, V. Salone, and W.D. Wang Stability of a competition model with two-stage structure Applied Mathematics and Computation 99 1999 221 231
30. R. Xu, M.A.J. Chaplain, and F.A. Davidson Persistence and global stability of a ratio-dependent predator-prey model with stage structure Applied Mathematics and Computation 158 2004 729 744
31. J. Zu, W.D. Wang, and B. Zu Evolutionary dynamics of prey-predator systems with Holling type II functional response Mathematical Biosciences and Engineering 4 2007 221 237
32. J. Zu, M. Mimura, and J.Y. Wakano The evolution of pheno typic traits in a predator-prey system subject to Allee effect Journal of Theoretical Biology 262 2010 528 543
33. J.F. Wang, J.P. Shi, and J.J. Wei Predator-prey system with strong Allee effect in prey Journal of Mathematical Biology 2010