Consequences of double Allee effect on the number of limit cycles in a predatorprey model

Computers and Mathematics with Applications

Volumn 62, Issue 9, 2011, Pages 3449-3463

  González Olivares, Eduardo ^a   González Yaez, Betsabé ^a   Mena Lorca, Jaime ^a   Rojas Palma, Alejandro ^a   Flores, José D ^b  

^a PONTIFICIA UNIVERSIDAD CATÓLICA DE VALPARAÍSO   (Chile)

^b W O Farber Center for Civic Leadership   (United States)

Author keywords

Allee effect; Bifurcation; Limit cycles; Predatorprey model; Separatrix curve; Stability

Indexed keywords

ALGEBRAIC EXPRESSION; ALLEE EFFECT; COMPARATIVE ANALYSIS; CONTINUOUS TIME; LIMIT CYCLE; LYAPUNOV; MATHEMATICAL EXPRESSIONS; MATHEMATICAL FORMS; MATHEMATICAL FORMULATION; OBSERVED DATA; OSCILLATORY BEHAVIORS; POSITIVE EQUILIBRIUM; PREDATOR-PREY MODELS; SEPARATRIX; VOLTERRA;

POPULATION STATISTICS;

DELTA SIGMA MODULATION;

EID: 80054709237     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2011.08.061     Document Type: Article

References (44)

1. P. Turchin Complex Population Dynamics: A Theoretical/Empirical Synthesis Monographs in Population Biology vol. 35 2003 Princeton University Press
2. A.A. Berryman, A.P. Gutierrez, and R. Arditi Credible, parsŠimonious and useful predatorprey modelsa reply to Abrams, Gleeson, and Sarnelle Ecology 76 1995 1980 1985
3. A.D. Bazykin Nonlinear Dynamics of Interacting Populations 1998 World Scientific
4. E.D. Conway, and J.A. Smoller Global analysis of a system of predatorprey equations SIAM Journal on Applied Mathematics 46 1986 630 642
5. J. Wang, J. Shi, and J. Wei Predatorprey system with strong Allee effect in prey Journal of Mathematical Biology 62 2011 291 331
6. J. Zu, and M. Mimura The impact of Allee effect on a predatorprey system with Holling type II functional response Applied Mathematics and Computation 217 2010 3542 3556
7. R.M. May Stability and Complexity in Model Ecosystems 2nd ed. 2001 Princeton University Press
8. K. Hasík On a predatorprey system of Gause type Journal of Mathematical Biology 60 2010 59 74
9. D. Xiao, and Z. Zhang On the uniqueness and nonexistence of limit cycles for predatorprey systems Nonlinearity 16 2003 1185 1201
10. K.S. Cheng Uniqueness of a limit cycle for a predatorprey system SIAM Journal on Applied Mathematics 12 1981 541 548
11. Y. Kuang, and H.I. Freedman Uniqueness of limit cycles in Gause-type models of predatorprey systems Mathematical Biosciences 88 1988 67 84
12. C.S. Coleman Hilbert's 16th problem: how many cycles? M. Braun, C.S. Coleman, D. Drew, Differential Equations Model 1983 Springer Verlag 279 297
13. P. Aguirre, E. González-Olivares, and E. Sáez Three limit cycles in a LeslieGower predatorprey model with additive Allee effect SIAM Journal on Applied Mathematics 69 2009 1244 1262
14. E. González-Olivares, H. Meneses-Alcay, B. González-Yaez, J. Mena-Lorca, A. Rojas-Palma, and Rodrigo Ramos-Jiliberto Multiple stability and uniqueness of limit cycle in a Gause-type predatorprey model considering Allee effect on prey Nonlinear Analysis. Real World Applications 12 2011 2931 2942
15. E. González-Olivares, and A. Rojas-Palma Multiple limit cycles in a Gause type predatorprey model with holling type III functional response and Allee effect on prey Bulletin of Mathematical Biology 73 2011 1378 1397
16. E. González-Olivares, J. Mena-Lorca, A. Rojas-Palma, and J.D. Flores Dynamical complexities in the LeslieGower predatorprey model considering a simple form to the Allee effect on prey Applied Mathematical Modelling 35 2011 366 381
17. B. González-Yaez, and E. González-Olivares Consequences of Allee effect on a Gause type predatorprey model with nonmonotonic functional response R. Mondaini, Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology vol. 2 2004 E-Papers Servios Editoriais Ltda. Río de Janeiro 358 373
18. C. Chicone Ordinary Differential Equations with Applications 2nd ed. Texts in Applied Mathematics vol. 34 2006 Springer
19. V.A. Gaiko Global Bifurcation Theory and Hilbert's Sixteenth Problem Mathematics and its Applications vol. 559 2003 Kluwer Academic Publishers
20. P.A. Stephens, and W.J. Sutherland Consequences of the Allee effect for behaviour, ecology and conservation Trends in Ecology & Evolution 14 1999 401 405
21. P.A. Stephens, W.J. Sutherland, and R.P. Freckleton What is the Allee effect? Oikos 87 1999 185 190
22. F. Courchamp, T. Clutton-Brock, and B. Grenfell Inverse dependence and the Allee effect Trends in Ecology & Evolution 14 1999 405 410
23. L. Berec, E. Angulo, and F. Courchamp Multiple Allee effects and population management Trends in Ecology & Evolution 22 2007 185 191
24. F. Courchamp, L. Berec, and J. Gascoigne Allee Effects in Ecology and Conservation 2008 Oxford University Press
25. E. Angulo, G.W. Roemer, L. Berec, J. Gascoigne, and F. Courchamp Double Allee effects and extinction in the island fox Conservation Biology 21 2007 1082 1091
26. A.D. Bazykin, F.S. Berezovskaya, A.S. Isaev, and R.G. Khlebopros Dynamics of forest insect density: bifurcation approach Journal of Theoretical Biology 186 1997 267 278
27. M. Kot Elements of Mathematical Ecology 2001 Cambridge University Press
28. G.A.K. van Voorn, L. Hemerik, M.P. Boer, and B.W. Kooi Heteroclinic orbits indicate overexploitation in predatorprey systems with a strong Allee effect Mathematical Biosciences 209 2007 451 469
29. M.-H. Wang, and M. Kot Speeds of invasion in a model with strong or weak Allee effects Mathematical Biosciences 171 2001 83 97
30. C.W. Clark Mathematical Bioeconomics: The Mathematics of Conservation 3rd ed. 2010 John Wiley and Sons Inc.
31. M. Liermann, and R. Hilborn Depensation: evidence, models and implications Fish and Fisheries 2 2001 33 58
32. D.S. Boukal, and L. Berec Single-species models and the Allee effect: extinction boundaries, sex ratios and mate encounters Journal of Theoretical Biology 218 2002 375 394
33. E. González-Olivares, B. González-Yaez, J. Mena-Lorca, and R. Ramos-Jiliberto Modelling the Allee effect: are the different mathematical forms proposed equivalents? R. Mondaini, Proceedings of the 2006 International Symposium on Mathematical and Computational Biology 2007 E-Papers Servios Editoriais Ltda. Río de Janeiro 53 71
34. 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 72 2007 36 147
35. S.-R. Zhou, Y.-F. Liu, and G. Wang The stability of predatorprey systems subject to the Allee effect Theoretical Population Biology 67 2005 23 31
36. E.F. Knipling Possibilities of insect control or eradication through the use of sexually sterile males Journal of Economic Entomology 48 4 1955 459 462
37. H. Barclay, and M. Mackauer The sterile insect release method for pest control: a density-dependent model Environmental Entomology 9 1980 810 817
38. J.D. Flores, J. Mena-Lorca, B. González-Yaez, and E. González-Olivares Consequences of depensation in a Smith's bioeconomic model for open-access fishery R. Mondaini, Proceedings of the 2006 International Symposium on Mathematical and Computational Biology 2007 E-Papers Servios Editoriais Ltda. Río de Janeiro 219 232
39. H.I. Freedman Deterministic Mathematical Model in Population Ecology 1980 Marcel Dekker
40. B.-S. Goh Management and Analysis of Biological Populations 1980 Elsevier Scientific Publishing Company
41. F. Dumortier, J. Llibre, and J.C. Artés Qualitative Theory of Planar Differential Systems 2006 Springer
42. D.K. Arrowsmith, and C.M. Place Dynamical System: Differential Equations, Maps and Chaotic Behaviour 1992 Chapman and Hall
43. S. Wolfram Mathematica: A System for Doing Mathematics by Computer 2nd ed. 1991 Wolfram Research, Addison Wesley
44. L. Perko Differential Equations and Dynamical Systems 3rd ed. Texts in Applied Mathematics vol. 7 2001 Springer-Verlag