Existence of Almost-Periodic Solutions for Lotka-Volterra Cooperative Systems with Time Delay

Abstract

This paper considers the existence of positive almost-periodic solutions for almost-periodic Lotka-Volterra cooperative system with time delay. By using Mawhin’s continuation theorem of coincidence degree theory, sufficient conditions for the existence of positive almost-periodic solutions are obtained. An example and its simulation figure are given to illustrate the effectiveness of our results.

1. Introduction

Lotka-Volterra system is one of the most celebrated models in mathematical biology and population dynamics. In recent years, it has also been found with successful and interesting applications in epidemiology, physics, chemistry, economics, biological science, and other areas (see [1–4]). Moreover, in [5], it was shown that the continuous-time recurrent neural networks can be embedded into Lotka-Volterra models by changing coordinates, which suggests that the existing techniques in the analysis of Lotak-Volterra systems can also be applied to recurrent neural networks.

Owing to its theoretical and practical significance, Lotka-Volterra system have been studied extensively (see [6–16] and the cites therein). Since biological and environmental parameters are naturally subject to fluctuation in time, the effects of a periodically varying environment (e.g., seasonal effects of weather, food supplies, mating habits, etc.) are considered as important selective forces on systems in a fluctuating environment. Therefore, on the one hand, models should take into account both the periodically changing environment and the effects of time delays. However, on the other hand, in fact, it is more realistic and reasonable to study almost-periodic system than periodic system. Recently, there are two main approaches to obtain sufficient conditions for the existence and stability of the almost-periodic solutions of biological models: one is using the fixed point theorem, Lyapunov functional method, and differential inequality techniques (see [17–19]); the other is using functional hull theory and Lyapunov functional method (see [14–16]). However, to the best of our knowledge, there are very few published letters considering the almost-periodic solutions for nonautonomous Lotka-Volterra cooperative system with time delay by applying the method of coincidence degree theory. Motivated by this, in this letter, we apply the coincidence theory to study the existence of positive almost-periodic solutions for Lotka-Volterra cooperative system with time delay as follows:𝑢̇_𝑖(𝑡)=𝑢_𝑖(𝑡)𝑟_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑢_𝑖𝑡−𝜏_𝑖(𝑡)+^𝑛_{𝑗=1,𝑗≠𝑖}𝑐_𝑖𝑗(𝑡)𝑢_𝑗𝑡−𝜏_𝑖𝑗(𝑡),𝑖=1,2,…,𝑛, (1.1) where 𝑢_𝑖(𝑡) stands for the 𝑖th species population density at time 𝑡∈ℝ, 𝑟_𝑖(𝑡) is the natural reproduction rate, 𝑏_𝑖(𝑡) represents the inner-specific competition, 𝑐_𝑖𝑗(𝑡)(𝑖≠𝑗) stands for the interspecific cooperation, 𝜏_𝑖(𝑡)>0 and 𝜏_𝑖𝑗(𝑡)>0 are all continuous almost-periodic functions on ℝ. Throughout this paper, we always assume that 𝑟_𝑖(𝑡), 𝑏_𝑖(𝑡), and 𝑐_𝑖𝑗(𝑡) are all nonegative almost periodic functions with respect to 𝑡∈ℝ.

The initial condition of (1.1) is of the form𝑢_𝑖(𝑠)=𝜙_𝑖(𝑠),𝑖=1,2,…,𝑛, (1.2) where 𝜙_𝑖(𝑠) is positive bounded continuous function on [−𝜏,0] and 𝜏=max_{1≤𝑖,𝑗≤𝑛}sup_𝑡∈𝑅{|𝜏_𝑖𝑗(𝑡)|}.

The organization of the rest of this paper is as follows. In Section 2, we introduce some preliminary results which are needed in later sections. In Section 3, we establish our main results for the existence of almost-periodic solutions of (1.1). Finally, an example and its simulation figure are given to illustrate the effectiveness of our results in Section 4.

2. Preliminaries

To obtain the existence of an almost-periodic solution of system (1.1), we first make the following preparations.

Definition 2.1 (see [20]). Let 𝑢(𝑡)∶ℝ→ℝ be continuous in 𝑡.𝑢(𝑡) is said to be almost-periodic on ℝ, if, for any 𝜖>0, the set 𝐾(𝑢,𝜖)={𝛿∶|𝑢(𝑡+𝛿)−𝑢(𝑡)|<𝜖,forany𝑡∈ℝ} is relatively dense, that is, for any 𝜖>0, it is possible to find a real number 𝑙(𝜖)>0, for any interval with length 𝑙(𝜖), there exists a number 𝛿=𝛿(𝜖) in this interval such that |𝑢(𝑡+𝛿)−𝑢(𝑡)|<𝜖, for any 𝑡∈ℝ.

Definition 2.2. A solution 𝑢(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑛(𝑡))^𝑇 of (1.1) is called an almost periodic solution if and only if for each 𝑖=1,2,…,𝑛, 𝑢_𝑖(𝑡) is almost periodic.

For convenience, we denote 𝐴𝑃(ℝ) the set of all real-valued, almost-periodic functions on ℝ and for each 𝑗=1,2,…,𝑛, let∧𝑓_𝑗=𝜆∈ℝ∶lim_𝑇→∞1𝑇^𝑇₀𝑓_𝑗(𝑠)𝑒^−𝑖𝜆_𝑠,𝑓d𝑠≠0mod_𝑗=^𝑁_𝑖=1𝑛_𝑖𝜆_𝑖∶𝑛_𝑖∈𝑍,𝑁∈𝑁⁺,𝜆_𝑖∈∧𝑓_𝑗 (2.1) be the set of Fourier exponents and the module of 𝑓_𝑗, respectively, where 𝑓_𝑗(⋅) is almost periodic. Suppose 𝑓_𝑗(𝑡,𝜙_𝑗) is almost periodic in 𝑡, uniformly with respect to 𝜙_𝑗∈𝐶([−𝜏,0],ℝ). 𝐾_𝑗(𝑓_𝑗,𝜖,𝜙_𝑗) denote the set of 𝜖-almost periods for 𝑓_𝑗 uniformly with respect to 𝜙_𝑗∈𝐶([−𝜏,0],ℝ). 𝑙_𝑗(𝜖) denote the length of inclusion interval. Let 𝑚(𝑓_𝑗)=(1/𝑇)∫^𝑇₀𝑓_𝑗(𝑠)d𝑠 be the mean value of 𝑓_𝑗 on interval [0,𝑇], where 𝑇>0 is a constant. clearly, 𝑚(𝑓_𝑗) depends on 𝑇.𝑚[𝑓_𝑗]=lim_𝑇→∞(1/𝑇)∫^𝑇₀𝑓_𝑗(𝑠)𝑑𝑠.

Lemma 2.3 (see [20]). Suppose that 𝑓 and 𝑔 are almost periodic. Then the following statements are equivalent.

(i)mod(𝑓)⊃mod(𝑔),(ii)for any sequence {𝑡^∗_𝑛}, if lim_𝑛→∞𝑓(𝑡+𝑡^∗_𝑛)=𝑓(𝑡) for each 𝑡∈ℝ, then there exists a subsequence {𝑡_𝑛}⊆{𝑡^∗_𝑛} such that lim_𝑛→∞𝑔(𝑡+𝑡_𝑛)=𝑓(𝑡) for each 𝑡∈ℝ.

Lemma 2.4 (see [21]). Let 𝑢∈𝐴𝑃(ℝ). Then ∫^𝑡_𝑡−𝜏𝑢(𝑠)d𝑠 is almost periodic.

Let 𝑋 and 𝑍 be Banach spaces. A linear mapping 𝐿∶dom(𝐿)⊂𝑋→𝑍 is called Fredholm if its kernel, denoted by ker(𝐿)={𝑋∈dom(𝐿)∶𝐿𝑥=0}, has finite dimension and its range, denoted by Im(𝐿)={𝐿𝑥∶𝑥∈dom(𝐿)}, is closed and has finite codimension. The index of 𝐿 is defined by the integer dim𝐾(𝐿)−codimdom(𝐿). If 𝐿 is a Fredholm mapping with index 0, then there exist continuous projections 𝑃∶𝑋→𝑋 and 𝑄∶𝑍→𝑍 such that Im(𝑃)=ker(𝐿) and ker(𝑄)=Im(𝐿). Then 𝐿|_{dom(𝐿)∩ker(𝑃)}∶Im(𝐿)∩ker(𝑃)→Im(𝐿) is bijective, and its inverse mapping is denoted by 𝐾_𝑃∶Im(𝐿)→dom(𝐿)∩ker(𝑃). Since ker(𝐿) is isomorphic to Im(𝑄), there exists a bijection 𝐽∶ker(𝐿)→Im(𝑄). Let Ω be a bounded open subset of 𝑋 and let 𝑁∶𝑋→𝑍 be a continuous mapping. If 𝑄𝑁(Ω) is bounded and 𝐾_𝑃(𝐼−𝑄)𝑁∶Ω→𝑋 is compact, then 𝑁 is called 𝐿-compact on Ω, where 𝐼 is the identity.

Let 𝐿 be a Fredholm linear mapping with index 0 and let 𝑁 be a 𝐿-compact mapping on Ω. Define mapping 𝐹∶dom(𝐿)∩Ω→𝑍 by 𝐹=𝐿−𝑁. If 𝐿𝑥≠𝑁𝑥 for all 𝑥∈dom(𝐿)∩𝜕Ω, then by using 𝑃, 𝑄, 𝐾_𝑃, 𝐽 defined above, the coincidence degree of 𝐹 in Ω with respect to 𝐿 is defined bydeg_𝐿(𝐹,Ω)=deg𝐼−𝑃−𝐽⁻¹𝑄+𝐾_𝑃(𝐼−𝑄)𝑁,Ω,0, (2.2) where deg(𝑔,Γ,𝑝) is the Leray-Schauder degree of 𝑔 at 𝑝 relative to Γ.

Then The Mawhin’s continuous theorem is given as follows.

Lemma 2.5 (see [22]). Let Ω⊂𝑋 be an open bounded set and let 𝑁∶𝑋→𝑍 be a continuous operator which is 𝐿-compact on Ω. Assume(a)for each 𝜆∈(0,1), 𝑥∈𝜕Ω∩dom(𝐿),𝐿𝑥≠𝜆𝑁𝑥;(b)for each 𝑥∈𝜕Ω∩𝐿, 𝑄𝑁𝑥≠0;(c)deg(𝐽𝑁𝑄,Ω∩ker(𝐿),0)≠0. Then 𝐿𝑥=𝑁𝑥 has at least one solution in Ω∩dom(𝐿).

In this paper, since we need some related properties of 𝑀-matrix we introduce them as follows. In addition, A matrix 𝐴=(𝑎_𝑖𝑗)≥0 means that each elements 𝑎_𝑖𝑗≥0.

Definition 2.6 (see [23]). If a real matrix 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 satisfies the following conditions (1) and (2):(1)𝑎_𝑖𝑖>0, 𝑖=1,2,…,𝑛, 𝑎_𝑖𝑗≤0, 𝑖≠𝑗, 𝑖,𝑗=1,2,…,𝑛,(2)𝐴 is a positive-definite matrix,then 𝐴 is called a 𝑀-matrix.

Lemma 2.7 (see [23]). If matrix 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 is a 𝑀-matrix, then 𝐴⁻¹ exists and its every element is nonnegative.

Lemma 2.8. Suppose that matrix 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 is a 𝑀-matrix, then 𝐴𝑋≤𝐵 implies 𝑋≤𝐴⁻¹𝐵.

Proof. In fact, there exists a nonnegative positive vector 𝜀₀=(𝜀₁,𝜀₂,…,𝜀_𝑛)^𝑇∈𝑅^𝑛 such that 𝐴𝑋−𝐵+𝜀₀=(0,0,…,0)^𝑇 which imply that 𝑋−𝐴⁻¹𝐵+𝐴⁻¹𝜀₀=(0,0,…,0)^𝑇. According to Lemma 2.4, there exists at least one positive element in the every row of 𝐴⁻¹, which imply 𝐴⁻¹𝜀₀≥(0,0,…,0)^𝑇. Thus, we obtain 𝑋≤𝐴⁻¹𝐵.

3. Main Result

In this section, we state and prove our main results of our this paper. By making the substitution𝑢_𝑖(𝑡)=𝑒^{𝑦𝑖(𝑡)},𝑖=1,2,…,𝑛, (3.1) Equation (1.1) can be reformulated as𝑦_𝑖̇(𝑡)=𝑟_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑒^{𝑦𝑖(𝑡−𝜏𝑖𝑖(𝑡))}+^𝑛_{𝑗=1,𝑖≠𝑗}𝑐_𝑖𝑗(𝑡)𝑒^{𝑦𝑗(𝑡−𝜏𝑖𝑗(𝑡))},𝑖=1,2,…,𝑛. (3.2) The initial condition (1.2) can be rewritten as follows: 𝑦_𝑖(𝑠)=ln𝜙_𝑖(𝑠)=∶𝜓_𝑖(𝑠),𝑖=1,2,…,𝑛. (3.3) Set 𝑋=𝑍=𝑉₁⊕𝑉₂, where𝑉₁=𝑦(𝑡)=𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡)^𝑇∈𝐶ℝ,ℝ^𝑛∶𝑦_𝑖(𝑡)(ℝ),𝑦∈𝐴𝑃mod_𝑖(𝑡)⊂mod𝐻_𝑖(𝑡),∀𝜆_𝑖∈∧𝑦_𝑖(𝑡)satisfies𝜆_𝑖,𝑉>𝛽,𝑖=1,2,…,𝑛₂=𝑦(𝑡)≡ℎ₁,ℎ₂,…,ℎ_𝑛^𝑇∈ℝ^𝑛,𝐻_𝑖(𝑡)=𝑟_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑒^{𝜓𝑖(−𝜏𝑖𝑖(0))}+^𝑛_{𝑗=1,𝑖≠𝑗}𝑐_𝑖𝑗(𝑡)𝑒^{𝜓𝑗(−𝜏𝑖𝑗(0))} (3.4) and 𝜓(⋅) is defined as (3.3), 𝑖=1,2,…,𝑛,𝛽>0is a given constant. For 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)^𝑇∈𝑍, define ‖𝑦‖=max_{1≤𝑖≤𝑛}sup_𝑡∈ℝ|𝑦_𝑖(𝑡)|.

Lemma 3.1. 𝑍 is a Banach space equipped with the norm ‖⋅‖.

Proof. If 𝑦^{𝑘}⊂𝑉₁ and 𝑦^{𝑘}=(𝑦₁^{𝑘},𝑦₂^{𝑘},…,𝑦_𝑛^{𝑘})^𝑇 converges to 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)^𝑇, that is, 𝑦_𝑗^{𝑘}→𝑦_𝑗, as 𝑘→∞,𝑗=1,2,…,𝑛. Then it is easy to show that 𝑦_𝑗∈𝐴𝑃(ℝ) and mod(𝑦_𝑗)∈mod(𝐻_𝑗). For any |𝜆_𝑗|≤𝛽, we have that lim_𝑇→∞1𝑇^𝑇₀𝑦_𝑗^{𝑘}(𝑡)𝑒^−𝑖𝜆_𝑗𝑡𝑑𝑡=0,𝑗=1,2,…,𝑛; (3.5) therefore, lim_𝑇→∞1𝑇^𝑇₀𝑦_𝑗(𝑡)𝑒^−𝑖𝜆_𝑗𝑡𝑑𝑡=0,𝑗=1,2,…,𝑛, (3.6) which implies 𝑦∈𝑉₁. Then it is not difficult to see that 𝑉₁ is a Banach space equipped with the norm ‖⋅‖. Thus, we can easily verify that 𝑋 and 𝑍 are Banach spaces equipped with the norm ‖⋅‖. The proof of Lemma 3.1 is complete.

Lemma 3.2. Let 𝐿∶𝑋→𝑍, 𝐿𝑦=𝑦̇, then 𝐿 is a Fredholm mapping of index 0.

Proof. Clearly, 𝐿 is a linear operator and ker(𝐿)=𝑉₂. We claim that Im(𝐿)=𝑉₁. Firstly, we suppose that 𝑧(𝑡)=(𝑧₁(𝑡),𝑧₂(𝑡),…,𝑧_𝑛(𝑡))^𝑇∈Im(𝐿)⊂𝑍. Then there exist 𝑧^{1}(𝑡)=(𝑧₁^{1}(𝑡),𝑧₂^{1}(𝑡),…,𝑧_𝑛^{1}(𝑡))^𝑇∈𝑉₁ and constant vector 𝑧^{2}=(𝑧₁^{2},𝑧₂^{2},…,𝑧_𝑛^{2})^𝑇∈𝑉₂ such that 𝑧(𝑡)=𝑧^{1}(𝑡)+𝑧^{2}, (3.7) that is, 𝑧_𝑖(𝑡)=𝑧_𝑖^{1}(𝑡)+𝑧_𝑖^{2},𝑖=1,2,…,𝑛. (3.8) From the definition of 𝑧_𝑖(𝑡) and 𝑧_𝑖^{1}(𝑡), we can easily see that ∫^𝑡_𝑡−𝜏𝑧_𝑖(𝑠)𝑑𝑠 and ∫^𝑡_𝑡−𝜏𝑧_𝑖^{1}(𝑠)𝑑𝑠 are almost-periodic functions. So we have 𝑧_𝑖^{2}≡0,𝑖=1,2,…,𝑛, then 𝑧^{2}≡(0,0,…,0)^𝑇, which implies 𝑧(𝑡)∈𝑉₁, that is Im(𝐿)⊂𝑉₁.
On the other hand, if 𝑢(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑛(𝑡))^𝑇∈𝑉₁⧵{0}, then we have ∫^𝑡₀𝑢_𝑗(𝑠)𝑑𝑠∈𝐴𝑃(ℝ), 𝑗=1,2,…,𝑛. If 𝜆_𝑗≠0, then we obtain lim_𝑇→∞1𝑇^𝑇₀^𝑡₀𝑢_𝑗(𝑠)𝑑𝑠𝑒^−𝑖𝜆_𝑗𝑡1𝑑𝑡=𝑖𝜆_𝑗lim_𝑇→∞1𝑇^𝑇₀𝑢_𝑗(𝑡)𝑒^−𝑖𝜆_𝑗𝑡𝑑𝑡,𝑗=1,2,…,𝑛. (3.9) It follows that ∧^𝑡₀𝑢_𝑗(𝑠)𝑑𝑠−𝑚^𝑡₀𝑢_𝑗(𝑠)𝑑𝑠=∧𝑢_𝑗(𝑡),𝑗=1,2,…,𝑛, (3.10) hence ^𝑡₀𝑢(𝑠)𝑑𝑠−𝑚^𝑡₀𝑢(𝑠)𝑑𝑠∈𝑉₁⊂𝑋. (3.11) Note that ∫^𝑡₀𝑢(𝑠)𝑑𝑠−𝑚(∫^𝑡₀𝑢(𝑠)𝑑𝑠) is the primitive of 𝑢(𝑡) in 𝑋, we have 𝑢(𝑡)∈Im(𝐿), that is, 𝑉₁⊂Im(𝐿). Therefore, Im(𝐿)=𝑉₁.
Furthermore, one can easily show that Im(𝐿) is closed in 𝑍 and dimker(𝐿)=𝑛=codimIm(𝐿); (3.12) therefore, 𝐿 is a Fredholm mapping of index 0. The proof of Lemma 3.2 is complete.

Lemma 3.3. Let 𝑁∶𝑋→𝑍, 𝑁𝑦=(𝐺^𝑦₁,𝐺^𝑦₂,…,𝐺^𝑦_𝑛)^𝑇, where 𝐺^𝑦_𝑖=𝑟_𝑖(𝑡)−𝑏_𝑖(𝑡)^{𝑦𝑖(𝑡−𝜏𝑖(𝑡))}+^𝑛_{𝑗=1,𝑖≠𝑗}𝑐_𝑖𝑗(𝑡)𝑒^{𝑦𝑗(𝑡−𝜏𝑖𝑗(𝑡))},𝑖=1,2,…,𝑛. (3.13) Set 𝑃∶𝑋⟶𝑍,𝑃𝑦=𝑚𝑦₁,𝑚𝑦₂,…,𝑚𝑦_𝑛^𝑇,𝑄∶𝑍⟶𝑍,𝑄𝑧=𝑚𝑧₁,𝑚𝑧₂,…,𝑚𝑧_𝑛^𝑇. (3.14) Then 𝑁 is 𝐿-compact on Ω, where Ω is an open bounded subset of 𝑋.

Proof. Obviously, 𝑃 and 𝑄 are continuous projectors such that Im𝑃=ker(𝐿),Im(𝐿)=ker(𝑄). (3.15) It is clear that (𝐼−𝑄)𝑉₂={(0,0,…,0)}, (𝐼−𝑄)𝑉₁=𝑉₁. Hence Im(𝐼−𝑄)=𝑉₁=Im(𝐿). (3.16) Then in view of Im(𝑃)=ker(𝐿),Im(𝐿)=ker(𝑄)=Im(𝐼−𝑄), (3.17) we obtain that the inverse 𝐾_𝑃∶Im(𝐿)→ker(𝑃)∩dom(𝐿) of 𝐿_𝑃 exists and is given by 𝐾_𝑃(𝑧)=^𝑡₀𝑧(𝑠)𝑑𝑠−𝑚^𝑡₀𝑧(𝑠)𝑑𝑠. (3.18) Thus, 𝑄𝑁𝑦=𝑚𝐺^𝑦₁,𝑚𝐺^𝑦₂,…,𝑚𝐺^𝑦_𝑛^𝑇𝐾_𝑃(𝐼−𝑄)𝑁𝑦=𝑓𝑦₁−𝑄𝑓𝑦₁,𝑓𝑦₂−𝑄𝑓𝑦₂,…,𝑓𝑦_𝑛−𝑄𝑓𝑦_𝑛^𝑇, (3.19) where 𝑓𝑦_𝑖=^𝑡₀𝐺^𝑦_𝑖−𝑚𝐺^𝑦_𝑖𝑑𝑠,𝑖=1,2,…,𝑛. (3.20)
Clearly, 𝑄𝑁 and (𝐼−𝑄)𝑁 are continuous. Now we will show that 𝐾_𝑃 is also continuous. By assumptions, for any 0<𝜖<1 and any compact set 𝜙_𝑖⊂𝐶([−𝜏,0],ℝ), let 𝑙_𝑖(𝜖_𝑖) be the length of the inclusion interval of 𝐾_𝑖(𝐻_𝑖,𝜖_𝑖,𝜙_𝑖), 𝑖=1,2,…,𝑛. Suppose that {𝑧^𝑘(𝑡)}⊂Im(𝐿)=𝑉₁ and 𝑧^𝑘(𝑡)=(𝑧^𝑘₁(𝑡),𝑧^𝑘₂(𝑡),…,𝑧^𝑘_𝑛(𝑡))^𝑇 uniformly converges to 𝑧(𝑡)=(𝑧₁(𝑡),𝑧₂(𝑡),…,𝑧_𝑛(𝑡))^𝑇, that is 𝑧^𝑘_𝑖→𝑧_𝑖, as 𝑘→∞, 𝑖=1,2,…,𝑛. Because of ∫^𝑡₀𝑧^𝑘(𝑠)d𝑠∈𝑍, 𝑘=1,2,…,𝑛, there exists 𝜎_𝑖(0<𝜎_𝑖<𝜖_𝑖) such that 𝐾_𝑖(𝐻_𝑖,𝜎_𝑖,𝜙_𝑖)⊂𝐾_𝑖(∫^𝑡₀𝑧^𝑘_𝑖(𝑠)d𝑠,𝜎_𝑖,𝜙_𝑖), 𝑖=1,2,…,𝑛. Let 𝑙_𝑖(𝜎_𝑖) be the length of the inclusion interval of 𝐾_𝑖(𝐻_𝑖,𝜎_𝑖,𝜙_𝑖) and 𝑙_𝑖=max𝑙_𝑖𝜖_𝑖,𝑙_𝑖𝜎_𝑖,𝑖=1,2,…,𝑛. (3.21) It is easy to see that 𝑙_𝑖 is the length of the inclusion interval of 𝐾_𝑖(𝐻_𝑖,𝜎_𝑖,𝜙_𝑖) and 𝐾_𝑖(𝐻_𝑖,𝜖_𝑖,𝜙_𝑖), 𝑖=1,2,…,𝑛. Hence, for any 𝑡∉[0,𝑙_𝑖], there exists 𝜉_𝑡∈𝐾_𝑖(𝐻_𝑖,𝜎_𝑖,𝜙_𝑖)⊂𝐾_𝑖(∫^𝑡₀𝑧^𝑘_𝑖(𝑠)𝑑𝑠,𝜎_𝑖,𝜙_𝑖) such that 𝑡+𝜉_𝑡∈[0,𝑙_𝑖], 𝑖=1,2,…,𝑛. Hence, by the definition of almost periodic function we have ^𝑡₀𝑧^𝑘(𝑠)𝑑𝑠=max_{1≤𝑖≤𝑛}sup_𝑡∈𝑅^𝑡₀𝑧^𝑘_𝑖(𝑠)𝑑𝑠≤max_{1≤𝑖≤𝑛}sup_{𝑡∈0,𝑙𝑖}^𝑡₀𝑧^𝑘_𝑖(𝑠)𝑑𝑠+max_{1≤𝑖≤𝑛}sup_{𝑡∉0,𝑙𝑖}^𝑡₀𝑧^𝑘_𝑖(𝑠)𝑑𝑠−^{𝑡+𝜉𝑡}₀𝑧^𝑘_𝑖(𝑠)𝑑𝑠+^{𝑡+𝜉𝑡}₀𝑧^𝑘_𝑖(𝑠)𝑑𝑠≤2max_{1≤𝑖≤𝑛}sup_{𝑡∈0,𝑙𝑖}^𝑡₀𝑧^𝑘_𝑖(𝑠)𝑑𝑠+max_{1≤𝑖≤𝑛}sup_{𝑡∉0,𝑙𝑖}^𝑡₀𝑧^𝑘_𝑖(𝑠)𝑑𝑠−^{𝑡+𝜉𝑡}₀𝑧^𝑘_𝑖(𝑠)𝑑𝑠≤2max_{1≤𝑖≤𝑛}^𝑙𝑖₀𝑧^𝑘_𝑖(𝑠)𝑑𝑠+max_{1≤𝑖≤𝑛}𝜖_𝑖. (3.22) From this inequality, we can conclude that ∫^𝑡₀𝑧(𝑠)𝑑𝑠 is continuous, where 𝑧(𝑡)=(𝑧₁(𝑡),𝑧₂(𝑡),…,𝑧_𝑛(𝑡))^𝑇∈Im(𝐿). Consequently, 𝐾_𝑃 and 𝐾_𝑃(𝐼−𝑄)𝑁𝑦 are continuous.
From (3.22), we also have ∫^𝑡₀𝑧(𝑠)𝑑𝑠 and 𝐾_𝑃(𝐼−𝑄)𝑁𝑦 also are uniformly bounded in Ω. Further, it is not difficult to verify that 𝑄𝑁(Ω) is bounded and 𝐾_𝑃(𝐼−𝑄)𝑁𝑦 is equicontinuous in Ω. By the Arzela-Ascoli theorem, we have immediately concluded that 𝐾_𝑃(𝐼−𝑄)𝑁(Ω) is compact. Thus 𝑁 is 𝐿-compact on Ω. The proof of Lemma 3.3 is complete.

Theorem 3.4. Assume that the following conditions (𝐻₁) and (𝐻₂) hold:(𝐻₁)𝑒_𝑖∶=𝑚[𝑟_𝑖(𝑡)]>0,𝑚[𝑏_𝑖(𝑡)]>0,𝑚[𝑐_𝑖𝑗(𝑡))]>0,𝑖,𝑗=1,2,…,𝑛; (𝐻₂)𝐷 is a positive-definite matrix, where 𝑑_𝑖𝑖=𝑚[𝑏_𝑖(𝑡)], 𝑑_𝑖𝑗=𝑚[𝑐_𝑖𝑗(𝑡)], 𝑖≠𝑗, 𝑖,𝑗=1,2,…,𝑛, 𝑑𝐷=¹¹−𝑑₁₂⋯−𝑑_1𝑛−𝑑₂₁𝑑₂₂⋯−𝑑_2𝑛⋮⋮⋮⋮−𝑑_𝑛1−𝑑_𝑛2⋯𝑑_𝑛𝑛. (3.23) Then (1.1) has at least one positive almost periodic solution.

Proof. To use the continuation theorem of coincidence degree theorem to establish the existence of a solution of (3.2), we set Banach space 𝑋 and 𝑍 the same as those in Lemma 3.1 and set mappings 𝐿, 𝑁, 𝑃,𝑄 the same as those in Lemmas 3.2 and 3.3, respectively. Then we can obtain that 𝐿 is a Fredholm mapping of index 0 and 𝑁 is a continuous operator which is 𝐿-compact on Ω.
Now, we are in the position of searching for an appropriate open, bounded subset Ω for the application of the continuation theorem. Corresponding to the operator equation 𝐿𝑦=𝜆𝑁𝑦,𝜆∈(0,1), (3.24) we obtain 𝑦_𝑖̇(𝑡)=𝜆𝑟_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑒^{𝑦𝑖(𝑡−𝜏𝑖(𝑡))}+^𝑛_{𝑗=1,𝑖≠𝑗}𝑐_𝑖𝑗(𝑡)𝑒^{𝑦𝑗(𝑡−𝜏𝑖𝑗(𝑡))},𝑖=1,2,…,𝑛. (3.25) Assume that 𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))^𝑇∈𝑋 is a solution of (3.25) for some 𝜆∈(0,1). Denote 𝑀_𝑖=sup_𝑡∈𝑅{𝑦_𝑖(𝑡)}, 𝑀_𝑖=inf_𝑡∈𝑅{𝑦_𝑖(𝑡)}.
On the one hand, by (3.25), we derive −𝑦_𝑖̇(𝑡)≤𝜆𝑟_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑒^{𝑦𝑖(𝑡−𝜏𝑖(𝑡))}+^𝑛_{𝑗=1,𝑖≠𝑗}𝑐_𝑖𝑗(𝑡)𝑒^{𝑦𝑗(𝑡−𝜏𝑖𝑗(𝑡))},𝑖=1,2,…,𝑛. (3.26) On the both sides of (3.26), integrating from 0 to 𝑇 and applying the mean value theorem of integral calculus, we have 0≤𝜆𝑚𝑟_i(𝑡)−𝑚𝑏_𝑖(𝑡)𝑒^{𝑦𝑖(𝜉𝑖−𝜏𝑖(𝜉𝑖))}+^𝑛_{𝑗=1,𝑖≠𝑗}𝑚𝑐_𝑖𝑗(𝑡)𝑒^{𝑦𝑗(𝜂𝑖𝑗−𝜏𝑖𝑗(𝜂𝑖𝑗))}+𝑚𝑦_𝑖̇(𝑡),𝑖=1,2,…,𝑛, (3.27) where 𝜉_𝑖∈[0,𝑇], 𝜂_𝑖𝑗∈[0,𝑇], 𝑖,𝑗=1,2,…,𝑛. In the light of (3.27), we get for 𝑖=1,2,…,𝑛, 𝜆𝑚𝑏_𝑖(𝑡)𝑒^{𝑦𝑖(𝜉𝑖−𝜏𝑖(𝜉𝑖))}≤𝜆𝑚𝑟_𝑖(𝑡)+^𝑛_{𝑗=1,𝑖≠𝑗}𝑚𝑐_𝑖𝑗(𝑡)𝑒^{𝑦𝑗(𝜂𝑖𝑗−𝜏𝑖𝑗(𝜂𝑖𝑗))}+𝑚𝑦_𝑖̇(𝑡). (3.28) On the both sides of (3.28), taking the supremum with respect to 𝜉_𝑖,𝜂_𝑖𝑗 and letting 𝑇→+∞, we obtain 𝑚𝑏_i(𝑡)𝑒^𝑀𝑖≤𝑚𝑟_𝑖(𝑡)+^𝑛_{𝑗=1,𝑗≠𝑖}+𝑚𝑐_𝑖𝑗(𝑡)𝑒^𝑀𝑗, (3.29) that is, 𝑑_𝑖𝑖𝑒^𝑀𝑖−^𝑛_{𝑗=1,𝑗≠𝑖}𝑑_𝑖𝑗𝑒^𝑀𝑗≤𝑒_𝑖,𝑖=1,2,…,𝑛. (3.30) Equation (3.30) can be written by the following matrix form 𝑑₁₁−𝑑₁₂⋯−𝑑_1𝑛−𝑑₂₁𝑑₂₂⋯−𝑑_2𝑛⋮⋮⋮⋮−𝑑_𝑛1−𝑑_𝑛2⋯𝑑_𝑛𝑛𝑒^𝑀1𝑒^𝑀2⋮𝑒^𝑀𝑛≤𝑒₁𝑒₂⋮𝑒_𝑛. (3.31) By Lemma 2.8 and assumption, we obtain 𝑒^𝑀1𝑒^𝑀2⋮𝑒^𝑀𝑛≤𝐷⁻¹𝑒₁𝑒₂⋮𝑒_𝑛𝐻=∶⁺¹𝐻⁺₂⋮𝐻⁺_𝑛, (3.32) which imply that 𝑀_𝑖≤ln𝐻⁺_𝑖,𝑖=1,2,…,𝑛. (3.33) On the two sides of (3.28), taking the infimum with respect to 𝜉_𝑖, 𝜂_𝑖𝑗, and letting 𝑇→+∞, we obtain 𝑀_𝑖≤ln𝐻⁺_𝑖,𝑖=1,2,…,𝑛. (3.34)
On the other hand, according to (3.25), we derive 𝜆𝑟_𝑖(𝑡)−𝑦_𝑖̇(𝑡)<𝜆𝑏_𝑖(𝑡)𝑒^{𝑦𝑖(𝑡−𝜏𝑖(𝑡))},𝑖=1,2,…,𝑛. (3.35) On the both sides of (3.35), integrating from 0 to 𝑇 and using the mean value theorem of integral calculus, we get 𝜆𝑚𝑟_𝑖(𝑡)<𝑚𝑦_𝑖̇(𝑡)+𝜆𝑚𝑏_𝑖(𝑡)𝑒^{𝑦𝑖(𝜁𝑖−𝜏𝑖(𝜁𝑖))},𝑖=1,2,…,𝑛, (3.36) where 𝜁_𝑖∈[0,𝑇], 𝑖=1,2,…,𝑛. On the both sides of (3.36), take the supremum and infimum with respect to 𝜁_𝑖, respectively, and let 𝑇→+∞, then we have for 𝑖=1,2,…,𝑛, 𝑚𝑟_𝑖(𝑡)<𝑚𝑏_𝑖𝑖(𝑡)𝑒^𝑀𝑖,𝑚𝑟_𝑖(𝑡)<𝑚𝑏_𝑖𝑖(𝑡)𝑒^𝑀𝑖, (3.37) namely, 𝑒^𝑀𝑖>𝑚𝑟_𝑖(𝑡)𝑚𝑏_𝑖(𝑡=𝑒)𝑑^𝑖_𝑖𝑖,𝑒^𝑀𝑖>𝑚𝑟_𝑖(𝑡)𝑚𝑏_𝑖(𝑡=𝑒)𝑑^𝑖_𝑖𝑖 (3.38) which imply that 𝑀_𝑖𝑒>ln^𝑖𝑑_𝑖𝑖,𝑀_𝑖𝑒>ln^𝑖𝑑_𝑖𝑖. (3.39) Combining with (3.33), (3.34), and (3.39), we derive for all 𝑡∈ℝ,𝑖=1,2,…,𝑛, min_{1≤𝑖≤𝑛}𝑒ln^𝑖𝑑_𝑖𝑖𝑒≤ln^𝑖𝑑_𝑖𝑖<𝑀_𝑖≤𝑦_𝑖(𝑡)≤𝑀_𝑖≤ln𝐻⁺_𝑖<max_{1≤𝑖≤𝑛}ln𝐻⁺_𝑖+1. (3.40) Denote 𝑀=max{|min_{1≤𝑖≤𝑛}{ln(𝑒_𝑖/𝑑_𝑖𝑖)}|,|max_{1≤𝑖≤𝑛}{ln𝐻⁺_𝑖}+1|}. Clearly, 𝑀 is independent of 𝜆. Take Ω=𝑦=𝑦₁,𝑦₂,…,𝑦_𝑛^𝑇∈𝑋∶𝑦<𝑀. (3.41) It is clear that Ω satisfies the requirement (a) in Lemma 2.5. When 𝑦∈𝜕Ω∩ker(𝐿),𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)^𝑇 is a constant vector in ℝ^𝑛 with ‖𝑦‖=𝑀. Then 𝑄𝑁𝑦=𝑚𝐺₁,𝑚𝐺₂,…,𝑚𝐺_𝑛^𝑇,𝑦∈𝑋, (3.42) where 𝐺_𝑖=𝑟_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑒^𝑦𝑖+^𝑛_{𝑗=1,𝑗≠𝑖}𝑐_𝑖𝑗(𝑡)𝑒^𝑦𝑗𝑚𝐺,𝑖=1,2,…,𝑛,_𝑖=𝑚𝑟_𝑖(𝑡)−𝑚𝑏_𝑖𝑖(𝑡)𝑒^𝑦𝑖+^𝑛_{𝑗=1,𝑗≠𝑖}𝑚𝑐_𝑖𝑗(𝑡)𝑒^𝑦𝑗=𝑒_𝑖−𝑑_𝑖𝑖𝑒^𝑦𝑖+^𝑛_{𝑗=1,𝑗≠𝑖}𝑑_𝑖𝑗𝑒^𝑦𝑗,𝑖=1,2,…,𝑛. (3.43) If 𝑄𝑁𝑦=(0,0,…,0])^𝑇, then we have 𝑑₁₁−𝑑₁₂⋯−𝑑_1𝑛−𝑑₂₁𝑑₂₂⋯−𝑑_2𝑛⋮⋮⋮⋮−𝑑_𝑛1−𝑑_𝑛2⋯𝑑_𝑛𝑛𝑒^𝑦1𝑒^𝑦2⋮𝑒^𝑦𝑛=𝑒₁𝑒₂⋮𝑒_𝑛, (3.44) which imply that 𝑦_𝑖=ln𝐻⁺_𝑖, 𝑖=1,2,…,𝑛. Thus, 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)^𝑇∈Ω, this contradicts the fact that 𝑦∈𝜕Ω∩ker(𝐿). Therefore, 𝑄𝑁𝑦≠(0,0,…,0)^𝑇, which implies that the requirement (b) in Lemma 2.5 is satisfied. If necessary, we can let 𝑀 be greater such that 𝑦^𝑇𝑄𝑁𝑦<0, for any 𝑦∈𝜕Ω∩ker(𝐿). Furthermore, take the isomorphism 𝐽∶Im(𝑄)→ker(𝐿), 𝐽𝑧≡𝑧 and let Φ(𝛾;𝑦)=−𝛾𝑦+(1−𝛾)𝐽𝑄𝑁𝑦, then for any 𝑦∈𝜕Ω∩ker(𝐿), 𝑦^𝑇Φ(𝛾;𝑦)<0, we have deg{𝐽𝑄𝑁,Ω∩ker(𝐿),0}=deg−𝑦,Ω∩ker(𝐿),0≠0. (3.45) So, the requirement (c) in Lemma 2.5 is satisfied. Hence, (3.2) has at least one almost-periodic solution in Ω, that is, (1.1) has at least one positive almost periodic solution. The proof is complete.

4. An Example and Simulation

Consider the following two species cooperative system with time delay:𝑥̇(𝑡)=𝑥(𝑡)𝑟₁(𝑡)−𝑏₁(𝑡)𝑥𝑡−𝜏₁(𝑡)+𝑐₁₂(𝑡)𝑦𝑡−𝜏₁₂(𝑡),𝑦̇(𝑡)=𝑦(𝑡)𝑟₂(𝑡)−𝑏₂(𝑡)𝑦𝑡−𝜏₂(𝑡)+𝑐₂₁(𝑡)𝑥𝑡−𝜏₂₁(𝑡), (4.1) where 𝑟₁(𝑡)=2+sin√2𝑡+sin√3𝑡, 𝑏₁(𝑡)=2+sin√3𝑡+sin√5𝑡, 𝑐₁₂(𝑡)=(2+cos√2𝑡+cos√3𝑡)/2, 𝜏₁(𝑡)=𝑒^{√sin√2𝑡+sin5𝑡}, 𝜏₁₂(𝑡)=𝑒^{√sin𝑡+cos2𝑡}, 𝑟₂(𝑡)=2−sin√2𝑡−sin√3𝑡, 𝑏₂(𝑡)=2+sin√3𝑡−sin√5𝑡, 𝑐₂₁(𝑡)=(2−cos√2𝑡+cos√3𝑡)/2, 𝜏₂(𝑡)=𝑒^{√sin√2𝑡+cos5𝑡}, 𝜏₂₁(𝑡)=𝑒^{√sin𝑡−cos2𝑡}. Since𝑒₁=𝑚𝑟₁(𝑡)=2,𝑑₁₁=𝑚𝑏₁(𝑡)=2,𝑑₁₂=𝑚𝑐₁₂(𝑡𝑒)=1,₂=𝑚𝑟₁(𝑡)=2,𝑑₂₂=𝑚𝑏₁(𝑡)=2,𝑑₂₁=𝑚𝑐₁₂(𝑡)=1,𝐷=2−1−12,det2−1−12=3>0,𝐷⁻¹=13,2112 (4.2) then, the matrix 𝐷 is positive definite, and𝑒ln¹𝑑₁₁𝑒=ln²𝑑₂₂𝐻=0,⁺¹𝐻⁺₂=𝐷⁻¹𝑒₁𝑒₂=22,ln𝐻⁺₁ln𝐻⁺₂=,ln2ln2𝑀=maxmin_{1≤𝑖≤𝑛}𝑒ln^𝑖𝑑_𝑖𝑖,max_{1≤𝑖≤𝑛}ln𝐻⁺_𝑖𝑦+1=1+ln2,Ω=𝑦=_1,𝑦₂,…,𝑦_𝑛^𝑇∈𝑋∶𝑦<1+In2. (4.3) Therefore, all conditions of Theorem 3.4 are satisfied. By Theorem 3.4, system (4.1) has one positive almost-periodic solution. The resulting numerical simulation is depicted in Figure 1.

Figure 1 

Acknowledgments

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant (no. 11161025), Yunnan Province natural scientific Research Fund Project (no. 2011FZ058), and Yunnan Province education department scientific Research Fund Project (no. 2001Z001).