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

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DOI: 10.4236/am.2015.64063    3,575 Downloads   4,443 Views  

ABSTRACT

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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

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.

References

[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.
http://dx.doi.org/10.1137/1.9780898719147
[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.
http://dx.doi.org/10.1137/1.9780898719147
[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.
http://dx.doi.org/10.1016/0022-247X(78)90139-7
[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.
http://dx.doi.org/10.1080/00036819608840467
[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.
http://dx.doi.org/10.1006/jmaa.1999.6612
[12] Lu, Z. (1998) Global Stability for a Lotka-Volterra System with a Weakly Diagonally Dominant Matrix. Applied Ma-thematics Letters, 11, 81-84.
http://dx.doi.org/10.1016/S0893-9659(98)00015-9
[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.
http://dx.doi.org/10.1007/BF00276098
[15] Takeuchi, Y. Adachi, N. (1984) Influence of Predation on Species Coexistence in Volterra Models. Journal of Mathe-matical Biology, 70, 65-90.
http://dx.doi.org/10.1016/0025-5564(84)90047-6
[16] Takeuchi, Y. (1996) Global Dynamical Properties of Lotka-Volterra Systems. World Scientific, Singapore City.
http://dx.doi.org/10.1142/9789812830548
[17] Rao, M. (1980) Ordinary Differential Equations Theory and Applications. Pitman Press, Bath.

  
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