Positive Periodic Solutions of a Periodic Discrete Competitive System Subject to Feedback Controls

Abstract

Species living in a fluctuating medium and human exploitation activities might result in the duration of continuous changes. Such changes can be well-approximated as feedback controls. In this contribution a periodic discrete competitive system subject to feedback controls is proposed. By using the methods of discrete inequality, fixed point theorem, and analysis techniques, a good understanding of the existence and global asymptotic stability of positive periodic solutions is obtained. Some numerical investigations are provided to verify our analytical results.

1. Introduction

Many mathematical models about bioecology play important roles for researchers to realize the interactions of ecological species. There has been increasing interest in studying the dynamical behaviors such as stability, permanence, and periodicity of competitive systems (see [1–5]). Recently, motivated by an autonomous competitive model of [6] in [7] we introduced a corresponding nonautonomous version 𝑦^′₁(𝑡)=𝑦₁(𝑡)𝑎₁(𝑡)−𝑏₁(𝑡)𝑦₁(𝑡)−𝑐₁(𝑡)𝑦₂(𝑡)−𝑑₁(𝑡)𝑦²₁(𝑡,𝑦_′2)(𝑡)=𝑦₂(𝑡)𝑎₂(𝑡)−𝑏₂(𝑡)𝑦₂(𝑡)−𝑐₂(𝑡)𝑦₁(𝑡)−𝑑₂(𝑡)𝑦²₂(𝑡), (1) where 𝑦₁(𝑡) and 𝑦₂(𝑡) can be interpreted as the density of two competing species at time 𝑡, respectively. 𝑎_𝑖(𝑡) stand for the growth rates of species, 𝑏_𝑖(𝑡) and 𝑑_𝑖(𝑡) represent the effects of intraspecific competition, and 𝑐_𝑖(𝑡) are the rates of interspecific competition. Furthermore, we took the influence of almost periodic environment and impulsive perturbations into account and established sufficient conditions for the uniformly asymptotic stability of a unique positive almost periodic solution for the above system (for details see [7]).

Note that ecosystems in the real world are often disturbed by outside continuous forces which can lead to changes in biological coefficients such as survival rates. In the language of control, we call the disturbance functions as feedback control variables. Many good results on this direction are deliberated (see [8–12]) and some similar works on the topic have been done (see [13, 14]). This paper is concerned with a discrete model and is a continuation of the work in [7]; in this contribution we search for certain schemes (such as harvesting procedure) to ensure the system coexists under appropriate conditions. For this reason, we consider a discrete version which corresponds to differential version (i.e., system (1)), meanwhile, replacing abrupt external perturbations (impulses) in [7] by continuous external perturbations (feedback control variables), a nonautonomous discrete controlled system can be described as follows: 𝑦₁(𝑘+1)=𝑦₁(𝑘)exp𝑎₁(𝑘)−𝑏₁(𝑘)𝑦₁(𝑘)−𝑐₁(𝑘)𝑦₂(𝑘)−𝑑₁(𝑘)𝑦²₁(𝑘)−𝑒₁(𝑘)𝑣₁(𝑘,𝑦₂)(𝑘+1)=𝑦₂(𝑘)exp𝑎₂(𝑘)−𝑏₂(𝑘)𝑦₂(𝑘)−𝑐₂(𝑘)𝑦₁(𝑘)−𝑑₂(𝑘)𝑦²₂(𝑘)−𝑒₂(𝑘)𝑣₂(𝑘),Δ𝑣₁=𝑣₁(𝑘+1)−𝑣₁(𝑘)=ℎ₁(𝑘)−𝑓₁(𝑘)𝑣₁(𝑘)+𝑔₁(𝑘)𝑦₁(𝑘),Δ𝑣₂=𝑣₂(𝑘+1)−𝑣₂(𝑘)=ℎ₂(𝑘)−𝑓₂(𝑘)𝑣₂(𝑘)+𝑔₂(𝑘)𝑦₂(𝑘),𝑘∈ℤ⁺. (2) Here 𝑦_𝑖(0)>0, 𝑣_𝑖(0)>0. Δ𝑣_𝑖(𝑘) are the first-order forward difference operators, 𝑦_𝑖(𝑘) stand for the densities of species 𝑦_𝑖 at the 𝑘th generation, 𝑎_𝑖(𝑘) represent the natural growth rates of species 𝑦_𝑖 at the 𝑘th generation, 𝑏_𝑖(𝑘) and 𝑑_𝑖(𝑘) stand for the intraspecific effects of the 𝑘th generation of species 𝑦_𝑖 on own population, and 𝑐_𝑖(𝑘) measure the interspecific effects of the 𝑘th generation of species 𝑦_𝑗 on species 𝑦_𝑖. The coefficients {𝑎_𝑖(𝑘)}, {𝑏_𝑖(𝑘)}, {𝑐_𝑖(𝑘)}, {𝑑_𝑖(𝑘)}, {𝑒_𝑖(𝑘)}, {𝑓_𝑖(𝑘)}, {𝑔_𝑖(𝑘)}, and {ℎ_𝑖(𝑘)} are 𝑇-periodic sequences with 0<𝑓_𝑖(𝑘)<1,𝑖,𝑗=1,2;𝑖≠𝑗. ℤ⁺ is the set of nonnegative integers.

It is well known that the discrete models governed by difference equations may be more appropriate than the continuous ones when populations have a short life expectancy or nonoverlapping generations. Also, discrete models can provide efficient computational methods of continuous models for numerical simulations. A very important ecological problem associated with the study of multispecies population interaction in a periodic environment is the positive periodic solution which plays the role played by the equilibrium of the autonomous model. In this paper, we will discuss the above discrete system (2) and focus on the existence and stability of positive periodic solution of system (2). To the best of our knowledge, there are few published papers concerning system (2).

The rest structure of this paper is as follows. In the next section, we establish sufficient conditions for the existence of positive periodic solutions. In Section 3, we further discuss the global asymptotic stability of positive periodic solutions. In Section 4, we carry out an example and its numerical simulations to substantiate our theoretical results. To simplify the reading of paper, we give the proofs of lemmas and theorems in appendices.

2. Existence

In this section, we establish sufficient conditions for the existence of positive periodic solutions of system (2). To do this, we first give two preliminary lemmas. For convenience, given a bounded sequence {𝑓(𝑘)}, let 𝑓^𝑈 and 𝑓^𝐿 be defined as 𝑓^𝑈=sup_𝑘∈ℤ⁺𝑓(𝑘), 𝑓^𝐿=inf_𝑘∈ℤ⁺𝑓(𝑘), and𝑀_𝑖=exp𝑎^𝑈_𝑖−1𝑏^𝐿_𝑖,𝑁_𝑖=ℎ^𝑈_𝑖+𝑔^𝑈_𝑖𝑀_𝑖𝑓^𝐿_𝑖,𝑚_𝑖=𝑎^𝐿_𝑖−𝑐^𝑈_𝑖𝑀_𝑗−𝑑^𝑈_𝑖𝑀²_𝑖−𝑒^𝑈_𝑖𝑁_𝑖𝑏^𝑈_𝑖×exp𝑎^𝐿_𝑖−𝑏^𝑈_𝑖𝑀_𝑖−𝑐^𝑈_𝑖𝑀_𝑗−𝑑^𝑈_𝑖𝑀²_𝑖−𝑒^𝑈_𝑖𝑁_𝑖,𝑛_𝑖=ℎ^𝐿_𝑖+𝑔^𝐿_𝑖𝑚_𝑖𝑓^𝑈_𝑖,𝑖≠𝑗;𝑖,𝑗=1,2. (3)

Lemma 1. Every positive solution {𝑦₁(𝑘),𝑦₂(𝑘),𝑣₁(𝑘),𝑣₂(𝑘)} of system (2) satisfies limsup_𝑘→+∞𝑦_𝑖(𝑘)≤𝑀_𝑖 and limsup_𝑘→+∞𝑣_𝑖(𝑘)≤𝑁_𝑖,𝑖=1,2.

The proof of Lemma 1 is given in Appendix A.

Lemma 2. Assuming that 𝑎^𝐿₁−𝑐^𝑈₁𝑀₂−𝑑^𝑈₁𝑀²₁−𝑒^𝑈₁𝑁₁𝑎>0,_𝐿2−𝑐^𝑈₂𝑀₁−𝑑^𝑈₂𝑀²₂−𝑒^𝑈₂𝑁₂>0 (4) hold, then every positive solution {𝑦₁(𝑘),𝑦₂(𝑘),𝑣₁(𝑘),𝑣₂(𝑘)} of system (2) satisfies liminf_𝑘→+∞𝑦_𝑖(𝑘)≥𝑚_𝑖 and liminf_𝑘→+∞𝑣_𝑖(𝑘)≥𝑛_𝑖,𝑖=1,2.

The proof of Lemma 2 is given in Appendix B.

Theorem 3. If the inequalities in (4) hold, then system (2) has at least one strictly positive 𝑇-periodic solution.

The proof of Theorem 3 is given in Appendix C.

3. Global Asymptotic Stability

In this section, we further investigate the stability of positive periodic solutions of system (2). Denote 𝜌₁=max1−𝑏^𝐿₁𝑚₁−2𝑑^𝐿₁𝑚²₁,1−𝑏^𝑈₁𝑀₁−2𝑑^𝑈₁𝑀²₁+𝑐^𝑈₁𝑀₂+𝑒^𝑈₁,𝜌₂=max1−𝑏^𝐿₂𝑚₂−2𝑑^𝐿₂𝑚²₂,1−𝑏^𝑈₂𝑀₂−2𝑑^𝑈₂𝑀²₂+𝑐^𝑈₂𝑀₁+𝑒^𝑈₂,𝜌₃=1−𝑓^𝐿₁+𝑔^𝑈₁𝑀₁,𝜌₄=1−𝑓^𝐿₂+𝑔^𝑈₂𝑀₂. (5)

Theorem 4. In addition to (4), further suppose that 𝜌_𝑖<1, 𝑖=1,2,3,4. Then the positive periodic solution of system (2) is globally asymptotically stable.

The proof of Theorem 4 is given in Appendix D.

4. An Example

In this section we give a numerical example and its corresponding simulations.

Example 1. Consider a discrete competitive system with feedback controls 𝑦₁(𝑘+1)=𝑦₁(𝑘)()−((𝑦exp0.98+0.003sin𝑘𝜋0.85−0.002cos𝑘𝜋))₁(𝑘)−(0.002+0.0001sin(𝑘𝜋))𝑦₂(𝑘)−(0.004−0.0002cos(𝑘𝜋))𝑦²₁(𝑘)−(0.003+0.0001sin(𝑘𝜋))𝑣₁(𝑘,𝑦₂)(𝑘+1)=𝑦₂(𝑘)()−((𝑦exp1.00−0.002sin𝑘𝜋0.92+0.001cos𝑘𝜋))₂(𝑘)−(0.003+0.0001cos(𝑘𝜋))𝑦₁(𝑘)−(0.005+0.0003sin(𝑘𝜋))𝑦²₂(𝑘)−(0.002+0.0002sin(𝑘𝜋))𝑣₂(𝑘),Δ𝑣₁(𝑘)()−((𝑣=0.005+0.0002sin𝑘𝜋0.95+0.003cos𝑘𝜋))₁(𝑘)+(0.006+0.0001sin(𝑘𝜋))𝑦₁(𝑘),Δ𝑣₂(𝑘)()−((𝑣=0.004+0.0001sin𝑘𝜋0.98+0.002sin𝑘𝜋))₂(𝑘)+(0.003+0.0002cos(𝑘𝜋))𝑦₂(𝑘). (6) A computation shows that 𝑎^𝐿₁=0.977,𝑎^𝑈₁=0.983,𝑏^𝐿₁𝑏=0.848,_𝑈1=0.852,𝑐^𝐿₁=0.0019,𝑐^𝑈₁𝑑=0.0021,_𝐿1=0.0038,𝑑^𝑈₁=0.0042,𝑒^𝐿₁𝑒=0.0029,_𝑈1=0.0031,𝑓^𝐿₁=0.947,𝑓^𝑈₁𝑔=0.953,_𝐿1=0.0059,𝑔^𝑈₁=0.0061,ℎ^𝐿₁ℎ=0.0048,_𝑈1=0.0052,𝑎^𝐿₂=0.998,𝑎^𝑈₂𝑏=1.002,_𝐿2=0.919,𝑏^𝑈₂=0.921,𝑐^𝐿₂𝑐=0.0029,_𝑈2=0.0031,𝑑^𝐿₂=0.0047,𝑑^𝑈₂𝑒=0.0053,_𝐿2=0.0018,𝑒^𝑈₂=0.0022,𝑓^𝐿₂𝑓=0.978,_𝑈2=0.982,𝑔^𝐿₂=0.0028,𝑔^𝑈₂ℎ=0.0032,_𝐿2=0.0039,ℎ^𝑈₂=0.0041. (7) Furthermore, it follows from (3) that 𝑀₁≈1.1594,𝑀₂𝑁≈1.0903,₁≈0.0130,𝑁₂𝑚≈0.0078,₁≈1.1162,𝑚₂𝑛≈1.0557,₁≈0.0119,𝑛₂𝑎≈0.0070,_𝐿1−𝑐^𝑈₁𝑀₂−𝑑^𝑈₁𝑀²₁−𝑒^𝑈₁𝑁₁𝑎≈0.9690>0,_𝐿2−𝑐^𝑈₂𝑀₁−𝑑^𝑈₂𝑀²₂−𝑒^𝑈₂𝑁₂≈0.9881>0. (8) Obviously, the assumptions in (4) are satisfied, and moreover, one has 𝜌₁=max1−𝑏^𝐿₁𝑚₁−2𝑑^𝐿₁𝑚²₁,1−𝑏^𝑈₁𝑀₁−2𝑑^𝑈₁𝑀²₁+𝑐^𝑈₁𝑀₂+𝑒^𝑈₁𝜌≈0.0494<1,₂=max1−𝑏^𝐿₂𝑚₂−2𝑑^𝐿₂𝑚²₂,1−𝑏^𝑈₂𝑀₂−2𝑑^𝑈₂𝑀²₂+𝑐^𝑈₂𝑀₁+𝑒^𝑈₂𝜌≈0.0251<1,₃=1−𝑓^𝐿₁+𝑔^𝑈₁𝑀₁𝜌≈0.0601<1,₄=1−𝑓^𝐿₂+𝑔^𝑈₂𝑀₂≈0.0255<1, (9) so the assumptions of Theorem 4 are also satisfied. Thus, there exists a globally asymptotically stable positive periodic solution of system (6). Figure 1 shows that system (6) exists a positive periodic solution(𝑦^∗₁(𝑘),𝑦^∗₂(𝑘),𝑣^∗₁(𝑘),𝑣^∗₂(𝑘)), and the two-dimensional phase portrait of periodic system (6) is displayed in Figure 2. From Figure 3, we can see that any positive solution (𝑦₁(𝑘),𝑦₂(𝑘),𝑣₁(𝑘),𝑣₂(𝑘)) tends to the periodic solution (𝑦^∗₁(𝑘),𝑦^∗₂(𝑘),𝑣^∗₁(𝑘),𝑣^∗₂(𝑘)), and the two-dimensional phase portrait reflects the fact in Figure 4.

(a)

(b)

(c)

(d)

Figure 1 

Positive periodic solution of system (6). (a)–(d) Time-series of 𝑦∗1(𝑘), 𝑦∗2(𝑘), 𝑣∗1(𝑘), and 𝑣∗2(𝑘) with initial values 𝑦∗1(0)=1.1560, 𝑦∗2(0)=1.0720, 𝑣∗1(0)=0.0128, and 𝑣∗2(0)=0.0078 for 𝑘∈[0,100], respectively.

(a)

(b)

Figure 2 

Two-dimensional phase portrait. (a) Phase portrait of 𝑦∗1(𝑘) and 𝑦∗2(𝑘) with initial values 𝑦∗1(0)=1.1560 and 𝑦∗2(0)=1.0720 for 𝑘∈[0,100], respectively. (b) Phase portrait of 𝑣∗1(𝑘) and 𝑣∗2(𝑘) with initial values 𝑣∗1(0)=0.0128 and 𝑣∗2(0)=0.0078 for 𝑘∈[0,100], respectively.

(a)

(b)

(c)

(d)

Figure 3 

Global asymptotic stability. (a)–(d) Time-series of 𝑦∗1(𝑘), 𝑦∗2(𝑘), 𝑣∗1(𝑘), and 𝑣∗2(𝑘) with initial values 𝑦∗1(0)=1.1560, 𝑦∗2(0)=1.0720, 𝑣∗1(0)=0.0128, and 𝑣∗2(0)=0.0078 and 𝑦1(𝑘), 𝑦2(𝑘), 𝑣1(𝑘), and 𝑣2(𝑘) with initial values 𝑦1(0)=1.1300, 𝑦2(0)=1.0850, 𝑣1(0)=0.0123, and 𝑣2(0)=0.0071 for 𝑘∈[0,100], respectively.

(a)

(b)

Figure 4 

Two-dimensional phase portrait. (a) Phase portrait of 𝑦∗1(𝑘) and 𝑦∗2(𝑘) with initial values 𝑦∗1(0)=1.1560 and 𝑦∗2(0)=1.0720 and 𝑦1(𝑘) and 𝑦2(𝑘) with initial values 𝑦1(0)=1.1300 and 𝑦2(0)=1.0850 for 𝑘∈[0,100], respectively. (b) Phase portrait of 𝑣∗1(𝑘) and 𝑣∗2(𝑘) with initial values 𝑣∗1(0)=0.0128 and 𝑣∗2(0)=0.0078 and 𝑣1(𝑘) and 𝑣2(𝑘) with initial values 𝑣1(0)=0.0123 and 𝑣2(0)=0.0071 for 𝑘∈[0,100], respectively.

Appendices

A. Proof of Lemma 1

We consider the following Cases I and II to show that limsup_𝑘→+∞𝑦_𝑖(𝑘)≤𝑀_𝑖,𝑖=1,2. (A.1)Case I. Suppose that there is a 𝑘₀∈ℤ⁺ such that 𝑦₁(𝑘₀)≤𝑦₁(𝑘₀+1), it follows from the first equation of system (2) that 𝑦₁𝑘₀+1=𝑦₁𝑘₀exp𝑎₁𝑘₀−𝑏₁𝑘₀𝑦₁𝑘₀−𝑐₁𝑘₀𝑦₂𝑘₀−𝑑₁𝑘₀𝑦²₁𝑘₀−𝑒₁𝑘₀𝑣₁𝑘₀≤𝑦₁𝑘₀exp𝑎^𝑈₁−𝑏^𝐿₁𝑦₁𝑘₀≤exp𝑎^𝑈₁−1𝑏^𝐿₁=𝑀₁. (A.2) Here we use the fact that max_𝑥∈ℝ⁺[𝑥exp(𝑎−𝑏𝑥)]=[exp(𝑎−1)]/𝑏 for 𝑎,𝑏>0 and ℝ⁺ is the set of all positive real numbers. Thus 𝑦₁(𝑘₀)≤𝑦₁(𝑘₀+1)≤𝑀₁.

One claims that 𝑦₁(𝑘)≤𝑀₁ for all 𝑘≥𝑘₀. Otherwise, if there is a 𝑙₀>𝑘₀ such that 𝑦₁(𝑙₀)>𝑀₁, then 𝑙₀≥𝑘₀+2. Set 𝑙₀=min𝑙₀∶𝑙₀≥𝑘₀+2,𝑦₁𝑙₀>𝑀₁; (A.3) namely, 𝑦₁(𝑙₀)>𝑀₁ and 𝑙₀≥𝑘₀+2; then 𝑦₁(𝑙₀)>𝑀₁≥𝑦₁(𝑙₀−1). From the above discussion one obtains that 𝑦₁(𝑙₀)≤𝑀₁, which is a contradiction. Hence, 𝑦₁(𝑘)≤𝑀₁ for all 𝑘≥𝑘₀; then limsup_𝑘→+∞𝑦₁(𝑘)≤𝑀₁. Case I is complete.

Case II. Assume that 𝑦₁(𝑘)>𝑦₁(𝑘+1) for 𝑘∈ℤ⁺. lim_𝑘→+∞𝑦₁(𝑘) especially exists, denoted by 𝑦₁. One claims that 𝑦₁≤𝑎^𝑈₁/𝑏^𝐿₁. Otherwise, if 𝑦₁>𝑎^𝑈₁/𝑏^𝐿₁, then it follows from the first equation of system (2) that lim_𝑘→+∞𝑎₁(𝑘)−𝑏₁(𝑘)𝑦₁(𝑘)−𝑐₁(𝑘)𝑦₂(𝑘)−𝑑₁(𝑘)𝑦²₁(𝑘)−𝑒₁(𝑘)𝑣₁(𝑘)=0, (A.4) which is a contradiction since lim_𝑘→+∞𝑎₁(𝑘)−𝑏₁(𝑘)𝑦₁(𝑘)−𝑐₁(𝑘)𝑦₂(𝑘)−𝑑₁(𝑘)𝑦²₁(𝑘)−𝑒₁(𝑘)𝑣₁(𝑘)≤lim_𝑘→+∞𝑎₁(𝑘)−𝑏₁(𝑘)𝑦₁(𝑘)≤𝑎^𝑈₁−𝑏^𝐿₁𝑦₁<0. (A.5) This proves the claim. By 𝑦₁≤𝑎^𝑈₁/𝑏^𝐿₁ and the fact that min_𝑥∈ℝ⁺[exp(𝑥−1)/𝑥]=1, one can show that 𝑦₁≤𝑎^𝑈₁𝑏^𝐿₁≤𝑎^𝑈₁𝑏^𝐿₁exp𝑎^𝑈₁−1𝑎^𝑈₁=𝑀₁. (A.6) Hence limsup_𝑘→+∞𝑦₁(𝑘)≤𝑀₁. We can prove that limsup_𝑘→+∞𝑦₂(𝑘)≤𝑀₂ in the same way.

In the following, we prove that limsup_𝑘→+∞𝑣₁(𝑘)≤𝑁₁. For any 𝜀>0, there exists an integer 𝑚₀∈ℤ⁺ such that 𝑦₁(𝑘)≤𝑀₁+𝜀 for all 𝑘≥𝑚₀. We have from the third equation of system (2) that 𝑣₁(𝑘)=^𝑘−1_𝑖=01−𝑓₁(𝑖)𝑣₁(0)+^𝑘−1_𝑖=0ℎ₁(𝑖)+𝑔₁(𝑖)𝑦₁(𝑖)∏^𝑖_𝑗=01−𝑓₁𝑗≤1−𝑓^𝐿₁^𝑘𝑣₁(0)+𝜇₁+ℎ^𝑈₁+𝑔^𝑈₁𝑀₁+𝜀^𝑘−1_𝑖=𝑚0^𝑘−1_𝑗=𝑖+11−𝑓₁𝑗≤1−𝑓^𝐿₁^𝑘𝑣₁(0)+𝜇₁+ℎ^𝑈₁+𝑔^𝑈₁𝑀₁+𝜀^𝑘−1_𝑖=𝑚01−𝑓^𝐿₁^{𝑘−𝑖−1}, (A.7) where 𝜇₁=∑^𝑚0−1_𝑖=0((ℎ₁(𝑖)+𝑔₁(𝑖)𝑦₁(𝑖))/(∏^𝑖_𝑗=0(1−𝑓₁(𝑗)))). Since 0<𝑓^𝐿₁<1, we can choose a constant 𝜃>0 such that 1−𝑓^𝐿₁=𝑒^−𝜃. By using the Stolz theorem, one has, as 𝑘→+∞, ^𝑘−1_𝑖=𝑚01−𝑓^𝐿₁^{𝑘−𝑖−1}=∑^𝑘−1_𝑖=𝑚0𝑒^𝜃(𝑖+1)𝑒^𝜃𝑘⟶11−𝑒^−𝜃=1𝑓^𝐿₁. (A.8) So limsup_𝑘→+∞𝑣₁(𝑘)≤(ℎ^𝑈₁+𝑔^𝑈₁(𝑀₁+𝜀))/𝑓^𝐿₁. Since 𝜀 is arbitrary, we obtain that limsup_𝑘→+∞𝑣₁(𝑘)≤𝑁₁.

Repeating a similar argument, we can prove that limsup_𝑘→+∞𝑣₂(𝑘)≤𝑁₂. The proof of Lemma 1 is complete.

B. Proof of Lemma 2

In view of Lemma 1, we obtain that there exist 𝜀>0 and 𝑘^∗∈ℤ⁺ such that, for 𝑘≥𝑘^∗, 𝑦_𝑖(𝑘)≤𝑀_𝑖+𝜀,𝑣_𝑖(𝑘)≤𝑁_𝑖+𝜀,𝑖=1,2. (B.1) The first inequality of (4) implies that 𝑎^𝐿₁−𝑐^𝑈₁𝑀₂+𝜀−𝑑^𝑈₁𝑀₁+𝜀²−𝑒^𝑈₁𝑁₁+𝜀>0. (B.2) Similar to Lemma 1, we consider the following Cases I and II to show that liminf_𝑘→+∞𝑦₁(𝑘)≥𝑚₁. (B.3)Case I. Suppose that there is a nonnegative integer 𝑘₀≥𝑘^∗ such that 𝑦₁(𝑘₀+1)≤𝑦₁(𝑘₀); we have from the first equation of system (2) that 𝑦₁𝑘₀+1=𝑦₁𝑘₀exp𝑎₁𝑘₀−𝑏₁𝑘₀𝑦₁𝑘₀−𝑐₁𝑘₀𝑦₂𝑘₀−𝑑₁𝑘₀𝑦²₁𝑘₀−𝑒₁𝑘₀𝑣₁𝑘₀≥𝑦₁𝑘₀exp𝑎^𝐿₁−𝑏^𝑈₁𝑦₁𝑘₀−𝑐^𝑈₁𝑀₂+𝜀−𝑑^𝑈₁𝑀₁+𝜀²−𝑒^𝑈₁𝑁₁+𝜀, (B.4) and hence 𝑎^𝐿₁−𝑏^𝑈₁𝑦₁𝑘₀−𝑐^𝑈₁𝑀₂+𝜀−𝑑^𝑈₁𝑀₁+𝜀²−𝑒^𝑈₁𝑁₁+𝜀≤0. (B.5) Note that (B.2) and 𝜀 is sufficiently small; one has from (B.5) that 𝑦₁𝑘₀≥𝑎^𝐿₁−𝑐^𝑈₁𝑀₂+𝜀−𝑑^𝑈₁𝑀₁+𝜀²−𝑒^𝑈₁𝑁₁+𝜀𝑏^𝑈₁>0. (B.6) Furthermore, it follows from (B.4) and (B.6) that 𝑦₁𝑘₀+1=𝑦₁𝑘₀exp𝑎₁𝑘₀−𝑏₁𝑘₀𝑦₁𝑘₀−𝑐₁𝑘₀𝑦₂𝑘₀−𝑑₁𝑘₀𝑦²₁𝑘₀−𝑒₁𝑘₀𝑣₁𝑘₀≥𝑎^𝐿₁−𝑐^𝑈₁𝑀₂+𝜀−𝑑^𝑈₁𝑀₁+𝜀²−𝑒^𝑈₁𝑁₁+𝜀𝑏^𝑈₁×exp𝑎^𝐿₁−𝑏^𝑈₁𝑀₁+𝜀−𝑐^𝑈₁𝑀₂+𝜀−𝑑^𝑈₁𝑀₁+𝜀²−𝑒^𝑈₁𝑁₁+𝜀>0. (B.7) Hence 𝑦₁(𝑘₀)≥𝑦₁(𝑘₀+1)≥𝑦^𝜀₁, where 𝑦^𝜀₁=𝑎^𝐿₁−𝑐^𝑈₁𝑀₂+𝜀−𝑑^𝑈₁𝑀₁+𝜀²−𝑒^𝑈₁𝑁₁+𝜀𝑏^𝑈₁×exp𝑎^𝐿₁−𝑏^𝑈₁𝑀₁+𝜀−𝑐^𝑈₁𝑀₂+𝜀−𝑑^𝑈₁𝑀₁+𝜀²−𝑒^𝑈₁𝑁₁+𝜀. (B.8)

Now, we claim that 𝑦₁(𝑘)≥𝑦^𝜀₁ for all 𝑘≥𝑘₀. Otherwise, if there exists a 𝑞₀>𝑘₀ such that 𝑦₁(𝑞₀)<𝑦^𝜀₁, then 𝑞₀≥𝑘₀+2. Set 𝑞₀=min𝑞₀∶𝑞₀≥𝑘₀+2,𝑦₁𝑞₀<𝑦^𝜀₁; (B.9) then 𝑦₁(𝑞₀−1)>𝑦₁(𝑞₀). The above argument produces that 𝑦₁(𝑞₀)≥𝑦^𝜀₁, which is a contradiction. Thus 𝑦₁(𝑘)≥𝑦^𝜀₁ for all 𝑘≥𝑘₀. Since 𝜀 is sufficiently small, this reduces to 𝑦₁(𝑘)≥𝑎^𝐿₁−𝑐^𝑈₁𝑀₂−𝑑^𝑈₁𝑀²₁−𝑒^𝑈₁𝑁₁𝑏^𝑈₁×exp𝑎^𝐿₁−𝑏^𝑈₁𝑀₁−𝑐^𝑈₁𝑀₂−𝑑^𝑈₁𝑀²₁−𝑒^𝑈₁𝑁₁=𝑚₁. (B.10) Hence, liminf_𝑘→+∞𝑦₁(𝑘)≥𝑚₁. This verifies Case I.

Case II. We assume that 𝑦₁(𝑘+1)≥𝑦₁(𝑘) for all large 𝑘≥𝑘^∗. In this case, lim_𝑘→+∞𝑦₁(𝑘) exists, denoted by 𝑦_1∗. One claims that 𝑦_1∗≥(𝑎^𝐿₁−𝑐^𝑈₁(𝑀₂+𝜀)−𝑑^𝑈₁(𝑀₁+𝜀)²−𝑒^𝑈₁(𝑁₁+𝜀))/𝑏^𝑈₁; otherwise, assume that 𝑦_1∗<(𝑎^𝐿₁−𝑐^𝑈₁(𝑀₂+𝜀)−𝑑^𝑈₁(𝑀₁+𝜀)²−𝑒^𝑈₁(𝑁₁+𝜀))/𝑏^𝑈₁. It follows from the first equation of system (2) that lim_𝑘→+∞𝑎₁(𝑘)−𝑏₁(𝑘)𝑦₁(𝑘)−𝑐₁(𝑘)𝑦₂(𝑘)−𝑑₁(𝑘)𝑦²₁(𝑘)−𝑒₁(𝑘)𝑣₁(𝑘)=0; (B.11) however, lim_𝑘→+∞𝑎₁(𝑘)−𝑏₁(𝑘)𝑦₁(𝑘)−𝑐₁(𝑘)𝑦₂(𝑘)−𝑑₁(𝑘)𝑦²₁(𝑘)−𝑒₁(𝑘)𝑣₁(𝑘)≥lim_𝑘→+∞𝑎₁(𝑘)−𝑏₁(𝑘)𝑦₁(𝑘)−𝑐₁(𝑘)𝑀₂+𝜀−𝑑₁(𝑘)𝑀₁+𝜀²−𝑒₁(𝑘)𝑁₁+𝜀≥𝑎^𝐿₁−𝑏^𝑈₁𝑦_1∗−𝑐^𝑈₁𝑀₂+𝜀−𝑑^𝑈₁𝑀₁+𝜀²−𝑒^𝑈₁𝑁₁+𝜀>0, (B.12) which is a contradiction. This fact implies that liminf_𝑘→+∞𝑦₁(𝑘)≥𝑦^𝜀₁. Since 𝜀 can be sufficiently small, one yields liminf_𝑘→+∞𝑦₁(𝑘)≥𝑚₁, where 𝑚₁ is defined as (B.10).

Similarly, one can derive that liminf_𝑘→+∞𝑦₂(𝑘)≥𝑚₂.

In the following, we prove that liminf_𝑘→+∞𝑣₁(𝑘)≥𝑛₁. For any 𝜀>0, there exists an integer 𝑝₀∈ℤ⁺ such that 𝑦₁(𝑘)≥𝑚₁−𝜀 for 𝑘≥𝑝₀. We have from the third equation of system (2) that 𝑣₁(𝑘)=^𝑘−1_𝑖=01−𝑓₁(𝑖)𝑣₁(0)+^𝑘−1_𝑖=0ℎ₁(𝑖)+𝑔₁(𝑖)𝑦₁(𝑖)∏^𝑖_𝑗=01−𝑓₁𝑗≥1−𝑓^𝑈₁^𝑘𝑣₁(0)+𝜇₂+ℎ^𝐿₁+𝑔^𝐿₁𝑚₁−𝜀^𝑘−1_𝑖=𝑝0^𝑘−1_𝑗=𝑖+11−𝑓₁𝑗≥1−𝑓^𝑈₁^𝑘𝑣₁(0)+𝜇₂+ℎ^𝐿₁+𝑔^𝐿₁𝑚₁−𝜀^𝑘−1_𝑖=𝑝01−𝑓^𝑈₁^{𝑘−𝑖−1}, (B.13) where 𝜇₂=∑^𝑝0−1_𝑖=0((ℎ₁(𝑖)+𝑔₁(𝑖)𝑦₁(𝑖))/(∏^𝑖_𝑗=0(1−𝑓₁(𝑗)))). Since 0<𝑓^𝑈₁<1, we can choose a constant 𝜂>0 such that 1−𝑓^𝑈₁=𝑒^−𝜂; then by Stolz’s theorem one yields ^𝑘−1_𝑖=𝑝01−𝑓^𝑈₁^{𝑘−𝑖−1}=∑^𝑘−1_𝑖=𝑝0𝑒^𝜂(𝑖+1)𝑒^𝜂𝑘⟶11−𝑒^−𝜂=1𝑓^𝑈₁as𝑘⟶+∞. (B.14) Then liminf_𝑘→+∞𝑣₁(𝑘)≥(ℎ^𝐿₁+𝑔^𝐿₁(𝑚₁−𝜀))/𝑓^𝑈₁. Since 𝜀 can be sufficiently small, we have liminf_𝑘→+∞𝑣₁(𝑘)≥𝑛₁=ℎ^𝐿₁+𝑔^𝐿₁𝑚₁𝑓^𝑈₁. (B.15) Analogously, liminf_𝑘→+∞𝑣₂(𝑘)≥𝑛₂=ℎ^𝐿₂+𝑔^𝐿₂𝑚₂𝑓^𝑈₂. (B.16) The proof of Lemma 2 is complete.

C. Proof of Theorem 3

It follows from Lemmas 1 and 2 that Θ=𝑚₁,𝑀₁×𝑚₂,𝑀₂×𝑛₁,𝑁₁×𝑛₂,𝑁₂ (C.1) is an invariant set of system (2). Define the continuous mapping Π on Θ by Π𝑦^∗₁(0),𝑦^∗₂(0),𝑣^∗₁(0),𝑣^∗₂(0𝑦)=Π^∗₁(𝑇),𝑦^∗₂(𝑇),𝑣^∗₁(𝑇),𝑣^∗₂(𝑇), (C.2) for {𝑦^∗₁(0),𝑦^∗₂(0),𝑣^∗₁(0),𝑣^∗₂(0)}∈Θ. Clearly, Π depends continuously on {𝑦^∗₁(0),𝑦^∗₂(0),𝑣^∗₁(0),𝑣^∗₂(0)}; then Π is continuous and maps the compact set Θ into itself. Hence, Π has a fixed point {𝑦₁,𝑦₂,𝑣̃₁,𝑣̃₂}. It is easy to see that the solution {𝑦^∗₁(𝑘),𝑦^∗₂(𝑘),𝑣^∗₁(𝑘),𝑣^∗₂(𝑘)} which passes through {𝑦₁,𝑦₂,𝑣̃₁,𝑣̃₂} is a 𝑇-periodic solution of system (2). The proof of Theorem 3 is complete.

D. Proof of Theorem 4

Let {𝑦^∗₁(𝑘),𝑦^∗₂(𝑘),𝑣^∗₁(𝑘),𝑣^∗₂(𝑘)} be positive periodic solution and let {𝑦₁(𝑘),𝑦₂(𝑘),𝑣₁(𝑘),𝑣₂(𝑘)} be any positive solution of system (2), respectively. Let us make the change of variables as follows: 𝑥₁(𝑘)=ln𝑦₁(𝑘)−ln𝑦^∗₁(𝑘),𝑥₂(𝑘)=ln𝑦₂(𝑘)−ln𝑦^∗₂(𝑘),𝑢₁(𝑘)=𝑣₁(𝑘)−𝑣^∗₁(𝑘),𝑢₂(𝑘)=𝑣₂(𝑘)−𝑣^∗₂(𝑘). (D.1) By the mean-value theorem, we have from the first equation of system (2) that 𝑥₁(𝑘+1)=ln𝑦₁(𝑘+1)−ln𝑦^∗₁(𝑘+1)=ln𝑦₁(𝑘)−ln𝑦^∗₁(𝑘)−𝑏₁(𝑘)𝑦₁(𝑘)−𝑦^∗₁(𝑘)−𝑐₁(𝑘)𝑦₂(𝑘)−𝑦^∗₂(𝑘)−𝑑₁(𝑘)𝑦²₁(𝑘)−𝑦₁^∗2(𝑘)−𝑒₁(𝑘)𝑣₁(𝑘)−𝑣^∗₁(𝑘)=ln𝑦₁(𝑘)−ln𝑦^∗₁(𝑘)−𝑏₁(𝑘)exp𝜃₁(𝑘)×ln𝑦₁(𝑘)−ln𝑦^∗₁(𝑘)−𝑐₁(𝑘)exp𝜃₂(𝑘)⋅ln𝑦₂(𝑘)−ln𝑦^∗₂(𝑘)−2𝑑₁(𝑘)exp2𝜃₃(𝑘)×ln𝑦₁(𝑘)−ln𝑦^∗₁(𝑘)−𝑒₁(𝑘)𝑣₁(𝑘)−𝑣^∗₁(𝑘)=1−𝑏₁(𝑘)exp𝜃₁(𝑘)−2𝑑₁(𝑘)exp2𝜃₃(𝑘)𝑥₁(𝑘)−𝑐₁(𝑘)exp𝜃₂(𝑘)𝑥₂(𝑘)−𝑒₁(𝑘)𝑢₁(𝑘). (D.2) Similarly, we can arrive at the following desired results: 𝑥₂(𝑘+1)=1−𝑏₂(𝑘)exp𝜃₄(𝑘)−2𝑑₂(𝑘)exp2𝜃₆(𝑘)𝑥₂(𝑘)−𝑐₂(𝑘)exp𝜃₅(𝑘)𝑥₁(𝑘)−𝑒₂(𝑘)𝑢₂(𝑘),𝑢₁(𝑘+1)=1−𝑓₁(𝑘)𝑢₁(𝑘)+𝑔₁(𝑘)exp𝜃₇(𝑘)𝑥₁(𝑘),𝑢₂(𝑘+1)=1−𝑓₂(𝑘)𝑢₂(𝑘)+𝑔₂(𝑘)exp𝜃₈(𝑘)𝑥₂(𝑘), (D.3) where 𝜃₁(𝑘),𝜃₃(𝑘),𝜃₅(𝑘), and 𝜃₇(𝑘) lie between ln𝑦^∗₁(𝑘) and ln𝑦₁(𝑘) and 𝜃₂(𝑘), 𝜃₄(𝑘), 𝜃₆(𝑘), and 𝜃₈(𝑘) lie between ln𝑦^∗₂(𝑘) and ln𝑦₂(𝑘).

By the inequalities in (5), we will now select a constant 𝜀 small enough such that 𝜌^𝜀₁=max1−𝑏^𝐿₁𝑚₁−𝜀−2𝑑^𝐿₁𝑚₁−𝜀²,1−𝑏^𝑈₁𝑀₁+𝜀−2𝑑^𝑈₁𝑀₁+𝜀²+𝑐^𝑈₁𝑀₂+𝜀+𝑒^𝑈₁𝜌<1,_𝜀2=max1−𝑏^𝐿₂𝑚₂−𝜀−2𝑑^𝐿₂𝑚₂−𝜀²,1−𝑏^𝑈₂𝑀₂+𝜀−2𝑑^𝑈₂𝑀₂+𝜀²+𝑐^𝑈₂𝑀₁+𝜀+𝑒^𝑈₂𝜌<1,_𝜀3=1−𝑓^𝐿₁+𝑔^𝑈₁𝑀₁𝜌+𝜀<1,_𝜀4=1−𝑓^𝐿₂+𝑔^𝑈₂𝑀₂+𝜀<1. (D.4) From Lemmas 1 and 2, we know that there exists a 𝑘∈ℤ⁺ such that, for 𝑘≥𝑘, 0<𝑚₁−𝜀≤𝑦^∗₁(𝑘)≤𝑀₁+𝜀,0<𝑚₁−𝜀≤𝑦₁(𝑘)≤𝑀₁+𝜀,0<𝑚₂−𝜀≤𝑦^∗₂(𝑘)≤𝑀₂+𝜀,0<𝑚₂−𝜀≤𝑦₂(𝑘)≤𝑀₂+𝜀,0<𝑛₁−𝜀≤𝑣^∗₁(𝑘)≤𝑁₁+𝜀,0<𝑛₁−𝜀≤𝑣₁(𝑘)≤𝑁₁+𝜀,0<𝑛₂−𝜀≤𝑣^∗₂(𝑘)≤𝑁₂+𝜀,0<𝑛₂−𝜀≤𝑣₂(𝑘)≤𝑁₂+𝜀. (D.5) It is easy to see that exp(𝜃₁(𝑘)),exp(𝜃₃(𝑘)),exp(𝜃₅(𝑘)), and exp(𝜃₇(𝑘)) are between 𝑦^∗₁(𝑘) and 𝑦₁(𝑘). Meanwhile, exp(𝜃₂(𝑘)),exp(𝜃₄(𝑘)),exp(𝜃₆(𝑘)), and exp(𝜃₈(𝑘)) are between 𝑦^∗₂(𝑘) and 𝑦₂(𝑘). It follows from the equation of (D.2) that 𝑥₁(𝑘+1)=1−𝑏₁(𝑘)exp𝜃₁(𝑘)−2𝑑₁(𝑘)exp2𝜃₃(𝑘)⋅𝑥₁(𝑘)+𝑐₁(𝑘)exp𝜃₂(𝑘)⋅𝑥₂(𝑘+𝑒₁)(𝑘)⋅𝑢₁(𝑘)≤max1−𝑏^𝐿₁𝑚₁−𝜀−2𝑑^𝐿₁𝑚₁−𝜀²,1−𝑏^𝑈₁𝑀₁+𝜀−2𝑑^𝑈₁𝑀₁+𝜀²×𝑥₁(𝑘)+𝑐^𝑈₁𝑀₂+𝜀𝑥₂(𝑘)+𝑒^𝑈₁𝑢₁(𝑘)≤𝜌^𝜀₁max𝑥₁(𝑘),𝑥₂(𝑘),𝑢₁(𝑘),𝑢₂(𝑘). (D.6) Analogously, 𝑥₂(𝑘+1)≤𝜌^𝜀₂max𝑥₁(𝑘),𝑥₂(𝑘),𝑢₁(𝑘),𝑢₂(𝑘,𝑢₁)(𝑘+1)≤𝜌^𝜀₃max𝑥₁(𝑘),𝑥₂(𝑘),𝑢₁(𝑘),𝑢₂(𝑘,𝑢₂)(𝑘+1)≤𝜌^𝜀₄max𝑥₁(𝑘),𝑥₂(𝑘),𝑢₁(𝑘),𝑢₂(𝑘). (D.7) Let 𝜌^𝜀=max{𝜌^𝜀₁,𝜌^𝜀₂,𝜌^𝜀₃,𝜌^𝜀₄}; then 𝜌^𝜀<1. Hence, when 𝑘≥𝑘 we have max𝑥₁(𝑘+1),𝑥₂(𝑘+1),𝑢₁(𝑘+1),𝑢₂(𝑘+1)≤𝜌^𝜀max𝑥₁(𝑘),𝑥₂(𝑘),𝑢₁(𝑘),𝑢₂(𝑘≤𝜌_{𝜀𝑘𝑘−})max𝑥₁𝑘,𝑥₂𝑘,𝑢₁𝑘,𝑢₂𝑘, (D.8) which implies lim_𝑘→+∞𝑦₁(𝑘)−𝑦^∗₁(𝑘)=0,lim_𝑘→+∞𝑦₂(𝑘)−𝑦^∗₂(𝑘)=0,lim_𝑘→+∞𝑣₁(𝑘)−𝑣^∗₁(𝑘)=0,lim_𝑘→+∞𝑣₂(𝑘)−𝑣^∗₂(𝑘)=0. (D.9) So the positive periodic solution {𝑦^∗₁(𝑘),𝑦^∗₂(𝑘),𝑣^∗₁(𝑘),𝑣^∗₂(𝑘)} of system (2) is globally asymptotically stable. The proof of Theorem 4 is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions that greatly improved the presentation of this work. The work is supported by the National Natural Science Foundation of China (no. 11261017), the Key Project of Chinese Ministry of Education (no. 212111), the Key Laboratory of Biological Resources Protection and Utilization of Hubei Province (PKLHB1329), and the Key Subject of Hubei Province (Forestry).