Predator-prey interactions with delays due to juvenile maturation

SIAM Journal on Applied Mathematics

Volumn 66, Issue 3, 2006, Pages 1050-1079

  Cooke, Kenneth L ^a   Elderkin, Richard H ^a   Huang, Wenzhang ^b  

^a POMONA COLLEGE   (United States)

^b University of Alabama in Huntsville ^*  (United States)

Author keywords

Age structure; Bifurcation; Delay differential equations; Host parasite; Maturation delay; Predator prey

Indexed keywords

AGE STRUCTURE; DELAY DIFFERENTIAL EQUATIONS; HOST-PARASITE; MATURATION DELAY; PREDATOR-PREY;

BIFURCATION (MATHEMATICS); LINEAR EQUATIONS; MONTE CARLO METHODS;

MATHEMATICAL MODELS;

EID: 33745657847     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/05063135     Document Type: Article

References (28)

1. BERETTA, E. AND Y. KUANG, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), pp. 1144-1165.
2. C. J. BRIGGS, R. M. NISBET, AND W. W. MURDOCH, Delayed feedback and multiple attractors in a host-parasitoid system, J. Math. Biol., 38 (1999), pp. 317-345.
3. C. J. BRIGGS AND H. C. J. GODFRAY, The dynamics of insect-pathogen interactions in stage-structured populations, Amer. Naturalist, 145 (1995), pp. 855-887.
4. M. CAVANI, M. LIZANA, AND H. SMITH, Stable periodic orbits for a predator-prey model with delay, J. Math. Anal. Appl., 249 (2000), pp. 324-339.
5. K. L. COOKE AND Z. GROSSMAN, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86 (1982), pp. 592-627.
6. K. L. COOKE, R. H. ELDERKIN, AND W. HUANG, Appendix of graphics, http://www.pages.pomona.edu/~relderkin/pub/GraphAppend.pdf (2004).
7. K. L. COOKE AND P. VAN DEN DRIESSCHE, On zeroes of some transcendental equations, Funkcial. Ekvac., 29 (1986), pp. 77-90.
8. K. L. COOKE, P. VAN DEN DRIESSCHE, AND X. ZOU, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol., 39 (1999), pp. 332-352.
9. J. M. CUSHING, Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomath. 20, Springer-Verlag, Berlin/Heidelberg/New York, 1977.
10. O. DIEKMANN, S. A. VAN GILS, S. M. V. LUNEL, AND H.-O. WALTHER, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995.
11. H. FREEDMAN, I. Deterministic Mathematical Models in Population Ecology, 2nd ed., HIFR Consulting, Edmonton, 1987.
12. K. GOPALSAMY, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, Dordrecht/Boston/London, 1992.
13. S. A. GOURLEY AND Y. KUANG, A stage structured predator-prey model and its dependence on maturation delay and death rate, J. Math. Biol., 49 (2004), pp. 188-200.
14. J. K. HALE AND S. M. V. LUNEL, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
15. M. P. HASSELL AND S. W. PACALA, Heterogeneity and the dynamics of host-parasitoid interactions, Philos. Trans. R. Soc. Lond. Ser. B, 330 (1990), pp. 203-220.
16. A. HASTINGS, Age-dependent predation is not a simple process. I. Continuous time models, Theoret. Pop. Biol., 23 (1983), pp. 347-362.
17. A. HASTINGS, Delays in recruitment at different trophic levels: Effects on stability, J. Math. Biol., 21 (1984), pp. 35-44.
18. K. ENGELBORGHS, T. LUZYANINA, AND D. ROOSE, Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Trans. Math. Software, 28 (2002), pp. 1-21.
19. Y. KUANG, Delay Differential Equations, Academic Press, Boston, 1993.
20. S. LEELA, AND V. LAKSHMIKANTHAM, Differential and Integral Inequalities, Academic Press, New York, 1969.
21. W. W. MURDOCH, C. J. BRIGGS, AND R. M. NISBET, Dynamical effects of host size- and parasitoid state-dependent attacks by parasitoids, J. Animal Ecol., 66 (1997), pp. 542-556.
22. W. W. MURDOCH, R. M. NISBET, S. P. BLYTHE, W. S. C. GURNEY, AND J. D. REEVE, An invulnerable age class and stability in delay-differential parasitoid-host models, Amer. Naturalist, 129 (1987), pp. 263-282.
23. J. D. MURRAY, Mathematical Biology, Springer-Verlag, Berlin, 1989.
24. R. M. NISBET AND W. S. C. GURNEY, Modelling Fluctuating Populations, John Wiley & Sons, New York, 1982.
25. L. NUNNEY, The effect of long time delays in predator-prey systems, Theoret. Pop. Biol., 27 (1985a), pp. 202-221.
26. L. NUNNEY, Absolute stability in predator-prey models, Theoret. Pop. Biol., 28 (1985b), pp. 209-232.
27. L. NUNNEY, Short time delays in population models: A role in enhancing stability, Ecology, 66 (1985c), pp. 1849-1858.
28. P. TURCHIN, A. D. TAYLOR, AND J. D. REEVE, Dynamical role of predators in population cycles of a forest insect: An experimental test, Science, 285 (1999), pp. 1068-1071.