Uniformly asymptotic stability of almost periodic solutions for a delay difference system of plankton allelopathy
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RESEARCH
Open Access
Uniformly asymptotic stability of almost periodic solutions for a delay difference system of plankton allelopathy Qinglong Wang and Zhijun Liu* * Correspondence: [email protected] Department of Mathematics, Hubei University for Nationalities, Enshi, Hubei 445000, P.R. China
Abstract In this contribution, we investigate a delayed difference almost periodic system for the growth of two species of plankton with competition and allelopathic effects on each other. By using the methods of Lyapunov function and preliminary lemmas, sufficient conditions which guarantee the existence and uniformly asymptotic stability of a unique positive almost periodic solution of the system are established. An example together with its numerical simulations is presented to verify the validity of the proposed criteria. Keywords: delay difference system; allelopathy; almost periodic solutions; uniformly asymptotic stability; Lyapunov function
1 Introduction Allelopathy is a biological phenomenon by which individuals of a population release one or more biochemicals that have an effect on the growth, survival, and reproduction of the individuals of another population. As an important factor for ecosystem functioning, allelopathic interactions have occurred in various aspects: between bacteria [], between bacteria and phytoplankton [, ], between phytoplankton and zooplankton [], and also between calanoid copepods []. Especially, allelopathic interactions are widespread in phytoplankton communities, which deeply attract the attention of researchers. Thus, in aquatic ecology, the study of tremendous fluctuations in abundance of many phytoplankton communities is a significant theme. Recently, many workers have been aware that the increased population of one species of phytoplankton might restrain the growth of one or several other species by the production of allelopathic toxins. For detailed literature studies, we can refer to [–] and the references cited therein. In [], Qin and Liu discussed the permanence and global attractivity of the following delay difference system with plankton allelopathy: ⎧ ⎪ x (n + ) = x (n) exp{r (n) – a (n)x (n) – a (n)x (n) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ – b (n)x (n) M ⎪ p= k (p)x (n – p)}, ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x (n + ) = x (n) exp{r (n) – a (n)x (n) – a (n)x (n) – b (n)x (n) M p= k (p)x (n – p)}, xi () ≥ ,
∈ [–p, ] ∩ Z;
xi () > ,
(.)
i = , ,
©2013 Wang and Liu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Wang and Liu Advances in Difference Equations 2013, 2013:283 http://www.advancesindifferenceequations.com/content/2013/1/283
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where xi (n) are the population densities of species xi at the nth generation, ri (n) stand for the intrinsic growth rates of species xi at the nth generation, aii (n) ar
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