SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Almost periodic solutions of a single-species system with feedback control on time scales

Meng Hu* and Haiyan Lv

Author Affiliations

School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan, 455000, People’s Republic of China

For all author emails, please log on.

Advances in Difference Equations 2013, 2013:196  doi:10.1186/1687-1847-2013-196

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/196


Received:30 January 2013
Accepted:19 June 2013
Published:3 July 2013

© 2013 Hu and Lv; 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.

Abstract

This paper is concerned with a single-species system with feedback control on time scales. Based on the theory of calculus on time scales, by using the properties of almost periodic functions and constructing a suitable Lyapunov functional, sufficient conditions which guarantee the existence of a unique globally attractive positive almost periodic solution of the system are obtained. Finally, an example and numerical simulations are presented to illustrate the feasibility and effectiveness of the results.

Keywords:
permanence; almost periodic solution; global attractivity; time scale

1 Introduction

In the past few years, different types of ecosystems with periodic coefficients have been studied extensively; see, for example, [1-5] and the references therein. However, if the various constituent components of the temporally nonuniform environment are with incommensurable (nonintegral multiples) periods, then one has to consider the environment to be almost periodic since there is no a priori reason to expect the existence of periodic solutions. Therefore, if we consider the effects of the environmental factors (e.g., seasonal effects of weather, food supplies, mating habits and harvesting), the assumption of almost periodicity is more realistic, more important and more general. Almost periodicity of different types of ecosystems has received more recently researchers’ special attention; see [6-10] and the references therein.

However, in the natural world, there are many species whose developing processes are both continuous and discrete. Hence, using the only differential equation or difference equation cannot accurately describe the law of their development; see, for example, [11,12]. Therefore, there is a need to establish correspondent dynamic models on new time scales.

To the best of the authors’ knowledge, there are few papers published on the existence of an almost periodic solution of ecosystems on time scales.

Motivated by the above, in the present paper, we shall study an almost periodic single-species system with feedback control on time scales as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M2">View MathML</a>, is an almost time scale. All the coefficients <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M4">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M5">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M6">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M7">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M8">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M9">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M10">View MathML</a> are continuous, almost periodic functions.

For convenience, we introduce the notation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M11">View MathML</a>

where f is a positive and bounded function. Throughout this paper, we assume that the coefficients of almost periodic system (1.1) satisfy

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M12">View MathML</a>

The initial condition of system (1.1) is in the form

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M13">View MathML</a>

(1.2)

The aim of this paper is, by using the properties of almost periodic functions and constructing a suitable Lyapunov functional, to obtain sufficient conditions for the existence of a unique globally attractive positive almost periodic solution of system (1.1).

In this paper, the time scale considered is unbounded above, and for each interval of , we denote <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M17">View MathML</a>.

2 Preliminaries

Let be a nonempty closed subset (time scale) of ℝ. The forward and backward jump operators <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M19">View MathML</a> and the graininess <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M20">View MathML</a> are defined, respectively, by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M21">View MathML</a>

A point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M2">View MathML</a> is called left-dense if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M23">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M24">View MathML</a>, left-scattered if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M25">View MathML</a>, right-dense if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M26">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M27">View MathML</a>, and right-scattered if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M28">View MathML</a>. If has a left-scattered maximum m, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M30">View MathML</a>; otherwise <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M31">View MathML</a>. If has a right-scattered minimum m, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M33">View MathML</a>; otherwise <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M34">View MathML</a>.

A function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M35">View MathML</a> is right-dense continuous provided it is continuous at a right-dense point in and its left-side limits exist at left-dense points in . If f is continuous at each right-dense point and each left-dense point, then f is said to be a continuous function on .

For the basic theories of calculus on time scales, one can see [13].

A function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M39">View MathML</a> is called regressive provided <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M40">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M41">View MathML</a>. The set of all regressive and rd-continuous functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M42">View MathML</a> will be denoted by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M43">View MathML</a>. Define the set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M44">View MathML</a>.

If r is a regressive function, then the generalized exponential function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M45">View MathML</a> is defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M46">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M47">View MathML</a>, with the cylinder transformation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M48">View MathML</a>

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M49">View MathML</a> be two regressive functions, define

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M50">View MathML</a>

Lemma 2.1 (see [13])

If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M49">View MathML</a>are two regressive functions, then

(i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M52">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M53">View MathML</a>;

(ii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M54">View MathML</a>;

(iii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M55">View MathML</a>;

(iv) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M56">View MathML</a>;

(v) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M57">View MathML</a>;

(vi) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M58">View MathML</a>.

Lemma 2.2 (see [14])

Assume that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M59">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M60">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M61">View MathML</a>. Then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M62">View MathML</a>

implies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M63">View MathML</a>

Lemma 2.3 (see [14])

Assume that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M59">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M60">View MathML</a>. Then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M66">View MathML</a>

implies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M67">View MathML</a>

Let be a time scale with at least two positive points, one of them being always one: <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M69">View MathML</a>. There exists at least one point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M70">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M71">View MathML</a>. Define the natural logarithm function on the time scale by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M73">View MathML</a>

Lemma 2.4 (see [15])

Assume that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M74">View MathML</a>is strictly increasing and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M75">View MathML</a>is a time scale. If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M76">View MathML</a>exists for<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M77">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M78">View MathML</a>

Lemma 2.5 (see [13])

Assume that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M79">View MathML</a>are differentiable at<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M77">View MathML</a>, then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M81">View MathML</a>is differentiable attwith

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M82">View MathML</a>

Definition 2.1 (see [16])

A time scale is called an almost periodic time scale if

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M84">View MathML</a>

Definition 2.2 (see [16])

Let be an almost periodic time scale. A function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M86">View MathML</a> is called an almost periodic function if the ε-translation set of f

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M87">View MathML</a>

is a relatively dense set in for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M89">View MathML</a>; that is, for any given <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M89">View MathML</a>, there exists a constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M91">View MathML</a> such that in any interval of length <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M92">View MathML</a>, there exists at least a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M93">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M94">View MathML</a>

τ is called the ε-translation number of f, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M92">View MathML</a> is called the inclusion length of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M96">View MathML</a>.

For relevant definitions and the properties of almost periodic functions, see [16-18]. Similar to the proof of Corollary 1.2 in [18], we can get the following lemma.

Lemma 2.6Letbe an almost periodic time scale. If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M98">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M10">View MathML</a>are almost periodic functions, then, for any<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M89">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M101">View MathML</a>is a nonempty relatively dense set in; that is, for any given<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M89">View MathML</a>, there exists a constant<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M91">View MathML</a>such that in any interval of length<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M92">View MathML</a>, there exists at least a<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M106">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M107">View MathML</a>

Remark 2.1 Lemma 2.6 is a special case of Theorem 3.22 in [16].

3 Main results

Assume that the coefficients of (1.1) satisfy

(H1) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M108">View MathML</a>.

Lemma 3.1Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M109">View MathML</a>be any positive solution of system (1.1) with initial condition (1.2). If (H1) holds, then system (1.1) is permanent, that is, any positive solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M109">View MathML</a>of system (1.1) satisfies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M111">View MathML</a>

(3.1)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M112">View MathML</a>

(3.2)

especially if<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M113">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M114">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M115">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M116">View MathML</a>

Proof Assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M109">View MathML</a> is any positive solution of system (1.1) with initial condition (1.2). From the first equation of system (1.1), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M118">View MathML</a>

(3.3)

By Lemma 2.3, we can get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M119">View MathML</a>

Then, for an arbitrarily small positive constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M89">View MathML</a>, there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M121">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M122">View MathML</a>

From the second equation of system (1.1), when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M123">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M124">View MathML</a>

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M125">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M126">View MathML</a>

(3.4)

By Lemma 2.2, we can get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M127">View MathML</a>

Then, for an arbitrarily small positive constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M89">View MathML</a>, there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M129">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M130">View MathML</a>

On the other hand, from the first equation of system (1.1), when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M131">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M132">View MathML</a>

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M125">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M134">View MathML</a>

(3.5)

By Lemma 2.3, we can get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M135">View MathML</a>

Then, for an arbitrarily small positive constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M89">View MathML</a>, there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M137">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M138">View MathML</a>

From the second equation of system (1.1), when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M139">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M140">View MathML</a>

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M125">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M142">View MathML</a>

(3.6)

By Lemma 2.2, we can get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M143">View MathML</a>

Then, for arbitrarily small positive constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M89">View MathML</a>, there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M145">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M146">View MathML</a>

In special case, if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M113">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M114">View MathML</a>, by Lemma 2.2 and Lemma 2.3, it follows from (3.3)-(3.6) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M149">View MathML</a>

This completes the proof. □

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M150">View MathML</a> be a set of all solutions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M151">View MathML</a> of system (1.1) satisfying <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M152">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M153">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M70">View MathML</a>.

Lemma 3.2<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M155">View MathML</a>.

Proof By Lemma 3.1, we see that for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M156">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M113">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M114">View MathML</a>, system (1.1) has a solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M151">View MathML</a> satisfying <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M152">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M153">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M162">View MathML</a>. Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M4">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M5">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M6">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M7">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M8">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M9">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M10">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M170">View MathML</a> are almost periodic, it follows from Lemma 2.6 that there exists a sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M171">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M172">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M173">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M174">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M175">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M176">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M177">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M178">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M179">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M180">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M181">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M173">View MathML</a> uniformly on .

We claim that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M184">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M185">View MathML</a> are uniformly bounded and equi-continuous on any bounded interval in .

In fact, for any bounded interval <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M187">View MathML</a>, when n is large enough, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M188">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M189">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M190">View MathML</a>. So, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M191">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M192">View MathML</a> for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M193">View MathML</a>, that is, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M184">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M185">View MathML</a> are uniformly bounded. On the other hand, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M196">View MathML</a>, from the mean value theorem of differential calculus on time scales, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M197">View MathML</a>

(3.7)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M198">View MathML</a>

(3.8)

Inequalities (3.7) and (3.8) show that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M184">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M200">View MathML</a> are equi-continuous on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M201">View MathML</a>. By the arbitrariness of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M201">View MathML</a>, the conclusion is valid.

By the Ascoli-Arzela theorem, there exists a subsequence of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M171">View MathML</a>, we still denote it as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M171">View MathML</a>, such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M205">View MathML</a>

as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M173">View MathML</a> uniformly in t on any bounded interval in . For any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M208">View MathML</a>, we can assume that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M209">View MathML</a> for all n. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M210">View MathML</a>, integrating both equations of system (1.1) from <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M211">View MathML</a> to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M212">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M213">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M214">View MathML</a>

Using the Lebesgue dominated convergence theorem, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M215">View MathML</a>

This means that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M216">View MathML</a> is a solution of system (1.1), and by the arbitrariness of θ, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M216">View MathML</a> is a solution of system (1.1) on . It is clear that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M219">View MathML</a>

This completes the proof. □

Lemma 3.3In addition to condition (H1), assume further that the coefficients of system (1.1) satisfy the following conditions:

(H2) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M220">View MathML</a>;

(H3) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M221">View MathML</a>.

Then system (1.1) is globally attractive.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M222">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M223">View MathML</a> be any two positive solutions of system (1.1). It follows from (3.1)-(3.2) that for a sufficiently small positive constant <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M224">View MathML</a><a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M225">View MathML</a>, there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M226">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M227">View MathML</a>

(3.9)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M228">View MathML</a>

(3.10)

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M229">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230">View MathML</a>, are positive, bounded and differentiable functions on , then there exists a positive, bounded and differentiable function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M232">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M2">View MathML</a>, such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M234">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230">View MathML</a>, are strictly increasing on . By Lemma 2.4 and Lemma 2.5, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M237">View MathML</a>

Here, we can choose a function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M232">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M239">View MathML</a> is bounded on , that is, there exist two positive constants <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M241">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M242">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M243">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M244">View MathML</a>.

Set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M245">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M246">View MathML</a> is a constant (if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M247">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M248">View MathML</a>; if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M249">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M250">View MathML</a>). It follows from the mean value theorem of differential calculus on time scales for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M251">View MathML</a> that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M252">View MathML</a>

(3.11)

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M253">View MathML</a>. We divide the proof into two cases.

Case I. If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M249">View MathML</a>, set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M255">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M256">View MathML</a>. Calculating the upper right derivatives of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M257">View MathML</a> along the solution of system (1.1), it follows from (3.9)-(3.11), (H2) and (H3) that for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M258">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M259">View MathML</a>

(3.12)

By the comparison theorem and (3.12), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M260">View MathML</a>

that is,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M261">View MathML</a>

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M262">View MathML</a>

(3.13)

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M256">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M264">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M265">View MathML</a>. It follows from (3.13) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M266">View MathML</a>

Case II. If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M247">View MathML</a>, set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M248">View MathML</a>, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M27">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M270">View MathML</a>. Calculating the upper right derivatives of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M257">View MathML</a> along the solution of system (1.1), it follows from (3.9)-(3.11), (H2) and (H3) that for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M251">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M273">View MathML</a>

(3.14)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M274">View MathML</a>. By the comparison theorem and (3.14), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M275">View MathML</a>

that is,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M276">View MathML</a>

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M277">View MathML</a>

(3.15)

It follows from (3.15) that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M278">View MathML</a>

From the above discussion, we can see that system (1.1) is globally attractive. This completes the proof. □

Theorem 3.1Assume that conditions (H1)-(H3) hold, then system (1.1) has a unique globally attractive positive almost periodic solution.

Proof By Lemma 3.2, there exists a bounded positive solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M279">View MathML</a>, then there exists a sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M280">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M281">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M282">View MathML</a>, such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M283">View MathML</a> is a solution of the following system:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M284">View MathML</a>

From the above discussion and Lemma 2.1, we have that not only <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M285">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230">View MathML</a>, but also <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M287">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230">View MathML</a>, are uniformly bounded, thus <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M289">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230">View MathML</a>, are uniformly bounded and equi-continuous. By the Ascoli-Arzela theorem, there exists a subsequence of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M291">View MathML</a> such that for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M89">View MathML</a>, there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M293">View MathML</a> with the property that if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M294">View MathML</a> then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M295">View MathML</a>

It shows that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M296">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230">View MathML</a>, are asymptotically almost periodic functions, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M298">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230">View MathML</a>, are the sum of an almost periodic function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M300">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230">View MathML</a>, and a continuous function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M302">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230">View MathML</a>, defined on , that is,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M305">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M306">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M307">View MathML</a> is an almost periodic function. It means that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M308">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230">View MathML</a>.

On the other hand,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M310">View MathML</a>

So, the limit <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M307">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230">View MathML</a>, exists.

Next, we shall prove that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M313">View MathML</a> is an almost solution of system (1.1).

From the properties of an almost periodic function, there exists a sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M171">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M172">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M173">View MathML</a>, such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M174">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M175">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M176">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M177">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M178">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M179">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M180">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M181">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M173">View MathML</a> uniformly on .

It is easy to know that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M327">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M230">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M173">View MathML</a>, then we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M330">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M331">View MathML</a>

This proves that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M313">View MathML</a> is a positive almost periodic solution of system (1.1). Together with Lemma 3.3, system (1.1) has a unique globally attractive positive almost periodic solution. This completes the proof. □

4 Example and simulations

Consider the following system on time scales:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M333">View MathML</a>

(4.1)

By a direct calculation, we can get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M334">View MathML</a>

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M335">View MathML</a>

that is, conditions (H1)-(H3) hold. According to Theorem 3.1, system (4.1) has a unique globally attractive positive almost periodic solution. For dynamic simulations of system (4.1) with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M336">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M337">View MathML</a>, see Figures 1 and 2, respectively.

thumbnailFigure 1. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M338">View MathML</a>. Dynamics behavior of system (4.1) with initial condition<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M339">View MathML</a>.

thumbnailFigure 2. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M340">View MathML</a>. Dynamics behavior of system (4.1) with initial condition<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/196/mathml/M341">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally and significantly in writing this paper. The authors read and approved the final manuscript.

Acknowledgements

This work was supported by the National Natural Sciences Foundation of China (Grant No. 11071143) and the Natural Sciences Foundation of Henan Educational Committee (Grant No. 2011A110001).

References

  1. Zhang, G, Shen, Y, Chen, B: Positive periodic solutions in a non-selective harvesting predator-prey model with multiple delays. J. Math. Anal. Appl.. 395(1), 298–306 (2012). Publisher Full Text OpenURL

  2. Zhang, X, Wang, M: Multiple periodic solutions of a ratio-dependent predator-prey discrete model. Discrete Dyn. Nat. Soc.. 2012, (2012) Article ID 713503

  3. Liu, G, Yan, J: Existence of positive periodic solutions for neutral delay Gause-type predator-prey system. Appl. Math. Model.. 35(12), 5741–5750 (2011). Publisher Full Text OpenURL

  4. Hu, D, Zhang, Z: Four positive periodic solutions of a discrete time delayed predator-prey system with nonmonotonic functional response and harvesting. Comput. Math. Appl.. 56(12), 3015–3022 (2008). Publisher Full Text OpenURL

  5. Shen, S, Weng, P: Positive periodic solution of a discrete predator-prey patch-system. Acta Math. Appl. Sin.. 24(4), 627–642 (2008). Publisher Full Text OpenURL

  6. Itokazu, T, Hamaya, Y: Almost periodic solutions of prey-predator discrete models with delay. Adv. Differ. Equ.. 2009, (2009) Article ID 976865

  7. Wu, W, Ye, Y: Existence and stability of almost periodic solutions of nonautonomous competitive systems with weak Allee effect and delays. Commun. Nonlinear Sci. Numer. Simul.. 14, 3993–4002 (2009). Publisher Full Text OpenURL

  8. Niu, C, Chen, X: Almost periodic sequence solutions of a discrete Lotka-Volterra competitive system with feedback control. Nonlinear Anal., Real World Appl.. 10, 3152–3161 (2009). Publisher Full Text OpenURL

  9. Liu, Q: Almost periodic solution of a diffusive mixed system with time delay and type III functional response. Discrete Dyn. Nat. Soc.. 2008, (2008) Article ID 706154

  10. Wang, C, Shi, J: Positive almost periodic solutions of a class of Lotka-Volterra type competitive system with delays and feedback controls. Appl. Math. Comput.. 193, 240–252 (2007). Publisher Full Text OpenURL

  11. Spedding, V: Taming nature’s numbers. New Sci.. 2404, 28–31 (2003)

  12. McKellar, R, Knight, K: A combined discrete-continuous model describing the lag phase of Listeria monocytogenes. Int. J. Food Microbiol.. 54(3), 171–180 (2000). PubMed Abstract | Publisher Full Text OpenURL

  13. Bohner, M, Peterson, A: Dynamic Equations on Time Scales: an Introduction with Applications, Birkhauser, Boston (2001)

  14. Hu, M, Wang, L: Dynamic inequalities on time scales with applications in permanence of predator-prey system. Discrete Dyn. Nat. Soc.. 2012, (2012) Article ID 281052

  15. Mozyrska, D, Torres, D: The natural logarithm on time scales. J. Dyn. Syst. Geom. Theories. 7, 41–48 (2009). Publisher Full Text OpenURL

  16. Li, Y, Wang, C: Uniformly almost periodic functions and almost periodic solutions to dynamic equations on time scales. Abstr. Appl. Anal.. 2011, (2011) Article ID 341520

  17. Li, Y, Wang, C: Almost periodic functions on time scales and applications. Discrete Dyn. Nat. Soc.. 2011, (2011) Article ID 727068

  18. He, C: Almost Periodic Differential Equations, Higher Eduction Publishing House, Beijing (1992)