GLOBAL STABILITY OF A DELAYED RATIO-DEPENDENT PREDATOR-PREY MODEL WITH GOMPERTZ GROWTH FOR PREY

Shihua Zhang; Rui Xu

doi:10.11948/2015003

2015 Volume 5 Issue 1

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Shihua Zhang, Rui Xu. GLOBAL STABILITY OF A DELAYED RATIO-DEPENDENT PREDATOR-PREY MODEL WITH GOMPERTZ GROWTH FOR PREY[J]. Journal of Applied Analysis & Computation, 2015, 5(1): 28-37. doi: 10.11948/2015003

Citation:

Shihua Zhang, Rui Xu. GLOBAL STABILITY OF A DELAYED RATIO-DEPENDENT PREDATOR-PREY MODEL WITH GOMPERTZ GROWTH FOR PREY[J]. Journal of Applied Analysis & Computation, 2015, 5(1): 28-37. doi: 10.11948/2015003

GLOBAL STABILITY OF A DELAYED RATIO-DEPENDENT PREDATOR-PREY MODEL WITH GOMPERTZ GROWTH FOR PREY

Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, No. 97 Heping West Road, Shijiazhuang 050003, Hebei Province, China

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Abstract

Abstract

A delayed ratio-dependent predator-prey model with Gompertz growth for prey is investigated. The local stability of a predator-extinction equilibrium and a coexistence equilibrium is discussed. Furthermore, the existence of Hopf bifurcation at the coexistence equilibrium is established. By constructing a Lyapunov functional, sufficient conditions are obtained for the global stability of the coexistence equilibrium.
- Predator-prey model /
- ratio-dependence /
- Gompertz growth /
- stability /
- bifurcation
MSC: 34D12

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References

[1]	R. Arditi, L.R. Ginzburg and H.R. Akcakaya, Variation in plankton densities among lakes:A case for ratio-dependent models, Am. Nat., 138(1991), 1287-1296. Google Scholar
[2]	R. Arditi, N. Perrin and H. Saiah, Founctional response and heterogeneities:An experiment test with cladocerans, OIKOS,60(1991), 69-75. Google Scholar
[3]	R. Arditi and L.R. Ginzbuig, Coupling in predator-prey dynamics:Ratiodependence, J. Theoretical Biol., 139(1989), 311-326. Google Scholar
[4]	J. Dieuonné, Foundations of Modern Analysis, Academic, 1960. Google Scholar
[5]	H.I. Freedman and V.S.H. Rao, The tradeoff between mutual interference and time lag in predator-prey models, Bull. Math. Biol., 45(1983), 991-1004. Google Scholar
[6]	H.I. Freedman, J. So and P. Waltman, Coexistnce in a model of competition in the chemostat incorporating discrete time delays, SIAM J. Appl.Math., 49(1989), 859-870. Google Scholar
[7]	G. Gompertz, On the nature of the function expressive of the law of human mortality, and on the new mode of determining the value of life contingencies, Philos. Trans. R. Soc. London, 115(1825), 513-585. Google Scholar
[8]	I. Hanski, The functional response of predator:Worries about scale, TREE, 6(1991), 141-142. Google Scholar
[9]	J.K. Hale and S. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. Google Scholar
[10]	Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. Google Scholar
[11]	A.K. Laird, Dynamics of tumour growth:comparison of growth rates and extrapolation of growth curve to one cell, Br. J. Cancer, 19(1965), 278-291. Google Scholar
[12]	A.K. Laird, S.A. Tyler and A.D. Barton, Dynamics of normal growth, Growth, 21(1965), 233-248. Google Scholar
[13]	R. Xu and L. Chen, Persistence and global stability for n-species ratiodependent predator-prey system with time delays, J. Math. Anal. Appl., 275(2002), 27-43. Google Scholar