Positive periodic solutions for a model of gene regulatory system with time-varying coefficients and delays
- Wei Chen^{1} and
- Wentao Wang^{2}Email author
https://doi.org/10.1186/s13662-016-0788-4
© Chen and Wang 2016
Received: 2 November 2015
Accepted: 22 February 2016
Published: 29 February 2016
Abstract
The paper is concerned with periodic solutions of a model of gene regulatory system with time-varying coefficients and delays. We establish some sufficient conditions for the existence, positivity, and permanence of solutions, which help to derive the global exponential stability of positive periodic solutions for this model. Our method depends on differential inequality technique and Lyapunov functional. At last, we give an example and its numerical simulations to verify theoretical results.
Keywords
gene regulatory system delay periodic solution global exponential stability1 Introduction
In the real-world phenomena, the periodic variation of the environment (e.g., temperature, moisture, pressure, seasonal effects of weather, reproduction, food supplies, mating habits, etc.) plays a pivotal role in determining the dynamics, so that some classic models, such as the Nicholson blowflies model [13, 14], hematopoiesis model [15, 16], etc., have been generalized to the nonautonomous nonlinear delay differential equation with time-varying coefficients and delays. Consequently, it is worth studying the model of gene regulatory system with time-varying coefficients and delays.
Let \(R^{n}\) (\(R_{+}^{n}\)) be the set of all (nonnegative) real vectors; by \(x=(x_{1},\ldots,x_{n})^{T}\in R^{n}\) we denote a column vector, in which the symbol \(({}^{T})\) denotes the transpose of a vector. We denote by \(|x|\) the absolute-value vector \(|x|=(|x_{1}|,\ldots,|x_{n}|)^{T}\) and define \(\|x\|=\max_{1\leq i\leq n}|x_{i}|\). For \(\tau^{+}>0\), we denote by \(C= C([-\tau^{+},0], R^{2})\) the Banach space equipped with the supremum norm, that is, \(\|\varphi\|=\sup_{-\tau^{+}\leq t\leq0}\max_{1\leq i\leq2}|\varphi_{i}(t)|\) for all \(\varphi(t)=(\varphi _{1}(t),\varphi_{2}(t))^{T} \in C\). Let \(C_{+}=\{\varphi\in C| \varphi(t)\in R_{+}^{2} \mbox{ for } t\in [-\tau^{+},0]\}\). If \(x_{i}(t)\) is defined on \([t_{0}-\tau^{+},\nu)\) with \(t_{0}, \nu\in R\) and \(i =1,2 \), then we define \(x_{t}\in C \) as \(x_{t}=(x_{t}^{1}, x_{t}^{2} )^{T}\) where \(x^{i}_{t}(\theta)=x_{i}(t+\theta)\) for all \(\theta\in[-\tau^{+} ,0]\) and \(i =1,2\).
Definition 1.1
The objective of this paper is twofold. The first is getting the attracting set for system (1.3). The other is deriving conditions on the existence, uniqueness, and global exponential stability of positive periodic solutions. Finally, we give an example and its numerical simulations to illustrate our main results.
2 Preliminary results
In this section, we derive the following lemmas, which will be used to prove our main results in Section 3.
Lemma 2.1
Proof
Since the initial value φ satisfies (1.4), Eq. (2.2) leads to \(x_{i}(t)>0 \) (\(i=1,2\)) for \(t\in[t_{0}, t_{0}+\tau^{+}]\). Then, by the method of steps we obtain that \(x(t;t_{0},\varphi)\) is positive and exists on \([t_{0}-\tau^{+},+\infty)\).
Lemma 2.2
Proof
3 Main results
In this section, we establish sufficient conditions on the existence, uniqueness, and global exponential stability of positive ω-periodic solutions for system (1.3).
Theorem 3.1
Suppose that all conditions in Lemma 2.2 are satisfied. Then system (1.3) has exactly one positive ω-periodic solution \(\widetilde {x}(t)\). Moreover, \(\widetilde{x}(t)\) is globally exponentially stable.
Proof
4 An example
In this section, we give an example and numerical simulations to demonstrate the results obtained in previous sections.
Example 4.1
Remark 4.1
To the best of our knowledge, rare authors studied the problems of the global exponential stability of positive periodic solutions for genetic regulatory system with time-varying coefficients and delays. It is obvious that all results in [3, 4] and the references therein cannot be applicable to prove that all solutions of system (4.1) converge exponentially to a positive periodic solution since the system has time-varying coefficients and delays. This implies that the results of this paper are generalization and complement of previously known results.
Declarations
Acknowledgements
We would like to thank the anonymous referees for valuable comments and suggestions that greatly helped us to improve the paper. Financial support by the National Natural Science Foundation of China (grant no. 11301341) and the Natural Scientific Research Fund of Zhejiang Provincial of China (grant no. LY16A010018) is gratefully acknowledged.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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