Dynamic behaviors of a delay differential equation model of plankton allelopathy

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 obta...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2007-09, Vol.206 (2), p.733-754
Main Authors: Chen, Fengde, Li, Zhong, Chen, Xiaoxing, Laitochová, Jitka
Format: Article
Language:English
Subjects:
Competition
Exact sciences and technology
Extinction
Global attractivity
Lyapunov functional
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Ordinary differential equations
Permanence
Sciences and techniques of general use
Toxicology
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Summary: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.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2006.08.020