Intra- and Inter-Specific Competitions of Two Stage-Structured Species in a Patchy Environment

Abstract

With metamorphosis or not, creatures have varying ability in their different life stages to compete for resource, space or mating. Interaction of species with environment and competition between species are key factors in the evolution of ecological population. Taking these concerns into account, we study a model with two life stages, immature and mature, and incorporate both intra- and inter-specific competitions between two species in a two-patch environment. The structure of monotone dynamics in such a model leads us to explore its local and global dynamics. The investigation starts with the single-species model on which we establish the threshold dynamics that either the species eventually goes extinction or exists on both patches, which is determined by the parameters. Then we study the two-species model and formulate the threshold competition strength which monotonously but oppositely depends on the maturation times of two species, and indicates how the competitor invade an environment. Moreover, we demonstrate two mechanisms which give rise to dominance dynamics, under competition-dependent and -independent criteria respectively. Finally, we conduct numerical simulations to show that the proposed model admits multiple positive equilibria due to the consideration of two life stages.

Appendices

Appendices

1.1 A.I Proofs of Lemma, Propositions and Theorems

Proof of Theorem 2.1

Equation (2.6) is a cooperative and irreducible delay differential equation, cf. Chapter 5 [32]. In addition, the corresponding semiflow is eventually strongly monotone, and then SOP, see Corollary 5.3.5 and Proposition 1.1.1 [32]. A SOP semiflow possesses an important property of generic convergence to equilibrium, and even the global convergence dynamics to an equilibrium if it is unique, cf. Theorem 2.3.1 [32].

All solutions of (2.6) are uniformly eventually bounded. Indeed, from equation (2.6) with (2.8), we have

$$\begin{aligned} \frac{dU(t)}{d t} \le \tilde{b}_0 -\mu _{m}U(t), \end{aligned}$$

where $\tilde{b}_0:=(\mu _{l}e^{-\mu _{l}\tau }\tilde{b})/[\mu _{l} + k_{l} (1-e^{-\mu _{l}\tau }) \tilde{b}]$, and $\tilde{b}$ is given in (2.8). Thus $\limsup _{t\rightarrow \infty } U(t) \le \tilde{b}_0/\mu _{m}$.

To observe the configuration of $B(\xi )$, we compute

$$\begin{aligned} B'(\xi ) = \frac{\mu _{l}^2e^{-\mu _{l}\tau } b'(\xi )}{[\mu _{l}+k_{l}(1-e^{-\mu _{l}\tau })b(\xi )]^2} >0, \end{aligned}$$

(A.1)

and

$$\begin{aligned} B''(\xi )= & {} [\mu _{l}+k_{l}(1-e^{-\mu _{l}\tau })b(\xi )]^{-4} \cdot \mu _{l}^2e^{-\mu _{l}\tau }[\mu _{l}+k_{li}(1-e^{-\mu _{l}\tau })b(\xi )] \nonumber \\{} & {} \cdot \left\{ b''(\xi )[\mu _{l}+k_{l}(1-e^{-\mu _{l}\tau })b(\xi )] - 2k_{l}(1-e^{-\mu _{l}\tau })(b'(\xi ))^2\right\} \nonumber \\< & {} 0, \end{aligned}$$

(A.2)

due to (2.8). Note that $B'(0)=b'(0)e^{-\mu _l\tau }$. From the configuration of $B(\xi )$, we can see that a unique positive equilibrium $\overline{U}$ exists if and only if $B'(0)>\mu _m$. The intersections for the graphs of functions $B(\xi )$ and $h(\xi ):=\mu _m \xi +k_m\xi ^2$ are located at $\xi =0,~\overline{U}$. From these graphs, we have

$$\begin{aligned} B'(\overline{U})<\mu _m+2k_m\overline{U}. \end{aligned}$$

(A.3)

When $b'(0)e^{-\mu _l\tau }<\mu _m$, the trivial solution $U \equiv \hat{0}$ is the unique equilibrium, and it attracts every element in $C([-\tau ,0],\mathbb {R}_+)$, according to Theorem 2.3.1 [32]. In addition, the corresponding characteristic equation is $\lambda + \mu _m-B'(0)e^{-\lambda \tau }=0$. By Theorem 2.3 [40], we see that the trivial solution is stable (resp., unstable) when $B'(0)<\mu _m$ (resp., $B'(0)>\mu _m$). Hence, assertion (i) is confirmed.

The condition $b'(0)e^{-\mu _l\tau }>\mu _m$, i.e., $B'(0)>\mu _m$, enforces the existence of positive equilibrium $\overline{U}$, and a sufficiently small $\rho ^*>0$ such that

$$\begin{aligned} B'(\rho ^*)-\mu _m-k_m\rho ^*>0. \end{aligned}$$

(A.4)

In order to conclude the global convergence to $\overline{U}$, we first claim the uniform persistence for (2.6) by using the persistence theory in Theorem 4.6 [39]. Define

$$\begin{aligned} \mathcal {X} = C([-\tau ,0],\mathbb {R}_+),~~\mathcal {X}_0 = \left\{ \varphi \in \mathcal {X}~\textrm{with} ~\varphi \ne \hat{0}\right\} . \end{aligned}$$

Then $\partial \mathcal {X}_0:= \mathcal {X}\setminus \mathcal {X}_0 =\{ \hat{0} \}$. As shown in Theorem 5.2.1 [32], a solution starting nonnegative remains nonnegative in future time. That is, $\mathcal {X}$ is positively invariant. In addition, from

$$\begin{aligned} \frac{dU(t)}{d t} \ge B(U(t-\tau )) -\mu _{m}U(t) - k_m \mathcal {U}U(t), \end{aligned}$$

where $\mathcal {U}:=\sup _{t\ge 0}U(t)<\infty$, we have

$$\begin{aligned} U(t) \ge e^{-(\mu _m+k_m \mathcal {U}) t}U(0) + \int ^t_0e^{-(\mu _m+k_m \mathcal {U})(t-\xi )} B(U(\xi -\tau ))d\xi >0, \end{aligned}$$

for $t >0$, whenever $U(0) \in \mathcal {X}_0$. Hence, $\mathcal {X}_0$ is positively invariant. Thus, $\mathcal {M}_{\partial }:=\{ \varphi \in \mathcal {X}~|~\Psi _t(\varphi )\in \partial \mathcal {X}_0,~\forall ~t\ge 0 \}=\{\hat{0}\}$, and note that $\mathcal {X}\setminus \mathcal {X}_0=\{ \hat{0}\}$ is relatively closed in $\mathcal {X}$. Next, we claim that

$$\begin{aligned} \limsup _{t\rightarrow \infty }U(t)>\rho ^*,~\mathrm{for~all}~\varphi \in \mathcal {X}_0. \end{aligned}$$

(A.5)

Suppose, on the contrary, that there exists an initial value $\varphi \in \mathcal {X}_0$ and a $t_1>0$ such that $U(t)\le \rho ^*$ for $t\ge t_1-\tau$. From (2.6) and the concavity of $B(\cdot )$, for $t\ge t_1$,

$$\begin{aligned} \frac{dU(t)}{d t} \ge B'(\rho ^*)U(t-\tau )-(\mu _m+k_m\rho ^*)U(t). \end{aligned}$$

Consider the auxiliary equation

$$\begin{aligned} \frac{dx(t)}{d t} = B'(\rho ^*)x(t-\tau )-(\mu _m+k_m\rho ^*)x(t). \end{aligned}$$

(A.6)

Equation (A.6) is cooperative and irreducible, and generates a semiflow of (A.6) which is eventually strong monotone, with $\hat{0}$ an equilibrium. Under (A.4), Eq. (A.6) admits a positive stability modulus associated with a positive eigenvector $\textbf{z}_0$, according to Theorem 5.5.1 and Corollary 5.5.2] [32]. Denote the solution semiflow of (A.6) by $\tilde{\Psi }_t$. For the initial value $\varphi$ of (2.6), there exist a $t_2>t_1$ and a small $\alpha _0>0$ such that

$$\begin{aligned} \hat{0}\ll \alpha _0\hat{\textbf{z}}_0\ll \Psi _{t_2}(\varphi ). \end{aligned}$$

Hence, we have

$$\begin{aligned} \tilde{\Psi }_t(\alpha _0\hat{\textbf{z}}_0) \le \Psi _t(\alpha _0\hat{\textbf{z}}_0) \ll \Psi _{t+t_2}(\varphi ), \end{aligned}$$

for $t\ge 0$, which is a contradiction to the boundedness of semiflow $\Psi _t$. This contradiction proves (A.5). Obviously, $\{\hat{0}\}$ is an isolated invariant set in $\partial \mathcal {X}_0$, and thus the set $\mathcal {M}_{\partial }$ consists of an acyclic equilibrium point. From (A.5), $W^s(\hat{0})\cap \mathcal {X}_0 = \emptyset$, where $W^s(\hat{0})$ denotes the stable manifold of $\hat{0}$. By the persistence theory in Theorem 4.6 [39], system (2.6) is uniformly persistent with respect to $(\mathcal {X}_0,\partial \mathcal {X}_0)$ under $b'(0)e^{-\mu _l\tau }>\mu _m$, which means that there exists a $\rho ^{**}>0$, with $\rho ^{**}<U^*$, such that $\liminf _{t\rightarrow \infty }U(t)>\rho ^{**}$. Accordingly, it suffices to consider (2.6) in the space $\mathcal {X}^*:=C([-\tau ,0],[\rho ^{**}/2,\infty ))$. In fact, the equilibrium $\overline{U}$ is the unique equilibrium in $\mathcal {X}^*$, and again Theorem 2.3.1 [32] implies its global attractivity. The characteristic equation at $\overline{U}$ is $\lambda + \mu _m+2k_m \overline{U}-B'(\overline{U})e^{-\lambda \tau }=0$. If it has a root with nonnegative real part, then

$$\begin{aligned} \mu _m+2k_m \overline{U} \le |\lambda + \mu _m+2k_m \overline{U}| = |B'(\overline{U})e^{-\lambda \tau }| < B'(\overline{U}) \end{aligned}$$

which contradicts to (A.3) for any $\tau >0$. Thus, it has only roots with negative real parts. Therefore, assertion (ii) is justified. $\square$

Proof of Theorem 2.4

System (2.11) is also cooperative and irreducible. By using a similar argument as in Theorem 2.1, we can show that the solutions of system (2.11) are ultimately bounded. The linearization of system (2.11) at an equilibrium $(\tilde{U}_1, \tilde{U}_2)$ reads

$$\begin{aligned} \frac{dx_1(t)}{dt}= & {} B_1'(\tilde{U}_1)x_1(t-\tau ) - \mu _{m1}x_1(t) - 2k_{m1}\tilde{U}_1x_1(t) + Dx_2(t) - Dx_1(t), \nonumber \\ \frac{dx_2(t)}{dt}= & {} B_2'(\tilde{U}_2)x_2(t-\tau ) - \mu _{m2}x_2(t) - 2k_{m2}\tilde{U}_2x_2(t) + Dx_1(t) - Dx_2(t). \end{aligned}$$

(A.7)

Suppose that $(\mathcal {S})$ does not hold. The trivial solution is the unique nonnegative equilibrium, and hence the property of SOP implies the global convergence dynamics. In addition, by a direct computation we obtain the characteristic equation at the trivial equilibrium

$$\begin{aligned}{} & {} \lambda ^2 + (p_1+p_2)\lambda + (p_1p_2-D^2) - (B_1'(0) + B_2'(0)) \lambda e^{-\tau \lambda } \nonumber \\{} & {} ~~~ - (p_1B_2'(0) + p_2B_1'(0)) e^{-\tau \lambda } + B_1'(0)B_2'(0) e^{-2\tau \lambda } =0, \end{aligned}$$

(A.8)

here

$$\begin{aligned} p_1 := \mu _{m1} +D,~~p_2 := \mu _{m2} +D. \end{aligned}$$

Since (2.11) is a cooperative and irreducible delay differential system, from Corollary 5.5.2 [32], the characteristic equation (A.8) has the stability modulus of the same sign as that of the characteristic equation for the associated ordinary differential equation of (A.7) at the trivial equilibrium, i.e.,

$$\begin{aligned}{} & {} \qquad \qquad \lambda ^2 + m_1 \lambda +m_0 =0, \end{aligned}$$

(A.9)

$$\begin{aligned}{} & {} m_1 := p_1+p_2 - B'_1(0) - B'_2(0), \nonumber \\{} & {} m_0 := (p_1- B'_1(0))(p_2 - B'_2(0)) - D^2. \end{aligned}$$

(A.10)

Obviously, both $m_1$ and $m_0$ are positive. Hence, both roots of (A.9) have negative real parts, and thus we conclude that all roots of (A.8) have negative real parts. Together with the global convergence dynamics from SOP, we conclude that the trivial equilibrium of (2.11) is GAS in $C([-\tau ,0],\mathbb {R}_+^2)$.

Suppose that $(\mathcal {S})$ holds. From Theorem 2.3, we have the uniform persistence for the (2.11). Hence, it suffices to consider the solutions with initial values in the set $C([-\tau ,0],[\rho ^*,\infty )^2)$. It attracts all solutions starting from $C([-\tau ,0],\mathbb {R}_+^2)\setminus \{(\hat{0}, \hat{0})\}$, and contains the unique equilibrium $(\overline{U}_1, \overline{U}_2)$. Together with the eventually uniform boundedness, the SOP property implies the global convergence to $(\overline{U}_1, \overline{U}_2)$. Thus, it remains to show that $(\overline{U}_1, \overline{U}_2)$ is stable. With a direct calculation, we obtain the characteristic equation at $(\overline{U}_1, \overline{U}_2)$

$$\begin{aligned}{} & {} \lambda ^2 + (\tilde{p}_1+\tilde{p}_2)\lambda + (\tilde{p}_1\tilde{p}_2-D^2) - (B_1'(\overline{U}_1) + B_2'(\overline{U}_2)) \lambda e^{-\tau \lambda } \nonumber \\{} & {} ~~~ - (\tilde{p}_1B_2'(\overline{U}_2) + \tilde{p}_2B_1'(\overline{U}_1)) e^{-\tau \lambda } + B_1'(\overline{U}_1)B_2'(\overline{U}_2) e^{-2\tau \lambda } =0, \end{aligned}$$

(A.11)

where

$$\begin{aligned} \tilde{p}_1:= & {} \mu _{m1} + 2k_{m1} \overline{U}_1 +D, \\ \tilde{p}_2:= & {} \mu _{m2} + 2k_{m2} \overline{U}_2 +D. \end{aligned}$$

Again, the characteristic equation (A.11) has stability modulus of the same sign as that of the characteristic equation for the corresponding ordinary differential equation of (A.7) at $(\overline{U}_1, \overline{U}_2)$, i.e.,

$$\begin{aligned}{} & {} \qquad \qquad \qquad \lambda ^2 + \tilde{m}_1 \lambda +\tilde{m}_0 =0,\nonumber \\{} & {} \tilde{m}_1 := \tilde{p}_1+\tilde{p}_2 - B_1'(\overline{U}_1) - B_2'(\overline{U}_2), \nonumber \\{} & {} \tilde{m}_0 := (\tilde{p}_1-B_1'(\overline{U}_1))(\tilde{p}_2-B_2'(\overline{U}_2)) - D^2. \end{aligned}$$

(A.12)

We claim that both roots of (A.12) have negative real parts. The equilibrium $(\overline{U}_1, \overline{U}_2)$ satisfies

$$\begin{aligned} B_1(\overline{U}_1)-\mu _{m1}\overline{U}_1-k_{m1}(\overline{U}_1)^2+D\overline{U}_2-D\overline{U}_1=0. \end{aligned}$$

Since $f_1$ has a positive derivative at $\overline{U}_1$, we see that

$$\begin{aligned} \mu _{m1}+2k_{m1}\overline{U}_1+D-B_1'(\overline{U}_1)>0, \end{aligned}$$

and then

$$\begin{aligned} \tilde{p}_1-B_1'(\overline{U}_1)>0. \end{aligned}$$

Similarly, it holds that $\tilde{p}_2-B_2'(\overline{U}_2)>0$. Thus, $\tilde{m}_1>0$. In addition, since the point $(\overline{U}_1,\overline{U}_2)$ is the intersection of the graphs $U_2=f_1(U_1)$ and $U_1=f_2(U_2)$, we have

$$\begin{aligned} f_1'(\overline{U}_1)= & {} \frac{1}{D}[\mu _{m1}+D + 2k_{m1}\overline{U}_1 -B_1'(\overline{U}_1) ] \\= & {} \frac{\overline{U}_2}{\overline{U}_1} + \frac{1}{D} \left( \frac{B_1(U_1)}{U_1}+ k_{m1}\overline{U}_1 -B_1'(\overline{U}_1) \right) \\> & {} \frac{\overline{U}_2}{\overline{U}_1}, \end{aligned}$$

due to the property of $B_1(\cdot )$ as in (A.1) and (A.2). Similarly, $f_2'(\overline{U}_2)> \frac{\overline{U}_1}{\overline{U}_2}$. Hence, we see that $f'_1(\overline{U}_1)f'_2(\overline{U}_2)>1$, which leads to

$$\begin{aligned} \tilde{m}_0=(\tilde{p}_1-B_1'(\overline{U}_1))(\tilde{p}_2-B_2'(\overline{U}_2)) - D^2 >0. \end{aligned}$$

Accordingly, we conclude that all roots of (A.11) have negative real parts, and therefore the positive equilibrium $(\overline{U}_1, \overline{U}_2)$ is GAS in $C([-\tau ,0],\mathbb {R}_+^2)\setminus \{(\hat{0},\hat{0})\}$, when $(\mathcal {S})$ holds. $\square$

Proof of Proposition 3.2

We only justify the assertion for $\mathbb {X}_u$, as the one for $\mathbb {X}_v$ is similar. (i) Suppose that $t_1>0$ is the first time such that the solution is out of $\mathbb {X}_u$, that is $U_1(t)=U_2(t)=0$ for $t\in [t_1-\tau _u,t_1]$. Then for $0<\epsilon <\tau _u$, there is an $s\in [t_1-\epsilon -\tau _u,t_1-\epsilon ]$ such that $U_{i_0}(s)>0$ for $i_0=$ 1 or 2. In fact, $s\in [t_1-\epsilon -\tau _u,t_1-\tau _u)$, and then $s+\tau _u\in [t_1-\epsilon ,t_1)\subset [t_1-\tau _u,t_1]$. Hence, $U'_{i_0}(s+\tau _u)=0$. However, $U'_{i_0}(s+\tau _u)=B_{ui_0}(U_{i_0}(s))>0$, which is a contradiction. Hence, $\mathbb {X}_u$ is positively invariant under the solution flow of (3.1).

To show the positive invariance of $\textrm{int}(\mathbb {X})$, we consider an initial condition $\phi \in \textrm{int}(\mathbb {X})$, which means that $\phi _i(\theta )>0$, $\theta \in [-\tau _u,0]$ for $i=1,2$, and $\phi _i(\theta )>0$, $\theta \in [-\tau _v,0]$ for $i=3,4$. From Proposition 3.1, we obtain, for $i=1,2$,

$$\begin{aligned} \frac{dU_i(t)}{dt} \ge B_{ui}(U_i(t-\tau _u)) - \tilde{b}U_i(t), \end{aligned}$$

where $\tilde{b}_i=\mu _{mui}+k_{mui}\mathcal {U}_i+c_{uv}\mathcal {V}_i+D_u$, $\mathcal {U}_i=\sup _{t\ge 0}U_i(t)<\infty$, and $\mathcal {V}_i=\sup _{t\ge 0}V_i(t)<\infty$. Hence, for $0<t\le \tau _u$,

$$\begin{aligned} U_i(t) \ge e^{-\tilde{b}t}U_i(0) + \int ^t_0e^{-\tilde{b}(t-\xi )} B_{ui}(\phi _i(\xi -\tau _u))d\xi , \end{aligned}$$

and, for $t>\tau _u$,

$$\begin{aligned} U_i(t) \ge e^{-\tilde{b}t}U_i(0) + \int ^{\tau _u}_0e^{-\tilde{b}(t-\xi )} B_{ui}(\phi _i(\xi -\tau _u))d\xi + \int ^t_{\tau _u}e^{-\tilde{b}(t-\xi )} B_{ui}(U_i(\xi -\tau _u))d\xi . \end{aligned}$$

Thus, $U_i(t)>0$ for $t>0$ because of $U_i(0)>0$. Similarly, $V_i(t)>0$ for $t>0$. We conclude that $\textrm{int}(\mathbb {X})$ is positively invariant.

• (ii) We observe that, for $i,j=1,2$ and $i\ne j$,

  $$\begin{aligned} \frac{dU_i(t)}{dt} \ge B_{ui}(U_i(t-\tau _u))-\tilde{b}_iU_i(t)+D_uU_j(t). \end{aligned}$$

  (A.13)

Now, suppose $\phi _{i_0}\ne \hat{0}$ for some $i_0=1,2$. Then, for $t\in [0,\tau _u]$,

$$\begin{aligned} U_{i_0}(t)\ge e^{-\tilde{b}_{i_0}t}\phi _{i_0}(0) + \int ^t_0 e^{-\tilde{b}_{i_0}(t-\xi )} [B_{ui_0}(\phi _{i_0}(\xi -\tau _u)) + D_uU_{j_0}(\xi )] d\xi . \end{aligned}$$

(A.14)

We first claim that there is an $t_2\in [0,\tau _u)$ such that $U_{i_0}(t_2)>0$. Otherwise, we have $U_{i_0}(t)=0$ for $t\in [0,\tau _u)$, and (A.14) enforces $\phi _{i_0}(0)=0$ and $\phi _{i_0}(\xi -\tau _u)=0$ for $\xi \in [0,\tau _u)$, which contradicts to $\phi _{i_0}\ne \hat{0}$. Next, we claim that $U_1(t)>0$ and $U_2(t)>0$ for $t>t_2$. From (A.13), we see that, for $t_2\le t\le \tau _u$,

$$\begin{aligned} U_{i_0}(t)\ge e^{-\tilde{b}_{i_0}(t-t_2)}U_{i_0}(t_2) + \int ^t_{t_2} e^{-\tilde{b}_{i_0}(t-\xi )} [B_{ui_0}(\phi _{i_0}(\xi -\tau _u)) + D_uU_{j_0}(\xi )] d\xi , \end{aligned}$$

and for $t> \tau _u$,

$$\begin{aligned} U_{i_0}(t)\ge & {} e^{-\tilde{b}_{i_0}(t-t_2)}U_{i_0}(t_2) + \int ^{\tau _u}_{t_2} e^{-\tilde{b}_{i_0}(t-\xi )} [B_{ui_0}(\phi _{i_0}(\xi -\tau _u)) + D_uU_{j_0}(\xi )] d\xi \\{} & {} + \int ^t_{\tau _u} e^{-\tilde{b}_{i_0}(t-\xi )} [B_{ui_0}(U_{i_0}(\xi -\tau _u)) + D_uU_{j_0}(\xi )] d\xi . \end{aligned}$$

Hence, $U_{i_0}(t)>0$ for $t\ge t_2$ since $U_{i_0}(t_2)>0$. From (A.13), we also have, for $t_2\le t\le \tau _u$,

$$\begin{aligned} U_{j_0}(t)\ge e^{-\tilde{b}_{j_0}(t-t_2)}U_{j_0}(t_2) + \int ^t_{t_2} e^{-\tilde{b}_{j_0}(t-\xi )} [B_{uj_0}(\phi _{j_0}(\xi -\tau _u)) + D_uU_{i_0}(\xi )] d\xi , \end{aligned}$$

and for $t> \tau _u$,

$$\begin{aligned} U_{j_0}(t)\ge & {} e^{-\tilde{b}_{j_0}(t-t_2)}U_{j_0}(t_2) + \int ^{\tau _u}_{t_2} e^{-\tilde{b}_{j_0}(t-\xi )} [B_{uj_0}(\phi _{j_0}(\xi -\tau _u)) + D_uU_{i_0}(\xi )] d\xi \\{} & {} + \int ^t_{\tau _u} e^{-\tilde{b}_{j_0}(t-\xi )} [B_{uj_0}(U_{i_0}(\xi -\tau _u)) + D_uU_{i_0}(\xi )] d\xi . \end{aligned}$$

Hence, $U_{j_0}(t)>0$ for $t> t_2$ since $U_{i_0}(t)>0$ for $t\ge t_2$. Finally, we conclude that $U_1(t)>0$ and $U_2(t)>0$ for $t\ge \tau _u(>t_2)$. This completes the proof. $\square$

Proof of Lemma 3.9

Let us justify the first assertion, and the second one is similar. Obviously, the first inequality in (3.17) is equivalent to $c_{uv}<\tilde{c}_{uv}$, where

$$\begin{aligned} \tilde{c}_{uv}:=\frac{B_{u1}'(0)-\mu _{mu1}+B_{u2}'(0)-\mu _{mu2}-2D_u}{\overline{V}_1+\overline{V}_2}. \end{aligned}$$

On the other hand, the second inequality in (3.17) is equivalent to $\mathcal {J}(c_{uv}):=(c_{uv})^2 +Pc_{uv} +Q <0$, where

$$\begin{aligned} P= & {} - \frac{(B_{u1}'(0)-\mu _{mu1}-D_u)\overline{V}_2 + (B_{u2}'(0)-\mu _{mu2}-D_u)\overline{V}_1}{\overline{V}_1\overline{V}_2}, \\ Q= & {} \frac{(B_{u1}'(0)-\mu _{mu1}-D_u)(B_{u2}'(0)-\mu _{mu2}-D_u)-D_u^2}{\overline{V}_1\overline{V}_2}. \end{aligned}$$

Note that the discriminant of the quadratic function $\mathcal {J}$ is $\frac{\Delta _u}{\overline{V}_1^2\overline{V}_2^2}$ which is positive. Hence, the second inequality in (3.17) holds for $c_{uv}^-< c_{uv} < c_{uv}^+$, where

$$\begin{aligned} c_{uv}^{\pm }:=\frac{(B_{u1}'(0)-\mu _{mu1}-D_u)\overline{V}_2 + (B_{u2}'(0)-\mu _{mu2}-D_u)\overline{V}_1 \pm \sqrt{\Delta _u}}{2\overline{V}_1\overline{V}_2}. \end{aligned}$$

A tedious calculation shows that $c_{uv}^-\le \tilde{c}_{uv}<c_{uv}^+$. In summary, the inequalities in (3.17) are equivalent to $c_{uv}<\tilde{c}_{uv}$ and $c_{uv}^-< c_{uv} < c_{uv}^+$, respectively. Thus, condition (3.17) is equivalent to their union, which is $c_{uv}<c_{uv}^+$. $\square$

Proof of Theorem 3.11

We only justify the assertion for u-species and assume that $(\mathcal {S}_u)$ holds. We will discuss the case that both $(\mathcal {S}_v)$ and $c_{uv}<c^+_{uv}$ hold, and the simpler case that $(\mathcal {S}_v)$ does not hold can be treated in a similar way. We will follow the persistence theory in Theorem 4.6 [39] for the justification. Since $c_{uv}<c^+_{uv}$, from the fact that $B_{ui}$, $i=1,2$, are continuously differentiable and Lemma 3.9, there is a sufficiently small $\varrho ^*>0$ such that

$$\begin{aligned}{} & {} B_{u1}'(\varrho ^*)+B_{u2}'(\varrho ^*) > \mu _{mu1}+k_{mu1}\varrho ^*+c_{uv}(\overline{V}_1+\varrho ^*) \nonumber \\{} & {} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad +\,\mu _{mu2}+k_{mu2}\varrho ^*+c_{uv}(\overline{V}_2+\varrho ^*)+2D_u,~\textrm{or} \nonumber \\{} & {} (B_{u1}'(\varrho ^*)-\mu _{mu1}-k_{mu1}\varrho ^*-c_{uv}(\overline{V}_1+\varrho ^*)-D_u)\cdot \nonumber \\{} & {} (B_{u2}'(\varrho ^*)-\mu _{mu2}-k_{mu2}\varrho ^*-c_{uv}(\overline{V}_2+\varrho ^*)-D_u)<D_u^2. \end{aligned}$$

(A.15)

We aim to justify the assertion with this positive $\varrho ^*$. Let us divide our proof into the following four parts:

(I) An auxiliary system: From Proposition 3.5, for this $\varrho ^*>0$, there is a $t_0>0$ such that $U_i(t)<\overline{U}_i+\varrho ^*$ and $V_i(t)<\overline{V}_i+\varrho ^*$ for $t\ge t_0$. From the U-equation, for $t\ge t_0$, we have

$$\begin{aligned} \frac{dU_1(t)}{dt}\ge & {} B_{u1}(U_1(t-\tau _u)) - \mu _{mu1}U_1(t) -k_{mu1}(U_1(t))^2 \nonumber \\{} & {} - c_{uv}(\overline{V}_1+\varrho ^*)U_1(t) + D_uU_2(t) - D_uU_1(t), \nonumber \\ \frac{dU_2(t)}{dt}\ge & {} B_{u2}(U_2(t-\tau _u)) - \mu _{mu2}U_2(t) -k_{mu2}(U_2(t))^2 \nonumber \\{} & {} - c_{uv}(\overline{V}_2+\varrho ^*)U_2(t) + D_uU_1(t) - D_uU_2(t). \nonumber \end{aligned}$$

Shift the time t by $\tilde{t}=t-t_0$ but retain the symbols. Then we need to justify the same assertion. Consider the auxiliary system

$$\begin{aligned} \frac{d\omega _1(t)}{dt}= & {} B_{u1}(\omega _1(t-\tau _u)) - \mu _{mu1}\omega _1(t) -k_{mu1}(\omega _1(t))^2 \nonumber \\{} & {} - c_{uv}(\overline{V}_1+\varrho ^*)\omega _1(t) + D_u\omega _2(t) - D_u\omega _1(t), \nonumber \\ \frac{d\omega _2(t)}{dt}= & {} B_{u2}(\omega _2(t-\tau _u)) - \mu _{mu2}\omega _2(t) -k_{mu2}(\omega _2(t))^2 \nonumber \\{} & {} - c_{uv}(\overline{V}_2+\varrho ^*)\omega _2(t) + D_u\omega _1(t) - D_u\omega _2(t). \end{aligned}$$

(A.16)

By the comparison principle, the solutions of (3.1) and (A.16) starting from the same initial condition satisfy $U_i(t)\ge \omega _i(t)$ for all $t>0$. Thus it suffices to show that system (A.16) is uniformly persistent. Define

$$\begin{aligned} X= & {} \mathcal {C}_u~(=C([-\tau _u,0],\mathbb {R}^2_+)~), \\ X_0= & {} \{ \varphi =(\varphi _1,\varphi _2) \in X~\textrm{with} ~\varphi _i\ne \hat{0}~\textrm{for}~i=1,2\}, \\ \partial X_0= & {} X\setminus X_0 =\{ \varphi \in X : \,\textrm{either}\,\,\varphi _1=\hat{0} \,\,\,\textrm{or}\,\,\,\varphi _2=\hat{0}\,\}. \end{aligned}$$

(II) Dynamics of (A.16) in subsets $X_0$ and $\partial X_0$: System (A.16) is point-dissipative, by arguments similar to those for Proposition 3.1. In addition, both X and $X_0$ are positively invariant under the semiflow of (A.16) and $X\setminus X_0$ is relatively closed in X. In fact, the solution of system (A.16) with initial value $\varphi \in X_0$ satisfies, for $i,j=1,2$, $i\ne j$,

$$\begin{aligned} \frac{d\omega _i(t)}{dt} \ge B_{ui}(\omega _i(t-\tau _u)) - [\mu _{mui}+k_{mui}\mathcal {W}_i+c_{uv}(\overline{V}_i+\varrho ^*)+D_u]\omega _i(t) + D_u\omega _j(t), \end{aligned}$$

where $\mathcal {W}_i=\sup _{t\ge 0}\omega _i(t)$, which is finite. By the comparison principle, we have, for $t\in (0,\tau _u]$,

$$\begin{aligned} \omega _i(t) \ge e^{-\tilde{a}t}\omega _i(0) + \int ^t_0e^{-\tilde{a}(t-\xi )} [B_{ui}(\varphi _i(\xi -\tau _u))+D_u\omega _j(\xi )]d\xi , \end{aligned}$$

(A.17)

where $\tilde{a}:=\mu _{mui}+k_{mui}\mathcal {W}_i+c_{uv}(\overline{V}_i+\varrho ^*)+D_u$. Suppose there is a $t_1\in (0,\tau _u]$ and some $i_0\in \{1,2\}$ such that $(\omega _{i_0})_{t_1}(\cdot )=\hat{0}$. Then $\varphi _{i_0}(\theta )=0$ for $\theta \in [t_1-\tau _u,0]$, and $\omega _{i_0}(t)=0$ for $t\in [0,t_1]$. The latter one together with (A.17) imply that $\varphi _{i_0}(\theta )=0$ for $\theta \in [-\tau _u,t_1-\tau _u]$, and then $\varphi _{i_0}(\cdot )=\hat{0}$. It contradicts to the assumption $\varphi \in X_0$. Hence $\omega _t(\cdot )$ stays in $X_0$ for $t\in (0,\tau _u]$. By proceeding the same arguments for $t\in [k\tau _u,(k+1)\tau _u]$, $k=1,2,3,\ldots$, we see that $X_0$ is positively invariant under (A.16).

(III) The maximal positively invariant subset of (A.16) in the boundary $\partial X_0$: We set

$$\begin{aligned} M_{\partial }=\{ \varphi \in X~|~\check{\Phi }_t(\varphi )\in \partial X_0,~\forall ~t\ge 0 \}, \end{aligned}$$

where $\check{\Phi }_t$ is the semiflow generated by (A.16). Then obviously $M_{\partial } \subset \partial X_0$. We claim that $M_{\partial }=\{ \hat{\textbf{0}} \}$, where $\textbf{0} =(0,0)$. It is clear that $\{\hat{\textbf{0}}\} \subset M_{\partial }$, so it suffices to show $M_{\partial } \subset \{\hat{\textbf{0}} \}$. Assume the opposite, that there is $\varphi =(\varphi _1, \varphi _2) \in M_{\partial }$ and $\varphi \ne \hat{\textbf{0}}$. Without loss of generality, we suppose $\varphi _1\ne \hat{0}$, $\varphi _2=\hat{0}$. Let us discuss three possibilities: (i) If $\varphi _1(0)>0$, we have $\frac{d\omega _2(0)}{dt}>0$, and then there is a small $t_0>0$ such that $\omega _2(t)>0$ for $t\in (0,t_0)$. Since $\varphi _1(0)>0$, the continuity of the solution implies the existence of a $t_1\le t_0$ such that $\omega _1(t)>0$ for $t\in (0,t_1)$. Thus, $\check{\Phi }_t(\varphi )\in X_0$ for $t\in (0,t_1)$, a contradiction to the fact that $\varphi \in M_{\partial }$. (ii) If $\varphi _1(0)=0$ and $\varphi _1(-\tau _u)>0$, we see that

$$\begin{aligned} \frac{d\omega _1(0)}{dt}=B_{u1}(\omega _1(-\tau _u))=B_{u1}(\varphi _1(-\tau _u))>0. \end{aligned}$$

By continuity of the solution to (A.16) and $\varphi \in X$, there is a $t_2>0$ such that

$$\begin{aligned} \frac{d\omega _1(t)}{dt}>B_{u1}(\varphi _1(-\tau _u))/2=:K_1>0, \end{aligned}$$

for $t\in [0,t_2]$. Then $\omega _1(t)\ge K_1t$, for $t\in [0,t_2]$. In addition,

$$\begin{aligned} \frac{d\omega _2(t)}{dt}\ge & {} -\mu _{mu2}\omega _2(t) -k_{mu2}(\omega _2(t))^2 -c_{uv}(\overline{V}_2+\varrho ^*)\omega _2(t) +D_uK_1t-D_u\omega _2(t) \\= & {} -(\mu _{mu2}+k_{mu2}\omega _2(t)+c_{uv}(\overline{V}_2+\varrho ^*)+D_u)\omega _2(t) +D_uK_1t \\\ge & {} -K_2\omega _2(t)+D_uK_1t, \end{aligned}$$

where

$$\begin{aligned} K_2:= \mu _{mu2}+k_{mu2}\mathcal {W}_2+c_{uv}(\overline{V}_2+\varrho ^*)+D_u < \infty . \end{aligned}$$

By the comparison principle,

$$\begin{aligned} \omega _2(t)\ge e^{-K_2t}\omega _2(0)+\int _0^te^{-K_2t}D_uK_1sds>0, \end{aligned}$$

for $t\in (0,t_2]$, also a contradiction to $\varphi \in M_{\partial }$. (iii) If $\varphi _1(0)=0$ and $\varphi _1(-\tau _u)=0$, we set $\tau _u^*=\sup \{ -\theta | \theta \in [-\tau _u,0],~ \varphi _1(\theta )\ne 0 \} \le \tau _u$. Then $\frac{d\omega _1(t)}{dt}=0$ and $\omega _1(t)=0$ for $t\in [0,\tau _u-\tau _u^*]$, allowing $\tau _u-\tau _u^*=0$. From the assumption $\varphi _2=\hat{0}$, it also holds that $\omega _2(t)=0$ for $t\in [0,\tau _u-\tau _u^*]$. Since $\varphi \in C$, there is a small $\epsilon _1>0$ such that $\varphi _1(-\tau _u^*+\epsilon _1)>0$. Define

$$\begin{aligned} \psi (\theta )=\left\{ \begin{array}{ll} \varphi (\theta +\tau _u-\tau _u^*+\epsilon _1), &{} \textrm{if}~\theta \in [-\tau _u,\tau _u^*-\epsilon _1-\tau _u), \\ \omega (\theta +\tau _u-\tau _u^*+\epsilon _1), &{} \textrm{if}~\theta \in [\tau _u^*-\epsilon _1-\tau _u,0], \end{array} \right. \end{aligned}$$

where $\omega =(\omega _1,\omega _2)$. Then $\psi _2(\theta )\ge \hat{0}$ for $\theta \in [-\tau _u,0]$ and

$$\begin{aligned}{} & {} \psi _1(0) = \omega _1(\tau _u-\tau _u^*+\epsilon _1)\ge 0, \\{} & {} \psi _1(-\tau _u) = \varphi _1(-\tau _u^*+\epsilon _1)>0. \end{aligned}$$

By the comparison principle (for $\psi _2\ge \tilde{\psi }_2:=\hat{0}$) and previous result in (ii) (for $\varphi _1=\psi _1$ and $\varphi _2=\tilde{\psi }_2$), it yields that $\psi \notin M_{\partial }$. Note that $\check{\Phi }_t(\psi )=\check{\Phi }_{t+\tau _u-\tau _u^*+\epsilon _1}(\varphi )$. By the positive invariance of the set $M_{\partial }$, it leads to $\varphi \notin M_{\partial }$, a contradiction again. From the contradictions in all three cases (i)-(iii), we conclude that $M_{\partial } \subset \{ \hat{\textbf{0}} \}$, and hence $M_{\partial } = \{ \hat{\textbf{0}} \}$. The claim is thus justified.

(IV) The trivial equilibrium is a weak repeller in system (A.16): We claim that

$$\begin{aligned} \limsup _{t\rightarrow \infty }\max _{i} \{\omega _i(t)\}>\varrho ^*,~\mathrm{for~all}~\varphi \in X_0, \end{aligned}$$

(A.18)

where $\varrho ^*$ is defined in (A.15). Suppose, on the contrary, there exist an initial value $\varphi \in X_0$ and an $t_3>0$ such that $|\omega _i(t)|\le \varrho ^*$, $i=1,2$, for $t\ge t_3-\tau _u$. From (A.16), for $t\ge t_3$,

$$\begin{aligned} \frac{d\omega _1(t)}{dt}&\ge B'_{u1}(\varrho ^*)\omega _1(t-\tau _u)-(\mu _{mu1}+k_{mu1}\varrho ^*+c_{uv}(\overline{V}_1+\varrho ^*)+D_u)\omega _1(t)+D_u\omega _2(t), \\ \frac{d\omega _2(t)}{dt}&\ge B'_{u2}(\varrho ^*)\omega _2(t-\tau _u)-(\mu _{mu2}+k_{mu2}\varrho ^*+c_{uv}(\overline{V}_2+\varrho ^*)+D_u)\omega _2(t)+D_u\omega _1(t). \end{aligned}$$

We consider an auxiliary equation

$$\begin{aligned} \frac{dx_1(t)}{dt}&= B'_{u1}(\varrho ^*)x_1(t-\tau _u)-(\mu _{mu1}+k_{mu1}\varrho ^* +c_{uv}(\overline{V}_1+\varrho ^*)+D_u)x_1(t)+D_ux_2(t), \nonumber \\ \frac{dx_2(t)}{dt}&= B'_{u2}(\varrho ^*)x_2(t-\tau _u)-(\mu _{mu2}+k_{mu2}\varrho ^* +c_{uv}(\overline{V}_2+\varrho ^*)+D_u)x_2(t)+D_ux_1(t).\nonumber \\ \end{aligned}$$

(A.19)

From the comparison theory in Theorem 5.5.1 [32], we have

$$\begin{aligned} \tilde{\Phi }_t(\varphi )\le \check{\Phi }_t(\varphi ),~\textrm{for}~t\ge 0, \end{aligned}$$

where $\tilde{\Phi }_{t}$ is the semiflow of (A.19). Consider the ordinary differential equation associated with (A.19),

$$\begin{aligned} \frac{dy_1(t)}{dt}&= B'_{u1}(\varrho ^*)y_1(t)-(\mu _{mu1}+k_{mu1} \varrho ^*+c_{uv}(\overline{V}_1+\varrho ^*)+D_u)y_1(t)+D_uy_2(t), \nonumber \\ \frac{dy_2(t)}{dt}&= B'_{u2}(\varrho ^*)y_2(t)-(\mu _{mu2}+k_{mu2} \varrho ^*+c_{uv}(\overline{V}_2+\varrho ^*)+D_u)y_2(t)+D_uy_1(t). \nonumber \\ \end{aligned}$$

(A.20)

The characteristic equation at the trivial equilibrium is

$$\begin{aligned} \lambda ^2 -(r_1+r_2) \lambda +r_1r_2-D_u^2=0, \end{aligned}$$

(A.21)

where, for $i=1,2$,

$$\begin{aligned} r_i = B'_{ui}(\varrho ^*)-[\mu _{mui}+k_{mui}\varrho ^* +c_{uv}(\overline{V}_i+\varrho ^*)+D_u]. \end{aligned}$$

Equation (A.21) has a solution with positive real part whenever $r_1+r_2>0$ or $r_1r_2-D_u^2<0$, which are valid because of (A.15). Therefore, the stability modulus of (A.20) is positive. Also note that (A.19) is a cooperative irreducible system. From Theorem 5.5.1 and Corollary 5.5.2 [32], system (A.19) also admits a positive stability modulus associated with a positive eigenvector $\textbf{z}$. Note that the semiflow of (A.16) is eventually strong monotone in $[\hat{\textbf{0}},\hat{\textbf{r}}^*]$, where $\textbf{r}^*=(\rho ^*,\rho ^*)$, see Corollary 5.3.5 [32], and $\hat{\textbf{0}}$ is an equilibrium therein. There exist a $t_4>t_3$ and a small $\alpha >0$ such that

$$\begin{aligned} \hat{\textbf{0}}\ll \alpha \hat{\textbf{z}}\ll \check{\Phi }_{t_4}(\varphi ). \end{aligned}$$

Hence, we have

$$\begin{aligned} \tilde{\Phi }_t(\alpha \hat{\textbf{z}}) \le \check{\Phi }_t(\alpha \hat{\textbf{z}}) \ll \check{\Phi }_{t+t_4}(\varphi ), \end{aligned}$$

for $t\ge 0$, which is a contradiction to boundedness of the semiflow $\check{\Phi }_t$, and this contradiction proves (A.18).

Obviously, $\{\hat{\textbf{0}}\}$ is an isolated invariant set in $\partial X_0$, and the set $M_{\partial }$ consists of an acyclic equilibrium point. From (A.18), $W^s(\hat{\textbf{0}})\cap X_0 = \emptyset$, where $W^s(\hat{\textbf{0}})$ denotes the stable manifold of $\hat{\textbf{0}}$. By the persistence theory in Theorem 4.6 [39], system (A.16) is uniformly persistent with respect to $(X_0,\partial X_0)$ and the assertion is proved.

As for the case when $(\mathcal {S}_v)$ does not hold, we see from $(\mathcal {S}_u)$ that there exists a $\varrho ^{**}>0$ such that

$$\begin{aligned}{} & {} B_{u1}'(\varrho ^{**})>\mu _{mu1}+D_u,~\textrm{or}~~B_{u2}'(\varrho ^{**})>\mu _{mu2}+D_u,~\textrm{or} \nonumber \\{} & {} 0 \le \left( \mu _{mu1}+D_u-B_{u1}'(\varrho ^{**}) \right) \left( \mu _{mu2}+D_u-B_{u1}'(\varrho ^{**}) \right) <D_u^2. \end{aligned}$$

(A.22)

In addition, from Proposition 3.5, there exists a $t_5>0$ such that $V_i(t)<\varrho ^{**}$ for $t\ge t_5$ and $i=1,2$. Therefore, the U-equation in (3.1) is bounded below by the auxiliary system

$$\begin{aligned} \frac{d\tilde{\omega }_1(t)}{dt}= & {} B_{u1}(\tilde{\omega }_1(t-\tau _u)) - \mu _{mu1}\tilde{\omega }_1(t) -k_{mu1}(\tilde{\omega }_1(t))^2 \\{} & {} - c_{uv}\varrho ^{**}\tilde{\omega }_1(t) + D_u\tilde{\omega }_2(t) - D_u\tilde{\omega }_1(t), \\ \frac{d\tilde{\omega }_2(t)}{dt}= & {} B_{u2}(\tilde{\omega }_2(t-\tau _u)) - \mu _{mu2}\tilde{\omega }_2(t) -k_{mu2}(\tilde{\omega }_2(t))^2 \\{} & {} - c_{uv}\varrho ^{**}\tilde{\omega }_2(t) + D_u\tilde{\omega }_1(t) - D_u\tilde{\omega }_2(t), \end{aligned}$$

or $t\ge t_5$. In this case, the uniform persistence of u-species can be established by the same arguments with condition (A.22). This completes the proof. $\square$

Proof of Proposition 3.13

We show the case of $c^+_{uv}$. Note that the value of $\overline{V}_i$ depends on $\tau _v$ but not on $\tau _u$. A direct calculation gives

$$\begin{aligned} \frac{\partial c^+_{uv}}{\partial \tau _u}= & {} \frac{1}{2\overline{V}_1\overline{V}_2} \left\{ -\mu _{lu1}b'_1(0)e^{-\mu _{lu1}\tau _u}\overline{V}_2 -\mu _{lu2}b'_2(0)e^{-\mu _{lu2}\tau _u}\overline{V}_1\right. \\{} & {} \left. +\frac{d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1}{\sqrt{\Delta _u}} (-\mu _{lu1}b'_1(0)e^{-\mu _{lu1}\tau _u}\overline{V}_2 +\mu _{lu2}b'_2(0)e^{-\mu _{lu2}\tau _u}\overline{V}_1) \right\} \\< & {} 0. \end{aligned}$$

The last inequality holds true because of the fact $|(d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1)/\sqrt{\Delta _u}|\le 1$.

Now, we only vary the value of $\tau _v$ in

$$\begin{aligned} c_{uv}^+ = \frac{d_{u1}\overline{V}_2 + d_{u2}\overline{V}_1+\sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 }}{2\overline{V}_1\overline{V}_2 }. \end{aligned}$$

Note that in the right hand side, only $\overline{V}_i=\overline{V}_i(\tau _v)$, $i=1,2$, depends on $\tau _v$, and from Remark 2.2, $(\overline{V}_i)':=\frac{\partial \overline{V}_i}{\partial \tau _v}<0$ for $i=1,2$. Thus, we obtain

$$\begin{aligned} \frac{\partial c^+_{uv}}{\partial \tau _v}= & {} \frac{1}{(2\overline{V}_1\overline{V}_2)^2} \Bigg \{ \Bigg [ d_{u1}(\overline{V}_2)'+d_{u2}(\overline{V}_1) \nonumber \\{} & {} +\frac{(d1\overline{V}_1-d_{u2}\overline{V}_1)(d1(\overline{V}_2)'-d_{u2}(\overline{V}_1)') + 2D_u^2((\overline{V}_1)'\overline{V}_2+\overline{V}_1(\overline{V}_2)')}{\sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 }}\Bigg ] \cdot 2\overline{V}_1\overline{V}_2 \nonumber \\{} & {} \left. - 2\left[ d_{u1}\overline{V}_2 + d_{u2}\overline{V}_1+\sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 }\right] ( (\overline{V}_1)'\overline{V}_2+\overline{V}_1(\overline{V}_2)' ) \right\} \nonumber \\= & {} \frac{1}{(2\overline{V}_1\overline{V}_2)^2\sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 }} \cdot \nonumber \\{} & {} \Bigg \{ (\overline{V}_1)' 2\overline{V}_2 \Bigg [ d_{u2}\overline{V}_1\sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 } - d_{u2}\overline{V}_1(d_{u1}\overline{V}_2-d_{u2}\overline{V}_1) \nonumber \\{} & {} - (d_{u1}\overline{V}_2+d_{u2}\overline{V}_1)\sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 } \nonumber \\{} & {} - (d_{u1}\overline{V}_2-d_{u2}\overline{V}_1)^2 - 2D_u^2\overline{V}_1\overline{V}_2 \Bigg ] \nonumber \\{} & {} + (\overline{V}_2)' 2\overline{V}_1 \Bigg [ d_{u1}\overline{V}_2\sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 } + d_{u1}\overline{V}_2(d_{u1}\overline{V}_2-d_{u2}\overline{V}_1) \nonumber \\{} & {} - (d_{u1}\overline{V}_2+d_{u2}\overline{V}_1)\sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 } \nonumber \\{} & {} - (d_{u1}\overline{V}_2-d_{u2}\overline{V}_1)^2 - 2D_u^2\overline{V}_1\overline{V}_2 \Bigg ] \Bigg \} \nonumber \\= & {} \frac{ (\overline{V}_1)' (\overline{V}_2)^2 \Lambda _1(\tau _v) + (\overline{V}_2)' (\overline{V}_1)^2 \Lambda _2(\tau _v)}{2(\overline{V}_1\overline{V}_2)^2\sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 }}, \end{aligned}$$

(A.23)

where

$$\begin{aligned} \Lambda _1(\tau _v)= & {} d_{u1}\left( d_{u2}\overline{V}_1-d_{u1}\overline{V}_2 - \sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 } \right) -2 D_u^2\overline{V}_1, \\ \Lambda _2(\tau _v)= & {} d_{u2}\left( d_{u1}\overline{V}_2 -d_{u2}\overline{V}_1 - \sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 } \right) -2 D_u^2\overline{V}_2. \end{aligned}$$

Suppose that $D_u^2\le d_{u1}d_{u2}$. There are two cases to discuss. If $d_{u1}>0$ and $d_{u2}>0$, from (A.23) we obtain $\frac{\partial c^+_{uv}}{\partial \tau _v} >0$. If $d_{u1}<0$ and $d_{u2}<0$, from Lemma 3.12, we have $\sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 } \le |d_{u1}\overline{V}_2+d_{u2}\overline{V}_1|=-d_{u1}\overline{V}_2-d_{u2}\overline{V}_1$, and then $c_{uv}^+\le 0$.

Suppose that $D_u^2> d_{u1}d_{u2}$. From Lemma 3.12, it holds that $\sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 } >|d_{u1}\overline{V}_2+d_{u2}\overline{V}_1|$. Hence $\sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 } +d_{u1}\overline{V}_2 +d_{u2}\overline{V}_1>0$ which means that $c_{uv}^+$ is always positive. In addition, from Lemma 3.12 we have

$$\begin{aligned} \left| d_{u1}\sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 }\right|< & {} \left| d_{u1}(d_{u2}\overline{V}_1-d_{u1}\overline{V}_2)-2D_u^2\overline{V}_1\right| , \qquad \qquad \end{aligned}$$

(A.24)

$$\begin{aligned} \left| d_{u2}\sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 }\right|< & {} \left| d_{u1}(d_{u1}\overline{V}_2-d_{u2}\overline{V}_1)-2D_u^2\overline{V}_2\right| . \qquad \qquad \end{aligned}$$

(A.25)

If $d_{u1}>0$ and $d_{u2}>0$, from (A.23) we obtain $\frac{\partial c^+_{uv}}{\partial \tau _v} >0$. If $d_{u1}\le 0$ and $d_{u2}>0$, (A.24) implies that

$$\begin{aligned} 0\ge d_{u1}\sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 } > d_{u1}(d_{u2}\overline{V}_1-d_{u1}\overline{V}_2)-2D_u^2\overline{V}_1, \end{aligned}$$

which means that

$$\begin{aligned} \Lambda _1(\tau _v)<0. \end{aligned}$$

In addition, we observe that

$$\begin{aligned} d_{u2}\sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 }>0>d_{u2}(d_{u1}\overline{V}_2-d_{u2}\overline{V}_1)-2D_u^2\overline{V}_2, \end{aligned}$$

which means that

$$\begin{aligned} \Lambda _2(\tau _v)<0. \end{aligned}$$

Together with the fact $(\overline{V}_i)'<0$ for $i=1,2$, we see that $\frac{\partial c^+_{uv}}{\partial \tau _v} >0$. If $d_{u1}> 0$ and $d_{u2}\le 0$, a similar argument under (A.25) also leads to $\frac{\partial c^+_{uv}}{\partial \tau _v} >0$. If $d_{u1}\le 0$ and $d_{u2}\le 0$, from (A.24) we obtain

$$\begin{aligned} 0\ge d_{u1}\sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 } > d_{u1}(d_{u2}\overline{V}_1-d_{u1}\overline{V}_2)-2D_u^2\overline{V}_1, \end{aligned}$$

which implies $\Lambda _1(\tau _v)<0$. From (A.25) we obtain

$$\begin{aligned} 0\ge d_{u2}\sqrt{ ( d_{u1}\overline{V}_2 - d_{u2}\overline{V}_1 )^2 +4\overline{V}_1\overline{V}_2D_u^2 } > d_{u1}(d_{u1}\overline{V}_2-d_{u2}\overline{V}_1)-2D_u^2\overline{V}_2, \end{aligned}$$

which implies $\Lambda _2(\tau _v)<0$. Again, we see that $\frac{\partial c^+_{uv}}{\partial \tau _v} >0$. This completes the proof. $\square$

Proof of Theorem 3.15

We first state conditions (H1)–(H4) in Theorem B [16], with the notations for system (3.1), as follows:

• (H1) The semiflow $\Phi _t$ is strictly order-preserving with respect to the order induced from the cone $\mathcal {C}_K$, that is $\Phi _t(\phi ) <_K \Phi _t(\psi )$ whenever $\phi <_K \psi$. In addition, for each $t>0$, $\Phi _t:\mathbb {X}\rightarrow \mathbb {X}$ is order compact, i.e., for each $(\tilde{\phi }_1,\tilde{\phi }_2)\in \mathbb {X}$, $\Phi _t([\hat{\textbf{0}},\tilde{\phi }_1]\times [\hat{\textbf{0}},\tilde{\phi }_2])$ has compact closure in $\mathbb {X}$.

• (H2) The trivial solution $E_0$ is a repelling equilibrium.

• (H3) $\Phi _t(\mathcal {C}_u\times \{\hat{\textbf{0}}\}) \subset \mathcal {C}_u\times \{\hat{\textbf{0}}\}$ for $t\ge 0$. $E_u$ attracts all solutions in $\mathcal {C}_u\times \{\hat{\textbf{0}}\}$ except the trivial solution. The symmetric conditions hold for $\Phi _t$ on $\{\hat{\textbf{0}}\} \times \mathcal {C}_v$.

• (H4) If $\phi =(\tilde{\phi }_1,\tilde{\phi }_2)\in \mathbb {X}$ and $\tilde{\phi }_i\ne \hat{\textbf{0}}$, $i=1,2$, then $\Phi _t(\phi )\gg \hat{\textbf{0}}$ for $t>\tau _m$. If $\phi ,\psi \in \mathbb {X}$ satisfy $\phi <_K \psi$, and either $\phi$ or $\psi$ belongs to $\textrm{int}(\mathbb {X})$, then $\Phi _t(\phi )\ll _K \Phi _t(\psi )$ for $t\ge 2\tau _m$.

Next, we show that all conditions hold under $(\mathcal {S}_u)$ and $(\mathcal {S}_v)$. For (H1), suppose $\phi <_K \psi$. From Proposition 3.3, it satisfies $\Phi _t(\phi ) \le _K \Phi _t(\psi )$. Hence, it suffices to show that $\Phi _t(\phi ) \ne \Phi _t(\psi )$ for $t>0$. Denote $\Phi _t(\phi )=(U^{\phi }_1(t),U^{\phi }_2(t),V^{\phi }_1(t),V^{\phi }_2(t))$ and $\Phi _t(\psi )=(U^{\psi }_1(t),U^{\psi }_2(t),V^{\psi }_1(t),V^{\psi }_2(t))$ for $t\ge 0$. If $\phi _{i_0}\ne \psi _{i_0}$ for $i_0=1$ or 2 (denote $j_0=2$ or 1, respectively), we claim that $\Phi _t(\phi ) \ne \Phi _t(\psi )$ for $t\in [0,\tau _u]$. Otherwise, there is a $t_1\in [0,\tau _u]$ such that $\Phi _{t_1}(\phi ) = \Phi _{t_1}(\psi )$, which means that

$$\begin{aligned} \phi (\theta )=\psi (\theta )~\textrm{for}~t_1-\tau _u\le \theta \le 0, \end{aligned}$$

(A.26)

and $U^{\phi }(t)=U^{\psi }(t)$ for $0\le t\le t_1$. In addition, from (3.1) we have

$$\begin{aligned} U^{\phi }_{i_0}(t_1)= & {} \int ^{t_1}_0[B_{ui_0}(U^{\phi }_{i_0}(\xi -\tau _u)) -\mu _{mui_0}U^{\phi }_{i_0}(\xi ) -k_{mui_0}(U^{\phi }_{i_0}(\xi ))^2 \\{} & {} ~~~~~~~~~~ -c_{uv} U^{\phi }_{i_0}(\xi )V^{\phi }_{i_0}(\xi ) +D_uU^{\phi }_{j_0}(\xi ) -D_uU^{\phi }_{i_0}(\xi )] d\xi , \\ U^{\psi }_{i_0}(t_1)= & {} \int ^{t_1}_0[B_{ui_0}(U^{\psi }_{i_0}(\xi -\tau _u)) -\mu _{mui_0}U^{\psi }_{i_0}(\xi ) -k_{mui_0}(U^{\psi }_{i_0}(\xi ))^2 \\{} & {} ~~~~~~~~~~ -c_{uv} U^{\psi }_{i_0}(\xi )V^{\psi }_{i_0}(\xi ) +D_uU^{\psi }_{j_0}(\xi ) -D_uU^{\psi }_{i_0}(\xi )] d\xi . \end{aligned}$$

Since $U^{\phi }_{i_0}(\xi -\tau _u)\le U^{\psi }_{i_0}(\xi -\tau _u)$, $U^{\phi }_{i_0}(\xi )= U^{\psi }_{i_0}(\xi )$, $U^{\phi }_{j_0}(\xi )\le U^{\psi }_{j_0}(\xi )$, and $V^{\phi }_{i_0}(\xi )\ge V^{\psi }_{i_0}(\xi )$ for $\xi \in [0,t_1]$, it enforces $U^{\phi }_{i_0}(\xi -\tau _u)= U^{\psi }_{i_0}(\xi -\tau _u)$ for $\xi \in [0,t_1]$, i.e.,

$$\begin{aligned} U^{\phi }_{i_0}(\theta )= U^{\psi }_{i_0}(\theta )~\textrm{for}~\theta \in [-\tau _u,t_1-\tau _u]. \end{aligned}$$

(A.27)

From (A.26) and (A.27), it leads to $\phi _{i_0}=\psi _{i_0}$, which is a contradiction. Hence, $\Phi _t(\phi ) \ne \Phi _t(\psi )$ for $t\in [0,\tau _u]$. Proceeding the arguments successively for $t\in [k\tau _u,(k+1)\tau _u]$, $k=1,2,\cdots$, we derive $\Phi _t(\phi ) \ne \Phi _t(\psi )$ for $t\ge 0$. When $\phi _{i_0}\ne \psi _{i_0}$ for $i_0=3$ or 4, the same result can be shown by the same argument. Therefore, $\Phi _t(\phi ) <_K \Phi _t(\psi )$ for $t\ge 0$. In addition, the second assertion is true because of eventually uniform boundedness shown in Theorem 3.1.

For (H2), when $(\mathcal {S}_u)$ holds, there exists a sufficiently small $\eta ^*>0$ such that

$$\begin{aligned} \left\{ \begin{array}{l} (\mathcal {S}_u^1)~~\mu _{mu1}+k_{mu1}\eta ^*+c_{uv}\eta ^*+D_u< e^{-\mu _{lu1}\tau _u}b'_{u1}(\eta ^*),~\textrm{or} \\ (\mathcal {S}_u^2)~~\mu _{mu2}+k_{mu2}\eta ^*+c_{uv}\eta ^*+D_u< e^{-\mu _{lu2}\tau _u}b'_{u2}(\eta ^*),~\textrm{or} \\ (\mathcal {S}_u^3)~~0 \le \left( \mu _{mu1}+k_{mu1}\eta ^*+c_{uv}\eta ^*+D_u-e^{-\mu _{lu1}\tau _u}b'_{u1}(\eta ^*) \right) \times \\ ~~~~~~~~~~~~~~~~~ \left( \mu _{mu2}+k_{mu2}\eta ^*+c_{uv}\eta ^*+D_u-e^{-\mu _{lu2}\tau _u}b'_{u2}(\eta ^*) \right) <D_u^2, \end{array} \right. \end{aligned}$$

due to the assumption on functions $b_{ui}$ and $b_{vi}$, $i=1,2$. We claim that $E_0$ repels the set $N_{\eta ^*}:=\{\phi \in \mathbb {X} | \hat{0}\le \phi _i\le \hat{\eta }^*,~i=1,2,3,4\}\setminus \{E_0\}$, i.e., for each $\phi \in N_{\eta ^*}$, there exists a $t_2>0$ such that $\Phi _{t_2}(\phi )\notin N_{\eta ^*}$. Suppose, for the sake of contradiction, $\Phi _t(\phi )\in N_{\eta ^*}$ for $t>0$. Then $0\le U_1(t),U_2(t),V_1(t),V_2(t)\le \eta ^*$ for $t>0$. However, from the U-equation of (3.1), we see that

$$\begin{aligned} \frac{dU_1(t)}{dt}\ge & {} B_{u1}'(\eta ^*)U_1(t-\tau _u) - \mu _{mu1}U_1(t) -k_{mu1}\eta ^*U_1(t) - c_{uv}\eta ^*U_1(t) \\ {}{} & {} + D_uU_2(t) - D_uU_1(t), \\ \frac{dU_2(t)}{dt}\ge & {} B_{u2}'(\eta ^*)U_1(t-\tau _u) - \mu _{mu2}U_2(t) -k_{mu2}\eta ^*U_2(t) - c_{uv}\eta ^*U_2(t) \\ {}{} & {} + D_uU_1(t) - D_uU_2(t). \end{aligned}$$

When $(\mathcal {S}_u^1)$ holds, we consider

$$\begin{aligned} \frac{dU_1(t)}{dt} \ge B_{u1}'(\eta ^*)U_1(t-\tau _u) - \mu _{mu1}U_1(t) -k_{mu1}\eta ^*U_1(t) - c_{uv}\eta ^*U_1(t) - D_uU_1(t). \end{aligned}$$

By Theorem 2.3 [40] and a comparison principle, it holds that $\lim _{t\rightarrow \infty } U_1(t)=\infty$, which is a contradiction. A similar contradiction also occurs when $(\mathcal {S}_u^2)$ holds. When $(\mathcal {S}_u^3)$ holds, we consider the auxiliary equation

$$\begin{aligned} \frac{dx_1(t)}{dt}&= B'_{u1}(\eta ^*)x_1(t-\tau _u)-(\mu _{mu1}+k_{mu1}\eta ^*+c_{uv}\eta ^*+D_u)x_1(t)+D_ux_2(t), \nonumber \\ \frac{dx_2(t)}{dt}&= B'_{u2}(\eta ^*)x_2(t-\tau _u)-(\mu _{mu2}+k_{mu2}\eta ^*+c_{uv}\eta ^*+D_u)x_2(t)+D_ux_1(t). \nonumber \\ \end{aligned}$$

(A.28)

Denote $\hat{\Upsilon }_t(\phi )=(U_1(t),U_2(t))$. From the comparison theory in Theorem 5.5.1 [32], we have

$$\begin{aligned} \Upsilon _t(\phi )\le \hat{\Upsilon }_t(\phi ),~\textrm{for}~t\ge 0,~\phi \in \mathcal {C}_u, \end{aligned}$$

where $\Upsilon _t$ is the semiflow of (A.28). Discussing (A.28) under condition $(\mathcal {S}_u^3)$ as in the proof (IV) of Theorem 3.11, a contradiction arises, as there exists a solution of system (3.1) with unbounded $(U_1(t),U_2(t))$. Therefore, $E_0$ repels the set $N_{\eta ^*}$. Note that with only condition $(\mathcal {S}_v)$, it also enforces $E_0$ to be repelling, once we manipulate a similar argument.

For (H3), obviously, the set $\mathcal {C}_u\times \{\hat{\textbf{0}}\}$ is positively invariant under $\Phi _t$ and Theorem 2.4 implies the global attractivity of $E_u$ in $\mathcal {C}_u\times \{\hat{\textbf{0}}\}\setminus \{\hat{\textbf{0}}\times \hat{\textbf{0}}\}$. Similarly, the symmetric case holds true.

For (H4), the first assertion is confirmed by Proposition 3.2(ii). We discuss the second one in two cases. If $\phi <_K \psi$ and $\phi \in \textrm{int}(\mathbb {X})$, Proposition 3.4 implies $\Phi _t(\phi )\ll _K \Phi _t(\psi )$ for $t>2\tau _m$. If $\phi <_K \psi$, $\phi \in \partial \mathbb {X}$ and $\psi \in \textrm{int}(\mathbb {X})$, we denote the non-empty subset $\mathcal {I}\subset \{1,2,3,4\}$ with $\phi _i\ne \psi _i$ for $i\in \mathcal {I}$. Define $\phi ^p$ by $\phi ^p_i=\psi _i$ for $i\notin \mathcal {I}$, and $\phi ^p_i=\frac{1}{2}(\phi _i+\psi _i)$ for $i\in \mathcal {I}$. Then $\hat{\textbf{0}}\ll \phi ^p$, where $\textbf{0}=(0,0,0,0)$, and $\phi<_K\phi ^p <_K\psi$ since $\mathcal {I}$ is non-empty. From Proposition 3.3 and Proposition 3.4, it reveals that

$$\begin{aligned} \Phi _t(\phi ) \le _K \Phi _t(\phi ^p) \ll _K\Phi _t(\psi ), \end{aligned}$$

for $t\ge 2\tau _m$. This completes the proof. $\square$

1.2 A.II Limit Semiflows of (Non-autonomous) Continuous Semiflows

We first define limit autonomous semiflows, and refer to [38] for more details. Let $\Phi _{t+t_0}(t_0,\psi )$ be a continuous semiflow on a metric space $\textbf{X}$, with initial time $t_0$ and initial value $\psi$.

Definition A.1

A (non-autonomous) continuous semiflow $\Phi$ is called asymptotically autonomous, with a limit-semiflow $\Theta$, if $\Phi _{t_j+s_j}(s_j,\psi ^{(j)})\rightarrow \Theta _t(\psi ),~\textrm{as}~j\rightarrow \infty$, for any sequences $t_j\rightarrow t$, $s_j\rightarrow \infty$, $\psi ^{(j)}\rightarrow \psi$ for $j\rightarrow \infty$, with $\psi ^{(j)}$, $\psi \in \textbf{X}$, $0\le t_j,~t<\infty$, and $s_j\ge 0$.

If $\Phi$ is an asymptotically autonomous continuous semiflow and $\Theta$ is its continuous limit-semiflow, the global convergence dynamics of $\Phi$ to an equilibrium can be obtained by checking its asymptotical stability and basin of attraction under the limit semiflow.

Theorem A.1

(Theorem 4.1 [38]) Let e be a locally asymptotically stable equilibrium of $\Theta _t$ and $\mathcal {B}(e)=\{x\in \textbf{X}~:~\Theta _t(x)\rightarrow e,~\textrm{as}~t\rightarrow \infty \}$ its basin of attraction. Then every pre-compact $\Phi$-orbit whose $\omega$-$\Phi$-limit set intersects $\mathcal {B}(e)$ converges to e.

1.3 A.III A Limit Semiflow of (3.1) When $(\mathcal {S}_v)$ Does Not Hold

We discuss the limit semiflow of system (3.1) based on the method in [29], where a scalar case was considered.

Theorem A.2

When $(\mathcal {S}_v)$ does not hold, the solution flow of system (3.22) is a limit semiflow of system (3.1).

Proof

Let $\textbf{W}=(W_1(t),W_2(t),W_3(t),W_4(t))$ and $\textbf{Z}=(Z_1(t),Z_2(t),Z_3(t),Z_4(t))$ be a solution for system (3.1) and system (3.22) respectively, where $W_1=U_1$, $W_2=U_2$, $W_3=V_1$ and $W_4=V_2$, and similarly for $Z_j$, $j=1,\ldots ,4$. With $\textbf{W}_t(\theta ):=\textbf{W}(t+\theta )$ and $\textbf{Z}_t(\theta ):=\textbf{Z}(t+\theta )$, $\textbf{W}(t)$ and $\textbf{Z}(t)$ satisfy

$$\begin{aligned} \textbf{W}'=\mathcal {F}(\textbf{W}_t) + \mathcal {G}(t)~\textrm{and}~\textbf{Z}'=\mathcal {F}(\textbf{Z}_t), \end{aligned}$$

respectively, where $\mathcal {F}=(\mathcal {F}_1,\mathcal {F}_2,\mathcal {F}_3,\mathcal {F}_4)$ with

$$\begin{aligned} \mathcal {F}_1(\textbf{W}_t)= & {} B_{u1}(W_1(t-\tau _u)) -\mu _{mu1}W_1(t) - k_{mu1}(W_1(t))^2 +D_uW_2(t)-D_uW_1(t), \\ \mathcal {F}_2(\textbf{W}_t)= & {} B_{u2}(W_2(t-\tau _u)) -\mu _{mu2}W_2(t) - k_{mu2}(W_2(t))^2 +D_uW_1(t)-D_uW_2(t), \\ \mathcal {F}_3(\textbf{W}_t)= & {} -D_vW_3(t), \\ \mathcal {F}_4(\textbf{W}_t)= & {} -D_vW_4(t), \end{aligned}$$

and $\mathcal {G}(t)=(\mathcal {G}_1,\mathcal {G}_2,\mathcal {G}_3,\mathcal {G}_4)=\mathcal {G}(\textbf{W}_t)$ with

$$\begin{aligned} \mathcal {G}_1(t)= & {} -c_{uv}W_1(t)W_3(t), \\ \mathcal {G}_2(t)= & {} -c_{uv}W_2(t)W_4(t), \\ \mathcal {G}_3(t)= & {} B_{v1}(W_3(t-\tau _v)) -\mu _{mv1}W_3(t) - k_{mv1}(W_3(t))^2 +D_vW_4(t)-c_{uv}W_1(t)W_3(t), \\ \mathcal {G}_4(t)= & {} B_{v2}(W_4(t-\tau _v)) -\mu _{mv2}W_4(t) - k_{mv2}(W_4(t))^2 +D_vW_3(t)-c_{uv}W_2(t)W_4(t). \end{aligned}$$

As in Definition A.1, we denote the semiflow of (3.1) by $\Phi _{t+t_0}(t_0,\psi )$, with initial time $t_0$ and initial value $\psi$, and by $\Theta _t(\psi )$ the semiflow of (3.22), with initial time 0 and initial value $\psi$. Then we may express

$$\begin{aligned} \Phi _{t+t_0}(t_0,\psi )=\textbf{W}_{t+t_0}(t_0,\psi ),~~\Theta _t(\psi )=\textbf{Z}_t(0,\psi ), \end{aligned}$$

which satisfy $\textbf{W}_{t_0}=\psi$ and $\textbf{Z}_0=\psi$. Note that $\textbf{Z}_t(0,\psi )=\textbf{Z}_{t+s}(s,\psi )$ for all $s\ge 0$. Given sequences $t_j\rightarrow t$, $s_j\rightarrow \infty$, $\psi ^{(j)}\rightarrow \psi$ as $j\rightarrow \infty$, with $\psi ^{(j)}$, $\psi \in \mathbb {X}$, and $s_j\ge 0$, we define

$$\begin{aligned} \mathcal {D}_k^{(j)}(\theta )=W_{k,t_j+s_j}\left( s_j,\psi ^{(j)}\right) (\theta ) - Z_{k,t}(0,\psi )(\theta ). \end{aligned}$$

Then

$$\begin{aligned} \mathcal {D}_k^{(j)}(\theta )= & {} W_{k,t_j+s_j}\left( s_j,\psi ^{(j)}\right) (\theta ) - Z_{k,t+s_j}(s_j,\psi )(\theta ) \\= & {} W_k^{(j)}(t_j+s_j+\theta ) - Z_k^{(j)}(t+s_j+\theta ), \end{aligned}$$

where $\textbf{W}_{t+s_j}^{(j)}$ and $\textbf{Z}_{t+s_j}^{(j)}$ are the solutions of (3.1) and (3.22) respectively, both starting from the initial time $s_j$ and respectively from the initial values $\psi ^{(j)}$ and $\psi$. It suffices to show that

$$\begin{aligned} \max _{-\tau _m\le \theta \le 0} \sum _{k=1}^4 \left| \mathcal {D}_k^{(j)}(\theta ) \right| \rightarrow 0,~\textrm{as}~ j\rightarrow \infty . \end{aligned}$$

Since $s_j\rightarrow \infty$ as $j\rightarrow \infty$, we assume, without loss of generality, that $t_j+s_j+\theta >0$ for all j. Define $\mathcal {G}^{(j)}(t)=\mathcal {G}(\textbf{W}_t^{(j)})$, and note that

$$\begin{aligned} W_k^{(j)}(t_j+s_j+\theta )= & {} \psi ^{(j)}_k(0) + \int ^{t_j+s_j+\theta }_{s_j} \left[ \mathcal {F}_k(\textbf{W}_{\xi }^{(j)}) + \mathcal {G}_k^{(j)}(\xi )\right] d\xi , \\ Z_k^{(j)}(t+s_j+\theta )= & {} \psi _k(0) + \int ^{t+s_j+\theta }_{s_j} \mathcal {F}_k\left( \textbf{Z}_{\xi }^{(j)}\right) d\xi . \end{aligned}$$

Thus

$$\begin{aligned} \mathcal {D}_k^{(j)}(\theta )= & {} \psi ^{(j)}_k(0)-\psi _k(0) + \int ^{t_j+s_j+\theta }_{s_j} \mathcal {G}_k^{(j)}(\xi ) d\xi \\{} & {} + \int ^{t_j+s_j+\theta }_{s_j} \mathcal {F}_k\left( \textbf{W}_{\xi }^{(j)}\right) d\xi - \int ^{t+s_j+\theta }_{s_j} \mathcal {F}_k\left( \textbf{Z}_{\xi }^{(j)}\right) d\xi \\= & {} \psi ^{(j)}_k(0)-\psi _k(0) + \int ^{t_j+s_j+\theta }_{s_j} \mathcal {G}_k^{(j)}(\xi ) d\xi \\{} & {} + \int ^{t_j+s_j+\theta }_{s_j} \left[ \mathcal {F}_k\left( \textbf{W}_{\xi }^{(j)}\right) - \mathcal {F}_k\left( \textbf{Z}_{\xi }^{(j)}\right) \right] d\xi - \int ^{t+s_j+\theta }_{t_j+s_j+\theta } \mathcal {F}_k\left( \textbf{Z}_{\xi }^{(j)}\right) d\xi . \end{aligned}$$

In addition, we see that

$$\begin{aligned}{} & {} \int ^{t_j+s_j+\theta }_{s_j} \left[ \mathcal {F}_1\left( \textbf{W}_{\xi }^{(j)}\right) - \mathcal {F}_1\left( \textbf{Z}_{\xi }^{(j)}\right) \right] d\xi \\{} & {} \quad \le L_1 \int ^{t_j+s_j+\theta }_{s_j} \left( \left| W_1^{(j)}(\xi -\tau _u)-Z_1^{(j)}(\xi -\tau _u)\right| +\left| W_1^{(j)}(\xi )-Z_1^{(j)}(\xi )\right| \right. \\{} & {} \quad \quad \left. +\left| W_2^{(j)}(\xi )-Z_2^{(j)}(\xi )\right| \right) d\xi \\{} & {} \quad \le L_1 \int ^{t_j+s_j+\theta }_{s_j-\tau _u} \left( 2\left| W_1^{(j)}(\xi )-Z_1^{(j)}(\xi )\right| +\left| W_2^{(j)}(\xi )-Z_2^{(j)}(\xi )\right| \right) d\xi , \\{} & {} \quad \le L_1 \int ^{\theta }_{-t_j-\tau _u} \left( 2\left| W_1^{(j)}(t_j+s_j+\xi )-Z_1^{(j)}(t_j+s_j+\xi )\right| \right. \\{} & {} \quad \quad \left. +\left| W_2^{(j)}(t_j+s_j+\xi )-Z_2^{(j)}(t_j+s_j+\xi )\right| \right) d\xi , \end{aligned}$$

where $L_1=\max \{B_{u1}'(0),\mu _{mu1}+k_{mu1}(\mathcal {U}_1^W+\mathcal {U}_1^Z)+D_u\}<\infty$, $\mathcal {U}_1^W=\sup _{t\ge 0}U_1(t)<\infty$ for the solution in (3.1), and $\mathcal {U}_1^Z=\sup _{t\ge 0}U_1(t)<\infty$ for the solution in (3.22). Similarly, we have

$$\begin{aligned}{} & {} \int ^{t_j+s_j+\theta }_{s_j} \left[ \mathcal {F}_2\left( \textbf{W}_{\xi }^{(j)}\right) - \mathcal {F}_2\left( \textbf{Z}_{\xi }^{(j)}\right) \right] d\xi \\{} & {} \quad \le L_2 \int ^{\theta }_{-t_j-\tau _u} \left( 2\left| W_2^{(j)}(t_j+s_j+\xi )-Z_2^{(j)}(t_j+s_j+\xi )\right| \right. \\{} & {} \quad \quad +\left. \left| W_1^{(j)}(t_j+s_j+\xi )-Z_1^{(j)}(t_j+s_j+\xi )\right| \right) d\xi , \end{aligned}$$

for some positive consttant $L_2<\infty$. Obviously, denoting $L_k=D_v$ for $k=3,4$, we have

$$\begin{aligned}{} & {} \int ^{t_j+s_j+\theta }_{s_j} \left[ \mathcal {F}_k\left( \textbf{W}_{\xi }^{(j)}\right) - \mathcal {F}_k\left( \textbf{Z}_{\xi }^{(j)}\right) \right] d\xi \\ {}{} & {} \quad \le L_k \int ^{\theta }_{-t_j} \left| W_k^{(j)}(t_j+s_j+\xi )-Z_k^{(j)}(t_j+s_j+\xi )\right| d\xi \\{} & {} \quad \le L_k \int ^{\theta }_{-t_j-\tau _u} \left| W_k^{(j)}(t_j+s_j+\xi )-Z_k^{(j)}(t_j+s_j+\xi )\right| d\xi . \end{aligned}$$

Therefore,

$$\begin{aligned} \sum _{k=1}^4 \left| \mathcal {D}_k^{(j)}(\theta )\right|\le & {} \sum _{k=1}^4 \left\{ \left| \psi ^{(j)}_k(0)-\psi _k(0)\right| + \int ^{t_j+s_j+\theta }_{s_j} \left| \mathcal {G}_k^{(j)}(\xi ))\right| d\xi \right. \\{} & {} \left. + \int ^{t_j+s_j+\theta }_{s_j} \left| \mathcal {F}_k\left( \textbf{W}_{\xi }^{(j)}\right) - \mathcal {F}_k\left( \textbf{Z}_{\xi }^{(j)}\right) \right| d\xi + \int ^{t+s_j+\theta }_{t_j+s_j+\theta } \left| \mathcal {F}_k\left( \textbf{Z}_{\xi }^{(j)}\right) \right| d\xi \right\} \\\le & {} \sum _{k=1}^4 \left\{ \left| \psi ^{(j)}_k(0)-\psi _k(0)\right| + \int ^{t_j+s_j+\theta }_{s_j} \left| \mathcal {G}_k^{(j)}(\xi ))\right| d\xi + \int ^{t+s_j+\theta }_{t_j+s_j+\theta } \left| \mathcal {F}_k\left( \textbf{Z}_{\xi }^{(j)}\right) \right| d\xi \right. \\{} & {} \left. + \tilde{L} \int ^{\theta }_{-t_j-\tau _u} \left| W_k^{(j)}(t_j+s_j+\xi )-Z_k^{(j)}(t_j+s_j+\xi )\right| d\xi \right\} \\= & {} \mathcal {H}(\theta ;j) + \tilde{L} \int ^{\theta }_{-t_j-\tau _u} \sum _{k=1}^4 \left| \mathcal {D}_k^{(j)}(\xi )\right| d\xi , \end{aligned}$$

where $\tilde{L}=\max \{2L_1+L_2,L_1+2L_2,L_3,L_4\}$ and

$$\begin{aligned} \mathcal {H}(\theta ;j) = \sum _{k=1}^4 \left\{ \left| \psi ^{(j)}_k(0)-\psi _k(0)\right| + \int ^{t_j+s_j+\theta }_{s_j} \left| \mathcal {G}_k^{(j)}(\xi ))\right| d\xi + \int ^{t+s_j+\theta }_{t_j+s_j+\theta } \left| \mathcal {F}_k(\textbf{Z}_{\xi }^{(j)})\right| d\xi \right\} . \end{aligned}$$

Using Gronwall’s inequality, it leads to

$$\begin{aligned} \sum _{k=1}^4 \left| \mathcal {D}_k^{(j)}(\theta )\right| \le \mathcal {H}(\theta ;j) + \tilde{L} \int ^{\theta }_{-t_j-\tau _u} \mathcal {H}(\xi ;j) \exp \left( \int ^{\theta }_{\xi }\tilde{L} d\tilde{\xi } \right) d\xi . \end{aligned}$$

Note that each $\mathcal {G}^{(j)}(t)$ exponentially decays in time when $(\mathcal {S}_v)$ does not hold. In fact, $(W_3(t),W_4(t))$ is dominated by the solution of

$$\begin{aligned} \frac{dx_3(t)}{dt}= & {} B_{v1}'(0)x_3(t-\tau ) - \mu _{mv1}x_1(t) + Dx_4(t) - Dx_3(t), \nonumber \\ \frac{dx_4(t)}{dt}= & {} B_{v2}'(0)x_4(t-\tau ) - \mu _{mv2}x_2(t) + Dx_3(t) - Dx_4(t), \end{aligned}$$

(A.29)

which exponentially decays in time due to its negative stability modulus when $(\mathcal {S}_v)$ does not hold, as in the proof in Theorem 2.4. Thus, $\lim _{j\rightarrow \infty }\int ^{t_j+s_j+\theta }_{s_j} |\mathcal {G}_k^{(j)}(\xi ))| d\xi = 0$. In addition, since $\psi ^{(j)}\rightarrow \psi$ as $j\rightarrow \infty$, and each $\mathcal {F}_k$ is continuous, it holds that $\mathcal {H}(\theta ;j)\rightarrow 0$, and then $\sum _{k=1}^4 |\mathcal {D}_k^{(j)}(\theta )|\rightarrow 0$, as $j\rightarrow \infty$. This completes the proof.