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Stability analysis of linear positive systems with time delays on time scales
Advances in Difference Equations volume 2012, Article number: 56 (2012)
Abstract
This article studies the necessary and sufficient conditions for asymptotic stability of the linear positive systems with bounded timevarying delays on time scales. Using Lyapunov functions, asymptotic stability conditions for positive systems with constant delays on time scales are established. The relations between the two types of systems have been found out, which bridged the gap between their stability. These results not only unify the existing ones of linear positive continuous and discrete systems but also contain more complicated time scales.
1 Introduction
In the real world, many physical, biological and medical systems involve variables that have nonnegative sign, e.g., population levels, absolute temperature, and concentration of substances. Such systems are referred to as positive systems. A dynamical system is called positive if any trajectory of the system starting from nonnegative initial states remains forever nonnegative for nonnegative controls. An overview of state of the art in positive systems theory is given in the monographs [1–3].
Stability is one of the most important properties of dynamic systems, and a massive monograph has been concentrated on this topic for positive systems [4–9]. In the last few years, conditions for stability and robust stability of linear positive discretetime systems with delays have been given in [10–19].
Firstly, we introduce some notations. \mathbb{N}\left({\mathbb{N}}_{0}\right) denotes the set of positive (nonnegative) integers, \mathbb{R}\left({\mathbb{R}}_{0,+}\right) denotes the set of real (nonnegative) numbers, {\mathbb{R}}^{n}\phantom{\rule{0.3em}{0ex}}\left({\mathbb{R}}_{0,+}^{n},\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\mathbb{R}}_{+}^{n}\right) is ndimensional linear vector space over real (nonnegative, positive) numbers with the maximum modulus norm ∥·∥ given by ∥x∥ = max_{i=1,2,···,n}x_{ i }, x\in {\mathbb{R}}^{n}, and {\mathbb{R}}^{n\times m}\left({\mathbb{R}}_{0,+}^{n\times m}\right) is the set of all real (nonnegative) matrices of (n × m)dimension. For A\in {\mathbb{R}}^{n\times m}, we write A\succcurlyeq 0\left(\preccurlyeq 0,\succ 0,\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{and}}\prec 0\right) to indicate that all elements of matrix A are nonnegative (nonpositive, positive, and negative) and the notation of matrices will be used throughout this brief: A = [a_{ lg }] and {A}_{i}=\left[{a}_{lg}^{\left(i\right)}\right]\cdot
In [9], Haddad and Chellaboina obtained a necessary and sufficient stability condition for the following continuoustime positive systems with constant delays:
where {A}_{d}\in {\mathbb{R}}_{0,+}^{n\times n},{a}_{lg}\ge 0 for l ≠ g, τ ≥ 0, and \phi :\left[\tau ,0\right]\to {\mathbb{R}}_{0,+}^{n} is the vector value initial function. They obtained the following result [9, Theorem 3.1]: the above continuoustime positive system is asymptotically stable if and only if there exists a vector \lambda \in {\mathbb{R}}_{+}^{n} such that
Recently, Liu et al. [19] employed a comparison result to establish some necessary and sufficient conditions for the following discretetime positive systems with timevarying delays:
where {A}_{i}\in {\mathbb{R}}_{0,+}^{n\times n} for i ∈ {0, 1, 2,..., p}, 0 ≤ τ_{ i }(t) ≤ τ_{ i } with {\tau}_{i}\in \mathbb{N}, τ = max{τ_{ i }i = 1, 2,..., p}, and \phi :\left\{\tau ,\phantom{\rule{2.77695pt}{0ex}}\dots ,0\right\}\to {\mathbb{R}}_{0,+}^{n} is the vectorvalued initial function. They found out the following similar result [19, Theorem 1]: the above discretetime positive system is asymptotically stable if and only if there exists a vector \lambda \in {\mathbb{R}}_{+}^{n} such that
Since time scales is a tool for establishing a unified framework for continuous and discrete analysis. In the present article, we try to unify the necessary and sufficient conditions for asymptotic stability of linear positive continuous and discretetime systems with delays into one form on time scales.
This article is organized as follows. In Section 2, necessary preliminaries are presented, and some lemmas are provided. Section 3 proposes the necessary and sufficient stability criteria for positive systems with timevarying delays on time scales.
2 Notations and preliminaries
In this section, some basic definitions and some fundamental results are introduced, which are necessary for developing the main results of this article. Let denote a time scales, that is, is a nonempty closed subset of Denote {T}_{0}=\left\{0\right\}\cup T and {T}_{0,+}={\mathbb{R}}_{0,+}\cap {T}_{0}. For simplicity, let \underset{}{N}=\left\{1,2,\dots ,n\right\}, \underset{}{P}=\left\{1,2,\dots ,p\right\}, and {\underset{}{P}}_{0}=\left\{0\right\}\cup \underset{}{P}, where n and p are positive integers.
Define the forward and backward jump operators \sigma ,\rho :T\to T by
where \text{inf}\varnothing =\text{sup}T,\text{sup}\varnothing =\text{inf}T. A point t\in T is called rightscattered, rightdense, leftscattered, and leftdense if σ(t) > t, σ(t) = t, ρ(t) < t, and ρ(t) = t, respectively. We put {T}^{k}=T if is unbounded above and {T}^{k}=T\backslash \left(\rho \left(\text{max}T\right),\text{max}T\right] otherwise. The graininess \mu :T\to \left[0,\infty \right) is defined by
Let f be a function defined on f is said to be (delta) differentiable at t\in {T}^{k} provided there exists a constant a such that for any ε > 0, there is a neighborhood U of t (i.e., U=\left(t\delta ,t+\delta \right)\cap T for some δ > 0) with
In this case, denote f^{Δ}(t) := a. If f is (delta) differentiable for every t\in {T}^{k}, then f is said to be (delta) differentiable on If f is differentiable at t\in {T}^{k}, then
If F^{Δ}(t) = f(t) for all t\in {T}^{k}, then F(t) is called an antiderivative of f on . In this case, define the delta integral by
Moreover, a function f defined on is said to be rdcontinuous if it is continuous at every rightdense point in and its leftsided limit exists at every leftdense point in .
Definition 1. A function f:T\to \mathbb{R} is said to be rightincreasing at {t}_{0}\in T\backslash \left\{\text{max}T\right\} provided

(i)
f(σ(t_{0})) > f(t_{0}) in the case that t_{0} is rightscattered;

(ii)
there is a neighborhood U of t_{0} such that f(t) > f(t_{0}) for all t ∈ U with t > t_{0} in the case that t_{0} is rightdense.
If the inequalities for f are reverse in (i) and (ii), f is said to be rightdecreasing at t_{0}.
The following result can be directly derived from (1).
Lemma 1. Assume that f:T\to \mathbb{R} is differentiable at {t}_{0}\in T\backslash \left\{\text{max}T\right\}. If f^{Δ}(t_{0}) > 0, then f is rightincreasing at t_{0}; and if f^{Δ}(t_{0}) < 0, then f is rightdecreasing at t_{0}.
Consider the following delayed system
where {A}_{i}\in {\mathbb{R}}^{n\times n} for i\in {\underset{}{P}}_{0}, \tau =\underset{i\in \underset{}{P}}{\text{max}}\mathsf{\text{{su}}{\mathsf{\text{p}}}_{t\in {T}_{0,+}}{\tau}_{i}\left(t\right)\}, t{\tau}_{i}\left(t\right)\in {T}_{0}, {\tau}_{i}\left(t\right)\in {T}_{0,+} for t\in {T}_{0,+}, i\in \underset{}{P}, and \phi :\left[\tau ,0\right]\cap {T}_{0}\to {\mathbb{R}}_{0,+}^{n} is the vector value initial function.
Definition 2. System (2) is said to be positive if and only if for any initial condition \phi \left(\cdot \right)\succcurlyeq 0, the corresponding trajectory x\left(t\right)\succcurlyeq 0 holds for all t\in {T}_{0,+}.
Lemma 2. If {A}_{i}\succcurlyeq 0 for all i\in {\underset{}{P}}_{0} and {a}_{jj}^{\left(0\right)}\ge 1 for all j\in \underset{}{N} then system (2) is positive.
Lemma 3. If system (2) is positive then {A}_{i}\succcurlyeq 0 for all i\in \underset{}{P} and {a}_{lg}^{\left(0\right)}\ge 0 for l≠ g.
Represent (2) componentwise, i.e.,
where x_{ l }(·) is the l th component of vector x(·).
Proof of Lemma 2. Assume that {A}_{i}\succcurlyeq 0 for all i\in {\underset{}{P}}_{0}, {a}_{jj}^{\left(0\right)}\ge 1 for all j\in \underset{}{N}, and the initial condition \phi \left(\cdot \right)\succcurlyeq 0. Then, we have {x}_{l}^{\Delta}\left(0\right)\ge 0 by (3) and therefore, x_{ l }(t) ≥ 0 for t\in \left[\tau ,\sigma \left(0\right)\right]\cap {T}_{0}.
Assume for contradiction that there exists {t}_{1}\in {T}_{0,+} such that x_{ l }(t_{1}) < 0. Thus, there exists {t}_{0}\in \left[0,{t}_{1}\right)\cap {T}_{0,+} such that x_{ j }(t_{0}) ≥ 0, x_{ j }(t_{0}  τ_{ i }(t_{0})) ≥ 0 for all j\in \underset{}{N}, i\in \underset{}{P}, and {x}_{l}^{\Delta}\left({t}_{0}\right)<0. It follows from (3) that
which contradicts to {x}_{l}^{\Delta}\left({t}_{0}\right)<0. Therefore system (2) is positive. This completes the proof.
Proof of Lemma 3. Suppose that system (2) is positive. By Definition 2, x\left(t\right)\succcurlyeq 0 for all t\in {T}_{0,+} if \phi \left(\cdot \right)\succcurlyeq 0.
Case 1. Assume for contradiction that there exists q\in \underset{}{P} such that {A}_{q}\succcurlyeq 0 does not hold. Thus, there exists an element {a}_{lg}^{\left(q\right)}<0. Denote φ(τ_{ i }(0)) = [φ_{i 1}, φ_{i 2},..., φ_{ in }]^{T}and x(0) = [φ_{01}, φ_{02},..., φ_{0n}]^{T}. Choose {\phi}_{ij}\left(i\in {\underset{}{P}}_{0},j\in \underset{}{N}\right) such that φ_{ qg } > 0 and φ_{ ij } = 0 for all i ≠ q or j ≠ g. Since q ≠ 0 then x(0) = [φ_{01}, φ_{02},..., φ_{0n}]^{T} = [0, 0,..., 0]^{T}. It follows from (3) that
Since x_{ l }(0) = 0 and {x}_{l}^{\Delta}\left(0\right)<0, then there exists {t}_{0}\in {T}_{0,+} such that x_{ l }(t_{0}) < 0, which contradicts the assumption that x\left(t\right)\succcurlyeq 0 for all t\in {T}_{0,+}.
Case 2. Assume for contradiction that {a}_{lg}^{\left(0\right)}<0 for l ≠ g. Let φ(τ_{ i }(0)) = [0, 0,..., 0]^{T} and x(0) = [0,..., 0, φ_{0g}, 0,..., 0]^{T} where φ_{0g}> 0. It follows from (3) that
Since x_{ l }(0) = 0 and {x}_{l}^{\Delta}\left(0\right)<0, then there exists {t}_{0}\in {T}_{0,+} such that x_{ l }(t_{0}) < 0, which contradicts the assumption that x\left(t\right)\succcurlyeq 0 for all t\in {T}_{0,+}. Therefore, {A}_{i}\succcurlyeq 0 for all i\in \underset{}{P} and {a}_{lg}^{\left(0\right)}\ge 0 for l ≠ g. This completes the proof.
Lemma 4. Assume that system (2) is positive. Let x_{ a }(t) and x_{ b }(t) be the trajectories solutions of (2) under the initial conditions φ_{ a }(·) and φ_{ b }(·), respectively. Then {\phi}_{a}\left(\cdot \right)\preccurlyeq {\phi}_{b}\left(\cdot \right) implies that {x}_{a}\left(t\right)\preccurlyeq {x}_{b}\left(t\right) for all t\in {T}_{0,+} .
Proof. It is easy to see that x(t) = x_{ b }(t)  x_{ a }(t) is the solution to system (2) under the initial condition \phi \left(\cdot \right)={\phi}_{b}\left(\cdot \right){\phi}_{a}\left(\cdot \right)\succcurlyeq 0. Since (2) is positive, x\left(t\right)\succcurlyeq 0 for all t\in {T}_{0,+}. This completes the proof.
3 Main results
The purpose of this section is to establish asymptotic stability criteria for the positive system with bounded timevarying delays on time scales. Consider the following positive system
where {A}_{i}\in {\mathbb{R}}_{0,+}^{n\times n} for i\in \underset{}{P}, {a}_{lg}^{\left(0\right)}\ge 0 for l\ne g, t{\tau}_{i}\left(t\right)\in {T}_{0} for t\in {T}_{0,+}, and the delays {\tau}_{i}\left(t\right)\in {T}_{0,+} satisfy
with constants {\alpha}_{i}\in {T}_{0,+}, {\tau}_{i}\in {T}_{0,+}, and
\phi :\left[\tau ,0\right]\cap {T}_{0}\to {\mathbb{R}}_{0,+}^{n} is the vectorvalued initial function. By Lemma 2, system (4) is positive.
Now, we consider the following system closely related to (4)
where all the system matrices are as in (4), and τ_{ i } in (7) is the supremum of τ_{ i }(t) in (4), as shown in (5), τ is defined by (6), and \varphi :\left[\tau ,0\right]\cap {T}_{0}\to {\mathbb{R}}_{0,+}^{n} is an arbitrary initial condition. Note that the delays in (7) are constant.
Lemma 5. Consider system (7). Suppose that there exists a vector \lambda \in {\mathbb{R}}_{+}^{n} satisfying
and that the initial condition is ϕ(·) ≡ λ. Then, for any t\in \left(\left[\tau ,0\right]\cap {T}_{0}\right)\cup {T}_{0,+}, the solution to system (7) satisfies
Proof. Note that the initial condition ϕ(·) ≡ λ. By (7), it yields that
It follows from y(t) = ϕ(t) ≡ λ for all t\in \left[\tau ,0\right]\cap {T}_{0}, (8), and (10) that {y}^{\Delta}\left(t\right)\preccurlyeq 0 for any t\in \left[\tau ,0\right]\cap {T}_{0}.
Now assume for contradiction that there exists {t}_{0}\in {T}_{0,+} such that y^{Δ}(t_{0}) ϕ 0 and {y}^{\Delta}\left(t\right)\preccurlyeq 0 for any t\in \left[\tau ,{t}_{0}\right]\cap {T}_{0}. By Lemma 1, y(t) is rightdecreasing for all t\in \left[\tau ,{t}_{0}\right]\cap {T}_{0}.
It follows from (7), (8), and y(t) = ϕ(t) ≡ λ for t\in \left[\tau ,{t}_{0}\right]\cap {T}_{0} that
which contradicts the assumption that y^{Δ}(t_{0}) ϕ 0. Therefore, {y}^{\Delta}\left(t\right)\preccurlyeq 0 for any t\in \left(\left[\tau ,0\right]\cap {T}_{0}\right)\cup {T}_{0,+}. This completes the proof.
The next lemma will show that, under certain conditions, the solution to system (4) is not greater than that to system (7), as stated in the following discussion.
Lemma 6. Suppose that there exists a vector \lambda \in {\mathbb{R}}_{+}^{n} satisfying (8) and the initial conditions for system (4) and (7) are the same, i.e., φ(·) ≡ ϕ(·) ≡ λ. Then
for all t\in {T}_{0,+}, where x(t) and y(t) are solutions to (4) and (7), respectively.
Proof. By (4) and (7), it follows that
and
Hence x^{Δ}(0) = y^{Δ}(0), which implies x(σ(0)) = y(σ(0)).
Moreover, there exists {t}_{0}\in {T}_{0,+} such that x(t) = y(t) for all t\in \left[\tau ,{t}_{0}\right]\cap {T}_{0}. By Lemma 5 and (5), we have {x}^{\Delta}\left({t}_{0}\right)\preccurlyeq 0, {y}^{\Delta}\left({t}_{0}\right)\preccurlyeq 0, and
which implies x\left(\sigma \left({t}_{0}\right)\right)\preccurlyeq y\left(\sigma \left({t}_{0}\right)\right). In the following, we can prove that x\left(t\right)\preccurlyeq y\left(t\right) for all t\in \left({t}_{0},+\infty \right)\cap {T}_{0,+}.
Assume for contradiction that there exists {t}_{2}\in \left({t}_{0},+\infty \right)\cap {T}_{0,+} such that x(t_{2}) ϕ y(t_{2}). Thus, there exists {t}_{1}\in \left[{t}_{0},{t}_{2}\right)\cap {T}_{0,+} such that x\left(t\right)\preccurlyeq y\left(t\right) for all t\in \left[0,{t}_{1}\right]\cap {T}_{0,+} and x^{Δ}(t_{1}) ϕ y^{Δ}(t_{1}). It follows from (4), (5), (7), and Lemma 5 that
which contradicts the assumption x^{Δ}(t_{1})  y^{Δ}(t_{1}) ϕ 0. Then (11) holds for all t\in {T}_{0,+}. This completes the proof.
Consider the system (7) satisfying the assumption y(t) = ϕ(t) ≢0 for all t\in \left[\tau ,0\right]\cap {T}_{0}.
Lemma 7. Let (7) be a positive system. Then the positive system (7) is asymptotically stable if and only if there exists a vector \lambda \in {\mathbb{R}}_{+}^{n} such that (8) holds.
Proof. Necessity: By integration, it follows from (7) that
Then
Since the system (7) is asymptotically stable, then y(t) → 0 for t → +∞. It follows from
that
From assumption that y(t) ≢ 0 for all t\in \left[\tau ,0\right]\cap {T}_{0} and (7) is a positive system, we have the left side of (12) is negative and
The condition (13) is equivalent to \left({\Sigma}_{i=0}^{p}{A}_{i}I\right)\lambda \prec 0 for \lambda ={\int}_{0}^{+\infty}y\left(\theta \right)\Delta \theta .
Sufficiency: Let us consider the dual system
which is positive and asymptotically stable.
As a Lyapunov function for the dual system (14) we may choose the following function
which is positive for nonzero x\left(t\right)\in {\mathbb{R}}_{0,+}^{n} and for positive \lambda \in {\mathbb{R}}_{+}^{n}.
From (14) and (15) we have
Hence, the condition \left({\Sigma}_{i=0}^{p}{A}_{i}I\right)\lambda \prec 0 implies {V[x(t)]}^{Δ} ≺ 0 and the positive system (7) is asymptotically stable. This completes the proof.
Theorem 1. Let (4) be a positive system. Then the positive system (4) is asymptotically stable if and only if there exists a vector \lambda \in {\mathbb{R}}_{+}^{n} such that (8) holds.
Proof. Sufficiency: Assume that there exists a vector \lambda \in {\mathbb{R}}_{+}^{n} such that (8) holds. By Lemma 7, system (7) is asymptotically stable.
Define \left\right\varphi \left\right\phantom{\rule{0.3em}{0ex}}=\underset{t\in \left[\tau ,0\right]\cap {T}_{0}}{\text{sup}}\left\right\varphi \left(t\right)\left\right. Since system (7) is asymptotically stable, for any given ε > 0, there exists a scalar δ > 0, such that if ∥ϕ∥ < δ, then the corresponding solution y_{ ϕ }(t) to (7) satisfies that ∥y_{ ϕ }(t)∥ < ε for all t\in {T}_{0,+} and lim_{t→+∞}∥y_{ ϕ }(t)∥ = 0.
In particular, choose the initial condition ϕ(·) = αλ, where α > 0, such that ∥αλ∥< δ. Then, the corresponding solution y_{ αλ }(t) must satisfy that ∥y_{ αλ }(t)∥ < ε for all t\in {T}_{0,+} and lim_{t→+∞}∥y_{ αλ }(t)∥ = 0.
For system (4), let the initial condition be φ(·) = αλ. By Lemma 6, the corresponding solution x_{ αλ }(t) satisfies {x}_{\alpha \lambda}\left(t\right)\preccurlyeq {y}_{\alpha \lambda}\left(t\right) for all t\in {T}_{0,+}. In addition, by Lemma 4, for any initial condition \phi \left(\cdot \right)\preccurlyeq \alpha \lambda, the corresponding solution satisfies {x}_{\phi}\left(t\right)\preccurlyeq {x}_{\alpha \lambda}\left(t\right) for all t\in {T}_{0,+}.
Therefore, for system (4), arbitrary given ε > 0, there exists δ_{1} = ∥αλ∥ > 0 such that ∥φ∥ < δ_{1}, implies ∥x_{ φ }(t)∥ ≤ ∥x_{ αλ }(t)∥ ≤ ∥y_{ αλ }(t)∥ < ε for all t\in {T}_{0,+}.
Furthermore, since ∥x_{ φ }(t)∥ ≤ ∥ y_{ αλ }(t)∥ for all t\in {T}_{0,+}, lim_{t→+∞}∥y_{ αλ }(t)∥ = 0 implies
and therefore, system (4) is asymptotically stable.
Necessity: Suppose that system (4) is asymptotically stable for any delays satisfying (5). Particularly, let τ_{ i }(t) = τ_{ i }, then system (7) is necessarily asymptotically stable. It follows from Lemma 7 that (8) must hold. This completes the proof.
For the continuoustime positive systems with constant delays and discretetime positive systems with timevarying delays, [9, 19] have established some necessary and sufficient stability criteria. Theorem 1 can be viewed as an extension to time scales.
Remark 1. In the continuous case: µ(t) ≡ 0 (i.e., T=\mathbb{R}). It follows from (1) that the positive system (4) can be written as:
where 0 ≤ τ_{ i }(t) ≤ τ_{ i }, τ = max{τ_{ i }i = 1, 2,..., p}, and \phi :\left[\tau ,0\right]\to {\mathbb{R}}_{0,+}^{n} is the vectorvalued initial function. Let A_{0}  I = A, p = 1, A_{1} = A_{ d }, and τ_{ i }(t) ≡ τ, then the system (16) is the same as the continuoustime positive system with constant delays in [9]. Hence, Theorem 1 is the same as that obtained by Haddad and Chellaboina [9, Theorem 3.1] (see Section 1).
Remark 2. In the discrete case: µ(t) ≡ 1 (i.e., T={\mathbb{N}}_{0}). It also follows from (1) that the positive system (4) can be written as:
where 0 ≤ τ_{ i }(t) ≤ τ_{ i }, τ = max{τ_{ i }i = 1, 2,..., p}, and \phi :\left\{\tau ,\dots ,0\right\}\to {\mathbb{R}}_{0,+}^{n} is the vectorvalued initial function. By Δx(t) = x(t + 1)  x(t), the above system can be rewritten as:
It is evident that system (17) is of the same form as the discretetime positive system with timevarying delays in [19]. Hence, Theorem 1 in this special case is the same as [19, Theorem 1] (see Section 1).
Remark 3. Since system (4) includes delayindependent case (i.e., τ_{ i }(t) ≡ 0), if positive system (4) is asymptotically stable, then the matrix \left({\sum}_{i=0}^{p}{A}_{i}I\right) in (8) is a Hurwitz matrix.
Example 1. The positive systems play a key role in understanding many processes in biological science. To illustrate this, a delayindependent model of glycolysis in yeast has been constructed. The glycolysis is the name of a cellular process in which glucose is converted into different carbon compounds in a number of reaction steps. This process plays an important role in the metabolism of most organisms, and its function is partly to produce energy rich molecules and partly to produce carbon compounds that can be used for biosynthesis. The glycolysis has been extensively explored in many organisms and several mathematical models of it have been made. In 2000, a model of glycolysis has been made by Teusink et al. [20]. In order to make this model suitable for our system, a few modifications have been done. Such as changing state variables to constants and changing certain constants into state variables. After these changes the model contains 13 state variables, x_{1},..., x_{13}, each representing the concentration of a carbon compound, except from one, x_{5}, which represents a module of two interconvertable carbon compounds. Since the biological process is nonlinear, the system was linearized around its steady state.
where x = (x_{1},..., x_{13})^{T} is the vector of the 13 state variables and
Since A is a Hurwitz matrix and all its eigenvalues lie in the lefthalf of the complex plane, see Table 1, the positive system (18) is asymptotically stable. Let
then we have
By Theorem 1, we can also deduce that (18) is an asymptotically stable positive system.
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Acknowledgements
Many thanks to Leonid Berezansky (the editor) and the anonymous reviewers for helpful comments and suggestions. This research was supported by the NNSF of China (Grant Nos. 11071143, 11101241, and 61174217), the NNSF of Shandong Province (Grant Nos. ZR2009AL003, ZR2010AL002, ZR2010AL016, ZR2011AL007, and JQ201119), the Scientific Research and Development Project of Shandong Provincial Education Department (J11LA01), and the NSF of University of Jinan (XKY0918).
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YS directed the study and helped inspection. CZ carried out the the main results of this article, established the model, and drafted the manuscript. All the authors read and approved the final manuscript.
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Zhang, C., Sun, Y. Stability analysis of linear positive systems with time delays on time scales. Adv Differ Equ 2012, 56 (2012). https://doi.org/10.1186/16871847201256
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DOI: https://doi.org/10.1186/16871847201256