Time Periodic Traveling Waves for a Periodic and Diffusive SIR Epidemic Model

Abstract

In this paper, we study the time periodic traveling wave solutions for a periodic SIR epidemic model with diffusion and standard incidence. We establish the existence of periodic traveling waves by investigating the fixed points of a nonlinear operator defined on an appropriate set of periodic functions. Then we prove the nonexistence of periodic traveling via the comparison arguments combined with the properties of the spreading speed of an associated subsystem.

1 Introduction

The investigation on traveling wave solutions for various evolution systems arising in biology, chemistry, epidemiology and physics has received increasing interest, see, e.g., [6, 16, 35, 36, 47, 51, 56] and references therein. As a basic but important subject, the existence of traveling wave solutions has been widely studied. For autonomous monotone evolution systems, by standard approaches such as monotone iteration, comparison arguments or monotone semiflow, the theory of traveling wave solutions has been well developed, see, e.g., [13, 14, 28, 42, 48] and references therein. Meanwhile, there are a few results on the existence of traveling wave solutions for nonautonomous (in particular, time periodic) monotone systems: Alikakos et al. [1] established the existence and global stability of time periodic traveling wave solutions (see the form of (1.3)) for periodic reaction–diffusion equations with bistable nonlinearities; Liang et al. [27] extended the theory of spreading speeds and traveling waves for monotone autonomous semiflows to periodic semiflows in the monostable case; Fang and Zhao [14] developed the theory of traveling waves for monotone semiflows with bistable structure and applied it to time-periodic evolution system; Zhao and Ruan [54, 55] studied the existence, uniqueness and asymptotic stability of time periodic travelling wave solutions to periodic reaction–diffusion, advection–reaction–diffusion Lotka–Volterra competition systems, respectively; For bistable periodic traveling waves of periodic and diffusive Lotka–Volterra competition system, we refer to Bao and Wang [2]. More recently, Fang et al. [15] developed the theory of traveling waves and spreading speeds for time-space periodic monotone semiflows with monostable structure and applied the abstract results to a two species competition reaction–advection–diffusion model.

It is well known that many nonlinear reaction–diffusion systems modeling interaction of multi-species, such as predator and prey, the disease transmission among the susceptible individuals and infective individuals, combustion and the chemical reaction, etc., are non-monotone. Due to the lack of the comparison principle and monotonic properties for such evolution systems, the study of traveling waves is very challenging, and the related research is very limited. In the pioneering work of Dunbar [10, 11], the shooting argument was applied to prove the existence of traveling waves for a classical Lotka–Volterra predator–prey model. This method is also used in [25, 30] for predator–prey systems with different functional response, and in [19, 20] for classical Kermack–McKendrick SIR models. Huang [21] further developed the method in [10, 11] to provide a more effective way to obtain traveling waves for a large class of predator–prey systems. Based on a fixed-point problem and the limiting argument, Ducrot and Magal [8] and Ducrot et al. [9] studied the existence of traveling waves for an infection-age structured Kermack–McKendrick model with diffusion. Motivated by the method in [8, 9], there were also some works involving in traveling waves for a bio-reaction model [44], an H5N1 arian influenza model [46], and nonlocal dispersal Kermack–Mckendrick models [26, 49, 50]. By constructing an invariant cone and applying Schauder’s fixed point theorem, Wang and Wu [45] obtained the existence of travelling waves for a class of diffusive Kermack–McKendrick SIR models with non-local and delayed disease transmission (see also [38, 39]). Schauder’s fixed point theorem is also applied for the existence of traveling waves for evolution systems without monotonicity, see, e.g., [24, 29, 33, 34, 37, 52]. More recently, Huang [22] presented a geometrical approach to investigate the existence of traveling waves and their minimum wave speed for non-monotone reaction–diffusion systems, which include the models of predator–prey interaction, the combustion, Belousov-Zhabotinskii reaction, SI-type of disease transmission, and biological flow reactor in chemostat. Zhang et al. [53] introduced the concept of weak traveling waves and obtained the necessary and sufficient conditions for the existence of such solutions for a class of non-cooperative diffusion-reaction systems. Fu and Tsai [18] employed an iteration process to construct a set of super/sub-solutions to establish the existence of a family of traveling waves with the minimum speed.

However, there are very few investigations on the time periodic traveling wave solutions for periodic non-monotone evolution systems. The purpose of this paper is to study time periodic traveling waves for the following periodic and diffusive SIR model with standard incidence:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial }{\partial t}S(t,x)=d_1\Delta S(t,x)-\frac{\beta (t)S(t,x)I(t,x)}{S(t,x)+I(t,x)},\\ \frac{\partial }{\partial t}I(t,x)=d_2\Delta I(t,x)+\frac{\beta (t)S(t,x)I(t,x)}{S(t,x)+I(t,x)}-\gamma (t)I(t,x),\\ \frac{\partial }{\partial t}R(t,x)=d_3\Delta R(t,x)+\gamma (t)I(t,x), \end{array}\right. } \end{aligned}$$

(1.1)

where S(t, x), I(t, x) and R(t, x) denote the densities of the susceptible, infected and removed individuals at time t and in location x,  respectively. Further, $d_1, d_2$ and $d_3$ are the diffusion rates for the susceptible, infected and removed individuals, respectively. The infection rate $\beta$ and the removal rate $\gamma$ are positive T-periodic continuous functions of t. Here the incidence reflects the recovered individuals are removed from the population, and not involved in the contact and disease transmission, see [5, 40]. Since the equation for R of system (1.1) is decoupled from the equations for S and I,  it suffices to consider a two-dimensional system for S and I:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial }{\partial t}S(t,x)=d_1\Delta S(t,x)-\frac{\beta (t)S(t,x)I(t,x)}{S(t,x)+I(t,x)},\\ \frac{\partial }{\partial t}I(t,x)=d_2\Delta I(t,x)+\frac{\beta (t)S(t,x)I(t,x)}{S(t,x)+I(t,x)}-\gamma (t)I(t,x). \end{array}\right. } \end{aligned}$$

(1.2)

Time periodic traveling waves to system (1.2) are defined to be solutions of the form

$$\begin{aligned} \begin{pmatrix} S(t,x)\\ I(t,x) \end{pmatrix} =\begin{pmatrix} \phi (t,x+ct)\\ \psi (t,x+ct) \end{pmatrix},\ \ \begin{pmatrix} \phi (t+T,z)\\ \psi (t+T,z) \end{pmatrix} =\begin{pmatrix} \phi (t,z)\\ \psi (t,z) \end{pmatrix} \end{aligned}$$

(1.3)

satisfying

$$\begin{aligned} \begin{pmatrix} \phi (t,\pm \infty )\\ \psi (t,\pm \infty ) \end{pmatrix}= \begin{pmatrix} \phi _\pm (t)\\ \psi _\pm (t) \end{pmatrix}, \end{aligned}$$

where c is called the wave speed, $z=x+ct$ is the moving coordinate, and $\begin{pmatrix}\phi _+(t)\\ \psi _+(t)\end{pmatrix}$ and $\begin{pmatrix}\phi _-(t)\\ \psi _-(t)\end{pmatrix}$ are two periodic solutions of the corresponding kinetic system:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{dS}{dt}=-\frac{\beta (t)S(t)I(t)}{S(t)+I(t)},\\ \frac{dI}{dt}=\frac{\beta (t)S(t)I(t)}{S(t)+I(t)}-\gamma (t)I(t). \end{array}\right. } \end{aligned}$$

(1.4)

The profile $(\phi ,\psi )$ then solves the following time periodic parabolic system:

$$\begin{aligned} {\left\{ \begin{array}{ll} \phi _t(t,z)=d_1\phi _{zz}(t,z)-c\phi _z(t,z)-\frac{\beta (t)\phi (t,z)\psi (t,z)}{\phi (t,z)+\psi (t,z)}, \ \ (t,z)\in \mathbb {R}\times \mathbb {R},\\ \psi _t(t,z)=d_2\psi _{zz}(t,z)-c\psi _z(t,z)+\frac{\beta (t)\phi (t,z)\psi (t,z)}{\phi (t,z)+\psi (t,z)}-\gamma (t)\psi (t,z), \ \ (t,z)\in \mathbb {R}\times \mathbb {R}. \end{array}\right. } \end{aligned}$$

(1.5)

Since the periodic system (1.2) does not admit the comparison principle, the theory and methods developed for monotone periodic systems (see, e.g., [14, 27, 54, 55]) cannot be used here. In view of the profile system (1.5), the shooting arguments (see, e.g., [10, 11, 21, 22]) for predator–prey systems do not apply to periodic system (1.2). Although Schauder’s fixed point theorem is a powerful tool to prove the existence of traveling wave solutions for autonomous evolution systems (see, e.g., [24, 33, 38, 39, 45, 53]), it may not be applied directly to periodic system (1.2). Our strategy is to reduce the existence of periodic traveling waves to a fixed point problem by constructing a non-monotone operator on an appropriate convex set of periodic functions. To obtain the nonexistence of time periodic traveling waves, we combine the comparison arguments for single equations and the properties of spreading speeds for periodic and monotone systems, which is of its own interest and may apply to other non-monotone models.

This paper is organized as follows. In Sect. 2, we first construct some appropriate sub- and super-solutions to obtain an invariant convex set, then define a nonlinear non-monotone operator on it, and finally apply Schauder’s fixed point theorem to get the existence of periodic traveling waves. More precisely, we prove that if the basic reproduction number $R_0:=\frac{\int _0^T\beta (t)dt}{\int _0^T\gamma (t)dt}$ of the periodic kinetic system (1.4) is greater than unity, then there exists a $c^*>0$ such that for any $c\in (c^*,\infty )$, system (1.2) for S and I admits a time periodic, non-trivial and non-negative traveling wave solution with speed c. Section 3 is devoted to the nonexistence of such traveling waves for two cases where $R_0\le 1$, or $R_0>1$ and $c\in (0, c^*)$.

2 The Existence of Periodic Traveling Waves

In this section, we focus on the non-trivial and time periodic travelling waves $(\phi (t,z),$ $\psi (t,z))$ of the form (1.3). Such solutions satisfy the following system:

$$\begin{aligned} {\left\{ \begin{array}{ll} \phi _t(t,z)=d_1\phi _{zz}(t,z)-c\phi _z(t,z)-\frac{\beta (t)\phi (t,z)\psi (t,z)}{\phi (t,z)+\psi (t,z)},\\ \psi _t(t,z)=d_2\psi _{zz}(t,z)-c\psi _z(t,z)+\frac{\beta (t)\phi (t,z)\psi (t,z)}{\phi (t,z)+\psi (t,z)}-\gamma (t)\psi (t,z). \end{array}\right. } \end{aligned}$$

(2.1)

This system is posed on $(t,x)\in \mathbb {R}_+\times \mathbb {R}$ and is supplemented with the following asymptotic boundary conditions

$$\begin{aligned} \phi (t,-\infty )=S_0,\, \phi (t,\infty )=S^\infty ,\, \psi (t,\pm \infty )=0\ \mathrm{uniformly\ in}\, t\in \mathbb {R}. \end{aligned}$$

(2.2)

Here $S_0>0$ is a constant, and $(S_0,0)$ is the initial disease-free steady state. The parameter $c>0$ is the wave speed, while constant $S^\infty \ge 0$ describes the density of susceptible individuals after the epidemic. Our investigation procedure is as follows. Firstly, we construct some appropriate sub- and super-solutions that will be essential to obtain a closed and convex set $\mathcal {D}$. Note that this set contains all the bounded and uniformly continuous functions which lie between the sub- and super-solutions. Secondly, for any $({\tilde{\phi }},{\tilde{\psi }})\in \mathcal {D},$ we find a unique T-periodic solution $(\phi ^*,\psi ^*)$ to a linear integral system, and then we define a nonlinear operator $\mathcal {F}$ such that $\mathcal {F}({\tilde{\phi }},{\tilde{\psi }})=(\phi ^*,\psi ^*).$ Finally, by applying Schauder’s fixed point theorem to $\mathcal {F},$ we establish the existence of periodic traveling waves.

2.1 Construction of Sub- and Super-solutions

Linearizing system (2.1) at the disease-free steady state $(S_0,0)$, we obtain the following equation for the infective variable:

$$\begin{aligned} J_t=d_2 J_{zz}(t,z)-cJ_{z}(t,z)+(\beta (t)-\gamma (t))J(t,z). \end{aligned}$$

(2.3)

Denote $\overline{H}=\frac{1}{T}\int _0^TH(t)dt$ for any T-periodic function $H(\cdot ).$ Define

$$\begin{aligned} \Lambda _c(\lambda ):= & {} d_2\lambda ^2-c\lambda +\kappa _0,\ \ \kappa _0:=\frac{1}{T}\int _0^T[\beta (t)-\gamma (t)]dt=\overline{\beta (\cdot )- \gamma (\cdot )},\ c\in \mathbb {R},\,\lambda \in \mathbb {R},\\ Q^\lambda (t)= & {} \exp \left( \int _0^t\left[ \beta (s)-\gamma (s)\right] ds-t\kappa _0\right) . \end{aligned}$$

Clearly,

$$\begin{aligned} \kappa _0 Q^\lambda (t)=\left[ \beta (t)-\gamma (t)\right] Q^\lambda (t)- \frac{d Q^\lambda (t)}{d t}. \end{aligned}$$

We also set

$$\begin{aligned} \kappa =d_2 \kappa _0,\ \ \lambda _c=\frac{c-\sqrt{c-4\kappa }}{2d_2}\ \ \mathrm{if}\ c>c^*:=2\sqrt{\kappa }. \end{aligned}$$

In the following, we always assume that $R_0:=\frac{\int _0^T\beta (t)dt}{\int _0^T\gamma (t)dt}>1,$ and fix $c>c^*:=2\sqrt{\kappa }.$ Let $K(t)\!:=\!\exp \left( \int _0^t\left[ d_2\lambda _c^2\!-\!c\lambda _c\!+\!(\beta (s)-\gamma (s)) \right] ds\right)$. We define four functions as follows:

$$\begin{aligned} \phi ^+(t,z):= & {} S_0,\ \phi ^-(t,z):=\max \left\{ S_0\left( 1-M_1e^{\epsilon _1z}\right) ,0\right\} ,\\ \psi ^+(t,z):= & {} K(t)e^{\lambda _cz},\ \psi ^-(t,z):=\max \left\{ K(t)e^{\lambda _c z}\left( 1-M_2e^{\epsilon _2z}\right) , 0\right\} , \end{aligned}$$

where $\epsilon _1, M_1, \epsilon _2$ and $M_2$ are all positive constants and will be determined below. Then we have the following results.

Lemma 2.1

The function $\psi ^+(t,z)=K(t)e^{\lambda _cz}$ satisfies the following linear equation:

$$\begin{aligned} \psi _t=d_2\psi _{zz}-c\psi _z+(\beta (t)-\gamma (t))\psi . \end{aligned}$$

(2.4)

Lemma 2.2

Suppose $\epsilon _1$ is sufficiently small such that $0<\epsilon _1<\lambda _c$ and $M_1>1$ is sufficiently large. Then the function $\phi ^-$ satisfies

$$\begin{aligned} \phi _t-d_1\phi _{zz}+c\phi _z\le -\frac{\beta (t)\psi ^+\phi }{\psi ^++\phi } \end{aligned}$$

(2.5)

for any $z\not = z_1:=-\epsilon _1^{-1}\ln M_1.$

Proof

If $z> -\epsilon _1^{-1}\ln M_1,$ then $\phi ^-(t,z)=0,$ which implies that the inequality (2.5) holds.

If $z< -\epsilon _1^{-1}\ln M_1,$ then $\phi ^-(t,z)=S_0\left( 1-M_1e^{\epsilon _1z}\right) .$ Hence, the inequality (2.5) is equivalent to

$$\begin{aligned} d_1S_0M_1\epsilon _1^2e^{\epsilon _1z}-cS_0M_1\epsilon _1e^{\epsilon _1z}\le -\frac{\beta (t)S_0(1-M_1e^{\epsilon _1z})K(t)e^{\lambda _cz}}{S_0(1-M_1e^{\epsilon _1z})+K(t)e^{\lambda _cz}} \end{aligned}$$

for any $z< z_1:=-\epsilon _1^{-1}\ln M_1.$ Rewriting the above inequality, we have

$$\begin{aligned} S_0M_1\epsilon _1(c-d_1\epsilon _1)\ge \frac{\beta (t)S_0(1-M_1e^{\epsilon _1z})K(t)e^{(\lambda _c-\epsilon _1)z}}{S_0(1-M_1e^{\epsilon _1z})+K(t)e^{\lambda _cz}}. \end{aligned}$$

So for $z< z_1:=-\epsilon _1^{-1}\ln M_1,$ it is sufficient to verify

$$\begin{aligned} S_0M_1\epsilon _1(c-d_1\epsilon _1)\ge \beta (t)K(t)e^{-\epsilon _1^{-1}(\lambda _c-\epsilon _1)\ln M_1}=\beta (t)K(t)M_1^{-\epsilon _1^{-1}(\lambda _c-\epsilon _1)},\ \ \forall t\in \mathbb {R}. \end{aligned}$$

Note that $\beta (t)$ and K(t) are positive T-periodic functions. Thus the above inequality holds true if we choose $M_1=1/\epsilon _1$ with $\epsilon _1>0$ sufficiently small. $\square$

Lemma 2.3

Suppose $\epsilon _2>0$ sufficiently small such that $\epsilon _2<\min \{\epsilon _1,\lambda '_c-\lambda _c\},$ where $\lambda '_c:=\frac{c+\sqrt{c-4\kappa }}{2d_2},$ and $M_2$ is sufficiently large such that $-\epsilon _2^{-1}\ln M_2<-\epsilon _1^{-1}\ln M_1$. Then the function $\psi ^-$ satisfies

$$\begin{aligned} \psi _t-d_2\psi _{zz}+c\psi _z\le -\gamma (t)\psi - A[\phi ^-,\psi ] \end{aligned}$$

(2.6)

for any $z\not =z_2:=-\epsilon _2^{-1}\ln M_2,$ where $A[\phi ,\psi ](t,z)$ is defined by

$$\begin{aligned} A[\phi ,\psi ](t,z)= {\left\{ \begin{array}{ll} \quad \quad 0,\ &{}\phi (t,z)\psi (t,z)=0,\quad \forall (t,z)\in \mathbb {R}\times \mathbb {R},\\ \frac{\beta (t)\phi (t,z)\psi (t,z)}{\phi (t,z)+\psi (t,z)},\ &{} \phi (t,z)\psi (t,z)\not =0,\quad \forall (t,z)\in \mathbb {R}\times \mathbb {R}. \end{array}\right. } \end{aligned}$$

Proof

We assume that $M_2$ is sufficiently large such that $-\epsilon _2^{-1}\ln M_2<-\epsilon _1^{-1}\ln M_1.$ When $z>z_2:=-\epsilon _2^{-1}\ln M_2,$ we see that $\psi ^-(t,z)=0,$ and hence, the inequality (2.6) holds.

Let $z<z_2:=-\epsilon _2^{-1}\ln M_2.$ Then $\psi ^-(t,z)=K(t)e^{\lambda _c z}\left( 1-M_2e^{\epsilon _2z}\right)$ and $\phi ^-(t,z)=S_0(1-M_1e^{\epsilon _1z}).$ It suffices to verify

$$\begin{aligned} \psi ^-_t-d_2\psi ^-_{zz}\!+\!c\psi ^-_z\!\le \!-\gamma (t)\psi ^- -\frac{\beta (t)\phi ^-\psi ^-}{\phi ^-+\psi ^-}\!\le \! \left( \beta (t)-\gamma (t)\right) \psi ^- -\frac{\beta (t)(\psi ^-)^2}{\phi ^-+\psi ^-}.\quad \end{aligned}$$

(2.7)

In view of the expression of K(t) and $\psi ^-(t,z)$, we have that

$$\begin{aligned}&\psi ^-_t-d_2\psi ^-_{zz}+c\psi ^-_z-\left( \beta (t)-\gamma (t)\right) \psi ^-\\&\quad =K'(t)e^{\lambda _cz}\left( 1-M_2e^{\epsilon _2z}\right) -d_2\left[ \lambda _c^2K(t) e^{\lambda _cz}\left( 1-M_2e^{\epsilon _2z}\right) -\lambda _c\epsilon _2 M_2K(t)e^{(\lambda _c+\epsilon _2)z}\right. \\&\qquad \left. -(\lambda _c+\epsilon _2)\epsilon _2 M_2K(t)e^{(\lambda _c+\epsilon _2)z}\right] +c\left[ \lambda _cK(t) e^{\lambda _cz}\left( 1-M_2e^{\epsilon _2z}\right) -\epsilon _2 M_2K(t)e^{(\lambda _c+\epsilon _2)z}\right] \\&\qquad -[\beta (t)-\gamma (t)]K(t) e^{\lambda _cz}\left( 1-M_2e^{\epsilon _2z}\right) \\&\quad = e^{\lambda _cz}\left\{ K'(t)-d_2\lambda _c^2K(t)+c\lambda _cK(t)- [\beta (t)-\gamma (t)]K(t) \right\} \\&\qquad -M_2e^{(\lambda _c+\epsilon _2)z}\left\{ K'(t)-d_2(\lambda _c+\epsilon _2)^2K(t) +c(\lambda _c+\epsilon _2)K(t)- [\beta (t)-\gamma (t)]K(t) \right\} \\&\quad =-M_2e^{(\lambda _c+\epsilon _2)z}K(t)\left\{ \left[ d_2\lambda _c^2-c\lambda _c\right] -\left[ d_2(\lambda _c+\epsilon _2)^2-c(\lambda _c+\epsilon _2)\right] \right\} .\\&\quad =M_2e^{(\lambda _c+\epsilon _2)z}K(t)\cdot \Lambda _c(\lambda _c+\epsilon _2) \end{aligned}$$

Then the inequality (2.7) is equivalent to

$$\begin{aligned} M_2e^{(\lambda _c+\epsilon _2)z}K(t)\cdot \Lambda _c(\lambda _c+\epsilon _2) \le -\frac{\beta (t)K(t)e^{2\lambda _cz}\left( 1-M_2e^{\epsilon _2z}\right) ^2}{S_0\left( 1-M_1e^{\epsilon _1z}\right) +K(t)e^{\lambda _cz}\left( 1-M_2e^{\epsilon _2z}\right) }. \end{aligned}$$

(2.8)

Since $\epsilon _1<\lambda '_c-\lambda _c$, it follows that $\lambda _c+\epsilon _2\in (\lambda _c,\lambda '_c),$ and hence

$$\begin{aligned} \Lambda _c(\lambda _c+\epsilon _2)=d_2(\lambda _c+\epsilon _2)^2-c(\lambda _c +\epsilon _2)+\kappa _0<0. \end{aligned}$$

Due to the positivity and periodicity of both K(t) and $\beta (t)$ in $\mathbb {R}$, we see that the inequality (2.8) is satisfied if and only if

$$\begin{aligned}&-M_2 \Lambda _c(\lambda _c+\epsilon _2)\left[ S_0\left( 1-M_1e^{\epsilon _1z}\right) +K(t)e^{\lambda _cz}\left( 1-M_2e^{\epsilon _2z}\right) \right] \\&\quad \ge \beta (t)e^{(\lambda _c-\epsilon _2)z}\left( 1-M_2e^{\epsilon _2z}\right) ^2 \end{aligned}$$

for all $t\in [0,T].$ In terms of $z<-\epsilon _2^{-1}\ln M_2,$ we only need to show

$$\begin{aligned} -M_2 \Lambda _c(\lambda _c+\epsilon _2)S_0\left( 1-M_1M_2^{-\epsilon _1/\epsilon _2}\right) \ge \beta (t)M_2^{-(\lambda _c-\epsilon _2)/\epsilon _2}\ \ \mathrm{for\ all\ }t\in [0,T]. \end{aligned}$$

Since $\lambda _c-\epsilon _2>\lambda _c-\epsilon _1>0,$ when $M_2$ tends to infinity, the right-hand side of the last inequality tends to zero and the left-hand side of the last inequality tends to infinity, which means the last inequality holds true for large $M_2.$ $\square$

2.2 Reduction to a Fixed Point Problem

Let $X=BUC(\mathbb {R},\mathbb {R})$ be the Banach space of all bounded uniformly continuous functions from $\mathbb {R}$ into $\mathbb {R}$ with the usual suppremum norm $\Vert \cdot \Vert _{X}.$ Let

$$\begin{aligned} X^+=\{w\in X: w(x)\ge 0, x\in \mathbb {R}\}. \end{aligned}$$

Then X is a Banach lattice under the partial ordering induced by $X^+.$ It follows from [7, Theorem 1.5] that the X-realization $d\Delta _X$ of $d\Delta$ generates a strongly continuous analytic semigroup T(t) on X and $T(t)X^+\subset X^+$ for $t\ge 0.$ In addition, we have

$$\begin{aligned} \left( T(t)w\right) (x)=\frac{1}{\sqrt{4\pi dt}}\int _{\mathbb {R}}e^{-\frac{(x-y)^2}{4dt}}w(y)dy,\ \ t>0,\,x\in \mathbb {R},\,w(\cdot )\in X. \end{aligned}$$

(2.9)

For a given positive constant $\mu ,$ denote the functional space $B_{\mu }\left( [0,T]\times \mathbb {R},\mathbb {R}^2\right)$ by

$$\begin{aligned} \begin{aligned}&B_{\mu }\left( [0,T]\,\times \mathbb {R},\mathbb {R}^2\right) :=\\&\quad \left\{ u=(u_1,u_2): \begin{aligned}&u_i\in BUC([0,T]\times \mathbb {R},\mathbb {R}), \sup _{t\in [0,T],x\in \mathbb {R}} e^{-\mu \vert x\vert }|u_i(t,x)|<\infty ,\\&u_i(0,x)=u_i(T,x),\ x\in \mathbb {R},\ i=1,2. \end{aligned}\right\} \end{aligned} \end{aligned}$$

equipped with the norm

$$\begin{aligned} \Vert u\Vert _{\mu }:=\max \left\{ \sup _{t\in [0,T],x\in \mathbb {R}}e^{-\mu \vert x\vert }\vert u_1(t,x)\vert ,\ \sup _{t\in [0,T],x\in \mathbb {R}}e^{-\mu \vert x\vert }\vert u_2(t,x)\vert \right\} . \end{aligned}$$

Define a convex cone $\mathcal {D}$ as

$$\begin{aligned} \mathcal {D}=\left\{ (\tilde{\phi },\tilde{\psi })\in B_{\mu }\left( [0,T]\times \mathbb {R},\mathbb {R}^2\right) :\ \phi ^-\le \tilde{\phi }\le \phi ^+,\ \psi ^-\le \tilde{\psi }\le \min \{\psi ^+,\Lambda \}\right\} , \end{aligned}$$

where $\Lambda >0$ is sufficiently large such that $\frac{\beta (t)S_0}{S_0+\Lambda }-\gamma (t)<0$ for $t\in [0,T].$ For any given $({\tilde{\phi }},{\tilde{\psi }})\in \mathcal {D},$ define maps

$$\begin{aligned} f_1[{\tilde{\phi }},{\tilde{\psi }}](t,z)=\alpha _1\tilde{\phi }(t,z) -A[{\tilde{\phi }},{\tilde{\psi }}](t,z) \end{aligned}$$

and

$$\begin{aligned} f_2[{\tilde{\phi }},{\tilde{\psi }}](t,z)=\alpha _2\tilde{\psi }(t,z) +A[{\tilde{\phi }},{\tilde{\psi }}](t,z)-\gamma (t)\tilde{\psi }(t,z), \end{aligned}$$

where the functional A is defined as in Lemma 2.3, $\alpha _1$ and $\alpha _2$ are positive constants and satisfy $\alpha _1>\max _{t\in [0,T]}\beta (t)$ and $\alpha _2>\max _{t\in [0,T]}$ $\gamma (t),$ respectively. Fix a $({\tilde{\phi }},{\tilde{\psi }})\in \mathcal {D}.$ Consider the following parabolic initial value problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} \phi _t-d_1\phi _{zz}+c\phi _z+\alpha _1\phi =f_1[{\tilde{\phi }},{\tilde{\psi }}](t,z),\ 0<t\le T,\,z\in \mathbb {R},\\ \psi _t-d_1\psi _{zz}+c\psi _z+\alpha _2\psi =f_2[{\tilde{\phi }},{\tilde{\psi }}](t,z),\ 0<t\le T,\,z\in \mathbb {R},\\ \phi (0,z)=\phi _0(z),\ \psi (0,z)=\psi _0(z),\ \ z\in \mathbb {R}. \end{array}\right. } \end{aligned}$$

(2.10)

Rewrite (2.10) as an integral system:

$$\begin{aligned} {\left\{ \begin{array}{ll} \phi (t,z)=\left( T_1(t)\phi _0\right) (z)+\int _0^t \left( T_1(t-s)f_1[{\tilde{\phi }},{\tilde{\psi }}](s)\right) (z)ds,\\ \psi (t,z)=\left( T_2(t)\psi _0\right) (z)+\int _0^t \left( T_2(t-s)f_2[{\tilde{\phi }},{\tilde{\psi }}](s)\right) (z)ds, \end{array}\right. } \end{aligned}$$

(2.11)

where $T_i(t)$ is the analytic semigroup (see, e.g., [7], [31]) generated by the linear differential operator $A_i:D(A_i)\rightarrow C(\mathbb {R})$ defined by

$$\begin{aligned} D(A_i)=\left\{ \bigcap _{1\le p<\infty } W_{\mathrm{loc}}^{2,p}\,:\, A_iu=d_iu_{zz}-cu_z-\alpha _iu\in C(\mathbb {R})\right\} ,\ \ i=1,2. \end{aligned}$$

Moreover, $\overline{D(A)}=UC(\mathbb {R})$ (see [31, Chapter 5]), and following from (2.9), it is not difficult to obtain that

$$\begin{aligned} \left( T_i(t)w\right) (x)=e^{-\alpha _it}\frac{1}{\sqrt{4\pi dt}}\int _{\mathbb {R}}e^{-\frac{(x-ct-y)^2}{4dt}}w(y)dy,\ \ t>0,\,x\in \mathbb {R},\,w(\cdot )\in X. \end{aligned}$$

(2.12)

We note that the solution of (2.11) is the mild solution of linear system (2.10).

In what follows, we intend to prove that for any given $({\tilde{\phi }},{\tilde{\psi }})\in \mathcal {D},$ there exists a unique $(\phi ^*,\psi ^*)\in \mathcal {D}$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} \phi ^*(t)=T_1(t)\phi ^*(0)+\int _0^t T_1(t-s)f_1[{\tilde{\phi }},{\tilde{\psi }}](s)ds,\\ \psi ^*(t)=T_2(t)\psi ^*(0)+\int _0^t T_2(t-s)f_2[{\tilde{\phi }},{\tilde{\psi }}](s)ds. \end{array}\right. } \end{aligned}$$

(2.13)

Given a positive number $\eta ,$ denote a functional space ${\tilde{B}}_\eta (\mathbb {R},\mathbb {R}^2)$ by

$$\begin{aligned} {\tilde{B}}_{\eta }\left( \mathbb {R},\mathbb {R}^2\right) :=\left\{ v=(v_1,v_2): v_i\in X,\ \sup _{z\in \mathbb {R}} e^{-\eta \vert z\vert }|v_i(z)|<\infty ,\ z\in \mathbb {R},\ i=1,2.\right\} \end{aligned}$$

equipped with the norm

$$\begin{aligned} \vert v\vert _{\eta }:=\max \left\{ \sup _{z\in \mathbb {R}}e^{-\eta \vert z\vert }\vert v_1(z)\vert ,\ \sup _{z\in \mathbb {R}}e^{-\eta \vert z\vert }\vert v_2(z)\vert \right\} . \end{aligned}$$

Define

$$\begin{aligned} \tilde{\mathcal {D}}:=\left\{ (\phi _0(\cdot ),\psi _0(\cdot ))\in {\tilde{B}}_{\mu }\left( \mathbb {R},\mathbb {R}^2\right) : \begin{aligned}&\phi ^-(0,z)\le \phi _0(z)\le \phi ^+(0,z)\\&\psi ^-(0,z)\le \psi _0(z)\le \min \{\psi ^+(0,z),\Lambda \} \end{aligned}\right\} . \end{aligned}$$

Clearly, ${\tilde{\mathcal {D}}}$ is convex and closed. For a given $({\tilde{\phi }},{\tilde{\psi }})\in \mathcal {D},$ $f_i[{\tilde{\phi }},{\tilde{\psi }}](t,\cdot ),i=1,2$ belong to $C([0,T];C(\mathbb {R})).$ Moreover, $f_1[{\tilde{\phi }},{\tilde{\psi }}]$ and $f_2[{\tilde{\phi }},{\tilde{\psi }}]$ admit uniform bounds with respect to $({\tilde{\phi }},{\tilde{\psi }})\in \mathcal {D},$ respectively, uniformly for $(t,x)\in [0,T]\times \mathbb {R}.$ Thus, with the aid of [31, Theorem 5.1.2], for any $(\phi _0,\psi _0)\in \tilde{D},$ it follows that $(\phi ,\psi )$ defined by (2.11) belongs to $C([0,T]\times \mathbb {R},\mathbb {R})\cap C^{\theta ,2\theta }([\epsilon ,T]\times \mathbb {R},\mathbb {R})$ for every $\epsilon \in (0,T)$ and $\theta \in (0,1),$ and there are $C_1(\epsilon ,\theta )>0, C_2(\epsilon ,\theta )>0$ such that

$$\begin{aligned} \Vert \phi (T,\cdot )\Vert _{C^{2\theta }(\mathbb {R})}\le C_1(\epsilon ,\theta )\Big (\epsilon ^{-\theta }\Vert \phi _0\Vert _{\infty }+\Vert f_1[{\tilde{\phi }},{\tilde{\psi }}]\Vert _\infty \Big ) \end{aligned}$$

(2.14)

and

$$\begin{aligned} \Vert \psi (T,\cdot )\Vert _{C^{2\theta }(\mathbb {R})}\le C_2(\epsilon ,\theta )\Big (\epsilon ^{-\theta }\Vert \psi _0\Vert _{\infty }+\Vert f_2[{\tilde{\phi }},{\tilde{\psi }}]\Vert _\infty \Big ). \end{aligned}$$

(2.15)

In view of Lemma 2.1, we have the following integral equality for the function $\psi ^+(t,z):$

$$\begin{aligned} \psi ^+(t)=T_2(t)\psi ^+(0)+\int _0^t T_2(t-s)\Big [\alpha _2\psi ^+(s)+(\beta (s)-\gamma (s))\psi ^+(s)\Big ]ds. \end{aligned}$$

(2.16)

By Lemmas 2.2 and 2.3, and similar arguments to [43, Lemma 3.2], we further show the integral inequalities for $\phi ^-(t,z)$ and $\psi ^-(t,z)$.

Lemma 2.4

The following inequalities for $\phi ^-$ and $\psi ^-$

$$\begin{aligned} \phi ^-(t)\le T_1(t)\phi ^-(0)+\int _0^tT_1(t-s)f_1[\phi ^-,\psi ^+](s)ds \end{aligned}$$

(2.17)

and

$$\begin{aligned} \psi ^-(t)\le T_2(t)\psi ^-(0)+\int _0^t T_2(t-s)f_2[\phi ^-,\psi ^-](s)ds \end{aligned}$$

(2.18)

are valid, respectively.

Proof

Let ${\hat{\phi }}^-(t,z)=\phi ^-(t,z+ct)$ and ${\hat{\psi }}^+(t,z)=\psi ^+(t,z+ct)$ for any $(t,z)\in [0,T]\times \mathbb {R}.$ Then for any $t\in [0,T],$ by Lemma 2.2,

$$\begin{aligned} {\hat{\phi }}^-_t(t,z)-d_1{\hat{\phi }}^-_{zz}(t,z)+\alpha _1{\hat{\phi }}^-(t,z) -f_1[{\hat{\phi }}^-,{\hat{\psi }}^+](t,z)\le 0 \end{aligned}$$

for any $z\not =z^-(t)=\frac{1}{\epsilon _1}(-\ln M_1-c\epsilon _1t).$ Clearly,

$$\begin{aligned} \frac{\partial {\hat{\phi }}^-(t,z^-(t_0)-0)}{\partial z}=\lim _{z\rightarrow z^-(t_0)-0}\left\{ -S_0M_1\epsilon _1e^{\epsilon _1(z+ct)}\right\} =-\epsilon _1S_0<0. \end{aligned}$$

Define

$$\begin{aligned} G(t,z):=-{\hat{\phi }}^-_t(t,z)+d_1{\hat{\phi }}^-_{zz}(t,z)-\alpha _1{\hat{\phi }}^-(t,z) +f_1[{\hat{\phi }}^-, {\hat{\psi }}^+](t,z)\ge 0 \end{aligned}$$

and

$$\begin{aligned} H({\hat{\phi }}^-)(t,z,r):=\frac{e^{-\alpha _1(t-r)}}{\sqrt{4\pi d_1(t-r)}}\int _{-\infty }^{\infty }e^{-\frac{(z-y)^2}{4d_1(t-r)}}{\hat{\phi }}^-(r,y)dy. \end{aligned}$$

Then, by a direct calculation, we have

$$\begin{aligned} \frac{\partial }{\partial r}H({\hat{\phi }}^-)(t,z,r)= & {} \frac{\alpha _1e^{-\alpha _1(t-r)}}{\sqrt{4\pi d_1(t-r)}}\int _{-\infty }^{\infty }e^{-\frac{(z-y)^2}{4d_1(t-r)}}{\hat{\phi }}^-(r,y)dy\\&+\frac{e^{-\alpha _1(t-r)}}{2(t-r)\sqrt{4\pi d_1(t-r)}}\int _{-\infty }^{\infty }e^{-\frac{(z-y)^2}{4d_1(t-r)}}{\hat{\phi }}^-(r,y)dy\\&-\frac{e^{-\alpha _1(t-r)}}{\sqrt{4\pi d_1(t-r)}}\int _{-\infty }^{\infty } \frac{(z-y)^2}{4d_1(t-r)^2}e^{-\frac{(z-y)^2}{4d_1(t-r)}}{\hat{\phi }}^-(r,y)dy\\&+\frac{d_1e^{-\alpha _1(t-r)}}{\sqrt{4\pi d_1(t-r)}}\int _{-\infty }^{\infty } e^{-\frac{(z-y)^2}{4d_1(t-r)}}\frac{\partial ^2{\hat{\phi }}^-(r,y)}{\partial y^2}dy\\&-\frac{e^{-\alpha _1(t-r)}}{\sqrt{4\pi d_1(t-r)}}\int _{-\infty }^{\infty }e^{-\frac{(z-y)^2}{4d_1(t-r)}}\alpha _1{\hat{\phi }}^-(r,y)dy\\&+\frac{e^{-\alpha _1(t-r)}}{\sqrt{4\pi d_1(t-r)}}\int _{-\infty }^{\infty } e^{-\frac{(z-y)^2}{4d_1(t-r)}}\left[ f_1[{\hat{\phi }}^-, {\hat{\psi }}^+](r,y)-G(r,y)\right] dy. \end{aligned}$$

Furthermore, integration by parts yields

$$\begin{aligned}&\frac{d_1e^{-\alpha _1(t-r)}}{\sqrt{4\pi d_1(t-r)}}\int _{-\infty }^{\infty } e^{-\frac{(z-y)^2}{4d_1(t-r)}}\frac{\partial ^2{\hat{\phi }}^-(r,y)}{\partial y^2}dy\\&\quad =\frac{d_1e^{-\alpha _1(t-r)}}{\sqrt{4\pi d_1(t-r)}}\int _{-\infty }^{z^-(r)} e^{-\frac{(z-y)^2}{4d_1(t-r)}}\frac{\partial ^2{\hat{\phi }}^-(r,y)}{\partial y^2}dy\\&\quad =\frac{d_1e^{-\alpha _1(t-r)}}{\sqrt{4\pi d_1(t-r)}}e^{-\frac{(z-z^-(r))^2}{4d_1(t-r)}}\frac{\partial {\hat{\phi }}^-(r,z^-(r)-0)}{\partial z}\\&\quad -\frac{e^{-\alpha _1(t-r)}}{\sqrt{4\pi d_1(t-r)}}\int _{-\infty }^{z^-(r)}\frac{1}{2(t-r)} e^{-\frac{(z-y)^2}{4d_1(t-r)}}{\hat{\phi }}^-(r,y)dy\\&\quad +\frac{e^{-\alpha _1(t-r)}}{\sqrt{4\pi d_1(t-r)}}\int _{-\infty }^{z^-(r)}\frac{(z-y)^2}{4d_1(t-r)^2} e^{-\frac{(z-y)^2}{4d_1(t-r)}}{\hat{\phi }}^-(r,y)dy. \end{aligned}$$

Here we have used the fact that $\hat{\phi }^-(t,z)=0$, $\forall z>z^-(t)$. In view of $\frac{\partial {\hat{\phi }}^-(r,z^-(r)-0)}{\partial z}=-\epsilon _1S_0,$ we have

$$\begin{aligned} \quad \frac{\partial }{\partial r}H({\hat{\phi }}^-)(t,z,r)= & {} -\epsilon _1S_0\frac{d_1e^{-\alpha _1(t-r)}}{\sqrt{4\pi d_1(t-r)}}e^{-\frac{(z-z^-(r))^2}{4d_1(t-r)}}\\&+\frac{e^{-\alpha _1(t-r)}}{\sqrt{4\pi d_1(t-r)}}\int _{-\infty }^{\infty } e^{-\frac{(z-y)^2}{4d_1(t-r)}}\left[ f_1[{\hat{\phi }}^-,{\hat{\psi }}^+](r,y)-G(r,y)\right] dy. \end{aligned}$$

Since

$$\begin{aligned} \frac{d_1e^{-\alpha _1(t-r)}}{\sqrt{4\pi d_1(t-r)}}\exp \left\{ -\frac{(z-z^-(r))^2}{4d_1(t-r)}\right\} \frac{\partial {\hat{\phi }}^-(r,z^-(r)-0)}{\partial z} \end{aligned}$$

is integrable in $r\in [0,t), \frac{\partial }{\partial r}H({\hat{\phi }}^-)(t,z,r)$ is continuous in $r\in [0,t),$ and

$$\begin{aligned} \lim _{r\rightarrow t-0}\frac{e^{-\alpha _1(t-r)}}{\sqrt{4\pi d_1(t-r)}}\int _{-\infty }^{\infty } e^{-\frac{(z-y)^2}{4d_1(t-r)}}{\hat{\phi }}^-(r,y)dy={\hat{\phi }}^-(t,z), \end{aligned}$$

we conclude that

$$\begin{aligned} {\hat{\phi }}^-(t,z)= & {} \lim _{\eta \rightarrow 0+0}H({\hat{\phi }}^-)(t,z,t-\eta )\\= & {} H({\hat{\phi }}^-)(t,z,0)+\lim _{\eta \rightarrow 0+0}\int _0^{t-\eta }\frac{\partial }{\partial r}H({\hat{\phi }}^-)(t,z,r)dr\\= & {} \frac{e^{-\alpha _1t}}{\sqrt{4\pi d_1t}}\int _{-\infty }^{\infty } e^{-\frac{(z-y)^2}{4d_1t}}{\hat{\phi }}^-(0,y)dy\\&-\epsilon _1S_0\int _0^t\frac{d_1e^{-\alpha _1(t-r)}}{\sqrt{4\pi d_1(t-r)}}e^{-\frac{(z-z^-(r))^2}{4d_1(t-r)}}dr\\&+\int _0^t\frac{e^{-\alpha _1(t-r)}}{\sqrt{4\pi d_1(t-r)}}\int _{-\infty }^{\infty } e^{-\frac{(z-y)^2}{4d_1(t-r)}}\left[ f_1[{\hat{\phi }}^-,{\hat{\psi }}^+ ](r,y)-G(r,y)\right] dydr. \end{aligned}$$

With the aid of $G(r,y)\ge 0,$ we see that

$$\begin{aligned} {\hat{\phi }}^-(t)\le \hat{T}_1(t){\hat{\phi }}^-(0)+\int _0^t\hat{T}_1(t-r)f_1[{\hat{\phi }}^-,{\hat{\psi }}^+](r)dr,\ t\in (0,T], \end{aligned}$$

where $\hat{T}_1(t)$ is defined by

$$\begin{aligned} \left( \hat{T}_1(t)\phi \right) (x)=\frac{e^{-\alpha _1t}}{\sqrt{4\pi d_1t}}\int _{-\infty }^{\infty } e^{-\frac{(z-y)^2}{4d_1t}}\phi (y)dy. \end{aligned}$$

Hence, it is not difficult to obtain the inequality (2.17) for $\phi ^-$. Similarly, we can show that the inequality (2.18) for $\psi ^-$ holds. $\square$

On the basis of the above integral equation and integral inequalities, we shall show the invariance for integral equations (2.11) (see, e.g., [32]).

Lemma 2.5

Let $(\phi (t,z;\phi _0,\psi _0),\psi (t,z;\phi _0,\psi _0))$ be the solutions of the system (2.11) with the initial value $(\phi _0,\psi _0)\in {\tilde{D}}.$ Then

$$\begin{aligned}&\phi ^-(t,z)\le \phi (t,z;\phi _0,\psi _0)\le \phi ^+(t,z),\\&\psi ^-(t,z)\le \psi (t,z;\phi _0,\psi _0)\le \min \{\psi ^+(t,z),\Lambda \} \end{aligned}$$

for $(t,z)\in [0,T]\times \mathbb {R}$.

Proof

Recall that $({\tilde{\phi }},{\tilde{\psi }})\in \mathcal {D}, (\phi _0,\psi _0)\in {\tilde{\mathcal {D}}}$ and $(\phi ,\psi )$ satisfies the system (2.11). Moreover,

$$\begin{aligned} \phi ^-(t,z)\le \tilde{\phi }(t,z)\le \phi ^+(t,z),\quad \psi ^-(t,z)\le \tilde{\psi }(t,z)\le \min \{\psi ^+(t,z),\Lambda \},\quad (t,z)\!\in \![0,T]\times \mathbb {R} \end{aligned}$$

and

$$\begin{aligned} \phi ^-(0,z)\le \phi _0(z)\le \phi ^+(0,z),\quad \psi ^-(0,z)\le \psi _0(z)\le \min \{\psi ^+(0,z),\Lambda \},\quad (t,z)\!\in \![0,T]\times \mathbb {R}. \end{aligned}$$

Since $\phi ^+(t,z)\equiv S_0,$ it is easy to see that

$$\begin{aligned} \phi ^+(t)=T_1(t)\phi ^+(0)+\alpha _1\int _0^t T_1(t-s)\phi ^+(s)ds. \end{aligned}$$

(2.19)

Due to the positivity of semigroup $T_1(\cdot ),$ we have

$$\begin{aligned} \int _0^tT_1(t-s)f_1[{\tilde{\phi }},{\tilde{\psi }}](s)ds\, \le \,\alpha _1\int _0^tT_1(t-s){\tilde{\phi }}(s)ds \end{aligned}$$

for any $t\in (0,T]$. By (2.19), it follows that

$$\begin{aligned} \int _0^tT_1(t-s)f_1[{\tilde{\phi }},{\tilde{\psi }}](s)ds\,\le \, \phi ^+(t)-T_1(t)\phi ^+(0)\,\le \,\phi ^+(t)-T_1(t)\phi _0 \end{aligned}$$

for any $t\in (0,T]$, which implies that $\phi (t,z)\le \phi ^+(t,z)$ for any $t\in [0,T]$ and $z\in \mathbb {R}.$ Let $w(t,z)=\phi (t,z)-\phi ^-(t,z), \forall (t,z)\in [0,T]\times \mathbb {R}.$ By (2.17), we have

$$\begin{aligned} \begin{aligned} w(t)=&T_1(t)[\phi _0-\phi ^-(0)]\\&+\int _0^tT_1(t-s)\left\{ \alpha _1[{\tilde{\phi }}(s)-\phi ^-(s)]- A[{\tilde{\phi }},{\tilde{\psi }}](s)+A[\phi ^-,\psi ^+](s)\right\} ds\\&\ge T_1(t)[\phi _0-\phi ^-(0)]\\&+\int _0^tT_1(t-s)\left\{ \alpha _1[{\tilde{\phi }}(s)-\phi ^-(s)] -\frac{\beta (s){\tilde{\phi }}(s)\psi ^+(s)}{{\tilde{\phi }}(s)+\psi ^+(s)}+ \frac{\beta (s)\phi ^-(s)\psi ^+(s)}{{\tilde{\phi }}(s)+\psi ^+(s)}\right\} ds\\ =&T_1(t)[\phi _0-\phi ^-(0)]+\int _0^tT_1(t-s) \left[ \alpha _1-\frac{\beta (s)\psi ^+(s)}{{\tilde{\phi }}(s)+\psi ^+(s)}\right] [{\tilde{\phi }}(s)-\phi ^-(s)] ds\\ \ge&T_1(t)[\phi _0-\phi ^-(0)]+\int _0^tT_1(t-s) [\alpha _1-\beta (s)][{\tilde{\phi }}(s)-\phi ^-(s)] ds. \end{aligned} \end{aligned}$$

Since $\alpha _1>\max _{t\in [0,T]}\beta (t)$, it follows that $w(t)\ge 0,\forall t\in [0,T],$ which implies that

$$\begin{aligned} \phi (t,z)\ge \phi ^-(t,z),\ \ \forall (t,z)\in [0,T]\times \mathbb {R}. \end{aligned}$$

In the following, we consider $\psi (t,z;\phi _0,\psi _0)$ for $t\in [0,T],\,z\in \mathbb {R}$. It is easy to see that

$$\begin{aligned} \begin{aligned} \int _0^tT_2(t-s)f_2[{\tilde{\phi }},{\tilde{\psi }}](s)ds=&\int _0^tT_2(t-s) \left[ \alpha _2{\tilde{\psi }}(s)+ A[{\tilde{\phi }},{\tilde{\psi }}](s)-\gamma (s){\tilde{\psi }}(s)\right] ds\\ \le&\int _0^tT_2(t-s)\left[ \alpha _2\psi ^+(s)+ (\beta (s)-\gamma (s))\psi ^+(s)\right] ds \end{aligned} \end{aligned}$$

for any $t\in (0,T]$. By virtue of (2.16), we have

$$\begin{aligned} \int _0^tT_2(t-s)f_2[{\tilde{\phi }},{\tilde{\psi }}](s)ds\,\le \,\psi ^+(t)-T_2(t)\psi ^+(0)\,\le \, \psi ^+(t)-T_2(t)\psi _0 \end{aligned}$$

for all $t\in (0,T]$. In addition, $\psi (0,z)=\psi _0(z)\le \psi ^+(0,z)$ for any $z\in \mathbb {R}.$ It then follows that

$$\begin{aligned} \psi (t,z;\tilde{\phi },\tilde{\psi })\le \psi ^+(t,z),\ \ \forall t\in [0,T],\, z\in \mathbb {R}. \end{aligned}$$

Recall that $\Lambda >0$ satisfies $\frac{\beta (t)S_0}{S_0+\Lambda }-\gamma (t)<0$ for $t\in [0,T]$. It is not difficult to prove that $\psi _\Lambda ^+(t,z)\equiv \Lambda$ satisfies that

$$\begin{aligned} \psi _\Lambda ^+(t,z)=T_2(t)\psi _\Lambda ^+(0,z)+\alpha _2\int _0^tT_2(t-s) \psi _\Lambda ^+(s,z)ds. \end{aligned}$$

By a similar argument to the proof for $\phi ^+(t,z),$ we can prove that

$$\begin{aligned} \psi (t,z;\tilde{\phi },\tilde{\psi })\le \psi _\Lambda ^+(t,z)\equiv \Lambda ,\ \ \forall t\in [0,T],\, z\in \mathbb {R}. \end{aligned}$$

Since $T_2(\cdot )$ is positive and $\alpha _2>\max _{t\in [0,T]}\gamma (t),$ it follows that

$$\begin{aligned} \int _0^tT_2(t-s)f_2[{\tilde{\phi }},{\tilde{\psi }}](s)ds\, \ge \,\int _0^tT_2(t-s)f_2[\phi ^-,\psi ^-](s)ds \end{aligned}$$

for any $t\in (0,T]$. According to (2.18), we have

$$\begin{aligned} \int _0^tT_2(t-s)f_2[{\tilde{\phi }},{\tilde{\psi }}](s)ds\ge & {} \psi ^-(t)-T_2(t)\psi ^-(0)\\\ge & {} \psi ^-(t)-T_2(t)\psi ,\ \ \forall t\in (0,T], \end{aligned}$$

which yields

$$\begin{aligned} \psi (t)=T_2(t)\psi _0+\int _0^tT_2(t-s)f_2[{\tilde{\phi }},{\tilde{\psi }}](s)ds\ge \psi ^-(t),\ \forall t\in (0,T]. \end{aligned}$$

Additionally, $\psi (0,z)=\psi _0(z)\ge \psi ^-(0,z), \forall z\in \mathbb {R}.$ Consequently, we have proved that

$$\begin{aligned} \psi (t,z;{\tilde{\phi }},{\tilde{\psi }})\ge \psi ^-(t,z),\ \ \forall t\in [0,T],\, z\in \mathbb {R}. \end{aligned}$$

This completes the proof. $\square$

For any given $({\tilde{\phi }},{\tilde{\psi }})\in \mathcal {D}$, we denote the time-T map of system (2.11): $(\phi _0(z),\psi _0(z))\mapsto \left( \phi (T,z;\phi _0,\psi _0),\psi (T,z;\phi _0,\psi _0) \right)$ by

$$\begin{aligned} F_{({\tilde{\phi }},{\tilde{\psi }})}(\phi _0(\cdot ),\psi _0(\cdot ))= \left( \phi (T,\cdot ;\phi _0,\psi _0),\psi (T,\cdot ;\phi _0,\psi _0) \right) . \end{aligned}$$

Thus, any fixed point of the T-map $F_{({\tilde{\phi }},{\tilde{\psi }})}$ gives a T-periodic solution of system (2.11).

Theorem 2.6

For any given $({\tilde{\phi }},{\tilde{\psi }})\in \mathcal {D},$ there exists a unique $(\phi ^*,\psi ^*)\in \mathcal {D}$ such that (2.13) holds.

Proof

In view of Lemma 2.5 and the definitions of $\phi ^{\pm }$ and $\psi ^{\pm }$, we assert that $F_{({\tilde{\phi }},{\tilde{\psi }})}$ maps ${\tilde{\mathcal {D}}}$ into ${\tilde{\mathcal {D}}}$. For any compact interval $I\subset \mathbb {R},$ due to the estimates (2.14) and (2.15), we can conclude that $\{\left( \phi (T,\cdot ;\phi _0,\psi _0), \psi (T,\cdot ;\phi _0,\psi _0)\right) :\, (\phi _0,\psi _0)\in \tilde{\mathcal {D}}\}$ is compact on $C(I,\mathbb {R}^2)$. We can further show that $F_{({\tilde{\phi }},{\tilde{\psi }})}: \tilde{\mathcal {D}}\rightarrow \tilde{\mathcal {D}}$ is compact with respect to $\vert \cdot \vert _\mu .$ In addition, it is not difficult to see that $F_{({\tilde{\phi }},{\tilde{\psi }})}: \tilde{\mathcal {D}}\rightarrow \tilde{\mathcal {D}}$ is continuous with respect to $\vert \cdot \vert _\mu .$ Thus, the Schauder’s fixed point theorem implies that $F_{({\tilde{\phi }},{\tilde{\psi }})}$ admits a fixed point $(\phi _0^*,\psi _0^*)\in \tilde{\mathcal {D}}.$ As a result, $\left( \phi (t,z;\phi _0^*,\psi _0^*),\psi (t,z;\phi _0^*,\psi _0^*)\right)$ satisfies $\phi (T,z;\phi _0^*,\psi _0^*)=\phi _0^*(z)$ and $\psi (T,z;\phi _0^*,\psi _0^*)=\psi _0^*(z), \forall z\in \mathbb {R}.$ Furthermore, we claim that such a fixed point $(\phi _0^*,\psi _0^*)$ is unique. Suppose that there exists $(\phi _0^{**},\psi _0^{**})\in \tilde{\mathcal {D}}$ such that $\left( \phi (t,z;\phi _0^{**},\psi _0^{**}), \psi (t,z;\phi _0^{**},\psi _0^{**}) \right)$ satisfies (2.11), and $\phi (T,z;\phi _0^{**},\psi _0^{**})=\phi _0^{**}(z)$, $\psi (T,z;\phi _0^{**},\psi _0^{**})=\psi _0^{**}(z), \forall z\in \mathbb {R}$. Then

$$\begin{aligned} \left| \phi (T,z;\phi _0^*,\psi _0^*)-\phi (T,z;\phi _0^{**},\psi _0^{**})\right|\le & {} e^{-\alpha _1T}\int _\mathbb {R}\frac{e^{-\frac{(z-x-cT)^2}{4d_1T}}}{\sqrt{4\pi d_1T}}\left| \phi ^*_0(x)-\phi ^{**}_0(x)\right| dx\\\le & {} \left\| \phi ^*_0(\cdot )-\phi ^{**}_0(\cdot )\right\| _{L^\infty } e^{-\alpha _1T}\int _\mathbb {R}\frac{e^{-\frac{(z-x-cT)^2}{4d_1T}}}{\sqrt{4\pi d_1T}}dx\\= & {} e^{-\alpha _1T}\left\| \phi ^*_0(\cdot )-\phi ^{**}_0(\cdot )\right\| _{L^\infty }. \end{aligned}$$

On the other hand, $\phi (T,\cdot ;\phi _0^*,\psi _0^*)=\phi _0^*(\cdot )$ and $\phi (T,\cdot ;\phi _0^{**},\psi _0^{**})=\phi _0^{**}(\cdot ),$ we then see that

$$\begin{aligned} \left\| \phi ^*_0(\cdot )-\phi ^{**}_0(\cdot )\right\| _{L^\infty }\le e^{-\alpha _1T} \left\| \phi ^*_0(\cdot )-\phi ^{**}_0(\cdot )\right\| _{L^\infty }. \end{aligned}$$

Since $e^{-\alpha _1T}<1,$ we have $\phi _0^*(\cdot )\equiv \phi _0^{**}(\cdot )$. Similarly, we can also obtain $\psi ^*_0(\cdot )\equiv \psi ^{**}_0(\cdot ).$ Hence, there exists a unique $(\phi ^*,\psi ^*)$ satisfying (2.13). $\square$

Let $(\phi ^*(t,z),\psi ^*(t,z))=\left( \phi (t,z;\phi _0^*,\psi _0^*), \psi (t,z;\phi _0^*,\psi _0^*)\right)$, where $(\phi _0^*,\psi _0^*)\in {\tilde{\mathcal {D}}}$ is the unique fixed point of the operator $F_{({\tilde{\phi }},{\tilde{\psi }})}$. In view of Theorem 2.6, we can define an operator $\mathcal {F}: \mathcal {D}\rightarrow B_\mu$ by $\mathcal {F}({\tilde{\phi }},{\tilde{\psi }})=(\phi ^*,\psi ^*).$ Thus, the existence of periodic traveling waves is reduced to the existence of a fixed point of the operator $\mathcal {F}.$

2.3 The Periodic Traveling Waves

In this section, we prove the existence of periodic traveling waves. As discussed in Sect. 2.2, we need to study the existence of fixed points of the operator $\mathcal {F}.$ We start with the properties of $\mathcal {F}.$ In view of Lemma 2.5, $\mathcal {F}$ maps $\mathcal {D}$ into $\mathcal {D}.$

Lemma 2.7

The map $\mathcal {F}:\mathcal {D}\rightarrow \mathcal {D}$ is continuous with respect to the norm $\Vert \cdot \Vert _\mu$ in $B_{\mu }([0,T]\times \mathbb {R},\mathbb {R}^2).$

Proof

For any $({\tilde{\phi }}_1,{\tilde{\psi }}_1)\in \mathcal {D}$ and $({\tilde{\phi }}_2,{\tilde{\psi }}_2)\in \mathcal {D},$ let $(\phi _i^*(t,z;{\tilde{\phi }}_i,{\tilde{\psi }}_i), \psi _i^*(t,z;{\tilde{\phi }}_i,{\tilde{\psi }}_i))=\mathcal {F}({\tilde{\phi }}_i,{\tilde{\psi }}_i), i=1,2.$ From the first equation of system (2.13) and (2.12), we see that

$$\begin{aligned} \phi _i^*(T,z;{\tilde{\phi }}_i,{\tilde{\psi }}_i)= & {} e^{-\alpha _1T}\int _\mathbb {R} \frac{1}{\sqrt{4\pi d_1T}}e^{-\frac{(z-y-cT)^2}{4d_1T}}\phi _i^*(0,y)dy\\&+\int _0^Te^{-\alpha _1s}\int _\mathbb {R} \frac{1}{\sqrt{4\pi d_1s}}e^{-\frac{(z-y-cs)^2}{4d_1s}}f_1[{\tilde{\phi }}_i,{\tilde{\psi }}_i](T-s,y)dyds. \end{aligned}$$

Let $\tilde{\beta }=\max _{t\in [0,T]}\beta (t)$ and choose $\mu$ sufficiently small such that $e^{d_1T\mu ^2+cT\mu -\alpha _1T}\le \frac{1}{4}.$ Consequently,

$$\begin{aligned}&\left| \phi _1^*(T,z;{\tilde{\phi }}_1,{\tilde{\psi }}_1)\right. -\left. \phi _2^*(T,z;{\tilde{\phi }}_2, {\tilde{\psi }}_2)\right| e^{-\mu \vert z\vert }\\&\quad \le e^{-\alpha _1T}\int _\mathbb {R}\frac{1}{\sqrt{4\pi d_1T}}e^{-\frac{(z-y-cT)^2}{4d_1T}}\left| \phi _1^*(0,y)-\phi _2^*(0,y)\right| dy e^{-\mu \vert z\vert }\\&\quad \quad +\int _0^Te^{-\alpha _1s}\int _\mathbb {R} \frac{1}{\sqrt{4\pi d_1s}}e^{-\frac{(z-y-cs)^2}{4d_1s}}\left( \left| {\tilde{\phi }}_1(T-s,y)-{\tilde{\phi }}_2(T-s,y)\right| \right. \\&\quad \quad +\left. \left| {\tilde{\psi }}_1(T-s,y)-{\tilde{\psi }}_2(T-s,y)\right| \right) (\alpha _1+{\tilde{\beta }})dyds e^{-\mu \vert z\vert }\\&\quad \le e^{-\alpha _1T}\int _\mathbb {R}\frac{1}{\sqrt{4\pi d_1T}}e^{-\frac{(z-y-cT)^2}{4d_1T}}\left| \phi _1^*(0,y)-\phi _2^*(0,y)\right| e^{-\mu \vert y\vert } e^{\mu \vert y-z\vert } dy\\&\quad \quad +\int _0^Te^{-\alpha _1s}\int _\mathbb {R} \frac{1}{\sqrt{4\pi d_1s}}e^{-\frac{(z-y-cs)^2}{4d_1s}}\left( \left| {\tilde{\phi }}_1(T-s,y)-{\tilde{\phi }}_2(T-s,y)\right| e^{-\mu \vert y\vert } \right. \\&\quad \quad +\left. \left| {\tilde{\psi }}_1(T-s,y)-{\tilde{\psi }}_2(T-s,y)\right| e^{-\mu \vert y\vert }\right) e^{\mu \vert y-z\vert } (\alpha _1+{\tilde{\beta }})dyds\\&\quad \le e^{-\alpha _1T+\mu cT}\vert \phi _1^*(0)-\phi _2^*(0)\vert _\mu \int _\mathbb {R}\frac{1}{\sqrt{4\pi d_1T}}e^{-\frac{(z-y-cT)^2}{4d_1T}} e^{\mu \vert z-y-cT\vert }dy\\&\quad \quad +(\alpha _1+{\tilde{\beta }})\left( \left\| {\tilde{\phi }}_1-{\tilde{\phi }}_2\right\| _\mu +\left\| {\tilde{\psi }}_1-{\tilde{\psi }}_2\right\| _\mu \right) \\ \end{aligned}$$

$$\begin{aligned}&\quad \quad \times \int _0^Te^{-\alpha _1s} e^{\mu cs}\int _\mathbb {R}\frac{1}{\sqrt{4\pi d_1s}}e^{-\frac{(z-y-cs)^2}{4d_1s}} e^{\mu \vert z-y-cs\vert }dyds\\&\quad =e^{-\alpha _1T+\mu cT}\vert \phi _1^*(0)-\phi _2^*(0)\vert _\mu \int _\mathbb {R}\frac{1}{\sqrt{4\pi d_1T}}e^{-\frac{y^2}{4d_1T}} e^{\mu \vert y\vert }dy\\&\quad \quad +(\alpha _1+{\tilde{\beta }})\left( \left\| {\tilde{\phi }}_1-{\tilde{\phi }}_2\right\| _\mu +\left\| {\tilde{\psi }}_1-{\tilde{\psi }}_2\right\| _\mu \right) \\&\quad \quad \times \int _0^Te^{-\alpha _1s} e^{\mu cs}\int _\mathbb {R}\frac{1}{\sqrt{4\pi d_1s}}e^{-\frac{y^2}{4d_1s}} e^{\mu \vert y\vert }dyds\\&\quad \le 2e^{(d_1\mu ^2+c\mu -\alpha _1)T}\vert \phi _1^*(0)-\phi _2^*(0)\vert _\mu \\&\quad \quad +(\alpha _1+{\tilde{\beta }})\left( \left\| {\tilde{\phi }}_1-{\tilde{\phi }}_2\right\| _\mu +\left\| {\tilde{\psi }}_1-{\tilde{\psi }}_2\right\| _\mu \right) \int _0^T 2 e^{(d_1\mu ^2+c\mu -\alpha _1)s}ds\\&\quad \le 2e^{(d_1\mu ^2+c\mu -\alpha _1)T}\vert \phi _1^*(0)-\phi _2^*(0)\vert _\mu \\&\quad \quad +\frac{2(\alpha _1+{\tilde{\beta }})\left( e^{(d_1\mu ^2+c\mu -\alpha _1)T}-1\right) }{d_1\mu ^2+c\mu -\alpha _1} \left( \left\| {\tilde{\phi }}_1-{\tilde{\phi }}_2\right\| _\mu +\left\| {\tilde{\psi }}_1-{\tilde{\psi }}_2\right\| _\mu \right) \\&\quad \le \frac{1}{2}\vert \phi _1^*(0)-\phi _2^*(0)\vert _\mu \\&\quad \quad +\frac{2(\alpha _1+{\tilde{\beta }})\left( e^{(d_1\mu ^2+c\mu -\alpha _1)T}-1\right) }{d_1\mu ^2+c\mu -\alpha _1} \left( \left\| {\tilde{\phi }}_1-{\tilde{\phi }}_2\right\| _\mu +\left\| {\tilde{\psi }}_1-{\tilde{\psi }}_2\right\| _\mu \right) \end{aligned}$$

Let

$$\begin{aligned} L:=\frac{4(\alpha _1+{\tilde{\beta }})\left( e^{(d_1\mu ^2+c\mu -\alpha _1)T}-1\right) }{d_1\mu ^2+c\mu -\alpha _1}. \end{aligned}$$

Since $\phi _i^*(T,z;{\tilde{\phi }}_i,{\tilde{\psi }}_i)=\phi _i^*(0,z),i=1,2,$ we obtain from the above inequalities that

$$\begin{aligned} \vert \phi _1^*(0)-\phi _2^*(0)\vert _\mu \le L(\Vert {\tilde{\phi }}_1-{\tilde{\phi }}_2\Vert _\mu +\Vert {\tilde{\psi }}_1-{\tilde{\psi }}_2\Vert _\mu ). \end{aligned}$$

On the other hand, $\phi _i^*(t,z;{\tilde{\phi }}_i,{\tilde{\psi }}_i)$ satisfies that

$$\begin{aligned} \phi _i^*(t,z;{\tilde{\phi }}_i,{\tilde{\psi }}_i)= & {} e^{-\alpha _1t}\int _\mathbb {R} \frac{1}{\sqrt{4\pi d_1t}}e^{-\frac{(z-y-ct)^2}{4d_1t}}\phi _i^*(0,y)dy\\&+\int _0^te^{-\alpha _1s}\int _\mathbb {R} \frac{1}{\sqrt{4\pi d_1s}}e^{-\frac{(z-y-cs)^2}{4d_1s}}f_1[{\tilde{\phi }}_i,{\tilde{\psi }}_i](t-s,y)dyds. \end{aligned}$$

Hence, by similar arguments to above, it is not difficult to conclude that $\phi ^*(t,z;{\tilde{\phi }},{\tilde{\psi }})$ is continuous in $({\tilde{\phi }},{\tilde{\psi }})$ with respect to the norm $\Vert \cdot \Vert _\mu .$ Similarly, we can prove that $\psi ^*(t,z;{\tilde{\phi }},{\tilde{\psi }})$ is continuous in $({\tilde{\phi }},{\tilde{\psi }})$ with respect to the norm $\Vert \cdot \Vert _\mu .$ $\square$

Lemma 2.8

The map $\mathcal {F}:\mathcal {D}\rightarrow \mathcal {D}$ is compact with respect to the norm $\Vert \cdot \Vert _\mu$ in $B_{\mu }([0,T]\times \mathbb {R},\mathbb {R}^2).$

Proof

For any $({\tilde{\phi }},{\tilde{\psi }})\in \mathcal {D}$, let $(\phi ^*,\psi ^*)=\mathcal {F}({\tilde{\phi }},{\tilde{\psi }})$, where $(\phi ^*(t,z),\psi ^*(t,z)), t\in [0,T], z\in \mathbb {R}$ is the solution of system (2.13). In particular, it follows from the estimates (2.14) and (2.15) that there exists $K'(\theta )>0$ independent of $({\tilde{\phi }},{\tilde{\psi }})$ such that $\Vert \phi ^*(0)\Vert _{C^{2\theta }(\mathbb {R})}=\Vert \phi ^*(T)\Vert _{C^{2\theta }(\mathbb {R})}\le K'$ and $\Vert \psi ^*(0)\Vert _{C^{2\theta }(\mathbb {R})}=\Vert \psi ^*(T)\Vert _{C^{2\theta }(\mathbb {R})}\le K'.$ Moreover, $f_1[{\tilde{\phi }},{\tilde{\psi }}]$ and $f_2[{\tilde{\phi }},{\tilde{\psi }}]$ admit uniform bounds with respect to $({\tilde{\phi }},{\tilde{\psi }})\in \mathcal {D},$ respectively, uniformly for $(t,x)\in [0,T]\times \mathbb {R}.$ Thanks to [31, Theorem 5.1.2], it follows that $\phi ^*,\psi ^*\in C^{\theta ,2\theta }([0,T]\times \mathbb {R},\mathbb {R})$ with some $\theta \in (0,1)$, and there exists $C_i(\theta )>0, i=1,2$ and $\tilde{K}(\theta )>0$ such that

$$\begin{aligned} \Vert \phi ^*\Vert _{C^{\theta ,2\theta }([0,T]\times \mathbb {R})}\le C_1\left( \Vert \phi ^*(0)\Vert _{C^{2\theta }(\mathbb {R})}+\Vert f_1[{\tilde{\phi }},{\tilde{\psi }}]\Vert _\infty \right) \le \tilde{K}(\theta ) \end{aligned}$$

(2.20)

and

$$\begin{aligned} \Vert \psi ^*\Vert _{C^{\theta ,2\theta }([0,T]\times \mathbb {R})}\le C_2\left( \Vert \psi ^*(0)\Vert _{C^{2\theta }(\mathbb {R})}+\Vert f_2[{\tilde{\phi }},{\tilde{\psi }}]\Vert _\infty \right) \le \tilde{K}(\theta ). \end{aligned}$$

(2.21)

Let $(\phi _n^*,\psi _n^*)=\mathcal {F}(\tilde{\phi }_n,\tilde{\psi }_n).$ Since $\phi _n^*$ and $\psi _n^*$ satisfy the estimations (2.20) and (2.21), respectively, there is a subsequence of $\{(\phi _n^*,\psi _n^*)\},$ without loss of generality, still labeled by $\{(\phi _n^*,\psi _n^*)\},$ such that it converges in $C_\mathrm{loc}([0,T]\times \mathbb {R},\mathbb {R}^2)$ to a function $(\phi ^{**},\psi ^{**})\in C([0,T]\times \mathbb {R},\mathbb {R}^*),$ that is, for any $N\in \mathbb {R}^+,$

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert \left( \phi _n^*,\psi _n^*\right) -\left( \phi ^{**},\psi ^{**}\right) \Vert _{C([0,T]\times [-N,N],\mathbb {R}^2)}=0. \end{aligned}$$

(2.22)

Clearly, $(\phi ^{**},\psi ^{**})\in \mathcal {D}.$

In the following, we are ready to prove that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert \left( \phi _n^*,\psi _n^*\right) -\left( \phi ^{**},\psi ^{**}\right) \Vert _{\mu }=0. \end{aligned}$$

Note that $\mathcal {D}$ is uniformly bounded with respect to the norm $\Vert \cdot \Vert _\mu .$ Accordingly, the norm $\Vert \left( \phi _n^*,\psi _n^*\right) -\left( \phi ^{**},\psi ^{**}\right) \Vert _{\mu }$ is uniformly bounded for all $n\in \mathbb {N}.$ Given any $\rho >0,$ it is not difficult to find an $M^*>0$ such that

$$\begin{aligned} e^{-\mu \vert z\vert }\vert \left( \phi _n^*(t,z),\psi _n^*(t,z)\right) -\left( \phi ^{**}(t,z),\psi ^{**}(t,z)\right) \vert <\rho \end{aligned}$$

for any $t\in [0,T], \vert z\vert >M^*$ and $n\in \mathbb {N}.$ On the other hand, by virtue of (2.22), there exists $H\in \mathbb {N}$ such that

$$\begin{aligned} e^{-\mu \vert z\vert }\vert \left( \phi _n^*(t,z),\psi _n^*(t,z)\right) -\left( \phi ^{**}(t,z),\psi ^{**}(t,z)\right) \vert <\rho \end{aligned}$$

for any $t\in [0,T], z\in [-M^*,M^*]$ and $n>H.$ As a consequence, it follows from the above two inequalities that $\left( \phi _n^*(t,z),\psi _n^*(t,z)\right) \rightarrow \left( \phi ^{**}(t,z),\psi ^{**}(t,z)\right)$ with respect to the norm $\Vert \cdot \Vert _{\mu }.$ $\square$

To complete the proof of this section, we also need the following powerful lemma on the Harnack inequalities of cooperative parabolic systems, which is from Földes and Poláčik [17] (see also [41, 54]).

Lemma 2.9

([17]) Let the differential operators

$$\begin{aligned} \mathbf {L}_k:=\sum _{i,j=1}^{n}a_{i,j}^{k}(t,x)\frac{\partial ^2}{\partial _{x_i}\partial _{x_j}}+\sum _{i=1}^nb_i^k\frac{\partial }{\partial _{x_i}} -\frac{\partial }{\partial t},\ \ k=1,2,\ldots ,l, \end{aligned}$$

be uniformly parabolic in an open domain $(\tau ,M)\times \Omega$ of $(t,x)\in \mathbb {R}\times \mathbb {R}^n,$ that is, there is $\alpha _0>0$ such that $a_{i,j}^{k}(t,x)\xi _i\xi _j\ge \alpha _0\sum _{i=1}^n\xi _i^2$ for any n-tuples of real numbers $(\xi _1,\xi _2,\ldots ,\xi _n),$ where $-\infty<\tau <M\le +\infty$ and $\Omega$ is open and bounded. Suppose that $a_{i,j}^k,b_{i,j}^k\in C((\tau ,M)\times \Omega ,\mathbb {R})$ and

$$\begin{aligned} \max _{(t,\mathbf {x})\in (\tau ,M)\Omega }\vert b_i^k(t,\mathbf {x})\vert + \vert a_{ij}^k(t,\mathbf {x})\vert \le \beta _0 \end{aligned}$$

for some $\beta _0>0.$ Assume that

$$\begin{aligned} \mathbf {w}=(w_1,w_2,\ldots ,w_l)\in C((\tau ,M)\times {\bar{\Omega }},\mathbb {R}^l)\cap C^{1,2}((\tau ,M)\times \Omega ,\mathbb {R}^l) \end{aligned}$$

satisfies

$$\begin{aligned} \sum _{s=1}^lc^{k,s}(t,\mathbf {x})w_s+\mathbf {L}_kw_k\le 0,\ \ (t,\mathbf {x})\in (\tau ,M)\times \Omega ,\ k=1,2,\ldots ,l, \end{aligned}$$

(2.23)

where $c^{k,s}\in C((\tau ,M)\times \Omega ,\mathbb {R})$ and $c^{k,s}\ge 0$ if $k\not =s,$ and

$$\begin{aligned} \max _{t,\mathbf {x}\in (\tau ,M)\times \Omega }\vert c^{k,s}(t,\mathbf {x})\vert \le \gamma _0 \end{aligned}$$

(k,s=1,2,...,l) for some $\gamma _0>0.$ Let D and U be domains in $\Omega$ such that $D\subset \subset U,\ \mathrm{dist}({\bar{D}},$ $\partial U)>\varrho ,$ and $\vert D\vert >\epsilon$ for certain positive constants $\varrho$ and $\epsilon .$ Let $\theta$ be a positive constant with $\tau +4\theta <M.$ Then there exist positive constants $p,\omega _1$ and $\omega _2$ determined only by $\alpha _0,\beta _0,\gamma _0,\varrho ,\epsilon ,n,\mathrm{diam}\,\Omega$ and $\theta ,$ such that

$$\begin{aligned} \inf _{(\tau +3\theta ,\tau +4\theta )\times D}w_k\ge \omega _1\Vert (w_k)^+\Vert _{L^p((\tau +\theta ,\tau +2\theta )\times D)}-\omega _2\max _{j=1,2,\ldots ,k}\sup _{\partial _P((\tau ,\tau +4\theta )\times U)}(w_j)^-. \end{aligned}$$

Here $(w_k)^+=\max \{w_k,0\}, (w_k)^-=\max \{-w_k,0\}$ and $\partial _P((\tau ,\tau +4\theta )\times U)=\tau \times U\cup [\tau ,\tau +4\theta )\times \partial U.$ Moreover, if all inequalities in (2.23) are replaced by equalities, then the conclusion holds with $p=\infty$ and with $\omega _1,\omega _2$ independent of $\epsilon .$

Now we are ready to prove the main result of this section.

Theorem 2.10

Assume that $R_0>1.$ For any $c>c^*,$ system (1.2) admits a time periodic travelling wave solution $\left( \phi ^*,\psi ^*\right)$ satisfying (2.2). Furthermore, $0<\frac{1}{T}\int _0^T\psi ^*(t,z)dt\le S_0-S^\infty$ for any $z\in \mathbb {R},$ and

$$\begin{aligned} \frac{1}{T}\int _{-\infty }^{\infty }\int _0^T\gamma (t)\psi ^*(t,z)dtdz= \frac{1}{T}\int _{-\infty }^\infty \int _0^T\frac{\beta (t)\phi ^*(t,z)\psi ^*(t,z)}{\phi ^*(t,z)+\psi ^*(t,z)}dtdz=c[S_0-S^\infty ]. \end{aligned}$$

Proof

In view of Lemmas 2.7 and 2.8, the operator $\mathcal {F}$ is continuous and compact on $\mathcal {D}$ with respect to the norm $\Vert \cdot \Vert _\mu .$ Additionally, it is easy to verify that $\mathcal {D}$ is closed and convex. Then, the Schauder’s fixed point theorem implies that $\mathcal {F}$ has a fixed point $(\phi ^*,\psi ^*)\in \mathcal {D}$. Moreover, $(\phi ^*(T,\cdot ),\psi ^*(T,\cdot ))=(\phi ^*(0,\cdot ),\psi ^*(0,\cdot ))$ and $(\phi ^*,\psi ^*)$ satisfies that

$$\begin{aligned} {\left\{ \begin{array}{ll} \phi ^*(t)=T_1(t)\phi ^*(0)+\int _0^t T_1(t-s)f_1[\phi ^*,\psi ^*](s)ds,\\ \psi ^*(t)=T_2(t)\psi ^*(0)+\int _0^t T_2(t-s)f_2[\phi ^*,\psi ^*](s)ds \end{array}\right. } \end{aligned}$$

(2.24)

for $t\in [0,T]$. Define $(\hat{\phi }^*(t,z),\hat{\psi }^*(t,z))=(\phi ^*(t-kT,z),\psi ^*(t-kT,z))$ for any $t\in \mathbb {R}$ and $z\in \mathbb {R},$ where $k\in \mathbb {Z}$ satisfies $kT\le t<(k+1)T.$ Then we get $(\hat{\phi }^*(t+T,z),\hat{\psi }^*(t+T,z))=(\hat{\phi }^*(t,z),\hat{\psi }^*(t,z)), \forall (t,z)\in \mathbb {R}\times \mathbb {R}.$ Since $(\phi ^*,\psi ^*)\in C^{\theta ,2\theta }([0,T]\times \mathbb {R},\mathbb {R}^2)$ for some $\theta \in (0,1),$ we have $(\hat{\phi }^*,\hat{\psi }^*)\in C^{\theta ,2\theta }(\mathbb {R}\times \mathbb {R},\mathbb {R}^2)$. Due to the T-periodicity of ${\hat{\phi }}^*$ and ${\hat{\psi }}^*$, we see that $(\hat{\phi }^*,\hat{\psi }^*)$ satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} {\hat{\phi }}^*(t)=T_1(t){\hat{\phi }}^*(0)+\int _0^t T_1(t-s)f_1[{\hat{\phi }}^*,{\hat{\psi }}^*](s)ds,\\ {\hat{\psi }}^*(t)=T_2(t){\hat{\psi }}^*(0)+\int _0^t T_2(t-s)f_2[{\hat{\phi }}^*,{\hat{\psi }}^*](s)ds \end{array}\right. } \end{aligned}$$

(2.25)

for $t\in \mathbb {R}$. Denote $(\hat{\phi }^*,\hat{\psi }^*)$ by $(\phi ^*,\psi ^*)$ again. It follows from [31, Theorem 5.1.2, 5.1.3 and 5.1.4] that $(\phi ^*,\psi ^*)\in C^{1,2+2\theta }(\mathbb {R}\times \mathbb {R},\mathbb {R}^2)$ satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \phi ^*_t(t,z)=d_1\phi ^*_{zz}(t,z)-c\phi ^*_z(t,z) -\frac{\beta (t)\phi ^*(t,z)\psi ^*(t,z)}{\phi ^*(t,z)+\psi ^*(t,z)},\ \ &{} t\in \mathbb {R},\, z\in \mathbb {R},\\ \psi ^*_t(t,z)=d_2\psi ^*_{zz}(t,z)-c\psi ^*_z(t,z) +\frac{\beta (t)\phi ^*(t,z)\psi ^*(t,z)}{\phi ^*(t,z)+\psi ^*(t,z)} -\gamma (t)\psi ^*(t,z),\,\ \ &{} t\in \mathbb {R},\, z\in \mathbb {R} \end{array}\right. } \end{aligned}$$

(2.26)

and

$$\begin{aligned} \Vert \phi ^*\Vert _{C^{1,2+2\theta }(\mathbb {R}\times \mathbb {R},\mathbb {R})}+ \Vert \psi ^*\Vert _{C^{1,2+2\theta }(\mathbb {R}\times \mathbb {R},\mathbb {R})}<\infty \end{aligned}$$

(2.27)

for some $\theta \in (0,1).$

Next, we need to verify that $(\phi ^*,\psi ^*)$ satisfies the boundary conditions (2.2). By the definitions of $\phi ^{\pm }$ and $\psi ^{\pm }$, it follows that $\phi ^*(t,z)\rightarrow S_0$ and $\psi ^*(t,z)\rightarrow 0$ uniformly for $t\in \mathbb {R}$, as $z\rightarrow -\infty .$ On the other hand, by the estimate (2.27) and Landau type inequalities (see, e.g., [23] or [4]), we have

$$\begin{aligned} \left| \phi ^*_z\right| _{L^\infty ([0,T]\times (-\infty ,M])}\le 2 \left| \phi ^*-S_0\right| _{L^\infty ([0,T]\times (-\infty ,M])}^{\frac{1}{2}}\left| \phi ^*_{zz}\right| _{L^\infty ([0,T]\times (-\infty ,M])}^{\frac{1}{2}} \end{aligned}$$

and

$$\begin{aligned} \left| \psi ^*_z\right| _{L^\infty ([0,T]\times (-\infty ,M])}\le 2 \left| \psi ^*\right| _{L^\infty ([0,T]\times (-\infty ,M])}^{\frac{1}{2}}\left| \psi ^*_{zz}\right| _{L^\infty ([0,T]\times (-\infty ,M])}^{\frac{1}{2}}. \end{aligned}$$

As a result,

$$\begin{aligned} \lim _{z\rightarrow -\infty }(\phi _z^*(t,z),\psi _z^*(t,z))=(0,0)\ \ \mathrm{uniformly\ for}\ t\in \mathbb {R}. \end{aligned}$$

We further discuss the asymptotic behavior of $\phi ^*_{zz}$ and $\psi ^*_{zz}$ when z tends to $-\infty .$ By the (strong) maximum principle, it follows that $\phi ^*(t,x)>0,\psi ^*(t,x)>0,\forall t>0, x\in \mathbb {R}.$ Differentiating two side of the first equation of (2.26) with respect to z yields

$$\begin{aligned} (\phi ^*_z)_t=d_1(\phi ^*_z)_{zz}-c(\phi ^*_z)_z -\frac{\beta (t)\phi ^*_z(\psi ^*)^2+\psi ^*_z(\phi ^*)^2}{(\phi ^*+\psi ^*)^2},\ \ t>0,\, z\in \mathbb {R}. \end{aligned}$$

(2.28)

Since $\phi ^*_z\in C^{\theta ,2\theta }(\mathbb {R}\times \mathbb {R},\mathbb {R}^2)$ for some $\theta \in (0,1),$ it follows from the T-periodicity of $\phi ^*$ and [31, Theorems 5.1.3 and 5.1.4] that $\phi ^*_z\in C^{1,2+2\theta }(\mathbb {R}\times \mathbb {R},\mathbb {R})$ and

$$\begin{aligned} \Vert \phi ^*_z\Vert _{C^{1,2+2\theta }(\mathbb {R}\times \mathbb {R},\mathbb {R})}<\infty \end{aligned}$$

for some $\theta \in (0,1).$ By a similar argument to $\phi ^*$, we can conclude from the Landau type inequality that

$$\begin{aligned} \lim _{z\rightarrow -\infty }\phi _{zz}^*(t,z)=0\ \ \mathrm{uniformly\ for}\ t\in \mathbb {R}. \end{aligned}$$

Similarly, we have

$$\begin{aligned} \lim _{z\rightarrow -\infty }\psi _{zz}^*(t,z)=0\ \ \mathrm{uniformly\ for}\ t\in \mathbb {R}. \end{aligned}$$

Consequently, we can see from the system (2.26) that

$$\begin{aligned} \lim _{z\rightarrow -\infty }(\phi _t^*(t,z),\psi _t^*(t,z))=(0,0)\ \ \mathrm{uniformly\ for}\ t\in \mathbb {R}. \end{aligned}$$

Define $\Phi (z)=\frac{1}{T}\int _0^T\phi ^*(t,z)dt$. Clearly, $\Phi _z(z)\rightarrow 0$ as $z\rightarrow -\infty .$ It then follows from the first equation of system (2.26) that

$$\begin{aligned} c\Phi _z=d_1\Phi _{zz}-\frac{1}{T}\int _0^T \frac{\beta (t)\phi ^*(t,z)\psi ^*(t,z)}{\phi ^*(t,z)+\psi ^*(t,z)}dt. \end{aligned}$$

(2.29)

Integrating two sides of (2.29) from y to z and letting $y\rightarrow -\infty$ yield

$$\begin{aligned} d_1\Phi _z(z)=c\left[ \Phi (z)-S_0\right] + \frac{1}{T}\int _{-\infty }^{z}\int _0^T \frac{\beta (t)\phi ^*(t,y)\psi ^*(t,y)}{\phi ^*(t,y)+\psi ^*(t,y)}dtdy. \end{aligned}$$

(2.30)

Due to the uniform boundedness of $\phi ^*(t,z)$ and $\phi _z^*(t,z)$, it is easy to see that $\Phi (z)=\frac{1}{T}\int _0^T\phi ^*(t,z)dt$ and $\Phi _z(z)=\frac{1}{T}\int _0^T\phi _z^*(t,z)dt$ are uniformly bounded, respectively, and hence, $\frac{1}{T}\int _0^T \frac{\beta (t)\phi ^*(t,z)\psi ^*(t,z)}{\phi ^*(t,z)+\psi ^*(t,z)}dt$ is integrable on $\mathbb {R}.$ From (2.29), we have

$$\begin{aligned} \left( e^{-cz/d_1}\Phi _z\right) _z=e^{-cz/d_1}\left( \Phi _{zz}-c\Phi _z/d_1 \right) =\frac{e^{-cz/d_1}}{d_1T}\int _0^T\frac{\beta (t)\phi ^*(t,z)\psi ^*(t,z)}{\phi ^*(t,z)+\psi ^*(t,z)}dt. \end{aligned}$$

For the above equality, an integration from z to $\infty$ gives

$$\begin{aligned} e^{-cz/d_1}\Phi _z(z)=-\frac{1}{d_1T} \int _z^\infty e^{-cy/d_1}\int _0^T \frac{\beta (t)\phi ^*(t,y)\psi ^*(t,y)}{\phi ^*(t,y)+\psi ^*(t,y)}dtdy, \end{aligned}$$

which implies that $\Phi _z(z)<0$ for $z\in \mathbb {R}.$ It follows that $\Phi (+\infty )$ exists and $\Phi (+\infty )<\Phi (-\infty )=S_0.$ With the aid of Barbălat’s lemma (see, e.g., [3, 12]), we have $\Phi _z(z)\rightarrow 0$ as $z\rightarrow \infty .$ Furthermore, letting $z\rightarrow \infty$ in (2.30) yields

$$\begin{aligned} \frac{1}{T}\int _{-\infty }^\infty \int _0^T\frac{\beta (t)\phi ^*(t,z)\psi ^*(t,z)}{\phi ^*(t,z)+\psi ^*(t,z)}dtdz=c[S_0-\Phi (\infty )]=c[S_0-S^{\infty }], \end{aligned}$$

where $S^\infty :=\Phi (\infty )<S^0.$

In the following, we explore the asymptotic behavior of $\psi ^*(t,z)$ as $z\rightarrow \infty .$ Let ${\hat{\gamma }}:=\min _{t\in [0,T]}\gamma (t)$ and $\tilde{\gamma }=\max _{t\in [0,T]}\gamma (t),$ and define $\Psi (z)=\frac{1}{T}\int _0^T\psi ^*(t,z)dt$. Then $\Psi (z)$ satisfies

$$\begin{aligned} -d_2\Psi _{zz}+c\Psi _z+{\hat{\gamma }}\Psi = \frac{1}{T}\int _0^T\frac{\beta (t)\phi ^*(t,z)\psi ^*(t,z)}{\phi ^*(t,z)+\psi ^*(t,z)}dt-\frac{1}{T}\int _0^T\left( \gamma (t)-{\hat{\gamma }}\right) \psi ^*(t,z)dt. \end{aligned}$$

(2.31)

Denote by

$$\begin{aligned} \hat{\lambda }^{\pm }:=\frac{c\pm \sqrt{c^2+4d_2{\hat{\gamma }}}}{2d_2} \end{aligned}$$

the two roots of the characteristic equation

$$\begin{aligned} -d_2\lambda ^2+c\lambda +{\hat{\gamma }}=0. \end{aligned}$$

In addition, denote

$$\begin{aligned} \hat{\rho }:=d_2\left( {\hat{\lambda }}^+-{\hat{\lambda }}^-\right) =\sqrt{c^2+4d_2{\hat{\gamma }}}. \end{aligned}$$

It is easy to see that ${\hat{\lambda }}^-<0<{\hat{\lambda }}^+$. Since $\frac{1}{T}\int _0^T\frac{\beta (t)\phi ^*(t,z)\psi ^*(t,z)}{\phi ^*(t,z)+\psi ^*(t,z)}dt\le \frac{S_0}{T}\int _0^T\beta (t)dt=\overline{\beta }S_0,$ we see from (2.31) that

$$\begin{aligned} \Psi (z)= & {} \frac{1}{{\hat{\rho }} T}\int _{-\infty }^ze^{\hat{\lambda }^-(z-y)}\left[ \int _0^T\frac{\beta (t)\phi ^*(t,y)\psi ^*(t,y)}{\phi ^*(t,y)+\psi ^*(t,y)}-\int _0^T\left( \gamma (t)-{\hat{\gamma }}\right) \psi ^*(t,z) \right] dtdy\\&+\frac{1}{{\hat{\rho }} T}\int ^{\infty }_ze^{\hat{\lambda }^+(z-y)}\left[ \int _0^T\frac{\beta (t)\phi ^*(t,y)\psi ^*(t,y)}{\phi ^*(t,y)+\psi ^*(t,y)}-\int _0^T\left( \gamma (t)-{\hat{\gamma }}\right) \psi ^*(t,z) \right] dtdy\\\le & {} \frac{1}{{\hat{\rho }} T}\int _{-\infty }^ze^{\hat{\lambda }^-(z-y)} \int _0^T\frac{\beta (t)\phi ^*(t,y)\psi ^*(t,y)}{\phi ^*(t,y)+\psi ^*(t,y)}dtdy\\&+\frac{1}{{\hat{\rho }} T}\int ^{\infty }_ze^{\hat{\lambda }^+(z-y)} \int _0^T\frac{\beta (t)\phi ^*(t,y)\psi ^*(t,y)}{\phi ^*(t,y)+\psi ^*(t,y)}dtdy\\= & {} \frac{1}{{\hat{\rho }} T}\int _{0}^{\infty }e^{\hat{\lambda }^-y} \int _0^T\frac{\beta (t)\phi ^*(t,z-y)\psi ^*(t,z-y)}{\phi ^*(t,z-y)+\psi ^*(t,z-y)}dtdy\\&+\frac{1}{{\hat{\rho }} T}\int _{-\infty }^0e^{\hat{\lambda }^+y} \int _0^T\frac{\beta (t)\phi ^*(t,z-y)\psi ^*(t,z-y)}{\phi ^*(t,z-y)+\psi ^*(t,z-y)}dtdy. \end{aligned}$$

Integrating $\Psi (z)$ from $\zeta$ to $\xi$, we obtain

$$\begin{aligned} \int _\zeta ^\xi \Psi (z)dz= & {} \frac{1}{{\hat{\rho }} T}\int _{0}^{\infty }e^{\hat{\lambda }^-y}\int _\zeta ^\xi \int _0^T\frac{\beta (t)\phi ^*(t,z-y)\psi ^*(t,z-y)}{\phi ^*(t,z-y)+\psi ^*(t,z-y)}dtdzdy\\&+\frac{1}{{\hat{\rho }} T}\int _{-\infty }^0e^{\hat{\lambda }^+y} \int _\zeta ^\xi \int _0^T \frac{\beta (t)\phi ^*(t,z-y)\psi ^*(t,z-y)}{\phi ^*(t,z-y)+\psi ^*(t,z-y)}dtdzdy. \end{aligned}$$

Note that $\int _0^T\frac{\beta (t)\phi ^*\psi ^*}{\phi ^*+\psi ^*}dt$ is integrable on $\mathbb {R}.$ It then follows from Fubini’s theorem that $\Psi (z)$ is integral on $\mathbb {R}$, and

$$\begin{aligned} \int _{-\infty }^\infty \Psi (z)dz\le \frac{1}{\hat{\gamma T}}\int _{-\infty }^\infty \int _0^T\frac{\beta (t)\phi ^*(t,z)\psi ^*(t,z)}{\phi ^*(t,z)+\psi ^*(t,z)}dtdz. \end{aligned}$$

In view of (2.27), it is easy to see that $\Psi _z(z)$ is uniformly bounded on $\mathbb {R}$, and hence, Barbălat’s lemma guarantees that $\Psi (z)\rightarrow 0$ as $z\rightarrow \infty .$ On the other hand, for the second equation of system (2.26), applying Lemma 2.9 with $\tau =-T, \theta =T$ and $D:=D_z=(z-\frac{1}{4},z+\frac{1}{4}), U=(z-\frac{1}{2},z+\frac{1}{2}), \Omega =(z-1,z+1)$ with $z\in \mathbb {R},$ we have

$$\begin{aligned} \sup _{(0,T)\times D}\psi ^*(t,y)\le & {} C_0\inf _{(2T,3T)\times D}\psi ^*(t,z)\\= & {} C_0\min _{[2T,3T]\times \overline{D}}\psi ^*(t,y)\\\le & {} C_0\min _{\overline{D}}\psi ^*(0,y), \end{aligned}$$

where $C_0$ is a positive constant independent of D. Due to the periodicity of $\psi ^*$ in time t, we see that $\psi ^*(t,z)\rightarrow 0$ uniformly for $t\in \mathbb {R},$ as $z\rightarrow \infty .$

We further prove that $\phi ^*(t,z)\rightarrow S^\infty$ uniformly for $t\in \mathbb {R}$, as $z\rightarrow \infty$. On the basis of the T-periodicity of $\phi ^*,$ it suffices to show

$$\begin{aligned} \limsup _{z\rightarrow \infty }\max _{t\in [0,T]}\phi ^*(t,z)=:S_+^\infty =S^\infty =S_-^\infty :=\liminf _{z\rightarrow \infty }\min _{t\in [0,T]}\phi ^*(t,z). \end{aligned}$$

It is clear that there exist $\{t_n\}$ and $\{z_n\}$ satisfying $\{t_n\}\subset [0,T]$ and $z_n\rightarrow \infty$ (as $n\rightarrow \infty$), respectively, such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\phi ^*(t_n,z_n)=S_+^\infty . \end{aligned}$$

Let $\phi _n(t,z)=\phi ^*(t+t_n,z+z_n), \psi _n(t,z)=\psi ^*(t+t_n,z+z_n), \forall n\in \mathbb {N}, t\in \mathbb {R}, z\in \mathbb {R}.$ Due to the estimation (2.27), there exists a subsequence of $(\phi _n(t,z),\psi _n(t,z))$, still denoted by $(\phi _n(t,z),\psi _n(t,z))$, converging to $(\phi _*(t,z),0)$ in $C_\mathrm{loc}^{\theta ,2\theta }(\mathbb {R}\times \mathbb {R})$ for some $\theta \in (0,1)$, as $n\rightarrow \infty .$ In particularly, we have $\phi _*(0,0)=S_+^\infty$ and

$$\begin{aligned} \phi _*(t+T,z)=\phi _*(t,z),\quad \phi _*(t,z)\le S_+^\infty , \quad \forall (t,z)\in \mathbb {R}\times \mathbb {R}. \end{aligned}$$

Since $\{t_n\}\subset [0,T]$, without loss of generality, let $t_n\rightarrow t^*\in [0,T].$ Then $\phi _*^+(t,z)=\phi _*(t-t^*,z)$ satisfies

$$\begin{aligned} \begin{aligned} \phi _*^+(t)=&T_1(t)\phi _*^+(0)+\int _0^tT_1(t-s)f_1[\phi _*^+,0](s)ds\\ =&T_1(t)\phi _*^+(0)+\int _0^tT_1(t-s)\alpha _1\phi _*^+(s)ds. \end{aligned} \end{aligned}$$

Consequently, $\phi _*^+(t,z)$ satisfies

$$\begin{aligned} \partial _t\phi _*^+(t,z)=d_1\partial _{zz}\phi _*^+(t,z) -c\partial _z\phi _*^+(t,z),\ \ (t,z)\in \mathbb {R}\times \mathbb {R}. \end{aligned}$$

Since $\phi _*^+(t^*,0)=S_+^\infty$ and $\phi _*^+(t,z)\le S_+^\infty$, it follows from the maximum principle that $\phi _*^+(t,z)\equiv S_+^\infty$ for $t<t^*.$ By the T-periodicity of $\phi _*^+(\cdot ,z)$, we have $\phi _*^+(t,z)\equiv S_+^\infty , \forall t\in \mathbb {R}$, and hence $\Phi _*^+(z):=\frac{1}{T}\int _0^T\phi _*^+(t,z)dt\equiv S_+^\infty .$ On the other hand,

$$\begin{aligned} \begin{aligned} \Phi _*^+(z)=&\frac{1}{T}\int _0^T\phi _*^+(t,z)dt =\frac{1}{T}\int _0^T\phi _*(t-t^*,z)dt\\ =&\lim _{n\rightarrow \infty }\frac{1}{T}\int _0^T \phi _n(t-t^*,z)dt\\ =&\lim _{n\rightarrow \infty }\frac{1}{T}\int _0^T \phi ^*(t-t^*+t_n,z+z_n)dt\\ =&S^\infty , \end{aligned} \end{aligned}$$

which implies $S_+^\infty =S^\infty .$ Therefore, $\limsup _{z\rightarrow \infty }\max _{t\in [0,T]}\phi ^*(t,z)=S^\infty .$ Similarly, we can prove $\liminf _{z\rightarrow \infty }\min _{t\in [0,T]}\phi ^*(t,z)=S^\infty .$ This implies that $\phi _*^+(t,z)$ converges to $S^\infty$ uniformly in $t\in \mathbb {R}$ as $z\rightarrow \infty$.

Moreover, since $\Psi (z)$ satisfies

$$\begin{aligned} -d_2\Psi _{zz}+c\Psi _z= \frac{1}{T}\int _0^T\frac{\beta (t)\phi ^*(t,z)\psi ^*(t,z)}{\phi ^*(t,z)+\psi ^*(t,z)}dt-\frac{1}{T}\int _0^T\gamma (t)\psi ^*(t,z)dt, \end{aligned}$$

(2.32)

by making an integration of (2.32) on $\mathbb {R}$, we get

$$\begin{aligned} \frac{1}{T}\int _{-\infty }^{\infty }\int _0^T\gamma (t)\psi ^*(t,z)dtdz= \frac{1}{T}\int _{-\infty }^\infty \int _0^T\frac{\beta (t)\phi ^*(t,z)\psi ^*(t,z)}{\phi ^*(t,z)+\psi ^*(t,z)}dtdz=c[S_0-S^\infty ]. \end{aligned}$$

It remains to prove that $0<\frac{1}{T}\int _0^T\psi ^*(t,z)dt\le S_0-S^\infty .$ In order to achieve this, we shall use a similar argument to the proof of [45, Theorem 2.9]. First, by similar arguments to the proof of the asymptotic behavior of $\phi ^*_z(t,z)$ and $\phi ^*_{zz}(t,z)$ as $z\rightarrow -\infty$, we can show that

$$\begin{aligned} \lim _{z\rightarrow \infty }\psi _z^*(t,z) =\lim _{z\rightarrow \infty }\psi _{zz}^*(t,z)=0 \end{aligned}$$

uniformly for $t\in \mathbb {R}$. Thus, we have

$$\begin{aligned} \lim _{z\rightarrow \pm \infty }\psi _z^*(t,z) =\lim _{z\rightarrow \pm \infty }\psi _{zz}^*(t,z)=0 \end{aligned}$$

(2.33)

uniformly for $t\in \mathbb {R}$. For any $z\in \mathbb {R},$ we define a function

$$\begin{aligned} \Psi ^*(z)=\frac{1}{cT}\int _{-\infty }^z\int _0^T\gamma (t)\psi ^*(t,y)dtdy +\frac{1}{cT}\int ^{\infty }_z e^{c/d_2(z-y)}\int _0^T\gamma (t)\psi ^*(t,y)dtdy. \end{aligned}$$

(2.34)

It is easy to see that $\Psi ^*(z)$ satisfies the following equation:

$$\begin{aligned} c\Psi ^*_z(z)=d_2\Psi ^*_{zz}(z)+\frac{1}{T}\int _0^T\gamma (t)\psi ^*(t,y)dt,\ \ \forall z\in \mathbb {R}. \end{aligned}$$

By means of (2.33) and L’Hôpital’s rule, it follows that

$$\begin{aligned} \lim _{z\rightarrow -\infty }\Psi ^*(z)=0,\ \lim _{z\rightarrow \infty }\Psi ^*(z) =\frac{1}{cT}\int _{-\infty }^\infty \int _0^T\gamma (t)\psi ^*(t,y)dy=S_0-S^\infty \end{aligned}$$

and

$$\begin{aligned} \lim _{z\rightarrow \pm \infty }\Psi ^*_z(z)=0. \end{aligned}$$

Recall that $\Psi (z)=\frac{1}{T}\int _0^T\psi ^*(t,z)dt.$ We further introduce a function

$$\begin{aligned} {\hat{\Psi }}(z):=\Psi (z)+\Psi ^*(z), \ \ \forall z\in \mathbb {R}. \end{aligned}$$

Consequently, it is not difficult to obtain from (2.32) and (2.34) that

$$\begin{aligned} c{{\hat{\Psi }}}_z(z)=d_2{{\hat{\Psi }}}_{zz}(z)+\frac{1}{T}\int _0^T\frac{\beta (t)\phi ^*(t,z)\psi ^*(t,z)}{\phi ^*(t,z)+\psi ^*(t,z)}dt,\ \ \forall z\in \mathbb {R}. \end{aligned}$$

Multiplying two sides of the above equation by $e^{-c/d_2z}$ and integrating from z to $\infty ,$ we have

$$\begin{aligned} {{\hat{\Psi }}}_{z}(z)=\frac{1}{d_2T}\int _z^\infty e^{c/d_2(z-y)}\int _0^T\frac{\beta (t)\phi ^*(t,z)\psi ^*(t,z)}{\phi ^*(t,z)+\psi ^*(t,z)}dt. \end{aligned}$$

This implies that ${{\hat{\Psi }}}(z)$ is non-decreasing in $\mathbb {R}.$ Note that $\lim _{z\rightarrow \infty }{{\hat{\Psi }}}(z)=S_0-S^\infty .$ Hence, ${{\hat{\Psi }}}(z)\le S_0-S^\infty$ for all $z\in \mathbb {R}.$ In view of the definition of ${{\hat{\Psi }}}(z)$ and $\Psi ^*(z)$, we conclude that $\Psi (z)\le {{\hat{\Psi }}}(z)\le S_0-S^\infty$ for all $z\in \mathbb {R},$ that is, $0\le \frac{1}{T}\int _0^T\psi ^*(t,z)dt\le S_0-S^\infty$ for any $z\in \mathbb {R}.$ $\square$

3 The Nonexistence of Periodic Traveling Waves

In this section, we prove the nonexistence of time periodic traveling waves for two cases. In the case where $R_0\le 1,$ there is no time periodic traveling wave. In the case where $R_0>1$ and $c<c^*,$ there is no time periodic, non-trivial and non-negative travelling waves.

Theorem 3.1

Assume that $R_0=\frac{\int _0^T\beta (t)dt}{\int _0^T\gamma (t)dt}\le 1.$ Then for any $c\ge 0,$ there is no time periodic traveling wave solutions $(\phi ,\psi )$ satisfying

$$\begin{aligned} \phi (t,-\infty )=S_0,\, \phi (t,\infty )=S^\infty <S_0,\, \psi (t,\pm \infty )=0\ \mathrm{uniformly\ in}\, t\in \mathbb {R}. \end{aligned}$$

(3.1)

Proof

Suppose, by way of contradiction, that there exists a time periodic, non-trivial and non-negative solution $\left( \phi (t,z),\psi (t,z)\right)$ of (2.1) with (3.1). Then there exists a positive constant b such that $0\le \phi (t,z)\le b, \forall t\ge 0, x\in \mathbb {R}$, and hence,

$$\begin{aligned} \psi _t(t,z)= & {} d_2\psi _{zz}(t,z)-c\psi _{z}(t,z)+\frac{\beta (t)\phi (t,z)\psi (t,z)}{\phi (t,z)+\psi (t,z)}-\gamma (t)\psi (t,z)\\\le & {} d_2\psi _{zz}(t,z)-c\psi _{z}(t,z)+\left[ \frac{b\beta (t)}{b+\psi (t,z)} -\gamma (t)\right] \psi (t,z). \end{aligned}$$

for any $t>0$ and $z\in \mathbb {R}.$ Let $\eta :=\sup _{z\in \mathbb {R}}\psi (0,z)<\infty .$ Then $\psi (0,z)\le \eta ,\forall z\in \mathbb {R}.$ By the comparison principle, we have

$$\begin{aligned} \psi (t,z)\le v(t;\eta ),\quad \forall t>0,\,z\in \mathbb {R}, \end{aligned}$$

where $v(t;\eta )$ is the solution of the following ordinary differential equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} v'(t)=\left[ \frac{b\beta (t)}{b+v(t)}-\gamma (t)\right] v(t),\ \ t>0,\\ v(0)=\eta . \end{array}\right. } \end{aligned}$$

Since $R_0\le 1,$ we have $\frac{1}{T}\int _0^T\left( \beta (t)-\gamma (t)\right) dt\le 0.$ Set

$$\begin{aligned} p(t,v)=\frac{b\beta (t)}{b+v(t)}-\gamma (t). \end{aligned}$$

Then we have

$$\begin{aligned} \int _0^Tp(t,0)dt=\frac{1}{T}\int _0^T\left( \beta (t)-\gamma (t)\right) dt\le 0. \end{aligned}$$

Hence, [56, Theorem 3.1.2] implies that $\lim _{t\rightarrow \infty }v(t;\eta )\!=\!0.$ It follows that $\lim _{t\rightarrow \infty }\psi (t,z)$ $=0, \forall z\in \mathbb {R},$ which contradicts to the time periodicity of $\psi (t,\cdot )$ in t. $\square$

Next, we prove the non-existence of periodic traveling waves for the case where $R_0>1$ and $c<c^*.$ We first consider the following scalar periodic reaction–diffusion equation:

$$\begin{aligned} \frac{\partial u}{\partial t}=du_{xx}+f(t,u),\ \ t>0,\ x\in \mathbb {R}, \end{aligned}$$

(3.2)

where $d>0, f\in C^1(\mathbb {R}_+\times \mathbb {R}_+,\mathbb {R}_+),$ and $f(t,\cdot )$ is T-periodic in t for some $T>0.$ Assume that

• (A1) $f(t,0)=0$ for $t\ge 0,$ and there is a real number $H>0$ such that $f(t,H)\le 0$, and for each $t\ge 0, f(t,\cdot )$ is strictly subhomogeneous on [0, H] in the sense that $f(t,\alpha u)>\alpha f(t,u)$ whenever $\alpha \in (0,1), u\in (0,H].$

• (A2) $\overline{f_u(t,0)}:=\frac{1}{T}\int _0^T\frac{\partial f(t,0)}{\partial u}dt>0.$

By [56, Theorem 3.1.2], it follows that the periodic ordinary differential equation

$$\begin{aligned} \frac{du}{dt}=f(t,u),\ \ t\ge 0 \end{aligned}$$

(3.3)

has a unique positive T-periodic solution q(t) with $q(t)\in [0,H], \forall t\in [0,T],$ and q(t) is globally asymptotically stable in (0, H]. By the same arguments as in [27, Sect. 4] (just letting $\tau =0$ in Theorems 4.1 and 4.2), we have the following two results.

Proposition 3.2

Assume that (A1) and (A2) hold. Let $c^*=2\sqrt{d\cdot \overline{f_u(t,0)}}$ and $u(t,x,\varphi )$ be the solution of equation (3.2) with the initial data $\varphi$. Then the following statements are valid:

• (1) For any $c>c^*,$ if $\varphi \in C_{q(0)}=\{\varphi \in C(\mathbb {R},\mathbb {R}): 0\le \varphi (x)\le q(0), \forall x\in \mathbb {R}\}$ with $\varphi (x)<q(0),\forall x\in \mathbb {R},$ and $\varphi (x)=0$ for x outside a bounded interval, then $\lim _{t\rightarrow \infty ,\vert x\vert \ge ct}u(t,x,\varphi )=0.$

• (2) For any $c<c^*,$ if $\varphi \in C_{q(0)}$ with $\varphi \not \equiv 0,$ then $\lim _{t\rightarrow \infty ,\vert x\vert \le ct}\left( u(t,x,\varphi )-q(t)\right) =0.$

Proposition 3.3

Assume that (A1) and (A2) hold. Let $c^*$ be defined as in Proposition 3.2. Then $c^*$ is the minimal wave speed for the monotone periodic traveling waves $U(t,x+ct)$ of equation (3.2) connecting q(t) to 0.

Now we are in a position to prove the non-existence of periodic traveling wave solutions in the case where $R_0>1$ and $0<c<c^*.$

Theorem 3.4

Assume that $R_0>1$ and $0<c<c^*.$ Then there is no time-periodic traveling waves $(\phi ,\psi )$ satisfying

$$\begin{aligned} \phi (t,-\infty )=S_0,\, \phi (t,\infty )=S^\infty >0,\, \psi (t,\pm \infty )=0\ \mathrm{uniformly\ in}\, t\in \mathbb {R}. \end{aligned}$$

(3.4)

Proof

Suppose, by contradiction, that there exists such a traveling wave satisfying (3.4) for some $c<c^*.$ Then there exists $a>0$ such that $\phi (t,x+ct)\ge a>0, \forall t\ge 0,x\in \mathbb {R}.$ It follows that $v(t,x):=\psi (t,x+ct)$ satisfies

$$\begin{aligned} v_t\ge d_2v_{xx}+\frac{a\beta (t)}{a+v(t,x)}v(t,x)-\gamma (t)v(t,x),\ \ t\ge 0,\,x\in \mathbb {R}. \end{aligned}$$

Note that $R_0>1$ implies that (A2) holds. Let $q^a(t)$ be the unique positive T-periodic solution of

$$\begin{aligned} u'(t)=-\gamma (t)u(t)+\frac{a\beta (t)}{a+u(t)}u(t),\ \ t>0 \end{aligned}$$

and choose a continuous function $\psi _0(x)$ such that $0\le \psi _0(x)\le q^a(0)$ and $\psi _0(x)\le \psi (0,x), \forall x\in \mathbb {R},$ and $\psi _0\not \equiv 0.$ Then the comparison principle implies that

$$\begin{aligned} v(t,x)=\psi (t,x+ct)\ge u(t,x,\psi _0),\ \ \forall t\ge 0,\,x\in \mathbb {R}, \end{aligned}$$

(3.5)

where $u(t,x,\psi _0)$ is the unique solution of the following scalar reaction–diffusion equation

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t=d_2u_{xx}+\frac{a\beta (t)}{a+u(t,x)}u(t,x)-\gamma (t)u(t,x),\ \ t>0,\,x\in \mathbb {R},\\ u(0,x)=\psi _0(x),\ \ x\in \mathbb {R}. \end{array}\right. } \end{aligned}$$

(3.6)

By Proposition 3.2, $c^*=2\sqrt{d_2\cdot \overline{\beta (t)-\gamma (t)}}$ is the spreading speed of system (3.6). Fix a real number ${\bar{c}}\in (c,c^*).$ It then follows from Proposition 3.2(2) that

$$\begin{aligned} \lim _{t\rightarrow \infty ,\vert x\vert \le {{\bar{c}}}t}\left( u(t,x,\psi _0)-q^a(t)\right) =0. \end{aligned}$$

(3.7)

Since $q^a(t)$ is T-periodic, letting $t=nT, x=-{\bar{c}}t$ in (3.7), we obtain

$$\begin{aligned} \lim _{n\rightarrow \infty }u(nT,-{\bar{c}}nT,\psi _0)=q^a(0). \end{aligned}$$

In view of (3.5), we have

$$\begin{aligned} \psi (nT,(c-{\bar{c}})nT)\ge u(nT,-{\bar{c}}nT,\psi _0),\ \ \forall n\ge 1. \end{aligned}$$

It then follows that

$$\begin{aligned} \psi (0,-\infty )=\lim _{n\rightarrow \infty }\psi (0,(c-{\bar{c}})nT)=\lim _{n\rightarrow \infty }\psi (nT,(c-{\bar{c}})nT)\ge q^a(0)>0, \end{aligned}$$

which contradicts $\psi (0,-\infty )=0.$ $\square$