Propagation direction of traveling waves for a class of bistable epidemic models

Abstract

Traveling waves of a reaction–diffusion (RD) system connecting two spatially uniform stable equilibria are termed as bistable waves. Due to the uniqueness of a bistable wave in RD systems, it is difficult to determine its propagation direction, and there are very few analytical results on this subject. In this study, we propose an approach to give a complete characterization of the propagation direction of bistable waves for a class of bistable epidemic models arising from the spread of a cholera epidemic. Moreover, this characterization also gives a parameter threshold above which the epidemic disease eventually tends to extinction, and below which the epidemic outbreak happens.

Appendix: Proof of Lemma 3.2

In this appendix, we will prove Lemma 3.2. To proceed the proof, let (U, V, c) be a traveling wave solution of (1.1). Then according to the general theory of the adjoint problem (Theorem 5.1 of Chapter 4 in the book of Volpert et al. 1994), for each $d_1, d_2 > 0$, the adjoint problem (3.8) with respect to (U, V, c) has a unique positive solution $(U^*,V^*)$. It remains to prove Lemma 3.2 for the degenerate cases where $d_1d_2 = 0$. We will only prove Lemma 3.2 for the cases (i) $d_2=0$ and $c=0$, and (ii) $d_2=0$ and $c\not =0$, since the proof for the cases where $d_1=0$ is similar to those for cases (i) and (ii).

1.1 The case that $d_2=0$ and $c=0$

We first prove Lemma 3.2 for the case (i) $d_2=0$ and $c=0$. Let (U, V, c) with $c=0$ be a traveling wave solution of (1.1) with $d_2=0$. Then from Eq. (2.3) it follows that $V=g(U)/\beta$, and hence that U is a solution of the following problem:

$$\begin{aligned} d_1U'' - U + (\alpha /\beta ) g(U)= & {} 0, \end{aligned}$$

(A.1a)

$$\begin{aligned} U(- \infty ) = 0, ~U(+\infty )= & {} b. \end{aligned}$$

(A.1b)

Note that by assumptions (A1)–(A2), the function $f(u) := -u + (\alpha /\beta ) g(u)$ admits three zeros: 0, a, and b. Moreover, $f'(0) < 0$ and $f'(b) < 0$. Thus, (U, c) with $c=0$ is a traveling wave solution of the following bistable parabolic equation:

$$\begin{aligned} u_t = d_1 u_{xx} + f(u). \end{aligned}$$

Then according to the general theory of the adjoint problem (Theorem 5.1 of Chapter 4 in the book of Volpert et al. 1994), the following adjoint problem with respect to the traveling wave solution (U, c) with $c=0$ admits a unique positive solution $U^*$:

$$\begin{aligned} \left\{ \begin{array}{l} {\mathcal {L}}^*(u) := d_1u'' - u + (\alpha /\beta ) G(\xi ) u = 0, \\ u(\pm \infty ) = 0, \; \sup _{\xi \in {\mathbb {R}}}u(\xi )=1, \end{array} \right. \end{aligned}$$

where $G(\cdot ) = g'(U(\cdot ))$. Set $V^*(\cdot ) = (\alpha /\beta )U^*(\cdot )$ on ${\mathbb {R}}$. Then $(U^*,V^*)$ is a solution of the adjoint problem (3.8) with $d_2=0$. This completes the proof of Lemma 3.2 for the case (i) $d_2=0$ and $c=0$.

1.2 The case that $d_2=0$ and $c\not =0$

Next, we prove Lemma 3.2 for the case (ii) $d_2=0$ and $c\not =0$.

1.2.1 A priori estimate

To proceed the proof, the following lemma gives a uniform bound for the wave speed of traveling wave solutions of (1.1) whose proof follows standard arguments for the bistable monotone system [see Lemma 4.4 of Fang and Zhao 2009].

Lemma A.1

Let (U, V, c) be a traveling wave solution of (1.1). Then there exists a constant $C > 0$, independent of $d_2 \in (0,1]$, such that

$$\begin{aligned} |c| \le C \quad \forall \, d_2 \in (0,1]. \end{aligned}$$

Lemma A.2

For $d_1, d_2 > 0$, let $(U^*,V^*)$ be the positive solution of the adjoint problem (3.8) with respect to the traveling wave solution (U, V, c) of (1.1). Then it holds that

$$\begin{aligned} \sup _{\xi \in {\mathbb {R}}}V^*(\xi ) \le \alpha /\beta . \end{aligned}$$

Proof

To begin with, recall the normalized condition $\sup _{\xi \in {\mathbb {R}}} U^*(\xi ) = 1$. Since $V^*(\xi ) \rightarrow 0$ as $\xi \rightarrow \pm \infty$, $V^*$ takes its global maximum at finite value, say $\xi = \xi _0$. From Eq. (3.8b), it follows that

$$\begin{aligned} 0 \ge d_2 (V^*)''(\xi _0) = - \alpha U^*(\xi _0) + \beta V^*(\xi _0), \end{aligned}$$

and so $V^*(\xi _0) \le \alpha U^*(\xi _0) / \beta$. Therefore,

$$\begin{aligned} \sup _{\xi \in {\mathbb {R}}}V^*(\xi ) = V^*(\xi _0) \le (\alpha /\beta ) U^*(\xi _0) \le \alpha /\beta . \end{aligned}$$

The proof of this lemma is completed. $\square$

The following lemma gives an approach on how to approximate a traveling wave solution of (1.1) for the degenerate case where $d_2=0$. Also note that in order to prove this lemma, we need two auxiliary lemmas: Lemmas A.4 and A.5 whose statements and proof are postponed after this lemma. In order to emphasize the dependence on the diffusivity parameter $d_2$, we denote by $(U(\cdot ;d_2), V(\cdot ;d_2),c_{d_2})$ the traveling wave solution of system (1.1).

Lemma A.3

Let (U, V, c) be the traveling wave solution of (1.1) with $d_2=0$ and $c\not =0$. Then there exists a sequence $\{ d_{2j} \}_{j = 1}^{\infty }$ of positive numbers such that $d_{2j} \rightarrow 0$ and $c_{d_{2j}} \rightarrow c$ as $j \rightarrow \infty$, and that either

$$\begin{aligned} (U(\cdot ;d_{2j}), V(\cdot ;d_{2j})) \rightarrow (U(\cdot ), V(\cdot ))\text { uniformly on }{\mathbb {R}}\text { as }j \rightarrow \infty , \end{aligned}$$

or

$$\begin{aligned} ({\hat{U}}_j (\cdot ), {\hat{V}}_j (\cdot )) \rightarrow (U(\cdot + \zeta _0), V(\cdot + \zeta _0))\text { uniformly on }{\mathbb {R}}\text { as }j \rightarrow \infty \end{aligned}$$

where $({\hat{U}}_j (\cdot ), {\hat{V}}_j (\cdot )) = (U(\cdot + \zeta _j;d_{2j}), V(\cdot + \zeta _j;d_{2j}))$ with $\zeta _j\in {\mathbb {R}}$ satisfying

$$\begin{aligned} U(\zeta _0) = \delta ^* /2 \; \text{ and } \; U(\zeta _j;d_{2j}) = \delta ^* /2 \quad \forall \, j\in {\mathbb {N}}. \end{aligned}$$

Proof

First, since c is nonzero, the traveling wave solution (U, V, c) is unique and the wave profile (U, V) is continuous on ${\mathbb {R}}$. Then following the same lines of the arguments of Theorem 4.1 of Fang and Zhao (2009), one can find the sequences $\{ d_{2j} \}_{j \in {\mathbb {N}}}$, $\{c_{d_{2j}} \}_{j \in {\mathbb {N}}}$, and $\{({\hat{U}}_j (\cdot ), {\hat{V}}_j (\cdot ))\}_{j\in {\mathbb {N}}}$ such that the convergence assertions of the lemma hold but the convergence is pointwise convergence on ${\mathbb {R}}$. On the other hand, from the fact that the wave profile (U, V) is continuous on ${\mathbb {R}}$ and Helly’s selection theorem (see Lemma A.4 for the statement), it follows that the convergence in this lemma is uniform on any compact subset of ${\mathbb {R}}$. Finally, from Lemma A.5, $(U(\xi ;d_{2j}), V(\xi ;d_{2j}))$ and $({\hat{U}}_j (\xi ), {\hat{V}}_j (\xi ))$ have the uniform decay estimates around $\xi =\pm \infty$. Taken together, we can conclude that the convergence in this lemma is uniform on ${\mathbb {R}}$. This proof of the lemma is completed. $\square$

For readers’ convenience, we state the classical Helly’s selection theorem in the following lemma.

Lemma A.4

[Helly’s selection theorem (Helly 1921)] Let $\{f_n\}_{n\in {\mathbb {N}}}$ be a sequence of monotone increasing real-valued functions such that the sequence $\{f_n (x)\}_{n \in {\mathbb {N}}}$ is uniformly bounded for each $x\in {\mathbb {R}}$. Then there is a subsequence $\{f_{n_k}\}_{k\in {\mathbb {N}}}$ that converges pointwise on ${\mathbb {R}}$. In addition, if the limiting function is continuous on ${\mathbb {R}}$, then this convergence is uniform on any compact subset of ${\mathbb {R}}$.

The following lemma gives uniform decay estimates of $(U(\xi ;d_{2j}), V(\xi ;d_{2j}))$ and $({\hat{U}}_j (\xi ), {\hat{V}}_j (\xi ))$ around $\xi =\pm \infty$, which ensure the convergence in Lemma A.3 is uniform on ${\mathbb {R}}$.

Lemma A.5

Let the assumptions and notations of Lemma A.3 be in force. Then there exist $M > 0$, $\lambda < 0$, $j_0\in {\mathbb {N}}$, and $\xi _0 > 0$, independent of $j\in {\mathbb {N}}$, such that the following hold:

• (i) if $(U(\cdot ;d_{2j}), V(\cdot ;d_{2j}))$ converges pointwise to $(U(\cdot ), V(\cdot ))$ on ${\mathbb {R}}$ as $j \rightarrow \infty$, then for each $j \ge j_0$, it holds that

  $$\begin{aligned} |U(\pm \xi ;d_{2j}) - U(\pm \infty )| \le M e^{\lambda \xi },~|V(\pm \xi ;d_{2j}) - V(\pm \infty )| \le M e^{\lambda \xi }~~\forall \, \xi \ge \xi _0. \end{aligned}$$

• (ii) if $({\hat{U}}_j (\cdot ), {\hat{V}}_j (\cdot ))$ converges pointwise to $(U(\cdot + \zeta _0), V(\cdot + \zeta _0))$ on ${\mathbb {R}}$ as $j \rightarrow \infty$, then for each $j \ge j_0$, it holds that

  $$\begin{aligned} |{\hat{U}}_j(\pm \xi ) - U(\pm \infty )| \le M e^{\lambda \xi },~|{\hat{V}}_j(\pm \xi ) - V(\pm \infty )| \le M e^{\lambda \xi }~~\qquad \forall \, \xi \ge \xi _0. \end{aligned}$$

Proof

We assume that the second convergence result of Lemma A.3 holds, and prove that the assertion (ii) holds. The other case can be proven similarly. (Also see the proof of Lemma A.6 for this case.) We divide this proof into five steps. The assertion (ii) for $\xi \le 0$ will be proven by Step 1-4, while that for $\xi \ge 0$ will be proven in Step 5. More precisely, the first and second steps give the construction of upper functions on $(-\infty , 0]$, while the third and fourth steps show that the quantities involved in the upper functions are uniformly bounded.

Step 1. We construct a pair of upper functions on $(-\infty ,0)$. For convenience, let $(u_j,v_j) = ({\hat{U}}_j, {\hat{V}}_j)$. For each $j \in {\mathbb {N}}$, it follows from the mean value theorem that there is a function ${\tilde{U}}_j: (-\infty , 0] \rightarrow {\mathbb {R}}$ such that $0 \le {\tilde{U}}_j \le u_j$ on $(-\infty , 0]$ and

$$\begin{aligned} g(u_j)= g(u_j) - g(0) = g'({\tilde{U}}_j) u_j < G^- u_j \quad \text {on }(-\infty ,0]. \end{aligned}$$

Note that the last inequality follows from the choice of $G^-$ and the fact that $u_j(\xi ) = {\hat{U}}_j(\xi ) \le {\hat{U}}_j(0) = \delta ^*/2$ for $\xi \le 0$. Thus, for each $\xi \le 0$ and $j \in {\mathbb {N}}$, it follows from (2.3) that

$$\begin{aligned} {\left\{ \begin{array}{l} \begin{aligned} &{} d_1u_j'' + c_ju_j' -u_j + \alpha v_j = 0, \\ &{} d_{2j} v_j'' + c_jv_j' + G^- u_j - \beta v_j > 0. \end{aligned} \end{array}\right. } \quad \xi < 0, \end{aligned}$$

(A.2)

Hence, by the comparison principle, for each $l < 0$ and $\xi \in (l,0)$, the inequalities

$$\begin{aligned} u_j \le {{\bar{u}}}_j^l \text{ and } v_j \le {{\bar{v}}}_j^l \end{aligned}$$

(A.3)

hold for $j \in {\mathbb {N}}$ where $({{\bar{u}}}_j^l, {{\bar{v}}}_j^l)$ is a positive solution of the problem

$$\begin{aligned} {\left\{ \begin{array}{l} \begin{array}{l} d_1u'' + c_ju' -u + \alpha v = 0, \\ d_{2j}v'' + c_jv' + G^- u - \beta v = 0, \end{array} \quad l< \xi < 0, \\ \; {{\bar{u}}}^l(0) = u^{l,-}_j, \quad {{\bar{v}}}^l(0) = v^{l,-}_j,\\ \; {{\bar{u}}}^l(l) = u^{l,+}_j, \quad {{\bar{v}}}^l(l) = v^{l,+}_j, \end{array}\right. } \end{aligned}$$

(A.4)

where $u_j^{l,\pm } \ge b$ and $v_j^{l,\pm } \ge b/\alpha$ are constants to be determined later.

Step 2. We seek a positive solution of problem (A.4). Assume that $({{\bar{u}}}_j^l, {{\bar{v}}}_j^l) = (u_{0,j}, v_{0,j})e^{\lambda _j \xi }$ is a solution of problem (A.4). Then with a direct computation, $(u_{0,j}, v_{0,j}, \lambda _j)$ satisfies

$$\begin{aligned} {\left\{ \begin{array}{l} \begin{aligned} &{} Q_j(\lambda _j) = P_{1j}(\lambda _j)P_{2j}(\lambda _j) - \alpha G^- \\ &{} \quad := (d_1 \lambda _j^2 + c_j\lambda _j - 1) (d_{2j} \lambda _j^2 + c_j \lambda _j -\beta ) - \alpha G^- = 0, \\ &{} P_{1j}(\lambda _j)u_{0,j} = -\alpha v_{0,j}, \quad P_{2j}(\lambda _j)v_{0,j} = -G^- u_{0,j}. \end{aligned} \end{array}\right. } \end{aligned}$$

(A.5)

For each $j \in {\mathbb {N}}$, set

$$\begin{aligned} \begin{aligned}&\lambda _{1,j}^\pm = \big (-c_j \pm \sqrt{c_j^2 + 4d_1} \big )/2d_1, \quad \lambda _{2,j}^\pm = \big (-c_j \pm \sqrt{c_j^2 + 4d_{2j} \beta } \big )/2d_{2j}, \\&\lambda _{M,j}^\pm = \max \big \{\lambda _{1,j}^\pm , \lambda _{2,j}^\pm \big \}, \qquad \lambda _{m,j}^\pm = \min \big \{\lambda _{1,j}^\pm , \lambda _{2,j}^\pm \big \}. \end{aligned} \end{aligned}$$

Note that for $i=1,2$, $\lambda _{i,j}^-< 0 < \lambda _{i,j}^+$, $P_{1j}(\lambda _{1,j}^\pm )=P_{2j}(\lambda _{2,j}^\pm )=0$, and

$$\begin{aligned} Q_j(\pm \infty ) = \infty , \; Q_j(0)= \beta - \alpha G^- > 0,\; Q_j(\lambda _{M,j}^\pm ) = Q_j(\lambda _{m,j}^\pm ) = - \alpha G^- < 0. \end{aligned}$$

Then $Q_j(\lambda )$ has four real roots in the following intervals:

$$\begin{aligned} (-\infty , \lambda _{m,j}^-), \; (\lambda _{M,j}^-, 0), \; (0, \lambda _{m,j}^+), \; (\lambda _{M,j}^+, \infty ). \end{aligned}$$

Let $\lambda _j^-$ and $\lambda _j^+$ be the roots of $Q_j(\lambda )$ in the interval $(\lambda _{M,j}^-, 0)$ and $(0, \lambda _{m,j}^+)$, respectively. Then $P_{1j}(\lambda _j^+) < 0$ and $P_{2j}(\lambda _j^-) < 0$, and

$$\begin{aligned} \left( \begin{array}{c} u_{0,j}^+ \\ v_{0,j}^+ \\ \lambda _j^+ \end{array} \right) = \left( \begin{array}{c} \alpha \\ -P_{1j}(\lambda _j^+) \\ \lambda _j^+ \end{array} \right) \text{ and } \left( \begin{array}{c} u_{0,j}^- \\ v_{0,j}^- \\ \lambda _j^- \end{array} \right) = \left( \begin{array}{c} -P_{2j}(\lambda _j^-) \\ G^- \\ \lambda _j^- \end{array} \right) \end{aligned}$$

(A.6)

are solutions of (A.5). Set

$$\begin{aligned} M_j^+ = \max \big \{ b/u_{0,j}^+, b/(\alpha v_{0,j}^+) \big \}, \quad M_j^- = \max \big \{ b/u_{0,j}^-, b/(\alpha v_{0,j}^-) \big \}. \end{aligned}$$

Then for each $j \in {\mathbb {N}}$ and $l<0$,

$$\begin{aligned} \begin{aligned} \left( \begin{array}{c} {{\bar{u}}}_j^l(\xi ) \\ {{\bar{v}}}_j^l(\xi ) \end{array} \right)&= M_j^+ \Big (\frac{1 - e^{\lambda _j^- l}}{e^{\lambda _j^+ l} - e^{\lambda _j^- l}} \Big ) e^{\lambda _j^+\xi } \left( \begin{array}{c} u_{0,j}^+ \\ v_{0,j}^+ \end{array} \right) \\&\quad +\, M_j^- \Big (\frac{ e^{\lambda _j^+ l} - 1}{e^{\lambda _j^+ l} - e^{\lambda _j^- l}} \Big ) e^{\lambda _j^-\xi } \left( \begin{array}{c} u_{0,j}^- \\ v_{0,j}^- \end{array} \right) \end{aligned} \end{aligned}$$

is a solution of problem (A.4), and ${{\bar{u}}}_j^l(0), {{\bar{u}}}_j^l(l) \ge b$ and ${{\bar{v}}}_j^l(0), {{\bar{v}}}_j^l(l) \ge b/\alpha$. Now for each $\xi \le 0$,

$$\begin{aligned} \left( \begin{array}{c} {{\bar{u}}}_j^l(\xi ) \\ {{\bar{v}}}_j^l(\xi ) \end{array} \right) \rightarrow M_j^+ e^{\lambda _j^+ \xi } \left( \begin{array}{c} u_{0,j}^+ \\ v_{0,j}^+ \end{array} \right) \text{ as } l \rightarrow -\infty . \end{aligned}$$

Hence, by fixing $\xi > 0$ and taking the limit in (A.3) as $l \rightarrow -\infty$,

$$\begin{aligned} u_j(\xi ) \le M_j^+ u_{0,j}^+ e^{\lambda _j^+\xi } \; \text{ and } \; v_j(\xi ) \le M_j^+ v_{0,j}^+ e^{\lambda _j^+\xi } \quad \forall \, \xi \le 0, \; j \in {\mathbb {N}}. \end{aligned}$$

(A.7)

Step 3. We show that there exist ${\underline{\lambda }}, {\bar{\lambda }} > 0$ such that

$$\begin{aligned} 0 < {\underline{\lambda }} \le \lambda _j^{+} \le {\bar{\lambda }} \quad \forall \, j\in {\mathbb {N}}. \end{aligned}$$

Indeed, recall from Lemma A.1 that there exists a constant $C > 0$ such that

$$\begin{aligned} |c_j| \le C \quad \forall \, j \in {\mathbb {N}}. \end{aligned}$$

Together with the fact that $\lambda _j^+ \in (0,\lambda _{m,j}^+)$ it follows that for each $j\in {\mathbb {N}}$,

$$\begin{aligned} \begin{array}{ll} \displaystyle \lambda _j^+ \le \lambda _{m,j}^+ \le \lambda _{1,j}^+ = \frac{-c_j+\sqrt{c_j^2+4d_1}}{2d_1} \le \frac{C+\sqrt{C^2+4d_1}}{2d_1} =: {\bar{\lambda }}. \\ \end{array} \end{aligned}$$

(A.8)

Then the ${\bar{\lambda }}$ gives a positive upper bound of the sequence $\{\lambda _j^+\}_{j\in {\mathbb {N}}}$. Now, since $\{d_{2j}\}_{j\in {\mathbb {N}}}$ and $\{c_{j}\}_{j\in {\mathbb {N}}}$ are bounded, $Q_j$ is continuous in $\lambda \in [-\Lambda ,\Lambda ]$ uniformly with respect to $j \in {\mathbb {N}}$ for any $\Lambda > 0$. Then from $Q_j(0) = \beta - \alpha G^- > 0$, there exists a small ${\underline{\lambda }} > 0$, independent of $j\in {\mathbb {N}}$, such that

$$\begin{aligned} Q_j(\lambda )> \frac{Q_j(0)}{2} > 0 \quad \forall \, |\lambda | \le {\underline{\lambda }} \ \text {and} \ j \in {\mathbb {N}}. \end{aligned}$$

Then the ${\underline{\lambda }}$ gives a positive lower bound of the sequence $\{\lambda _j^+\}_{j\in {\mathbb {N}}}$. Together with (A.8), this establishes the assertion of the claim.

Step 4. We claim that the sequences $\{ M_j^+ \}_{j\in {\mathbb {N}}}$, $\{ u_{0,j}^+ \}_{j\in {\mathbb {N}}}$ and $\{ v_{0,j}^+ \}_{j\in {\mathbb {N}}}$ are uniformly bounded. By Step 3 and the fact that $\{ c_j \}_{j\in {\mathbb {N}}}$ is bounded, it suffices to prove that there exists a $\varepsilon _0 > 0$, independent of $j\in {\mathbb {N}}$, such that

$$\begin{aligned} P_{1j}(\lambda _j^+)< -\varepsilon _0 \quad \forall \, j \in {\mathbb {N}}. \end{aligned}$$

For contradiction, we assume that there exists a subsequence $\{\lambda _{j_n}^+\}_{n\in {\mathbb {N}}}$ of $\{\lambda _j^+\}_{j\in {\mathbb {N}}}$ such that $P_{1j}(\lambda _{j_n}^+) \rightarrow 0$ as $n \rightarrow \infty$. Since the sequence $\{\lambda _j^+\}_{j\in {\mathbb {N}}}$ is uniformly bounded, by choosing a further subsequence if necessary, we assume that $\lambda _{j_n}^+ \rightarrow \lambda _0$ as $n\rightarrow \infty$ for some $\lambda _0 \ge 0$. These two facts, together with (A.5) and the uniform boundedness of $\{d_{2j}\}_{j\in {\mathbb {N}}}$ and $\{c_{j}\}_{j\in {\mathbb {N}}}$, imply that $Q_{j_n}(\lambda _{j_n}^+) \rightarrow -\alpha G^-$ as $n \rightarrow \infty$. This in turn implies that

$$\begin{aligned} Q_{j_n}(\lambda _{j_n}^-) \le \frac{-\alpha G^-}{2} \quad \forall \, n \ge N_0 \end{aligned}$$

for some large $N_0 \in {\mathbb {N}}$. This contradicts to the fact that $Q_j(\lambda _j^+) = 0$ for all $j \in {\mathbb {N}}$, and so the assertion of the claim is proven. Hence the assertion (ii) of this lemma for $\xi \le 0$ is established.

Step 5. We will prove the assertion (ii) of this lemma for $\xi \ge 0$. To do this, let $({\tilde{u}}_j,{\tilde{v}}_j) = (b - {\hat{U}}_j, b/\alpha - {\hat{V}}_j)$. Then using (2.3), the governing equations of $({\tilde{u}}_j,{\tilde{v}}_j)$ read

$$\begin{aligned} {\left\{ \begin{array}{l} \begin{aligned} &{} d_1{\tilde{u}}_j'' + c_j{\tilde{u}}_j' -{\tilde{u}}_j + \alpha {\tilde{v}}_j = 0, \\ &{} d_{2j} {\tilde{v}}_j'' + c_j{\tilde{v}}_j' + [g(b)- g({\hat{U}}_j)] - \beta {\tilde{v}}_j = 0. \end{aligned} \end{array}\right. } \end{aligned}$$

(A.9)

Since $(u_j, v_j)$ converges pointwise to $(U(\cdot + \zeta _0), V(\cdot + \zeta _0))$ on ${\mathbb {R}}$ as $j \rightarrow \infty$ and $U(\xi ) \rightarrow b$ as $\xi \rightarrow \infty$, there exist $j_0 \in {\mathbb {N}}$ and $\xi _0 > 0$ such that

$$\begin{aligned} \displaystyle b - \frac{\delta ^*}{2}< u_j(\xi _0) < b ~~~\forall \, j \ge j_0. \end{aligned}$$

Together with the monotonicity of $u_j$ and the fact that $u_j < b$ on ${\mathbb {R}}$, the above inequality implies

$$\begin{aligned} \displaystyle b - \frac{\delta ^*}{2}< u_j(\xi ) < b ~~~\forall \, \xi \ge \xi _0 \,\text { and }\, j \ge j_0. \end{aligned}$$

On the other hand, from the mean value theorem it follows that

$$\begin{aligned} g(b) - g({\hat{U}}_j(\xi )) = g'(W_j(\xi )){\tilde{u}}_j(\xi ) ~~~\forall \, \xi \ge \xi _0 \,\text { and }\, j \ge j_0. \end{aligned}$$

for some function $W_j: [\xi _0, \infty ) \rightarrow {\mathbb {R}}$ satisfying $b - \delta ^*/2< {\hat{U}}_j(\xi ) = u_j(\xi ) \le W_j(\xi ) < b$ for all $\xi \ge \xi _0$ and $j \ge j_0$. Then by the choice of $\delta ^*$ and $G^+$ [see (2.5) and (2.6)], (A.9) gives the system of inequalities

$$\begin{aligned} {\left\{ \begin{array}{l} \begin{aligned} &{} d_1{\tilde{u}}_j'' + c_j{\tilde{u}}_j' -{\tilde{u}}_j + \alpha {\tilde{v}}_j = 0, \\ &{} d_{2j} {\tilde{v}}_j'' + c_j{\tilde{v}}_j' + G^+ {\tilde{u}}_j - \beta {\tilde{v}}_j > 0 \end{aligned} \end{array}\right. } \xi \ge \xi _0, \end{aligned}$$

for all $j \ge j_0$. Now using the argument of Step 1–4 and the above system, one can establish the assertion (ii) of this lemma for $\xi \ge \xi _0$ and $j\ge j_0$. The proof of this lemma is thus completed. $\square$

1.2.2 Compactness

Lemma A.6

Let the assumptions and notations of Lemma A.3 be in force. Then there exist $M > 0$, $\lambda < 0$, $j_0\in {\mathbb {N}}$, and $\xi _0 > 0$, independent of $d_{2j}$, $j\in {\mathbb {N}}$, such that

$$\begin{aligned}&\text {either} \quad |U^*(\xi ;d_{2j})| \le M e^{\lambda |\xi |},~|V^*(\xi ;d_{2j})| \le M e^{\lambda |\xi |}~~\forall \, |\xi | \ge \xi _0 \,\text { and }\, j \ge j_0, \\&\quad \text {or} \quad |{\hat{U}}^*(\xi ;d_{2j})| \le M e^{\lambda |\xi |},~|{\hat{V}}^*(\xi ;d_{2j})| \le M e^{\lambda |\xi |}~~\forall \, |\xi | \ge \xi _0 \,\text { and }\, j \ge j_0. \end{aligned}$$

Proof

We assume that the first convergence result of Lemma A.3 holds, and prove that the first inequality of the assertion holds. The other case can be proven similarly. (Also see the proof of Lemma A.5 for this case.) For simplicity, we write $(U^*(\cdot ;d_{2j}),V^*(\cdot ;d_{2j}))$ as $(u_j,v_j)$. Recall the normalized condition $\sup _{\xi \in {\mathbb {R}}} u_j(\xi ) = 1$, and $\sup _{\xi \in {\mathbb {R}}} v_j(\xi ) \le \alpha /\beta$ by Lemma A.2. The proof consists of five steps. The first convergence result for $\xi \ge 0$ will be proven by Step 1–4, while that for $\xi \le 0$ will be proven in Step 5. More precisely, the first and second steps give the construction of upper functions on $[0,\infty )$, while the third and fourth steps show that the quantities involved in the upper functions are uniformly bounded. Finally, in the fifth step, the fact that the uniform convergence of the sequence $\{U(\cdot ;d_{2j})\}_{j\in {\mathbb {N}}}$ to $U(\cdot )$ and the arguments in Step1–4 give the bound for $(u_j,v_j)$ on $(-\infty ,0]$.

Step 1. The governing equations for upper functions $({{\bar{u}}}_j^l, {{\bar{v}}}_j^l)$ of $(u_j,v_j)$ in the interval [0, l] for each $l>0$. Since $b-\delta ^*/ 2 \le U(\cdot ;d_{2j})(\xi ) \le b$ for all $\xi \ge 0$ and $j \in {\mathbb {N}}$, by the choice of $G^+$ [see (2.5)] we have

$$\begin{aligned} G_j(\xi ) = g'(U(\cdot ;d_{2j})(\xi )) < G^+~~~~\forall \, \xi \ge 0 \text{ and } j \in {\mathbb {N}}. \end{aligned}$$

This yields that for all $\xi \ge 0$ and $j \in {\mathbb {N}}$,

$$\begin{aligned} {\left\{ \begin{array}{l} \begin{aligned} &{} d_1u_j'' - c_ju_j' -u_j + G^+ v_j > 0, \\ &{} d_{2j} v_j'' - c_jv_j' + \alpha u_j - \beta v_j = 0. \end{aligned} \end{array}\right. } \end{aligned}$$

(A.10)

Then, by comparison principle, for each $l > 0$ and $\xi \in (0,l)$, the inequalities

$$\begin{aligned} u_j \le {{\bar{u}}}_j^l \text{ and } v_j \le {{\bar{v}}}_j^l \end{aligned}$$

(A.11)

hold for $j \in {\mathbb {N}}$ where $({{\bar{u}}}_j^l, {{\bar{v}}}_j^l)$ is a positive solution of the problem

$$\begin{aligned} {\left\{ \begin{array}{l} \begin{array}{l} d_1u'' - c_ju' -u + G^+ v = 0, \\ d_{2j}v'' - c_jv' + \alpha u - \beta v = 0, \end{array} \quad 0< \xi < l \\ \; {{\bar{u}}}^l(0) = u^{l,-}_j, \quad {{\bar{v}}}^l(0) = v^{l,-}_j,\\ \; {{\bar{u}}}^l(l) = u^{l,+}_j, \quad {{\bar{v}}}^l(l) = v^{l,+}_j, \end{array}\right. } \end{aligned}$$

(A.12)

where $u_j^{l,\pm } \ge 1$ and $v_j^{l,\pm } \ge \alpha /\beta$ are constants to be determined later.

Step 2. Construction of upper functions $({{\bar{u}}}_j^l, {{\bar{v}}}_j^l)$. Now we seek a positive solution of problem (A.12). Assume that $({{\bar{u}}}_j^l, {{\bar{v}}}_j^l) = (u_{0,j}, v_{0,j})e^{\lambda _j \xi }$ is a solution of problem (A.12). Then with a direct computation, $(u_{0,j}, v_{0,j}, \lambda _j)$ satisfies

$$\begin{aligned} {\left\{ \begin{array}{l} \begin{aligned} &{} Q_j(\lambda _j) = P_{1j}(\lambda _j)P_{2j}(\lambda _j) - \alpha G^+ \\ &{} \quad := (d_1 \lambda _j^2 - c_j\lambda _j - 1) (d_{2j} \lambda _j^2 - c_j \lambda _j -\beta ) - \alpha G^+ = 0, \\ &{} P_{1j}(\lambda _j)u_{0,j} = -G^+ v_{0,j}, \quad P_{2j}(\lambda _j)v_{0,j} = -\alpha u_{0,j}. \end{aligned} \end{array}\right. } \end{aligned}$$

(A.13)

For each $j \in {\mathbb {N}}$, set

$$\begin{aligned} \begin{aligned}&\lambda _{1,j}^\pm = \big (c_j \pm \sqrt{c_j^2 + 4d_1} \big )/2d_1, \quad \lambda _{2,j}^\pm = \big (c_j \pm \sqrt{c_j^2 + 4d_{2j} \beta } \big )/2d_{2j} \\&\lambda _{M,j}^\pm = \max \big \{\lambda _{1,j}^\pm , \lambda _{2,j}^\pm \big \}, \qquad \lambda _{m,j}^\pm = \min \big \{\lambda _{1,j}^\pm , \lambda _{2,j}^\pm \big \}. \end{aligned} \end{aligned}$$

Note that for $i=1,2$, $\lambda _{i,j}^-< 0 < \lambda _{i,j}^+$, $P_{1j}(\lambda _{1,j}^\pm )=P_{2j}(\lambda _{2,j}^\pm )=0$, and

$$\begin{aligned} Q_j(\pm \infty ) = \infty , \; Q_j(0)= \beta - \alpha G^+ > 0,\; Q_j(\lambda _{M,j}^\pm ) = Q_j(\lambda _{m,j}^\pm ) = - \alpha G^+ < 0. \end{aligned}$$

Then $Q_j(\lambda )$ has four real roots in the following intervals:

$$\begin{aligned} (-\infty , \lambda _{m,j}^-), \; (\lambda _{M,j}^-, 0), \; (0, \lambda _{m,j}^+), \; (\lambda _{M,j}^+, \infty ). \end{aligned}$$

Let $\lambda _j^-$ and $\lambda _j^+$ be the roots of $Q_j(\lambda )$ in the interval $(\lambda _{M,j}^-, 0)$ and $(0, \lambda _{m,j}^+)$, respectively. Then $P_{1j}(\lambda _j^+) < 0$ and $P_{2j}(\lambda _j^-) < 0$, and

$$\begin{aligned} \left( \begin{array}{c} u_{0,j}^+ \\ v_{0,j}^+ \\ \lambda _j^+ \end{array} \right) = \left( \begin{array}{c} G^+ \\ -P_{1j}(\lambda _j^+) \\ \lambda _j^+ \end{array} \right) \text{ and } \left( \begin{array}{c} u_{0,j}^- \\ v_{0,j}^- \\ \lambda _j^- \end{array} \right) = \left( \begin{array}{c} -P_{2j}(\lambda _j^-) \\ \alpha \\ \lambda _j^- \end{array} \right) \end{aligned}$$

(A.14)

are solutions of (A.13). Set

$$\begin{aligned} M_j^+ = \max \big \{ 1/u_{0,j}^+, \alpha /(\beta v_{0,j}^+) \big \}, \quad M_j^- = \max \big \{ 1/u_{0,j}^-, \alpha /(\beta v_{0,j}^-) \big \}. \end{aligned}$$

Then for each $j \in {\mathbb {N}}$ and $l>0$,

$$\begin{aligned} \begin{aligned} \left( \begin{array}{c} {{\bar{u}}}_j^l(\xi ) \\ {{\bar{v}}}_j^l(\xi ) \end{array} \right)&= M_j^+ \Big (\frac{1 - e^{\lambda _j^- l}}{e^{\lambda _j^+ l} - e^{\lambda _j^- l}} \Big ) e^{\lambda _j^+\xi } \left( \begin{array}{c} u_{0,j}^+ \\ v_{0,j}^+ \end{array} \right) \\&\quad +\, M_j^- \Big (\frac{ e^{\lambda _j^+ l} - 1}{e^{\lambda _j^+ l} - e^{\lambda _j^- l}} \Big ) e^{\lambda _j^-\xi } \left( \begin{array}{c} u_{0,j}^- \\ v_{0,j}^- \end{array} \right) \end{aligned} \end{aligned}$$

is a solution of problem (A.12), and ${{\bar{u}}}_j^l(0), {{\bar{u}}}_j^l(l) \ge 1$ and ${{\bar{v}}}_j^l(0), {{\bar{v}}}_j^l(l) \ge \alpha /\beta$. Now for each $\xi \ge 0$,

$$\begin{aligned} \left( \begin{array}{c} {{\bar{u}}}_j^l(\xi ) \\ {{\bar{v}}}_j^l(\xi ) \end{array} \right) \rightarrow M_j^- e^{\lambda _j^- \xi } \left( \begin{array}{c} u_{0,j}^- \\ v_{0,j}^- \end{array} \right) \text{ as } l \rightarrow \infty . \end{aligned}$$

Hence, by fixing $\xi > 0$ and taking the limit in (A.11) as $l \rightarrow \infty$,

$$\begin{aligned} u_j(\xi ) \le M_j^- u_{0,j}^- e^{\lambda _j^-\xi } \; \text{ and } \; v_j(\xi ) \le M_j^- v_{0,j}^- e^{\lambda _j^-\xi } \quad \forall \, \xi \ge 0, \; j \in {\mathbb {N}}. \end{aligned}$$

(A.15)

Step 3. We show that there exist ${\underline{\lambda }}, {\bar{\lambda }} > 0$ such that

$$\begin{aligned} 0 < {\underline{\lambda }} \le |\lambda _j^{-}| \le {\bar{\lambda }} \quad \forall \, j\in {\mathbb {N}}. \end{aligned}$$

The proof of this step is similar to that of Step 3 of Lemma A.5, and thus is omitted.

Step 4. We claim that the sequences $\{ M_j^- \}_{j\in {\mathbb {N}}}$, $\{ u_{0,j}^- \}_{j\in {\mathbb {N}}}$ and $\{ v_{0,j}^- \}_{j\in {\mathbb {N}}}$ are uniformly bounded. By Step 3 and the fact that $\{ c_j \}_{j\in {\mathbb {N}}}$ is bounded, it suffices to prove that there exists a $\varepsilon _0 > 0$, independent of $j\in {\mathbb {N}}$, such that

$$\begin{aligned} P_{2j}(\lambda _j^-)< -\varepsilon _0 \quad \forall \, j \in {\mathbb {N}}. \end{aligned}$$

Then following the lines of Step 4 of Lemma A.5, we can deduce that

$$\begin{aligned} Q_{j_n}(\lambda _{j_n}^-) \le \frac{-\alpha G^+}{2} \quad \forall \, n \ge N_0 \end{aligned}$$

for some large $N_0 \in {\mathbb {N}}$. This contradicts to the fact that $Q_j(\lambda _j^-) = 0$ for all $j \in {\mathbb {N}}$. Thus the assertion of the claim is proven.

Step 5. We finish the proof. Since $(U(\cdot ;d_{2j}), V(\cdot ;d_{2j}))$ converges to $(U(\cdot ), V(\cdot ))$ as $j\rightarrow \infty$ uniformly on ${\mathbb {R}}$ and $U(\xi ) \rightarrow 0$ as $\xi \rightarrow -\infty$ (see Lemma A.3), there exist $j_0 \in {\mathbb {N}}$ and $\xi _0 < 0$ such that

$$\begin{aligned} 0< U(\xi ;d_{2j}) < \delta ^*/2 ~~~\forall \, \xi \le \xi _0 \,\text { and }\, j \ge j_0. \end{aligned}$$

With the choice of $G^-$ [see (2.5) and (2.6)], this implies that

$$\begin{aligned} G_j(\xi ) = g'(U(\xi ;d_{2j})) < G^- ~~~\forall \, \xi \le \xi _0 \,\text { and }\, j \ge j_0. \end{aligned}$$

Then proceeding along the same line of reasoning as in the previous steps, we can conclude that there exists a $M>0$ and $\lambda _0 > 0$, independent of $j\in {\mathbb {N}}$, such that

$$\begin{aligned} u_j(\xi ) \le M e^{\lambda _0 \xi } \,\text { and }\, v_j(\xi ) \le M e^{\lambda _0 \xi } \quad \forall \, \xi \le \xi _0 \,\text { and }\, j \ge j_0. \end{aligned}$$

The proof of this lemma is completed. $\square$

Lemma A.7

Let $d_1, d_2 > 0$. Suppose that $(U^*,V^*)$ is the positive solution of the adjoint problem (3.8) with respect to the traveling wave solution (U, V, c) of (1.1). Then there exists a constant K, independent of $d_2 > 0$, such that

$$\begin{aligned} \Vert U^*\Vert _{C^3 ({\mathbb {R}})} \le K, \quad \Vert V^*\Vert _{C^2 ({\mathbb {R}})} \le K \quad \forall \, d_2 > 0. \end{aligned}$$

Proof

For simplicity, we write $(U^*,V^*)$ as (u, v). Recall the normalized condition $\sup _{\xi \in {\mathbb {R}}} |u(\xi )| = 1$ and $\sup _{\xi \in {\mathbb {R}}}v(\xi ) \le \alpha /\beta$ by Lemma A.2. Since $u'(\xi ) \rightarrow 0$ as $|z| \rightarrow \infty$, $|u'|$ takes its global maximum at finite value, say $\xi = \xi _ 1$. Choose $\xi _3 < \xi _4$ such that $\xi _1 \in (\xi _3 , \xi _4)$, $u'(\xi _3) = u'(\xi _4) = 0$, and $u'(\xi ) \not =0$ for $z\in (\xi _3 , \xi _4)$. Note that $\xi _3$ can be $-\infty$ and $\xi _4$ can be $+\infty$. Then for any $\xi \in (\xi _3 , \xi _4)$, it follows from the mean-value theorem of integrals that

$$\begin{aligned} \displaystyle \left| \int _{\xi _3}^{\xi } G(\zeta )vu' d \zeta \right|= & {} \left| G(\xi ^*)v(\xi ^*)\displaystyle \int _{\xi _3}^{\xi } u' d \zeta \right| \nonumber \\ \displaystyle= & {} \left| G(\xi ^*)v(\xi ^*) \big ( u(\xi )-u(\xi _3) \big )\right| \nonumber \\\le & {} G(\xi ^*)v(\xi ^*) \big ( u(\xi ) + u(\xi _3) \big ) \nonumber \\\le & {} (2 \alpha /\beta ) \displaystyle \sup _{0 \le u \le b} g'(u) = : K_1 \end{aligned}$$

(A.16)

where $\xi ^* \in (\xi _3 ,\xi )$. Next, multiplying Eq. (3.8a) by $u'$ and integrating from $\xi _3$ to $\xi _4$, we obtain

$$\begin{aligned} 0 = \frac{d_1}{2}(u'(\xi ))^2 \Big | _{\xi _3}^{\xi _4} - c \int _{\xi _3}^{\xi _4} (u'(\zeta ))^2 d \zeta - \frac{1}{2} \big (u(\xi _4)^2 - u(\xi _3)^2\big ) + \int _{\xi _3}^{\xi _4} G(\zeta ) vu' d \zeta , \end{aligned}$$

which, together with (A.16), implies

$$\begin{aligned} \displaystyle |c| \int _{\xi _3}^{\xi _4} (u'(\zeta ))^2 d \zeta= & {} \Big | \displaystyle -\frac{1}{2} \big (u(\xi _4)^2 - u(\xi _3)^2 \big ) + \int _{\xi _3}^{\xi _4} G(\zeta ) vu' d \zeta \Big | \\\le & {} \displaystyle \frac{1}{2} \big (u(\xi _4)^2 + u(\xi _3)^2\big )+ \left| \int _{\xi _3}^{\xi _4} G(\zeta ) vu' d \zeta \right| \\\le & {} \displaystyle \frac{1}{2}(1 + 1) + K_1 = 1 + K_1. \end{aligned}$$

This, together with (A.16), in turn gives

$$\begin{aligned} \displaystyle \frac{d_1}{2} \big (u'(\xi _1) \big )^2= & {} \displaystyle \frac{d_1}{2} \big (u'(\xi ) \big )^2 \Big | _{\xi _3}^{\xi _1} \\ \displaystyle= & {} c \int _{\xi _3}^{\xi _1} (u'(\zeta ))^2 d \zeta + \frac{1}{2} \big ( u(\xi _1)^2 - u(\xi _3)^2 \big ) - \int _{\xi _3}^{\xi _1}G(\zeta ) vu' d \zeta \nonumber \\ \displaystyle\le & {} |c| \int _{\xi _3}^{\xi _4} (u'(\zeta ))^2 d \zeta + 1 + K_1 \displaystyle \le 2(1+K_1), \end{aligned}$$

and so

$$\begin{aligned} \displaystyle \sup _{z\in {\mathbb {R}}}|u'(\xi )| = |u'(\xi _1)| \le \displaystyle 2 \sqrt{1+K_1}/\sqrt{d_1} =: K_2. \end{aligned}$$

(A.17)

Let C be given in Lemma A.1. Next, from Eq. (3.8a), Lemmas A.2 and A.3, and the estimate (A.17), it follows that for any $\xi \in {\mathbb {R}}$,

$$\begin{aligned} | u''(\xi ) | = \frac{1}{d_1}|cu' + u - G(\xi ) v| \le \displaystyle \frac{1}{d_1} \big (\displaystyle CK_2 +1 + \frac{ \alpha }{\beta } \sup _{0 \le u \le b} g'(u) \big )=: K_3.\nonumber \\ \end{aligned}$$

(A.18)

Now, we estimate $v'$. Since $v'(\xi ) \rightarrow 0$ as $|\xi | \rightarrow \infty$, $v'$ takes its global minimum and global maximum at finite values, say $\xi = \xi _m$ and $\xi = \xi _M$, respectively. Differentiating Eq. (3.8b) with respect to $\xi$ and using $v''(\xi _m) = 0$ and $v''(\xi _M) = 0$, we have

$$\begin{aligned} 0 \le d_2 v'''(\xi _m) = \beta v'(\xi _m) - \alpha u'(\xi _m), \end{aligned}$$

and

$$\begin{aligned} 0 \ge d_2 v'''(\xi _M) = \beta v'(\xi _M) - \alpha u'(\xi _M). \end{aligned}$$

These two inequalities, together with (A.17), give

$$\begin{aligned} v'(\xi _m) \ge \frac{\alpha }{\beta } u'(\xi _m) \ge - \frac{\alpha }{\beta }K_2 \,\text { and }\, v'(\xi _M) \le \frac{\alpha }{\beta } u'(\xi _M) \le \frac{\alpha }{\beta }K_2. \end{aligned}$$

Thus, we obtain

$$\begin{aligned} \sup _{\xi \in {\mathbb {R}}}|v'(\xi )| \le \displaystyle \frac{\alpha }{\beta }K_2. \end{aligned}$$

(A.19)

Next, differentiating Eq. (3.8b) with respect to $\xi$ and using the estimate (A.18), it follows from similar arguments as above that

$$\begin{aligned} \sup _{\xi \in {\mathbb {R}}}|v''(\xi )| \le \displaystyle \frac{\alpha }{\beta }K_3. \end{aligned}$$

(A.20)

Finally, differentiating Eq. (3.8) with respect to $\xi$ twice and using the estimates (A.17), (A.18), (A.19), and (A.20), we obtain uniform estimates for $u'''$ and $v'''$. Hence the proof of this lemma is completed. $\square$

1.2.3 Final proof of Lemma 3.2

Let (U, V, c) be the traveling wave solution of (1.1) with $d_2=0$, $c\not =0$, and $U(0)=b - \delta ^*/2$. Using Lemma A.3, we can choose a sequence of traveling wave solutions $(U_j,V_j,c_j)$ of (1.1) with $d_2=d_{2j}$ satisfying that $d_{2j} \rightarrow 0$ and $c_j \rightarrow c$ as $j \rightarrow \infty$, and that there exists a $\zeta _0\in {\mathbb {R}}$ such that

$$\begin{aligned}&(U_j(\cdot ), V_j(\cdot )) \rightarrow (U(\cdot +\zeta _0), V(\cdot +\zeta _0))\text { uniformly on }{\mathbb {R}}\text { as }j \rightarrow \infty \text {, and} \\&\quad \text {either }\ U(\zeta _0) = U_j(0) = \delta ^*/2 \, \forall \, j\in {\mathbb {N}},\\&\quad \text {or} \, U(\zeta _0) = U_j(0) = b - \delta ^*/2 \, \forall \, j\in {\mathbb {N}}. \end{aligned}$$

Note that $\zeta _0 = 0$ in the latter case. We assume that the former case holds since the latter one can be handled similarly. For each $j\in {\mathbb {N}}$, let $(U^*_j,V^*_j)$ be the positive solution of the adjoint problem (3.8) with respect to the traveling wave solution $(U_j,V_j,c_j)$. Also let $\xi _j\in {\mathbb {R}}$ be such that $U_j^*(\xi _j)=\sup _{\xi \in {\mathbb {R}}}U_j^*(\xi )=1$. Now from Lemma A.6, it follows that the sequence $\{\xi _j\}_{j\in {\mathbb {N}}}$ is uniformly bounded. Then by choosing a subsequence of $\{\xi _j\}_{j\in {\mathbb {N}}}$ if necessary, we may assume that $\xi _j \rightarrow \xi _0$ as $j \rightarrow \infty$ for some $\xi _0\in {\mathbb {R}}$. Next, by Lemma A.7 and the Arzelà-Ascoli theorem, $\{(U_j^*,V_j^*)\}_{j\in {\mathbb {N}}}$ is precompact in $C^2([-L,L])$ for each $L>0$. Hence, there exist a pair of nonnegative functions $(U^*,V^*)$ and a subsequence $\{(U_{j_n}^*,V_{j_n}^*)\}_{n\in {\mathbb {N}}}$ of $\{(U_j^*,V_j^*)\}_{j\in {\mathbb {N}}}$ such that as $n\rightarrow \infty$, $(U_{j_n}^*,V_{j_n}^*)$ converges to $(U^*,V^*)$ uniformly on any compact set of ${\mathbb {R}}$ together with its derivatives up to second order. Using Lemma A.7 and passing to the limit in Eqs. (3.8a)–(3.8b) (with $d_2=d_{2j_n}$ and $G(\cdot )=g'(U_{j_n}(\cdot ))$), $(U^*,V^*)$ is a solution of the following degenerate adjoint problem:

$$\begin{aligned} d_1u'' - cu' - u + g'(U(\xi +\zeta _0)) v= & {} 0, \end{aligned}$$

(A.21a)

$$\begin{aligned} - cv' + \alpha u - \beta v= & {} 0, \end{aligned}$$

(A.21b)

subject to the boundary value condition

$$\begin{aligned} (u,v)(- \infty ) = 0 ,~(u,v)(+\infty ) = 0 ,~u(\xi _0) = 1, \end{aligned}$$

(A.21c)

where the prime denotes differentiation with respect to $\xi$. In the following, we will show that $U^* > 0$ and $V^* >0$ in ${\mathbb {R}}$.

Indeed, recall that $U^* \ge 0$ and $V^* \ge 0$ on ${\mathbb {R}}$. Suppose that there exists $z _0 \in {\mathbb {R}}$ such that

$$\begin{aligned} U^*(z _0) = 0 \, \text{ or } \, V^*(z _0) = 0. \end{aligned}$$

For the former case where $U^*(z _0) = 0$, we have $(U^*)' (z _0) = 0$ and $(U^*)''(z _0) \ge 0$. From the nonnegativity of each term in Eq. (A.21a), it follows that $(U^*)''(z _0) = g'(U(z_0+\zeta _0)) V^*(z_0) = 0$. We have two disjoint subcases: (i) $g'(U(z_0+\zeta _0))=0$ and (ii) $g'(U(z_0+\zeta _0))\not =0$. For case (i), since $U > 0$ on ${\mathbb {R}}$, $g'(U(z_0+\zeta _0)) > 0$. This is a contradiction. For case (ii), from Eq. (A.21a), we have $V^*(z _0) = 0$.

For the later case where $V^*(z _0) = 0$, we have $(V^*)'(z _0) = 0$ since $V^*\ge 0$ on ${\mathbb {R}}$. Then from Eq. (A.21b), it follows that $U^*(z _0) = 0$, and thus that $(U^*)'(z _0) = 0$. Therefore, both of the cases give that

$$\begin{aligned} U^*(z _0) = 0, ~ (U^*)' (z _0) = 0,~ V^*(z _0) = 0. \end{aligned}$$

It then follows from the uniqueness theorem of differential equations that $(U^*,V^*) \equiv (0,0)$ on ${\mathbb {R}}$. This contradicts to the fact that $U^*(\xi _0) = 1$. Hence we have $U^* > 0$ and $V^* > 0$ on ${\mathbb {R}}$. This completes the proof of Lemma 3.2.