In this section, we consider system (1.2) where breeding is continuous. That is, for the remainder of this section, we assume that \(b(t)=b\) is a positive constant. Thus, system (1.2) can be written as the following difference equation:

$$ \begin{aligned} x(t+1)=B\bigl(x(t)\bigr)x(t). \end{aligned} $$

(2.1)

Here, \(x(t)=(E(t),L(t),P(t),A(t))^{T}\), and the projection matrix *B* that maps the density at time *t* to the density at time \(t+1\) has the form

B(x)=\left(\begin{array}{cccc}(1-{\gamma}_{1}){s}_{1}(E)& 0& 0& b\\ {\gamma}_{1}{s}_{1}(E)& (1-{\gamma}_{2}){s}_{2}(L)& 0& 0\\ 0& {\gamma}_{2}{s}_{2}(L)& (1-{\gamma}_{3}){s}_{3}& 0\\ 0& 0& {\gamma}_{3}{s}_{3}& {s}_{4}(A)\end{array}\right).

(2.2)

This allows us to determine the steady state values by solving the system of equations

In population dynamic applications, we are interested in solutions with non-negative components \(E(t)\geq 0\), \(L(t)\geq 0\), \(P(t)\geq 0\), \(A(t)\geq 0\). Let \({\mathbb{R}_{+}^{4}}\doteq (0,+\infty )\times (0,+ \infty )\times (0,+\infty )\times (0,+\infty )\). It must be pointed out that the solution of system (1.2) remains non-negative. System (1.2) will always have a trivial steady state \(E_{0}=(0,0,0,0)\), where all four of the population classes are zero. From \((H_{1})\), we obtain that system (1.2) or (2.1) has the property: if \(x\leq y\), \(B(x)\geq B(y)\), where vector and matrix inequalities hold componentwise.

We shall use the techniques in [15,16,17] to find the net reproductive number \(\Re _{0}\) of the population. Notice that the inherent projection matrix is \(B(0)=G+T(0)\), where the transition matrix \(T(0)\) is

\begin{array}{r}T(0)=\left(\begin{array}{cccc}(1-{\gamma}_{1}){a}_{1}& 0& 0& 0\\ {\gamma}_{1}{a}_{1}& (1-{\gamma}_{2}){a}_{2}& 0& 0\\ 0& {\gamma}_{2}{a}_{2}& (1-{\gamma}_{3}){s}_{3}& 0\\ 0& 0& {\gamma}_{3}{s}_{3}& {a}_{4}\end{array}\right),\end{array}

and the fertility matrix *G* is

\begin{array}{r}G=\left(\begin{array}{cccc}0& 0& 0& b\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right).\end{array}

Thus, the net reproductive number is the positive, simple, and strictly dominant eigenvalue of the matrix \(G(I-T(0))^{-1}\). Through direct calculations, we know that

\begin{array}{rl}& {(I-T(0))}^{-1}\\ & \phantom{\rule{1em}{0ex}}=\frac{1}{{\eta}_{1}{\eta}_{2}{\eta}_{3}(1-{a}_{4})}\\ & \phantom{\rule{2em}{0ex}}\times \left(\begin{array}{cccc}{\eta}_{2}{\eta}_{3}(1-{a}_{4})& 0& 0& 0\\ {\gamma}_{1}{a}_{1}{\eta}_{3}(1-{a}_{4})& {\eta}_{1}{\eta}_{3}(1-{a}_{4})& 0& 0\\ {\gamma}_{1}{\gamma}_{2}{a}_{1}{a}_{2}(1-{a}_{4})& {\eta}_{1}{\gamma}_{2}{a}_{2}(1-{a}_{4})& {\eta}_{1}{\eta}_{2}(1-{a}_{4})& {\eta}_{1}{\eta}_{2}(1-{a}_{4})\\ {\gamma}_{1}{\gamma}_{2}{\gamma}_{3}{a}_{1}{a}_{2}{s}_{3}& {\eta}_{1}{\gamma}_{2}{\gamma}_{3}{a}_{2}{s}_{3}& {\gamma}_{3}{s}_{3}{\eta}_{1}{\eta}_{2}& {\eta}_{1}{\eta}_{2}{\eta}_{3}\end{array}\right),\end{array}

where \(\eta _{1}=1-(1-\gamma _{1})a_{1}\), \(\eta _{2}=1-(1-\gamma _{2})a _{2}\), \(\eta _{3}=1-(1-\gamma _{3})s_{3}\). Thus,

$$ \begin{aligned} G\bigl(I-T(0)\bigr)^{-1} = \frac{b\gamma _{1}\gamma _{2}\gamma _{3}a_{1}a_{2}s_{3}}{ \eta _{1}\eta _{2}\eta _{3}(1-a_{4})}. \end{aligned} $$

It is easily seen that

$$ \begin{aligned} \Re _{0}(\gamma _{1},\gamma _{2},\gamma _{3})=\frac{b\gamma _{1}\gamma _{2} \gamma _{3}a_{1}a_{2}s_{3}}{(1-(1-\gamma _{1})a_{1})(1-(1-\gamma _{2})a _{2})(1-(1-\gamma _{3})s_{3})(1-a_{4})}. \end{aligned} $$

(2.3)

It is worth noting that we use the notation \(\Re _{0}(\gamma _{1},\gamma _{2},\gamma _{3})\) to indicate the dependency of \(\Re _{0}\) on \(\gamma _{1}\), \(\gamma _{2}\), and \(\gamma _{3}\). Furthermore, if system (1.2) has a nontrivial interior steady state \(E_{1}=( \widetilde{E},\widetilde{L},\widetilde{P},\widetilde{A})\), where all stages have positive density, then *Ẽ*-component of a nontrivial steady state must satisfy

$$ \begin{aligned} 1=(1-\gamma _{1})s_{1}( \widetilde{E})+\frac{b\gamma _{1}\gamma _{2}\gamma _{3}s_{1}(\widetilde{E})s_{2}(\widetilde{L})s_{3}}{(1-(1-\gamma _{2})s _{2}(\widetilde{L}))(1-(1-\gamma _{3})s_{3})(1-s_{4}(\widetilde{A}))} \equiv F(\widetilde{E}), \end{aligned} $$

(2.4)

where

$$\begin{aligned}& \begin{aligned} &\widetilde{L}=\frac{1}{1-(1-\gamma _{2})s_{2}(\widetilde{L})}\gamma _{1}s_{1}(\widetilde{E})\widetilde{E}, \\ &\widetilde{P}=\frac{\gamma _{2}s_{2}(\widetilde{L})\widetilde{L}}{1-(1- \gamma _{3})s_{3}}=\frac{\gamma _{1}\gamma _{2}s_{1}(\widetilde{E})s_{2}( \widetilde{L})\widetilde{E}}{(1-(1-\gamma _{2})s_{2}(\widetilde{L}))(1-(1- \gamma _{3})s_{3})}, \\ &\widetilde{A}=\frac{1}{1-s_{4}(\widetilde{A})}\gamma _{3}s_{3} \widetilde{P}=\frac{\gamma _{1}\gamma _{2}\gamma _{3}s_{1}(\widetilde{E})s _{2}(\widetilde{L})s_{3}\widetilde{E}}{(1-(1-\gamma _{2})s_{2}( \widetilde{L}))(1-(1-\gamma _{3})s_{3})(1-s_{4}(\widetilde{A}))}. \end{aligned} \end{aligned}$$

Since \(F'(\widetilde{E})<0\) and \(\lim_{\widetilde{E}\rightarrow \infty }F(\widetilde{E})=0\), we see that if

$$ \begin{aligned} F(0)=(1-\gamma _{1})a_{1}+ \frac{b\gamma _{1}\gamma _{2}\gamma _{3}a_{1}a _{2}s_{3}}{(1-(1-\gamma _{2})a_{2})(1-(1-\gamma _{3})s_{3})(1-a_{4})}>1, \end{aligned} $$

which is equivalent to \(\Re _{0}(\gamma _{1},\gamma _{2},\gamma _{3})>1\), then system (1.2) has a nontrivial unique interior steady state \(E_{1}=(\widetilde{E},\widetilde{L},\widetilde{P},\widetilde{A})\). Furthermore, we shall show that system (1.2) is point dissipative. In fact, with assumption \((H_{1})\), we obtain

$$ P(t+1)=\gamma _{2}s_{2}\bigl(L(t)\bigr)L(t)+(1-\gamma _{3})s_{3}P(t) \leq \gamma _{2} \widehat{a_{2}}+s_{3}P(t). $$

Let \(\overline{P(t+1)}\) satisfy the recursion \(\overline{P(t+1)}=\gamma _{2} \widehat{a_{2}}+s_{3}\overline{P(t)}\). Then it is easy to see that \(\overline{P(t+1)}=\gamma _{2} \widehat{a_{2}}\sum_{j=0}^{t-1}s_{3}^{j}+s_{3}^{t} \overline{P(0)}\) and \(\overline{P(t)}\) converges at \(t\rightarrow \infty \). Since \(0\leq P(t)\leq \overline{P(t)}\), \(P(t)\) is bounded as \(t\rightarrow \infty \). Assume that \(P(t)\leq M\), \(\forall t=0,1,2, \ldots \) . Then, it follows from system (1.2) that

$$\begin{aligned}& \begin{aligned} &L(t)\leq \gamma _{1}\widehat{a_{1}}+(1- \gamma _{2})\widehat{a_{2}}, \quad \forall t\geq 0, \\ &A(t)\leq \widehat{a_{4}}+\gamma _{3}s_{3}M:= \delta _{1},\quad \forall t=1,2, \ldots, \\ &E(t)\leq b(\widehat{a_{4}}+\gamma _{3}s_{3}M)+(1- \gamma _{1}) \widehat{a_{1}}:=\delta _{2}, \quad \forall t=2,3,\ldots . \end{aligned} \end{aligned}$$

Upon the positivity and boundedness of solutions for model (1.2), we claim the following result.

### Lemma 2.1

*Let*
\((E(t), L(t), P(t), A(t))\)*be the solution of model* (1.2). *Then the positive octant*
\(\{(E(t)>0, L(t)>0, P(t)>0, A(t)>0) \}\)*is invariant*, *the sequence*
\((E(t), L(t), P(t), A(t))\), \(t\geq 0\)*is ultimately bounded for as*
\(t\rightarrow \infty \). *That is to say*, *there is a compact set*
\(\varGamma \in \mathbb{R}_{+}^{4}\)*such that every forward solution sequence of* (1.2) *enters**Γ**in at most two time steps*, *and remain in**Γ**forever after*.

### Proof

The first part is obvious. As for the boundedness of solutions to model (1.2), we denote

$$\begin{aligned} \varGamma =&\bigl\{ (E,L,P,M)\in \mathbb{R}_{+}^{4}: E\in [0,\delta _{2}],L \in \bigl[0,\gamma _{1} \widehat{a_{1}}+(1-\gamma _{2})\widehat{a_{2}} \bigr], \\ &{}P \in [0,M],A\in [0,\delta _{1}]\bigr\} . \end{aligned}$$

(2.5)

This implies that the compact set *Γ* is positively invariant and all the solutions are non-negative and ultimately bounded.

The following results can be verified by direct calculations.

- (i)
If \(\Re _{0}(\gamma _{1},\gamma _{2},\gamma _{3})<1\), then (1.2) always has a unique extinction equilibrium \(E_{0}=(0,0,0,0)\).

- (ii)
If \(\Re _{0}(\gamma _{1},\gamma _{2},\gamma _{3})>1\), then (1.2) has a nontrivial equilibria, denoted by \(E_{1}=( \widetilde{E},\widetilde{L},\widetilde{P},\widetilde{A})\).

Thus, \(\Re _{0}(\gamma _{1},\gamma _{2},\gamma _{3})\), which indicates the average number of offspring produced per individual, plays a key role in determining the existence and stability of equilibria of model (1.2). Now, we prove the following stability result for (1.2). Notice that an equilibrium is globally asymptotically stable on \(\mathbb{R}^{4}_{+}\) if it is locally asymptotically stable on \(\mathbb{R}^{4}_{+}\) and if \((E(0),L(0),P(0),A(0)) \in \mathbb{R}_{+}^{4}\) implies that \((E(t),L(t),P(t),A(t))\) tends to the equilibrium as \(t\rightarrow \infty \). □

### Theorem 2.1

*If*
\(\Re _{0}(\gamma _{1},\gamma _{2},\gamma _{3})<1\), *then the extinction equilibrium*
\(E_{0}\)*of model* (1.2) *is globally asymptotically stable*. *Moreover*, \(E_{0}\)*is unstable if*
\(\Re _{0}(\gamma _{1},\gamma _{2},\gamma _{3})>1\).

### Proof

The proof is similar to that of [18,19,20] with some minor modifications. Since \(\Re _{0}(\gamma _{1},\gamma _{2},\gamma _{3})<1\), system (1.2) only has a trivial steady state \(E_{0}=(0, 0, 0, 0)\). Define the map \(P:\mathbb{R}_{+}^{4}\rightarrow \mathbb{R}_{+}^{4}\) for the right-hand side of system (1.2). To calculate the stability of \(E_{0}\), we need to linearize model (1.2) about steady state \(E_{0}\) and evaluate the resulting Jacobian matrix

JP({E}_{0})=\left(\begin{array}{cccc}(1-{\gamma}_{1}){a}_{1}& 0& 0& b\\ {\gamma}_{1}{a}_{1}& (1-{\gamma}_{2}){a}_{2}& 0& 0\\ 0& {\gamma}_{2}{a}_{2}& (1-{\gamma}_{3}){a}_{3}& 0\\ 0& 0& {\gamma}_{3}{a}_{3}& {a}_{4}\end{array}\right).

(2.6)

Then the eigenvalues of \(JP(E_{0})\) have magnitude less than one, and hence, equilibrium \(E_{0}\) is locally asymptotically stable.

We next establish global asymptotic stability of \(E_{0}\). Note that the inherent projection matrix \(B(0)\) of system (1.2) is non-negative, irreducible, and primitive. It has positive, simple, and strictly dominant eigenvalues *r*. Moreover, since \(\Re _{0}(\gamma _{1},\gamma _{2},\gamma _{3})<1\), it follows from [16] that \(r<1\) and \(\lim_{t\rightarrow \infty }B^{t}(0)=0\). Thus, for any \(x(0)\), we have \(0\leq x(1)=B(x(0))x(0)\leq B(0)x(0)\), and repeating this we get that \(0\leq x(t)\leq B^{t}(0)x(0)\rightarrow 0\) as \(t\rightarrow \infty \). Hence, \(E_{0}\) is globally asymptotically stable. That is to say, if the extinction steady state is stable, the mosquito population can not persist.

In addition, when \(\Re _{0}(\gamma _{1},\gamma _{2},\gamma _{3})>1\), it follows from Theorem 1.1.3 in [16] that \(B(0)\) has a positive strictly dominant eigenvalue greater than one. That is, the linearization of (1.2) at \(E_{0}\) has a positive eigenvalue greater than one, which means trivial fixed point \(E_{0}\) is unstable. □

### Theorem 2.2

*If*
\(\Re _{0}(\gamma _{1},\gamma _{2},\gamma _{3})>1\), *then system* (1.2) *is uniformly persistent*.

### Proof

Assume \(\Re _{0}(\gamma _{1},\gamma _{2},\gamma _{3})>1\), let *f* be the map on the right-hand side of (1.2) from \(\mathbb{R}_{+}^{4}\) to \(\mathbb{R}_{+}^{4}\), and *D* denotes the boundary of *Γ* defined in (2.5). Since \(\operatorname{int} \mathbb{R}_{+}^{4}\) is positively invariant for system (1.2), it follows from Lemma 2.1 that \(f^{t}(\varGamma \backslash D)\subset \varGamma \backslash D\), where \(f^{t}(x)\) denotes the *t*th iteration of *x* under *f*. Moreover, employing Theorem 2.1 in [21] and Lemma 2.1, we know that there exists a global attractor *X* in *Γ*.

Let \(M={(0,0,0,0)}\) be the maximal compact invariant set in *X*, and \(\varGamma \backslash M\) is positively invariant. In order to prove if \(r>1\), system (1.2) is uniformly persistent, which is equivalent to saying that *M* is a uniform repeller.

In the following, we will construct a continuous function \(Q: \mathbb{R}_{+}^{4}\rightarrow \mathbb{R}_{+}\) satisfying the conditions:

- (1)
\(Q(x)=0\) for \(x\in M\);

- (2)
there exists a neighborhood *U* of *M* such that \(\forall x\in U\backslash M\), \(\exists n>0\), such that \(Q(f^{n}(x))>Q(x)\).

Since \(B(0)\) is non-negative and irreducible, it has a dominant eigenvalue \(r>1\), which has a corresponding positive left eigenvector \(\theta >0\),

$$ \theta ^{T}B(0)=r\theta ^{T}. $$

Let \(r^{*}\in (1,r)\) such that \(\theta ^{T}B(0)>r^{*}\theta ^{T}\). Furthermore, there exists a neighbourhood *U* of *M* by the continuity of \(B(x)\) such that

$$ \theta ^{T}B(x)>r^{*}\theta ^{T}. $$

Define \(Q:\mathbb{R}_{+}^{4}\rightarrow \mathbb{R}_{+}\) as follows:

Then \(Q(x)=0\) for \(x\in U\) iff \(x\in M\), and positive elsewhere in *U*. Moreover,

$$ Q\bigl(f(x)\bigr)=\theta ^{T}B(x)x>r^{*}\theta ^{T}x>\theta ^{T}x=Q(x), \quad \forall x \in U \backslash M. $$

Hence, (1.2) is uniformly persistent. This implies that there exists a positive number \(\rho \in \mathbb{R}_{+}^{4}\) such that, for every solution \((E(t),L(t),P(t),A(t))\), we have

$$ \liminf_{t\rightarrow \infty }\bigl(E(t),L(t),P(t),A(t)\bigr)\geq \rho >0 $$

for all non-zero orbits in \(\mathbb{R}_{+}^{4}\). This completes the proof. □

In the following, we describe the situation when all translating rates equal one. Namely, \(\gamma _{1}=\gamma _{2}=\gamma _{3}=1\). That is, we include a term that describes the survivorship of population from generation to generation. To proceed further, we review the relevant existing results on the \(k+1\)-order nonlinear difference equation before we formulate the problem that we subsequently study.

$$ x_{n+1}=F(x_{n},x_{n-1},\ldots ,x_{n-k}), \quad n=0,1,2,\ldots , $$

(2.7)

where \(F\in C(I^{k+1},{\mathbb{R}})\) and *I* is an open interval of \(\mathbb{R}\).

### Lemma 2.2

*Let*
\(x^{*}\in I\)*be an equilibrium of* (2.7). *Suppose that**F**satisfies the following two conditions*:

- (1)
*F**is non*-*decreasing in each of its arguments*;

- (2)
*F**satisfies*
\((u-x^{*})[F(u,u,\ldots ,u)-u]<0\)*for all*
\(u\in I\backslash {x^{*}}\).

*Then equilibrium point*
\(x^{*}\)*is a global attractor of all solutions of equation* (2.7).

The analysis given below focuses on the case of the following system of difference equations:

$$ \textstyle\begin{cases} E(t+1)=bA(t), \\ L(t+1)=s_{1}(E(t))E(t), \\ P(t+1)=s_{2}(L(t))L(t), \\ A(t+1)=s_{3}P(t)+s_{4}(A(t))A(t). \end{cases} $$

(2.8)

We first observe that (2.8) has a trivial steady state \(E_{0}\). It is easily seen that a nontrivial constant equilibrium \((\overline{E}, \overline{L},\overline{P},\overline{A})\) must be an equilibrium that has all positive components. The A-component of a nontrivial steady state must satisfy

$$ 1=bs_{3}s_{2}\bigl(s_{1}(b\overline{A})b \overline{A}\bigr)s_{1}(b\overline{A})+s _{4}( \overline{A}). $$

(2.9)

Consequently, equation (2.8) has a coexistence steady state \(E^{*}=(\overline{E},\overline{L},\overline{P},\overline{A})\), where all stages have positive density if and only if

$$ \frac{bs_{3}a_{1}a_{2}}{1-a_{4}}>1, $$

where \(\overline{E}=b\overline{A}\), \(\overline{L}=s_{1}(b\overline{A})b \overline{A}\), \(\overline{P}=s_{2}(s_{1}(b\overline{A})b\overline{A})s _{1}(b\overline{A})b\overline{A}\). The interior steady state is unique whenever it exists. Specially, if we consider the inhibition of larvae density on the egg hatching, namely \(s_{1}=s_{1}(L)\), then system (2.8) is the same as (1.1) in [12]. Consequently, we can use the stability information about the steady states to understand the asymptotic dynamics of our model.

### Theorem 2.3

*If*
\(\Re _{0}(1,1,1)=\frac{bs_{3}a_{1}a_{2}}{1-a_{4}}>1\), *then* (2.8) *has a unique equilibrium*
\(E^{*}=(\overline{E},\overline{L}, \overline{P},\overline{A})\)*which is globally asymptotically stable in the interior of*
\(\mathbb{R}_{+}^{4}\).

### Proof

Since \(\Re _{0}(1,1,1)>1\), it is clear that \(E^{*}\) exists by the above analysis. We now turn our attention to equilibrium point \(E^{*}\). First, we prove that \(E^{*}\) is locally asymptotically stable. To calculate the local stability of \(E^{*}\), the linearization of (2.8) about \(E^{*}\) yields the Jacobian matrix

\begin{array}{r}J\left({E}^{\ast}\right)=\left(\begin{array}{cccc}0& 0& 0& b\\ {J}_{21}& 0& 0& 0\\ 0& {J}_{32}& 0& 0\\ 0& 0& {s}_{3}& {J}_{44}\end{array}\right),\end{array}

where \(J_{21}=s_{1}^{\prime }(\overline{E})\overline{E}+s_{1}(\overline{E})>0\), \(J_{32}=s_{2}^{\prime }(\overline{L})\overline{L}+s_{2}(\overline{L})>0\), and \(J_{44}=s_{4}^{\prime }(\overline{A})\overline{A}+s_{4}(\overline{A})>0\). Furthermore, at equilibrium \(E^{*}\), the characteristic equation for the corresponding linearized model of (2.8) is

$$ f(\lambda )=\lambda ^{4}-J_{44}\lambda ^{3}-bs_{3}J_{21}J_{32}=0. $$

(2.10)

To show local asymptotic stability of \(E^{*}\), we need to show that the following inequalities hold:

$$\begin{aligned}& \mbox{(i)}\quad f(1)>0, \\& \mbox{(ii)}\quad f(-1)>0, \\& \mbox{(iii)}\quad \vert {-}bs_{3}J_{21}J_{32} \vert < 1, \\& \mbox{(iv)}\quad \bigl\vert 1-b^{2}s_{3}^{2}J_{21}^{2}J_{32}^{2} \bigr\vert > \vert {-}bs_{3}J_{21}J_{32}J_{44} \vert , \\& \mbox{(v)}\quad \bigl\vert \bigl(1-b^{2}s_{3}^{2}J_{21}^{2}J_{32}^{2} \bigr)^{2}-b^{2}s_{3}^{2}J_{21}^{2}J _{32}^{2}J_{44}^{2} \bigr\vert > \bigl\vert -bs_{3}J_{21}J_{32}J_{44}^{2} \bigr\vert . \end{aligned}$$

Since *A̅* satisfies (2.9), \(\overline{E}=b\overline{A}\), \(\overline{L}=s_{1}(\overline{E})\overline{E}\), and \(\overline{P}=s _{2}(\overline{L})\overline{L}\), we have by (2.9) that

$$ 1=bs_{3}s_{1}(\overline{E})s_{2}( \overline{L})+s_{4}(\overline{A}). $$

(2.11)

Substituting the above expression of 1 into \(f(1)\), and with the assumptions of \((H_{1})\), we have

$$ \begin{aligned} f(1) &=1-J_{44}-bs_{3}J_{21}J_{32} \\ &=1-\bigl(s_{4}^{\prime }(\overline{A})\overline{A}+s_{4}( \overline{A})\bigr) -bs _{3}\bigl(s_{1}^{\prime }( \overline{E})\overline{E}+s_{1}(\overline{E})\bigr) \bigl(s_{2} ^{\prime }(\overline{L})\overline{L}+s_{2}( \overline{L})\bigr) \\ &=-s_{4}^{\prime }(\overline{A})\overline{A} -bs_{3}s_{2}(\overline{L})s _{1}^{\prime }( \overline{E})\overline{E} -bs_{3}s_{2}^{\prime }( \overline{L}) \overline{L} \bigl(s_{1}^{\prime }(\overline{E}) \overline{E}+s_{1}(\overline{E})\bigr)>0. \end{aligned} $$

It is also clear that

$$ f(-1)=1+J_{44}-bs_{3}J_{21}J_{32}>2J_{44}>0, $$

as \(J_{44}>0\). Next, we prove the third inequality

$$ \vert {-}bs_{3}J_{21}J_{32} \vert < 1. $$

(2.12)

Note that the term inside the absolute value on the left-hand side of equation (2.12) is negative, yielding

$$ \begin{aligned} \vert {-}bs_{3}J_{21}J_{32} \vert &=bs_{3}J_{21}J_{32}=bs_{3} \bigl(s_{1}^{\prime }( \overline{E})\overline{E}+s_{1}( \overline{E})\bigr) \bigl(s_{2}^{\prime }( \overline{L}) \overline{L}+s_{2}(\overline{L})\bigr) \\ &< bs_{3}s_{1}(\overline{E})s_{2}( \overline{L}) < bs_{3}s_{1}( \overline{E})s_{2}( \overline{L})+s_{4}(\overline{A})=1. \end{aligned} $$

Because of equation (2.12) and \(b,s_{3},J_{21},J_{32},J_{44}>0\), condition (iv) becomes

$$ 1-b^{2}s_{3}^{2}J_{21}^{2}J_{32}^{2}-bs_{3}J_{21}J_{32}J_{44}>0. $$

Since (i) has been satisfied, then

$$ \begin{aligned} 1-b^{2}s_{3}^{2}J_{21}^{2}J_{32}^{2}-bs_{3}J_{21}J_{32}J_{44} &= 1-bs _{3}J_{21}J_{32}(bs_{3}J_{21}J_{32}+J_{44}), \\ &= 1+bs_{3}J_{21}J_{32}\bigl(f(1)-1\bigr) \\ &= 1-bs_{3}J_{21}J_{32}+bs_{3}J_{21}J_{32}f(1)>0. \end{aligned} $$

At last, we proceed to verifying the last inequality (v). For convenience, we denote \(m=-bs_{3}J_{21}J_{32}\), \(n=-J_{44}\). Then (v) is equal to

$$ \begin{aligned} & \bigl\vert \bigl(1-b^{2}s_{3}^{2}J_{21}^{2}J_{32}^{2} \bigr)^{2}-b^{2}s_{3}^{2}J_{21} ^{2}J_{32}^{2}J_{44}^{2} \bigr\vert - \bigl\vert -bs_{3}J_{21}J_{32}J_{44}^{2} \bigr\vert \\ &\quad =\bigl(1-m^{2}\bigr)^{2}-n^{2}m^{2}+n^{2}m \\ &\quad =(1-m) \bigl((1+m)^{2}(1-m)+n^{2}m\bigr). \end{aligned} $$

Since \(-1< m<0\), \(n<0\), we have \(1-m>0\). Then (v) is satisfied if \((1+m)^{2}(1-m)+n^{2}m>0\). However, by \(f(1)=1+m+n>0\), we obtain

$$ (1+m)^{2}(1-m)+n^{2}m >n^{2}(1-m)+n^{2}m =n^{2}>0. $$

This implies that \(E^{*}\) is locally asymptotically stable.

In what follows, we show the convergence of equilibrium \(E^{*}\). Notice that (2.8) can be converted into the following scalar difference equation:

$$ A(t+4)=s_{3}s_{2}\bigl(s_{1}\bigl(bA(t) \bigr)bA(t)\bigr)s_{1}\bigl(bA(t)\bigr)bA(t)+s_{4} \bigl(A(t+3)\bigr)A(t+3). $$

(2.13)

Since \(bs_{3}a_{1}a_{2}+a_{4}>1\), (2.13) admits a unique positive steady state *A̅*. It is sufficient to prove that *A̅* is globally attracting for (2.13) in \((0,\infty )\).

Let

$$ g(E,L,P,A)=s_{3}s_{2}\bigl(s_{1}(bE)bE \bigr)s_{1}(bE)bE+s_{4}(A)A, $$

we obtain

$$\begin{aligned}& \frac{\partial g}{\partial E} =s_{3}\bigl[s_{2} \bigl(s_{1}(bE)bE\bigr)s_{1}(bE)bE+s _{2}^{\prime } \bigl(s_{1}(bE)bE\bigr)s_{1}(bE)bE\bigr] \bigl[s_{1}^{\prime }(bE)bE+s_{1}(bE)\bigr]>0, \\& \frac{\partial g}{\partial L} =\frac{\partial g}{\partial P}=0, \\& \frac{\partial g}{\partial A} =s_{4}^{\prime }(A)A+s_{4}(A)>0 \end{aligned}$$

for all \(E,L,P,A>0\). Moreover,

$$ \begin{aligned} & (A-\overline{A})\bigl[g(A,A,A,A)-A\bigr] \\ &\quad =(A-\overline{A})\bigl[s_{3}s_{2} \bigl(s_{1}(bA)bA\bigr)s_{1}(bA)bA+s_{4}(A)A-A \bigr] \\ &\quad =(A-\overline{A})\bigl[s_{3}s_{2} \bigl(s_{1}(bA)bA\bigr)s_{1}(bA)b+s_{4}(A)-1 \bigr]A< 0 \end{aligned} $$

for \(A>0\) and \(A\neq \overline{A}\). Hence *A̅* is globally attracting for (2.13) in the interior of \(\mathbb{R}_{+}^{4}\). By using Lemma 2.2, equilibrium \(E^{*}\) is globally attracting for (2.8). This concludes the proof of this theorem. □

Furthermore, we also know that system (2.8) is uniform in the parameters for a perturbation around \(\gamma _{1}=\gamma _{2}=\gamma _{3}=1\).

### Theorem 2.4

*If*
\(\Re _{0}(1,1,1)=\frac{bs_{3}a_{1}a_{2}}{1-a_{4}}>1\), *then the extinction equilibrium for system* (2.8) *is a repeller*, *uniform in the parameter*
\(\gamma =(\gamma _{1},\gamma _{2}, \gamma _{3})\)*near*
\((1,1,1)\). *That is*, *there is an open neighbourhood*
\(U_{0}\)*of* 0 *in*
\(\mathbb{R}_{+}^{4}\), *and positive constants*
\(c_{i}\in (0,1)\), \(i=1,2,3\), *such that*, *for every*
\(x(0)\neq 0\)*and every*
\(\gamma \in [c_{1},1]\times [c_{2},1]\times [c_{3},1]\), *there is some*
\(N(x(0),\gamma )\geq 0\)*such that*

$$ f^{n}\bigl(x(t)\bigr)\notin U_{0}, \quad \forall n\geq N, $$

*where*
\(x(t)=(E(t),L(t),P(t),A(t))\), \(f(x(t))\)*denotes the right*-*hand side of system* (2.8), *and*
\(f^{n}(x)\)*denotes the**nth iteration of**x**under**f*.

### Proof

The proof is similar to that of Ackleh ([19], Lemma 5) with some minor modifications, so we omit it here. □