Exploration on dynamics in a discrete predator–prey competitive model involving feedback controls

Abstract

In this work, we set up a new discrete predator–prey competitive model with time-varying delays and feedback controls. By virtue of the difference inequality knowledge, a sufficient condition which guarantees the permanence of the established discrete predator–prey competitive model with time-varying delays and feedback controls is derived. Under some appropriate parameter conditions, we have proved that the periodic solution of the system without delay exists and globally attractive. To verify the correctness of the derived theoretical fruits, we give two examples and execute computer simulations. Our obtained results are novel and complement previous known results.

Keywords:

• Competitive model
• permanence
• feedback control
• delay
• global attractivity

Mathematics Subject Classification (2000):

• 34K20
• 34C25
• 92D25

1. Introduction

In recent few decades, population dynamical models have received considerable attention [13, 19, 31, 43, 57]. In the research of population dynamical models, competitive systems play an important role in describing the interaction among the multi-species. A lot of competitive systems have been explored by numerous scholars. For example, Hou [16] analysed the permanence of a competitive Lotka–Volterra system with delays, Balbus [2] addressed the attractivity and stability in a competitive system of PDEs of Kolmogorov type, Shi et al. [39] focused on the extinction of a nonautonomous Lotka–Volterra competitive system with infinite delay and feedback controls, Liu and Wang [32] discussed the asymptotic behaviour of a stochastic nonautonomous Lotka–Volterra competitive system with impulsive perturbations, Kulenovi$\acute{c}$ and Nurkanovi$\acute{c}$ [20] made a theoretical discussion on the global behaviour of a two-dimensional competitive system of difference equations with stocking. For more detailed publications on the permanence and global attractivity behaviour of predator–prey models, one can see [4, 6, 7, 9, 11, 12, 15, 17, 18, 21, 23–25, 28–30, 33–36, 40, 41, 44, 45, 47–52, 55, 56].

In 1992, Gopalsamy [14] explored the following competitive model on predator species and prey species: (1) $\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{w}_1(t)}{\mathrm{d}t}=w_1(t)[{\alpha }_1-{\beta }_1{w}_1(t)-{\gamma }_1{w}_2(t)-{\delta }_1{w}_1^2(t)],}\\ {\displaystyle \frac{\mathrm{d}{w}_2(t)}{\mathrm{d}t}=w_2(t)[{\alpha }_2-{\beta }_2{w}_2(t)-{\gamma }_2{w}_1(t)-{\delta }_2{w}_2^2(t)],}\end{array}$(1) where $w_1(t),w_2(t)$ stand for the densities of two competing species at time t, respectively. ${\alpha }_1,{\alpha }_2$ denote the intrinsic growth rates of predator species and prey species, ${\beta }_1,{\delta }_1,{\beta }_2,{\delta }_2$ represent the effects of intra-specific competition, and ${\gamma }_1,{\gamma }_2$ are the effects of inter-specific competition. Considering that the parameters, in natural world, often vary due to the change of surroundings, we think that it is important to introduce the time-varying parameters into the predator–prey model. Varying parameters will lead to much more different dynamical behaviour in predator–prey models than that of predator–prey models with fixed parameters. Motivated by this idea, we can modify system (1(1) $\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{w}_1(t)}{\mathrm{d}t}=w_1(t)[{\alpha }_1-{\beta }_1{w}_1(t)-{\gamma }_1{w}_2(t)-{\delta }_1{w}_1^2(t)],}\\ {\displaystyle \frac{\mathrm{d}{w}_2(t)}{\mathrm{d}t}=w_2(t)[{\alpha }_2-{\beta }_2{w}_2(t)-{\gamma }_2{w}_1(t)-{\delta }_2{w}_2^2(t)],}\end{array}$(1) ) as the form (2) $\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{w}_1(t)}{\mathrm{d}t}=w_1(t)[{\alpha }_1(t)-{\beta }_1(t)w_1(t)-{\gamma }_1(t)w_2(t)-{\delta }_1(t)w_1^2(t)],}\\ {\displaystyle \frac{\mathrm{d}{w}_2(t)}{\mathrm{d}t}=w_2(t)[{\alpha }_2(t)-{\beta }_2(t)w_2(t)-{\gamma }_2(t)w_1(t)-{\delta }_2(t)w_2^2(t)],}\end{array}$(2) where ${\alpha }_i(t),{\beta }_i(t),{\gamma }_i(t),{\delta }_i(t)(i=1,2)$ are functions with respect to the time t. Since predator species and prey species live in a real fluctuating environment and many exploitation activities of people might lead to abrupt vary, Tan et al. [42] set up the following impulsive competitive model: (3) $\{\begin{array}{rl}& \begin{array}{rl}{\displaystyle \frac{\mathrm{d}{w}_1(t)}{\mathrm{d}t}}& =w_1(t)[{\alpha }_1(t)-{\beta }_1(t)w_1(t)-{\gamma }_1(t)w_2(t)-{\delta }_1(t)w_1^2(t)],\\ {\displaystyle \frac{\mathrm{d}{w}_2(t)}{\mathrm{d}t}}& =w_2(t)[{\alpha }_2(t)-{\beta }_2(t)w_2(t)-{\gamma }_2(t)w_1(t)-{\delta }_2(t)w_2^2(t)],\end{array}\}t\ne t_k,\\ & \begin{array}{rl}{\displaystyle w_1(t_k^{+})}& =(1+{\vartheta }_{1k})w_1(t_k),\\ {\displaystyle w_2(t_k^{+})}& =(1+{\vartheta }_{2k})w_2(t_k),\end{array}\}t=t_k,k\in N,\end{array}$(3) where $w_1(0^{+})=w_1(0)>0,w_2(0^{+})=w_2(0)>0$ and N is the set of positive integers, all the coefficients ${\alpha }_i(t),{\beta }_i(t),{\gamma }_i(t),{\delta }_i(t)(i=1,2)$ are all continuous almost periodic functions which are bounded above and below by positive constants, ${\vartheta }_{1k}>-1$ and ${\vartheta }_{2k}>-1$ are constants and $0<t_1<t_2<\cdots <t_k<t_{k+1}<\cdots$ are impulse points with $\underset{k\to +\mathrm{\infty }}{lim}{t}_k=+\mathrm{\infty }$.

Considering that two species are constantly in the competition, and when a species suffers damage from another one by competition, another one could benefit, the duration time of density for species would also play an important role, we modify system (1(1) $\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{w}_1(t)}{\mathrm{d}t}=w_1(t)[{\alpha }_1-{\beta }_1{w}_1(t)-{\gamma }_1{w}_2(t)-{\delta }_1{w}_1^2(t)],}\\ {\displaystyle \frac{\mathrm{d}{w}_2(t)}{\mathrm{d}t}=w_2(t)[{\alpha }_2-{\beta }_2{w}_2(t)-{\gamma }_2{w}_1(t)-{\delta }_2{w}_2^2(t)],}\end{array}$(1) ) as the following form (4) $\{\begin{array}{rl}& {\displaystyle \frac{\mathrm{d}{w}_1(t)}{\mathrm{d}t}}\\ & =w_1(t)[{\alpha }_1(t)-{\beta }_1(t)w_1(t-\rho (t))-{\gamma }_1(t)w_2(t-\rho (t))-{\delta }_1(t)w_1^2(t-\rho (t))],\\ & {\displaystyle \frac{\mathrm{d}{w}_2(t)}{\mathrm{d}t}}\\ & =w_2(t)[{\alpha }_2(t)-{\beta }_2(t)w_2(t-\rho (t))-{\gamma }_2(t)w_1(t-\rho (t))-{\delta }_2(t)w_2^2(t-\rho (t))],\end{array}$(4) where $\rho (t)\ge 0$ stands for the hunting delay. Many scholars [1, 3, 5, 8, 10, 22, 26, 27, 37, 38, 53, 54] argue that discrete dynamical models which are described by difference equations are often regarded as more suitable tools to depict the dynamical relationship among different species than continuous ones since the species owns non-overlapping generations. Discrete dynamical model is also thought to be a very useful tool to carry out numerical simulations for the continuous ones. In addition, we would like to point out that external force often make the parameters of biological systems vary. For example, the competition and cooperation model of two enterprises is usually interfered by manpower, material resources, financial resources and so on. What we are interested in is how to real the dynamical behaviour of biological systems which are affected by external force. In control language, the external force can be called as control variables. From a mathematical point of view, it is of great interest to investigate the effect of control variables on the dynamics for biological systems. Stimulated by the analysis above, we can modify system (4(4) $\{\begin{array}{rl}& {\displaystyle \frac{\mathrm{d}{w}_1(t)}{\mathrm{d}t}}\\ & =w_1(t)[{\alpha }_1(t)-{\beta }_1(t)w_1(t-\rho (t))-{\gamma }_1(t)w_2(t-\rho (t))-{\delta }_1(t)w_1^2(t-\rho (t))],\\ & {\displaystyle \frac{\mathrm{d}{w}_2(t)}{\mathrm{d}t}}\\ & =w_2(t)[{\alpha }_2(t)-{\beta }_2(t)w_2(t-\rho (t))-{\gamma }_2(t)w_1(t-\rho (t))-{\delta }_2(t)w_2^2(t-\rho (t))],\end{array}$(4) ) as follows (5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) where $w_1(n)$ and $w_2(n)$ denote the densities of two competing species at the generation, respectively, and ${\mu }_i(n)(i=1,2)$ is the control variable. ${\alpha }_i(n),{\beta }_i(n),{\gamma }_i(n),{\delta }_i(n),{\vartheta }_i(n),b_i(n),{\xi }_i(n)(i=1,2)$ are bounded nonnegative sequences and $\rho (n)$ are integer-valued sequences. As far as I know, it is first time to probe into system (5(5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) ) involving feedback control. We believe that this work on the permanence and global attractivity of the discrete competitive model has significant theoretical meaning and tremendous potential application in preserving population coexistence and maintaining ecological balance.

The chief task of this study is to probe into the permanence and global attractivity of system (5(5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) ). In order to set up the key conclusions, the following assumptions are needed: $(\text{H1})0<{\alpha }_i^l\le {\alpha }_i^u,0<{\beta }_i^l\le {\beta }_i^u,0<{\gamma }_i^l\le {\gamma }_i^u,0<{\delta }_i^l\le {\delta }_i^u,0<b_i^l\le b_i^u(i=1,2).$Denote $h^u=\underset{n\in N}{sup}\{h(n)\}$ and $h^l=\underset{n\in N}{inf}\{h(n)\}$ where $\{h(n)\}$ is bounded sequence. Let ${\rho }^m=\underset{n\in Z}{sup}\{\rho (n)\},{\rho }^l=\underset{n\in Z}{inf}\{\rho (n)\}$. For system (5(5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) ), we give the initial value as follows: (6) $w_i(\sigma )={\varphi }_i(\sigma )\ge 0,\sigma \in N[-\rho ,0]=\{-\rho ,-\rho +1,\dots ,0\},{\varphi }_i(0)>0.$(6) It is not difficult to see that solutions of (5(5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) ) and (6(6) $w_i(\sigma )={\varphi }_i(\sigma )\ge 0,\sigma \in N[-\rho ,0]=\{-\rho ,-\rho +1,\dots ,0\},{\varphi }_i(0)>0.$(6) ) are well defined for all $n\ge 0$ and satisfy $w_i(n)>0,\text{for}n\in Z,i=1,2.$We plan the structure of this work as follows. Section 2 presents the necessary definitions and lemmas and states the permanence result of system (5(5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) ) Section 3 deals with the existence of periodic solution and global attractivity of periodic solution for system (5(5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) ) without delay. Section 4 carries out computer simulation to illustrate the feasibility and effectiveness of our results derived is Sections 2 and 3. A brief conclusion is drawn in Section 5.

2. Permanence

In this section, we firstly introduce the definition of permanence and several lemmas which is needed to set up the key results.

Definition 2.1

We say that system (5(5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) ) is permanent provided that ∃  two positive constants $\mathcal{M}$ and m such that for each positive solution $(w_1(n),w_2(n),{\mu }_1(n),{\mu }_2(n))$ of system (5(5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) ) satisfies $\begin{array}{rl}m& \le \underset{n\to +\mathrm{\infty }}{lim}inf{w}_i(n)\le \underset{n\to +\mathrm{\infty }}{lim}sup{w}_i(n)\le \mathcal{M}(i=1,2),\\ m& \le \underset{n\to +\mathrm{\infty }}{lim}inf{\mu }_i(n)\le \underset{n\to +\mathrm{\infty }}{lim}sup{\mu }_i(n)\le \mathcal{M}(i=1,2).\end{array}$Give the single species discrete model as follows: (7) $N(n+1)=N(n)\exp (\alpha (n)-\beta (n)N(n)),$(7) where $\{\alpha (n)\}$ and $\{\beta (n)\}$ are strictly positive sequences of real numbers defined for $n\in N=\{0,1,2,\dots \}$ and $0<{\alpha }^l\le {\alpha }^u,0<{\beta }^l\le {\beta }^u$. Similarly to the proofs of Propositions 1 and 3 in [3], one can obtain the following result.

Lemma 2.1

Every solution of system (7(7) $N(n+1)=N(n)\exp (\alpha (n)-\beta (n)N(n)),$(7) ) with initial value $N(0)>0$ satisfies $m\le \underset{n\to +\mathrm{\infty }}{lim}infN(n)\le \underset{n\to +\mathrm{\infty }}{lim}supN(n)\le \mathcal{M},$where $\mathcal{M}=\frac{1}{{\beta }^l}\exp ({\alpha }^u-1),m=\frac{{\alpha }^l}{{\beta }^u}\exp ({\alpha }^l-{\beta }^u\mathcal{M}).$

Give the first order difference equation (8) $v(n+1)=\mathcal{A}v(n)+\mathcal{B},n=1,2,\dots ,$(8) where $\mathcal{A}$ and $\mathcal{B}$ are positive constants. Following Theorem 6.2 of Wang and Wang [46, page 125], one has the following result.

Lemma 2.2

[46]

Assume that $|\mathcal{A}|<1$, for any initial value $v(0)$, there exists a unique solution $v(n)$ of (8(8) $v(n+1)=\mathcal{A}v(n)+\mathcal{B},n=1,2,\dots ,$(8) ) which can be expressed as follows: $v(n)={\mathcal{A}}^n(v(0)-v^{\ast })+v^{\ast },$where $v^{\ast }=\frac{\mathcal{B}}{1-\mathcal{A}}$. Thus, for any solution $\{v(n)\}$ of system (8(8) $v(n+1)=\mathcal{A}v(n)+\mathcal{B},n=1,2,\dots ,$(8) ), $\underset{n\to +\mathrm{\infty }}{lim}v(n)=v^{\ast }$.

Lemma 2.3

[46]

Let $n\in N_{n_0}^{+}=\{n_0,n_0+1,\dots ,n_0+l,\dots \},r\ge 0$. For any fixed n, $h(n,r)$ is a nondecreasing function with respect to r, and for $n\ge n_0$, the following inequalities hold: $v(n+1)\le h(n,v(n)),\mu (n+1)\ge h(n,\mu (n)).$If $v(n_0)\le \mu (n_0)$, then $v(n)\le \mu (n)$ for all $n\ge n_0$.

Proposition 2.1

For system (5(5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) ), assume that (H1) holds, then $\underset{n\to +\mathrm{\infty }}{lim}sup{w}_i(n)\le {\mathcal{M}}_i,\underset{n\to +\mathrm{\infty }}{lim}sup{\mu }_i(n)\le {\mathcal{U}}_i,i=1,2,$where ${\mathcal{M}}_i=\frac{1}{{\beta }_i^l}\exp \{{\alpha }_i^u(\rho +1)-1\},{\mathcal{U}}_i=\frac{{\xi }_i^u{\mathcal{M}}_i}{{\vartheta }_i^l}(i=1,2).$

Proof.

Let $(w_1(n),w_2(n),{\mu }_1(n),{\mu }_2(n))$ be any positive solution of system (5(5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) ) with the initial condition $(w_1(0),w_2(0),{\mu }_1(0),{\mu }_2(0))$. It follows from system (5(5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) ) that (9) $w_i(n+1)\le w_i(n)\exp \{{\alpha }_i(n)\}(i=1,2).$(9) Set $w_i(n)=\exp \{v_i(n)\}(i=1,2)$, then (9(9) $w_i(n+1)\le w_i(n)\exp \{{\alpha }_i(n)\}(i=1,2).$(9) ) is equivalent to (10) $v_i(n+1)-v_i(n)\le {\alpha }_i(n).$(10) By virtue of (10(10) $v_i(n+1)-v_i(n)\le {\alpha }_i(n).$(10) ), one has $\sum _{j=n-\rho (n)}^{n-1}(v_i(j+1)-v_i(j))\le \sum _{j=n-\rho (n)}^{n-1}{\alpha }_i(j)\le {\alpha }_i^u{\rho }^m,$which leads to (11) $v_i(n-\rho (n))\ge v_i(n)-{\alpha }_i^u{\rho }^m.$(11) Then (12) $w_i(n-\rho (n))\ge w_i(n)\exp \{-{\alpha }_i^u{\rho }^m\}.$(12) By means of (12(12) $w_i(n-\rho (n))\ge w_i(n)\exp \{-{\alpha }_i^u{\rho }^m\}.$(12) ) and system (5(5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) ), one gets (13) $w_i(n+1)\le w_i(n)\exp \{{\alpha }_i(n)-{\beta }_i(n)\exp \{-{\alpha }_i^u{\rho }^m\}{w}_i(n)\}.$(13) It follows from (13(13) $w_i(n+1)\le w_i(n)\exp \{{\alpha }_i(n)-{\beta }_i(n)\exp \{-{\alpha }_i^u{\rho }^m\}{w}_i(n)\}.$(13) ) and Lemma 2.1 that (14) $\underset{n\to +\mathrm{\infty }}{lim}sup{w}_i(n)\le \frac{1}{{\beta }_i^l}\exp \{{\alpha }_i^u({\rho }^m+1)-1\}:={\mathcal{M}}_i.$(14) ∀  $\epsilon >0$, it follows (14(14) $\underset{n\to +\mathrm{\infty }}{lim}sup{w}_i(n)\le \frac{1}{{\beta }_i^l}\exp \{{\alpha }_i^u({\rho }^m+1)-1\}:={\mathcal{M}}_i.$(14) ) that ∃  $N_1>0$ such that ∀  $n>N_1+\rho$ (15) $w_i(n)\le {\mathcal{M}}_i+\epsilon .$(15) In view of the third and fourth equations of the system (5(5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) ), one can easily get (16) $\mathrm{\Delta }{\mu }_i(n)\le -{\vartheta }_i(n){\mu }_i(n)+{\xi }_i(n)({\mathcal{M}}_i+\epsilon )(i=1,2).$(16) Then (17) ${\mu }_i(n+1)\le (1-{\vartheta }_i^l){\mu }_i(n)+{\xi }_i^u({\mathcal{M}}_i+\epsilon )(i=1,2).$(17) Applying Lemmas 2.2 and 2.3, we have (18) $\underset{n\to +\mathrm{\infty }}{lim}sup{\mu }_i(n)\le \frac{{\xi }_i^u({\mathcal{M}}_i+\epsilon )}{{\vartheta }_i^l}(i=1,2).$(18) Setting $\epsilon \to 0$, we get (19) $\underset{n\to +\mathrm{\infty }}{lim}sup{\mu }_i(n)\le \frac{{\xi }_i^u{M}_i}{{\vartheta }_1^l}:={\mathcal{U}}_i(i=1,2),$(19) which completes the proof.

Theorem 2.1

Let ${\mathcal{M}}_i$ and ${\mathcal{U}}_i$ be defined by (14(14) $\underset{n\to +\mathrm{\infty }}{lim}sup{w}_i(n)\le \frac{1}{{\beta }_i^l}\exp \{{\alpha }_i^u({\rho }^m+1)-1\}:={\mathcal{M}}_i.$(14) ) and (19(19) $\underset{n\to +\mathrm{\infty }}{lim}sup{\mu }_i(n)\le \frac{{\xi }_i^u{M}_i}{{\vartheta }_1^l}:={\mathcal{U}}_i(i=1,2),$(19) ), respectively. Assume that (H1) and (H2) ${\alpha }_1^l>{\gamma }_1^u{\mathcal{M}}_2+{\delta }_1^u{\mathcal{M}}_1^2+b_1^u{\mathcal{U}}_1,{\alpha }_2^l>{\gamma }_2^u{\mathcal{M}}_1+{\delta }_2^u{\mathcal{M}}_2^2+b_2^u{\mathcal{U}}_2$ hold, then system (5(5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) ) is permanent.

Proof.

By applying Proposition 2.1, we are easy to see that to end the proof of Theorem 2.1, it is enough to show that under the conditions of Theorem 2.1, $\begin{array}{rl}& \underset{n\to +\mathrm{\infty }}{lim}inf{w}_1(n)\ge m_1,\underset{n\to +\mathrm{\infty }}{lim}inf{w}_2(n)\ge m_2,\\ & \underset{n\to +\mathrm{\infty }}{lim}inf{\mu }_1(n)\ge {\nu }_1,\underset{n\to +\mathrm{\infty }}{lim}inf{\mu }_2(n)\ge {\nu }_2.\end{array}$In view of Proposition 2.1, ∀  $\epsilon >0$, ∃  $N_2>0,N_2\in N,$ for all $n>N_2$, (20) $w_i(n)\le {\mathcal{M}}_i+\epsilon ,{\mu }_i(n)\le {\mathcal{U}}_i+\epsilon ,i=1,2.$(20) It follows from system (5(5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) ) and (20(20) $w_i(n)\le {\mathcal{M}}_i+\epsilon ,{\mu }_i(n)\le {\mathcal{U}}_i+\epsilon ,i=1,2.$(20) ) that ∀  $n>N_2+\rho$, (21) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& \ge w_1(n)\exp \{{\alpha }_1^l-{\beta }_1^u({\mathcal{M}}_1+\epsilon )-{\gamma }_1^u({\mathcal{M}}_2+\epsilon )-{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2\},\\ {\displaystyle w_2(n+1)}& \ge w_2(n)\exp \{{\alpha }_2^l-{\beta }_2^u({\mathcal{M}}_2+\epsilon )-{\gamma }_2^u({\mathcal{M}}_1+\epsilon )-{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2\}.\end{array}$(21) Set $w_i(n)=\exp \{v_i(n)\}$, then (21(21) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& \ge w_1(n)\exp \{{\alpha }_1^l-{\beta }_1^u({\mathcal{M}}_1+\epsilon )-{\gamma }_1^u({\mathcal{M}}_2+\epsilon )-{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2\},\\ {\displaystyle w_2(n+1)}& \ge w_2(n)\exp \{{\alpha }_2^l-{\beta }_2^u({\mathcal{M}}_2+\epsilon )-{\gamma }_2^u({\mathcal{M}}_1+\epsilon )-{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2\}.\end{array}$(21) ) is equivalent to (22) $\{\begin{array}{rl}{\displaystyle v_1(n+1)-v_1(n)}& \ge {\alpha }_1^l-{\beta }_1^u({\mathcal{M}}_1+\epsilon )-{\gamma }_1^u({\mathcal{M}}_2+\epsilon )-{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2,\\ {\displaystyle v_2(n+1)-v_2(n)}& \ge {\alpha }_2^l-{\beta }_2^u({\mathcal{M}}_2+\epsilon )-{\gamma }_2^u({\mathcal{M}}_1+\epsilon )-{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2.\end{array}$(22) In view of (22(22) $\{\begin{array}{rl}{\displaystyle v_1(n+1)-v_1(n)}& \ge {\alpha }_1^l-{\beta }_1^u({\mathcal{M}}_1+\epsilon )-{\gamma }_1^u({\mathcal{M}}_2+\epsilon )-{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2,\\ {\displaystyle v_2(n+1)-v_2(n)}& \ge {\alpha }_2^l-{\beta }_2^u({\mathcal{M}}_2+\epsilon )-{\gamma }_2^u({\mathcal{M}}_1+\epsilon )-{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2.\end{array}$(22) ), we obtain (23) $\{\begin{array}{rl}& {\displaystyle \sum _{j=n-\rho (n)}^{n-1}(v_1(j+1)-v_1(j))}\\ & \ge \sum _{j=n-\rho (n)}^{n-1}[{\alpha }_1^l-{\beta }_1^u({\mathcal{M}}_1+\epsilon )-{\gamma }_1^u({\mathcal{M}}_2+\epsilon )-{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2]\\ & {\displaystyle \ge [{\alpha }_1^l-{\beta }_1^u({\mathcal{M}}_1+\epsilon )-{\gamma }_1^u({\mathcal{M}}_2+\epsilon )-{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2]{\rho }^m,}\\ & {\displaystyle \sum _{j=n-\rho (n)}^{n-1}(v_2(j+1)-v_2(j))}\\ & \ge \sum _{j=n-\rho (n)}^{n-1}[{\alpha }_2^l-{\beta }_2^u({\mathcal{M}}_2+\epsilon )-{\gamma }_2^u({\mathcal{M}}_1+\epsilon )-{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2]\\ & {\displaystyle \ge [{\alpha }_2^l-{\beta }_2^u({\mathcal{M}}_2+\epsilon )-{\gamma }_2^u({\mathcal{M}}_1+\epsilon )-{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2]{\rho }^m.}\end{array}$(23) Then (24) $\{\begin{array}{rl}{\displaystyle v_1(n-\rho (n))}& \le v_1(n)-[{\alpha }_1^l-{\beta }_1^u({\mathcal{M}}_1+\epsilon )-{\gamma }_1^u({\mathcal{M}}_2+\epsilon )-{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2]{\rho }^m,\\ {\displaystyle v_2(n-\rho (n))}& \le v_2(n)-[{\alpha }_2^l-{\beta }_2^u({\mathcal{M}}_2+\epsilon )-{\gamma }_2^u({\mathcal{M}}_1+\epsilon )-{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2]{\rho }^m.\end{array}$(24) Thus (25) $\{\begin{array}{rl}& {\displaystyle w_1(n-\rho (n))}\\ & \le w_1(n)\exp \{-[{\alpha }_1^l-{\beta }_1^u({\mathcal{M}}_1+\epsilon )-{\gamma }_1^u({\mathcal{M}}_2+\epsilon )-{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2]{\rho }^m\},\\ & {\displaystyle w_2(n-\rho (n))}\\ & \le w_2(n)\exp \{-[{\alpha }_2^l-{\beta }_2^u({\mathcal{M}}_2+\epsilon )-{\gamma }_2^u({\mathcal{M}}_1+\epsilon )-{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2]{\rho }^m\}.\end{array}$(25) Applying (25(25) $\{\begin{array}{rl}& {\displaystyle w_1(n-\rho (n))}\\ & \le w_1(n)\exp \{-[{\alpha }_1^l-{\beta }_1^u({\mathcal{M}}_1+\epsilon )-{\gamma }_1^u({\mathcal{M}}_2+\epsilon )-{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2]{\rho }^m\},\\ & {\displaystyle w_2(n-\rho (n))}\\ & \le w_2(n)\exp \{-[{\alpha }_2^l-{\beta }_2^u({\mathcal{M}}_2+\epsilon )-{\gamma }_2^u({\mathcal{M}}_1+\epsilon )-{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2]{\rho }^m\}.\end{array}$(25) ) and (5(5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) ), one has (26) $\{\begin{array}{rl}& {\displaystyle w_1(n+1)}\\ & \ge w_1(n)\exp \{{\alpha }_1^l-{\gamma }_1^u({\mathcal{M}}_2+\epsilon )-{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2-b_1^u({\mathcal{U}}_1+\epsilon )\\ & -{\displaystyle {\beta }_1^u{w}_1(n)\exp \{-[{\alpha }_1^l-{\beta }_1^u({\mathcal{M}}_1+\epsilon )-{\gamma }_1^u({\mathcal{M}}_2+\epsilon )-{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2]{\rho }^m\}}\},\\ & {\displaystyle w_2(n+1)}\\ & \ge w_2(n)\exp \{{\alpha }_2^l-{\gamma }_2^u({\mathcal{M}}_1+\epsilon )-{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2-b_2^u({\mathcal{U}}_2+\epsilon )\\ & -{\displaystyle {\beta }_2^u{w}_2(n)\exp \{-[{\alpha }_2^l-{\beta }_2^u({\mathcal{M}}_2+\epsilon )-{\gamma }_2^u({\mathcal{M}}_1+\epsilon )-{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2]{\rho }^m\}}\}.\end{array}$(26) In view of Lemmas 2.1 and 2.3, we get (27) $\underset{n\to +\mathrm{\infty }}{lim}inf{w}_1(n)\ge m_1^{\epsilon },\underset{n\to +\mathrm{\infty }}{lim}inf{w}_2(n)\ge m_2^{\epsilon },$(27) where (28) $\{\begin{array}{rl}{\displaystyle m_1^{\epsilon }}& =\frac{{\alpha }_1^l-{\gamma }_1^u({\mathcal{M}}_2+\epsilon )-{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2-b_1^u({\mathcal{U}}_1+\epsilon )}{b_1^u\exp \{-[{\alpha }_1^l-{\beta }_1^u({\mathcal{M}}_1+\epsilon )-{\gamma }_1^u({\mathcal{M}}_2+\epsilon )-{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2]{\rho }^m\}}\\ & {\displaystyle \times \exp \{{\alpha }_1^l-{\gamma }_1^u({\mathcal{M}}_2+\epsilon )-{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2-b_1^u({\mathcal{U}}_1+\epsilon )}\\ & -{\displaystyle {\beta }_1^u\exp \{-[{\alpha }_1^l-{\beta }_1^u({\mathcal{M}}_1+\epsilon )-{\gamma }_1^u({\mathcal{M}}_2+\epsilon )-{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2]{\rho }^m\}{\mathcal{M}}_1}\},\\ {\displaystyle m_2^{\epsilon }}& =\frac{{\alpha }_2^l-{\gamma }_2^u({\mathcal{M}}_1+\epsilon )-{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2-b_2^u({\mathcal{U}}_2+\epsilon )}{{\beta }_2^u\exp \{-[{\alpha }_2^l-{\beta }_2^u({\mathcal{M}}_2+\epsilon )-{\gamma }_2^u({\mathcal{M}}_1+\epsilon )-{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2]{\rho }^m\}}\\ & {\displaystyle \times \exp \{{\alpha }_2^l-{\gamma }_2^u({\mathcal{M}}_1+\epsilon )-{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2-b_2^u({\mathcal{U}}_2+\epsilon )}\\ & -{\displaystyle {\beta }_2^u\exp \{-[{\alpha }_2^l-{\beta }_2^u({\mathcal{M}}_2+\epsilon )-{\gamma }_2^u({\mathcal{M}}_1+\epsilon )-{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2]{\rho }^m\}{\mathcal{M}}_2}\}.\end{array}$(28) Setting $\epsilon \to 0$ in (28(28) $\{\begin{array}{rl}{\displaystyle m_1^{\epsilon }}& =\frac{{\alpha }_1^l-{\gamma }_1^u({\mathcal{M}}_2+\epsilon )-{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2-b_1^u({\mathcal{U}}_1+\epsilon )}{b_1^u\exp \{-[{\alpha }_1^l-{\beta }_1^u({\mathcal{M}}_1+\epsilon )-{\gamma }_1^u({\mathcal{M}}_2+\epsilon )-{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2]{\rho }^m\}}\\ & {\displaystyle \times \exp \{{\alpha }_1^l-{\gamma }_1^u({\mathcal{M}}_2+\epsilon )-{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2-b_1^u({\mathcal{U}}_1+\epsilon )}\\ & -{\displaystyle {\beta }_1^u\exp \{-[{\alpha }_1^l-{\beta }_1^u({\mathcal{M}}_1+\epsilon )-{\gamma }_1^u({\mathcal{M}}_2+\epsilon )-{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2]{\rho }^m\}{\mathcal{M}}_1}\},\\ {\displaystyle m_2^{\epsilon }}& =\frac{{\alpha }_2^l-{\gamma }_2^u({\mathcal{M}}_1+\epsilon )-{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2-b_2^u({\mathcal{U}}_2+\epsilon )}{{\beta }_2^u\exp \{-[{\alpha }_2^l-{\beta }_2^u({\mathcal{M}}_2+\epsilon )-{\gamma }_2^u({\mathcal{M}}_1+\epsilon )-{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2]{\rho }^m\}}\\ & {\displaystyle \times \exp \{{\alpha }_2^l-{\gamma }_2^u({\mathcal{M}}_1+\epsilon )-{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2-b_2^u({\mathcal{U}}_2+\epsilon )}\\ & -{\displaystyle {\beta }_2^u\exp \{-[{\alpha }_2^l-{\beta }_2^u({\mathcal{M}}_2+\epsilon )-{\gamma }_2^u({\mathcal{M}}_1+\epsilon )-{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2]{\rho }^m\}{\mathcal{M}}_2}\}.\end{array}$(28) ), we have (29) $\underset{n\to +\mathrm{\infty }}{lim}inf{w}_1(n)\ge m_1,\underset{n\to +\mathrm{\infty }}{lim}inf{w}_2(n)\ge m_2,$(29) where (30) $\{\begin{array}{rl}{\displaystyle m_1}& =\frac{{\alpha }_1^l-{\gamma }_1^u{\mathcal{M}}_2-{\delta }_1^u{\mathcal{M}}_1^2-b_1^u{\mathcal{U}}_1}{{\beta }_1^u\exp \{-[{\alpha }_1^l-{\beta }_1^u{\mathcal{M}}_1-{\gamma }_1^u{\mathcal{M}}_2-{\delta }_1^u{\mathcal{M}}_1^2]{\rho }^m\}}\\ & {\displaystyle \times \exp \{{\alpha }_1^l-{\pi }_1^u{\mathcal{M}}_2-{\delta }_1^u{\mathcal{M}}_1^2-b_1^u{\mathcal{U}}_1}\\ & -{\displaystyle {\beta }_1^u\exp \{-[{\alpha }_1^l-{\beta }_1^u{\mathcal{M}}_1-{\gamma }_1^u{\mathcal{M}}_2-{\delta }_1^u{\mathcal{M}}_1^2]{\rho }^m\}{\mathcal{M}}_1}\},\\ {\displaystyle m_2}& =\frac{{\alpha }_2^l-{\gamma }_2^u{M}_1-{\delta }_2^u{\mathcal{M}}_2^2-b_2^u{\mathcal{U}}_2}{{\beta }_2^u\exp \{-[{\alpha }_2^l-{\beta }_2^u{\mathcal{M}}_2-{\gamma }_2^u{\mathcal{M}}_1-{\delta }_2^u{\mathcal{M}}_2^2]{\rho }^m\}}\\ & {\displaystyle \times \exp \{{\alpha }_2^l-{\gamma }_2^u{\mathcal{M}}_1-{\delta }_2^u{\mathcal{M}}_2{+}^2-b_2^u{\mathcal{U}}_2}\\ & -{\displaystyle {\beta }_2^u\exp \{-[{\alpha }_2^l-{\beta }_2^u{\mathcal{M}}_2-{\gamma }_2^u{\mathcal{M}}_1-{\delta }_2^u{\mathcal{M}}_2^2]{\rho }^m\}{\mathcal{M}}_2}\}.\end{array}$(30) Without loss of generality, we assume that $\epsilon <\frac{1}{2}min\{m_1,m_2\}$. For any positive constant ε small enough, it follows from (29(29) $\underset{n\to +\mathrm{\infty }}{lim}inf{w}_1(n)\ge m_1,\underset{n\to +\mathrm{\infty }}{lim}inf{w}_2(n)\ge m_2,$(29) ) that there exists enough large $N_3>N_2+\rho$ which satisfies (31) $w_1(n)\ge m_1-\epsilon ,w_2(n)\ge m_2-\epsilon$(31) ∀  $n\ge N_3$. By means of (5(5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) ) and (31(31) $w_1(n)\ge m_1-\epsilon ,w_2(n)\ge m_2-\epsilon$(31) ), one gets (32) $\mathrm{\Delta }{\mu }_i(n)\ge -{\vartheta }_i(n){\mu }_i(n)+{\xi }_i(n)(m_i-\epsilon ),i=1,2.$(32) Hence (33) ${\mu }_i(n+1)\ge (1-{\vartheta }_i^u){\mu }_i(n)+{\xi }_i^l(m_i-\epsilon ),i=1,2.$(33) By Lemmas 2.1 and 2.2, we get (34) $\underset{n\to +\mathrm{\infty }}{lim}inf{\mu }_i(n)\ge \frac{{\xi }_i^l(m_i-\epsilon )}{{\vartheta }_i^u},i=1,2.$(34) Setting $\epsilon \to 0$ in the above inequality leads to (35) $\underset{n\to +\mathrm{\infty }}{lim}inf{\mu }_i(n)\ge \frac{{\xi }_i^l{m}_i}{{\vartheta }_i^u}:={\mathcal{U}}_i^l,i=1,2,$(35) which completes the proof.

3. Existence and stability of periodic solution of system (5) without delay

In this section, we are to discuss the stability of system (5(5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) ) with ${\rho }_i(n)=0(i=1,2)$, that is, we discuss the following system (36) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& =w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n)-{\gamma }_1(n)w_2(n)\\ & -{\displaystyle {\delta }_1(n)w_1^2(n)-b_1(n){\mu }_1(n)}\},\\ {\displaystyle w_2(n+1)}& =w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n)-{\gamma }_2(n)w_1(n)\\ & -{\displaystyle {\delta }_2(n)w_2^2(n)-b_2(n){\mu }_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)}& =-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)}& =-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n).\end{array}$(36) Throughout this section we always assume that ${\alpha }_i(n),{\beta }_i(n),{\gamma }_i(n),{\delta }_i(n),{\vartheta }_i(n),{\xi }_1(n)(i=1,2)$ are all bounded negative periodic sequences with a common periodic ω and satisfy (37) $0<{\vartheta }_i(n)<1,n\in N\cap [0,\varpi ],i=1,2.$(37) Also it is assumed that the initial values of (36(36) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& =w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n)-{\gamma }_1(n)w_2(n)\\ & -{\displaystyle {\delta }_1(n)w_1^2(n)-b_1(n){\mu }_1(n)}\},\\ {\displaystyle w_2(n+1)}& =w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n)-{\gamma }_2(n)w_1(n)\\ & -{\displaystyle {\delta }_2(n)w_2^2(n)-b_2(n){\mu }_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)}& =-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)}& =-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n).\end{array}$(36) ) are given by (38) $w_i(0)>0,{\mu }_i(0)>0,i=1,2.$(38) Applying the similar approach, under some conditions, we can obtain the permanence of system (36(36) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& =w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n)-{\gamma }_1(n)w_2(n)\\ & -{\displaystyle {\delta }_1(n)w_1^2(n)-b_1(n){\mu }_1(n)}\},\\ {\displaystyle w_2(n+1)}& =w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n)-{\gamma }_2(n)w_1(n)\\ & -{\displaystyle {\delta }_2(n)w_2^2(n)-b_2(n){\mu }_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)}& =-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)}& =-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n).\end{array}$(36) ). We still let ${\mathcal{M}}_i$ and ${\mathcal{U}}_i$ be the upper bound of $\{w_i(n)\}$ and $\{{\mu }_i(n)\}$, and $m_i$ and ${\mathcal{U}}_i^l$ be the lower bound of $\{w_i(n)\}$ and $\{{\mu }_i(n)\}$.

Theorem 3.1

In addition to (37(37) $0<{\vartheta }_i(n)<1,n\in N\cap [0,\varpi ],i=1,2.$(37) ), suppose that (H1) and (H2) ${\alpha }_1^l>{\gamma }_1^u{\mathcal{M}}_2+{\delta }_1^u{\mathcal{M}}_1^2+b_1^u{\mathcal{U}}_1,{\alpha }_2^l>{\gamma }_2^u{\mathcal{M}}_1+{\delta }_2^u{\mathcal{M}}_2^2+b_2^u{\mathcal{U}}_2$ hold, then system (36(36) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& =w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n)-{\gamma }_1(n)w_2(n)\\ & -{\displaystyle {\delta }_1(n)w_1^2(n)-b_1(n){\mu }_1(n)}\},\\ {\displaystyle w_2(n+1)}& =w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n)-{\gamma }_2(n)w_1(n)\\ & -{\displaystyle {\delta }_2(n)w_2^2(n)-b_2(n){\mu }_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)}& =-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)}& =-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n).\end{array}$(36) ) owns a periodic solution $\{{\overline{w}}_1(n),{\overline{w}}_2(n),{\overline{\mu }}_1(n),{\overline{\mu }}_2(n)\}$.

Proof.

Let $\mathrm{\Omega }=\{(w_1,w_2,{\mu }_1,{\mu }_2)|m_i\le w_i\le {\mathcal{M}}_i,{\mathcal{U}}_i^l\le {\mu }_i\le {\mathcal{U}}_i,i=1,2\}$. It is easy to see that Ω is an invariant set of system (36(36) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& =w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n)-{\gamma }_1(n)w_2(n)\\ & -{\displaystyle {\delta }_1(n)w_1^2(n)-b_1(n){\mu }_1(n)}\},\\ {\displaystyle w_2(n+1)}& =w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n)-{\gamma }_2(n)w_1(n)\\ & -{\displaystyle {\delta }_2(n)w_2^2(n)-b_2(n){\mu }_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)}& =-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)}& =-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n).\end{array}$(36) ). Then we can define a mapping F on Ω by (39) $F(w_1(0),w_2(0),{\mu }_1(0),{\mu }_2(0))=(w_1(\varpi ),w_2(\varpi ),{\mu }_1(\varpi ),{\mu }_2(\varpi ))$(39) for $(w_1(0),w_2(0),{\mu }_1(0),{\mu }_2(0))\in \mathrm{\Omega }$. Clearly, F depends continuously on $(w_1(0),w_2(0),{\mu }_1(0),{\mu }_2(0))$, then F is continuous and maps a compact set Ω into itself. So, F owns a fixed point $({\overline{w}}_1(n),{\overline{w}}_2(n),{\overline{\mu }}_1(n),$ ${\overline{\mu }}_2(n))$. Thus one can know that system (36(36) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& =w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n)-{\gamma }_1(n)w_2(n)\\ & -{\displaystyle {\delta }_1(n)w_1^2(n)-b_1(n){\mu }_1(n)}\},\\ {\displaystyle w_2(n+1)}& =w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n)-{\gamma }_2(n)w_1(n)\\ & -{\displaystyle {\delta }_2(n)w_2^2(n)-b_2(n){\mu }_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)}& =-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)}& =-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n).\end{array}$(36) ) owns the solution $({\overline{w}}_1(n),{\overline{w}}_2(n),{\overline{\mu }}_1(n),{\overline{\mu }}_2(n))$ which passes through $({\overline{w}}_1,{\overline{w}}_2,{\overline{\mu }}_1,{\overline{\mu }}_2)$. This ends the proof of Theorem 3.1.

Next, we are to deal with the globally stability property of the periodic solution derived in Theorem 3.1.

Theorem 3.2

In addition to the conditions of Theorem 3.1, suppose that the following condition (H3) holds, $(\mathrm{H}3)\{\begin{array}{rl}{\mathrm{\Pi }}_1& =max\{|1-{\beta }_1^l{m}_1-2{\delta }_1^l{m}_1^2|,|1-{\beta }_1^u{\mathcal{M}}_1+2{\delta }_1^u{\mathcal{M}}_1^2|\}\\ & +{\gamma }_1^u{\mathcal{M}}_2+b_1^u<1,\\ {\mathrm{\Pi }}_2& =max\{|1-{\beta }_2^l{m}_2-2{\delta }_2^l{m}_2^2|,|1-{\beta }_2^u{\mathcal{M}}_2+2{\delta }_2^u{\mathcal{M}}_2^2|\}\\ & +{\gamma }_2^u{\mathcal{M}}_1+b_2^u<1,\\ {\mathrm{\Pi }}_3& =(1-{\vartheta }_1^l)+{\xi }_1^u{\mathcal{M}}_1<1,\\ {\mathrm{\Pi }}_4& =(1-{\vartheta }_2^l)+{\xi }_2^u{\mathcal{M}}_2<1,\end{array}$then the ω periodic solution $({\overline{w}}_1(n),{\overline{w}}_2(n),{\overline{\mu }}_1(n),{\overline{\mu }}_2(n))$ obtained in Theorem 3.1 is globally attractive.

Proof.

Assume that $(w_1(n),w_2(n),{\mu }_1(n),{\mu }_2(n))$ is any positive solution of system (36(36) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& =w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n)-{\gamma }_1(n)w_2(n)\\ & -{\displaystyle {\delta }_1(n)w_1^2(n)-b_1(n){\mu }_1(n)}\},\\ {\displaystyle w_2(n+1)}& =w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n)-{\gamma }_2(n)w_1(n)\\ & -{\displaystyle {\delta }_2(n)w_2^2(n)-b_2(n){\mu }_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)}& =-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)}& =-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n).\end{array}$(36) ). Let (40) $w_i(n)={\overline{w}}_i(n)\exp \{v_i(n)\},{\mu }_i(n)={\overline{\mu }}_i(n)+{\varrho }_i(n),i=1,2.$(40) To finish the proof, we only need to prove that (41) $\underset{n\to \mathrm{\infty }}{lim}{v}_i(n)=0,\underset{n\to \mathrm{\infty }}{lim}{\varrho }_i(n)=0,i=1,2.$(41) Notice that (42) $\begin{array}{rl}{v}_1(n+1)& =v_1(n)-{\beta }_1(n){\overline{w}}_1(n)[\exp (v_1(n))-1]\\ & -{\gamma }_1(n){\overline{w}}_2(n)[\exp (v_2(n))-1]\\ & -{\delta }_1(n){\overline{w}}_1^2(n)[\exp (2v_1(n))-1]-b_1(n){\varrho }_1(n)\\ & =v_1(n)-{\beta }_1(n){\overline{w}}_1(n)\exp \{{\theta }_1(n)v_1(n)\}{v}_1(n)\\ & -{\gamma }_1(n){\overline{w}}_2(n)\exp \{{\theta }_2(n)v_2(n)\}{v}_2(n)\\ & -{\delta }_1(n){\overline{w}}_1^2(n)\exp \{2{\theta }_3(n)v_1(n)\}2v_1(n)-b_1(n){\varrho }_1(n),\end{array}$(42) where ${\theta }_i(n)\in (0,1),i=1,2,3$. In a similar approach, one gets (43) $\begin{array}{rl}{v}_2(n+1)& =v_2(n)-{\beta }_2(n){\overline{w}}_2(n)\exp \{{\theta }_4(n)v_2(n)\}{v}_2(n)\\ & -{\gamma }_2(n){\overline{w}}_1(n)\exp \{{\theta }_5(n)v_1(n)\}{v}_1(n)\\ & -{\delta }_2(n){\overline{w}}_2^2(n)\exp \{2{\theta }_6(n)v_2(n)\}2v_2(n)-b_2(n){\varrho }_2(n),\end{array}$(43) where ${\theta }_j(n)\in (0,1),j=4,5,6$. Also, one has (44) $\begin{array}{rl}{\varrho }_1(n+1)& =(1-{\vartheta }_1(n)){\varrho }_1(n)+{\xi }_1(n){\overline{w}}_1(n)[\exp \{v_1(n)\}-1]\\ & =(1-{\vartheta }_1(n)){\varrho }_1(n)+{\xi }_1(n){\overline{w}}_1(n)\exp \{{\theta }_7(n)v_1(n)\}{v}_1(n),\end{array}$(44) (45) $\begin{array}{rl}{\varrho }_2(n+1)& =(1-{\vartheta }_2(n)){\varrho }_2(n)+{\xi }_2(n){\overline{w}}_2(n)[\exp \{v_2(n)\}-1]\\ & =(1-{\vartheta }_2(n)){\varrho }_2(n)+{\xi }_2(n){\overline{w}}_2(n)\exp \{{\theta }_8(n)v_2(n)\}{v}_2(n).\end{array}$(45) By (H3), we can choose a $\epsilon >0$ such that (46) $\{\begin{array}{rl}{\mathrm{\Pi }}_1^{\epsilon }& =max\{|1-{\beta }_1^l(m_1-\epsilon )-2{\delta }_1^l(m_1-\epsilon {)}^2|,\\ & |1-{\beta }_1^u({\mathcal{M}}_1+\epsilon )+2{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2|\phantom{{\delta }_1^l}\}\\ & +{\gamma }_1^u({\mathcal{M}}_2+\epsilon )+b_1^u<1\\ {\mathrm{\Pi }}_2^{\epsilon }& =max\{|1-{\beta }_2^l(m_2-\epsilon )-2{\delta }_2^l(m_2-\epsilon {)}^2|,\\ & |1-{\beta }_2^u({\mathcal{M}}_2+\epsilon )+2{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2|\phantom{{\delta }_1^l}\}\\ & +{\gamma }_2^u({\mathcal{M}}_1+\epsilon )+b_2^u<1,\\ {\mathrm{\Pi }}_3^{\epsilon }& =(1-{\vartheta }_1^l)+{\xi }_1^u({\mathcal{M}}_1+\epsilon )<1,\\ {\mathrm{\Pi }}_4^{\epsilon }& =(1-{\vartheta }_2^l)+{\xi }_2^u({\mathcal{M}}_2+\epsilon )<1,\end{array}$(46) In view of Proposition 2.1 and Theorem 2.1, ∃  $N_4>N_3$ which satisfies (47) $m_i-\epsilon \le w_i(n),{\overline{w}}_i(n)\le {\mathcal{M}}_i+\epsilon ,\text{for}n\ge N_5,i=1,2.$(47) It follows from (42(42) $\begin{array}{rl}{v}_1(n+1)& =v_1(n)-{\beta }_1(n){\overline{w}}_1(n)[\exp (v_1(n))-1]\\ & -{\gamma }_1(n){\overline{w}}_2(n)[\exp (v_2(n))-1]\\ & -{\delta }_1(n){\overline{w}}_1^2(n)[\exp (2v_1(n))-1]-b_1(n){\varrho }_1(n)\\ & =v_1(n)-{\beta }_1(n){\overline{w}}_1(n)\exp \{{\theta }_1(n)v_1(n)\}{v}_1(n)\\ & -{\gamma }_1(n){\overline{w}}_2(n)\exp \{{\theta }_2(n)v_2(n)\}{v}_2(n)\\ & -{\delta }_1(n){\overline{w}}_1^2(n)\exp \{2{\theta }_3(n)v_1(n)\}2v_1(n)-b_1(n){\varrho }_1(n),\end{array}$(42) )m and (43(43) $\begin{array}{rl}{v}_2(n+1)& =v_2(n)-{\beta }_2(n){\overline{w}}_2(n)\exp \{{\theta }_4(n)v_2(n)\}{v}_2(n)\\ & -{\gamma }_2(n){\overline{w}}_1(n)\exp \{{\theta }_5(n)v_1(n)\}{v}_1(n)\\ & -{\delta }_2(n){\overline{w}}_2^2(n)\exp \{2{\theta }_6(n)v_2(n)\}2v_2(n)-b_2(n){\varrho }_2(n),\end{array}$(43) ) that (48) $\begin{array}{rl}{v}_1(n+1)& \le max\{|1-{\beta }_1^l(m_1-\epsilon )-2{\delta }_1^l(m_1-\epsilon {)}^2|,\\ & |1-{\beta }_1^u({\mathcal{M}}_1+\epsilon )+2{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2|\phantom{{\delta }_1^l}\}|v_1(n)|\\ & +{\gamma }_1^u({\mathcal{M}}_2+\epsilon )|v_2(n)|+b_1^u|{\varrho }_1(n)|,\end{array}$(48) (49) $\begin{array}{rl}{v}_2(n+1)& \le max\{|1-{\beta }_2^l(m_2-\epsilon )-2{\delta }_2^l(m_2-\epsilon {)}^2|,\\ & |1-{\beta }_2^u({\mathcal{M}}_2+\epsilon )+2{\delta }_2^u({\mathcal{M}}_2+\epsilon {)}^2|\phantom{{\delta }_1^l}\}|v_2(n)|\\ & +{\gamma }_2^u({\mathcal{M}}_1+\epsilon )|v_1(n)|+b_2^u|{\varrho }_2(n)|,\end{array}$(49) Also, for $n>N_4$, one has (50) $\begin{array}{rl}{\varrho }_1(n+1)& \le (1-{\vartheta }_1^l)|{\varrho }_1(n)|+{\xi }_1^u({\mathcal{M}}_1+\epsilon )|v_1(n)|,\end{array}$(50) (51) $\begin{array}{rl}{\varrho }_2(n+1)& \le (1-{\vartheta }_2^l)|{\varrho }_2(n)|+{\xi }_2^u({\mathcal{M}}_2+\epsilon )|v_2(n)|.\end{array}$(51) Let $\chi =max\{{\chi }_1^{\epsilon },{\chi }_2^{\epsilon },{\chi }_3^{\epsilon },{\chi }_4^{\epsilon }\}$, then $0<\chi <1$. It follows from (48(48) $\begin{array}{rl}{v}_1(n+1)& \le max\{|1-{\beta }_1^l(m_1-\epsilon )-2{\delta }_1^l(m_1-\epsilon {)}^2|,\\ & |1-{\beta }_1^u({\mathcal{M}}_1+\epsilon )+2{\delta }_1^u({\mathcal{M}}_1+\epsilon {)}^2|\phantom{{\delta }_1^l}\}|v_1(n)|\\ & +{\gamma }_1^u({\mathcal{M}}_2+\epsilon )|v_2(n)|+b_1^u|{\varrho }_1(n)|,\end{array}$(48) )–(51(51) $\begin{array}{rl}{\varrho }_2(n+1)& \le (1-{\vartheta }_2^l)|{\varrho }_2(n)|+{\xi }_2^u({\mathcal{M}}_2+\epsilon )|v_2(n)|.\end{array}$(51) ) that (52) $\begin{array}{rl}& max\{|w_1(n+1)|,|w_2(n+1)|,|{\varrho }_1(n+1)|,|{\varrho }_2(n+1)|\}\\ & \le \chi max\{|w_1(n)|,|w_2(n)|,|{\varrho }_1(n)|,|{\varrho }_2(n)|\}\end{array}$(52) for $n>N_4$. Then we get (53) $\begin{array}{rl}& max\{|w_1(n)|,|w_2(n)|,|{\varrho }_1(n)|,|{\varrho }_2(n)|\}\\ & \le {\chi }^{n-N_4}max\{|w_1(N_4)|,|w_2(N_4)|,|{\varrho }_1(N_4)|,|{\varrho }_2(N_4)|\}.\end{array}$(53) Thus (54) $\underset{n\to \mathrm{\infty }}{lim}{w}_i(n)=0,\underset{n\to \mathrm{\infty }}{lim}{\varrho }_i(n)=0,i=1,2.$(54) This completes the proof.

Remark 3.1

In [14, 42], the authors dealt with the continuous or pulsing competitive model without time delays and feedback controls. In this paper, we consider the practical situation and introduce time delays and feedback controls. Based on this viewpoint, the acquired outcomes of our work are new and replenish the outcomes of [14, 42].

Remark 3.2

We do not investigate the existence of periodic solution and global attractivity of system (5(5) $\{\begin{array}{l}{\displaystyle w_1(n+1)=w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))}\\ {\displaystyle -{\delta }_1(n)w_1^2(n-\rho (n))-b_1(n)u_1(n)}\},\\ {\displaystyle w_2(n+1)=w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))}\\ {\displaystyle -{\delta }_2(n)w_2^2(n-\rho (n))-b_2(n)u_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)=-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),}\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)=-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),}\end{array}$(5) ) since the introduction of delay leads to the difficulties in analysis methods. We leave this aspect for future work.

4. Examples

In this section, we will execute computer simulations via Matlab software to confirm the rationality of our derived key conclusions.

Example 4.1

Consider the following system (55) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& =w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))\\ & -{\displaystyle {\delta }_1(n)w_1^2(n-\rho (n))-{\beta }_1(n){\mu }_1(n)}\},\\ {\displaystyle w_2(n+1)}& =w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))\\ & -{\displaystyle {\delta }_2(n)w_2^2(n-\rho (n))-b_2(n){\mu }_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)}& =-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)}& =-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),\end{array}$(55) where ${\alpha }_1(n)=15+\cos (n),{\alpha }_2(n)=16+\sin (n),{\beta }_1(n)=1.3+\sin (n),{\beta }_2(n)=1.2+\cos (n),{\gamma }_1(n)=0.2+\cos (n),{\gamma }_2(n)=0.3+\sin (n),{\delta }_1(n)=0.3+\sin (n),{\delta }_2(n)=0.1+\cos (n),b_1(n)=1.1+\sin (n),b_2(n)=1.3+\cos (n),{\vartheta }_1(n)=0.4\cos (n),{\vartheta }_2(n)=0.3\sin (n),{\xi }_1(n)=1.4+\sin (n),{\xi }_2(n)=1.6+\cos (n),\rho (n)=0.5$. Then ${\alpha }_1^l=14,{\alpha }_2^l=15,{\gamma }_1^u=1.2,{\gamma }_2^u=1.3,{\delta }_1^u=1.3,{\delta }_2^u=1.1,b_1^u=2.1,b_2^u=2.3$. We can verify that all the hypotheses in Theorem 2.1 are true. Thus one can easily know that system (55(55) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& =w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))\\ & -{\displaystyle {\delta }_1(n)w_1^2(n-\rho (n))-{\beta }_1(n){\mu }_1(n)}\},\\ {\displaystyle w_2(n+1)}& =w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))\\ & -{\displaystyle {\delta }_2(n)w_2^2(n-\rho (n))-b_2(n){\mu }_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)}& =-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)}& =-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),\end{array}$(55) ) is permanent. The computer simulation figures are clearly presented in Figures 1–4.

Figure 1. Computer simulation figure of system (55(55) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& =w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))\\ & -{\displaystyle {\delta }_1(n)w_1^2(n-\rho (n))-{\beta }_1(n){\mu }_1(n)}\},\\ {\displaystyle w_2(n+1)}& =w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))\\ & -{\displaystyle {\delta }_2(n)w_2^2(n-\rho (n))-b_2(n){\mu }_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)}& =-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)}& =-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),\end{array}$(55) ): the relation between the time k and the variable $w_1$.

Figure 2. Computer simulation figure of system (55(55) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& =w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))\\ & -{\displaystyle {\delta }_1(n)w_1^2(n-\rho (n))-{\beta }_1(n){\mu }_1(n)}\},\\ {\displaystyle w_2(n+1)}& =w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))\\ & -{\displaystyle {\delta }_2(n)w_2^2(n-\rho (n))-b_2(n){\mu }_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)}& =-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)}& =-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),\end{array}$(55) ): the relation between the time k and the variable $w_2$.

Figure 3. Computer simulation figure of system (55(55) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& =w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))\\ & -{\displaystyle {\delta }_1(n)w_1^2(n-\rho (n))-{\beta }_1(n){\mu }_1(n)}\},\\ {\displaystyle w_2(n+1)}& =w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))\\ & -{\displaystyle {\delta }_2(n)w_2^2(n-\rho (n))-b_2(n){\mu }_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)}& =-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)}& =-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),\end{array}$(55) ): the relation between the time k and the variable ${\mu }_1$.

Figure 4. Computer simulation figure of system (55(55) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& =w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n-\rho (n))-{\gamma }_1(n)w_2(n-\rho (n))\\ & -{\displaystyle {\delta }_1(n)w_1^2(n-\rho (n))-{\beta }_1(n){\mu }_1(n)}\},\\ {\displaystyle w_2(n+1)}& =w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n-\rho (n))-{\gamma }_2(n)w_1(n-\rho (n))\\ & -{\displaystyle {\delta }_2(n)w_2^2(n-\rho (n))-b_2(n){\mu }_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)}& =-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)}& =-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),\end{array}$(55) ): the relation between the time k and the variable ${\mu }_2$.

Example 4.2

Consider the following system (56) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& =w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n)-{\gamma }_1(n)w_2(n)\\ & -{\displaystyle {\delta }_1(n)w_1^2(n)-b_1(n){\mu }_1(n)}\},\\ {\displaystyle w_2(n+1)}& =w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n)-{\gamma }_2(n)w_1(n)\\ & -{\displaystyle {\delta }_2(n)w_2^2(n)-b_2(n){\mu }_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)}& =-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)}& =-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),\end{array}$(56) where ${\alpha }_1(n)=17+\sin (n),{\alpha }_2(n)=17+\cos (n),{\beta }_1(n)=1.3+\cos (n),{\beta }_2(n)=1.2+\sin (n),{\gamma }_1(n)=1.3+\sin (n),{\gamma }_2(n)=1.2+\cos (n),{\delta }_1(n)=1.2+\cos (n),{\delta }_2(n)=1.2+\sin (n),b_1(n)=1.2+\cos (n),b_2(n)=1.3+\sin (n),{\vartheta }_1(n)=1-0.8\sin (n),{\vartheta }_2(n)=1-0.7\cos (n),{\xi }_1(n)=0.2-0.1\cos (n),{\xi }_2(n)=0.2-0.1\sin (n)$. Then ${\alpha }_1^l=16,{\alpha }_2^l=16,{\gamma }_1^u=2.3,{\pi }_2^u=2.2,{\delta }_1^u=2.2,{\delta }_2^u=2.2,b_1^u=2.2,b_2^u=2.3,{\vartheta }_1^l=0.2,{\vartheta }_2^l=0.3,{\xi }_1^u=0.3,{\xi }_2^u=0.3$. We can verify that all the hypotheses in Theorem 3.1 are true. Thus one can easily know that system (56(56) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& =w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n)-{\gamma }_1(n)w_2(n)\\ & -{\displaystyle {\delta }_1(n)w_1^2(n)-b_1(n){\mu }_1(n)}\},\\ {\displaystyle w_2(n+1)}& =w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n)-{\gamma }_2(n)w_1(n)\\ & -{\displaystyle {\delta }_2(n)w_2^2(n)-b_2(n){\mu }_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)}& =-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)}& =-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),\end{array}$(56) ) owns globally attractive periodic solution. The computer simulation figures are clearly presented in Figures 5–8.

Figure 5. Computer simulation figure of system (56(56) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& =w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n)-{\gamma }_1(n)w_2(n)\\ & -{\displaystyle {\delta }_1(n)w_1^2(n)-b_1(n){\mu }_1(n)}\},\\ {\displaystyle w_2(n+1)}& =w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n)-{\gamma }_2(n)w_1(n)\\ & -{\displaystyle {\delta }_2(n)w_2^2(n)-b_2(n){\mu }_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)}& =-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)}& =-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),\end{array}$(56) ): the relation between the time k and the variable $w_1$.

Figure 6. Computer simulation figure of system (56(56) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& =w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n)-{\gamma }_1(n)w_2(n)\\ & -{\displaystyle {\delta }_1(n)w_1^2(n)-b_1(n){\mu }_1(n)}\},\\ {\displaystyle w_2(n+1)}& =w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n)-{\gamma }_2(n)w_1(n)\\ & -{\displaystyle {\delta }_2(n)w_2^2(n)-b_2(n){\mu }_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)}& =-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)}& =-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),\end{array}$(56) ): the relation between the time k and the variable $w_2$.

Figure 7. Computer simulation figure of system (56(56) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& =w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n)-{\gamma }_1(n)w_2(n)\\ & -{\displaystyle {\delta }_1(n)w_1^2(n)-b_1(n){\mu }_1(n)}\},\\ {\displaystyle w_2(n+1)}& =w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n)-{\gamma }_2(n)w_1(n)\\ & -{\displaystyle {\delta }_2(n)w_2^2(n)-b_2(n){\mu }_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)}& =-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)}& =-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),\end{array}$(56) ): the relation between the time k and the variable ${\mu }_1$.

Figure 8. Computer simulation figure of system (56(56) $\{\begin{array}{rl}{\displaystyle w_1(n+1)}& =w_1(n)\exp \{\phantom{w_1^2}{\alpha }_1(n)-{\beta }_1(n)w_1(n)-{\gamma }_1(n)w_2(n)\\ & -{\displaystyle {\delta }_1(n)w_1^2(n)-b_1(n){\mu }_1(n)}\},\\ {\displaystyle w_2(n+1)}& =w_2(n)\exp \{\phantom{w_1^2}{\alpha }_2(n)-{\beta }_2(n)w_2(n)-{\gamma }_2(n)w_1(n)\\ & -{\displaystyle {\delta }_2(n)w_2^2(n)-b_2(n){\mu }_2(n)}\},\\ {\displaystyle \mathrm{\Delta }{\mu }_1(n)}& =-{\vartheta }_1(n){\mu }_1(n)+{\xi }_1(n)w_1(n),\\ {\displaystyle \mathrm{\Delta }{\mu }_2(n)}& =-{\vartheta }_2(n){\mu }_2(n)+{\xi }_2(n)w_2(n),\end{array}$(56) ): the relation between the time k and the variable ${\mu }_2$.

5. Conclusions

In this current work, we propose a new discrete predator–prey competitive model involving delays and feedback controls. By virtue of the difference inequality knowledge, a novel sufficient condition guaranteeing the permanence of the system is set up. We find that under a suitable parameter conditions, two species will keep a state of coexistence. The study reveals that feedback control effect and time delays play a vital role in remaining the co-existence of two species In addition, we also obtain the sufficient conditions which ensure the existence and stability of unique globally attractive periodic solution of the system without time delays. The derived results own significant theoretical guiding value in keeping a balance of biological population. Meanwhile, our results are new and supplement the existed results in [14, 42].

Data availability statement

No data were used to support this study.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by National Natural Science Foundation of China [grant numbers 12261015, 62062018], Project of High-level Innovative Talents of Guizhou Province [grant number [2016]5651], Guizhou Key Laboratory of Big Data Statistical Analysis [grant number [2019]5103], Key Project of Hunan Education Department [grant number 17A181], University Science and Technology Top Talents Project of Guizhou Province [grant number KY[2018]047], Foundation of Science and Technology of Guizhou Province [grant number [2019]1051], Guizhou University of Finance and Economics [grant number 2018XZD01].