Persistence and bifurcation analysis on a predator-prey system of holling type
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 2, p. 339-344

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.

DOI : https://doi.org/10.1051/m2an:2003029
Classification:  34D23,  34D45,  92D25
Keywords: persistance, bifurcation, stability, holling type II
@article{M2AN_2003__37_2_339_0,
     author = {Mukherjee, Debasis},
     title = {Persistence and bifurcation analysis on a predator-prey system of holling type},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {2},
     year = {2003},
     pages = {339-344},
     doi = {10.1051/m2an:2003029},
     zbl = {1029.34040},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2003__37_2_339_0}
}
Mukherjee, Debasis. Persistence and bifurcation analysis on a predator-prey system of holling type. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 2, pp. 339-344. doi : 10.1051/m2an:2003029. http://www.numdam.org/item/M2AN_2003__37_2_339_0/

[1] V.D. Adams, D.L. Deangelis and R.A. Goldstein, Stability analysis of the time delay in a Host-Parasitoid Model. J. Theoret. Biol. 83 (1980) 43-62.

[2] E. Beretta and Y. Kuang, Convergence results in a well known delayed predator-prey system. J. Math. Anal. Appl. 204 (1996) 840-853. | Zbl 0876.92021

[3] A.A. Berryman, The origins and evolution of predator-prey theory. Ecology 73 (1992) 1530-1535.

[4] Y. Cao and H.I. Freedman, Global attractivity in time delayed predator-prey system. J. Austral. Math. Soc. Ser. B. 38 (1996) 149-270. | Zbl 0882.92029

[5] B.W. Dale, L.G. Adams and R.T. Bowyer, Functional response of wolves preying on barren ground caribou in a multiple prey ecosystem. J. Anim. Ecology 63 (1994) 644-652.

[6] M. Farkas and H.I. Freedman, The stable coexistence of competing species on a renewable resource. 138 (1989) 461-472. | Zbl 0661.92021

[7] H.I. Freedman and V.S.H. Rao, The trade-off between mutual interface and time lags in predator-prey systems. Bull. Math. Biol. 45 (1983) 991-1004. | Zbl 0535.92024

[8] J.K. Hale and P. Waltman, Persistence in infinite dimensional systems. SIAM J. Math. Anal. 20 (1989) 388-395. | Zbl 0692.34053

[9] Y. Kuang, Non uniqueness of limit cycles of Gause type predator-prey systems. Appl. Anal. 29 (1988) 269-287. | Zbl 0629.34036

[10] Y. Kuang, On the location and period of limit cycles in Gause type predator-prey systems. J. Math. Anal. Appl. 142 (1989) 130-143. | Zbl 0675.92017

[11] Y. Kuang, Limit cycles in a chemostat related model. SIAM J. Appl. Math. 49 (1989) 1759-1767. | Zbl 0683.34021

[12] Y. Kuang, Global stability of Gause type predator-prey systems. J. Math. Biol. 28 (1990) 463-474. | Zbl 0742.92022

[13] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics. Academic Press, San Diego (1993). | MR 1218880 | Zbl 0777.34002

[14] Y. Kuang and H.I. Freedman, Uniqueness of limit cycles in Gause type predator-prey systems. Math. Biosci. 88 (1988) 67-84. | Zbl 0642.92016

[15] R.M. May, Time-delay versus stability in population models with two and three trophic levels. Ecology 54 (1973) 315-325.

[16] D. Mukherjee and A.B. Roy, Uniform persistence and global attractivity theorem for generalized prey-predator system with time delay. Nonlinear Anal. 38 (1999) 59-74. | Zbl 0958.34058

[17] R.E. Ricklefs and G.L. Miller, Ecology. W.H. Freeman and Company, New York (2000).

[18] C.E. Taylor and R.R. Sokal, Oscillations of housefly population sizes due to time lags. Ecology 57 (1976) 1060-1067.

[19] B.G. Vielleux, An analysis of the predatory interactions between Paramecium and Didinium, J. Anim. Ecol. 48 (1979) 787-803.

[20] W.D. Wang and Z.E. Ma, Harmless delays for uniform persistence. J. Math. Anal. Appl. 158 (1991) 256-268. | Zbl 0731.34085