Long-time dynamics of an epidemic model with nonlocal diffusion and free boundaries

1.   Introduction

To model the 1973 cholera epidemic in the European Mediterranean region, Capasso and Paveri-Fontana ^[3] proposed the following ODE system

where

● $u(t)$ and $v(t)$ represent respectively the average population concentration of the infectious agents and the infective humans in the infected area at time $t$,

● $a$, $b$, $c$ are all positive constants such that $1/a$ represented the mean lifetime of the agents in the environment, $1/b$ the mean infectious period of the infective humans, $c$ the multiplicative factor of the infectious agents due to the infective humans, and

● the function $G(u)$ is the infection rate of the human population, assuming that the total number of susceptible humans remain constant during the epidemic. The function $G$ is assumed to satisfy the following:

$\textbf{(G1)} \; G\in C^1([0, \infty]), G(0) = 0, G'(z) > 0$ for all $z\ge 0$;

$\textbf{(G2)} \; \left(\frac{G(z)}{z}\right)' < 0$ for $z > 0$ and $\lim_{z\rightarrow +\infty} \frac{G(z)}{z} < \frac{ab}{c}$.

A simple example of such a function is given by $G(z) = \frac{\alpha z}{1+z}$ with $\alpha \in (0, ab/c)$.

They were able to establish the following result for the long-time dynamics of (1.1): Let $R_0: = \frac{cG'(0)}{ab}$; then regardless of the positive initial values $u(0)$ and $v(0)$,

(i) the epidemic tends to extinction if $R_0 < 1$, namely $\lim_{t\to\infty}(u(t), v(t))\rightarrow (0, 0)$ if $R_0 < 1$,

(ii) the epidemic tends to a positive equilibrium state if $R_0 > 1$, namely $\lim_{t\to\infty}(u(t), v(t))\rightarrow (u^*, v^*)$ if $R_0 > 1$, where $u^*, v^*$ are uniquely determined by

To include the mobility of the infectious agents (assuming the mobility of the infective human population is small and thus ignored), Capasso and Maddalena ^[4] proposed the following spatial reaction-diffusion model with Robin (or Neumann) boundary conditions

where $d$ denotes the diffusion rate of $u$, the epidemic region $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain and $\alpha \ge 0$. They proved that the long-time behaviour of (1.3) is similar to the ODE model (1.1) with $R_0$ there replaced by $\tilde R_0: = \frac{cG'(0)}{(a+d\lambda_1)b}$, where $\lambda_1$ is the first eigenvalue of the eigenvalue problem

In the literature, $R_0$ and $\tilde R_0$ are often called the reproduction number of the epidemic being modelled.

To describe the spatial spreading of an epidemic, it is important to know how the front of the epidemic propagates. In ^[5], Ahn, Baek, and Lin regarded the epidemic region as a changing interval and used the following free boundary problem to model the evolution and spreading of the epidemic:

where $h(t)$ and $g(t)$ are the moving boundaries of the infected region, $\mu$ is a positive constant and the initial data $(u_0, v_0)$ satisfy

The equations for $h'(t)$ and $g'(t)$ mean that the expanding rate of the infected region is proportional to the spatial gradient of $u$ at the front. This is known as the Stefan condition which was first used to describe the melting of ice (see, e.g., ^[6]). It has been extensively used in the study of the spread of population since Du and Lin ^[7].

The long-time dynamics of (1.4) can be described by a spreading-vanishing dichotomy; more precisely, Ahn et al. showed that the unique solution $(u, v, g, h)$ to (1.4) satisfies either

$\text{(i)}$ (vanishing)

$\text{(ii)}$ (spreading)

Furthermore, using the reproduction number of (1.1), namely

the dichotomy is determined as follows: If $R_0\le 1$, then vanishing always happens; in the case where $R_0 > 1$, there exists a critical length, $l^* : = \frac{\pi}{2}\sqrt{\frac{d_1}{a (R_0-1)}}$, such that

● if $h_0 \ge l^*$ (i.e., the initial size of the infected region is no less than $2l^*$), then spreading always happens, and

● if $h_0 < l^*$, then vanishing (resp. spreading) happens if the initial functions $(u_0, v_0)$ are sufficiently small (resp. large).

In the case of an epidemic spreading predicted by (1.4), it was shown by Zhao, Li, and Ni ^[8] that there exists a uniquely determined $c_0 > 0$ such that

which means the epidemic region $[g(t), h(t)]$ expands with asymptotic speed $c_0$.

In (1.4) (as well as in (1.3)), the spatial dispersal of the infectious agents is assumed to follow the rules of random walk, which ignores any nonlocal effect in the dispersal process. Such nonlocal effect can be included if the local diffusion operator is replaced by a nonlocal diffusion operator of the form

with an appropriate kernel function $J$. Here $J(x)$ can be interpreted as the probability that an individual of the species moves from location 0 to $x$. A widely used class of kernel functions consists of $J: \mathbb{R} \rightarrow \mathbb{R}$ satisfying

One recent paper by Cao, Du, Li, and Li ^[9] extended many basic results of ^[7] to the corresponding nonlocal model with the above kernel. Following the fashion of ^[9], Zhao, Zhang, Li, and Du ^[1] considered the corresponding nonlocal version of (1.4), which has the following form

It was shown in ^[1] that (1.7) has a unique solution defined for all $t > 0$, and its long-time dynamics is determined by a spreading-vanishing dichotomy, in a similar fashion to (1.4) (with some subtle differences though). A striking difference of (1.7) to (1.4) is revealed by ^[2], which shows that the spreading determined by (1.7) may have infinite asymptotic spreading speed, a phenomenon known as "accelerated spreading". More precisely, if the kernel function $J$ satisfies

and spreading happens, then

Moreover, if $J(x)\simeq |x|^{-\gamma}$ for some $\gamma\in (1, 2]$ and all large $|x| > 0$, then for all large $t > 0$,

Here, and in what follows, $\eta(t) \simeq \xi(t)$ means $C_1\xi(t) \le \eta (t) \le C_2 \xi(t)$ for some positive constants $C_1 \le C_2$ and all $t$ in the specified range.

In this paper, to understand the effect of the mobility of the infective host on the epidemic spreading, we examine a full version of (1.7) ^*, where the dispersal of infective host is included. Before giving this full version, let us note that, in (1.7), since $u(x, t) = 0$ for $x\not\in (g(t), h(t))$ and $\int_{\mathbb R}J(x)dx = 1$,

^*A full version of the local diffusion model (1.4) was recently investigated in ^[10], which showed that its long-time dynamics is similar to that of (1.4) though some differences occur in the criteria governing the spreading-vanishing dichotomy. In particular, when spreading happens, there exists a finite asymptotic spreading speed.

For $i = 1, 2$, suppose $J_i: \mathbb{R}\rightarrow \mathbb{R}$ satisfy (J). Let $a, b, c, d_1, d_2, \mu_1, \mu_2$ and $h_0$ be constants, all positive except $\mu_1$ and $\mu_2$, which are assumed to be nonnegative with $\mu_1+\mu_2 > 0$, and let the initial functions $u_{0}(x)$ and $v_0(x)$ satisfy (1.5). Then the full version of (1.7) can be written in the following form

We will prove the following results.

Theorem 1.1 (Existence and Uniqueness). The problem $(1.8)$ admits a unique positive solution $(u, v, g, h)$ defined for $t\ge 0$.

Theorem 1.2 (Spreading-Vanishing Dichotomy). Let $(u, v, g, h)$ be the solution to $(1.8)$ and denote $h_\infty : = \lim_{t\rightarrow \infty} h(t)$ and $g_\infty : = \lim_{t\rightarrow \infty} g(t)$. Then either

$\text{(i)}$ (vanishing) $(g_\infty, h_ \infty)$ is a finite interval and

$\text{(ii)}$ (spreading) $(g_\infty, h_\infty) = \mathbb{R}$ and

Theorem 1.3 (Spreading-Vanishing Criteria). Let $(u, v, g, h)$ be the solution of $(1.8)$ and $R_0$ be given by $(1.6)$.

$\text{(a)}$ If $R_0\le 1$, then vanishing always occurs.

$\text{(b)}$ If $R_0 > 1$, then spreading always occurs if one of the following holds:

where $L^* > 0$ is a certain critical length depending on $a, b, c, d_1, d_2, J_1, J_2$ but independent of the initial data $(u_0, v_0)$.

$\text{(c)}$ If $R_0 > 1$ and

then

$\text{(i)}$ for any given initial datum $(u_0, v_0)$ satisfying $(1.5)$, and any given constants $\sigma_1^0, \sigma_2^0$ nonnegative satisfying $\sigma_1^0+\sigma_2^0 > 0$, there exists $\mu^* > 0$ such that

$(\alpha)$ if $(\mu_1, \mu_2) = (\mu\sigma_1^0, \mu\sigma_2^0)$ and $0 < \mu\le \mu^*$, then vanishing occurs, and

$(\beta)$ if $(\mu_1, \mu_2) = (\mu\sigma_1^0, \mu\sigma_2^0)$ and $\mu > \mu^*$, then spreading occurs.

$\rm (ii)$ for fixed $(\mu_1, \mu_2)$ and sufficiently small initial datum $(u_0, v_0)$, vanishing occurs.

Remark 1.4. $\rm (i)$ The constant $L^*$ in Theorem 1.3 is uniquely determined by an eigenvalue problem; see Proposition 3.4(iii) below.

$\rm (ii)$ In the case of (1.7) considered in ^[1], which is equivalent to (1.8) with $d_2 = \mu_2 = 0$, the long-time dynamics is also governed by a spreading-vanishing dichotomy, and the spreading-vanishing criteria coincide with those in Theorem 1.3 but with $d_2$ and $\mu_2$ replaced by 0.

$\rm (iii)$ If $\mu_2 = 0$ and all the other parameters in (1.8) are positive and fixed except $d_2$, which is allowed to vary in $[0, \infty)$, then from Lemmas 3.2 and 3.3 below it is easily seen that $L^* = L^*(d_2)$ is strictly increasing in $d_2$. Therefore part (b) of Theorem 1.3 and the result in ^[1] indicate that the range of parameters $(a, b, c, d_1, h_0)$ for which spreading happens regardless of the size of the initial function pare $(u_0, v_0)$, i.e., (I) or (II) above holds, is enlarged as $d_2$ is decreased, and such a range is maximized when $d_2 = 0$. Biologically, this means that reducing the mobility of the infective host increases the chance of successful spreading of the disease, which appears counter-intuitive at first look. However, such a phenomenon is not new; it arises in the local diffusion models considered in ^[4,10] already.

When spreading happens, the spreading profile of (1.8) can be determined by using the general results in ^[2]. For this purpose, we will need the following condition

$\textbf{(J1):} \; \int_0^\infty x J_i (x) dx < \infty$ for $i = 1, 2$ with $\mu_i > 0$.

Theorem 1.5 (Spreading Speed). In Theorem 1.2, if spreading happens, then

where $c_0 > 0$ is uniquely determined by the associated semi-wave problem to $(1.8)$ (see [2,Section 1.2]) .

Furthermore, in the case of accelerated spreading, we can determine the rate of accelerated spreading for a rather general class of kernel functions.

Theorem 1.6 (Rate of Accelerated Spreading). In Theorem 1.5 suppose additionally that for $i = 1, 2$ with $\mu_i > 0$, the kernel function $J_i$ satisfies $J_i(x) \simeq |x|^{-\gamma}$ for some $\gamma \in (1, 2]$ and $|x|\gg1$. Then for $t \gg 1$, we have

Let us note that when $J_i$ satisfies $J_i(x) \simeq |x|^{-\gamma}$ for $|x|\gg 1$ and for $i\in\{1, 2\}$ such that $\mu_i > 0$, (J1) holds if and only if $\gamma > 2$. Thus Theorem 1.6 covers the exact range of $\gamma$ such that accelerated spreading is possible. Note also that in condition (J1) as well as in Theorem 1.6, the condition only applies to the kernel function $J_i$ where $\mu_i > 0$. For example, if $\mu_2 = 0$, then no extra condition on $J_2$ is needed apart from satisfying (J).

Problem (1.8) has an entire space version where no free boundary is involved, which has the form

Problem (1.12) has been successfully used to determine the spreading speed of the epidemic; see ^[11] and the references therein for many interesting results on this and related problems. For the entire space version of (1.7), see ^[2,12] and the references therein for more details. The local diffusion counterparts of these entire space problems have been studied more extensively; see, for example, ^[13,14,15]. As mentioned above, the corresponding free boundary models have the advantage of providing the exact location of the spreading front of the concerned epidemics.

The rest of the paper is organized as follows. In Section 2, we introduce the preparatory results relating to (1.8) and use them to prove Theorem 1.1. In Section 3, we gather the necessary results associated with the corresponding fixed boundary problems, which will be used to determine the long-time dynamical behavior of (1.8). In Section 4, we use the results of the previous sections to establish the vanishing-spreading dichotomy as related to the reproduction number $R_0$, proving Theorems 1.2 and 1.3. Finally, Section 5 is devoted to proving the assumptions required in ^[2] for Theorems 1.5 and 1.6.

We would like to point out that, although (1.8) has some significantly different features from the West Nile virus model studied in ^[16], for example, the nature of the reaction terms in (1.8) makes any nonnegative initial function admissible while the model in ^[16] only allows initial functions taken from a certain bounded order interval, but many techniques of ^[16] can be adapted to treat (1.8), which has helped to considerably reducing the length of the current paper. Here we only provide the details of the proofs when they are very different from ^[16].

2.   Existence and uniqueness, and comparison principle

In this section, we prove the well-posedness of (1.8) and some associated comparison principles.

We first recall a maximum principle from [16,Lemma 3.1], which is more general than needed in this paper, but in view of possible applications elsewhere, we state it in the general form as in ^[16].

Let $T > 0$ and $\xi \in C([0, T])$. We define the set of strict local semi-maximum points of $\xi$ by

Similarly, the set of strict local semi-minimum points of $\xi$ is given by

If $\xi$ is strictly increasing, then $\Sigma_{\max}^\xi = (0, T]$. If $\xi$ is nondecreasing, then $\Sigma_{\min}^\xi = \emptyset$.

Lemma 2.1 (Maximum Principle). Let $T, h_0 > 0, g, h\in C([0, T])$ satisfy $g(t) < h(t)$ and $-g(0) = h(0) = h_0$. Denote $D_T: = \{(x, t) : t \in (0, T], g(t) < x < h(t)\}$ and suppose that for $i, j\in \{1, 2, ..., n\}$, $\phi_i, \partial_t \phi_i \in C(\overline{D_T}), d_i, c_{ij}\in L^\infty(D_T)$, $d_i\ge 0$, and

where $J_i$ satisfies (J). Then the following holds:

$\text{(i)}$ If $c_{ij}\ge 0$ on $D_T$ for $i, j \in \{1, 2, ...n\}$ and $i \neq j$, then $\phi_i \ge 0$ on $\overline{D_T}$ for $i \in \{1, 2, ..., n\}$.

$\text{(ii)}$ If for some $i_0\in\{1, ..., n\}$ we assume additionally that $d_{i_0} > 0$ in $D_T$ and $\phi_{i_0}(x, 0) \not\equiv 0$ in $[-h_0, h_0]$, then $\phi_{i_0}(x, t) > 0$ in $D_T$.

Lemma 2.2 (Comparison Principle I). For $T\in (0, +\infty)$, suppose that $\overline{g}, \overline{h} \in C([0, T])$, $D = \{(x, t) : t \in (0, T], \overline g(t) < x < \overline h(t)\}$, \ $\overline{u}, \overline{v} \in C(\overline{D})$, $\overline{u}, \overline{v}\ge 0$. If $(\overline{u}, \overline{v}, \overline{g}, \overline{h})$ satisfies

then the unique solution $(u, v, g, h)$ of $(1.8)$ satisfies

for $0 < t\le T$ and $g(t)\le x\le h(t)$.

Proof. By (G1), we have that $G(\overline{u}) = G'(\xi)\overline{u}$ with $\xi = \xi(x, t)\in (0, \overline{u}(x, t)]$. From (1.5) and Lemma 2.1, we infer that $\overline{u}, \overline{v} > 0$, for $0 < t\le T$ and $\overline{g}(t) < x < \overline{h}(t)$. Therefore, $-\overline{g}$ and $\overline{h}$ are strictly increasing.

For small $\epsilon > 0$, let $(u_\epsilon, v_\epsilon, g_\epsilon, h_\epsilon)$ denote the unique solution of (1.8) with $h_0$ replaced by $h_0^\epsilon: = h_0(1-\epsilon)$, $\mu_i$ by $\mu_i^\epsilon: = \mu_i(1-\epsilon)$ for $i = 1, 2$, and $(u_0, v_0)$ by $(u_0^\epsilon, v_0^\epsilon)$ satisfying

We claim that $h_\epsilon(t) < \overline{h}(t)$ and $g_\epsilon(t) > \overline{g}(t)$ for all $t\in(0, T]$. It is clear that these hold for small $t > 0$. Suppose that there exists $t_1\le T$ such that

Without loss of generality, assume that $h_\epsilon(t_1) = \overline{h}(t_1)$ and $g_\epsilon(t_1)\ge \overline{g}(t_1)$. Let $w: = \overline{u}-u_\epsilon$ and $z: = \overline{v}-v_\epsilon$; then $(w, z)$ satisfies

where $\eta = \eta(x, t)$ is between $\overline{u}(x, t)$ and $u_\epsilon (x, t)$. Therefore we can apply Lemma 2.1 to conclude that $w(x, t) > 0$ and $z(x, t) > 0$ for $0 < t\le t_1$ and $g_\epsilon(t) < x < h_\epsilon(t)$.

However, by definition of $t_1$, we have $h_\epsilon '(t_1) \ge \overline{h}'(t_1)$, giving us that

This contradiction proves our claim, namely, $h_\epsilon (t) < \overline{h}(t)$ and $g_\epsilon(t) > \overline{g}(t)$ for all $t \in (0, T]$. Hence (2.4) holds with $t_1$ replaced by $T$, which yields that $\overline{u}(x, t) > u_\epsilon(x, t)$ and $\overline{v}(x, t) > v_\epsilon(x, t)$ for $0 < t\le T$ and $g_\epsilon(t) < x < h_\epsilon(t)$. Letting $\epsilon \rightarrow 0$, we obtain the desired result from the continuous dependence of $(u_\epsilon, v_\epsilon, g_\epsilon, h_\epsilon)$ on $\epsilon$.

We introduce a second comparison principle where the boundaries are regarded as given.

Lemma 2.3 (Comparison Principle II). Assume (J) holds, $T > 0$, $g, h \in C([0, T])$ satisfying $g(t) < h(t)$, and $D_T$ defined as in Lemma 2.1. If for $i = 1, 2$, $u_i, \tilde{u}_i \in C(\overline{D_T})$ satisfy the following conditions:

$\text{(i)} \; \phi_t \in C(\overline{D_T})$ for $\phi \in \{u_1, u_2, \tilde{u}_1, \tilde{u}_2\}$,

$\text{(ii)}$ for $(x, t) \in D_T$,

$\text{(iii)}$ for $(x, t) \in D_T$, $(u_1, u_2)$ satisfies (2.5) but with the inequalities reversed,

$\text{(iv)}$ at the boundary,

$\text{(v)}$ and at the initial time, $u_i (x, 0) \le \tilde{u}_i (x, 0)$ for $x \in [g(0), h(0)]$ and $i = 1, 2$.

Then for $i = 1, 2$, we must have

Proof. For $i = 1, 2$, define $\phi_i : = \tilde {u_i} - u_i$ and

Then by the maximum principle in Lemma 2.1, we obtain that $\phi_i \ge 0$ in $D_T$ for $i = 1, 2$.

Lemma 2.4 (A Priori Bound). For $T\in(0, +\infty)$, let $(u, v, g, h)$ be a solution of $(1.8)$ for $t\in (0, T]$. Then there exists constants $C_1$ and $C_2$ independent of $T$ such that

Proof. By assumption (G2), there exist $C_1$ and $C_2$ such that

and

Let $\left(U_1(x, t), U_2(x, t)\right) \equiv (C_1, C_2)$. For $i = 1, 2$, let us denote

Clearly $\mathcal{L}_i[U_i](x, t)\leq 0$ and $(U_i)_t = 0$. It now follows from (2.6) and (2.7) that

By Lemma 2.3, we obtain $u (x, t)\le C_1$ and $v (x, t)\le C_2$ for all $0 < t\le T$ and $g(t)\le x\le h(t)$.

Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1. Let us first consider the following slightly modified problem, with $C_2$ taken from Lemma 2.4:

By taking $f_1(u, v) = -a u + c \min \{v, C_2\}$ and $f_2(u, v) = -bv+G(u)$, in view of the conditions on $G$, we see that they satisfy the conditions of Theorem 4.1 in ^[16]. Therefore (2.8) has a unique solution $(u, v, g, h)$ defined for all $t > 0$.

Since $f_1(u, v)\leq -au+cv$, from the proof of Lemma 2.4 we see that $u(x, t)\leq C_1$ and $v(x, t)\leq C_2$, and therefore $f_1(u(x, t), v(x, t)) = -au(x, t)+c v(x, t)$. Thus $(u, v, g, h)$ solves (1.8).

Conversely, by Lemma 2.4 any solution of (1.8) satisfies $v(x, t)\leq C_2$ and hence it solves (2.8). Thus global existence and uniqueness holds for (1.8).

3.   Some associated fixed boundary problems

3.1.   Eigenvalue problem

For any $L > 0$, we consider the eigenvalue problem

By Theorems 2.2 and 2.3 of ^[17], we have the following result:

Proposition 3.1. The eigenvalue problem $(3.1)$ has a principal eigenvalue $\lambda = \lambda_1 (L)$ with positive eigenfunction pair $(\phi, \psi) = (\phi_1, \psi_1)\in C([-L, L]) \times C([-L, L])$.

Then by Lemma 2.2 and Proposition 2.3 of ^[16], we have the following two results on the properties of the eigenvalue $\lambda_1(L)$.

Lemma 3.2. Let $\lambda_1(L)$ be the principal eigenvalue of $(3.1)$. Let $\Phi, \Psi \in C([-L, L])$ be two functions such that $\Phi, \Psi \ge 0$ and $\Phi, \Psi \not\equiv 0$ in $[-L, L]$, and $\tilde{\lambda}$ be a constant such that

then $\lambda_1 (L) \ge (\le) \tilde{\lambda}$. Moreover, $\lambda_1 (L) = \tilde{\lambda}$ only if equalities hold in $(3.2)$.

Lemma 3.3. Let $\lambda_1(L)$ be the principal eigenvalue of $(3.1)$. Then

$\text{(i)} \; \lambda_1(L)$ is strictly decreasing with respect to $L\in (0, \infty)$,

$\text{(ii)} \; \lambda_1(L)$ is continuous for $L\in (0, \infty)$.

The following proposition is essential for establishing the spreading and vanishing criteria in Theorem 1.3.

Proposition 3.4. The principal eigenvalue $\lambda_1(L)$ of $(3.1)$ has the following properties:

$\text{(i)}$ If $R_0 \le 1$, then $\lambda_1(L) > 0$ for any $L > 0$.

$\text{(ii)}$ If $R_0 > 1$ and

then $\lambda_1(L) < 0$ for any $L > 0$.

$\text{(iii)}$ If $R_0 > 1$ and

then there exists $L^*$ such that

Proof. The proof follows that of [16,Proposition 2.4], and the details are omitted.

Corollary 3.5. Let $l_1 < l_2$ and $\lambda_1(l_1, l_2)$ be the principal eigenvalue of $(3.1)$ with $[-L, L]$ replaced by $[l_1, l_2]$. Then

$\text{(i)} \; \lambda_1 (l_1, l_2)$ is strictly decreasing with respect to $l_2-l_1$ and is continuous in $l_1$ and $l_2$.

$\text{(ii)}$ If $R_0 \le 1$, then $\lambda_1(l_1, l_2) > 0$ for any $l_1$ and $l_2$.

$\text{(iii)}$ If $R_0 > 1$ and $(3.3)$ holds, then $\lambda_1(l_1, l_2) < 0$ for any $l_1$ and $l_2$.

$\text{(iv)}$ If $R_0 > 1$ and $(3.4)$ holds, then $\lambda_1(l_1, l_2) = 0$ for $l_2-l_1 = 2L^*$ and

where $L^* > 0$ is given by Proposition 3.4(iii).

3.2.   Fixed boundary problem

For $L > 0$, we define $Q_L = (-L, L)\times (0, \infty)$ and consider the corresponding fixed boundary problem of (1.8):

where $u_{0}, v_0 \in C([-L, L])$ are nonnegative and not identically 0 simultaneously. It is well-known that fixed boundary problems such as (3.6) has a unique positive solution which is defined for all $t > 0$ (see, for example, Remark 4.3 in ^[16]).

The corresponding steady state problem of (3.6) is

Definition 3.6. A function pair $({\phi}, {\psi}) \in C([-L, L]) \times C([-L, L])$ is said to be an upper solution of (3.7) if

It is called a lower solution of (3.7) if these inequalities are reversed.

Proposition 3.7. Suppose that $R_0 > 1$ and $(u, v)$ is the unique positive solution of $(3.6)$. Let $\lambda_1(L)$ be the principal eigenvalue of $(3.1)$ and $(u^*, v^*)$ be as defined in $(1.2)$. Then the following conclusions hold:

$\text{(i)}$ The fixed boundary problem $(3.6)$ has a unique positive steady state solution $(\tilde{u}, \tilde{v})\in C([-L, L]) \times C([-L, L])$ if $\lambda_1(L) < 0$, and $(0, 0)$ is the only nonnegative steady state when $\lambda_1(L) \ge 0$. Moreover, $0 < \tilde{u}(x) \le u^*$ and $0 < \tilde{v}(x) \le v^*$ in $[-L, L]$ when $\lambda_1(L) < 0$.

$\text{(ii)}$ If $\lambda_1 (L) \ge 0$, then $(u(x, t), v(x, t))$ converges to $(0, 0)$ as $t\rightarrow \infty$ uniformly for $x\in[-L, L]$.

$\text{(iii)}$ If $\lambda_1 (L) < 0$, then $(u(x, t), v(x, t))$ converges to $(\tilde{u}, \tilde{v})$ as $t\rightarrow \infty$ uniformly for $x\in[-L, L]$.

Proof. Due to the different nature of the reaction terms in (1.8) from the model in ^[16], our proof here uses rather different techniques. In particular, we will make use of the monotonicity (in time $t$) of the to-be-constructed lower and upper solutions and Dini's theorem.

(i) Suppose that $\lambda_1(L) < 0$. Then we easily see that $(Mu^*, Mv^*)$ and $(\epsilon \phi_1, \epsilon \psi_1)$ are respectively upper and lower solutions of (3.7) for small enough $\epsilon > 0$ and any $M\geq 1$, where $(\phi_1, \psi_1)$ is a positive eigenfunction pair corresponding to $\lambda_1(L)$.

Let $(\underline{u}, \underline{v})$ be the unique positive solution of (3.6) with initial function pair $(\epsilon \phi_1, \epsilon \psi_1)$. Using $(\epsilon \phi_1, \epsilon \psi_1)$ as a lower solution of (3.6), we can use the comparison principle in Lemma 2.3 with $(g(t), h(t))\equiv (-L, L)$ to conclude that $(\epsilon \phi_1(x), \epsilon \psi_1(x))\le (\underline{u}(x, t), \underline{v}(x, t))\leq (u^*, v^*)$ for $(x, t)\in Q_L$. In particular, for any fixed $s > 0$,

It is easily seen that $(\hat u(x, t), \hat v(x, t): = (\underline{u}(x, s+t), \underline{v}(x, s+t))$ is a solution of (3.6) with initial data $(\underline{u}(x, s), \underline{v}(x, s))$. Therefore we can use the comparison principle to deduce

Since $s > 0$ is arbitrary, this implies that $(\underline{u}(x, t), \underline{v}(x, t))$ is nondecreasing in $t$ and hence

and $(\epsilon \phi_1(x), \epsilon \psi_1(x))\leq (U(x), V(x))\leq (u^*, v^*)$ in $[-L, L]$. Moreover, it is easily seen that $(U, V)$ solves (3.7). Thus there exists at least one positive steady state solution.

We show next that $(U, V)$ as well as any other positive solution of (3.7) are continuous in $[-L, L]$. Indeed, from the continuity of $J_1$ and $J_2$, we easily see that

are continuous in $[-L, L]$. From (3.7), we obtain

From the conditions on $G$, we see that $F(z): = \frac{(a+d_1)(b+d_2)}c z-G(z)$ satisfies

Thus from $F(U(x)) = G_2(x)+\frac{b+d_2}c G_1(x)$, $F'(z) > 0$ and the fact that $G_1(x)$ and $G_2(x)$ are continuous, we obtain $U(x)$ is continuous, which in turn implies that $V(x) = \frac{a+d_1}{c}U(x)-\frac{G_1(x)}c$ is continuous.

To prove uniqueness, let $(\hat U, \hat V)$ be another positive solution of (3.7). By choosing $\epsilon > 0$ sufficiently small, we may assume that $(\epsilon \phi_1(x), \epsilon \psi_1(x))\leq (\hat U(x), \hat V(x))$ in $[-L, L]$. Thus the above obtained $(U, V)$ satisfies $(U, V)\leq (\hat U, \hat V)$.

We define

From $(U, V)\leq (\hat U, \hat V)$ and the definition of $k^*$, we have immediately $k^*(U, V)\geq (\hat U, \hat V)$ and $k^* \ge 1$. If $k^* = 1$, then we immediately obtain $(U, V) = (\hat U, \hat V)$ and the uniqueness is proved. If $k^* > 1$, we show that a contradiction arises. So suppose $k^* > 1$. Since $G(z)/z$ is decreasing by (G2), it is easily checked that $(k^* U, k^* V)$ is an upper solution of (3.7). We now consider $(\Phi(x), \Psi(x)): = (k^* U(x)-\hat U(x), k^*V(x)-\hat V(x))$. It can be shown that it satisfies a pair of inequalities of the form in Lemma 2.1, and $\Phi(x)\not\equiv 0$, $\Psi(x)\not\equiv 0$ (due to $k^* > 1$ and $(U, V)\leq (\hat U, \hat V)$). Thus, by Lemma 2.1 we deduce $\Phi(x) > 0$ and $\Psi(x) > 0$ in $[-L, L]$. Since they are continuous functions on $[-L, L]$, this implies that $k(U, V) \ge (\hat U, \hat V) \text{ in } [-L, L]$ for some $k < k^*$, which is a contradiction to the definition of $k^*$. Thus we must have that $k^* = 1$, and uniqueness is proved.

Now let us assume that $(\tilde{u}, \tilde{v})$ is a positive steady state solution of (3.6) and $\lambda_1(L)\ge 0$. By (G2), we see that $(\tilde{u}, \tilde{v})$ satisfies

By applying Lemma 3.2 with $\tilde{\lambda} = 0$, we obtain that $\lambda_1(L) < 0$. This contradicts our assumption, proving the nonexistence.

Next we prove (ii) and (iii) simultaneously. By choosing $\epsilon > 0$ sufficiently small and $M > 1$ sufficiently large, we can ensure that $(\epsilon\phi_1(x), \epsilon\psi_1(x))\leq (u(x, 1), v(x, 1))\leq (Mu^*, Mv^*)$ in $[-L, L]$. Let $(\overline u(x, t), \overline v(x, t))$ be the unique solution of (3.6) with initial data $(Mu^*, Mv^*)$. Then an analogous reasoning to that for $(\underline u(x, t), \underline v(x, t))$ shows that $(\overline u(x, t), \overline v(x, t))$ is non-increasing in $t$ and hence

If $\lambda_1(L)\geq 0$, then by (i) we know that necessarily $(\tilde U, \tilde V) = (0, 0)$. By Dini's theorem, the monotonicity in $t$ implies that the convergence in the above limit is uniform in $x\in [-L, L]$. The comparison principle implies that $(0, 0)\leq (u(x, t+1), v(x, t+1))\leq (\overline u(x, t), \overline v(x, t))$. Letting $t\to\infty$, we thus obtain $\lim_{t\to\infty} (u(x, t), v(x, t)) = (0, 0)$ uniformly in $[-L, L]$ when $\lambda_1(L)\geq 0$. This proves (ii).

Note that the comparison principle implies

When $\lambda_1(L) < 0$, letting $t\to\infty$, we deduce $(U, V)\leq (\tilde U, \tilde V)$, and the uniqueness result obtained in (i) implies $(U, V) = (\tilde U, \tilde V)$. This in turn implies, by the above inequalities, $\lim_{t\to\infty} (u(x, t), v(x, t)) = (U(x), V(x))$. Moreover, this convergence is uniform in $[-L, L]$ (by Dini's theorem again) since the convergences of $(\underline u(x, t), \underline v(x, t))$ and $(\overline u(x, t), \overline v(x, t))$ to $(U(x), V(x))$ are uniform due to their monotonicity in $t$ and the continuity of $U(x)$ and $V(x)$.

Remark 3.8. By Proposition 3.4, if $R_0 > 1$, then $\lambda_1(L) < 0$ for large enough $L$. Thus the steady state problem (3.7) has a unique positive solution for all large $L$, which we will denote by $(\tilde{u}_L, \tilde{v}_L)$ to stress its dependence on $L$.

Following the proof of [16,Proposition 3.5], we have the following result.

Proposition 3.9. Assume (J) holds and that $R_0 > 1$. Then

where $(u^*, v^*)$ is defined in $(1.2)$.

4.   Spreading-vanishing dichotomy and critera

In this section, we prove Theorems 1.2 and 1.3. It is clear that $h(t)$ and $g(t)$ are respectively monotonically increasing and decreasing. Therefore, their limits

are well-defined.

4.1.   Vanishing

Here we look at cases where vanishing happens: either when the reproduction number $R_0 \le 1$ or for sufficiently small initial data $(u_0, v_0)$.

Lemma 4.1. If $h_\infty - g_\infty < \infty$, then

Proof. We claim that $\lambda_1 (g_\infty, h_\infty)\ge 0$ where $\lambda_1(g_\infty, h_\infty)$ is the principal eigenvalue of (3.1) with $[-L, L]$ replaced by $[g_\infty, h_\infty]$.

Suppose by contradiction that $\lambda_1 (g_\infty, h_\infty) < 0$. Then by Lemma 3.3, there exists a constant $T > 0$ large enough such that $\lambda_1(g(T), h(T)) < 0$. We may assume that $g (T)$ and $h(T)$ satisfy $|g(T)-g_\infty| < \epsilon$ and $|h(T)-h_\infty| < \epsilon$ with $\epsilon\in(0, h_0)$ small enough such that $J_1(x), J_2(x) > 0$ for $x\in[-4\epsilon, 4\epsilon]$. Clearly $[h(t)-2\epsilon, h(t)-\epsilon] \subset [g(T), h(T)]$ for $t \ge T$.

Let $(u_1(x, t), v_1(x, t))$ be the solution of (3.6) with $Q_L$ replaced by $(g(T), h(T))\times (0, \infty)$, and initial functions $(u_0, v_0) = (u(x, T), v(x, T))$. By the comparison principle, we obtain that

By Proposition 3.7(iii), we obtain uniform convergence of $(u_1, v_1)$ for $x\in[g(T), h(T)]$, giving

Therefore, there exists $T_1\ge T$ such that

Let $c_1, c_2, c_3$ and $c_4$ be constants as defined below

Then from the above, we can calculate that

which implies $h_\infty = \infty$, contradicting to $h_\infty < \infty$. Therefore $\lambda_1(g_\infty, h_\infty)\ge 0$.

Now, let $(u_2(x, t), v_2(x, t))$ be the solution of (3.6) with $Q_L$ replaced by $(0, \infty)\times (g_\infty, h_\infty)$ and with the same initial data $(u_0, v_0)$ as $(u, v)$. By the comparison principle, we have $0 \le u(x, t) \le u_2(x, t)$ and $0 \le v(x, t) \le v_2(x, t)$ for $t > 0$ and $x\in [g(t), h(t)]$. Since $\lambda_1(g_\infty, h_\infty)\ge 0$, Proposition 3.7(ii) gives that

which implies the vanishing result.

Lemma 4.2. If $R_0\le 1$, then

and hence vanishing happens, where $m_0: = \min\{d_1, \frac{d_2c}{b}\}$.

Proof. Since $\int_{\mathbb{R}} J_i (x)dx = 1$ and $J_i(x)$ is even for $i = 1, 2$, a straightforward calculation shows that

Moreover, we can calculate that

and the same holds when $(u, J_1)$ is replaced by $(v, J_2)$. By the above, we obtain that

Using (G2), it follows from $R_0\le 1$ that $-au(x, t)+\frac{c}{b}G (u(x, t)) \le 0$ for $x\in [g(t), h(t)]$ and $t\ge 0$. Hence,

Integrating the above from $0$ to $t$ gives us (4.4). Then by Lemma 4.1, we obtain the vanishing result.

Now for initial data $(u_0, v_0)$ small enough, we show that vanishing also occurs.

Lemma 4.3. Let $\lambda_1(h_0)$ be the principal eigenvalue of $(3.1)$ with $L = h_0$. If $R_0 > 1$, $\lambda_1(h_0) > 0$ and $\| u_{0}\|_{C([-h_0, h_0])}+\| v_{0}\|_{C([-h_0, h_0])}$ is sufficiently small, then vanishing happens.

Proof. Since $\lambda_1(h_0) > 0$, there exists $h_1 > h_0$ but close to $h_0$ such that $\lambda_1(h_1) > 0$. Let $(\phi, \psi)$ be a positive eigenfunction pair corresponding to $\lambda_1(h_1)$ and

Then define, for $t\ge 0$, $x\in[-h_1, h_1]$,

We see that $\overline{h}(t) \in [h_0, h_1)$ for $t\ge 0$ and if we let $\sigma: = \min\left\{\min_{x\in[-h_0, h_0]}\phi(x), \min_{x\in[-h_0, h_0]}\psi(x)\right\}$ and

then we have

Clearly, we can calculate

By (G2), we obtain $G(\overline u)\leq G'(0)\overline u$ and hence

Moreover, for $x\in\{\overline g (t), \overline h(t)\}$, we have that $(\overline u(x, t), \overline v(x, t))\geq (0, 0)$ and

In view of $\overline g (t) = -\overline h(t)$, we can now use the the comparison principle in Lemma 2.2 to conclude that $h(t)\leq \overline h(t)\leq h_1$ for all $t > 0$, and hence vanishing happens.

Remark 4.4. If $(\mu_1, \mu_2) = (\mu\sigma_1^0, \mu\sigma_2^0)$ with $\sigma_1^0$ and $\sigma_2^0$ fixed, nonnegative and $\sigma_1^0+\sigma_2^0 > 0$, then by the proof of Lemma 4.3, we see that $M\rightarrow \infty$ when $\mu \rightarrow 0$. Thus, for any given initial data $(u_0, v_0)$, there exists $\mu_0 > 0$ such that (4.5) holds for all $\mu \in (0, \mu_0)$. Thus if $0 < \mu \le \mu_0$, then vanishing must happen for (1.8) for this given initial data $(u_0, v_0)$.

We note that the following lemma implies that if $h_\infty - g_\infty = \infty$ holds, then we must have that $h_\infty = -g_\infty = +\infty$.

Lemma 4.5. The inequality $h_\infty < +\infty$ if and only if $g_\infty > -\infty$.

Proof. The proof of this lemma is similar to the proof of Lemma 4.10 in ^[16]. Since the modifications are obvious, we omit the details.

4.2.   Spreading

In this section, we look at cases where spreading happens.

Lemma 4.6. If $\lambda_1(g(t_0), h(t_0)) < 0$ for some $t_0\ge 0$, then $h_\infty = -g_\infty = +\infty$ and

where $\lambda_1(g(t_0), h(t_0))$ is the eigenvalue of $(3.1)$ with $[-L, L]$ replaced by $[g(t_0), h(t_0)]$, and $(u^*, v^*)$ are as defined in $(1.2)$.

Remark 4.7. If for $R_0 > 1$ and for some $t_0 > 0$, we have $h(t_0)-g(t_0)\ge 2 L^*$, then by Proposition 3.4, we obtain that $\lambda_1(h(t_0), g(t_0))\leq 0$. Hence for any $t_1 > t_0$ we have $\lambda_1(h(t_1), g(t_1)) < 0$. Then by Lemma 4.6, we have that spreading occurs. This implies that when $R_0 > 1$ and vanishing happens, we must have $h(t)-g(t) < 2L^*$ for all $t\geq 0$.

Proof of Lemma 4.6. We see that $h_\infty = -g_\infty = \infty$. Suppose by contradiction that $h_\infty - g_\infty < \infty$. Since $[g(t_0), h(t_0)] \subset [g(t), h(t)]$ for some $t\ge t_0$, by Corollary 3.5, we have $\lambda_1(g(t), h(t)) < 0$ for $t\ge t_0$. Then we derive the contradiction as in Lemma 4.1. By Lemma 4.5, we further obtain that $-g_\infty = h_\infty = \infty$. Then by Lemma 4.2, we must have $R_0 > 1$, which guarantees the existence of the positive equilibrium $(u^*, v^*)$. It remains to show (4.6).

Let us first consider the limit superior of the solution. Let $(\overline{u}, \overline{v})$ be the unique positive solution of the following ODE problem:

Since $R_0 > 1$, we have $\lim_{t \rightarrow \infty} (\overline{u}(t), \overline{v}(t)) = (u^*, v^*)$. We then note that

and $\overline{u}(0)\ge u_0(x), \overline{v}(0)\ge v_0(x)$; so by the comparison principle in Lemma 2.3, we have

Thus we have that

Then following [16,Lemma 4.11], we can make use of Propositions 3.7 and 3.9 to show

Thus (4.6) holds.

Lemma 4.8. If $R_0 > 1$, $h_0 < L^*$ and $(\mu_1, \mu_2) = (\mu\sigma_1^0, \mu\sigma_2^0)$ with $\sigma_1^0, \ \sigma_2^0$ nonnegative and $\sigma_1^0+\sigma_2^0 > 0$, then there exists $\mu^0 > 0$ depending on the initial data $(u_0, v_0)$ such that spreading happens if $\mu > \mu^0$.

Proof. Similar to the calculations in the proof of Lemma 4.2, by setting $m^0: = \max\{d_1, \frac{d_2c}{b}\}$, we obtain for $t > 0$,

Suppose that $h_\infty - g_\infty < \infty$; then in view of Lemma 4.1 and Remark 4.7, by letting $t \rightarrow \infty$ in the above inequality, we get

However, this is patently false in the case

This completes the proof.

4.3.   Proof of Theorems 1.2 and 1.3

Proof of Theorem 1.2. If $h_\infty-g_\infty < \infty$, then (1.9) holds by Lemma 4.2. On the other hand, if $h_\infty - g_\infty = \infty$, then $R_0 > 1$ by Lemma 4.2. By Corollary 3.5, we find that $\lambda_1(g(t_0), h(t_0)) < 0$ for some large $t_0 > 0$. Hence (4.6) holds by Lemma 4.6.

Proof of Theorem 1.3. $\text{(a)}$ This follows from Lemma 4.2.

$\text{(b)}$ By Proposition 3.4, we obtain that $\lambda_1(h_0)\leq 0$. Since $h(t)$ is strictly increasing in $t$ and $\lambda_1(L)$ is strictly decreasing in $L$, we obtain $\lambda_1(h(1)) < \lambda_1(h_0)\leq 0$. Thus by Lemma 4.6, we have that spreading occurs.

$\text{(c)} \; \text{(i)}$ It follows from Remark 4.7 that if vanishing occurs, then $h_\infty-g_\infty \le 2L^*$. Define

Then by Remark 4.4 and Lemma 4.8, we respectively have that $(0, \mu_0] \subset \Gamma$ and $\Gamma \cap (\mu^0, \infty) = \emptyset$. Denoting by $\mu^*: = \sup \Gamma \in [\mu_0, \mu^0]$, we have by definition that $h_\infty-g_\infty > 2L^*$ for $\mu > \mu^*$ and hence spreading happens for $\mu > \mu^*$ by Theorem 1.2.

Suppose that $\mu^* \notin \Gamma$. Then we have $h_\infty-g_\infty = \infty$ when $\mu = \mu^*$ and there exists a $T > 0$ such that $h(T)-g(T) > 2L^*$. Let us emphasis the dependence of the solution $(u, v, g, h)$ of (1.8) on $\mu$ by rewriting it as $(u_\mu, v_\mu, g_\mu, h_\mu)$. Then we have $h_{\mu^*}(T)-g_{\mu^*}(T) > 2L^*$. By the continuity of the solution in $\mu$, hence there exists $\epsilon > 0$ such that for $|\mu-\mu^*| < \epsilon$, we have $h_{\mu}(T)-g_{\mu}(T) > 2L^*$. Then for every $\mu$ such that $|\mu-\mu^*| < \epsilon$, by the monotonicity of $h(t)$ and $-g(t)$ in $t$, we have that $\lim_{t\rightarrow \infty} h_\mu (t) - g_\mu (t) > h_\mu(T) - g_\mu(T) > 2L^*$. Thus we get the contradiction that $\sup \Gamma \le \mu^* - \epsilon$. Hence we must have $\mu^* \in \Gamma$.

It remains to show that vanishing also occurs for $\mu < \mu^*$. For every $\mu\in (0, \mu^*)$, $(u_{\mu^*}, v_{\mu^*}, g_{\mu^*}, h_{\mu^*})$ is an upper solution to (1.8). Thus by the comparison principle, we see that $h_\mu(t) \le h_{\mu^*}(t)$ and $g_\mu (t) \ge g_{\mu^*}(t)$ for $t > 0$. It follows that $\lim_{t\rightarrow \infty}(h_\mu (t) - g_\mu(t)) \le \lim_{t\rightarrow \infty} (h_{\mu^*} (t) - g_{\mu^*}(t)) \le 2L^*$. This proves our assertion.

$\text{(ii)}$ From the assumptions, we obtain that $\lambda_1(h_0) > 0$. Thus the assertion follows directly from Lemma 4.3.

The proof is now complete.

5.   Spreading speed

In this section, we consider the asymptotic spreading speed when spreading happens in our system (1.8). As such, we necessarily have that $R_0 > 1$.

Let $F(u, v) = (f_1(u, v), f_2(u, v))$ with $f_1(u, v): = -au+cv$ and $f_2(u, v): = -bv+G(u)$. We now check that $F$ satisfies the assumptions $\bf{(f_1)}-\bf{(f_4)}$ in ^[2] with $\hat{\bf u} = \infty$ and $m = n = 2$, namely

$\bf{(f_1)}$: $F(u, v) = (0, 0)$ has only two nonnegative solutions $(0, 0)$ and $(u^*, v^*)$, and the Jacobian matrix of $F$ evaluated at (0, 0), denoted by $\nabla F(0, 0)$, is irreducible with principal eigenvalue positive.

$\bf{(f_2)}$: $F(ku, kv)\geq kF(u, v)$ for $k\in [0, 1]$ and all $u, v\geq 0$.

$\bf{(f_3)}$: $\nabla F(u^*, v^*)$ is invertible, $(u^*, v^*)\nabla F(u^*, v^*)\leq (0, 0)$ component wise, and for $i\in \{1, 2\}$,

$\bf{(f_4)}$: The solution of the corresponding problem (1.12) with initial function pair $(u_0, v_0)$ nonnegative, bounded and not identically $(0, 0)$ is positive and globally defined, and as time $t\to\infty$, it converges to $(u^*, v^*)$ locally uniformly for $x\in\mathbb R$.

It is straightforward to check that $\bf{(f_1)}$, $\bf{(f_2)}$ and $\bf{(f_3)}$ are satisfied. It now remains for us to prove $\bf{(f_4)}$, namely the following lemma.

Lemma 5.1. Let $(U(x, t), V(x, t))$ satisfy

If $(U(\cdot, 0), V(\cdot, 0)) \in L^\infty(\mathbb R)^2\cap C(\mathbb R)^2$ is nonnegative, then $(U(x, t), V(x, t))\in [0, \infty)\times [0, \infty)$ for every $t > 0$ and $x\in \mathbb{R}$. Moreover, it holds that $\lim_{t\rightarrow \infty}(U(x, t), V(x, t)) = (u^*, v^*)$ in $L_{loc}^\infty(\mathbb{R})$ if additionally $(U(x, 0), V(x, 0)) \not\equiv (0, 0)$.

Proof. Let $(U(\cdot, 0), V(\cdot, 0)) \in L^\infty(\mathbb R)^2\cap C(\mathbb R)^2$ be nonnegative. If $(U(x, 0), V(x, 0))\equiv (0, 0)$, then clearly $(U, V)\equiv (0, 0)$ is the unique solution of (5.1). In the following we assume that $(U(x, 0), V(x, 0)) \not\equiv (0, 0)$. For $L > 0$, let $(u_L(x, t), v_L(x, t))$ be the solution to (3.6) with initial data $(u_L(x, 0), v_L(x, 0)) = (U(x, 0), V(x, 0))|_{[-L, L]}$. By the comparison principle in Lemma 2.3, we get that

Let $(\overline{u}(t), \overline{v}(t))$ be the unique positive solution of the following system of ordinary differential equations:

By the comparison principle in Lemma 2.3, we obtain $(U(x, t), V(x, t))\leq (\overline u(t), \overline v(t))$ for $(x, t) \in [-L, L]\times (0, \infty)$. Since $L\geq L_0$ is arbitrary, this and the earlier estimates imply that

Moreover, it follows from $R_0 > 1$ that $\lim_{t\to\infty} (\overline u(t), \overline v(t)) = (u^*, v^*)$. Therefore we must have

Since $(U(x, 0), V(x, 0)) \not\equiv (0, 0)$ for $x\in\mathbb R$, there exists $L_0 > L^*$ large enough such that

By Proposition 3.7(iii), we obtain

where $(\tilde u_L, \tilde v_L)$ is the unique positive steady-state of (3.6). It follows that

Letting $L\to\infty$, by Proposition 3.9, we obtain

This and (5.3) imply

The proof is complete. Since $F = (f_1, f_2)$ satisfies $\bf{(f_1)-(f_4)}$ in ^[2], Theorems 1.3 and 1.5 (as well as two results on the associated semi-wave problem: Theorems 1.1 and 1.2) in ^[2] can be applied to obtain Theorems 1.5 and 1.6 here.

Acknowledgments

Y. Du's research was supported by the Australian Research Council. We thank the referees for their useful suggestions which helped to improve the presentation of the paper.

Conflict of interest

All authors declare no conflicts of interest in this paper.