Persistence in Non-Autonomous Lotka-Volterra System with Predator-Prey Ratio-Dependence and Density Dependence

Haiyin Li
DOI: 10.4236/am.2011.29159                

Abstract

The main purpose of this article is considering the persistence non-autonomous Lotka-Volterra system with predator-prey ratio-dependence and density dependence. We get the sufficient conditions of persistence of system, further have the necessary conditions, also the uniform persistence condition, which can be easily checked for the model is obtained.

Keywords

Uniform Persistence, Density Dependent Predator, Ratio-Dependent, Non-Autonomous Lotka-Volterra System

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	J. Cui, “Permanence of Predator-Prey System with Periodic Coef?cients,” Mathematical and Computer Modelling, Vol. 42, No. 1-2, 2005, pp. 87-98. doi:10.1016/j.mcm.2005.03.001
[2]	Z. Teng, “Uniform Persistence of the Periodic Predator-Prey Lotka-Volterra Systems,” Applied Analysis, Vol. 72, 1998, pp. 339-352.
[3]	Y. Kuang and H. I. Freedman, “Uniqueness of Limit Cycles in Cause Type Models of Predator-Prey System,” Mathematical Biosciences, Vol. 88, 1988, pp. 67-84. doi:10.1016/0025-5564(88)90049-1
[4]	K. S. Cheng, “Uniqueness of a Limit Cycle for a Predator-Prey System,” SIAM Journal on Mathematical Analysis, Vol. 12, 1981, pp. 541-548. doi:10.1137/0512047
[5]	H. I. Freedman, “Deterministic Mathematical Models in Population Ecology,” Marcel Dekker, New York, 1980.
[6]	S. B. Hsu and T. W. Huang, “Global Stability for a Class of Predator-Prey Systems,” SIAM Journal on Mathematical Analysis, Vol. 55, No. 3, 1995, pp. 763-783. doi:10.1137/S0036139993253201
[7]	H. R. Akcakaya, “Population Cycles of Mammals: Evidence for a Ratio-Dependent Predation Hypothesis,” Ecological Monographs, Vol. 62, No. 1, 1992, pp. 119- 142. doi:10.2307/2937172
[8]	R. Arditi and L. R. Ginzburg, “Coupling in Predator-Prey Dynamics: Ratio-Dependence,” Journal of Theoretical Biology, Vol. 139, No. 3, 1989, pp. 311-326. doi:10.1016/S0022-5193(89)80211-5
[9]	Y. Kuang and E. Beretta, “Global Qualitative Analysis of a Ratio-Dependence Predator-Prey System,” Journal of Mathematical Biology, Vol. 36, No. 4, 1998, pp. 389-406. doi:10.1007/s002850050105
[10]	H. Li and Y. Takeuchi, “Dynamics of a Nonautonomous Density Dependent and Ratio-Dependent Predator-Prey System,” Journal of Mathematical Analysis and Applications, Vol. 374, No. 1, 2011, pp. 644-654. doi:10.1016/j.jmaa.2010.08.029
[11]	H. Li and Y. Takeuchi, “Stability for Ratio-Dependent Predator-Prey System with Density Dependent,” Proceedings of the 7th Conference on Biological Dynamic System and Stability of Differential Equation, Chongqing, 14-16 May 2010, pp. 144-147.
[12]	Z. Lu and H. Li, “Stability of Ratio-Dependent Delayed Predator-Prey System with Density Regulation,” Journal of Biomathematics, Vol. 20, No. 3, 2005, pp. 264-272.
[13]	P. Kratina, M. Vos, A. Bateman and B. R. Anholt, “Functional Responses Modified by Predator Density,” Oecologia, Vol. 159, No. 2, 2009, pp. 425-433.