1 Introduction

During the last decades, many scholars investigated the dynamic behaviors of the cooperative system, or mutualism model, see [1–36] and the references cited therein. Specially, in a series of their studies, Chen and his coauthors ([1–6, 28]) argued that a suitable cooperative model should consider the saturating effect of the relationship between the species, and they gave a thorough investigation on the dynamic behaviors of the cooperative system with Holling II type functional response.

Stimulated by the works of [1–6], in [28], Yang et al. studied the dynamic behaviors of the following autonomous discrete cooperative system:

$$\begin{gathered} x_{1}(k+1)=x_{1}(k)\exp \biggl\{ r_{1} \biggl[\frac{K_{1}+\alpha _{1}x_{2}(k)}{1+x_{2}(k)}-x_{1}(k) \biggr] \biggr\} , \\ x_{2}(k+1)=x_{2}(k)\exp \biggl\{ r_{2} \biggl[ \frac{K_{2}+\alpha _{2}x_{1}(k)}{1+x_{1}(k)}-x_{2}(k) \biggr] \biggr\} , \end{gathered} $$

where \(x_{i}(k)\) (\(i=1,2\)) is the population density of the ith species at k-generation. They showed that if

\((H_{1})\) :

\(r_{i}\), \(K_{i}\), \(\alpha_{i}\) (\(i=1, 2\)) are all positive constants and \(\alpha_{i}>K_{i}\) (\(i=1,2\));

and

\((H_{2})\) :

\(r_{i}\alpha_{i}\leq1\) (\(i=1,2\))

hold, then the above system admits a unique positive equilibrium \((x_{1}^{*},x_{2}^{*})\), which is globally asymptotically stable.

Recently, Chen [36] argued that it is interesting to investigate the dynamic behaviors of the above system under the assumption \(K_{i}>\alpha_{i}\). He showed that if \(r_{i}K_{i}\leq1\), \(i=1, 2\), the system could also admit a unique globally attractive positive equilibrium.

In [35], Chen further proposed the following discrete competition model:

$$\begin{aligned}& x_{1}(k+1) = x_{1}(k)\exp \biggl\{ r_{1}(k) \biggl[ \frac{K_{1}(k) +\alpha_{1}(k)\sum_{s=0}^{+\infty}J_{2}(s)x_{2}(k-s)}{1+\sum_{s=0}^{+\infty}J_{2}(s)x_{2}(k-s)} -x_{1} \bigl(k-\delta_{1}(k) \bigr) \biggr] \biggr\} , \\& x_{2}(k+1) = x_{2}(k)\exp \biggl\{ r_{2}(k) \biggl[ \frac{K_{2}(k) +\alpha_{2}(k)\sum_{s=0}^{+\infty}J_{1}(s)x_{1}(k-s)}{1+\sum_{s=0}^{+\infty}J_{1}(s)x_{1}(k-s)} -x_{2} \bigl(k-\delta_{2}(k) \bigr) \biggr] \biggr\} , \end{aligned}$$

where \(r_{i}\), \(K_{i}\), \(\alpha_{i}\), \(\tau_{i}\) and \(\delta_{i}\), \(i=1,2\), are all nonnegative sequences bounded above and below by positive constants, and \(K_{i}>\alpha_{i}\), \(i=1,2\). Sufficient conditions are obtained for the permanence of the above system.

The success of [35, 36] motivated us to consider the continuous case. Xie et al. [14] have already investigated the stability property of the following two species mutualism model:

$$ \begin{gathered} \frac{dN_{1}(t)}{dt} = r_{1}N_{1}(t) \biggl[\frac{K_{1}+\alpha_{1} \int_{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds}{ 1+\int_{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds}-N_{1}(t) \biggr], \\ \frac{dN_{2}(t)}{dt} = r_{2}N_{2}(t) \biggl[ \frac{K_{2}+\alpha_{2} \int_{0}^{\infty}J_{1}(s)N_{1}(t-s)\,ds}{ 1+\int_{0}^{\infty}J_{1}(s)N_{1}(t-s)\,ds}-N_{2}(t) \biggr], \end{gathered} $$
(1.1)

where \(r_{i}\), \(K_{i}\) and \(\alpha_{i}\), \(i=1,2\), are all continuous positive constants, \(J_{i}\in C([0,+\infty), [0,+\infty))\) and \(\int_{0}^{\infty}J_{i}(s)\,ds=1\), \(i=1,2\). Under the assumption \(\alpha _{i}>K_{i}\), \(i=1,2\), by using the iterative method, Xie et al. [14] showed that the system admits a unique globally attractive positive equilibrium. For more background of system (1.1), one could refer to [1, 14, 20] and the references cited therein.

As far as system (1.1) is concerned, one interesting issue is proposed: What would happen if \(\alpha_{i}< K_{i}\), \(i=1, 2\)? Is it possible that the system admits dynamic behaviors similar to those of the case \(\alpha_{i}>K_{i}\), \(i=1, 2\)?

Note that under the assumption \(\alpha_{i}< K_{i}\), \(i=1, 2\), the first equation in system (1.1) can be rewritten as follows:

$$ \begin{aligned}[b] \frac{dN_{1}(t)}{dt}&=r_{1}N_{1}(t) \biggl[\frac{K_{1}+\alpha_{1} \int_{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds}{ 1+\int_{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds}-N_{1}(t) \biggr] \\ &=r_{1}N_{1}(t) \biggl[\frac{K_{1} (1+ \int_{0}^{\infty }J_{2}(s)N_{2}(t-s)\,ds )}{ 1+\int_{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds}-N_{1}(t) \\ &\quad{} -\frac{ (K_{1}-\alpha_{1})\int_{0}^{\infty }J_{2}(s)N_{2}(t-s)\,ds )}{ 1+\int_{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds} \biggr] \\ &= r_{1}N_{1}(t) \biggl[K_{1}-N_{1}(t)- \frac{ (K_{1}-\alpha_{1})\int _{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds }{ 1+\int_{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds} \biggr]. \end{aligned} $$
(1.2)

Similarly, the second equation in system (1.1) can be rewritten as follows:

$$ \frac{dN_{2}(t)}{dt} = r_{2}N_{2}(t) \biggl[K_{2}-N_{2}(t)-\frac{ (K_{2}-\alpha_{2})\int _{0}^{\infty}J_{1}(s)N_{1}(t-s)\,ds }{ 1+\int_{0}^{\infty}J_{1}(s)N_{1}(t-s)\,ds} \biggr]. $$
(1.3)

From (1.2) and (1.3) one could easily see that the first species has a negative effect on the second species, and the second species has a negative effect on the first species. Therefore, under the assumption \(\alpha_{i}< K_{i}\), \(i=1, 2\), the relationship between the two species is competition.

From the point of view of biology, in the sequel, we shall consider (1.2)-(1.3) together with the initial conditions

$$ N_{i}(s)=\phi_{i}(s), \quad s\in(-\infty,0], i=1,2, $$
(1.4)

where \(\phi_{i}\in \mathit{BC}^{+}\) and

$$\mathit{BC}^{+}=\bigl\{ \phi\in C\bigl((-\infty,0], [0,+\infty)\bigr): \phi(0)>0 \text{ and } \phi \text{ is bounded}\bigr\} , \quad i=1,2. $$

One could easily prove that \(N_{i}(t)>0\) for all \(i=1,2\) in a maximal interval of the existence of solution.

The aim of this paper is to give an affirmative answer to the above issue. More precisely, we will prove the following result.

Theorem 1.1

Under the assumption \(\alpha_{i}< K_{i}\), \(i=1, 2\), system (1.1) with the initial conditions (1.4) admits a unique positive equilibrium \((N_{1}^{*}, N_{2}^{*})\), which is globally attractive, that is, for any positive solution \((N_{1}(t), N_{2}(t))\) of system (1.1) with the initial conditions (1.4), one has

$$\lim _{t\rightarrow+\infty}N_{i}(t)=N_{i}^{*}, \quad i=1,2. $$

2 Proof of the main result

Now let us state several lemmas which will be useful in proving the main result.

Lemma 2.1

System (1.1) admits a unique positive equilibrium \((N_{1}^{*}, N_{2}^{*})\).

Proof

The positive equilibrium of system (1.1) satisfies the following equation:

$$ \begin{gathered} \frac{K_{1}+\alpha_{1}N_{2}}{1+N_{2}}-N_{1}=0, \\ \frac{K_{2}+\alpha_{2}N_{1}}{1+N_{1}}-N_{2}=0, \end{gathered} $$
(2.1)

which is equivalent to

$$ \begin{gathered} A_{1}N_{1}^{2}+A_{2}N_{1}+A_{3}=0, \\ B_{1}N_{2}^{2}+B_{2}N_{2}+B_{3}=0, \end{gathered} $$
(2.2)

where

$$\begin{gathered} A_{1}=1+\alpha_{2},\quad\quad A_{2}=K_{2}-K_{1}-\alpha_{2} \alpha_{1}+1, \quad\quad A_{3}=-\alpha_{1}K_{2}-K_{1}; \\ B_{1}=1+\alpha_{1}, \quad\quad B_{2}=K_{1}-K_{2}- \alpha_{1}\alpha_{2}+1, \quad\quad B_{3}=-K_{2}- \alpha_{2}K_{1}. \end{gathered} $$

Noting that \(A_{1}>0\), \(A_{3}<0\), \(B_{1}>0\), \(B_{3}<0\), it immediately follows that system (2.1) admits a unique positive solution \((N_{1}^{*}, N_{2}^{*})\), where

$$N_{1}^{*}=\frac{-A_{2}+\sqrt{A_{2}^{2}-4A_{1}A_{3}}}{2A_{1}}, \quad\quad N_{2}^{*}= \frac{-B_{2}+\sqrt{B_{2}-4B_{1}B_{3}}}{2B_{1}}. $$

This ends the proof of Lemma 2.1. □

Lemma 2.2

[27]

Let \(x:R\rightarrow R\) be a bounded nonnegative continuous function, and let \(k:[0,+\infty)\rightarrow(0,+\infty)\) be a continuous kernel such that \(\int_{0}^{\infty}k(s)\,ds=1\). Then

$$\begin{aligned} \liminf _{t\rightarrow+\infty}x(t) \leq& \liminf _{t\rightarrow +\infty} \int_{-\infty}^{t}k(t-s)x(s)\,ds \\ \leq&\limsup _{t\rightarrow +\infty} \int_{-\infty}^{t}k(t-s)x(s)\,ds\leq \limsup _{t\rightarrow+\infty}x(t). \end{aligned}$$

Lemma 2.3

[7]

If \(a>0\), \(b>0\) and \(\dot{x}\geq x(b-ax)\), when \(t\geq{0}\) and \(x(0)>0\), we have

$$\liminf _{t\rightarrow+\infty} x(t)\geq\frac{b}{a}. $$

If \(a>0\), \(b>0\) and \(\dot{x}\leq x(b-ax)\), when \(t\geq{0}\) and \(x(0)>0\), we have

$$\limsup _{t\rightarrow+\infty} x(t)\leq\frac{b}{a}. $$

Now we are in a position to prove the main result of this paper.

Proof of Theorem 1.1

Let \((N_{1}(t), N_{2}(t))\) be any positive solution of system (1.1) with initial conditions (1.4). From (1.2) it follows that

$$ \frac{dN_{1}(t)}{dt}\leq r_{1}N_{1}(t) \bigl(K_{1}-N_{1}(t) \bigr). $$
(2.3)

Thus, as a direct corollary of Lemma 2.3, according to (2.3), one has

$$ \limsup _{t\rightarrow+\infty}N_{1}(t)\leq K_{1}, $$
(2.4)

and so, from Lemma 2.2 we have

$$ \limsup _{t\rightarrow+\infty} \int_{-\infty }^{t}J_{1}(t-s)N_{1}(s)\,ds \leq K_{1}. $$
(2.5)

Hence, for \(\varepsilon>0\) small enough, it follows from (2.4) and (2.5) that there exists \(T_{11}>0\) such that for all \(t\geq T_{11}\),

$$ \begin{gathered} N_{1}(t)< K_{1} +\varepsilon \stackrel{\mathrm{def}}{=}M_{1}^{(1)}, \\ \int_{0}^{\infty}J_{1}(s)N_{1}(t-s)\,ds = \int_{-\infty }^{t}J_{1}(t-s)N_{1}(s)\,ds< K_{1} +\varepsilon\stackrel{\mathrm{def}}{=}M_{1}^{(1)}. \end{gathered} $$
(2.6)

Similarly, for above \(\varepsilon>0\), it follows from (1.3) that

$$\begin{gathered} \limsup _{t\rightarrow+\infty}N_{2}(t)\leq K_{2}, \\ \limsup _{t\rightarrow+\infty} \int_{-\infty }^{t}J_{2}(t-s)N_{2}(s)\,ds \leq K_{2}, \end{gathered} $$

and so, there exists \(T_{12}>T_{11} \) such that for all \(t\geq T_{12}\),

$$ \begin{gathered} N_{2}(t)< K_{2}+\varepsilon \stackrel{\mathrm{def}}{=}M_{2}^{(1)}, \\ \int_{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds= \int_{-\infty }^{t}J_{2}(t-s)N_{2}(s)\,ds< K_{2} +\varepsilon\stackrel{\mathrm{def}}{=}M_{2}^{(1)}. \end{gathered} $$
(2.7)

Also, from (1.2) we have

$$ \begin{aligned}[b] \frac{dN_{1}(t)}{dt}& = r_{1}N_{1}(t) \biggl[K_{1}-N_{1}(t)- \frac{ (K_{1}-\alpha_{1})\int _{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds }{ 1+\int_{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds} \biggr] \\ & \geq r_{1}N_{1}(t) \biggl[K_{1}-N_{1}(t)- \frac{ (K_{1}-\alpha_{1}) (1+\int_{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds ) }{ 1+\int_{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds} \biggr] \\ & = r_{1}N_{1}(t) \bigl[K_{1}-N_{1}(t)-(K_{1}- \alpha_{1}) \bigr] \\ & = r_{1}N_{1}(t) \bigl[\alpha_{1}-N_{1}(t) \bigr]. \end{aligned} $$
(2.8)

Thus, as a direct corollary of Lemma 2.3, according to (2.8), one has

$$ \liminf _{t\rightarrow+\infty}N_{1}(t)\geq \alpha_{1}, $$
(2.9)

and so, from Lemma 2.2 we have

$$ \liminf _{t\rightarrow+\infty} \int_{-\infty }^{t}J_{1}(t-s)N_{1}(s)\,ds \geq \alpha_{1}. $$
(2.10)

Hence, for \(\varepsilon>0 \) small enough, without loss of generality, we may assume that \(\varepsilon<\frac{1}{2}\min\{\alpha_{1},\alpha _{2}\}\). It follows from (2.9) and (2.10) that there exists \(T_{13}>T_{12}\) such that for all \(t\geq T_{13}\),

$$ \begin{gathered} N_{1}(t)>\alpha_{1} - \varepsilon\stackrel{\mathrm{def}}{=}m_{1}^{(1)}, \\ \int_{0}^{\infty}J_{1}(s)N_{1}(t-s)\,ds = \int_{-\infty }^{t}J_{1}(t-s)N_{1}(s)\,ds> \alpha_{1} -\varepsilon\stackrel{\mathrm {def}}{=}m_{1}^{(1)}. \end{gathered} $$
(2.11)

Similarly, for above \(\varepsilon>0\), it follows from (1.3) that

$$\begin{gathered} \liminf _{t\rightarrow+\infty}N_{2}(t)\geq \alpha_{2}, \\ \liminf _{t\rightarrow+\infty} \int_{-\infty }^{t}J_{2}(t-s)N_{2}(s)\,ds \geq \alpha_{2}, \end{gathered} $$

and so, there exists \(T_{14}>T_{13} \) such that for all \(t\geq T_{14}\),

$$ \begin{gathered} N_{2}(t)>\alpha_{2}-\varepsilon \stackrel{\mathrm{def}}{=}m_{2}^{(1)}, \\ \int_{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds= \int_{-\infty }^{t}J_{2}(t-s)N_{2}(s)\,ds> \alpha_{2} -\varepsilon\stackrel{\mathrm {def}}{=}m_{2}^{(1)}. \end{gathered} $$
(2.12)

It follows from (2.6), (2.7), (2.11) and (2.12) that for all \(t\geq T_{14}\),

$$ 0< m_{1}^{(1)}< x(t)< M_{1}^{(1)},\quad\quad 0< m_{2}^{(1)} < y(t)< M_{2}^{(1)}. $$
(2.13)

Note that the function \(g(x)=\frac{ x}{1+x}\) (\(x\geq0\)) is a strictly increasing function, hence (2.12) together with (1.2) implies

$$ \frac{dN_{1}(t)}{dt} < r_{1}N_{1}(t) \biggl[ K_{1} - N_{1}(t)-\frac {(K_{1}-\alpha_{1})m_{2}^{(1)}}{1+m_{2}^{(1)}} \biggr] \quad \text{for } t>T_{14}. $$
(2.14)

Therefore, by Lemma 2.3, we have

$$ \limsup _{t\rightarrow+\infty}N_{1}(t)\leq K_{1}- \frac {(K_{1}-\alpha _{1})m_{2}^{(1)}}{1+m_{2}^{(1)}}= \frac{K_{1}+\alpha_{1}m_{2}^{(1)}}{ 1+m_{2}^{(1)}}. $$
(2.15)

From Lemma 2.2 and (2.15) we have

$$ \limsup _{t\rightarrow+\infty} \int_{-\infty }^{t}J_{1}(t-s)N_{1}(s)\,ds \leq\frac{K_{1}+\alpha_{1}m_{2}^{(1)}}{ 1+m_{2}^{(1)}}. $$
(2.16)

That is, there exists \(T_{21}>T_{14} \) such that for all \(t\geq T_{21}\),

$$ \begin{gathered} N_{1}(t)< \frac{K_{1}+\alpha_{1}m_{2}^{(1)}}{ 1+m_{2}^{(1)}}+ \frac{\varepsilon}{2}\stackrel{\mathrm {def}}{=}M_{1}^{(2)}, \\ \int_{0}^{\infty}J_{1}(s)N_{1}(t-s)\,ds= \int_{-\infty }^{t}J_{1}(t-s)N_{1}(s)\,ds < \frac{K_{1}+\alpha_{1}m_{2}^{(1)}}{ 1+m_{2}^{(1)}}+\frac{\varepsilon}{2}\stackrel{\mathrm{def}}{=}M_{1}^{(2)}. \end{gathered} $$
(2.17)

Similar to the analysis of (2.14)-(2.17), from (2.11) and (1.3), there exists \(T_{22}>T_{21}\) such that for all \(t\geq T_{22}\),

$$ \begin{gathered} N_{2}(t)< \frac{K_{2}+\alpha_{2}m_{1}^{(1)}}{ 1+m_{1}^{(1)}}+ \frac{\varepsilon}{2}\stackrel{\mathrm {def}}{=}M_{2}^{(2)}, \\ \int_{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds < \frac{K_{2}+\alpha_{2}m_{1}^{(1)}}{ 1+m_{1}^{(1)}}+\frac{\varepsilon}{2}\stackrel{\mathrm{def}}{=}M_{2}^{(2)}. \end{gathered} $$
(2.18)

It follows from (2.6), (2.7), (2.17) and (2.18) that

$$ M_{1}^{(2)}< M_{1}^{(1)},\quad\quad M_{2}^{(2)}< M_{2}^{(1)}. $$
(2.19)

Again from the strictly increasing function \(g(x)=\frac{ x}{1+x}\) (\(x\geq0\)) and (1.2), it follows that

$$ \frac{dN_{1}(t)}{dt} > r_{1}N_{1}(t) \biggl[ K_{1} - N_{1}(t)-\frac {(K_{1}-\alpha_{1})M_{2}^{(1)}}{1+M_{2}^{(1)}} \biggr] \quad \text{for } t>T_{22}. $$
(2.20)

Therefore, by Lemma 2.3, we have

$$ \liminf _{t\rightarrow+\infty}N_{1}(t)\geq K_{1}- \frac {(K_{1}-\alpha _{1})M_{2}^{(1)}}{1+M_{2}^{(1)}}= \frac{K_{1}+\alpha_{1}M_{2}^{(1)}}{ 1+M_{2}^{(1)}}. $$
(2.21)

Thus, from Lemma 2.2 we have

$$ \liminf _{t\rightarrow+\infty} \int_{-\infty }^{t}J_{1}(t-s)N_{1}(s)\,ds \geq\frac{K_{1}+\alpha_{1}M_{2}^{(1)}}{ 1+M_{2}^{(1)}}. $$
(2.22)

That is, there exists \(T_{23}>T_{22} \) such that for all \(t\geq T_{23}\),

$$ \begin{gathered} N_{1}(t)>\frac{K_{1}+\alpha_{1}M_{2}^{(1)}}{ 1+M_{2}^{(1)}}- \frac{\varepsilon}{2}\stackrel{\mathrm {def}}{=}m_{1}^{(2)}, \\ \int_{0}^{\infty}J_{1}(s)N_{1}(t-s)\,ds= \int_{-\infty }^{t}J_{1}(t-s)N_{1}(s)\,ds >\frac{K_{1}+\alpha_{1}M_{2}^{(1)}}{ 1+M_{2}^{(1)}}-\frac{\varepsilon}{2}\stackrel{\mathrm{def}}{=}m_{1}^{(2)}. \end{gathered} $$
(2.23)

Similar to the analysis of (2.20)-(2.23), from (2.13) and (1.3), there exists \(T_{24}>T_{23}\) such that for all \(t\geq T_{24}\),

$$ \begin{gathered} N_{2}(t)>\frac{K_{2}+\alpha_{2}M_{1}^{(1)}}{ 1+M_{1}^{(1)}}- \frac{\varepsilon}{2}\stackrel{\mathrm {def}}{=}m_{2}^{(2)}, \\ \int_{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds >\frac{K_{2}+\alpha_{2}M_{1}^{(1)}}{ 1+M_{1}^{(1)}}-\frac{\varepsilon}{2}\stackrel{\mathrm{def}}{=}m_{2}^{(2)}. \end{gathered} $$
(2.24)

Obviously,

$$K_{1}-\alpha_{1}>(K_{1}-\alpha_{1}) \frac{M_{2}^{(1)}}{1+M_{2}^{(1)}}; $$

consequently,

$$K_{1}-(K_{1}-\alpha_{1})< K_{1}-(K_{1}- \alpha_{1})\frac{M_{2}^{(1)}}{1+M_{2}^{(1)}}, $$

and so, it follows from (2.11) and (2.23) that

$$ m_{1}^{(2)}>m_{1}^{(1)}. $$
(2.25)

Similarly, it follows from (2.12) and (2.24) that

$$ m_{2}^{(2)}>m_{2}^{(1)}. $$
(2.26)

Repeating the above procedure, we get four sequences \(M_{i}^{(n)}\), \(m_{i}^{(n)}\), \(i=1,2\), \(n=1,2,\ldots \) , such that for \(n\geq2\)

$$ \begin{gathered} M_{1}^{(n)}= \frac{K_{1}+\alpha_{1}m_{2}^{(n-1)}}{ 1+m_{2}^{(n-1)}}+ \frac{\varepsilon}{n}; \quad\quad M_{2}^{(n)}=\frac{K_{2}+\alpha_{2}m_{1}^{(n-1)}}{ 1+m_{1}^{(n-1)}}+ \frac{\varepsilon}{n}; \\ m_{1}^{(n)}=\frac{K_{1}+\alpha_{1}M_{2}^{(n-1)}}{ 1+M_{2}^{(n-1)}} -\frac{\varepsilon}{n};\quad\quad m_{2}^{(n)}=\frac{K_{2}+\alpha_{2}M_{1}^{(n-1)}}{ 1+M_{1}^{(n-1)}}-\frac{\varepsilon}{n}. \end{gathered} $$
(2.27)

Obviously,

$$m_{i}^{(n)}< N_{i}(t)< M_{i}^{(n)}\quad \text{for } t\geq T_{n4}, i=1,2. $$

We claim that sequences \(M_{i}^{(n)}\), \(i=1,2\), are nonincreasing, and sequences \(m_{i}^{(n)}\), \(i=1,2\), are nondecreasing. To prove this claim, we will carry on by induction. Firstly, from (2.25), (2.26) and (2.19), we have

$$M_{i}^{(2)}< M_{i}^{(1)},\quad\quad m_{i}^{(2)}> m_{i}^{(1)}, \quad i=1,2. $$

Let us assume now that our claim is true for n, that is,

$$M_{i}^{(n)}< M_{i}^{(n-1)},\quad\quad m_{i}^{(n)}> m_{i}^{(n-1)}, \quad i=1,2. $$

Let us consider the function \(g_{i}(x)=\frac{K_{i}+\alpha_{i} x}{1+x}\) (\(K_{i}>\alpha_{i}\), \(i=1, 2\)) since

$$g_{i}^{'}(x)=-\frac{K_{i}-\alpha_{i}}{(1+x)^{2}}< 0, $$

then \(g_{i}(x)\), \(i=1, 2\), is a strictly decreasing function of x. From the monotonic property of \(g_{i}(x)\), \(i=1, 2\), it immediately follows that

$$\begin{aligned}& M_{1}^{(n+1)}= \frac{K_{1}+\alpha_{1}m_{2}^{(n)}}{ 1+m_{2}^{(n)}}+ \frac{\varepsilon}{n+1} < \frac{K_{1}+\alpha_{1}m_{2}^{(n-1)}}{ 1+m_{2}^{(n-1)}} +\frac{\varepsilon}{n}=M_{1}^{(n)}; \\& M_{2}^{(n+1)}=\frac{K_{2}+\alpha_{2}m_{1}^{(n)}}{ 1+m_{1}^{(n)}}+\frac{\varepsilon}{n+1} < \frac{K_{2}+\alpha_{2}m_{1}^{(n-1)}}{ 1+m_{1}^{(n-1)}}+\frac{\varepsilon}{n}=M_{2}^{(n)}; \\& m_{1}^{(n+1)}=\frac{K_{1}+\alpha_{1}M_{2}^{(n)}}{ 1+M_{2}^{(n)}} - \frac{\varepsilon}{n+1} >\frac{K_{1}+\alpha_{1}M_{2}^{(n-1)}}{ 1+M_{2}^{(n-1)}} -\frac{\varepsilon}{n}=m_{1}^{(n)}; \\& m_{2}^{(n+1)}=\frac{K_{2}+\alpha_{2}M_{1}^{(n)}}{ 1+M_{1}^{(n)}}-\frac{\varepsilon}{n+1} >\frac{K_{2}+\alpha_{2}M_{1}^{(n-1)}}{ 1+M_{1}^{(n-1)}} -\frac{\varepsilon}{n}=m_{2}^{(n)}. \end{aligned}$$

Therefore,

$$\lim _{t\rightarrow+\infty}M_{i}^{(n)}=\overline{N}_{i},\quad\quad \lim _{t\rightarrow+\infty}m_{i}^{(n)}=\underline{N}_{i},\quad i=1,2. $$

Letting \(n\rightarrow+\infty\) in (2.27), we obtain

$$ \begin{gathered} \overline{N}_{1}= \frac{K_{1}+\alpha_{1}\underline{N}_{2}}{ 1+\underline{N}_{2}}; \quad\quad \overline{N}_{2}=\frac{K_{2}+\alpha_{2}\underline{N}_{1}}{ 1+\underline{N}_{1}}; \\ \underline{N}_{1}= \frac{K_{1}+\alpha_{1}\overline{N}_{2}}{ 1+\overline{N}_{2}}; \quad\quad \underline{N}_{2}=\frac{K_{2}+\alpha_{2}\overline{N}_{1}}{ 1+\overline{N}_{1}}; \end{gathered} $$
(2.28)

(2.28) shows that \((\overline{N}_{1},\underline{N}_{2})\) and \((\underline{N}_{1},\overline{N}_{2})\) are solutions of (2.1). By Lemma 2.1, (2.1) has a unique positive solution \(E^{*}(N_{1}^{*},N_{2}^{*})\). Hence, we conclude that

$$\overline{N}_{i}=\underline{N}_{i}=N_{i}^{*},\quad i=1,2, $$

that is,

$$\lim_{t\rightarrow+\infty}N_{i}(t)=N_{i}^{*}, \quad i=1,2. $$

Thus, the unique interior equilibrium \(E^{*}(N_{1}^{*},N_{2}^{*})\) is globally attractive. This completes the proof of Theorem 1.1. □

3 Numerical simulations

In this section we will give several examples to show the feasibility of Theorem 1.1. Firstly, let us consider the weakly integral kernel case.

Example 3.1

$$ \begin{gathered} \frac{dx(t)}{dt} = x(t) \biggl[ \frac{2+\int_{0}^{\infty }e^{-s}y(t-s)\,ds}{1+\int_{0}^{\infty}e^{-s}y(t-s)\,ds}-x(t) \biggr], \\ \frac{dy(t)}{dt} = y(t) \biggl[\frac{2+\int_{0}^{\infty }e^{-s}x(t-s)\,ds}{1+\int_{0}^{\infty}e^{-s}x(t-s)\,ds}-y(t) \biggr]. \end{gathered} $$
(3.1)

Corresponding to system (1.1), one has

$$r_{1} =r_{2}=\alpha_{1}=\alpha_{2}=1,\quad\quad K_{1}=K_{2}=2. $$

By a simple computation, system (3.1) admits a unique positive equilibrium \((\sqrt{2}, \sqrt{2})\). It follows from Theorem 1.1 that \((\sqrt{2}, \sqrt{2})\) is globally attractive. Figures 1 and 2 also support this assertion.

Figure 1
figure 1

Dynamic behavior of the first species in system ( 3.1 ) with the initial conditions \(\pmb{(x(s), y(s))=(0.5,0.5)}\) , \(\pmb{(0.1, 1)}\) , \(\pmb{(2, 2)}\) , \(\pmb{(2.5,2.5)}\) and \(\pmb{(3, 3)}\) , \(\pmb{s\in(-\infty,0]}\) , respectively.

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Figure 2
figure 2

Dynamic behavior of the second species in system ( 3.1 ) with the initial conditions \(\pmb{(x(s), y(s))=(0.5,0.5)}\) , \(\pmb{(0.1, 1)}\) , \(\pmb{(2, 2)}\) , \(\pmb{(2.5,2.5)}\) and \(\pmb{(3, 3)}\) , \(\pmb{s\in(-\infty,0]}\) , respectively.

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Example 3.2

$$ \begin{gathered} \frac{dx(t)}{dt} = x(t) \biggl[ \frac{3+3\int_{0}^{\infty }e^{-3s}y(t-s)\,ds}{1+\int_{0}^{\infty}e^{-s}y(t-s)\,ds}-x(t) \biggr], \\ \frac{dy(t)}{dt} = y(t) \biggl[\frac{3+3\int_{0}^{\infty }e^{-3s}x(t-s)\,ds}{1+\int_{0}^{\infty}e^{-s}x(t-s)\,ds}-y(t) \biggr]. \end{gathered} $$
(3.2)

Corresponding to system (1.1), one has

$$r_{1} =r_{2}=\alpha_{1}=\alpha_{2}=1,\quad\quad K_{1}=K_{2}=3. $$

By a simple computation, system (3.2) admits a unique positive equilibrium \((\sqrt{3}, \sqrt{3})\). It follows from Theorem 1.1 that \((\sqrt{3}, \sqrt{3})\) is globally attractive. Figures 3 and 4 also support this assertion.

Figure 3
figure 3

Dynamic behavior of the first species in system ( 3.2 ) with the initial conditions \(\pmb{(x(s), y(s))=(0.5,0.5)}\) , \(\pmb{(0.1, 1)}\) , \(\pmb{(2, 2)}\) , \(\pmb{(2.5,2.5)}\) and \(\pmb{(3, 3)}\) , \(\pmb{s\in(-\infty,0]}\) , respectively.

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Figure 4
figure 4

Dynamic behavior of the second species in system ( 3.2 ) with the initial conditions \(\pmb{(x(s), y(s))=(0.5,0.5)}\) , \(\pmb{(0.1, 1)}\) , \(\pmb{(2, 2)}\) , \(\pmb{(2.5,2.5)}\) and \(\pmb{(3, 3)}\) , \(\pmb{s\in(-\infty,0]}\) , respectively.

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Now let us consider the strong integral kernel case.

Example 3.3

$$ \begin{gathered} \frac{dx(t)}{dt} = x(t) \biggl[ \frac{4+\int_{0}^{\infty }se^{-s}y(t-s)\,ds}{1+\int_{0}^{\infty}e^{-s}y(t-s)\,ds}-x(t) \biggr], \\ \frac{dy(t)}{dt} = y(t) \biggl[\frac{4+\int_{0}^{\infty }se^{-s}x(t-s)\,ds}{1+\int_{0}^{\infty}e^{-s}x(t-s)\,ds}-y(t) \biggr]. \end{gathered} $$
(3.3)

Corresponding to system (1.1), one has

$$r_{1} =r_{2}=\alpha_{1}=\alpha_{2}=1,\quad\quad K_{1}=K_{2}=4. $$

By a simple computation, system (3.3) admits a unique positive equilibrium \((2, 2)\). It follows from Theorem 1.1 that \((2, 2)\) is globally attractive. Figures 5 and 6 also support this assertion.

Figure 5
figure 5

Dynamic behavior of the first species in system ( 3.3 ) with the initial conditions \(\pmb{(x(s), y(s))=(0.5,0.5)}\) , \(\pmb{(0.1, 1)}\) , \(\pmb{(2, 2)}\) , \(\pmb{(2.5,2.5)}\) and \(\pmb{(3, 3)}\) , \(\pmb{s\in(-\infty,0]}\) , respectively.

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Figure 6
figure 6

Dynamic behavior of the second species in system ( 3.3 ) with the initial conditions \(\pmb{(x(s), y(s))=(0.5,0.5)}\) , \(\pmb{(0.1, 1)}\) , \(\pmb{(2, 2)}\) , \(\pmb{(2.5,2.5)}\) and \(\pmb{(3, 3)}\) , \(\pmb{s\in(-\infty,0]}\) , respectively.

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4 Discussion

Xie et al. [14] studied the stability property of the integro-differential model of mutualism. Under the assumption \(\alpha_{i}>K_{i}\), \(i=1, 2\), by using the iterative technique, they showed that the system admits a unique globally attractive positive equilibrium. In this paper, we focus our attention on the case \(\alpha_{i}< K_{i}\), \(i=1, 2\). We first show that this case represents a population system of competition type. Then, by applying the iterative technique, we also show that the system admits a unique globally attractive positive equilibrium.

It is well known that the competitive exclusion principle is the most important rule for a population system. It says that two species competing for the same resource cannot coexist at constant population values. However, as we can see from Theorem 1.1, for all of the parameters which satisfy \(K_{i}>\alpha_{i}\), \(i=1, 2\), two species could coexist in a stable state. Also, numeric simulations (Figures 1-6) support this assertion. Why did this phenomenon happen? Maybe the reason relies on the term

$$\frac{ (K_{1}-\alpha_{1})\int_{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds }{ 1+\int_{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds} \quad \text{and}\quad \frac{ (K_{2}-\alpha_{2})\int_{0}^{\infty}J_{1}(s)N_{1}(t-s)\,ds }{ 1+\int_{0}^{\infty}J_{1}(s)N_{1}(t-s)\,ds}, $$

since \(K_{1}>K_{1}-\alpha_{1}\), \(K_{2}>K_{2}-\alpha_{2}\) and

$$\frac{\int_{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds }{ 1+\int_{0}^{\infty}J_{2}(s)N_{2}(t-s)\,ds}< 1, \quad\quad \frac{\int _{0}^{\infty}J_{1}(s)N_{1}(t-s)\,ds }{ 1+\int_{0}^{\infty}J_{1}(s)N_{1}(t-s)\,ds}< 1. $$

One could see that the influence of interspecific competition is less than the influence of intrinsic competition, and this leads to stable coexistence of the two species.