Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III

Journal of Differential Equations

Volumn 249, Issue 9, 2010, Pages 2316-2356

  Etoua, Remy Magloire ^a,b   Rousseau, Christiane ^b  

^a Ecole National Superieure Polytechnique Cameroon   (Cameroon)

^b UNIVERSITÉ DE MONTRÉAL   (Canada)

Author keywords

Generalized Gause model with prey harvesting; Generalized Holling response function of type III; Heteroclinic bifurcation; Hopf bifurcation; Nilpotent saddle bifurcation; Saddle node bifurcation

Indexed keywords

EID: 77956185641     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2010.06.021     Document Type: Article

References (31)

1. Arnold V.I. Geometrical methods in the theory of ordinary differential equations 1983, Springer-Verlag, New York.
2. Annik Martin, Predator-prey models with delays and prey harvesting, Master of Science thesis, Dalhousie University Halifax, Nova Scotia, 1999.
3. Bazykin A.D. Nonlinear Dynamics of Interacting Populations. World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises 1998, vol. 11. World Scientific.
4. Brauer F., Castillo-Chavez C. Mathematical Models in Population Biology and Epidemiology 2001, Springer-Verlag, New York, Heidelberg, Berlin.
5. Broer H.W., Naudot V., Roussarie R., Saleh K. Dynamics of a predator-prey model with non-monotonic response function. Discrete Contin. Dyn. Syst. 2007, 18:221-251.
6. Clark C.W. Mathematical Bioeconomics: The Optimal Management of Renewable Resources 1990, John Wiley and Sons, New York, Toronto. 2nd ed.
7. Caubergh M., Dumortier F. Hopf-Takens bifurcations and centres. J. Differential Equations 2004, 202:1-31.
8. Chow S.N., Hale J.K. Methods of Bifurcation Theory 1982, Springer-Verlag, New York, Heidelberg, Berlin.
9. Chow S.N., Li C., Wang D. Normal Forms and Bifurcation of Planar Vector Fields 1994, Cambridge University Press, New York.
10. Dumortier F., Roussarie R., Rousseau C. Elementary graphics of cyclicity one or two. Nonlinearity 1994, 7:1001-1043.
11. Dumortier F., Roussarie R., Sotomayor J. Generic 3-parameter families of planar vector fiels: Unfolding of saddle, focus and elliptic singularities with nilpotent linear parts. Lecture Notes in Math. 1991, vol. 1480:1-164. Springer-Verlag, New York.
12. Dai G.R., Tang M. Coexistence region and global dynamics of a harvested predator-prey system. SIAM J. Appl. Math. 1998, 58(1):193-210.
13. R.M.D. Etoua, Étude d'un modèle de Gause généralisé avec récolte de proies et fonction de Holling type III généralisée, Thèse de PhD, Université de Montréal, Novembre 2008.
14. R.M.D. Etoua, Étude des familles standard des déploiements du col nilpotent dont l'axe des abscisses est invariant, preprint, 2010.
15. Freedman H.I. Deterministic Mathematical Models in Population Ecology 1980, Marcel Dekker, New York.
16. Freedman H.I., Wolkowicz G.S.K. Predator-prey systems with group defence: The paradox of enrichment revisited. Bull. Math. Biol. 1986, 48(5/6):493-508.
17. Gause G.F. The Struggle for Existence 1935, Williams and Wilkins, Baltimore.
18. Godeau R. Algèbre Supérieure 1962, Université Libre de Bruxelles. deuxième édition.
19. Kuznetsov Y.A. Elements of Applied Bifurcation Theory. Appl. Math. Sci. 2004, vol. 112. Springer-Verlag, New York. third edition.
20. Lotka A. Elements of Physical Biology 1925, Williams and Wilkins, Baltimore.
21. Lamontagne Y., Coutu C., Rousseau C. Bifurcation analysis of a predator-prey system with generalised Holling type III function response. J. Dynam. Differential Equations 2008, 20(3):535-571.
22. Molles Manuel C. Ecology: Concepts and Applications 2002, McGraw-Hill Book Company, New York, 586 pp. International ed.
23. Perko L. Differential Equations and Dynamical Systems. Texts Appl. Math. 2002, vol. 7. Springer-Verlag. third edition.
24. Shi Shongling A method of constructing cycles without contact around a weak focus. J. Differential Equations 1981, 41:301-312.
25. Volterra V. Fluctuations in the abundance of species considered mathematically. Nature 1926, CXVIII:558-560.
26. Wolkowicz G.S.K. Bifurcation analysis of a predator-prey system involving group defence. SIAM J. Appl. Math. 1988, 48(3):592-606.
27. Xiao D., Jennings L.S. Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting. SIAM J. Appl. Math. 2005, 65(3):737-753.
28. Xiao D., Ruan S. Global analysis in predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math. 2001, 61(4):1445-1472.
29. Hsu Sze-Bi, Huang Tzy-Wei Global stability for a class of predator-prey systems. SIAM J. Appl. Math. 1995, 55(3):763-783.
30. Zhu H., Campbell S.A., Wolkowicz G.S.K. Bifurcation analysis of a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math. 2002, 63(2):636-682.
31. Rousseau C., Zhu H. PP-graphics with a nilpotent elliptic singularity in quadratic systems and Hilbert's 16th problem. J. Differential Equations 2004, 196:169-208.