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

Debasis Mukherjee

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 37, Issue: 2, page 339-344
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Mukherjee, Debasis. "Persistence and bifurcation analysis on a predator–prey system of holling type." ESAIM: Mathematical Modelling and Numerical Analysis 37.2 (2010): 339-344. <http://eudml.org/doc/194166>.

@article{Mukherjee2010,
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. },
author = {Mukherjee, Debasis},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Persistance; bifurcation; stability; holling type II.; persistence; Holling type II},
language = {eng},
month = {3},
number = {2},
pages = {339-344},
publisher = {EDP Sciences},
title = {Persistence and bifurcation analysis on a predator–prey system of holling type},
url = {http://eudml.org/doc/194166},
volume = {37},
year = {2010},
}

TY - JOUR
AU - Mukherjee, Debasis
TI - Persistence and bifurcation analysis on a predator–prey system of holling type
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 2
SP - 339
EP - 344
AB - 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.
LA - eng
KW - Persistance; bifurcation; stability; holling type II.; persistence; Holling type II
UR - http://eudml.org/doc/194166
ER -

References

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  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.  Zbl0876.92021
  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.  Zbl0882.92029
  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. Ecology63 (1994) 644-652.  
  6. M. Farkas and H.I. Freedman, The stable coexistence of competing species on a renewable resource. 138 (1989) 461-472.  Zbl0661.92021
  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.  Zbl0535.92024
  8. J.K. Hale and P. Waltman, Persistence in infinite dimensional systems. SIAM J. Math. Anal.20 (1989) 388-395.  Zbl0692.34053
  9. Y. Kuang, Non uniqueness of limit cycles of Gause type predator-prey systems. Appl. Anal.29 (1988) 269-287.  Zbl0629.34036
  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.  Zbl0675.92017
  11. Y. Kuang, Limit cycles in a chemostat related model. SIAM J. Appl. Math. 49 (1989) 1759-1767.  Zbl0683.34021
  12. Y. Kuang, Global stability of Gause type predator-prey systems. J. Math. Biol.28 (1990) 463-474.  Zbl0742.92022
  13. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics. Academic Press, San Diego (1993).  Zbl0777.34002
  14. Y. Kuang and H.I. Freedman, Uniqueness of limit cycles in Gause type predator-prey systems. Math. Biosci.88 (1988) 67-84.  Zbl0642.92016
  15. R.M. May, Time-delayversus stability in population models with two and three trophic levels. Ecology54 (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.  Zbl0958.34058
  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. Ecology57 (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.  Zbl0731.34085

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