The bifurcation structure of the Holling-Tanner model for predator-prey interactions using two-timing

SIAM Journal on Applied Mathematics

Volumn 63, Issue 3, 2003, Pages 889-904

  Braza, Peter A ^a  

^a UNIVERSITY OF NORTH FLORIDA   (United States)

Author keywords

Degenerate Hopf bifurcations; Holling Tanner; Limit cycles; Outbreaks; Predator prey; Two timing

Indexed keywords

ECOLOGY; ENVIRONMENTAL IMPACT; MATHEMATICAL MODELS;

HOPF BIFURCATIONS;

BIFURCATION (MATHEMATICS);

EID: 0037490631     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036139901393494     Document Type: Article

References (19)

1. R. M. MAY, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, 1973.
2. R. M. MAY, Limit cycles in predator-prey communities, Science, 177 (1972), pp. 900-902.
3. W. S. C. GURNEY AND R. M. NISBET, Ecological Dynamics, Oxford University Press, New York, 1998.
4. A. N. KOLMOGOROV, Sulla, Teoria di Volterra della Lotto, per l'Esisttenza, Giorna. Instituto Ital. Attuarri, 7 (1936), pp. 74-80.
5. W. E. HUTCHINSON, An Introduction to Population Ecology, Yale University Press, New Haven, CT, 1978.
6. C. S. HOLLING, The functional response of invertebrate predators to prey density, Mem. Ent. Soc. Can., 45 (1965), pp. 3-60.
7. M. P. HASSELL, The Dynamics of Arthropod Predator-Prey Systems, Princeton University Press, Princeton, NJ, 1978.
8. P. H. LESLIE, Some Further Notes on the Use of Matrices in Population Mathematics, Biometrica, 35 (1948), pp. 213-245.
9. E. SÁEZ AND E. GONZÁLEZ-OLIVARES, Dynamics of a predator-prey model, SIAM J. Appl. Math., 59 (1999), pp. 1867-1878.
10. J. T. TANNER, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56 (1975), pp. 855-867.
11. D. J. WOLLKIND, J. B. COLLINGS, AND J. A. LOGAN. Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit flies, Bull. Math. Biol., 50 (1988), pp. 379-409.
12. S.-B. Hsu AND T.-W. HUANG, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), pp. 763-783.
13. E. J. DOEDEL AND J. P. KERNÉVEZ, AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations, Technical report, California Institute of Technology, Pasadena, CA, 1986.
14. J. B. COLLINGS, Bifurcation and stability analysis of a temperature-dependent mite predator-prey interaction model incorporating a prey refuge, Bull. Math. Biol., 57 (1995), pp. 63-76.
15. W. BALTENSWEILER, The relevance of changes in the composition of larch bud moth populations for the dynamics of its numbers, in Dynamics of Populations, P. J. den Boer and G. R. Gradwell, eds., Center for Agricultural Publication and Documentation, Wageningen, The Netherlands, 1971, pp. 208-219.
16. R. A. DOLBEER, Population Dynamics of the Snowshoe Hare in Colorado, Ph.D. Thesis, Colorado State University, Fort Collins, CO, 1972.
17. L. S. LUCKINBILL, Coexistence in laboratory populations of Paramecium aurelia and its predator Didinium nasutum, Ecology, 54 (1973), pp. 1320-1327.
18. S. WOLFRAM, Mathematica 4.0, Wolfram Research Inc., Champaign, IL, 1999.
19. M. GOLUBITSKY AND W. F. LANGFORD, Classification and unfoldings of degenerate Hopf bifurcations, J. Differential Equations, 41 (1981), pp. 375-415.