Dynamic behaviors of a delay differential equation model of plankton allelopathy

Authors:
Fengde Chen;Zhong Li;Xiaoxing Chen;Jitka Laitochová
Affiliations:
College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, PR China;College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, PR China;College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, PR China;Department of Mathematics, Faculty of Education, Palacký University, Olomouc, Czech Republic
Venue:
Journal of Computational and Applied Mathematics
Year:
2007

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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.