Permanence and Global Stability for a Non-Autonomous Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes with Delays

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DOI: 10.4236/am.2011.21006   PDF   HTML     4,609 Downloads   8,786 Views   Citations

Abstract

In this paper, a nonautonomous predator-prey system based on a modified version of the Leslie-Gower scheme and Holling-type II scheme with delayed effect is investigated. The general criteria of integrable form on the permanence are established. By constructing suitable Lyapunov functionals, a set of easily verifiable sufficient conditions are derived for global stability of any positive solutions to the model

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L. Hu and L. Nie, "Permanence and Global Stability for a Non-Autonomous Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes with Delays," Applied Mathematics, Vol. 2 No. 1, 2011, pp. 47-56. doi: 10.4236/am.2011.21006.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] M. Fan, Q. Wang and X. F. Zou, “Dynamics of Non-autonomous Ratio-Dependent Predator-Prey System,” Proceedings of the Royal Society of Edinburgh: Section A, Vol. 133, No. 1, 2003, pp. 97-118. doi:10.1017/S0308 210500002304
[2] Y. Kuang, “Delay Differential Equations, with Applications in Population Dynamics,” Academic Press, New York, 1993.
[3] Z. Teng, “Persistence and Stability in General Non-autonomous Single-Species Kolmogorov Systems with Delays,” Nonlinear Analysis: Real World Applications, Vol. 8, No. 1, 2007, pp. 230-248. doi:10.1016/j.nonrwa. 2005.08.003
[4] Z. Teng and M. Rehim, “Persistence in Nonautonomous Predator-Prey Systems with Infinite Delays,” Journal of Computational and Applied Mathematics, Vol. 197, No. 2, 2006, pp. 302-321. doi:10.1016/j.cam.2005.11.006
[5] R. K. Upadhyay and S. R. K. Iyengar, “Effect of Seasonality on the Dynamics of 2 and 3 Species Prey-Predator System,” Nonlinear Analysis: Real World Applications, Vol. 6, No. 3, 2005, pp. 509-530. doi:10.1016/j.nonrwa. 2004.11.001
[6] R. Xu, M. A. J. Chaplain and F. A. Davidson, “Periodic Solutions for a Predator-Prey Model with Holling-Type Functional Response and Time Delays,” Applied Mathematics and Computation, Vol. 161, No. 2, 2005, pp. 637-654. doi:10.1016/j.amc.2003.12.054
[7] M. A. Aziz-Alaoui, “Study of a Leslie-Gower-Type Tritrophic Population,” Chaos, Solitons & Fractals, Vol. 14, No. 8, 2002, pp. 1275-1293. doi:10.1016/S0960-0779(02) 00079-6
[8] E. Beretta and Y. Kuang, “Global Analyses in Some Delayed Ratio-Depended Predator-Prey Systems,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 32, No. 3, 1998, pp. 381-408.
[9] C. Letellier and M. A. Aziz-Alaoui, “Analysis of the Dynamics of a Realistic Ecological Model,” Chaos, Solitons & Fractals, Vol. 13, No. 1, 2002, pp. 95-107. doi:10.1016/S0960-0779(00)00239-3
[10] M. A. Aziz-Alaoui and M. D. Okiye, “Boundedness and Global Stability for a Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes,” Applied Mathematics Letters, Vol. 16, No. 7, 2003, pp. 1069-1075. doi:10.1016/S0893-9659(03)90096-6
[11] A. F. Nindjin and M. A. Aziz-Alaoui, “Analysis of a Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes with Time Delay,” Nonlinear Analysis: Real World Applications, Vol. 7, No. 5, 2006, pp. 1104-1118. doi:10.1016/j.nonrwa.2005.10.003
[12] R. K. Upadhyay and V. Rai, “Crisis-Limited Chaotic Dynamics in Ecological Systems,” Chaos, Solitons & Fractals, Vol. 12, No. 2, 2001, pp. 205-218. doi:10.1016/ S0960-0779(00)00141-7
[13] R. R. Vance and E. A. Coddington, “A Nonautonomous Model of Population Growth,” Journal of Mathematical Biology, Vol. 27, No. 5, 1989, pp. 491-506. doi:10.1007/ BF00288430
[14] Z. Teng and Z. Li, “Permanence Criteria in Non-autonomous Predator-Prey Kolmogorov Systems and Its Applications,” Dynamical Systems, Vol. 19, No. 2, 2004, pp. 171-194. doi:10.1080/14689360410001698851
[15] Z. Teng and L. Chen, “Uniform Persistence and Existence of Strictly Positive Solutions in Nonautonomous Lotka-Volterra Competitive Systems with Delays,” Computers & Mathematics with Applications, Vol. 37, No. 7, 1999, pp. 61-71.
[16] W. Wang and Z. Ma. “Harmless Delays for Uniform Persistence,” Journal of Mathematical Analysis and Applications, Vol. 158, No. 1, 1991, pp. 256-268. doi:10. 1016/0022-247X(91)90281-4
[17] K. Gopalsamy, “Stability and Oscillations in Delay Different Equations of Population Dynamics,” Kluwer Academic, Norwell, 1992.

  
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