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Uniformly asymptotic stability of almost periodic solutions for a competitive system with impulsive perturbations
Advances in Difference Equations volume 2014, Article number: 2 (2014)
Abstract
Impulsive differential models play an important role in modeling population systems. In this article, we consider an almost periodic competitive model subject to impulsive perturbations and establish sufficient conditions for the uniformly asymptotic stability of a unique positive almost periodic solution for the system. The example and its numerical simulations are carried out to illustrate the feasibility of our main results.
1 Introduction
In [1], Gopalsamy introduced the following autonomous twospecies competitive system
where {x}_{1}(t), {x}_{2}(t) can be interpreted as the density of two competing species at time t, respectively. {a}_{1}, {a}_{2} stand for the intrinsic growth rates of two species, {b}_{1}, {d}_{1}, {b}_{2}, {d}_{2} represent the effects of intraspecific competition, and {c}_{1}, {c}_{2} are the effects of interspecific competition. Notice that the coefficients, in the real world, are not unchanged constants owing to the variation of environment, and the effect of a varying environment is significant for evolutionary theory as the selective forces on systems in such a fluctuating environment differ from those in a stable environment. So it is realistic to consider a corresponding nonautonomous version with the form
Here, all the coefficients {a}_{i}(t), {b}_{i}(t), {c}_{i}(t), {d}_{i}(t) (i=1,2) are subject to fluctuation in time. Furthermore, it is known that the assumption of almost periodicity of the coefficients is a way of incorporating the timedependent variability of the environment, and especially, if the various components of the environment are with incommensurable periods, then it is reasonable to consider the environment to be almost periodic, which leads to the almost periodicity of the coefficients of system (1.2). On the other hand, species live in a real fluctuating medium, and human exploitation activities might result in the duration of abrupt changes. Such changes can be well approximated as impulses, and these processes tend to be reasonably modeled by impulsive differential equations.
Motivated by the above facts, we establish the following almost periodic competitive system with impulsive perturbations:
Here, {x}_{1}({0}^{+})={x}_{1}(0)>0, {x}_{2}({0}^{+})={x}_{2}(0)>0, ℕ is the set of positive integers, the coefficients {a}_{i}(t), {b}_{i}(t), {c}_{i}(t), {d}_{i}(t) are all continuous almost periodic functions which are bounded above and below by positive constants, {\gamma}_{1k}>1 and {\gamma}_{2k}>1 are constants and 0<{\tau}_{1}<{\tau}_{2}<\cdots <{\tau}_{k}<{\tau}_{k+1}<\cdots are impulse points with {lim}_{k\to +\mathrm{\infty}}{\tau}_{k}=+\mathrm{\infty}. The jump conditions reflect the possibility of impulsive effects on two species. From biological viewpoints, when {\gamma}_{ik}>0, the perturbations may stand for stocking, while {\gamma}_{ik}<0 the perturbations mean harvesting.
In the research of population ecology, competitive systems are very important to describe the interactions in the multispecies population dynamics. Many competitive systems have been studied recently by many authors and there is quite extensive literature concerned with the dynamics such as stability of equilibrium [2], persistence [3], permanence or partial extinction [4–7], positive periodic solution [8–11], positive almost periodic solution [12–15]etc. However, there are not many papers considering the stability of positive almost periodic solutions for impulsive competitive systems [14–16]. In this article, we make an attempt to discuss such an issue by considering system (1.3). The rest of this paper is arranged as follows. In Section 2, we present some notations, definitions and lemmas. In Section 3, we give the main result on the uniformly asymptotic stability of a unique positive almost periodic solution for system (1.3). In Section 4, an example together with its numerical simulations is presented to verify the validity of the proposed criteria.
2 Preliminaries
In this section, we give some notations, definitions, lemmas which are useful for establishing our main result (i.e., Theorem 3.1).
Denote by {\mathbb{R}}^{+}, ℝ and ℤ the sets of nonnegative real numbers, real numbers and integers, respectively. {\mathbb{R}}^{2} and {\mathbb{R}}^{n} denote the cone of a twodimensional and ndimensional real Euclidean space, respectively.
Definition 2.1 (see [17])
A sequence \{{\tau}_{k}\} is called almost periodic if for any \u03f5>0 there exists a relatively dense set of its ϵperiods, i.e., there exists such a positive integer N=N(\u03f5) that, for any arbitrary k\in \mathbb{Z}, there is at least an integer p in the segment [k,k+N], for which {\tau}_{k+p}{\tau}_{k}<\u03f5 holds.
Definition 2.2 (see [17])
The set of sequences \{{\tau}_{k}^{j}={\tau}_{k+j}{\tau}_{k}\}, k,j\in \mathbb{Z}, is said to be uniformly almost periodic if for arbitrary \u03f5>0, there exists a relatively dense set of ϵalmost periodic common for any sequences.
Let PC(\mathbb{R},\mathbb{R}) = {u:\mathbb{R}\to \mathbb{R}: u is continuous for t\in \mathbb{R}, t\ne {\tau}_{k}, continuous from the left for t\in \mathbb{R} and discontinuities of the first kind occur at the point {\tau}_{k}\in \mathbb{R}, k\in \mathbb{N}}.
Definition 2.3 (see [17])
The function \phi \in PC(\mathbb{R},\mathbb{R}) is said to be almost periodic if the following conditions hold:

(1)
The set of sequences \{{\tau}_{k}^{j}\}, k,j\in \mathbb{Z} is uniformly almost periodic.

(2)
For any \u03f5>0, there exists a positive number \delta =\delta (\u03f5) such that if the points {t}^{\prime} and {t}^{\u2033} belong to the same interval of continuity and {t}^{\prime}{t}^{\u2033}<\delta, then \phi ({t}^{\prime})\phi ({t}^{\u2033})<\u03f5.

(3)
For any \u03f5>0, there exists a relative dense set T of ϵalmost periods such that if \tau \in T, then \phi (t+\tau )\phi (t)<\u03f5 for all t\in \mathbb{R}, satisfying the condition t{\tau}_{k}>\u03f5, k\in \mathbb{Z}.
Consider the following nonimpulsive system which corresponds to system (1.3)
where
The following Lemma 2.1 is obvious.
Lemma 2.1 Any solution ({y}_{1}(t),{y}_{2}(t)) of system (2.1) satisfies {y}_{i}(t)>0 for all t\ge 0.
Lemma 2.2 For systems (1.3) and (2.1), we have the following conclusions.

(1)
If ({y}_{1}(t),{y}_{2}(t)) is a solution of system (2.1), then ({x}_{1}(t),{x}_{2}(t))=({\prod}_{0<{\tau}_{k}<t}(1+{\gamma}_{1k}){y}_{1}(t),{\prod}_{0<{\tau}_{k}<t}(1+{\gamma}_{2k}){y}_{2}(t)) is a solution of system (1.3).

(2)
If ({x}_{1}(t),{x}_{2}(t)) is a solution of system (1.3), then ({y}_{1}(t),{y}_{2}(t))=({\prod}_{0<{\tau}_{k}<t}{(1+{\gamma}_{1k})}^{1}{x}_{1}(t),{\prod}_{0<{\tau}_{k}<t}{(1+{\gamma}_{2k})}^{1}{x}_{2}(t)) is a solution of system (2.1).
Proof (1) Assume that ({y}_{1}(t),{y}_{2}(t)) is a solution of system (2.1). It is easy to see that {x}_{i}(t)={\prod}_{0<{\tau}_{k}<t}(1+{\gamma}_{ik}){y}_{i}(t) are continuous on the interval ({\tau}_{k},{\tau}_{k+1}], then for any t\ne {\tau}_{k}, k\in \mathbb{N}, one has
and
On the other hand, for every t={\tau}_{k}, k\in \mathbb{N}, we get
and
Thus ({x}_{1}(t),{x}_{2}(t)) is a solution of system (1.3).

(2)
Since {y}_{1}(t) and {y}_{2}(t) are continuous on each interval ({\tau}_{k},{\tau}_{k+1}]. From system (2.1), one can easily check the continuity of {y}_{i}(t) at the impulse points t={\tau}_{k}, k\in \mathbb{N}. Recalling system (1.3), we have
\begin{array}{r}{y}_{1}\left({\tau}_{k}^{+}\right)=\prod _{0<{\tau}_{j}\le {\tau}_{k}}{(1+{\gamma}_{1j})}^{1}{x}_{1}\left({\tau}_{k}^{+}\right)=\prod _{0<{\tau}_{j}<{\tau}_{k}}{(1+{\gamma}_{1j})}^{1}{x}_{1}({\tau}_{k})={y}_{1}({\tau}_{k}),\\ {y}_{2}\left({\tau}_{k}^{+}\right)=\prod _{0<{\tau}_{j}\le {\tau}_{k}}{(1+{\gamma}_{2j})}^{1}{x}_{2}\left({\tau}_{k}^{+}\right)=\prod _{0<{\tau}_{j}<{\tau}_{k}}{(1+{\gamma}_{2j})}^{1}{x}_{2}({\tau}_{k})={y}_{2}({\tau}_{k}).\end{array}(2.7)
Also, by the basic theory of impulsive differential equations in [18, 19], we know that {x}_{i}({\tau}_{k}^{})={x}_{i}({\tau}_{k}). So we get
Equations (2.7) and (2.8) imply that {y}_{1}(t) and {y}_{2}(t) are continuous on {\mathbb{R}}^{+}. It is easy to see that ({y}_{1}(t),{y}_{2}(t)) is a solution of system (2.1). The proof of Lemma 2.2 is complete. □
Consider the following differential equation:
where f(t,X)\in C({\mathbb{R}}^{+}\times \mathbb{D},{\mathbb{R}}^{n}) and \mathbb{D} is an open set in {\mathbb{R}}^{n}, f(t,X) is almost periodic in t uniformly with respect to X\in \mathbb{D}. The following associate product system of system (2.9) can be expressed as
Lemma 2.3 (see Theorem 6.3 in [20])
Suppose that there exists a Lyapunov function V(t,X,Y) defined on [0,+\mathrm{\infty})\times \mathbb{D}\times \mathbb{D}, which satisfies the following conditions:

(1)
a(\parallel XY\parallel )\le V(t,X,Y)\le b(\parallel XY\parallel ), where a(\kappa ) and b(\kappa ) are continuous, increasing and positive definite functions.

(2)
V(t,{X}_{1},{Y}_{1})V(t,{X}_{2},{Y}_{2})\le K\{\parallel {X}_{1}{X}_{2}\parallel +\parallel {Y}_{1}{Y}_{2}\parallel \}, where K>0 is a constant.

(3)
{V}_{\text{(2.10)}}^{\prime}(t,X,Y)\le \mu V(t,X,Y), where \mu >0 is a constant.
Moreover, suppose that system (2.9) has a solution that remains in a compact set \mathbb{S}\subset \mathbb{D} for all t\ge 0. Then system (2.9) has a unique almost periodic solution in \mathbb{S}, which is uniformly asymptotically stable in \mathbb{D}.
Lemma 2.4 (see [21])

(1)
If a>0, b>0 and {x}^{\prime}(t)\ge x(t)(abx(t)), when t\ge 0 and x(0)>0, we have {lim\hspace{0.17em}inf}_{t\to +\mathrm{\infty}}x(t)\ge a/b. (2) If a>0, b>0 and {x}^{\prime}(t)\le x(t)(abx(t)), when t\ge 0 and x(0)>0, we have {lim\hspace{0.17em}sup}_{t\to +\mathrm{\infty}}x(t)\le a/b.
For convenience, given an almost periodic function g(t) defined on {\mathbb{R}}^{+}, let {g}^{L} and {g}^{U} be defined as {g}^{L}={inf}_{t\in {\mathbb{R}}^{+}}g(t), {g}^{U}={sup}_{t\in {\mathbb{R}}^{+}}g(t).
Lemma 2.5 Assume that the following two conditions
(A1) there exist positive constants {\alpha}_{i}, {\beta}_{i} such that {\alpha}_{i}\le {\prod}_{0<{\tau}_{k}<t}(1+{\gamma}_{ik})\le {\beta}_{i}, i=1,2,
(A2) {a}_{1}^{L}{c}_{1}^{U}{\beta}_{2}{M}_{2}>0 and {a}_{2}^{L}{c}_{2}^{U}{\beta}_{1}{M}_{1}>0
hold, then any solution ({y}_{1}(t),{y}_{2}(t)) of system (2.1) satisfies
where {m}_{i}=({a}_{i}^{L}{c}_{i}^{U}{\beta}_{j}{M}_{j})/({b}_{i}^{U}{\beta}_{i}+{d}_{i}^{U}{\beta}_{i}^{2}{M}_{i}), {M}_{i}={a}_{i}^{U}/({b}_{i}^{L}{\alpha}_{i}), 1\le i,j\le 2; i\ne j.
Proof Let ({y}_{1}(t),{y}_{2}(t)) be any solution of system (2.1). It follows from system (2.1) and (A1) that we have
Using (2) in Lemma 2.4, one has
Hence, for any small constant \epsilon >0, there exists {T}_{0}>0 such that for t\ge {T}_{0},
Together with system (2.1), we can derive that
Thus from (A2), (1) in Lemma 2.4 and \epsilon >0 is arbitrarily small, one has
The proof of Lemma 2.5 is complete. □
By (2.12) and (2.13), we denote by Θ the set of all solutions ({y}_{1}(t),{y}_{2}(t)) of system (2.1) satisfying {m}_{i}\le {y}_{i}(t)\le {M}_{i}, that is,
Lemma 2.6 Assume that (A1) and (A2) are satisfied. Suppose further that
(A3) the set of sequences \{{\tau}_{k}^{j}={\tau}_{k+j}{\tau}_{k}\}, k,j\in \mathbb{Z} is uniformly almost periodic,
(A4) {\prod}_{0<{\tau}_{k}<t}(1+{\gamma}_{ik}) is an almost periodic function.
Then \mathrm{\Theta}\ne \varphi.
Proof The almost periodicity of \{{a}_{i}(t)\}, \{{B}_{i}(t)\}, \{{C}_{i}(t)\}, \{{D}_{i}(t)\} implies that there exists a sequence \{{t}_{n}\}, {t}_{n}\to +\mathrm{\infty} as n\to +\mathrm{\infty} such that
as n\to +\mathrm{\infty} for t\in {\mathbb{R}}^{+}. It follows from (2.11) that, for any small enough \epsilon >0, there exists {T}_{1}>0 such that for t\ge {T}_{1},
It is obvious that the sequence \{{y}_{i}(t+{t}_{n})\} is uniformly bounded and equicontinuous on each bounded subset of {\mathbb{R}}^{+}, i=1,2. By Ascoli’s theorem, we obtain that \{{y}_{i}(t+{t}_{n})\} exists a subsequence, also denoted by \{{y}_{i}(t+{t}_{n})\}, converging on each bounded subset of {\mathbb{R}}^{+} as n\to +\mathrm{\infty}. Therefore, there is a continuous function {\overline{y}}_{i}(t) satisfying
For any {T}_{2}\in {\mathbb{R}}^{+} such that {t}_{n}+{T}_{2}\ge {T}_{1} for large enough n. Let t\ge 0, we have
Using Lebesgue’ dominated convergence theorem, and letting n\to +\mathrm{\infty} in (2.17), one has
Since {T}_{2}\in {\mathbb{R}}^{+} is arbitrary, ({\overline{y}}_{1}(t),{\overline{y}}_{2}(t)) is a solution of system (2.1) on {\mathbb{R}}^{+}. We easily obtain that {m}_{i}\epsilon \le {\overline{y}}_{i}(t)\le {M}_{i}+\epsilon for t\in {\mathbb{R}}^{+}, i=1,2. Furthermore, since ε is arbitrarily small, we get that {m}_{i}\le {\overline{y}}_{i}(t)\le {M}_{i}, i=1,2, for t\in {\mathbb{R}}^{+}. The proof of Lemma 2.6 is complete. □
3 The main result
In this section, we give our main result and establish the uniformly asymptotic stability of a unique positive almost periodic solution for system (1.3).
Theorem 3.1 Assume that (A1)(A4) hold. Furthermore, assume that
(A5) there exist positive constants {\theta}_{1}, {\theta}_{2}, σ, where \sigma =min\{{s}_{1},{s}_{2}\}, and
Then system (1.3) has a unique uniformly asymptotically stable positive almost periodic solution.
Proof Let us make the change of variables
then system (2.1) can be rewritten as
Obviously, the existence of a unique almost periodic solution of system (2.1) is equivalent to that of system (3.2). By Lemma 2.6, there is a bounded solution ({z}_{1}(t),{z}_{2}(t)) of system (3.2) satisfying
Define the norm \parallel ({z}_{1}(t),{z}_{2}(t))\parallel ={z}_{1}(t)+{z}_{2}(t), where ({z}_{1}(t),{z}_{2}(t))\in {\mathbb{R}}^{2}. Consider the associate product system of system (3.2)
Here, Z(t)=({z}_{1}(t),{z}_{2}(t)) and W(t)=({w}_{1}(t),{w}_{2}(t)) are any two solutions of system (3.2) defined on \mathbb{S}, and \mathbb{S}=\{({z}_{1}(t),{z}_{2}(t))\in {\mathbb{R}}^{2}ln{m}_{i}\le {z}_{i}(t)\le ln{M}_{i},i=1,2,t\in {\mathbb{R}}^{+}\}.
Next, let us consider a Lyapunov function defined on {\mathbb{R}}^{+}\times \mathbb{S}\times \mathbb{S} as follows:
It is obvious that
Let a(\kappa )=min\{{\theta}_{1},{\theta}_{2}\}\kappa, b(\kappa )=max\{{\theta}_{1},{\theta}_{2}\}\kappa, then condition (1) in Lemma 2.3 is satisfied.
In addition, for any (t,Z,W),(t,\tilde{Z},\tilde{W})\in {\mathbb{R}}^{+}\times \mathbb{S}\times \mathbb{S}, one has
where \tilde{Z}(t)=({\tilde{z}}_{1}(t),{\tilde{z}}_{2}(t)), \tilde{W}(t)=({\tilde{w}}_{1}(t),{\tilde{w}}_{2}(t)), \lambda =max\{{\theta}_{1},{\theta}_{2}\}. Hence, condition (2) in Lemma 2.3 is satisfied.
Finally, calculating the right derivative {D}^{+}V(t) of V(t) along the solutions of system (3.3), one has
By the meanvalue theorem, we have
i=1,2, where {\xi}_{i}(t) and {\eta}_{i}(t) lie between exp\{{z}_{i}(t)\} and exp\{{w}_{i}(t)\}, respectively. Substituting (3.8) into (3.7), one has
where \sigma =min\{{s}_{1},{s}_{2}\} and \mu =\sigma min\{1/{\theta}_{1},1/{\theta}_{2}\}. It follows from condition (A5) in Theorem 3.1 that we have \mu >0, that is, condition (3) in Lemma 2.3 is also satisfied. Therefore, it follows from Lemma 2.3 that system (3.2) has a unique almost periodic solution ({z}_{1}^{\ast}(t),{z}_{2}^{\ast}(t)) which is uniformly asymptotically stable in \mathbb{S}. That is, system (2.1) has a unique uniformly asymptotically stable positive almost periodic solution ({y}_{1}^{\ast}(t),{y}_{2}^{\ast}(t))=(exp\{{z}_{1}^{\ast}(t)\},exp\{{z}_{2}^{\ast}(t)\}).
Finally, we will prove that system (1.3) has a unique uniformly asymptotically stable positive almost periodic solution. It follows from Lemma 2.2 that
is a solution of system (1.3). By conditions (A3) and (A4), we can prove that {x}_{i}^{\ast}(t)={\prod}_{0<{\tau}_{k}<t}(1+{\gamma}_{ik}){y}_{i}^{\ast}(t) is an almost periodic function based on the proofs of Lemma 31 and Theorem 79 in [17]. Thus ({x}_{1}^{\ast}(t),{x}_{2}^{\ast}(t)) is a unique uniformly asymptotically stable positive almost periodic solution of system (1.3). The proof of Theorem 3.1 is complete. □
4 An example and numerical simulations
In this section, to illustrate the feasibility of our analytical results, we give the following example.
Example 4.1 Consider the competitive system with impulsive perturbations
Let {\gamma}_{ik}=exp\{{(1)}^{k+1}\frac{1}{{k}^{2}}\}1, {\tau}_{k}=k, k\in \mathbb{N}, we obtain that 1<{\prod}_{0<{\tau}_{k}<t}(1+{\gamma}_{ik})<e, i=1,2. So we can choose {\alpha}_{1}={\alpha}_{2}=1, {\beta}_{1}={\beta}_{2}=e. A computation shows that
Obviously, (A1) and (A2) in Lemma 2.5 are satisfied; moreover, letting {\theta}_{1}={\theta}_{2}=1, one has
that is, \sigma =min\{{s}_{1},{s}_{2}\}\approx 0.1429>0, so (A3)(A5) in Theorem 3.1 are satisfied. Thus, system (4.1) has a unique uniformly asymptotically stable positive almost periodic solution. From Figure 1, we can easily see that system (4.1) with initial value (0.3,0.5) has a positive almost periodic solution denoted by ({x}_{1}^{\ast}(t),{x}_{2}^{\ast}(t)). Figure 2 shows that a positive solution with initial value (0.5,1.2), denoted by ({x}_{1}(t),{x}_{2}(t)), tends to the above positive almost periodic solution ({x}_{1}^{\ast}(t),{x}_{2}^{\ast}(t)).
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Acknowledgements
The work is supported by the National Natural Science Foundation of China (No. 11261017) and the Project of Key Laboratory of Biological Resources Protection and Utilization of Hubei Province (Forestry). We would like to thank anonymous reviewers for their helpful comments which improved the presentation of this work.
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Tan, R., Liu, W., Wang, Q. et al. Uniformly asymptotic stability of almost periodic solutions for a competitive system with impulsive perturbations. Adv Differ Equ 2014, 2 (2014). https://doi.org/10.1186/1687184720142
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DOI: https://doi.org/10.1186/1687184720142