Dynamic consequences of prey refuges in a simple model system: More prey, fewer predators and enhanced stability

Ecological Modelling

Volumn 166, Issue 1-2, 2003, Pages 135-146

  González Olivares, Eduarde ^a   Ramos Jiliberto, Rodrigo ^a  

^a PONTIFICIA UNIVERSIDAD CATÓLICA DE VALPARAÍSO   (Chile)

Author keywords

Bifurcation; Limit cycles; Predator prey models; Refuge; Stability

Indexed keywords

ECOLOGICAL MODELING; HABITAT USE; PREDATOR-PREY INTERACTION; REFUGE;

EID: 0348077455     PISSN: 03043800     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-3800(03)00131-5     Document Type: Article

References (39)

1. Andronov, A.A., Leontovich, E.A., Gordon, I., Maier, A.G., 1973. Qualitative Theory of Second-Order Dynamic Systems. A Halsted Press Book. Wiley, New York, p. 524.
2. Ardito, A., Ricciardi, P., 1995. Lyapunov functions for a generalized Gause-type model. J. Math. Biol. 33, 816-828.
3. Arrowsmith, D.K., Place, C.M., 1992. Dynamical System. Differential Equations, Maps and Chaotic Behaviour. Chapman and Hall, London, p. 330.
4. Berryman, A.A., Gutierrez, A.P., Arditi, R., 1995. Credible, parsimonious and useful predator-prey models - a reply to Abrams, Gleeson, and Sarnelle. Ecology 76 (6), 1980-1985.
5. Clark, C.W., 1990. Mathematical Bioeconomic: The Optimal Management of Renewable Resources, 2nd ed. Wiley, New York, p. 291.
6. Collings, J.B., 1995. Bifurcation and stability analysis of a temperature-dependent mite predator-prey interaction model incorporating a prey refuge. Bull. Math. Biol. 57 (1), 63-76.
7. Cheng, K.-S., 1981. Uniqueness of a limit cycle for predator-prey system. SIAM J. Math. Anal. 12 (4), 541-548.
8. o 32, Instituto de Matemática Pura e Aplicada IMPA, Río de Janeiro, p. 191.
9. Edelstein-Keshet, L., 1988. Mathematical Models in Biology. McGraw-Hill, New York, p. 586.
10. Freedman, H.I., 1980. Deterministic Mathematical Model in Population Ecology. Marcel Dekker, New York, p. 254.
11. Getz, W.M., 1996. A hypothesis regarding the abruptness of density dependence and the growth rate populations. Ecology 77 (7), 2014-2026.
12. Ginzburg, L.R., 1988. Assuming reproduction to be a function of consumption raises doubts about some popular predator-prey models. J. Anim. Ecol. 67, 325-327.
13. Goh, B.-S., 1980. Management and Analysis of Biological Populations. Elsevier, Amsterdam, p. 279.
14. González-Olivares, E., 1998. Una clase de modelos bioeconómicos continuos. Investigaciones Marinas 26, 109-117.
15. Guckenheimer, J., Holmes, P., 1983. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, vol. 42. Springer, New York, p. 206.
16. Ives, A.R., Dobson, A.P., 1987. Antipredator behavior and the population dynamics of simple predator-prey systems. The Am. Naturalist 130, 431-447.
17. Harrison, G.W., 1979. Global stability of predator-prey interactions. J. Math. Biol. 8, 39-171.
18. Hasík, K., 2000. Uniqueness of limit cycle in the predator-prey system with symmetric prey isocline. Math. Biosci. 164, 203-215.
19. Hesaaraki, M., Moghadas, S.M., 2001. Existence of limit cycles for predator-prey systems with a class of functional responses. Ecol. Model. 142, 1-9.
20. Holling, C.S., 1959. Some characteristics of simple types of predation and parasitism. The Can. Entomol. 91, 385-398.
21. May, R.M., 1974. Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton, p. 261.
22. Maynard Smith, J., 1974. Models in Ecology. Cambridge University Press, Cambridge, p. 25.
23. McNair, J.M., 1986. The effects of refuges on predator-prey interactions: a reconsideration. Theor. Popul. Biol. 29, 38-63.
24. Murdoch, W.W., Oaten, A., 1975. Predation and population stability. Adv. Ecol. Res. 9, 2-132.
25. Murray, J.D., 1989. Mathematical Biology. Springer, Berlin, p. 767.
26. Real, L.A., Levin, S.A., 1991. The role of theory in the rise of modern ecology. In: Real, L.A., Brown, J.H. (Eds.), Foundations of Ecology. Classic Papers with Commentaries. The University of Chicago Press, Chicago, pp. 177-191.
27. Rosenzweig, M.L., 1971. Paradox of enrichment: destabilization of exploitation ecosystem in ecological time. Science 171, 385-387.
28. Ruxton, G.D., 1995. Short term refuge use and stability of predator-prey models. Theor. Popul. Biol. 47, 1-17.
29. Sáez, E., González-Olivares, E., 1999. Dynamics of a predator-prey model. SIAM J. Appl. Math. 59 (5), 1867-1878.
30. Scheffer, M., de Boer, R.J., 1995. Implications of spatial heterogeneity for the paradox of enrichment. Ecology 76 (7), 2270-2277.
31. Sih, A., 1987. Prey refuges and predator-prey stability. Theor. Popul. Biol. 31, 1-12.
32. Sih, A., Petranka, J.W., Kats, L.B., 1988. The dynamics of prey refuge use: a model and tests with sunfish and salamanders larvae. The Am. Naturalist 132 (4), 463-483.
33. Sotomayor, J., 1979. Liçoes de Equaçoes Diferenciais Ordinarias. Projeto Euclides IMPA. CNPq, Rio de Janeiro, p. 327.
34. Strogatz, S.H., 1994. Nonlinear Dynamics and Chaos. With Applications to Physics, Biology, Chemistry, and Engineering. Perseus Publishing, Cambridge, p. 498.
35. Sugie, J., Kohno, R., Miyazaki, R., 1997. On a predator-prey system of Holling type. Proc. Am. Math. Soc. 125 (7), 2041-2059.
36. Sugie, J., Katayama, M., 1999. Global asymptotic stability of a predator-prey system of Holling type. Nonlinear Anal. 38, 105-121.
37. Sunhong, D., 1989. On a kind of predator-prey system. SIAM J. Math. Anal. 20 (6), 1426-1435.
38. Taylor, R.J., 1984. Predation. Chapman and Hall, New York, p. 166.
39. Yodzis, P., 1989. Introduction to Theoretical Ecology. Harper and Row, New York, p. 384.