AM  Vol.6 No.4 , April 2015
Asymptotic Stability of Solutions of Lotka-Volterra Predator-Prey Model for Four Species
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.

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.

 
 
Top