In the real world, from the principle of ecosystem balance and saving resources, we only need to control the prey under the economic threshold level, and not to eradicate the prey totally. Thus we focus on the permanence of system (1.1).
First, we give the definition of permanence.
Definition 4.1 System (1.1) is said to be permanent if there exist positive constants m and M such that each positive solution ({x}_{1}(t),{x}_{2}(t),{y}_{1}(t),{y}_{2}(t)) of system (1.1) satisfies m\le {x}_{i}(t), {y}_{i}(t)\le M, i=1,2, for t large enough.
Theorem 4.1 Assume that:
\begin{array}{rl}({\mathrm{A}}_{2})& \phantom{\rule{1em}{0ex}}r{e}^{{d}_{1}{\tau}_{1}}\frac{kq}{c}{d}_{2}{d}_{3}\frac{M}{\lambda}>0,\\ ({\mathrm{A}}_{3})& \phantom{\rule{1em}{0ex}}{d}_{5}\frac{\lambda k{m}_{2}^{\ast}}{c+{m}_{2}^{\ast}}>0,\\ ({\mathrm{A}}_{4})& \phantom{\rule{1em}{0ex}}{m}_{2}\frac{M}{\lambda}{e}^{{d}_{1}{\tau}_{1}}>0,\\ ({\mathrm{A}}_{5})& \phantom{\rule{1em}{0ex}}\frac{{m}_{2}}{c+{m}_{2}}{m}_{4}\frac{{M}^{2}}{\lambda c+M}{e}^{{d}_{4}{\tau}_{2}}>0,\end{array}
where M, {m}_{2}, {m}_{4}, q are defined in (2.4), (4.7), (4.10), (4.13), respectively, then system (1.1) is permanent.
Proof Firstly, we will prove that there exists a constant {m}_{2}>0 such that {x}_{2}(t)>{m}_{2} for t sufficiently large. The second equation of (1.1) is equivalent to the following equality:
\begin{array}{rcl}{\dot{x}}^{2}(t)& =& (r{e}^{{d}_{1}{\tau}_{1}}\frac{k{y}_{2}(t)}{c+{x}_{2}(t)}{d}_{2}{d}_{3}{x}_{2}(t)){x}_{2}(t)\\ r{e}^{{d}_{1}{\tau}_{1}}\frac{d}{dt}{\int}_{t{\tau}_{1}}^{t}{x}_{2}(s)\phantom{\rule{0.2em}{0ex}}ds.\end{array}
(4.1)
According to (4.1), we define
V(t)={x}_{2}(t)+r{e}^{{d}_{1}{\tau}_{1}}{\int}_{t{\tau}_{1}}^{t}{x}_{2}(s)\phantom{\rule{0.2em}{0ex}}ds.
Calculating the derivative of V(t), we obtain
\dot{V}(t)=(r{e}^{{d}_{1}{\tau}_{1}}\frac{k{y}_{2}(t)}{c+{x}_{2}(t)}{d}_{2}{d}_{3}{x}_{2}(t)){x}_{2}(t).
(4.2)
Applying Lemma 2.5, (4.2) can be rewritten as follows:
\dot{V}(t)>(r{e}^{{d}_{1}{\tau}_{1}}\frac{k}{c}{y}_{2}(t){d}_{2}{d}_{3}\frac{M}{\lambda}){x}_{2}(t).
(4.3)
By hypothesis (A_{2}), there is an arbitrary small positive {\epsilon}_{4} such that
r{e}^{{d}_{1}{\tau}_{1}}>\frac{k}{c}(q+{\epsilon}_{4})+{d}_{2}+{d}_{3}\frac{M}{\lambda},
(4.4)
where q=\frac{\mu}{1{e}^{({d}_{5}\frac{\lambda k{m}_{2}^{\ast}}{c+{m}_{2}^{\ast}})}}.
Let {m}_{2}^{\ast} be determined as follows:
\frac{c}{k}(r{e}^{{d}_{1}{\tau}_{1}}{d}_{2}{d}_{3}\frac{M}{\lambda})=\frac{\mu}{1{e}^{({d}_{5}\frac{\lambda k{m}_{2}^{\ast}}{c+{m}_{2}^{\ast}})}}.
Then, for any {t}_{4}>0, it is impossible that {x}_{2}(t)<{m}_{2}^{\ast} for all t>{t}_{4}. Suppose that the claim is invalid, then there is {t}_{4}>0 such that {x}_{2}(t)<{m}_{2}^{\ast} for all {t}_{4}>0. It follows from the fourth and the eight equations of system (1.1) that
\{\begin{array}{l}{\dot{y}}_{2}(t)<({d}_{5}\frac{\lambda k{m}_{2}^{\ast}}{c+{m}_{2}^{\ast}}){y}_{2}(t),\phantom{\rule{1em}{0ex}}t\ne nT,\\ {y}_{2}({t}^{+})={y}_{2}(t)+\mu ,\phantom{\rule{1em}{0ex}}t=nT\end{array}
(4.5)
for all t>{t}_{4}+{\tau}_{2}. Consider the following auxiliary impulsive system of (4.5):
\{\begin{array}{l}{\dot{z}}_{6}(t)={z}_{6}(t)({d}_{5}\frac{\lambda k{m}_{2}^{\ast}}{c+{m}_{2}^{\ast}}),\phantom{\rule{1em}{0ex}}t\ne nT,\\ {z}_{6}({t}^{+})={z}_{6}(t)+\mu ,\phantom{\rule{1em}{0ex}}t=nT.\end{array}
(4.6)
By using Lemma 2.3, the unique positive periodic solution of (4.6) is
{z}_{6}(t)=\frac{\mu {e}^{({d}_{5}\frac{\lambda k{m}_{2}^{\ast}}{c+{m}_{2}^{\ast}})(tnT)}}{1{e}^{({d}_{5}\frac{\lambda k{m}_{2}^{\ast}}{c+{m}_{2}^{\ast}})}},\phantom{\rule{1em}{0ex}}nT<t\le (n+1)T.
This is globally asymptotically stable by hypothesis (A_{3}). Taking into account the comparison theorem of an impulsive differential equation, there exists {t}_{5} (>{t}_{4}+{\tau}_{2}) such that
{y}_{2}(t)\le {z}_{6}(t)+{\epsilon}_{4}.
For t>{t}_{5}, we have
{z}_{6}(t)\le \frac{\mu}{1{e}^{({d}_{5}\frac{\lambda k{m}_{2}^{\ast}}{c+{m}_{2}^{\ast}})}}\triangleq q.
(4.7)
Then
{y}_{2}(t)\le q+{\epsilon}_{4}\triangleq \sigma ,\phantom{\rule{1em}{0ex}}t\ge {t}_{5}.
(4.8)
According to (4.4), we have
r{e}^{{d}_{1}{\tau}_{1}}>\frac{k\sigma}{c}+{d}_{2}+{d}_{3}\frac{M}{\lambda}.
By (4.3) and (4.8), we get
\dot{V}(t)>(r{e}^{{d}_{1}{\tau}_{1}}\frac{k\sigma}{c}{d}_{2}{d}_{3}\frac{M}{\lambda}){x}_{2}(t),\phantom{\rule{1em}{0ex}}t\ge {t}_{5}.
(4.9)
Let {x}_{2}^{m}={min}_{t\in [{t}_{1},{t}_{1}+\tau ]}{x}_{2}(t).
We will show that {x}_{2}(t)\ge {x}_{2}^{m} for all t\ge {t}_{5}. Otherwise, there exists a {T}_{0}>0 such that {x}_{2}(t)\ge {x}_{2}^{m} for {t}_{5}\le t\le {t}_{5}+\tau +{T}_{0}, {x}_{2}({t}_{5}+\tau +{T}_{0})\ge {x}_{2}^{m} and {\dot{x}}_{2}({t}_{5}+\tau +{T}_{0})<0. From the second equation of system (1.1) and (4.8), we have
{\dot{x}}_{2}({t}_{5}+\tau +{T}_{0})>(r{e}^{{d}_{1}{\tau}_{1}}\frac{k\sigma}{c}{d}_{2}{d}_{3}\frac{M}{\lambda}){x}_{2}^{m}>0.
This is a contradiction. Thus, we have {x}_{2}(t)\ge {x}_{2}^{m}, t\ge {t}_{5}.
By (4.4) and (4.9), we have
\dot{V}(t)>(r{e}^{{d}_{1}{\tau}_{1}}\frac{k\sigma}{c}{d}_{2}{d}_{3}\frac{M}{\lambda}){x}_{2}^{m},\phantom{\rule{1em}{0ex}}t\ge {t}_{5}.
This means that V(t)\to \mathrm{\infty} as t\to \mathrm{\infty}. It is a contradiction with V(t)\le \frac{M}{\lambda}(1+r{\tau}_{1}{e}^{{d}_{1}{\tau}_{1}}).
Therefore, for any {t}_{4}>0, the inequality {x}_{2}(t)<{m}_{2}^{\ast} cannot hold for all t>{t}_{4}. So there exist the following two possibilities.

(i)
If {x}_{2}(t)\ge {m}_{2}^{\ast} holds for all t large enough, then our goal is obtained.

(ii)
If {x}_{2}(t) is oscillatory about {m}_{2}^{\ast}. Setting
{m}_{2}=min\{\frac{{m}_{2}^{\ast}}{2},{m}_{2}^{\ast}{e}^{(kM+{d}_{2}+{d}_{3}{m}_{2}^{\ast}){\tau}_{1}}\},
(4.10)
we prove that {x}_{2}(t)\ge {m}_{2} for all t large enough. Suppose that there exist two positive constants γ, η such that {x}_{2}(\gamma )={x}_{2}(\gamma +\eta ) and {x}_{2}(t)<{m}_{2}^{\ast} for all \gamma <t<\gamma +\eta, where γ is large enough, and the inequality (4.8) holds true for \gamma <t<\gamma +\eta. Since {x}_{2}(t) is continuous, bounded, and is not affected by impulses, we conclude that {x}_{2}(t) is uniformly continuous. Hence, there exists a constant {T}_{1} (0<{T}_{1}<{\tau}_{1} and {T}_{1} is independent of the choice of γ) such that {x}_{2}(\gamma )>\frac{{m}_{2}^{\ast}}{2} for \gamma \le t\le \gamma +{T}_{1}. If \eta \le {T}_{1}, our aim is obtained. If {T}_{1}<\eta \le {\tau}_{1}, from the second equation of (1.1), we obtain, for \gamma <t<\gamma +\eta, {\dot{x}}_{2}(t)\ge \frac{k}{c}{x}_{2}(t){y}_{2}(t){d}_{2}{x}_{2}(t){d}_{3}{x}_{2}^{2}(t). According to the assumption {x}_{2}(\gamma )={m}_{2}^{\ast} and {x}_{2}(t)<{m}_{2}^{\ast} for \gamma <t<\gamma +\eta, we have {\dot{x}}_{2}(t)\ge (\frac{k}{c}M+{d}_{2}+{d}_{3}{m}_{2}^{\ast}){x}_{2}(t) for \gamma <t\le \gamma +\eta \le \gamma +{\tau}_{1}. Then we derive that {x}_{2}(t)\ge {m}_{2}^{\ast}{e}^{(\frac{k}{c}M+{d}_{2}+{d}_{3}{m}_{2}^{\ast}){\tau}_{1}}. It is clear that {x}_{2}(t)\ge {m}_{2} for \gamma <t<\gamma +\eta. If \eta \ge {\tau}_{1}, then we have {x}_{2}(t)\ge {m}_{2} for \gamma <t<\gamma +{\tau}_{1}. The same arguments can be continued. We obtain {x}_{2}(t)\ge {m}_{2} for \gamma +{\tau}_{1}<t<\gamma +\eta. Since the interval [\gamma ,\gamma +\eta ] is arbitrarily chosen, we get {x}_{2}(t)\ge {m}_{2} for t large enough. In view of our arguments above, the choice of {m}_{2} is independent of the positive solution of (1.1), which satisfies {x}_{2}(t)\ge {m}_{2} for t large enough.
Next, by the first and the fifth equations of system (1.1), we have
\{\begin{array}{l}{\dot{x}}_{1}(t)\ge r({m}_{2}\frac{M}{\lambda}{e}^{{d}_{1}{\tau}_{1}}){d}_{1}{x}_{1}(t),\phantom{\rule{1em}{0ex}}t\ne nT,\\ {x}_{1}({t}^{+})=(1p){x}_{1}(t),\phantom{\rule{1em}{0ex}}t=nT.\end{array}
(4.11)
Consider the auxiliary system of (4.11) as follows:
\{\begin{array}{l}{\dot{z}}_{7}(t)=r({m}_{2}\frac{M}{\lambda}{e}^{{d}_{1}{\tau}_{1}}){d}_{1}{z}_{7}(t),\phantom{\rule{1em}{0ex}}t\ne nT,\\ {z}_{7}({t}^{+})=(1p){z}_{7}(t),\phantom{\rule{1em}{0ex}}t=nT.\end{array}
(4.12)
By hypothesis (A_{4}), and applying Lemma 2.4, we have
{z}_{7}(t)=\frac{r({m}_{2}\frac{M}{\lambda}{e}^{{d}_{1}{\tau}_{1}})}{{d}_{1}}(1\frac{p{e}^{{d}_{1}(tnT)}}{(1p){e}^{{d}_{1}T}}).
By the comparison theorem, there exists a positive constant {\epsilon}_{5} sufficiently small such that {\dot{x}}_{1}(t)\ge {z}_{7}(t){\epsilon}_{5} as t is large enough. Taking into account the comparison theorem of an impulsive differential equation, we obtain
{x}_{1}(t)\ge \frac{r({m}_{2}\frac{M}{\lambda}{e}^{{d}_{1}{\tau}_{1}})}{{d}_{1}}(1\frac{p}{(1p){e}^{{d}_{1}T}}){\epsilon}_{5}\triangleq {m}_{1}.
From (3.3), let \rho \triangleq {m}_{4}, then {y}_{2}(t)\ge {m}_{4}.
Finally, by the third equation of system (1.1), we have
{\dot{y}}_{1}(t)\ge \lambda k(\frac{{m}_{2}}{c+{m}_{2}}{m}_{4}\frac{{M}^{2}}{\lambda c+M}{e}^{{d}_{4}{\tau}_{2}}){d}_{4}{y}_{1}(t).
(4.13)
Consider the auxiliary system of (4.13),
{\dot{z}}_{8}(t)=\lambda k(\frac{{m}_{2}}{c+{m}_{2}}{m}_{4}\frac{{M}^{2}}{\lambda c+M}{e}^{{d}_{4}{\tau}_{2}}){d}_{4}{z}_{8}(t).
(4.14)
It is easy to calculate that
\begin{array}{rcl}{z}_{8}(t)& =& \frac{\lambda k(\frac{{m}_{2}}{c+{m}_{2}}{m}_{4}\frac{{M}^{2}}{\lambda c+M}{e}^{{d}_{4}{\tau}_{2}})}{{d}_{4}}\\ (\lambda k(\frac{{m}_{2}}{c+{m}_{2}}{m}_{4}\frac{{M}^{2}}{\lambda c+M}{e}^{{d}_{4}{\tau}_{2}}){z}_{8}\left({0}^{+}\right)){e}^{{d}_{4}t}.\end{array}
Applying the comparison theorem, by hypothesis (A_{5}), there exists a positive constant {\epsilon}_{6} small enough when t is large enough, such that
{y}_{1}(t)\ge {z}_{8}(t){\epsilon}_{6}\ge \frac{\lambda k(\frac{{m}_{2}}{c+{m}_{2}}{m}_{4}\frac{{M}^{2}}{\lambda c+M}{e}^{{d}_{4}{\tau}_{2}})}{{d}_{4}}{\epsilon}_{6}\triangleq {m}_{3}.
Then taking m=min\{{m}_{1},{m}_{2},{m}_{3},{m}_{4}\}, we have {x}_{i}(t),{y}_{i}(t)\ge m, i=1,2. Considering Lemma 2.5 and the above discussion, we can find that system (1.1) is permanent. This completes the proof. □