Let (*x*_{1}(*t*), *x*_{2}(*t*), *x*_{3}(*t*)) be any positive solution of the system (1.2). From the second and the third equation of system (1.1), we have

$$\begin{array}{}{\displaystyle \frac{d{x}_{2}}{d\tau}={x}_{3}-{x}_{2},}\\ {\displaystyle \frac{d{x}_{3}}{d\tau}\ge -c{x}_{3}+e{x}_{2}-f{x}_{3}^{2},}\end{array}$$(2.6)

Now let’s consider the system

$$\begin{array}{}{\displaystyle \frac{d{u}_{2}}{d\tau}={u}_{3}-{u}_{2},}\\ {\displaystyle \frac{d{u}_{3}}{d\tau}=-c{u}_{3}+e{u}_{2}-f{u}_{3}^{2},}\end{array}$$(2.7)

Noting that condition (1.3) implies that *e* > *c*, and so, from Lemma 2.2, (2.7) admits a unique globally asymptotically stable positive equilibrium $\begin{array}{}E({\displaystyle \frac{e-c}{f},\frac{e-c}{f})}\end{array}$. That is, let (*u*_{1}(*t*), *u*_{2}(*t*)) be any positive solution of the system (2.7), one has

$$\begin{array}{}{\displaystyle \underset{t\to +\mathrm{\infty}}{lim}{u}_{2}(t)=\underset{t\to +\mathrm{\infty}}{lim}{u}_{3}(t)=\frac{e-c}{f}.}\end{array}$$(2.8)

Let (*x*_{1}(*t*), *x*_{2}(*t*), *x*_{3}(*t*)) be any positive solution of system (1.2) with initial condition (*x*_{1}(0), *x*_{2}(0), *x*_{3}(0)) = (*x*_{10}, *x*_{20}, *x*_{30}), and let (*u*_{1}(*t*), *u*_{2}(*t*)) be the positive solution of system (2.7) with the initial condition (*u*_{2}(0), *u*_{3}(0)) = (*x*_{20},*x*_{30}), it then follows from the differential inequality theory that

$$\begin{array}{}{\displaystyle {x}_{i}(t)\ge {u}_{i}(t)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{f}\mathrm{o}\mathrm{r}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{l}\mathrm{l}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}t\ge 0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i=1,2.}\end{array}$$(2.9)

The positivity of the solution of system (1.2), (2.8) and (2.9) lead to

$$\begin{array}{}{\displaystyle \underset{t\to +\mathrm{\infty}}{lim\u2006inf}{x}_{i}(t)\ge \underset{t\to +\mathrm{\infty}}{lim}{u}_{i}(t)=\frac{e-c}{f},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i=1,2.}\end{array}$$(2.10)

Condition (1.3) implies that for enough small positive constant *ε* > 0, the following inequality holds.

$$\begin{array}{}{\displaystyle af+bc-be+bf\epsilon <0,}\end{array}$$(2.11)

which is equivalent to

$$\begin{array}{}a-b({\displaystyle \frac{e-c}{f}-\epsilon )<0.}\end{array}$$(2.12)

For *ε* > 0 enough small, which satisfies (2.12), it then follows from (2.10) that there exists an enough large *T*_{1} > 0 such that

$$\begin{array}{}{x}_{i}(t)>{\displaystyle \frac{e-c}{f}-\epsilon ,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i=1,2,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{f}\mathrm{o}\mathrm{r}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{l}\mathrm{l}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}t>{T}_{1}.}\end{array}$$(2.13)

Hence, for *t* > *T*_{1}, from the first equation of system (1.2) and (2.13), we have

$$\begin{array}{}{\displaystyle \frac{d{x}_{1}}{d\tau}=a{x}_{1}-{x}_{1}^{2}-b{x}_{1}{x}_{3},}\\ \phantom{\rule{2em}{0ex}}\le a{x}_{1}-{x}_{1}^{2}-b{x}_{1}({\displaystyle \frac{e-c}{f}-\epsilon )}\\ \phantom{\rule{2em}{0ex}}\le (a-b({\displaystyle \frac{e-c}{f}-\epsilon )){x}_{1}.}\end{array}$$

Consequently,

$$\begin{array}{}{x}_{1}(t)\le {x}_{1}({T}_{1})\mathrm{exp}\{(a-b({\displaystyle \frac{e-c}{f}-\epsilon ))(t-{T}_{1})\}\to 0\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{s}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}t\to \mathrm{\infty}.}\end{array}$$(2.14)

That is,

$$\begin{array}{}\underset{t\to +\mathrm{\infty}}{lim}{x}_{1}(t)=0.\end{array}$$(2.15)

For *ε* > 0 enough small, it follows from (2.15) that there exists a *T*_{2} > *T*_{1} such that

$$\begin{array}{}{x}_{1}(t)<\epsilon \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{f}\mathrm{o}\mathrm{r}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{l}\mathrm{l}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}t\ge {T}_{2}.\end{array}$$(2.16)

For *t* > *T*_{2}, from the second and third equation of system (1.2) and (2.16), we have

$$\begin{array}{}{\displaystyle \frac{d{x}_{2}}{d\tau}={x}_{3}-{x}_{2},}\\ {\displaystyle \frac{d{x}_{3}}{d\tau}\le -c{x}_{3}+d\epsilon {x}_{3}+e{x}_{2}-f{x}_{3}^{2},}\end{array}$$(2.17)

Now let’s consider the system

$$\begin{array}{}{\displaystyle \frac{d{v}_{2}}{d\tau}={v}_{3}-{v}_{2},}\\ {\displaystyle \frac{d{v}_{3}}{d\tau}=-c{v}_{3}+d\epsilon {v}_{3}+e{v}_{2}-f{v}_{3}^{2},}\end{array}$$(2.18)

it follows from *e* > *c* and Lemma 2.1 that (2.18) admits a unique globally asymptotically stable positive equilibrium $\begin{array}{}{E}_{1}({\displaystyle \frac{e-c+d\epsilon}{f},\frac{e-c+d\epsilon}{f}),}\end{array}$ That is, let (*v*_{1}(*t*), *v*_{2}(*t*)) be any positive solution of the system (2.18), one has

$$\begin{array}{}\underset{t\to +\mathrm{\infty}}{lim}{v}_{1}(t)={\displaystyle \frac{e-c+d\epsilon}{f},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\underset{t\to +\mathrm{\infty}}{lim}{v}_{2}(t)=\frac{e-c+d\epsilon}{f}.}\end{array}$$(2.19)

Let (*x*_{1}(*t*), *x*_{2}(*t*), *x*_{3}(*t*)) be any positive solution of system (1.2) with initial condition (*x*_{1}(*T*_{2}), *x*_{2}(*T*_{2}), *x*_{3}(*T*_{2})) = (*x*_{10}, *x*_{20}, *x*_{30}), and let (*v*_{2}(*t*), *v*_{3}(*t*)) be the positive solution of system (2.18) with the initial condition (*v*_{2}(*T*_{2}), *v*_{3}(*T*_{2})) = (*x*_{20},*x*_{30}), it then follows from the differential inequality theory that

$$\begin{array}{}{x}_{i}(t)\le {v}_{i}(t)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{f}\mathrm{o}\mathrm{r}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{l}\mathrm{l}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}t\ge {T}_{2}.\end{array}$$(2.20)

The positivity of the solution of system (1.2), (2.19) and (2.20) lead to

$$\begin{array}{}\underset{t\to +\mathrm{\infty}}{lim\u2006sup}{x}_{i}(t)\le \underset{t\to +\mathrm{\infty}}{lim}{v}_{i}(t)={\displaystyle \frac{e-c+d\epsilon}{f}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i=1,2.}\end{array}$$(2.21)

(2.10) and (2.18) show that

$$\begin{array}{}{\displaystyle \frac{e-c}{f}\le \underset{t\to +\mathrm{\infty}}{lim\u2006inf}{x}_{i}(t)\le \underset{t\to +\mathrm{\infty}}{lim\u2006sup}{x}_{i}(t)\le \frac{e-c+d\epsilon}{f},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i=1,2.}\end{array}$$(2.22)

Since *ε* could be any enough small positive constant, now, letting *ε* → 0 in (2.22) leads to

$$\begin{array}{}\underset{t\to +\mathrm{\infty}}{lim}{x}_{i}(t)={\displaystyle \frac{e-c}{f},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i=1,2.}\end{array}$$(2.23)

(2.23) together with (2.15) shows that $\begin{array}{}{P}_{3}(0,{\displaystyle \frac{e-c}{f},\frac{e-c}{f})}\end{array}$ is globally asymptotically stable. This completes the proof of Theorem 2.1.

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