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Dynamic behaviors of a delay differential equation model of plankton allelopathy

Department of Mathematics , Palacký University of Olomouc, Olmütz, Olomoucký, Czech Republic
Journal of Computational and Applied Mathematics (Impact Factor: 1.08). 09/2007; 206(2):733-754. DOI: 10.1016/j.cam.2006.08.020

ABSTRACT In this paper, we consider a modified delay differential equation model of the growth of n-species of plankton having competitive and allelopathic effects on each other. We first obtain the sufficient conditions which guarantee the permanence of the system. As a corollary, for periodic case, we obtain a set of delay-dependent condition which ensures the existence of at least one positive periodic solution of the system. After that, by means of a suitable Lyapunov functional, sufficient conditions are derived for the global attractivity of the system. For the two-dimensional case, under some suitable assumptions, we prove that one of the components will be driven to extinction while the other will stabilize at a certain solution of a logistic equation. Examples show the feasibility of the main results.

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