A diffusive virus model with a fixed intracellular delay and combined drug treatments

Abstract

The method of administration of an effective drug treatment to eradicate viruses within a host is an important issue in studying viral dynamics. Overuse of a drug can lead to deleterious side effects, and overestimating the efficacy of a drug can result in failure to treat infection. In this study, we proposed a reaction-diffusion within-host virus model which incorporated information regarding spatial heterogeneity, drug treatment, and the intracellular delay to produce productively infected cells and viruses. We also calculated the basic reproduction number ${\mathcal {R}}_0$ under the assumptions of spatial heterogeneity. We have shown that the infection-free periodic solution is globally asymptotically stable when ${\mathcal {R}}_0<1$, whereas when ${\mathcal {R}}_0>1$ there is an infected periodic solution and the infection is uniformly persistent. By conducting numerical simulations, we also revealed the effects of various parameters on the value of ${\mathcal {R}}_0$. First, we showed that the dependence of ${\mathcal {R}}_0$ on the intracellular delay could be monotone or non-monotone, depending on the death rate of infected cells in the immature stage. Second, we found that the mobility of infected cells or virions could facilitate the virus clearance. Third, we found that the spatial fragmentation of the virus environment enhanced viral infection. Finally, we showed that the combination of spatial heterogeneity and different sets of diffusion rates resulted in complicated viral dynamic outcomes.

Appendix

In this appendix, we will finish the proofs of Lemma 2.1, Lemma 2.2, Lemma 2.3, and Theorem 3.5.

Proof of Lemma 2.1

In view of system (2.16), we see that (2.13)–(2.14) admits a mild solution in the following integral form:

$$\begin{aligned} {\left\{ \begin{array}{ll} u(\cdot ,t) = {\mathbb {T}}(t,0)\phi (0) + \int ^t_0{\mathbb {T}}(t,s)F(s,u_s)ds,~\phi \in {\mathbb {C}}_{\tau }^+,~t>0, \\ u_0(\theta )=\phi (\theta )~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,~\theta \in [-\tau ,0]. \end{array}\right. } \end{aligned}$$

Clearly, F is locally Lipschitz continuous. Set $\bar{a}=\max _{x\in {\bar{\Omega }},t\in [0,\omega ]} a(x,t)$, ${\bar{\beta }}_j=\max _{x\in {\bar{\Omega }},t\in [0,\omega ]} \beta _j(x,t)$ for $j=V,~I$, and ${\underline{K}}=\min _{x\in {\bar{\Omega }},t\in [0,\omega ]} K(x,t)$. For any $(t,\phi )\in {\mathbb {R}}_+\times {\mathbb {C}}_{\tau }^+$ and $\xi >0$, we observe that

$$\begin{aligned}&\phi (x,0) + \xi F(t,\phi )(x) \nonumber \\&~~~=\left( \begin{array}{l} \phi _1(x,0) + \xi \left[ \Lambda (x,t) + a(x,t)\phi _1(x,0)\left( 1 - \frac{\phi _1(x,0)}{K(x,t)} \right) \right. \\ ~~~~~~~~~~~~~~~~~~~ - \left. \frac{\beta _V(x,t)\phi _1(x,0)\phi _3(x,0)}{1+\alpha _V(x,t)\phi _3(x,0)} - \frac{\beta _I(x,t)\phi _1(x,0) \phi _2(x,0)}{1+\alpha _I(x,t)\phi _2(x,0)}\right] \\ \phi _2(x,0) + \xi \int _{\Omega } {\mathcal {G}}(t,t-\tau ,x,y) \\ ~~~~~~~~~~~~~~~~~~~ \times \left[ \frac{\beta _V(y,t-\tau )\phi _1(y,-\tau )\phi _3(y,-\tau )}{1+\alpha _V(y,t)\phi _3(y,-\tau )} + \frac{\beta _I(y,t-\tau )\phi _1(y,-\tau )\phi _2(y,-\tau )}{1+\alpha _I(y,t)\phi _2(y,-\tau )}\right] dy \\ \phi _3(x,0) + \xi \gamma (x,t)\phi _2(x,0) \end{array} \right) \\&~~~\ge \left( \begin{array}{l} \phi _1(x,0) \left[ 1 - \xi \left( \frac{\bar{a}}{{\underline{K}}} \phi _1(x,0) + {\bar{\beta }}_V\phi _3(x,0) + {\bar{\beta }}_I\phi _2(x,0)\right) \right] \\ \phi _2(x,0) \\ \phi _3(x,0) \end{array} \right) , \end{aligned}$$

which implies that

$$\begin{aligned} \lim _{\xi \rightarrow 0^+} \frac{1}{\xi }\mathrm{dist}\left( \phi (x,0) + \xi F(t,\phi )(x),~{\mathbb {X}}^+ \right) = 0,~\forall ~(t,\phi )\in {\mathbb {R}}_+\times {\mathbb {C}}_{\tau }^+. \end{aligned}$$

In addition, from (Hess 1991), the operator ${\mathbb {T}}(t,s):{\mathbb {X}}^+\rightarrow {\mathbb {X}}^+$ is positive, for all $t\ge s$. Consequently, Corollary 4 in (Martin and Smith 1990) implies that system (2.13)–(2.14) admits a unique non-continuable mild solution $u(\cdot ,t;\phi )$ on its maximal existence interval $[0,t_{\phi })$ with $u_0=\phi$, where $t_{\phi }\le +\infty$, and $u(\cdot ,t;\phi )\in {\mathbb {C}}_{\tau }^+$ for $t\in [0,t_{\phi })$. Furthermore, by the analyticity of ${\mathbb {T}}(t,s)$ with respective to $(t,s)\in {\mathbb {R}}^2$ with $t>s$, we see that $u(\cdot ,t;\phi )$ is a classical solution for $t>\tau$.

Proof of Lemma 2.2

The proof is similar to those in (Zhang et al. 2015, Lemma 2.1), and we only sketch our arguments. For any ${\hat{\varphi }}\in {\mathbb {Y}}^+$, we can show that system (2.18) admits a unique solution $w(\cdot ,t;{\hat{\varphi }})$ on its maximal existence interval $[0,t_{{\hat{\varphi }}})$, where $t_{{\hat{\varphi }}}\le +\infty$, and $w(\cdot ,t;{\hat{\varphi }})\in {\mathbb {Y}}^+$ for $t\in [0,t_{{\hat{\varphi }}})$. Let

$$\begin{aligned} \bar{p}=\max _{x\in {\bar{\Omega }},t\in [0,\omega ]}p(x,t),\ \forall \ p=\Lambda ,\ a,\ K\ ; {\underline{p}}=\min _{x\in {\bar{\Omega }},t\in [0,\omega ]}p(x,t),\ \forall \ p=n_T,\ a. \end{aligned}$$

Then it follows from (2.18) that

$$\begin{aligned}&\frac{\partial w(x,t)}{\partial t}\le \bigtriangledown \cdot \left[ d_1(x,t)\bigtriangledown w(x,t)\right] + {\bar{\Lambda }}-{\underline{n}}_Tw(x,t)+\bar{a} w(x,t)-{\underline{a}} \frac{(w(x,t))^2}{\bar{K}} \nonumber \\&\quad = \bigtriangledown \cdot \left[ d_1(x,t)\bigtriangledown w(x,t)\right] + {\bar{\Lambda }}-{\underline{n}}_Tw(x,t)+\bar{a} w(x,t)\left( 1 - \frac{w(x,t)}{\hat{K}} \right) , \end{aligned}$$

(6.1)

where $\hat{K}=\frac{\bar{a}\bar{K}}{{\underline{a}}}$. In view of (6.1) and the comparison principle, we see that

$$\begin{aligned} \limsup _{t\rightarrow \infty }w(x,t)\le \zeta _0,\ \forall \ x\in {\bar{\Omega }}, \end{aligned}$$

where $\zeta _0>0$ is the unique positive root of the following quadratic equation

$$\begin{aligned} {\bar{\Lambda }}-{\underline{n}}_T\zeta +\bar{a} \zeta \left( 1 - \frac{\zeta }{\hat{K}}\right) =0. \end{aligned}$$

Thus, solutions of system (2.18) are ultimately bounded, and hence, $t_{{\hat{\varphi }}}= +\infty$.

Therefore, we may define the solution maps $S(t):{\mathbb {Y}}^+\rightarrow {\mathbb {Y}}^+$ associated with system (2.18). Since system (2.18) is scalar, we see that S(t) is strongly monotone for $t>0$, see, e. g., pages 56 and 133 in (Smith 1995). Observing that the reaction term in (2.18)

$$\begin{aligned} f(x,t,w):=\Lambda (x,t)-n_T(x,t)w + a(x,t)w\left( 1 - \frac{w}{K(x,t)} \right) \end{aligned}$$

is strictly subhomogeneous in the sense that $f(x,t,\vartheta w)>\vartheta f(x,t,w)$ for $\vartheta \in (0,1)$ and $w>0$. Then the rest of the proofs are same to those in (Zhang et al. 2015, Lemma 2.1), and we omit the details.

Remark 6.1

In this remark, we further discuss the global dynamics of system (2.18) on $C([-\tau ,0],{\mathbb {Y}}^+)$. For any $\varphi \in C([-\tau ,0],{\mathbb {Y}}^+)$, we let $w(\cdot ,t;\varphi (\cdot ,0))$ be the solution of system (2.18) with initial value $w(\cdot ,0;\varphi (\cdot ,0))=\varphi (\cdot ,0)\in {\mathbb {Y}}^+$. Define

$$\begin{aligned} w_t(x,\theta ;\varphi )= \left\{ \begin{array}{llll} w\left( x,t+\theta ;\varphi (\cdot ,0)\right) ,\quad \text {if}\ t+\theta>0,\ t>0,\ x\in {\bar{\Omega }},\ \theta \in [-\tau ,0],\\ \varphi (\cdot ,t+\theta ),\quad ~~~~~~~~~~~\text {if} \ t+\theta \le 0,\ t>0,\ x\in {\bar{\Omega }},\ \theta \in [-\tau ,0]. \end{array} \right. \!\!\!\nonumber \\ \end{aligned}$$

(6.2)

Then $w_t$ defines the solution map of system (2.18) on $C([-\tau ,0],{\mathbb {Y}}^+)$. Let ${\bar{P}}(\varphi )=w_\omega (\varphi )$ be the Poincaré map associated with system (2.18) on $C([-\tau ,0],{\mathbb {Y}}^+)$. By Lemma 2.2, it follows that for any $\varphi \in C([-\tau ,0],{\mathbb {Y}}^+)$, we see that the omega limit set of $\varphi$ for ${\bar{P}}$ is $\{u^*_{1,0}\}$, where $u^*_{1,0}\in C([-\tau ,0],{\mathbb {Y}}^+)$ is defined by $u^*_{1,0}(\cdot ,\theta )=u^*_1(\cdot ,0+\theta )=u^*_1(\cdot ,\theta )$, for $\theta \in [-\tau ,0]$.

Proof of Lemma 2.3

In view of Lemma 2.1, we see that for any $\phi \in {\mathbb {C}}_{\tau }^+$, system (2.13)–(2.14) has a unique solution $u(\cdot ,t;\phi )$ on its maximal existence interval $[0,t_{\phi })$ with $u_0=\phi$, where $t_{\phi }\le +\infty$. In view of Lemma 2.2, we see that system (2.18) admits a unique positive $\omega$-periodic solution $u_1^*(x,t)$ which is globally attractive in ${\mathbb {Y}}^+$. We further observe that the $u_1$-equation in system (2.13) is dominated by the equation (2.18). Thus, the comparison principle implies that there is a constant $B_1>0$ such that for any $\phi \in {\mathbb {C}}_{\tau }^+$, there exists a $t_1(\phi )$ such that

$$\begin{aligned} u_1(x,t)\le B_1,\ \forall \ x\in {\bar{\Omega }},\ t\ge t_1(\phi ). \end{aligned}$$

(6.3)

We are ready to adopt the similar ideas in (Thieme and Zhao 2001), see also (Guo et al. 2012; Zhang et al. 2015), to establish the ultimate boundedness of $u_i(x,t)$ for $i=2,3$. Set $U_i(t)=\int _{\Omega }u_i(x,t)dx$ for $i=1,2,3$. Integrating the first equation in (2.13) over $x\in \Omega$, and using the boundary condition, we see that

$$\begin{aligned}&\frac{dU_1(t)}{dt} \le \int _{\Omega }\Lambda (x,t)dx- \int _{\Omega }n_T(x,t)u_1(x,t)dx+\int _{\Omega }a(x,t)u_1(x,t)dx\nonumber \\&\quad - \int _{\Omega } \left[ \frac{\beta _V(x,t)u_1(x,t)u_3(x,t)}{1+\alpha _V(x,t)u_3(x,t)} + \frac{\beta _I(x,t)u_1(x,t) u_2(x,t)}{1+\alpha _I(x,t)u_2(x,t)} \right] dx,\ \forall \ t>0.\nonumber \\ \end{aligned}$$

(6.4)

Let

$$\begin{aligned} \Lambda _1:= & {} \max _{t\in [0,\omega ]}\int _{\Omega }\Lambda (x,t)dx+B_1\max _{t\in [0,\omega ]}\int _{\Omega }a(x,t)dx>0,\\ {\underline{p}}= & {} \min _{x\in {\bar{\Omega }},t\in [0,\omega ]}p(x,t),\ \forall \ p=n_T,\ b,\ c, \end{aligned}$$

and

$$\begin{aligned} {\bar{\gamma }}=\max _{x\in {\bar{\Omega }},t\in [0,\omega ]}\gamma (x,t). \end{aligned}$$

Then (6.3) and (6.4) imply that

$$\begin{aligned}&\int _{\Omega } \left[ \frac{\beta _V(x,t)u_1(x,t)u_3(x,t)}{1+\alpha _V(x,t)u_3(x,t)} + \frac{\beta _I(x,t)u_1(x,t) u_2(x,t)}{1+\alpha _I(x,t)u_2(x,t)} \right] dx\nonumber \\&\quad \le \Lambda _1 -{\underline{n}}_T U_1(t) - \frac{dU_1(t)}{dt},\ \forall \ t\ge t_1(\phi ). \end{aligned}$$

(6.5)

Integrating the $u_2$-equation in (2.13), and using the facts that ${\mathcal {G}}(t,t-\tau ,x,y)$ is bounded and (6.5), we obtain

$$\begin{aligned} \frac{d}{dt}U_2(t) \le -{\underline{b}}U_2(t) + k_1 - k_2U_1(t-\tau ) -k_3\frac{d}{dt}U_1(t-\tau ),\ \forall \ t\ge t_1(\phi )+\tau ,\nonumber \\ \end{aligned}$$

(6.6)

where $k_i$, $i=1,2,3$, are some positive values independent of $\phi$. Similarly, integrating the $u_3$-equation in (2.13) over $\Omega$ yields

$$\begin{aligned} \frac{d}{dt}U_3(t) \le {\bar{\gamma }}U_2(t) - {\underline{c}}U_3(t),\ \forall \ t>0. \end{aligned}$$

(6.7)

Let $m:=\frac{2{\bar{\gamma }}}{{\underline{b}}}$ and $k_4:=\min \{\frac{k_2}{k_3},\frac{1}{2}{\underline{b}},{\underline{c}}\}>0$. Then it follows from (6.6) and (6.7) that

$$\begin{aligned}&\frac{d}{dt}\left[ mk_3U_1(t-\tau )+mU_2(t)+U_3(t) \right] \nonumber \\&\quad \le mk_3\frac{d}{dt}U_1(t-\tau ) - 2{\bar{\gamma }}U_2(t)+mk_1-mk_2U_1(t-\tau )-mk_3\frac{d}{dt}U_1(t-\tau )\nonumber \\&\qquad +{\bar{\gamma }}U_2(t)-{\underline{c}}U_3(t) \nonumber \\&\quad \le mk_1-k_4\left[ mk_3U_1(t-\tau )+mU_2(t)+U_3(t) \right] ,\ \forall \ t\ge t_1(\phi )+\tau . \end{aligned}$$

This implies that we can find a $t_2(\phi )$ with $t_2(\phi )\ge t_1(\phi )$ such that

$$\begin{aligned} mk_3U_1(t-\tau )+mU_2(t)+U_3(t) \le \frac{mk_1}{k_4}+1,\ \forall \ t\ge t_2(\phi )+\tau . \end{aligned}$$

(6.8)

Using the positivity of solutions, (6.3), (6.8) and the facts that ${\mathcal {G}}(t,t-\tau ,x,y)$, $\beta _V(x,t)$, $\beta _I(x,t)$ are bounded, it derives from the second equation of (2.13) that there exist positive constants $k_5$ and $k_6$ such that

$$\begin{aligned} \frac{\partial u_2(x,t)}{\partial t}\le & {} \bigtriangledown \cdot [d_2(x,t)\bigtriangledown u_2(x,t)] - {\underline{b}}u_2(x,t) + \int _{\Omega } {\mathcal {G}}(t,t-\tau ,x,y) \times \nonumber \\&\left[ \beta _V(y,t-\tau )u_1(y,t-\tau )u_3(y,t-\tau )\right. \nonumber \\&\left. + \beta _I(y,t-\tau )u_1(y,t-\tau )u_2(y,t-\tau )\right] dy \nonumber \\\le & {} \bigtriangledown \cdot \left[ d_2(x,t)\bigtriangledown u_2(x,t)\right] \nonumber \\&- {\underline{b}}u_2(x,t) + k_5 U_3(t-\tau )+k_6U_2(t-\tau ) \nonumber \\\le & {} \bigtriangledown \cdot \left[ d_2(x,t)\bigtriangledown u_2(x,t)\right] \nonumber \\&- {\underline{b}}u_2(x,t) + \left( k_5+\frac{k_6}{m}\right) \left( \frac{mk_1}{k_4}+1\right) ,\ \forall \ t\ge t_2(\phi )+2\tau . \end{aligned}$$

(6.9)

From (6.9) and the comparison principle, it follows that

$$\begin{aligned} \limsup _{t\rightarrow \infty }u_2(x,t)\le \frac{mk_5+k_6}{m{\underline{b}}}\left( \frac{mk_1}{k_4}+1\right) ,\ \forall \ x\in {\bar{\Omega }}. \end{aligned}$$

Then there exists $t_3(\phi )\ge t_2(\phi )+2\tau$ and $B_2>\frac{mk_5+k_6}{m{\underline{b}}}\left( \frac{mk_1}{k_4}+1\right)$ such that

$$\begin{aligned} u_2(x,t)\le B_2,\ \forall \ x\in {\bar{\Omega }},\ t\ge t_3(\phi ). \end{aligned}$$

(6.10)

In view of the third equation in (2.13), and (6.10), we see that

$$\begin{aligned} \frac{\partial u_3(x,t)}{\partial t} \le \bigtriangledown \cdot \left[ d_3(x,t)\bigtriangledown u_3(x,t)\right] + B_2{\bar{\gamma }}- {\underline{c}}u_3(x,t),\ \forall \ x\in \Omega ,\ t\ge t_3(\phi ).\nonumber \\ \end{aligned}$$

(6.11)

From (6.11) and the comparison principle, it follows that

$$\begin{aligned} \limsup _{t\rightarrow \infty }u_3(x,t)\le \frac{B_2{\bar{\gamma }}}{{\underline{c}}},\ \forall \ x\in {\bar{\Omega }}. \end{aligned}$$

(6.12)

In view of (6.3), (6.10) and (6.12), we see that solutions of system (2.13)–(2.14) are ultimately bounded. Thus, $t_{\phi }= +\infty$ for any $\phi \in {\mathbb {C}}_{\tau }^+$.

Define the solution maps $\Phi (t):{\mathbb {C}}_{\tau }^+\rightarrow {\mathbb {C}}_{\tau }^+$ associated with system (2.13)–(2.14) by

$$\begin{aligned} \Phi (t)(\phi )(x,\theta )\!=\!u_t(\phi )(x,\theta )\!=\!u(x,t\!+\!\theta ;\phi ),\ \forall \ \phi \in {\mathbb {C}}_{\tau }^+ ,\ t\ge 0, \ \theta \in [\!-\!\tau ,0],\ x\in \Omega . \end{aligned}$$

We can further show that $\{\Phi (t)\}_{t\ge 0}$ is an $\omega$-periodic semiflow on ${\mathbb {C}}_{\tau }^+$ in the sense defined in (Zhao 2017a, Sect. 3.1). From the ultimate boundedness of $u_i(x,t)$, for $i=1,2,3$, it follows that $\Phi (t):{\mathbb {C}}_{\tau }^+\rightarrow {\mathbb {C}}_{\tau }^+$ is point dissipative. In view of (Wu 1996, Theorem 2.1.8), we see that $\Phi (t):{\mathbb {C}}_{\tau }^+\rightarrow {\mathbb {C}}_{\tau }^+$ is compact for each $t>\tau$. Then it follows from (Magal and Zhao 2005, Theorem 2.9) that the Poincaré map $\Phi (\omega )$ admits a global compact attractor in ${\mathbb {C}}_{\tau }^+$.

Theorem 3.5 states a threshold-type result on the global dynamics of system (2.13)–(2.14) in terms of ${\mathcal {R}}_0$. In order to prove Theorem 3.5, we first give a primary estimation for the solution.

Lemma 6.2

Let $u(\cdot ,t;\phi )$ be the solution of system (2.13)–(2.14) on $[0,\infty )$ with $u_0=\phi \in {\mathbb {C}}_{\tau }^+$. Then the following statements are valid:

• (i) If there exists some $t_0\ge 0$ such that $u_i(\cdot ,t_0;\phi )\not \equiv 0$ for some $i\in \{2,3\}$, then $u_i(x,t;\phi )>0$, $\forall \ t>t_0$, $x\in {\bar{\Omega }}$.

• (ii) For any $\phi \in {\mathbb {C}}_{\tau }^+$, we have

  $$\begin{aligned} u_1(x,t;\phi )>0,\ \forall t>0,\ x\in {\bar{\Omega }}, \end{aligned}$$

  (6.13)

  and

  $$\begin{aligned} \liminf \limits _{t\rightarrow \infty }u_1(x,t;\phi )\ge {\hat{\varrho }}, \ \text{ uniformly } \text{ for }\ x\in {\bar{\Omega }}, \end{aligned}$$

  (6.14)

  where ${\hat{\varrho }}$ is a positive constant independent of $\phi$.

Proof

Part (i). From the second and third equations of (2.13) and the positivity of solutions, one can easily see that for a given $\phi \in {\mathbb {C}}_{\tau }^+$, $u_2(x,t;\phi )$ and $u_3(x,t;\phi )$, respectively, satisfy

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial u_2(x,t)}{\partial t} \ge \bigtriangledown \cdot \left[ d_2(x,t)\bigtriangledown u_2(x,t)\right] - b(x,t)u_2(x,t),\ (x,t)\in \Omega \times [0,\infty ),\\ \left[ d_2(x,t)\bigtriangledown u_2(x,t)\right] \cdot \nu =0,\ x\in \partial \Omega ,\ t>0, \end{array}\right. }\nonumber \\ \end{aligned}$$

(6.15)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial u_3(x,t)}{\partial t} \ge \bigtriangledown \cdot \left[ d_3(x,t)\bigtriangledown u_3(x,t)\right] - c(x,t) u_3(x,t),\ (x,t)\in \Omega \times [0,\infty ),\\ \left[ d_3(x,t)\bigtriangledown u_3(x,t)\right] \cdot \nu =0,\ x\in \partial \Omega ,\ t>0. \end{array}\right. }\nonumber \\ \end{aligned}$$

(6.16)

By the parabolic maximum principle, see, e.g. (Hess 1991), it follows that the results in Part (i) hold.

Part (ii). By the positivity of solutions (see Lemma 2.1), it follows that

$$\begin{aligned} u_1(x,t;\phi )\ge 0,\ \forall t>0,\ x\in {\bar{\Omega }}. \end{aligned}$$

Suppose that (6.13) was not true, that is, there exists $x_0\in {\bar{\Omega }}$ and $t_0\in (0,\infty )$ such that $u_1(x_0,t_0;\phi )=0$. Let ${\mathcal {T}}_0>0$ be such that $t_0<{\mathcal {T}}_0$. Then $(x_0, t_0)\in {\bar{\Omega }} \times [0,{\mathcal {T}}_0]$ and $u_1$ attains its minimum on ${\bar{\Omega }} \times [0,{\mathcal {T}}_0]$ at the point $(x_0, t_0)$. In view of the first equation of (2.13), we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial u_1(x,t)}{\partial t}\ge \bigtriangledown \cdot \left[ d_1(x,t)\bigtriangledown u_1(x,t)\right] - \left[ n_T(x,t)+\frac{ a(x,t)u_1(x,t)}{K(x,t)}\right. \\ \ \ \ \ \ \ \ \ \ \ \ \ \quad \left. +\beta _V(x,t)u_3(x,t)\! \quad +\!\beta _I(x,t)u_2(x,t)\right] u_1(x,t),\ x\in \Omega ,\ t\in (0,{\mathcal {T}}_0],\\ \left[ d_1(x,t)\bigtriangledown u_1(x,t)\right] \cdot \nu =0,\ x\in \partial \Omega ,\ t\in (0,{\mathcal {T}}_0]. \end{array}\right. }\!\!\!\!\nonumber \\ \end{aligned}$$

(6.17)

In case $x_0\in \partial \Omega$, we apply the Hopf boundary lemma, see e.g. (Hess 1991, Sect. II.13), and get $\frac{\partial u_1}{\partial \nu }(x_0, t_0)<0$, which contradicts the boundary condition of $u_1$. In case $x_0\in \Omega$, we apply the strong maximum principle, see e.g. (Hess 1991, Section II.13), to obtain that $u_1(x,t)\equiv u_1(x_0, t_0)=0,\ \forall \ (x,t)\in {\bar{\Omega }}\times [0,{\mathcal {T}}_0]$. Inserting this result into the first equation of (2.13), it leads to $\Lambda (x,t)\equiv 0,\ \forall \ (x, t)\in {\bar{\Omega }} \times [0,{\mathcal {T}}_0]$, a contradiction. Thus, (6.13) is true.

Next, we establish the result in (6.14). By Lemma 2.3, it follows that solutions of (2.13)–(2.14) are ultimately bounded. Thus, there is a constant $\hat{B}>0$ such that for any $\phi \in {\mathbb {C}}_{\tau }^+$, there exists a $t_1>0$ such that

$$\begin{aligned} u_i(x,t)\le \hat{B},\ \forall \ x\in {\bar{\Omega }},\ t\ge t_1,\ i=1,2,3. \end{aligned}$$

For $x\in \Omega ,\ t\ge t_1$, it follows from the first equation of (2.13) that

$$\begin{aligned} \frac{\partial u_1(x,t)}{\partial t}\ge & {} \bigtriangledown \cdot \left[ d_1(x,t)\bigtriangledown u_1(x,t)\right] + \Lambda (x,t) \nonumber \\&- \left[ n_T(x,t) +a(x,t)\frac{u_1(x,t)}{K(x,t)}+\beta _V(x,t)u_3(x,t)+\beta _I(x,t)u_2(x,t)\right] u_1(x,t) \nonumber \\\ge & {} \bigtriangledown \cdot \left[ d_1(x,t)\bigtriangledown u_1(x,t)\right] + \Lambda (x,t) \nonumber \\&- \left[ n_T(x,t) +a(x,t)\frac{\hat{B}}{K(x,t)}+\beta _V(x,t)\hat{B}+\beta _I(x,t)\hat{B}\right] u_1(x,t)\nonumber \\\ge & {} \bigtriangledown \cdot \left[ d_1(x,t)\bigtriangledown u_1(x,t)\right] + {\underline{\Lambda }}-C_0u_1(x,t), \end{aligned}$$

(6.18)

where

$$\begin{aligned} C_0=\max _{x\in {\bar{\Omega }},t\in [0,\omega ]}\left[ n_T(x,t) +a(x,t)\frac{\hat{B}}{K(x,t)}+\beta _V(x,t)\hat{B}+\beta _I(x,t)\hat{B}\right] , \end{aligned}$$

and ${\underline{\Lambda }}=\min _{x\in {\bar{\Omega }},t\in [0,\omega ]}\Lambda (x,t)$. By (6.18) and the comparison principle, we see that

$$\begin{aligned} \liminf \limits _{t\rightarrow \infty }u_1(x,t;\phi )\ge \frac{{\underline{\Lambda }}}{C_0}, \ \text{ uniformly } \text{ for }\ x\in {\bar{\Omega }}, \end{aligned}$$

which establishes (6.14). $\square$

Proof of Theorem 3.5

Recall that $\Phi (t)(\cdot ):=u_t(\cdot ):{\mathbb {C}}_{\tau }^+\rightarrow {\mathbb {C}}_{\tau }^+$ is an $\omega$-periodic semiflow of system (2.13)–(2.14) (see Lemma 2.3), and let $Q:=\Phi (\omega )$ be the associated Poincaré map on ${\mathbb {C}}_{\tau }^+$.

Part (i). In the case where ${\mathcal {R}}_0<1$, Lemma 3.1 implies that $r(P(\omega ))<1$, and hence $\mu =\frac{\ln [r(P(\omega ))]}{\omega }<0$. Consider the following system with a parameter $\varepsilon >0$:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial v_1(x,t)}{\partial t} = \bigtriangledown \cdot [d_2(x,t)\bigtriangledown v_1(x,t)] - b(x,t)v_1(x,t) \\ \ \ \ \ \ \ \ \ \ \ + \int _{\Omega } {\mathcal {G}}(t,t-\tau ,x,y)(u_1^*(y,t-\tau )+\varepsilon ) \left[ \beta _V(y,t-\tau )v_2(y,t-\tau ) + \beta _I(y,t-\tau )v_1(y,t-\tau ) \right] dy, \\ \frac{\partial v_2(x,t)}{\partial t} = \bigtriangledown \cdot [d_3(x,t)\bigtriangledown v_2(x,t)] + \gamma (x,t)v_1(x,t) - c(x,t) v_2(x,t),\ \ (x,t)\in \Omega \times [0,\infty ), \\ \left[ d_2(x,t)\bigtriangledown v_1(x,t)\right] \cdot \nu =\left[ d_3(x,t)\bigtriangledown v_2(x,t)\right] \cdot \nu =0,\ \ x\in \partial \Omega ,\ t>0. \end{array}\right. }\nonumber \\ \end{aligned}$$

(6.19)

For any $\psi \in C([-\tau ,0],{\mathbb {E}})$, let $v^{\varepsilon }(x,t;\psi )=(v^{\varepsilon }_1(x,t;\psi ),v^{\varepsilon }_2(x,t;\psi ))$ be the unique solution of system (6.19) with $v^{\varepsilon }_0(\psi )(x,\theta )=\psi (x,\theta )$ for all $\theta \in [-\tau ,0],\ x\in \Omega$, where

$$\begin{aligned} v_t^{\varepsilon }(\psi )(x,\theta )=(v_1^{\varepsilon }(x,t+\theta ;\psi ),v_2^{\varepsilon }(x,t+\theta ;\psi )), \ \theta \in [-\tau ,0], \end{aligned}$$

for any $t\ge 0$ and $x\in \Omega$. Let $P_{\varepsilon }(\omega ):C([-\tau ,0],{\mathbb {E}})\rightarrow C([-\tau ,0],{\mathbb {E}})$ be the Poincaré map of system (6.19), i.e., $P_{\varepsilon }(\omega )\psi =v_{\omega }^{\varepsilon }(\psi )$, $\forall \psi \in C([-\tau ,0],{\mathbb {E}})$, and let $r(P_{\varepsilon }(\omega ))$ be the spectral radius of $P_{\varepsilon }(\omega )$. Since $\lim \limits _{\varepsilon \rightarrow 0}r(P_{\varepsilon }(\omega ))=r(P(\omega ))<1$, we can fix a sufficiently small number $\varepsilon _0>0$ such that $r(P_{\varepsilon _0}(\omega ))<1$. By Lemma 3.2, there is a positive $\omega$-periodic function $v^*_{\varepsilon _0}(x,t)$ such that $v^{\varepsilon _0}(x,t)=e^{\mu _{\varepsilon _0} t}v^*_{\varepsilon _0}(x,t)$ is a solution of system (6.19), where $\mu _{\varepsilon _0}=\frac{\ln [r(P_{\varepsilon _0}(\omega ))]}{\omega }<0$. In view of the first equation in (2.13), we see that

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial u_1(x,t)}{\partial t}\le \bigtriangledown \cdot \left[ d_1(x,t)\bigtriangledown u_1(x,t)\right] + \Lambda (x,t)-n_T(x,t)u_1(x,t)\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \quad + a(x,t)u_1(x,t)\left( 1 - \frac{u_1(x,t)}{K(x,t)} \right) ,\ (x,t)\in \Omega \times [0,\infty ),\\ \left[ d_1(x,t)\bigtriangledown u_1(x,t)\right] \cdot \nu =0,\ x\in \partial \Omega ,\ t>0. \end{array}\right. } \end{aligned}$$

By the global attractivity of $u_1^*(x,t)$ for system (2.18) (see Lemma 2.2) and the comparison principle, there exists a sufficiently large integer $n_1>0$ such that $n_1\omega >\tau$ and

$$\begin{aligned} u_1(x,t)\le u_1^*(x,t)+\varepsilon _0, \ \forall \ t\ge n_1\omega -\tau ,\ x\in {{\bar{\Omega }}}. \end{aligned}$$

From the second and third equations in (2.13), it follows that for any $t\ge n_1\omega$, $x\in \Omega$, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial u_2(x,t)}{\partial t}\le \bigtriangledown \cdot [d_2(x,t)\bigtriangledown u_2(x,t)] - b(x,t)u_2(x,t) \\ \ \ \ \ \ \ \ \ \ \ + \int _{\Omega } {\mathcal {G}}(t,t-\tau ,x,y) (u_1^*(y,t-\tau )+\varepsilon _0)\left[ \beta _V(y,t-\tau )u_3(y,t-\tau )+\beta _I(y,t-\tau )u_2(y,t-\tau )\right] dy, \\ \\ \frac{\partial u_3(x,t)}{\partial t} = \bigtriangledown \cdot [d_3(x,t)\bigtriangledown u_3(x,t)] + \gamma (x,t)u_2(x,t) - c(x,t) u_3(x,t), \end{array}\right. }\!\!\!\!\nonumber \\ \end{aligned}$$

(6.20)

with the no flux boundary condition. For any $\phi \in {\mathbb {C}}_{\tau }^+$, we may find a $m_1>0$ such that

$$\begin{aligned} \left( u_2(x,t;\phi ),u_3(x,t;\phi )\right) \le m_1e^{\mu _{\varepsilon _0} t}v^*_{\varepsilon _0}(x,t), \ \forall \ t\in \left[ n_1\omega -\tau ,n_1\omega \right] ,\ x\in {\bar{\Omega }}. \end{aligned}$$

By the comparison theorem for abstract functional differential equation, see e.g. (Martin and Smith 1990, Proposition 1), it follows that

$$\begin{aligned} \left( u_2(x,t;\phi ),u_3(x,t;\phi )\right) \le m_1e^{\mu _{\varepsilon _0} t}v^*_{\varepsilon _0}(x,t), \ \forall t\ge n_1\omega , \ x\in {\bar{\Omega }}. \end{aligned}$$

Since $\mu _{\varepsilon _0}<0$, we see that $\lim \limits _{t\rightarrow \infty }(u_2(x,t;\phi ),u_3(x,t;\phi ))=(0,0)$ uniformly for $x\in {\bar{\Omega }}$. Then the $u_1$-equation in (2.13) is asymptotic to system (2.18).

Next, we are ready to use the theory of internally chain transitive sets, see e.g. (Hirsch et al. 2001; Zhao 2017b), to prove the following result

$$\begin{aligned} \lim \limits _{t\rightarrow \infty }\left( u_1(x,t;\phi )-u_1^*(x,t)\right) =0,\ \text{ uniformly } \text{ for }\ x\in {\bar{\Omega }}, \end{aligned}$$

where $u_1^*(x,t)$ is a globally attractive solution of system (2.18). Let ${\mathcal {J}}$ be the omega limit set of $\phi \in {\mathbb {C}}_{\tau }^+$ for $Q:=\Phi (\omega )$. Since $\lim \limits _{t\rightarrow \infty }u_i(x,t;\phi )=0 \ (i=2,3)$ uniformly for $x\in {\bar{\Omega }}$, we have ${\mathcal {J}}={\mathcal {J}}_1\times \{\hat{0}\}\times \{\hat{0}\}$, where ${\hat{0}}(\cdot ,\theta )=0$, $\forall \ \theta \in [-\tau ,0]$. By (Hirsch et al. 2001, Lemma 2.1), see also (Zhao 2017b, Lemma 1.2.1), we note that ${\mathcal {J}}$ is an internally chain transitive set for Q, and we can further show that ${\mathcal {J}}_1$ is an internally transitive chain set for ${\bar{P}}$, where ${\bar{P}}$ is the Poincaré map associated with system (2.18) on $C([-\tau ,0],{\mathbb {Y}}^+)$ (see Remark 6.1). By (6.14) in Lemma 6.2, we see that ${\mathcal {J}}_1\ne \{\hat{0}\}$. Since $u^*_{1,0}$ defined in Remark 6.1 is globally attractive in $C([-\tau ,0],{\mathbb {Y}}^+)$ for ${\bar{P}}$, we have ${\mathcal {J}}_1\cap W^s(\{u^*_{1,0}\})\ne \emptyset$, where $W^s(\{u^*_{1,0}\})$ is the stable set of $\{u^*_{1,0}\}$. By (Hirsch et al. 2001, Theorem 3.1), see also (Zhao 2017b, Theorem 1.2.1), we get ${\mathcal {J}}_1\subseteq \{u^*_{1,0}\}$, that is, ${\mathcal {J}}_1=\{u^*_{1,0}\}$. This proves

$$\begin{aligned} {\mathcal {J}}={\mathcal {J}}_1\times \{\hat{0}\}\times \{\hat{0}\}=\left\{ \left( u^*_{1,0},\hat{0},\hat{0}\right) \right\} . \end{aligned}$$

Thus, $(u_1^*(x,t),\hat{0},\hat{0})$ is globally attractive for system (2.13)–(2.14) in ${\mathbb {C}}_{\tau }^+$.

Part (ii). In the case where ${\mathcal {R}}_0>1$, Lemma 3.1 implies that $r(P(\omega ))>1$, and hence $\mu =\frac{\ln [r(P(\omega ))]}{\omega }>0$.

Let

$$\begin{aligned} {\mathbb {W}}_0:=\left\{ \phi \in {\mathbb {C}}_{\tau }^+ : \phi _2(\cdot ,0)\not \equiv 0 \ \text {and} \ \phi _3(\cdot ,0)\not \equiv 0 \right\} , \end{aligned}$$

and

$$\begin{aligned} \partial {\mathbb {W}}_0:={\mathbb {C}}_{\tau }^+\backslash {\mathbb {W}}_0=\left\{ \phi \in {\mathbb {C}}_{\tau }^+ : \phi _2(\cdot ,0)\equiv 0 \ \text {or} \ \phi _3(\cdot ,0)\equiv 0 \right\} . \end{aligned}$$

Note that for any $\phi \in {\mathbb {W}}_0$, Lemma 6.2 implies that $u_i(x,t;\phi )>0, \forall \ t>0,\ x\in {{\bar{\Omega }}} \ (i=1,2,3)$. It follows that $Q^n{\mathbb {W}}_0\subset {\mathbb {W}}_0$, $\forall n\in {\mathbb {N}}$. In view of Lemma 2.3, we know that Q admits a global attractor in ${\mathbb {C}}_{\tau }^+$.

Let

$$\begin{aligned} {\mathbb {M}}_{\partial }:=\{\phi \in \partial {\mathbb {W}}_0 : Q^n\phi \in \partial {\mathbb {W}}_0,\forall n\in {\mathbb {N}}\}, \end{aligned}$$

and $\omega (\phi )$ be the omega limit set of the orbit $\gamma ^+=\{Q^n\phi : \forall n\in {\mathbb {N}}\}$. Set $M=(u^*_{1,0},{\hat{0}},{\hat{0}})$, where $u^*_{1,0}\in C([-\tau ,0],{\mathbb {Y}}^+)$ is defined by $u^*_{1,0}(\cdot ,\theta )=u^*_1(\cdot ,0+\theta )=u^*_1(\cdot ,\theta )$, for $\theta \in [-\tau ,0]$ (see Remark 6.1). Then the following claim indicates that $\{M\}$ cannot form a cycle for Q in $\partial {\mathbb {W}}_0$.

Claim 1

For any ${\hat{\phi }}\in {\mathbb {M}}_{\partial }$, the omega limit set $\omega ({\hat{\phi }})=\{M\}$.

For any given ${\hat{\phi }}\in {\mathbb {M}}_{\partial }$, $Q^n({\hat{\phi }})\in \partial {\mathbb {W}}_0$, $\forall \ n\in {\mathbb {N}}$. Thus, for each $n\in {\mathbb {N}}$, either $u_2(\cdot ,n\omega ;{\hat{\phi }})\equiv 0$ or $u_3(\cdot ,n\omega ;{\hat{\phi }})\equiv 0$. It then follows that for each $t\ge 0$, $u_2(\cdot ,t;{\hat{\phi }})\equiv 0$ or $u_3(\cdot ,t;{\hat{\phi }})\equiv 0$. Otherwise, it contradicts Lemma 6.2. In case where $u_3(\cdot ,t;{\hat{\phi }})\equiv 0$ for $t\ge 0$. Then it follows from the third equation of (2.13) that $u_2(\cdot ,t;{\hat{\phi }})\equiv 0$ for $t\ge 0$. Thus, $u_1(\cdot ,t;{\hat{\phi }})$ satisfies system (2.18) for $t\ge 0$, and hence, Lemma 2.2 implies that $\lim \limits _{t\rightarrow \infty }(u_1(x,t;{\hat{\phi }})-u_1^*(x,t))=0$ uniformly for $x\in {\bar{\Omega }}$. In case where $u_3(\cdot ,t_0;{\hat{\phi }})\not \equiv 0$ for some $t_0\ge 0$. Then it follows from Lemma 6.2 that $u_3(\cdot ,t;{\hat{\phi }})> 0 ,\ \forall \ t >t_0$. This implies that $u_2(\cdot ,t;{\hat{\phi }})\equiv 0,\ \forall \ t >t_0$. Inserting this equality into the second equation of (2.13), we derive

$$\begin{aligned} u_1\left( \cdot ,t-\tau ;{\hat{\phi }}\right) u_3\left( \cdot ,t-\tau ;{\hat{\phi }}\right) \equiv 0,\ \forall \ t >t_0, \end{aligned}$$

which is a contradiction. From the above discussions, Claim 1 holds.

Consider the following system with a parameter $\delta >0$:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial v_1(x,t)}{\partial t} = \bigtriangledown \cdot [d_2(x,t)\bigtriangledown v_1(x,t)] - b(x,t)v_1(x,t) \\ \ \ \ \ \ \ \ \ \ \ + \int _{\Omega } {\mathcal {G}}(t,t-\tau ,x,y)(u_1^*(y,t-\tau )-\delta ) \left[ \frac{ \beta _V(y,t-\tau )v_2(y,t-\tau )}{1+\alpha _V(y,t)\delta } + \frac{\beta _I(y,t-\tau )v_1(y,t-\tau )}{1+\alpha _I(y,t)\delta } \right] dy, \\ \frac{\partial v_2(x,t)}{\partial t} = \bigtriangledown \cdot [d_3(x,t)\bigtriangledown v_2(x,t)] + \gamma (x,t)v_1(x,t) - c(x,t) v_2(x,t),\ \ (x,t)\in \Omega \times [0,\infty ), \\ \left[ d_2(x,t)\bigtriangledown v_1(x,t)\right] \cdot \nu =[d_3(x,t)\bigtriangledown v_2(x,t)]\cdot \nu =0,\ \ x\in \partial \Omega ,\ t>0. \end{array}\right. } \end{aligned}$$

(6.21)

For any $\psi \in C([-\tau ,0],{\mathbb {E}})$, let $v^{\delta }(x,t;\psi )=(v^{\delta }_1(x,t;\psi ),v^{\delta }_2(x,t;\psi ))$ be the unique solution of system (6.21) with $v^{\delta }_0(\psi )(x,\theta )=\psi (x,\theta )$ for all $\theta \in [-\tau ,0],\ x\in \Omega$, where

$$\begin{aligned} v_t^{\delta }(\psi )(x,\theta )=(v_1^{\delta }(x,t+\theta ;\psi ),v_2^{\delta }(x,t+\theta ;\psi )), \ \theta \in [-\tau ,0], \end{aligned}$$

for any $t\ge 0$ and $x\in \Omega$. Let $P_{\delta }(\omega ):C([-\tau ,0],{\mathbb {E}})\rightarrow C([-\tau ,0],{\mathbb {E}})$ be the Poincáre map of system (6.21) , i.e., $P_{\delta }(\omega )\psi =v_{\omega }^{\delta }(\psi )$, $\forall \psi \in C([-\tau ,0],{\mathbb {E}})$, and let $r(P_{\delta }(\omega ))$ be the spectral radius of $P_{\delta }(\omega )$. Since $\lim \limits _{\delta \rightarrow 0}r(P_{\delta }(\omega ))=r(P(\omega ))>1$, we can fix a sufficiently small number $\delta _0>0$ such that

$$\begin{aligned} \delta _0<\min \limits _{x\in {\bar{\Omega }},t\in [0,\omega ]}u_1^*(x,t) \ \text {and} \ r(P_{\delta _0}(\omega ))>1. \end{aligned}$$

From the continuous dependence of solutions on the initial values, for the fixed $\delta _0 >0$, there exists a $\delta ^*>0$ such that for any $\phi \in {\mathbb {C}}_{\tau }^+$ with $\Vert \phi -M\Vert \le \delta ^*$, we have $\Vert \Phi (t)\phi -\Phi (t)M\Vert <\delta _0$ for all $t\in [0,\omega ]$. We now prove the following claim.

Claim 2

$\lim \sup _{n\rightarrow \infty }\Vert Q^n(\phi )-M\Vert \ge \delta ^*,\ \forall \ \phi \in {\mathbb {W}}_0$.

Suppose, by contradiction, that $\lim \sup _{n\rightarrow \infty }\Vert Q^n(\phi _0)-M\Vert < \delta ^*$ for some $\phi _0\in {\mathbb {W}}_0$. Then there exists $n_2\ge 1$ such that $\Vert Q^n(\phi _0)-M\Vert < \delta ^*$ for $n\ge n_2$. For any $t\ge n_2\omega$, letting $t=n\omega +t'$ with $n=[t/\omega ]$ and $t'\in [0,\omega )$, we have

$$\begin{aligned} \Vert \Phi (t)\phi _0-\Phi (t)M\Vert =\Vert \Phi (t')(\Phi (\omega )^n(\phi _0))-\Phi (t')M\Vert <\delta _0. \end{aligned}$$

(6.22)

We also note that

$$\begin{aligned} \Phi (t)\phi _0=u_t(\phi _0)=u\left( x,t+\theta ;\phi _0\right) ,\ \Phi (t)M=\left( u^*_{1,t},0,0\right) =\left( u^*_1(x,t+\theta ),0,0\right) ,\ \forall \ \theta \in [-\tau ,0].\nonumber \\ \end{aligned}$$

(6.23)

In view of (6.22), (6.23) and Lemma 6.2, it follows that

$$\begin{aligned} u_1\left( x,t;\phi _0\right) >u_1^*(x,t)-\delta _0 \ \text {and} \ 0<u_j\left( x,t;\phi _0\right) <\delta _0 \ (j=2,3), \end{aligned}$$

(6.24)

for any $t\ge n_2\omega -\tau$, and $x\in {\bar{\Omega }}$. By the second and third equations in (2.13) and using (6.24), for $t\ge n_2\omega$, we see that $u_2(x,t;\phi _0)$ and $u_3(x,t;\phi _0)$ satisfy

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial u_2(x,t)}{\partial t} \ge \bigtriangledown \cdot \left[ d_2(x,t)\bigtriangledown u_2(x,t)\right] - b(x,t)u_2(x,t) \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \quad + \int _{\Omega } {\mathcal {G}}(t,t-\tau ,x,y) \left( u_1^*(y,t-\tau )-\delta _0\right) \left[ \frac{\beta _V(y,t-\tau )u_3(y,t-\tau )}{1+\alpha _V(y,t)\delta _0} + \frac{\beta _I(y,t-\tau )u_2(y,t-\tau )}{1+\alpha _I(y,t)\delta _0}\right] dy, \\ \frac{\partial u_3(x,t)}{\partial t} = \bigtriangledown \cdot [d_3(x,t)\bigtriangledown u_3(x,t)] + \gamma (x,t)u_2(x,t) - c(x,t) u_3(x,t),\\ \left[ d_i(x,t)\bigtriangledown u_i(x,t)\right] \cdot \nu =0,\ x\in \partial \Omega ,\ t>0,\ i=2,3. \end{array}\right. }\quad \end{aligned}$$

(6.25)

By Lemma 3.2, there is a positive $\omega$-periodic function $v^*_{\delta _0}(x,t)$ such that $v^{\delta _0}(x,t)=e^{\mu _{0} t}v^*_{\delta _0}(x,t)$ is a solution of system (6.21), where $\mu _{0}=\frac{\ln [r(P_{\delta _0}(\omega ))]}{\omega }>0$. From Lemma 6.2, we see that $u(x,t;\phi _0)\gg 0$ for all $t\ge 0$ and $x\in {\bar{\Omega }}$. Then there exists an $m_2>0$ such that

$$\begin{aligned} \left( u_2\left( x,t;\phi _0\right) ,u_3\left( x,t;\phi _0\right) \right) \ge m_2e^{\mu _{0} t}v^*_{\delta _0}(x,t), \ \forall t\in \left[ n_2\omega -\tau ,n_2\omega \right] , \ x\in {\bar{\Omega }}.\nonumber \\ \end{aligned}$$

(6.26)

From (6.25), (6.26), and the comparison principle, it follows that

$$\begin{aligned} \left( u_2\left( x,t;\phi _0\right) ,u_3\left( x,t;\phi _0\right) \right) \ge m_2e^{\mu _{0} t}v^*_{\delta _0}(x,t), \ \forall t\ge n_2\omega , \ x\in {\bar{\Omega }}. \end{aligned}$$

Since $\mu _{0}>0$, it is easy to see that $u_i(\cdot ,t;\phi _0)\rightarrow +\infty \ (i=2,3)$ as $t\rightarrow +\infty$. This contradicts (6.24).

The above claim implies that M is an isolated invariant set for Q in ${\mathbb {C}}_{\tau }^+$, and $W^s(M)\bigcap {\mathbb {W}}_0=\emptyset$, where $W^s(M)$ is the stable set of M for Q. By (Magal and Zhao 2005, Theorem 3.7), as applied to Q, we know that Q admits a global attractor $A_0$ in ${\mathbb {W}}_0$. It then follows from (Zhao 2017b, Theorem 1.3.1) that Q is uniformly persistent with respect to $({\mathbb {W}}_0,\partial {\mathbb {W}}_0)$ in the sense that there exists ${\tilde{\varrho }}>0$ such that

$$\begin{aligned} \liminf \limits _{n\rightarrow \infty }\text {d}(Q^n(\phi ),\partial {\mathbb {W}}_0)\ge {\tilde{\varrho }}, \ \forall \phi \in {\mathbb {W}}_0. \end{aligned}$$

(6.27)

Next, we establish the practical persistence. Since $A_0=QA_0=\Phi (\omega )A_0$, we have that $\phi _2(\cdot ,0)>0$ and $\phi _3(\cdot ,0)>0$ for all $\phi \in A_0$. Let $B_0:=\bigcup \limits _{t\in [0,\omega ]}\Phi (t)A_0$. Then $B_0\subset {\mathbb {W}}_0$ and $\lim \limits _{t\rightarrow \infty }\text {d}(\Phi (t)\phi ,B_0)=0$, $\forall \phi \in {\mathbb {W}}_0$. Define a continuous function $p:{\mathbb {C}}_{\tau }^+\rightarrow {\mathbb {R}}_+$ by

$$\begin{aligned} p(\phi )=\min \left\{ \min \limits _{x\in {\bar{\Omega }}}\phi _2(x,0),\min \limits _{x\in {\bar{\Omega }}}\phi _3(x,0)\right\} , \ \forall \phi =\left( \phi _1,\phi _2,\phi _3\right) \in {\mathbb {C}}_{\tau }^+. \end{aligned}$$

Since $B_0$ is a compact subset of ${\mathbb {W}}_0$, it follows that $\inf \limits _{\phi \in B_0}p(\phi )=\min \limits _{\phi \in B_0}p(\phi )>0$. Consequently, there exists a $\varrho ^*>0$ such that

$$\begin{aligned} \liminf \limits _{t\rightarrow \infty }p(\Phi (t)\phi )\!=\!\liminf \limits _{t\rightarrow \infty }\min \left\{ \min \limits _{x\in {\bar{\Omega }}}u_2(x,t;\phi ),\min \limits _{x\in {\bar{\Omega }}}u_3(x,t;\phi )\right\} \ge \varrho ^*, \ \forall \phi \in {\mathbb {W}}_0.\!\!\!\!\nonumber \\ \end{aligned}$$

(6.28)

Let $\varrho :=\min \{{\hat{\varrho }},\ \varrho ^*\}$, where ${\hat{\varrho }}$ is given in Lemma 6.2 (ii). Then it follows from Lemma 6.2 and (6.28) that

$$\begin{aligned} \liminf \limits _{t\rightarrow \infty }\min \limits _{x\in {\bar{\Omega }}}u_i(x,t;\phi )\ge \varrho , \ \forall \phi \in {\mathbb {W}}_0 \ (1\le i\le 3). \end{aligned}$$

It remains to prove the existence of a positive periodic solution. By (Zhao 2017b, Theorem 3.5.1 and Remark 3.5.1), it follows that for each $t>0$, the solution map $\Phi (t):{\mathbb {C}}_{\tau }^+\rightarrow {\mathbb {C}}_{\tau }^+$ is an $\alpha$-contraction with respect to an equivalent norm on ${\mathbb {C}}_{\tau }^+$. Applying (Magal and Zhao 2005, Theorem 4.5) to $Q=\Phi (\omega )$, it reveals that Q has a fixed point in $A_0$. Thus, system (2.13)–(2.14) has an $\omega$-periodic solution $(\bar{u}_1(\cdot ,t),\bar{u}_2(\cdot ,t),$ $\bar{u}_3(\cdot ,t))$ with $(\bar{u}_{1t},\bar{u}_{2t},\bar{u}_{3t})\in {\mathbb {W}}_0$. By Lemma 6.2, we see that $(\bar{u}_1(\cdot ,t),\bar{u}_2(\cdot ,t),$ $\bar{u}_3(\cdot ,t))$ is a (componentwise) positive solution of system (2.13)–(2.14). The proof is finished.