Global stability of mutualistic interactions among three species system
Main Article Content
Abstract
In this paper, a mathematical model consisting of mutualistic interactions among three species is proposed and analyzed. Local and global stabilities are discussed by using Routh-Hurwitz criteria and constructing Lyapunov function respectively. Finally, we illustrate our results by some numerical examples
Downloads
Article Details
References
Addicott John, F.: Mutualistic interactions in populations and community processes. In: A new ecology: Novel approaches to interactive systems (Eds.: P.W. Price, W.S. Gaud and C.N. Slobodchikoff) 437-455, John Wiley and sons, New
York, (1984).
Boucher, D. M.: The Biology of Mutualism: Ecology and Evolution. London, Croom Helm 1985.
Brauer, F., Soudack, A. C.: Mutualism models with nonlinear growth rates. Internat. J. Control 41, 1601-1612 (1985).
Bronstein, J. L.: The exploitation of mutualisms. Ecology letters 4, 277-287 (2001).
Freedman, H. I., Bindhyachal, R.: Uniform persistence and global stability in models involving mutualism. II. Cometitor-competitor-mutualism system. Indian J. Math. 30, 175-186 (1988).
Goh, B.S.: Stability in models of mutualism. Amer. Natur. 13, 261-275 (1979).
Gurel, O., Lapidus, L.: Stability via Lyapunov’s second method. Ind. and Eng. Chem. 60, 13-26 (1968).
Hale, J. K.: Ordinary Differential Equations. New York. Wiley-Interscience 1969.
Mats, G., Yan, p., Wang, Y.: Limit cycles for competitor-competitor-mutualist Lotka-Volterra systems. Physica D 221, 135-145 (2006).
May, R. M.: Stability and Complexity in Model Ecosystems. Princeton 1973.
May, R. M.: Models for two interactring populations. Theoretical ecology: principles and applications. Blackwell, Oxford 49-70 (1976).
May, R. M.: Mutualistic interactions among species. Nature 296, 803-804 (1982).
Odum, E. P.: Fundamentals of Ecology, 3d ed. Saunders, Philadelphia, 1971.
Perko, L.: Differential Equations and Dynamical Systems. New York. Springer-Verlag 1991.
Rish, S., Boucher, D. H.: What ecologist look for. Bulletin of the Ecological Society of America 57, 8-9 (1976).
Robbins, C.: Birds of North America. Racine, WI: Western Publishing Co. 1966.
Simpson, K., Day N.: Birds of Australia. Princeton: Princeton University Press 1999.
Smith, J. M.: Models in Ecology. Cambridge. Cambridge University Press 1974.
Wolin, C. L.: The Population Dynamics of Mutualistic System. In: The Biology of Mutualism (Eds.: D.H. Boucher) 248-269, Oxford University Press, New York,
(1985).
Yuanshi, W., Hong Wu: A mutualism-competition model characterizing competitors with mutualism at low density. Mathematical and Computer Modelling
, 1654-1663 (2011).