Asymptotic Stability of Solutions of Lotka-Volterra Predator-Prey Model for Four Species

DOI: 10.4236/am.2015.64063   PDF   HTML   XML   3,928 Downloads   5,240 Views   Citations


In this paper, we consider Lotka-Volterra predator-prey model between one and three species. Two cases are distinguished. The first is Lotka-Volterra model of one prey-three predators and the second is Lotka-Volterra model of one predator-three preys. The existence conditions of nonnega-tive equilibrium points are established. The local stability analysis of the system is carried out.

Share and Cite:

Soliman, A. and Al-Jarallah, E. (2015) Asymptotic Stability of Solutions of Lotka-Volterra Predator-Prey Model for Four Species. Applied Mathematics, 6, 684-693. doi: 10.4236/am.2015.64063.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] May, R.M. (1973) Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton.
[2] Edelstein-Keshet, L. (2005) Mathematical Models in Biology. Society for Industrial and Applied Mathematics, New York.
[3] Farkas, M. Dynamical Models in Biology. Elsevier Science and Technology Books, 200.
[4] Freedman, H.I. (1980) Deterministic Mathematical Models in Population Ecology. Marcel Dekker, Inc., New York.
[5] Murray, J.D. (2002) Mathematical Biology, Interdisciplinary Applied Mathematics. Springer, Berlin.
[6] Perthame, B. (2007) Transport Equations in Biology. Birkh?user Verlag, Basel.
[7] Solimano, F. and Berettra, E. (1982) Graph Theoretical Criteria for Stability and Boundedness of Predator-Prey System. Bulletin of Mathematical Biology, 44, 579-585.
[8] Takeuchi, Y., Adachi, N. and Tokumaru, H. (1978) The Stability of Generalized Volterra Equations. Journal of Mathe-matical Analysis and Applications, 62, 453-473.
[9] Ji, X.-H. (1996) The Existence of Globally Stable Equilibria of N-Dimensional Lotka-Volterra Systems. Applicable Analysis: An International Journal, 62, 11-28.
[10] Arrowsmith, D.K. and Place, C.M. (1982) Ordinary Differential Equation. Chapman and Hall, New York.
[11] Li, X.-H., Tang, C.-L and Ji, X.-H. (1999) The Criteria for Globally Stable Equilibrium in N-Dimensional Lotka-Vol-terra Systems. Journal of Mathematical Analysis and Applications, 240, 600-606.
[12] Lu, Z. (1998) Global Stability for a Lotka-Volterra System with a Weakly Diagonally Dominant Matrix. Applied Ma-thematics Letters, 11, 81-84.
[13] Liu, J. (2003) A First Course in the Qualitative Theory of Differential Equations. Person Education, Inc., New York.
[14] Takeuchi, Y. and Adachi, N. (1980) The Existence of Globally Stable Equilibria of Ecosystems of the Generalized Volterratyp. Journal of Mathematical Biology, 10, 401-415.
[15] Takeuchi, Y. Adachi, N. (1984) Influence of Predation on Species Coexistence in Volterra Models. Journal of Mathe-matical Biology, 70, 65-90.
[16] Takeuchi, Y. (1996) Global Dynamical Properties of Lotka-Volterra Systems. World Scientific, Singapore City.
[17] Rao, M. (1980) Ordinary Differential Equations Theory and Applications. Pitman Press, Bath.

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.