Hostname: page-component-7c8c6479df-5xszh Total loading time: 0 Render date: 2024-03-28T14:56:53.530Z Has data issue: false hasContentIssue false

Persistence and bifurcation analysis on a predator–prey system of holling type

Published online by Cambridge University Press:  15 November 2003

Debasis Mukherjee*
Affiliation:
Department of Mathematics, Vivekananda College, Thakurpukur, Kolkata 700063, India. debasis_mukherjee2000@yahoo.co.in.
Get access

Abstract

We present a Gause type predator–prey model incorporating delay due to response of prey population growth to density and gestation. The functional response of predator is assumed to be of Holling type II. In absence of prey, predator has a density dependent death rate. Sufficient criterion for uniform persistence is derived. Conditions are found out for which system undergoes a Hopf–bifurcation.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adams, V.D., DeAngelis, D.L. and Goldstein, R.A., Stability analysis of the time delay in a Host-Parasitoid Model. J. Theoret. Biol. 83 (1980) 43-62. CrossRef
Beretta, E. and Kuang, Y., Convergence results in a well known delayed predator-prey system. J. Math. Anal. Appl. 204 (1996) 840-853.
Berryman, A.A., The origins and evolution of predator-prey theory. Ecology 73 (1992) 1530-1535. CrossRef
Cao, Y. and Freedman, H.I., Global attractivity in time delayed predator-prey system. J. Austral. Math. Soc. Ser. B. 38 (1996) 149-270. CrossRef
Dale, B.W., Adams, L.G. and Bowyer, R.T., Functional response of wolves preying on barren ground caribou in a multiple prey ecosystem. J. Anim. Ecology 63 (1994) 644-652. CrossRef
M. Farkas and H.I. Freedman, The stable coexistence of competing species on a renewable resource. 138 (1989) 461-472.
Freedman, H.I. and Rao, V.S.H., The trade-off between mutual interface and time lags in predator-prey systems. Bull. Math. Biol. 45 (1983) 991-1004. CrossRef
Hale, J.K. and Waltman, P., Persistence in infinite dimensional systems. SIAM J. Math. Anal. 20 (1989) 388-395. CrossRef
Kuang, Y., Non uniqueness of limit cycles of Gause type predator-prey systems. Appl. Anal. 29 (1988) 269-287. CrossRef
Kuang, Y., On the location and period of limit cycles in Gause type predator-prey systems. J. Math. Anal. Appl. 142 (1989) 130-143. CrossRef
Kuang, Y., Limit cycles in a chemostat related model. SIAM J. Appl. Math. 49 (1989) 1759-1767. CrossRef
Kuang, Y., Global stability of Gause type predator-prey systems. J. Math. Biol. 28 (1990) 463-474. CrossRef
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics. Academic Press, San Diego (1993).
Kuang, Y. and Freedman, H.I., Uniqueness of limit cycles in Gause type predator-prey systems. Math. Biosci. 88 (1988) 67-84. CrossRef
Time-delay, R.M. May versus stability in population models with two and three trophic levels. Ecology 54 (1973) 315-325.
Mukherjee, D. and Uniform, A.B. Roy persistence and global attractivity theorem for generalized prey-predator system with time delay. Nonlinear Anal. 38 (1999) 59-74. CrossRef
R.E. Ricklefs and G.L. Miller, Ecology. W.H. Freeman and Company, New York (2000).
Taylor, C.E. and Sokal, R.R., Oscillations of housefly population sizes due to time lags. Ecology 57 (1976) 1060-1067. CrossRef
Vielleux, B.G., An analysis of the predatory interactions between Paramecium and Didinium, J. Anim. Ecol. 48 (1979) 787-803. CrossRef
Wang, W.D. and Harmless de, Z.E. Malays for uniform persistence. J. Math. Anal. Appl. 158 (1991) 256-268.