2. Preliminaries and main results

Before stating the main results of this paper, we give some preliminaries as follows. A matrix $\mathbf {A}=(a_{i,j})_{n\times n}$ is essentially nonnegative if $\mathbf {A}-\min _{1\le i\le n} (a_{i,i}) \mathbf {I}_n$ is nonnegative, where $\mathbf {I}_n$ is the $n\times n$ identity matrix. Let the Banach space

\[ X:=BUC(\mathbb{R},\mathbb{R}^{3}) \]

denote the space of all bounded and uniformly continuous functions from $\mathbb {R}$ into $\mathbb {R}^{3}$, which is equipped with the usual supremum norm $\|\cdot \|_X$.

Let $m:=\max \{\sup _{x\in \mathbb {R}}V_{i,0}(x),\sup _{x\in \mathbb {R}}H_{i,0}(x)\}>0$ be given. Define the set of initial data $Y\subset X$ as

\[ Y=\{(V_{s,0},V_{i,0},H_{i,0})\in X\mid V_{s,0}=\beta/\mu,\quad 0\le V_{i,0},H_{i,0}\le m,\ V_{i,0},H_{i,0}\not\equiv 0\}. \]

We first recall a convergence result of the scalar Fisher-KPP equation [Reference Smoller45, Reference Ye, Li, Wang and Wu61].

Lemma 2.1 Consider the Fisher-KPP equation

(2.1)\begin{equation} \begin{cases} \partial_t v =\partial_{xx} v +v(\beta-\mu v), & x\in\mathbb{R},\ t>0,\\ v(x,0)=v_0(x), & x\in\mathbb{R}, \end{cases} \end{equation}

where $\beta,$ $\mu >0$ are constants and $v_0(x)\ge 0$ is a bounded, continuous function satisfying $\liminf _{|x|\to \infty }v_0(x)>0$. Then

\[ \lim_{t\to\infty}v(x,t)=\frac{\beta}{\mu}\quad \text{uniformly in }x\in\mathbb{R}. \]

Moreover, assume that $v(x,t)=\zeta (x+ct)$ is a positive bounded travelling wave solution of (2.1) such that

\[ \liminf_{\xi \to -\infty}\zeta (\xi) >0,\quad \liminf_{\xi \to +\infty}\zeta (\xi) >0, \]

then $\zeta (\xi )=\frac {\beta }{\mu },$ $\xi \in \mathbb {R}.$

Applying lemma 2.1 to the equation of $V_s+V_i$, we obtain the following boundedness result.

Lemma 2.2 For any initial data $(V_{s,0},V_{i,0},H_{i,0})\in Y,$ system (1.2) admits a globally classical, positive and bounded solution; i.e. there exists a constant $M>0$ independent of $(x,t)$ such that

\[ \|(V_s,V_i,H_i)({\cdot},t;V_{s,0},V_{i,0},H_{i,0})\|_X\le M, \quad t\in [0,\infty). \]

Proof. The local existence of (1.2) is evident. For any given $A\ge \frac {\beta }{\mu }$, we define $B,C>0$ by

\[ \sigma_1AC=\mu B^{2},\quad \sigma_2H_sB=\rho C. \]

Clearly, $A\to \infty$ implies that $B\to \infty,C\to \infty.$ Then $(0,0,0)$ and $(A,B,C)$ are a pair of generalized upper-lower solutions of (1.2) by selecting $A\ge \beta /\mu$ large enough such that the initial condition holds. Then there exists a constant $M(m)>0$ such that $0\le V_s,V_i, H_i\le M$ and the main result is clear.

Moreover, we show further bounds of solutions. For given $m>0$, adding the $V_s$-equation to the $V_i$-equation implies that $V_s+V_i$ satisfies

\[ \begin{cases} \partial_t (V_s+V_i) =\partial_{xx} (V_s+V_i) +(V_s+V_i)[\beta-\mu (V_s+V_i)],\quad & x\in\mathbb{R},t>0,\\ \frac{\beta}{\mu}\le (V_{s,0}+V_{i,0})(x)\le \frac{\beta}{\mu}+m, & x\in\mathbb{R}. \end{cases} \]

Lemma 2.1 ensures that

(2.2)\begin{align} & \lim_{t\to\infty}(V_{s }+V_{i})(x,t)=\frac{\beta}{\mu} \text{ uniformly in } x\in\mathbb{R}, \end{align}

(2.3)\begin{align} & \frac{\beta}{\mu}\le (V_{s }+V_{i})(x,t)\le \frac{\beta}{\mu}+m \quad \text{for all }(x,t)\in \mathbb{R}\times[0,\infty). \end{align}

From the $V_s$-equation, we also have

(2.4)\begin{equation} 0\le V_{s }(x,t)\le\frac{\beta}{\mu}\quad \text{for all }(x,t)\in \mathbb{R}\times[0,\infty). \end{equation}

The proof is complete.

Based on the boundedness of solutions and the standard estimates in [Reference Ladyženskaya, Solonnikov and Ural'ceva23], we have the following uniform estimates:

Lemma 2.3 Assume that $(V_s,V_i,H_i)$ is the solution of (1.2) satisfying $|V_s|$, $|V_i|,$ $|H_i|\le M_0$ for some constant $M_0>0$. Then there exists a constant $M_1>0$ such that

\[ |\partial_t u(x,t)|,\ |\partial_x u(x,t)|,\ |\partial_{tt} u(x,t)|,\ |\partial_{xx} u(x,t)|,\ |\partial_{xxx} u(x,t)|\le M_1, \]

where $u\in \{V_s,V_i,H_i\}$, $x\in \mathbb {R},$ $t\ge 1$.

In the corresponding kinetic system of (1.2), the threshold

\[ \mathcal{R}_{0}=\frac{H_s\sigma_1\sigma_2}{\rho\mu} \]

is called the basic reproduction number [Reference Fitzgibbon, Morgan and Webb16, Reference Magal, Webb and Wu33]. If $\mathcal {R}_0\le 1$, then there exists a disease-free equilibrium $E_1= (\beta /\mu,0,0)$. When $\mathcal {R}_0>1$, it also admits a positive equilibrium [Reference Fitzgibbon, Morgan and Webb16, Reference Magal, Webb and Wu33]

\[ E^{*}=(V^{*}_s,V^{*}_i, H^{*}_i)= \left( \frac{\beta}{\mu\mathcal{R}_0},\frac{\beta(\mathcal{R}_0-1)}{\mu\mathcal{R}_0}, \frac{\beta(\mathcal{R}_0-1)}{\sigma_1}\right). \]

To state our conclusion, we consider the following cooperative system

(2.5)\begin{equation} \begin{cases} \partial_t v_i=\partial_{xx} v_i +\sigma_1 h_i(\frac{\beta}{\mu}-v_i)-\beta v_i, & x\in\mathbb{R}, \ t>0,\\ \partial_t h_i =d \partial_{xx} h_i +\sigma_2 H_s v_i -\rho h_i, & x\in\mathbb{R}, \ t>0,\\ (v_i,h_i)(x,0)=(\psi_1,\psi_2)(x):=\Psi, & x\in\mathbb{R}, \end{cases} \end{equation}

where $\psi _i (i=1,2)$ are bounded, uniformly continuous functions, and $0\le \psi _1\le V_i^{*}$, $0\le \psi _2\le H_i^{*}$. Note that (2.5) is a cooperative, irreducible system such that it generates a monotone semiflow. Thus the result in Liang and Zhao [Reference Liang and Zhao28] implies that (2.5) has the following propagation properties.

We now analyse the definition of $c^{*}$ in order to estimate it in applications. Since the system also satisfies the subhomogeneous property, results in Liang and Zhao [Reference Liang and Zhao28, sections 3 and 5] imply that $c^{*}$ is determined by the following linear system

\[ \begin{cases} \partial_t v_i=\partial_{xx} v_i +\frac{\sigma_1\beta}{\mu} h_i-\beta v_i, & x\in\mathbb{R}, \ t>0,\\ \partial_t h_i =d \partial_{xx} h_i +\sigma_2 H_s v_i -\rho h_i, & x\in\mathbb{R}, \ t>0. \end{cases} \]

For such a linear cooperative system, the threshold $c^{*}$ has been investigated in several studies, see e.g., Hsu and Yang [Reference Hsu and Yang20], Wu and Hsu [Reference Wu and Hsu57]. To determine $c^{*},$ we further consider the following eigenvalue problem

\begin{align*} \lambda(\gamma)\left(\begin{array}{@{}c@{}} \varphi_1 \\ \varphi_2 \end{array}\right) & =\left[\gamma^{2}\left( \begin{array}{cc} 1 & 0\\ 0 & d \\ \end{array} \right) +\left( \begin{array}{cc} -\beta & \dfrac{\beta\sigma_1}{\mu} \\ \sigma_2H_s & -\rho\\ \end{array} \right)\right]\left(\begin{array}{@{}c@{}} \varphi_1 \\ \varphi_2 \end{array}\right) \\ & =:(\gamma^{2}\mathbf{D}+\mathbf{L})\left(\begin{array}{@{}c@{}} \varphi_1 \\ \varphi_2 \end{array}\right). \end{align*}

From [Reference Liang and Zhao28], $c^{*}$ is the threshold such that

\[ \lambda (\gamma )\left( \begin{array}{@{}c@{}} \varphi _{1} \\ \varphi _{2} \end{array} \right) =c\gamma\left( \begin{array}{@{}c@{}} \varphi _{1} \\ \varphi _{2} \end{array} \right) \]

has a positive eigenvector $(\varphi _1,\varphi _2)^{T}$ if $c\ge c^{*}.$

Note that $\mathbf {L}$ and $\mathbf {A}(\gamma ):=(\gamma ^{2}\mathbf {D}+\mathbf {L})$ are essentially nonnegative and irreducible matrices, we can estimate $c^{*}$ by using the idea in [Reference Girardin19, pp. 110]. Calculating the above eigenvalue problem, for any given $\gamma \ge 0$, the larger eigenvalue or the so-called Perron-Frobenius eigenvalue (see [Reference Girardin19]) of $\mathbf {A}(\gamma )$ is given by

\begin{align*} \lambda (\gamma ) & :=\frac{(d+1)\gamma ^{2}-(\beta +\rho )+\sqrt{\left[ (d-1)\gamma ^{2}-\rho +\beta \right] ^{2}+4\beta \rho \mathcal{R}_{0}}}{2} \\ & \geq \lambda (0)=\frac{-(\beta +\rho )+\sqrt{(\beta -\rho )^{2}+4\beta \rho \mathcal{R}_{0}}}{2} \\ & >\frac{-(\beta +\rho )+\sqrt{(\beta -\rho )^{2}+4\beta \rho }}{2} =0,\quad\gamma \geq 0. \end{align*}

Moreover, $\lambda (\gamma )$ is simple and has a positive eigenvector $(\varphi _1,\varphi _2)^{T}$. In fact, $(\varphi _1,\varphi _2)^{T}$ is also the corresponding eigenvector of the principal eigenvalue (the larger eigenvalue) of the following irreducible and positive matrix

\[ \mathbf{A}(\gamma )+\left( \begin{array}{cc} \beta +\rho & 0 \\ 0 & \beta +\rho \end{array} \right) , \]

for which we can use the Perron-Frobenius theorem. Evidently, regarding $\gamma$ as the unique variable, there exists $\gamma ^{\prime }>0$ such that

\[ \frac{d^{2}[\lambda(\gamma)]}{d\gamma ^{2}}> \frac{d+1}{2}, \gamma > \gamma', \]

which further implies that

\[ \lim_{\gamma \rightarrow 0}\frac{\lambda (\gamma )}{\gamma }=\lim_{\gamma \rightarrow \infty }\frac{\lambda (\gamma )}{\gamma }={+}\infty . \]

Using these two limits, we know that

(2.6)\begin{equation} c^{{\ast} }:=\min_{\gamma >0}\frac{\lambda (\gamma )}{\gamma }=\min_{\gamma >0} \frac{(d+1)\gamma ^{2}-(\beta +\rho )+\sqrt{\left[ (d-1)\gamma ^{2}-\rho +\beta \right] ^{2}+4\beta \rho \mathcal{R}_{0}}}{2\gamma } \end{equation}

is positive and finite, which is also attained for some finite $\gamma ^{*} >0$. By direct calculation, we know that $\frac {\lambda (\gamma )}{\gamma }$ is strictly convex (also see [Reference Girardin19, lemma 6.2]), so $\gamma ^{*}>0$ is unique.

At the same time, due to the special form of travelling wave solutions, (2.5) has a travelling wave solution $(\widetilde {\varphi },\widetilde {\psi })$ if and only if $\widetilde {\varphi }$ satisfies the following quasimonotone equation

(2.7)\begin{equation} c\widetilde{\varphi} ^{\prime }(\xi )=\widetilde{\varphi} ^{\prime \prime }(\xi )+\sigma _{1}(\beta /\mu -\widetilde{\varphi} (\xi ))(J\ast \widetilde{\varphi} )(\xi )-\beta \widetilde{\varphi} (\xi ) \end{equation}

with $\widetilde {\varphi }(-\infty )=0$, $\widetilde {\varphi }(\infty )=V_i^{*}$, where

(2.8)\begin{equation} \widetilde{\psi} (\xi )=\sigma _{2}H_{s}\int_0^{\infty}\frac{e^{-\rho t}}{\sqrt{4\pi d t}}\int_{\mathbb{R}}e^{-\frac{y^{2}}{4dt}}\widetilde{\varphi}(\xi-ct-y)\textrm{d}y \textrm{d}t:=(J*\widetilde{\varphi})(\xi) \end{equation}

is obtained by using the elementary solution of heat equations.

Linearizing (2.7) at zero and plugging $e^{\gamma \xi }$ into it, we obtain the corresponding characteristic equation

(2.9)\begin{equation} \Lambda (\gamma, c):=\gamma^{2} -c \gamma -\beta -\frac{\beta\rho\mathcal{R}_0}{d\gamma^{2}-c\gamma-\rho}=0,\quad \gamma\in[0,\gamma^{+}), \end{equation}

where $\gamma ^{+}=\frac {c+ \sqrt {c^{2} + 4d\rho }}{2d}$. By direct calculation, we obtain that $\Lambda (0,c)= \beta (\mathcal {R}_{0}-1)>0$, $\Lambda (\gamma,c)\to \infty$ for any $c\ge 0$ as $\gamma \to \gamma ^{+}-$, and $\partial _{\gamma \gamma }\Lambda (\gamma,c)>0$ for any $c\ge 0$, $\gamma \in [0,\gamma ^{+})$, which implies that $\Lambda (\gamma,c)$ is convex in $\gamma \in [0,\gamma ^{+})$. Moreover, since $\partial _{c}\Lambda (\gamma,c)<0$ for any given $\gamma \in [0,\gamma ^{+})$, $\Lambda (\gamma,c)$ is continuous and strictly decreasing in $c\in [0,\infty )$ such that for any $\gamma \in [0,\gamma ^{+})$, $\Lambda (\gamma,0)\ge \beta (\mathcal {R}_{0}-1)>0$, $\lim_{c\to \infty }\Lambda (\gamma,c)=-\infty$. We now define the following bounded constant

(2.10)\begin{equation} c^{*}:=\inf\left\{c>0~|~\Lambda(\gamma,c)=0\text{ has exact one positive root for }\gamma\in[0,\gamma^{+})\right\}, \end{equation}

which is the minimal wave speed of (2.7) ([Reference Xu and Xiao60, Reference Zhao and Xiao65]). Since the travelling wave in lemma 2.4 is equivalent to that of (2.7), lemma 2.4 implies that the two definitions of $c^{*}$ are equivalent.

With the above constants, we now state our main results as follows.

Theorem 2.5 Suppose that $\mathcal {R}_0>1$ holds. Then the solution of (1.2) satisfies the following spreading properties:

1. (i) For any given $c>c^{*},$ one has

   \[ \limsup\limits_{t\to\infty}\sup\limits_{|x|\ge ct}\left[\left|V_s(x,t)-\frac{\beta}{\mu}\right|+V_i(x,t)+H_i(x,t)\right]=0; \]

2. (ii) For any given $0< c< c^{*},$ one has

   \[ \limsup\limits_{t\to\infty}\sup\limits_{|x|\le ct} (|V_s(x,t)-V_s^{*}|+|V_i(x,t)-V_i^{*}|+|H_i(x,t)-H_i^{*}|)= 0. \]

Moreover, (1.3)–(1.4) has a monotone solution if and only if $c\ge c^{*}.$ In particular, if (1.3)–(1.4) has a monotone solution, then it must satisfy

(2.11)\begin{equation} \lim_{\xi\to \infty}\phi(\xi) =V_s^{*}, \quad\lim_{\xi\to \infty}\varphi(\xi) =V_i^{*}, \quad \lim_{\xi\to \infty}\psi(\xi) =H_i^{*} \end{equation}

and it is unique in the sense of phase shift; that is, if $(\phi _1(\xi ),\varphi _1(\xi ), \psi _1(\xi ))$ is a positive solution of (1.3)–(1.4), then there exists $h\in \mathbb {R}$ such that

\[ (\phi(\xi),\varphi(\xi), \psi(\xi))=(\phi_1(\xi+h),\varphi_1(\xi+h), \psi_1(\xi+h)), \quad\xi\in\mathbb{R}. \]

In addition, $\phi (\xi )$ is strictly decreasing in $\xi$ while $\varphi (\xi ),$ $\psi (\xi )$ are strictly increasing in $\xi$.

When $\mathcal {R}_0\le 1$, we state the following result on the convergence of solutions and the nonexistence of travelling waves.

Theorem 2.7 If $\mathcal {R}_0\le 1,$ then the solution of (1.2) satisfies

\[ \lim_{t\to\infty}(V_s,V_i,H_i)(x,t)=\left(\frac{\beta}{\mu},0,0\right)\text{ uniformly in }x\in\mathbb{R}. \]

Moreover, (1.3)–(1.4) does not have a positive solution for any $c\in \mathbb {R}.$

3. Spreading properties

In this section, we are devoted to proving the main part of theorem 2.5 if $\mathcal {R}_0>1$. To complete the proof of theorem 2.5 (i), we use an auxiliary system as the upper control system, and establish our results by constructing an upper solution and applying the comparison principle. We deal with theorem 2.5 (ii) in a weak sense by using the idea of uniform persistence [Reference Ducrot9] from dynamical system theory [Reference Magal and Zhao35]. In this procedure, we introduce the generalized principal eigenvalue problem [Reference Girardin19] of a weakly coupled elliptic system in the whole space to estimate the lower bounds of the spreading speed.

3.1 Upper bounds on the spreading speed

In this subsection, we prove theorem 2.5 (i) by showing the following lemma.

Lemma 3.1 For any given $\epsilon >0,$ the solution of (1.2) satisfies

\[ \limsup_{t \to\infty} \sup _{|x|\ge (c^{*}+\epsilon)t}\left[\left| V_s(x,t)-\frac{\beta}{\mu}\right|+V_i(x,t)+H_i (x,t)\right]=0. \]

Proof. It follows from lemma 2.2, (2.3)–(2.4) that $V_i(x,t)$ and $H_i(x,t)$ satisfy

(3.1)\begin{equation} \begin{cases} \partial_t V_i(x,t)\le\partial_{xx} V_i(x,t)+\frac{\beta\sigma_1}{\mu} H_i(x,t) -\beta V_i(x,t), & x\in\mathbb{R}, \ t>0,\\ \partial_t H_i(x,t)=d \partial_{xx} H_i(x,t)+\sigma_2H_s V_i(x,t)-\rho H_i(x,t), & x\in\mathbb{R}, \ t>0,\\ (V_i,H_i)(x,0)=(V_{i,0}(x),H_{i,0}(x)), & x\in\mathbb{R}. \end{cases} \end{equation}

Let $c^{*},\gamma ^{*},\varphi ^{*}=(\varphi _1,\varphi _2)^{T}$ be given in § 2, we define a positive vector function

\[ (\overline{V_i},\overline{H_i})(x,t)=\min\left\{e^{\gamma^{*}(x+c^{*}t+t_1)} ,\ e^{\gamma^{*}({-}x+c^{*}t+t_1)}\right\}\varphi^{*}, \]

where $t_1>0$ is sufficiently large such that $(\overline {V_i},\overline {H_i})(x,0)\ge (V_i,H_i)(x,0)$. Based on the above arguments, we verify that if $(\overline {V_i},\overline {H_i})(x,t)$ is differentiable, then

(3.2)\begin{equation} \begin{cases} \partial_t \overline{V_i}(x,t)=\partial_{xx} \overline{V_i}(x,t)+\frac{\beta\sigma_1}{\mu} \overline{H_i}(x,t) -\beta \overline{V_i}(x,t), & x\in\mathbb{R}, \ t>0,\\ \partial_t \overline{H_i}(x,t)=d \partial_{xx} \overline{H_i}(x,t)+\sigma_2H_s \overline{V_i}(x,t)-\rho \overline{H_i}(x,t), & x\in\mathbb{R}, \ t>0,\\ (\overline{V_i},\overline{H_i})(x,0)=\min\left\{e^{\gamma^{*}(x+t_1)} ,\ e^{\gamma^{*}({-}x+t_1)}\right\}\varphi^{*}, & x\in\mathbb{R} \end{cases} \end{equation}

and $(V_i,H_i)$ is a lower solution of (3.2). Applying the classical parabolic comparison principle [Reference Smoller45, Reference Ye, Li, Wang and Wu61] to the cooperative system (3.2), we obtain

\[ \limsup_{t \rightarrow \infty} \sup _{|x| \geq (c^{*}+\epsilon)t}(V_i+ H_i)(x,t)\le \limsup_{t \rightarrow \infty} \sup _{|x| \geq (c^{*}+\epsilon)t}(\overline{V_i}+\overline{H_i})(x,t)=0 \]

for any $\epsilon >0$.

Next we show the convergence of $V_s$ by contradiction. Assume that for any $\epsilon >0$, there exist $\delta >0$, a sequence $\{t_n\}_{n\ge 0}$ tending to infinity, and a sequence $\{x_n\}_{n\ge 0}\subset \mathbb {R}$ such that

\[ |x_n|\ge (c^{*}+\epsilon)t_n,\quad \left| V_s(x_n,t_n)-\frac{\beta}{\mu}\right|\ge \delta\quad\text{for all } n\ge 0. \]

Consider a sequence of functions

\[ (V_{s,n}, V_{i,n},H_{i,n})(x,t)=(V_{s},V_i,,H_i)(x+x_n,t+t_n). \]

By lemma 2.3 and the parabolic estimates, we obtain that $( V_{s,n},V_{i,n},H_{i,n})$ has a subsequence, still denoted by $( V_{s,n},V_{i,n},H_{i,n}),$ converging to some entire solution $(V_{s,\infty },V_{i,\infty },H_{i,\infty })$ of (1.2) locally uniformly. Thus $V_{s,\infty }$ satisfies

(3.3)\begin{equation} \left|V_{s,\infty}(0,0)-\frac{\beta}{\mu}\right|\ge \delta. \end{equation}

Due to the convergence of $V_i$ and $H_i$, we have $(V_{i,\infty },H_{i,\infty })(0,0)=(0,0)$. Then the strong maximum principle implies that $(V_{i,\infty },H_{i,\infty })(x,t)\equiv (0,0)$. Thus

\[ (V_{s}+V_i)(x+x_n,t+t_n)\to V_{s,\infty}(x,t),\quad n\to\infty. \]

By (2.2), we have $(V_{s}+V_i)(x+x_n,t+t_n) \to \frac {\beta }{\mu }$ as $n\to \infty$, which further implies that $V_{s,\infty }\equiv \frac {\beta }{\mu }$ by the uniqueness of the entire solution for the scalar Fisher-KPP equation [Reference Berestycki, Hamel and Nadin4]. It contradicts (3.3) and completes the proof.

3.2 Lower bounds on the spreading speed

In this subsection, we first prove a lemma as a weak version of (theorem 2.5(ii), which relies on the uniform persistence theory in dynamical systems. The final convergence in theorem 2.5(ii) will be completed in § 4.

Lemma 3.2 Suppose that $\mathcal {R}_0>1$. Then for any given $0< c< c^{*},$ the solution of (1.2) satisfies

\begin{align*} 0<& \liminf\limits_{t\to\infty}\inf\limits_{|x|\le ct}V_s(x,t)\le\limsup\limits_{t\to\infty}\sup\limits_{|x|\le ct} V_s(x,t)<\frac{\beta}{\mu},\\ 0<& \liminf\limits_{t\to\infty}\inf\limits_{|x|\le ct}V_i(x,t),\quad 0<\liminf\limits_{t\to\infty}\inf\limits_{|x|\le ct}H_i(x,t). \end{align*}

In what follows, we divide the proof of the above lemma into three steps: (1) point-wise weak spreading property, (2) point-wise spreading property, and (3) uniform spreading property. Throughout this subsection, we assume that $c^{0}$ is an arbitrarily fixed constant such that $0\le c^{0}< c^{*}.$

3.2.1 An eigenvalue problem.

In order to prove the weak point-wise spreading property, we first investigate an eigenvalue problem of weakly coupled elliptic systems by the corresponding generalized principal eigenvalue. For $0< R$, $0<\eta <\beta$, and $c\in \mathbb {R}$, we consider the following eigenvalue problem:

(3.4)\begin{equation} \begin{cases} \lambda\psi_1={-}\partial_{xx}\psi_1 -c\partial_x \psi_1 +(\beta+\eta)\psi_1-\sigma_1(\frac{\beta-\eta}{\mu} )\psi_2, & \quad x\in ({-}R,R),\\ \lambda\psi_2={-}d\partial_{xx}\psi_2-c\partial_x \psi_2-\sigma_2H_s\psi_1+\rho\psi_2, & \quad x\in ({-}R,R),\\ \psi_1(x)=\psi_2(x)=0, & \quad x\in\{{-}R,R\}. \end{cases} \end{equation}

Lemma 3.3 For any given $|c|< c^{*},$ there exist $\eta _c>0$ small enough and $\overline {R}_c>0$ large enough such that the principal eigenvalue of (3.4) satisfies $\Lambda _{R}(\eta )<0$ for any $R\ge \overline {R}_c$ and $0\le \eta \le \eta _c$.

Proof. Since this weak coupled elliptic system is cooperative and irreducible, the celebrated Krein-Rutman theorem [Reference Du7] implies that there exists a unique principal eigenvalue $\Lambda _{R}(\eta )$ associated with a positive eigenfunction pair $(\psi _1 ,\psi _2 )$ for (3.4).

We first consider the case $0\le c< c^{*}$. From [Reference Girardin19, theorem 4.2], we obtain that as $R\to +\infty$, $\Lambda _R(\eta )$ converges to a generalized principal eigenvalue of the operator

\[ \mathcal{L}[\psi_1,\psi_2]:=\left(\begin{array}{@{}c@{}} -\partial_{xx}\psi_1 -c\partial_x \psi_1+(\beta+\eta)\psi_1-\sigma_1\left(\dfrac{\beta-\eta}{\mu} \right)\psi_2\\ -d\partial_{xx}\psi_2-c\partial_x \psi_2-\sigma_2H_s\psi_1+\rho\psi_2 \end{array}\right), \]

in which the generalized principal eigenvalue is defined by

\[ \Lambda_1(\eta):=\sup \left\{\lambda\in\mathbb{R}\mid \exists (\psi_1,\psi_2)\in C^{2}(\mathbb{R},\mathbb{R}^{2}_+),\quad \mathcal{L}[\psi_1,\psi_2]\ge\lambda[\psi_1,\psi_2]\right\}, \]

where $C^{2}(\mathbb {R},\mathbb {R}^{2}_+)$ represents the space of all positive twice continuously differentiable vector functions. It follows from [Reference Girardin19, lemma 6.4] that

\[ \lim_{R\to\infty}\Lambda_R(\eta)=\Lambda_1(\eta)=\max_{\gamma\ge 0} [-\lambda(\gamma,\eta)+c\gamma], \]

where $\lambda (\gamma,\eta )$ denotes the unique Perron-Frobenius eigenvalue of the matrix

\[ \mathbf{A}(\gamma,\eta):=\left( \begin{array}{cc} \gamma^{2}-(\beta+\eta) & \dfrac{\sigma_1(\beta-\eta)}{\mu} \\ \sigma_2H_s & d\gamma^{2}-\rho \end{array} \right). \]

Recalling the first definition of $c^{*}$ in § 2, $\mathbf {A}(\gamma,0)$ is essentially nonnegative and irreducible, and admits the unique Perron-Frobenius eigenvalue $\lambda (\gamma,0)>0$ associated with a positive eigenvector (see [Reference Girardin19]). Moreover,

\[ c^{*}=\min_{\gamma>0}\frac{\lambda(\gamma,0)}{\gamma}>0. \]

For any given $c\in [0,c^{*})$, one obtains that there exist $\eta _c>0$ small enough and $\overline {R}_c$ large enough such that

\[ \Lambda_R(\eta)<0,\quad R\in[ \overline{R}_c,\infty),\quad\eta\in[0,\eta_c]. \]

By a symmetric argument on $-x$ and the uniqueness of this eigenvalue, the conclusion for the case $-c^{*}< c\le 0$ follows. This completes the proof.

3.2.2 The first step: point-wise weak spreading property

Lemma 3.4 There exists $\varepsilon _1(c^{0})>0$ such that for any $c\in [0,c^{0}]$ and $x\in \mathbb {R},$ the solution $(V_s,V_i,H_i)$ of (1.2) with initial data in $Y$ satisfies

(3.5)\begin{equation} \limsup_{t\to\infty} (V_i+H_i)(x+ct,t)\ge \varepsilon_1(c^{0}). \end{equation}

Proof. Suppose by contradiction that there exist sequences

\[ \{(V^{n}_{s,0},V^{n}_{i,0},H^{n}_{i,0})(x)\}_{n\ge 0}\subset Y, \quad \{c_n\}_{n\ge 0}\subset[0,c^{0}], \quad\{x_n\}_{n\ge 0}\subset\mathbb{R} \]

such that the solution $(V_{s}^{n},V_{i}^{n},H_{i}^{n})$ of (1.2) with initial data $(V^{n}_{s,0},V^{n}_{i,0},H^{n}_{i,0})$ satisfies

\[ \limsup_{t\to\infty}(V_{i}^{n}+H_{i}^{n})(x_n+c_nt,t)\le \frac{1}{n+1}\text{ for all }n\ge 0. \]

Then it implies that there exists $\{t_n\}_{n\ge 0}\subset [0,\infty )$ tending to $\infty$, and

(3.6)\begin{equation} \max\left\{V_{i}^{n}(x_n+c_nt,t),H_{i}^{n}(x_n+c_nt,t)\right\}\le \frac{2}{n+1} \text{ for all } t\ge t_n. \end{equation}

By (3.6), we claim that for any $R>0$, there exists a sequence $t'_n\ge t_n$ such that

(3.7)\begin{equation} \lim_{n\to\infty}\sup_{t\ge 0,|x|\le R} \left|V_{s}^{n}(x_n+c_n(t'_n+t)+x,t'_n+t)-\frac{\beta}{\mu}\right|=0. \end{equation}

In fact, it suffices to verify (3.7) with $\{t_n'=t_n\}$. We suppose by contradiction that there exist $\delta >0$, sequences $s_n\ge t_n$, $c_n\to c_{\infty }\in [0,c^{0}]$ and $x'_n\to x'_{\infty }\in [-R,R]$ such that

\[ \left| V_{s}^{n}(x_n+c_ns_n +x_n',s_n )-\frac{\beta}{\mu}\right|\ge\delta. \]

Using lemma 2.3, the standard parabolic estimates imply that up to a subsequence,

\[ (V_{s}^{n},V_{i}^{n},H_{i}^{n})(x_n+c_n(s_n+t)+x,s_n+t) \to (u_{\infty},v_{\infty},w_{\infty})(x,t)\text{ as }n\to\infty \]

locally uniformly for $(x,t)\in \mathbb {R}^{2}$, where $(u_{\infty },v_{\infty },w_{\infty })$ is an entire solution of

\[ \begin{cases} (\partial_t-\partial_{xx}-c_{\infty}\partial_x) u_{\infty}=(u_{\infty} +v_{\infty})(\beta-\mu u_{\infty})-\sigma_1 u_{\infty}w_{\infty},\\ (\partial_t-\partial_{xx}-c_{\infty}\partial_x)v_{\infty}= \sigma_1 u_{\infty}w_{\infty}-\mu(u_{\infty} +v_{\infty})v_{\infty},\\ (\partial_t-d\partial_{xx}-c_{\infty}\partial_x)w_{\infty}=\sigma_2 H_s v_{\infty} -\rho w_{\infty}. \end{cases} \]

Note that (3.6) yields $(v_{\infty },w_{\infty })(0,0)=(0,0)$. Applying the strong maximum principle, we obtain that $(u_{\infty },v_{\infty },w_{\infty })(x,t)\equiv (\beta /\mu,0,0)$. However, since the sequence $\{x_n'\}\subset [-R,R]$ is relatively compact, $u_{\infty }(x,t)\equiv \frac {\beta }{\mu }$ contradicts the fact that

\[ \left| u_{\infty}(x_{\infty}',0 )-\frac{\beta}{\mu}\right|\ge\delta, \]

which proves (3.7).

Now we return to prove lemma 3.4. Consider a sequence of functions $(u_n,v_n,w_n)$ with moving frames defined by

\[ (u_n,v_n,w_n)(x,t):=(V_{s}^{n},V_{i}^{n},H_{i}^{n})(x_n+c_n(t_n+t)+x,t_n+t), \]

then (3.6) becomes

(3.8)\begin{equation} \max\{v_n(0,t),w_n(0,t)\}\le \frac{2}{n+1}\text{ for all }n\ge 0,\ t\ge 0. \end{equation}

Let us fix small $\eta >0$ and large $R>0$ such that lemma 3.3 holds for $c^{0}$. It follows from (2.3) and (3.7) that for any $n$ large enough, one has

\[ u_n(x,t)\ge \frac{\beta-\eta}{\mu},\quad \frac{\beta}{\mu}\le(u_n+v_n)(x,t)\le \frac{\beta+\eta}{\mu}\text{ for all } t\ge 0,\ |x|\le R. \]

Then $(v_n,w_n)(x,t)$ satisfies

\[ \begin{cases} (\partial_t-\partial_{xx}-c_n\partial_x) v_n\ge \sigma_1\frac{\beta-\eta}{\mu} w_n-(\beta+\eta) v_n, & t\ge 0,\ |x|\le R,\\ (\partial_t-d\partial_{xx}-c_n\partial_x)w_n\ge \sigma_2H_s v_n-\rho w_n, & t\ge 0,\ |x|\le R. \end{cases} \]

Thus the comparison principle yields that

\[ (v_n,w_n)(x,t)\ge \tau e^{-\Lambda_R t}(\psi_1,\psi_2)(x)\quad\text{for all }|x|\le R,\ t\ge 0, \]

where $\Lambda _R$ and $(\psi _1,\psi _2)$ are the principal eigenvalue and the associated positive eigenfunction pair defined in lemma 3.3, $\tau >0$ is small enough so that $(v_n,w_n)(x,0)\ge \tau (\psi _1,\psi _2)(x)$ for all $|x|\le R$.

Using lemma 3.3, we have $\Lambda _R<0$ since $\eta$ is small enough and $R$ is large enough. Thus, $v_n(0,t)\ge \tau e^{-\Lambda _R t}\psi _1(0)\to \infty$ as $t\to \infty$, which contradicts (3.8). The proof is complete.

3.2.3 The second step: point-wise strong spreading property

In the following lemma, we shall improve the weak spreading properties stated in the previous subsection.

Lemma 3.5 There exists $\varepsilon _2(c^{0})>0$ such that for any $c\in [0,c^{0}]$ and $x\in \mathbb {R},$ the solution $(V_s,V_i,H_i)$ of (1.2) with initial data in $Y$ satisfies

(3.9)\begin{equation} \liminf_{t\to\infty} (V_i+H_i)(x+ct,t)\ge \varepsilon_2(c^{0}). \end{equation}

Proof. We prove (3.9) by contradiction. Assume that there exist sequences

\[ \{(V^{0}_{s,n},V^{0}_{i,n},H^{0}_{i,n})(x)\}_{n\ge 0}\subset Y, \quad \{c_n\}_{n\ge 0}\subset[0,c^{0}],\quad\{x_n\}_{n\ge 0}\subset\mathbb{R} \]

such that

\[ \liminf\limits_{t\to\infty} (V_{i,n}+H_{i,n})(x_n+c_nt,t)<\frac{1}{n+1},\quad n\in\mathbb{N}. \]

Without loss of generality, we further assume that $c_n\to c^{\infty }\in [0,c^{0}]$ as $n\to \infty$. Lemma 3.4 implies that there exist two sequences $t_n\to \infty$ and $\tau _n\in \mathbb {R}_+$ such that for each $n\ge 0$

\begin{align*} & [V_{i,n}+H_{i,n}](x_n+c_nt_n,t_n)=\frac{\varepsilon_1(c^{0})}{2},\\ & [V_{i,n}+H_{i,n}](x_n+c_nt,t)\le\frac{\varepsilon_1(c^{0})}{2},\quad t\in[t_n,t_n+\tau_n],\\ & [V_{i,n}+H_{i,n}](x_n+c_n(t_n+\tau_n),t_n+\tau_n)=\frac{1}{n+1}, \end{align*}

where $\varepsilon _1(c^{0})$ is given in lemma 3.4.

By lemma 2.3, standard parabolic estimates and up to a sequence, we have

\[ (V_{s,n},V_{i,n},H_{i,n})(x_n+c_nt_n+x,t_n+t) \to(V^{\infty}_{s},V^{\infty}_{i},H^{\infty}_{i})(x,t) \]

locally uniformly in $(x,t)\in \mathbb {R}^{2}$ as $n\to \infty$, where $(V^{\infty }_{s},V^{\infty }_{i},H^{\infty }_{i})$ is an entire solution of (1.2). Due to the choice of $t_n$, we have

\[ V_i^{\infty}(0,0)+H_i^{\infty}(0,0)=\frac{\varepsilon_1(c^{0})}{2}, \]

which further yields $V_i^{\infty }(x,t)+H_i^{\infty }(x,t)>0$ by the strong maximum principle.

We find that $\tau _n\to \infty$ as $n\to \infty$. Indeed, suppose by contradiction that its subsequence converges to some $t_0\in \mathbb {R}_+$, then one has

\[ (V_i^{\infty}+H_i^{\infty})(c^{\infty}t_0,t_0)=0. \]

Then it follows that $V_i^{\infty }(c^{\infty }t_0,t_0)=H_i^{\infty }(c^{\infty }t_0,t_0)=0$ since these solutions are nonnegative. The strong maximum principle implies that $V_i^{\infty }=H_i^{\infty }\equiv 0$, a contradiction. Hence by the second construction we have

\[ V_i^{\infty}(c^{\infty}t,t)+H_i^{\infty}(c^{\infty}t,t)\le \frac{\varepsilon_1(c^{0})}{2} \quad\text{for all }t\ge 0, \]

and it contradicts (3.5) in lemma 3.4 as $x=0$ and $c=c^{\infty }$. This completes the proof.

3.2.4 The third step: uniform spreading property

Lemma 3.6 For any $c\in [0,c^{0}]$, the solution of (1.2) with initial data in $Y$ satisfies

(3.10)\begin{align} & \liminf_{t\to\infty}\inf_{0\le x \le ct} V_i(x,t)>0,\quad \liminf_{t\to\infty}\inf_{0\le x \le ct}H_i(x,t)>0, \end{align}

(3.11)\begin{align} & 0<\liminf_{t\to\infty}\inf_{0\le x \le ct} V_s(x,t)\le\limsup_{t\to\infty}\sup_{0\le x \le ct} V_s(x,t)<\frac{\beta}{\mu}. \end{align}

Proof. We first prove that

(3.12)\begin{equation} \liminf_{t\to\infty}\inf_{0\le x\le ct} (V_i+H_i)(x,t)>0. \end{equation}

For any given $\widehat {c}\in (0,c^{0})$, we suppose by contradiction that there exist sequences $t_n\to \infty$ and $c_n\in [0,\widehat {c})$ such that

\[ \lim_{n\to\infty}(V_i+H_i)(c_nt_n,t_n)=0. \]

Up to a subsequence, we assume, without loss of generality, that $c_n\to \widetilde {c}_{\infty }\in [0,\widehat {c}]$ as $n\to \infty$. Select $c'>0$ such that $\widetilde {c}_{\infty }< c'\le c^{0}$, and define the sequence

\[ t'_n:=\frac{c_nt_n}{c'}\in[0,t_n)\quad\text{for all } n\ge 0. \]

We first consider the case that the sequence $\{c_nt_n\}_{n\ge 0}$ is bounded, which may occur if $\widetilde {c}_{\infty }=0$. Up to a subsequence, it follows from the strong maximum principle that as $n\to \infty$, $c_nt_n\to x_{\infty }$ and

\[ (V_i+H_i)(c_nt_n+x,t_n+t)\to 0\quad\text{locally uniformly for }(x,t)\in\mathbb{R}^{2}. \]

This implies in particular that $(V_i+H_i)(0,t_n)\to 0$ as $n\to \infty$, which contradicts the case $c=0$ in lemma 3.5. Thus $\widetilde {c}_{\infty }>0$ implies that the sequence $\{c_nt_n\}$ has no bounded subsequence.

Now we assume that $t'_n\to \infty$ as $n\to \infty$. Since $c'\in (0,c^{0}]$, lemma 3.5 implies that

\[ (V_i+H_i)(c_nt_n,t'_n)=(V_i+H_i)(c't'_n,t'_n)\ge \varepsilon_2(c^{0}) \]

for each $n$ large enough.

Now we define the third time sequence $\{t_n''\}$ as follows

\[ t_n'':=\inf\left\{t\le t_n\mid (V_i+H_i)(c_nt_n,s)\le \frac{\min\{\varepsilon_1(c^{0}),\varepsilon_2(c^{0})\}}{2}\ \text{for any }s\in(t,t_n)\right\}, \]

then $t_n''\in (t_n',t_n)$. Since $(V_i+H_i)(c_nt_n,t_n)\to 0$ as $n\to \infty$, we have

\[ (V_i+H_i)(c_nt_n,t_n'')=\frac{\min\{\varepsilon_1(c^{0}),\varepsilon_2(c^{0})\}}{2}. \]

By using a similar limiting argument and a strong maximum principle, one also has

\[ t_n-t_n''\to\infty\quad\text{as }n\to\infty. \]

Then by lemma 2.3 and standard parabolic estimates and up to a subsequence, we obtain that

\[ (V_s,V_i,H_i)(c_nt_n+x,t''_n+t)\to (\widetilde{u}_{\infty},\widetilde{v}_{\infty},\widetilde{w}_{\infty})(x,t),\quad n\to\infty \]

locally uniformly in $(x,t)$ such that

\begin{align*} & (\widetilde{v}_{\infty}+\widetilde{w}_{\infty})(0,0)>0,\\ & (\widetilde{v}_{\infty}+\widetilde{w}_{\infty})(0,t)\le \frac{\min\{\varepsilon_1(c^{0}),\varepsilon_2(c^{0})\}}{2}\quad \text{for all }t\ge 0, \end{align*}

which contradicts (3.9) in lemma 3.5 as $c=0$ and $x=0$. Thus (3.12) holds.

Next we show (3.10) by contradiction. Assume that there exist sequences $t_n\to \infty$ and $0\le x_n\le ct_n$ such that $V_i(x_n,t_n)\to 0$ as $n\to \infty$, then up to a subsequence, lemma 2.3, parabolic estimates and the strong maximum principle imply that as $n\to \infty$,

\[ (V_s,V_i,H_i)(x+x_n,t+t_n)\to (u^{\infty},0,w^{\infty})(x,t)\quad \text{locally uniformly in }(x,t)\in\mathbb{R}^{2}, \]

where $(u^{\infty },0,w^{\infty })$ is an entire solution of (1.2). But (3.12) yields $w^{\infty }>0$, which contradicts (1.2). Then it follows that for any $c\in [0,c^{0}]$,

\[ \liminf_{t\to\infty}\inf_{0\le x\le ct}V_i(x,t)>0. \]

Thus we finish the proof of (3.10) by using a similar argument for $H_i$.

To deal with (3.11), we first assume by contradiction that there exist sequences $t_n\to \infty$ and $0\le x_n\le ct_n$ such that $V_s(x_n,t_n)\to \frac {\beta }{\mu }$ as $n\to \infty$, then $(V_s,V_i,H_i)(x+x_n,t+t_n)$ converges to an entire solution $(u_{\infty },v_{\infty },w_{\infty })(x,t)$ of (1.2), which satisfies $u_{\infty }(0,0)=\frac {\beta }{\mu }$. Using the strong maximum principle, $u_{\infty }\equiv \frac {\beta }{\mu }$ follows. However, it follows from (3.10) that $w_{\infty }>0$, which contradicts (1.2). For the remaining part, if $V_s(x_n,t_n)\to 0$ as $n\to \infty$, then $V_i(x_n,t_n)\to 0$ as $n\to \infty$, which contradicts (3.10). This completes the proof.

Recall that the aforementioned three lemmas addressed the rightward speed case of lemma 3.2, the leftward part of spreading follows by using a symmetric argument. Thus lemma 3.2 is proved.

4. Convergence of solutions

In this section, we show some convergence results to complete the proofs of theorem 2.5 (ii) and theorem 2.7. If $\mathcal {R}_0>1$, we show that the solution converges to the positive equilibrium locally uniformly, which gives the final convergence in theorem 2.5 (ii). Then the disease will persist eventually. If $\mathcal {R}_0\le 1$, then the solution tends to the disease-free equilibrium, which indicates that the vector-borne disease will die out.

Before proving the final convergence in theorem 2.5 (ii), we state a convergence result for the bounded, persistent entire solutions of (1.2), which is established by constructing two auxiliary monotone systems.

Lemma 4.1 Suppose that $\mathcal {R}_0>1$. If $(V_s,V_i,H_i)$ is a bounded entire solution of (1.2) such that

(4.1)\begin{equation} \inf\limits_{(x,t)\in\mathbb{R}^{2}}V_s(x,t)>0,\quad \inf\limits_{(x,t)\in\mathbb{R}^{2}}V_i(x,t)>0, \ \inf\limits_{(x,t)\in\mathbb{R}^{2}}H_i(x,t)>0, \end{equation}

then $(V_s,V_i,H_i)(x,t)\equiv (V_s^{*},V_i^{*},H_i^{*})$ for all $(x,t)\in \mathbb {R}^{2}.$

Proof. We set $V=V_s+V_i$, then $V(x,t)$ satisfies the following Cauchy problem

(4.2)\begin{equation} \begin{cases} \partial_t V(x,t)=\partial_{xx} V(x,t) +V(x,t)[\beta-\mu V(x,t)], & x\in\mathbb{R}, \ t>0,\\ V(x,0)=(V_{s}+V_{i})(x,0)>0, & x\in\mathbb{R}. \end{cases} \end{equation}

It follows from (2.2) that for any $\varepsilon >0$, there exists $T>0$ such that

\[ \left|(V_s+V_i)(x,t)-\frac{\beta}{\mu}\right|<\varepsilon, \quad t\ge T,\ x\in\mathbb{R}. \]

Due to (4.1), we reset the initial function as $V(x,0)=(V_{s}+V_{i})(x,-t_0)$ for any $t_0>0$, we similarly have

\[ \left|(V_s+V_i)(x,t-t_0)-\frac{\beta}{\mu}\right|<\varepsilon, \quad t\ge T,\ x\in\mathbb{R}. \]

Taking $t_0\to \infty$, since the problem (4.2) is autonomous, we actually obtain that

\[ \left|(V_s+V_i)(x,t+T)-\frac{\beta}{\mu}\right|<\varepsilon, \quad t\in\mathbb{R},\ x\in\mathbb{R}. \]

Due to the arbitrariness of $\varepsilon$, the above inequality implies that $V_s+V_i\equiv \frac {\beta }{\mu }$ for all $(x,t)\in \mathbb {R}^{2}$. Thus the bounded entire solution $(V_s,V_i,H_i)$ satisfies the following subsystem:

(4.3)\begin{equation} \begin{cases} \partial_t V_i=\partial_{xx} V_i +\sigma_1 H_i(\frac{\beta}{\mu}-V_i)-\beta V_i, & x\in\mathbb{R}, \ t>0,\\ \partial_t H_i =d \partial_{xx} H_i +\sigma_2 H_s V_i -\rho H_i, & x\in\mathbb{R}, \ t>0,\\ (V_i,H_i)(x,0)=(V_{i,0},H_{i,0})(x), & x\in\mathbb{R}, \end{cases} \end{equation}

where the initial data $(V_{i,0}(x),H_{i,0}(x))$ satisfies

\[ \inf_{x\in\mathbb{R}}V_{i,0}(x)>0,\quad \inf_{x\in\mathbb{R}}H_{i,0}(x)>0. \]

Now we deal with the long time behaviour of a spatial homogeneous solution to (4.3). Let $(\underline {v},\underline {h})(t)$ be the solution of

(4.4)\begin{equation} \begin{cases} \dfrac{\textrm{d}\underline{v}(t)}{\textrm{d}t}= \sigma_1\underline{h}\left(\dfrac{\beta}{\mu}-\underline{v}\right)-\beta\underline{v}, & t>0,\\ \dfrac{\textrm{d}\underline{h}(t)}{\textrm{d}t}=\sigma_2H_s\underline{v} - \rho \underline{h}, & t>0,\\ (\underline{v},\underline{h})(0)=(\inf_{x\in\mathbb{R}}|V_{i,0}(x)|, \inf_{x\in\mathbb{R}}|H_{i,0}(x)|), \end{cases} \end{equation}

then $(\underline {v},\underline {h})(t)$ is a spatially homogeneous lower solution of (4.3). Since (4.4) is cooperative and $\mathcal {R}_0>1$ holds, it is evident that

(4.5)\begin{equation} (\underline{v},\underline{h})(t)\to \left(V^{*}_i, H^{*}_i\right),\quad t\to\infty. \end{equation}

Similarly, we consider a spatial homogeneous upper solution $(\overline {v},\overline {h})(t)$ of (4.3) with initial data $(\sup _{x\in \mathbb {R}}|V_{i,0}(x)|, \sup _{x\in \mathbb {R}}|H_{i,0}(x)|)$. Since $\mathcal {R}_0>1$, we have

(4.6)\begin{equation} (\overline{v},\overline{h})(t)\to \left(V^{*}_i, H^{*}_i\right),\quad t\to\infty. \end{equation}

It follows from (4.5)–(4.6) and the comparison principle that for any $\widetilde {\varepsilon }>0$, there exists $\widetilde {T}>0$ such that

\[ |V_i(x,t)-V_i^{*}|+|H_i(x,t)-H_i^{*}|<\widetilde{\varepsilon},\quad t\ge\widetilde{T},\ x\in\mathbb{R}. \]

Applying a similar argument as at the beginning of the proof to (4.3), we actually obtain that

\[ \left|V_i(x,t+\widetilde{T})-V_i^{*}\right| +\left|H_i(x,t+\widetilde{T})-H_i^{*}\right|<\widetilde{\varepsilon}, \quad t\in\mathbb{R},\ x\in\mathbb{R}. \]

Due to the arbitrariness of $\widetilde {\varepsilon }$, the above inequality together with $V_s+V_i\equiv \frac {\beta }{\mu }$ yields that $(V_s,V_i,H_i)\equiv (V_s^{*},V_i^{*},H_i^{*})$ for all $(x,t)\in \mathbb {R}^{2}$, which completes the proof.

Proof of the convergence in theorem 2.5 (ii): We suppose by contradiction that there exist $c\in [0,c^{*})$, a sequence $\{t_n\}_{n\ge 0}\subset (0,\infty )$ tending to $+\infty$, a sequence $\{x_n\}_{n\ge 0}\subset \mathbb {R}$ and a $\delta >0$ such that

(4.7)\begin{equation} |x_n|\le ct_n,\quad |V_s(x_n,t_n)-V_s^{*}|+|V_i(x_n,t_n)-V_i^{*}|+|H_i(x_n,t_n)-H_i^{*}|>\delta,\ n\ge 0. \end{equation}

Let us define a sequence of functions $(V_s,V_i,H_i)$ as follows:

\[ (V_s^{n},V_i^{n},H_i^{n})(x,t):=(V_s,V_i,H_i)(x+x_n,t+t_n). \]

We fix $c'>0$ such that $c< c'< c^{*}$. Note that $(V_s,V_i,H_i)$ is bounded, it follows from lemma 3.2 that there exist $T_1>0$ large enough and $\epsilon >0$ small enough such that for all $n\ge 0$, $(x,t)\in \mathbb {R}^{2}$, if $t+t_n\ge T_1$ and $|x|\le c't+(c'-c)t_n$, then

(4.8)\begin{equation} \epsilon\le V_s^{n}(x,t)\le \frac{1}{\epsilon},\quad \epsilon\le V_i^{n}(x,t)\le \frac{1}{\epsilon},\ \epsilon\le H_i^{n}(x,t)\le \frac{1}{\epsilon}. \end{equation}

Due to lemma 2.3 and the parabolic estimates, up to a subsequence, one has

\[ (V_s^{n},V_i^{n},H_i^{n})(x,t)\to (V_s^{\infty},V_i^{\infty},H_i^{\infty})(x,t) \text{ locally uniformly for }(x,t)\in\mathbb{R}^{2} \]

as $n\to \infty$, where $(V_s^{\infty },V_i^{\infty },H_i^{\infty })$ is a bounded entire solution of (1.2). Moreover, (4.8) yields

\[ \inf_{(x,t)\in\mathbb{R}^{2}}V_s^{\infty}>0,\quad\inf_{(x,t)\in\mathbb{R}^{2}}V_i^{\infty}>0, \quad \inf_{(x,t)\in\mathbb{R}^{2}}H_i^{\infty}>0. \]

Therefore, lemma 4.1 implies that $(V_s^{\infty },V_i^{\infty },H_i^{\infty })(x,t)\equiv (V_s^{*},V_i^{*},H_i^{*})$ for all $(x,t)\in \mathbb {R}^{2}$.

But (4.7) implies that

\[ |V_s^{\infty}(0,0)-V_s^{*}|+|V_i^{\infty}(0,0) -V_i^{*}|+|H_i^{\infty}(0,0)-H_i^{*}|>\delta, \]

which is a contradiction. This completes the proof of theorem 2.5 (ii).

To prove theorem 2.7, we show another convergence result for bounded entire solutions of (1.2) when $\mathcal {R}_0\le 1$.

Proof. Since the solution is bounded and nonnegative, there exists a sequence $\{(x_n,t_n)\}_{n\ge 0}\subset \mathbb {R}^{2}$ such that

\[ \lim_{n\to\infty}V_s(x_n,t_n)=\inf_{(x,t)\in\mathbb{R}^{2}}V_s(x,t)\ge 0. \]

Then we consider the sequence of functions

\[ (V_s^{n},V_i^{n},H_i^{n})(x,t):=(V_s,V_i,H_i)(x+x_n,t+t_n). \]

By the parabolic estimates and up to a subsequence, one has

\[ (V_s^{n},V_i^{n},H_i^{n})(x,t)\to(\widehat{V}_s,\widehat{V}_i,\widehat{H}_i)(x,t)\text{ locally uniformly for } (x,t)\in \mathbb{R}^{2} \]

as $n\to \infty$, where $(\widehat {V}_s,\widehat {V}_i,\widehat {H}_i)$ is a bounded entire solution of (1.2) and satisfies

(4.9)\begin{equation} \begin{cases} \partial_t \widehat{V}_s =\partial_{xx} \widehat{V}_s + (\widehat{V}_s +\widehat{V}_i )(\beta-\mu \widehat{V}_s) -\sigma_1 H_i\widehat{V}_s, & (x,t)\in\mathbb{R}^{2},\\ \partial_t \widehat{V}_i=\partial_{xx} \widehat{V}_i +\sigma_1 \widehat{H}_i\widehat{V}_s-\mu(\widehat{V}_s +\widehat{V}_i )\widehat{V}_i, & (x,t)\in\mathbb{R}^{2},\\ \partial_t \widehat{H}_i =d \partial_{xx} \widehat{H}_i +\sigma_2 H_s \widehat{V}_i -\rho \widehat{H}_i, & (x,t)\in\mathbb{R}^{2}. \end{cases} \end{equation}

The definition of $(x_n,t_n)$ implies that

\[ \widehat{V}_s(0,0)=\inf_{(x,t)\in\mathbb{R}^{2}}V_s(x,t). \]

Note that $0\le \widehat {V}_s\le \frac {\beta }{\mu }$ for all $(x,t)\in \mathbb {R}^{2}$. Then we apply the strong comparison principle to the $\widehat {V}_s$-equation and obtain that

(4.10)\begin{equation} \widehat{V}_s(x,t)\equiv\inf_{(x,t)\in\mathbb{R}^{2}}V_s(x,t):=V_{s}^{0}\ge 0, \end{equation}

which is a constant.

Next we consider two cases: (i) $V_{s}^{0}=0$, (ii) $V_{s}^{0}>0.$ For case (i), $V_{s}^{0}=0$ implies that $\widehat {V}_s(x,t)=\widehat {V}_i(x,t)=\widehat {H}_i(x,t)\equiv 0$ for all $(x,t)\in \mathbb {R}^{2}$. But lemma 2.1 yields that $(V_s+V_i)(x,t)$ with initial data $(V_{s,0}+V_{i,0})(x)\ge \frac {\beta }{\mu }$ converges to $\frac {\beta }{\mu }$ uniformly in $x\in \mathbb {R}$ as $t\to \infty$. It is a contradiction. Thus $V_{s}^{0}=0$ is impossible.

For case (ii), $V_{s}^{0}>0$ ensures that $\inf _{(x,t)\in \mathbb {R}^{2}}(V_s+V_i)(x,t)>0$. Applying [Reference Berestycki, Hamel and Nadin4, proposition 1.8] to $V_s+V_i$ satisfying a classical Fisher-KPP equation, we immediately obtain that $(V_s+V_i)(x,t)\equiv \frac {\beta }{\mu }$ for all $(x,t)\in \mathbb {R}^{2}$, which implies $\widehat {V}_s+\widehat {V}_i\equiv \frac {\beta }{\mu }$. Together with (4.9)–(4.10), one obtains that $(\widehat {V}_{s},\widehat {V}_{i},\widehat {H}_{i})(x,t)\equiv (V_{s}^{0},V_{i}^{0},H_{i}^{0})$ satisfies the system of stationary equations

\[ \begin{cases} (V_s^{0} +V_i^{0} )(\beta-\mu V_s^{0}) -\sigma_1 H_i^{0}V_s^{0}=0,\\ \sigma_1 H_i^{0}V_s^{0}-\mu(V_s^{0} +V_i^{0} )V_i^{0}=0,\\ \sigma_2 H_s V_i^{0} -\rho H_i^{0}=0. \end{cases} \]

Due to $\mathcal {R}_0\le 1$, $(V_{s}^{0},V_{i}^{0},H_{i}^{0})=(\frac {\beta }{\mu },0,0)$ follows. Combining (4.10) with the fact $0\le V_s\le \frac {\beta }{\mu }$ for all $(x,t)\in \mathbb {R}^{2}$, we obtain $(V_{s},V_{i},H_{i})\equiv (\frac {\beta }{\mu },0,0)$, which completes the proof of this lemma.

Proof of theorem 2.7: Note that the travelling wave solution is also a special entire solution of (1.2), lemma 4.2 directly shows that if $\mathcal {R}_0\le 1$, (1.3) and (1.4) does not have a positive solution for any $c\in \mathbb {R}.$

Now we prove that when $\mathcal {R}_0\le 1$, the solution of (1.2) goes to the disease-free equilibrium as time tending $\infty$. Suppose by contradiction that there exist a constant $\widetilde {\delta }>0$, a sequence $\{(x_n,t_n)\}_{n\ge 0}$ with $t_n\to \infty$, and $x_n\in \mathbb {R}$ such that

(4.11)\begin{equation} \left|V_s(x_n,t_n)-\frac{\beta}{\mu}\right|+|V_i(x_n,t_n)|+|H_i(x_n,t_n)|\ge\widetilde{\delta}, \quad n\ge 0. \end{equation}

We consider the sequence of functions $(V_s,V_i,H_i)$ as follows:

\[ (V_s^{n},V_i^{n},H_i^{n})(x,t):=(V_s,V_i,H_i)(x+x_n,t+t_n). \]

By lemma 2.3 and the parabolic estimates, up to a subsequence, one has

\[ (V_s^{n},V_i^{n},H_i^{n})(x,t)\to (u,v,h)(x,t) \text{ locally uniformly for }(x,t)\in\mathbb{R}^{2} \]

as $n\to \infty$, where $(u,v,h)$ is a bounded entire solution of (1.2). Moreover, it follows from (4.11) that

(4.12)\begin{equation} \left|u(0,0)-\frac{\beta}{\mu}\right|+|v(0,0)|+|h(0,0)|\ge\widetilde{\delta}>0. \end{equation}

However, due to $\mathcal {R}_0\le 1$, lemma 4.2 implies that $(u,v,h)(x,t)\equiv (\frac {\beta }{\mu },0,0)$ for all $(x,t)\in \mathbb {R}^{2}$, which contradicts (4.12). We complete the proof of theorem 2.7.

5. Minimal wave speed

In this section, we prove the travelling wave results given in theorem 2.5. Let $\chi (\xi )=\phi (\xi )+\varphi (\xi ),$ then

\[ c\chi ^{\prime }(\xi )=\chi ^{\prime \prime }(\xi )+\chi (\xi )\left[ \beta -\mu \chi (\xi )\right] ,\quad\xi \in \mathbb{R} \]

such that

\[ \liminf_{\xi \to -\infty}\chi (\xi) >0,\quad \liminf_{\xi \to \infty}\chi(\xi) >0 \]

by (1.4). From lemma 2.1, we have the following conclusion.

Lemma 5.1 A solution of (1.3) and (1.4) must satisfy

\[ \phi (\xi )+\varphi (\xi )=\beta /\mu,\quad \phi (\xi ),\varphi(\xi)\in (0, \beta/\mu), \ \xi\in\mathbb{R}. \]

By lemma 5.1, it suffices to study the monotone solutions of the following coupled system

(5.1)\begin{equation} \begin{cases} c\varphi ^{\prime }(\xi )=\varphi ^{\prime \prime }(\xi )+\sigma _{1}(\beta /\mu -\varphi (\xi ))\psi (\xi )-\beta \varphi (\xi ), \\ c\psi ^{\prime }(\xi )=d\psi ^{\prime \prime }(\xi )+\sigma _{2}H_{s}\varphi (\xi )-\rho \psi (\xi ),\\ \lim\limits_{\xi\to-\infty} (\varphi (\xi ),\psi (\xi ))=(0,0),\quad\liminf\limits_{\xi\to\infty}\varphi (\xi ) >0,\quad \liminf\limits_{\xi\to\infty}\psi (\xi ) >0 . \end{cases} \end{equation}

Note that (5.1) is the wave profile system of (2.5). It directly follows from lemma 2.4 that the existence, nonexistence and monotonicity of solutions for (5.1) are obtained if $\mathcal {R}_0>1$.

To better show the uniqueness of travelling wave solutions and provide a simple method to compute $c^{*},$ we further reduce the problem (5.1) to a scalar equation. Since a travelling wave solution is a special entire solution, it follows that $\psi (\xi ):=(J*\varphi )(\xi )$ satisfies (2.8). Therefore, $\varphi (\xi )$ satisfies the following differential-integral equation

(5.2)\begin{equation} c\varphi ^{\prime }(\xi )=\varphi ^{\prime \prime }(\xi )+\sigma _{1}(\beta /\mu -\varphi (\xi ))(J\ast \varphi )(\xi )-\beta \varphi (\xi ). \end{equation}

From the second definition of $c^{*}$ in § 2, we recall

\begin{align*} c^{*}& :=\inf\left\{ \vphantom{(0,(c+ \sqrt{c^{2} + 4d\rho})/(2d))} c>0: \Lambda (\gamma, c)=0 \text{ has exact one positive root if}\right.\\ & \qquad\qquad \left.\gamma\in(0,(c+ \sqrt{c^{2} + 4d\rho})/(2d))\right\}, \end{align*}

where $\Lambda (\gamma,c)$ is defined by (2.9).

Due to the equivalence between (5.1) and (5.2), it suffices to solve (5.2) with asymptotic boundary condition $\lim \limits _{\xi \to -\infty } \varphi (\xi )=0$, $\liminf \limits _{\xi \to \infty } \varphi (\xi )>0$. By the theory of travelling wave solutions in nonlocal delayed equations [Reference Fang and Zhao13, Reference Ruan and Xiao42, Reference Wang, Li and Ruan50, Reference Xu and Xiao60, Reference Zhao and Xiao65], we immediately have the following existence, nonexistence, monotonicity and uniqueness of travelling wave solutions (for very recent results, see [Reference Xu and Xiao60, theorems 2.1 and 2.2]).

Lemma 5.2 Suppose that $\mathcal {R}_0>1$. For any $c\ge c^{*},$ (5.2) has a positive solution $\varphi (\xi )$ with

(5.3)\begin{equation} \lim_{\xi\to -\infty}\varphi(\xi) =0,\quad \lim_{\xi\to \infty}\varphi(\xi) =V_i^{*}. \end{equation}

In particular, such a travelling wave solution is strictly increasing and unique in the sense of phase shift. For any $0< c< c^{*},$ (5.2) has no positive solution $\varphi (\xi )$ satisfying (5.3). Moreover, $c^{*}$ is the minimal wave speed of (5.2) and (5.3).

Combining lemmas 5.1 and 5.2 with (5.1), we obtain the existence, nonexistence, uniqueness and monotonicity of $(\phi,\psi )$. This completes the proof of travelling wave problem in theorem 2.5.

6. Numerical simulations

In this section, we illustrate the above theoretical results by performing numerical simulations in two examples. We select $\beta =\mu =H_s=1$ and consider the following system

(6.1)\begin{equation} \begin{cases} \partial_t V_s =\partial_{xx} V_s +(V_s +V_i )(1-V_s) -\sigma_1 H_iV_s & x\in\mathbb{R}, \ t>0,\\ \partial_t V_i=\partial_{xx} V_i +\sigma_1 H_iV_s-(V_s +V_i)V_i, & x\in\mathbb{R}, \ t>0,\\ \partial_t H_i =d \partial_{xx} H_i +\sigma_2 V_i -\rho H_i, & x\in\mathbb{R}, \ t>0,\\ V_s(x,0)=1,\qquad V_i(x,0)=H_i(x,0)=\mathbf{1}_{x\in[{-}5,0]}, & x\in\mathbb{R}. \end{cases} \end{equation}

By results in § 2, we obtain that

\[ \mathcal{R}_0=\frac{ \sigma_1\sigma_2}{\rho},\quad (V_s^{*},V_i^{*},H_i^{*})=\left(\frac{\rho}{\sigma_1\sigma_2},\ 1-\frac{\rho}{\sigma_1\sigma_2}, \frac{\sigma_2}{\rho}-\frac{1}{\sigma_1}\right),\quad E_1=(1,0,0). \]

Example 6.1 Consider the following special case of (6.1)

(6.2)\begin{equation} \begin{cases} \partial_t V_s =\partial_{xx} V_s +(V_s +V_i )(1-V_s) -1.5 H_iV_s & x\in\mathbb{R}, \ t>0,\\ \partial_t V_i=\partial_{xx} V_i +1.5 H_iV_s-(V_s +V_i)V_i, & x\in\mathbb{R}, \ t>0,\\ \partial_t H_i =1.2\partial_{xx} H_i +1.2 V_i -0.5 H_i, & x\in\mathbb{R}, \ t>0. \end{cases} \end{equation}

We have

\[ \mathcal{R}_0=3.6>1,\quad c^{*}\approx 1.6597,\ (V_s^{*},V_i^{*},H_i^{*})\approx (0.2778,0.7222,1.7333), \]

where $c^{*}$ is defined by (see Fig. 1)

\[ c^{*}:=\min_{\gamma>0}\frac{2.2\gamma^{2}-1.5+\sqrt{\left[0.2\gamma^{2} +0.5\right]^{2}+ 7.2 }}{2\gamma}:=\min_{\gamma>0}f(\gamma) \]

and

\begin{align*} c^{*}& :=\inf\left\{c>0:\Lambda(\gamma,c)=\gamma^{2} -c \gamma -1 -\frac{1.8}{(1.2\gamma^{2}-c\gamma-0.5)}=0\right.\\ & \quad\left.\text{ has exact one positive root if }\gamma\in(0,(c+ \sqrt{c^{2} + 2.4})/2.4)\vphantom{\frac{1.8}{(1.2\gamma^{2}-c\gamma-0.5)}}\right\}. \end{align*}

Fig. 1. The plot of the functions $f(\gamma )$ and $\Lambda (\gamma,1.6597 )$.

Figure 2 shows the spatial-temporal evolution of $V_s$, $V_i$ and $H_i$ defined by (6.2). We show the distributions of three components at $t=200$ in the first figure of figure 3. From figure 2, we find that $V_i$, $H_i$ almost invade at a constant speed. In order to estimate the invasion speed, we introduce the level set to describe the expansion speed of fronts. Denote

\begin{align*} & X_1(t)=\sup\left\{x~|~V_i(x,t)>0.005\right\},\quad X_2(t)=\sup\left\{x~|~H_i(x,t)>0.005\right\},\\ & c_{num}^{V_i}:=X_1(200)/200,\quad c_{num}^{H_i}:=X_2(200)/200. \end{align*}

Fig. 2. The spatio-temporal plots of the solution $(V_s,V_i,H_i)$ of system (6.2).

Fig. 3. The graphs of the solution $(V_s,V_i,H_i)(x,200)$, $X_1(t)/t$ and $X_2(t)/t$ for (6.2).

We estimate the invasion speeds of $V_i$ and $H_i$ by $X_1(t)/t$ and $X_2(t)/t$ in figure 3, which indicates that if $t$ is large, then $X_1(t)/t$, $X_2(t)/t$ are close to $c^{*}$ (see table I). From figures 2–3, we also see the solution $(V_s,V_i,H_i)$ on any compact interval converges to $(V_s^{*},V_i^{*},H_i^{*})$, which illustrates theorem 2.5.

Example 6.2 Consider the following special case of (6.1)

(6.3)\begin{equation} \begin{cases} \partial_t V_s =\partial_{xx} V_s +(V_s +V_i )(1-V_s) -1.5 H_iV_s & x\in\mathbb{R}, \ t>0,\\ \partial_t V_i=\partial_{xx} V_i +1.5H_iV_s-(V_s +V_i)V_i, & x\in\mathbb{R}, \ t>0,\\ \partial_t H_i =1.8\partial_{xx} H_i +1.5 V_i -\rho H_i, & x\in\mathbb{R}, \ t>0. \end{cases} \end{equation}

Table 1. Numerical speed $c_{num}$ vs. theoretical speed $c^{*}$

We have two cases.

1. (i) When $\rho =2.5$, $\mathcal {R}_0=0.9<1.$ Simulations of this case are presented in figures 4 and 5.

2. (ii) When $\rho =2.25$, $\mathcal {R}_0=1.$ Simulations of this case are presented in figures 6 and 7.

Fig. 4. The spatio-temporal plots of the solution $(V_s,V_i,H_i)$ of (6.3) with $\rho =2.5$.

Fig. 5. The spatial plots of the solution $(V_s,V_i,H_i)$ of (6.3) with $\rho =2.5$ when $t=100,200.$

Fig. 6. The spatio-temporal plots of the solution $(V_s,V_i,H_i)$ of (6.3) with $\rho =2.25$.

Fig. 7. The spatial plots of the solution $(V_s,V_i,H_i)$ of (6.3) with $\rho =2.25$ when $t=100,200.$

Figures 4–7 simulate the spatial-temporal evolutions of $V_s$, $V_i$ and $H_i$ defined by (6.3). They show the spatial distributions of the three components, which demonstrate that when the basic reproduction number $\mathcal {R}_0\le 1$, $V_i$ and $H_i$ cannot invade successfully and will vanish as $t$ tends to infinity. This illustrates theorem 2.7.