Abstract
In this paper, we study the time periodic traveling wave solutions for a Kermack–McKendrick SIR epidemic model with individuals diffusion and environment heterogeneity. In terms of the basic reproduction number \(R_0\) of the corresponding periodic ordinary differential model and the minimal wave speed \(c^*\), we establish the existence of periodic traveling wave solutions by the method of super- and sub-solutions, the fixed-point theorem, as applied to a truncated problem on a large but finite interval, and the limiting arguments. We further obtain the nonexistence of periodic traveling wave solutions for two cases involved with \(R_0\) and \(c^*\).
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1 Introduction
In this paper, we are interested in the following time periodic reaction–diffusion epidemic system
which describes the evolution of an epidemic within a spatially distributed population of individuals in a seasonal forcing environment. Here, S(t, x), I(t, x) and R(t, x) denote the densities of the susceptible, infected and recovered/removed individuals at time t and located at the spatial position \(x\in {\mathbb {R}}\), respectively. The positive constants \(d_1, d_2\) and \(d_3\) are the diffusion rates for the susceptible, infected and recovered/removed individuals, respectively. The disease transmission rate \(\beta (t)\) and the recovery rate \(\gamma (t)\) are all positive T-periodic continuous functions in t.
The kinetic system of (1.1) is
which has been deeply studied by Bacaër and Gomes [2], where they observed somewhat counterintuitive conclusions quite different from what is in a constant environment, that is, the classic Kermack–McKendrick SIR epidemic model [26] (see also [1, 5]):
but almost compatible with occurrence. The consequence in [2] implies that the behavior of epidemics under the seasonal forcing is not a straightforward generalization of the known results in a constant environment. In fact, it was reported that the transmission rates and the recovery rates of many epidemics can be significantly impacted by seasonality, see London and Yorke [30] for the yearly outbreaks of measles, chickenpox and mumps, and Hethcote and Yorke [21] for the seasonal oscillation of gonorrhea. In particular, London and Yorke [30] pointed out that there are two significant factors influencing the dynamics of epidemics and contributing to the one-year periodicity of the contact rate: (i) weather/climatic factors such as temperature and relative humidity; (ii) social behavior (contact patterns) influenced by public holidays (children due to school terms), vacations. Figure 1 in [8] also states that most human respiratory pathogens exhibit and annual increase in incidence each winter, although there are variations in the timing of onset and magnitude of the increase. For more on the impact of the seasonality in epidemic models, we refer to [19, 20, 32] and a review paper [6]. Here, we would like to emphasize that models (1.2) and (1.3) are usually used to describe the transmission of disease whose time scale is rather fast with respect to the vital dynamic of the population. Therefore, the vital dynamics is not incorporated into (1.2) and (1.3) and the total population number remains invariant in the transmission process of the epidemic. As mentioned above, the transmission dynamics of many epidemics such as measles, chickenpox, mumps and gonorrhea [21, 30] are significantly influenced by seasonality. On the other hand, the total number of the population usually remains (almost) invariant within several years. Thus, if we neglect (or do not consider) the effect of the vital dynamics of the population, then the system such as (1.2) (and (1.1)) is rather reasonable and should do duty for an admonition to interpret the epidemics influenced by seasonality.
To consider the propagation dynamics of (1.1), in which the random walk of individuals and the seasonality are incorporated, traveling wave solution is a key topic. For the autonomous version of system (1.1), namely
there has been extensive research on the traveling waves for the first two equations of (1.4) (the R equation can be decoupled). Kallen [24] and Kallen et al. [25] have studied the existence of traveling wave solutions when \(d_1=0\). Particularly, Hosnono and Ilyas [22] proved that there admits a pair of traveling wave solution \((S(x+ct),I(x+ct))\) satisfying \(S(-\infty )=S_0>0, S(+\infty )=S^\infty <S_0, I(\pm \infty )=0\) for each \(c\ge c^*=2\sqrt{\beta S_0d_2(1-\gamma /\beta S_0)}\) when the basic reproduction number \(R_0:=\frac{\beta S_0}{\gamma }\) of system (1.3) is larger than unit, which represents the transition from the initial disease-free equilibrium \((S_0,0,0)\) to another disease- free state \((S^\infty ,0,0)\) with \(S^\infty \) being determined by the model coefficients. Since then, there have been extensive investigations on traveling wave solutions of system (1.4) (see, e.g., [18, 23, 40] and references therein), and its variants such as age-infection structure [10, 11], delays or non-local delays [34], spatially discrete structure [17] and non-local dispersal case [37]. We also refer to [9] for the long-term behavior of (1.4) with spatial heterogeneity (\(d_1=0\)).
In the current work, we are concerned with time periodic traveling wave solutions (see the definition in the next section) for problem (1.1). Since system (1.1) involved with the same non-monotone structure as system (1.4), which implies that (1.1) does not have comparison principle, the theory and methods for monotone periodic systems (see, e.g., [13, 28, 41, 42]) are no longer effective. In addition, differently from system (1.4), problem (1.1) gives rise to a periodic parabolic system of wave profile, which leads to failure for the approaches in the aforementioned literatures to system (1.4). Recently, Wang et al. [35] studied time periodic traveling wave solutions for the following periodic and diffusive SIR model with standard incidence:
Here, S(t, x), I(t, x) and R(t, x) denote the densities of the susceptible, infected and recovered individuals at time t and in location x, respectively. The coefficients in (1.5) represent the same meaning as in system (1.1). It should be pointed out that the incidence in (1.5) reflects the recovered individuals is removed from the population and not involved in the contact and disease transmission (see [33]). They proved that if the basic reproduction number \(R_0:=\frac{\int _0^T\beta (t)dt}{\int _0^T\gamma (t)dt}\) of kinetic system of (1.5) is larger than unit, there exists a critical value \(c^*=2\sqrt{\frac{1}{d_2T}\int _0^T[\beta (t)-\gamma (T)]dt}\) such that for any wave speed \(c>c^*,\) system (1.5) admits a time periodic traveling wave solution. Furthermore, they obtained the nonexistence of periodic traveling wave solutions for two cases:(i) \(R_0\le 1;\) (ii) \(R_0>1\) and \(c<c^*.\) The literature [35] makes an elementary attempt and provides a novel train of thought to solve the existence of time periodic traveling wave solutions for periodic and non-monotone systems.
Note that mass action in (1.1) and standard incidence infection mechanism in (1.5) are widely adopted in modeling infectious diseases transmission. From the epidemiological perspective, the mass action is appropriate for modeling contact between infectious individuals and susceptible individuals in small population size, while utilizing the standard incidence frequently depends on population size, that is, it is suitable for larger population size. Another observation is that the basic reproduction number of the kinetic system associated with (1.1) is dependent on population size (see Sect. 2), while the basic reproduction number of kinetic system of (1.5) is independent of population size. The aforementioned difference on two incidence functions leads to some distinction on mathematical analysis in the corresponding models. In addition, in view of the bilinear incidence (or mass action infection mechanism) in system (1.1), the derivation of existence of periodic traveling wave solutions to (1.1) becomes much more challenging. Precisely speaking, it is difficult to verify the boundedness of I. On the other hand, the method on the nonexistence of periodic traveling wave solutions of (1.5) when \(R_0:=\frac{\int _0^T\beta (t)dt}{\int _0^T\gamma (t)dt}>1\) and \(c<c^*,\) can be hardly applied to system (1.1). Motivated by the ideas in [10, 35, 39], we shall consider the truncated problem on a finite interval and apply the limiting arguments to deal with the periodic traveling wave problem associated with (1.1). This will extend the research strategy on periodic traveling wave solutions for periodic and non-monotone systems. Here, we emphasize that in [39] a similar argument was used to establish the existence of periodic traveling wave solution for a time periodic and delayed reaction–diffusion equation without quasi-monotonicity, which describes the growth of mature population of a single species living in a fluctuating environment.
The rest of this paper is organized as follows. In the next section, by constructing a suitable pair of super- and sub-solutions and applying the Schauder’s fixed-point theorem to a similar problem on a bounded domain, we then use some a priori estimations and a limiting procedure to establish the existence of the periodic traveling wave solutions. Section 3 is devoted to the study of the nonexistence of periodic traveling wave solutions for two cases. A brief discussion completes the paper.
2 Existence of periodic traveling waves
In this section, we focus on the existence of the non-trivial and time periodic traveling waves \((\phi (t,z),\)\(\psi (t,z))\) of system (1.1). Since the R equation of system (1.1) can be decoupled, it is sufficient to consider the following system
Time periodic traveling waves to system (2.1) are defined to be solutions of the form
satisfying
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:
Such solutions \((\phi ,\psi )\) must satisfy the following system:
This system is posed on \((t,x)\in {\mathbb {R}}_+\times {\mathbb {R}}\) and is supplemented with the following asymptotic boundary conditions
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 basic procedure to prove the existence of periodic traveling wave solutions is as follows. Firstly, by constructing some suitable super- and sub-solutions for (2.4), we obtain a closed and convex set \(\Gamma _N\) of initial functions lying between the sub- and super-solutions. Secondly, we consider the truncated problem posed on the bounded domain and define a nonlinear solution operator \({\mathcal {F}}\) on \(\Gamma _N\), and then, we apply the Schauder’s fixed- point theorem to \({\mathcal {F}}\) after verifying the complete continuity of it. Finally, on the basis of some proposed a priori estimations of the obtained fixed point of \({\mathcal {F}},\) a limiting procedure can be used to extend the bounded interval to \({\mathbb {R}},\) and then, the existence of periodic traveling wave solutions is established. By similar arguments to [35], we further verify the asymptotic boundary conditions for periodic traveling wave solutions.
2.1 Construction of sub- and super-solutions
Linearizing system (2.4) at the disease-free steady state \((S _0 ,0)\), we have the following equation:
Define
where \(\varrho :=\frac{1}{T}\int _0^T \left( S_0\beta (t)-\gamma (t)\right) \mathrm{d}t\). Clearly, \(\varrho >0\) if the basic reproduction number \(R_0:=\frac{S_0{\int _0^T\beta (t)\mathrm{d}t}}{\int _0^T\gamma (t)\mathrm{d}t}>1\). In what follows, we always assume that \(R_0>1\). Let
if \(c>c^*:=2\sqrt{ d_2\varrho }\). Then, we have \(\Theta _c(\lambda _1 )=\Theta _c(\lambda _2 )=0\) and \(\Theta _c(\lambda )<0, \forall \lambda \in (\lambda _1,\lambda _2)\).
Fixing \(c>c^*\), we set
It is easy to see that K(t) is T-periodic. We further define the following functions
where \(M_i\) and \(\epsilon _i\), \(i=1,2\) are all positive constants and will be determined below. Then, we can inductively establish the following results.
Lemma 2.1
The function \(\psi ^+(t,z)=K(t)e^{\lambda _1 z}\) satisfies the following linear equation:
Lemma 2.2
For sufficiently small \(\epsilon _1\) such that \(0<\epsilon _1<\lambda _1\) and sufficiently large \(M_1>1\), the function \(\phi ^-\) satisfies
for any \(z\not =z_1:=-\epsilon _1^{-1}\ln M_1.\)
Proof
In case where \(z>-\epsilon _1^{-1}\ln M_1,\) we have \(\phi ^-(t,z)=0,\) which implies (2.8) holds.
In case where \(z<-\epsilon _1^{-1}\ln M_1\), then \(\phi ^-(t,z)=S_0\left( 1-M_1e^{\epsilon _1z}\right) .\) Thus, we need only to prove that
That is,
So for \(z<z_1:=-\epsilon _1^{-1}\ln M_1,\) it is sufficient to verify
Since both \(\beta (t)\) and K(t) are positive T-periodic functions, the above inequality is valid as long as we choose \(M_1=1/\epsilon _1\) with \(\epsilon _1>0\) sufficiently small and \(0<\epsilon _1<\lambda _1\). \(\square \)
Lemma 2.3
Suppose \(\epsilon _2>0\) is sufficiently small such that \(\epsilon _2<\min \{\epsilon _1,\lambda _2-\lambda _1\}\), 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
for any \(z\not =z_2:=-\epsilon _2^{-1}\ln M_2.\)
Proof
Choose \(M_2\) large enough to ensure that \(-\epsilon _2^{-1}\ln M_2<-\epsilon _1^{-1}\ln M_1.\) For \(z>z_2:=-\epsilon _2^{-1}\ln M_2,\) one has \(\psi ^-(t,z)=0,\) and hence, the inequality (2.9) holds.
When \(z<z_2:=-\epsilon _2^{-1}\ln M_2\), \(\psi ^-(t,z)=K(t)e^{\lambda _1 z}\left( 1-M_2e^{\epsilon _2z}\right) \) and \(\phi ^-(t,z)=S_0(1-M_1e^{\epsilon _1z}).\) In order to obtain (2.9), we only need to verify the following inequality
By the expression of K(t) and \(\psi ^-,\) it follows that
Then, the inequality (2.10) is equivalent to
Owing to \(\epsilon _1<\lambda _2-\lambda _1,\) we have \(\lambda _1+\epsilon _2\in (\lambda _1,\lambda _2),\) and hence,
Since \(\beta (t)\) is positive and T-periodic in \({\mathbb {R}},\) the inequality (2.11) is true if and only if
Thus, when \(z<-\epsilon _2^{-1}\ln M_2\), we need to show
for all \(t\in [0,T].\) The last inequality holds true when we choose sufficiently small \(\epsilon _2<\epsilon _1\) and \(M_2\) large enough. \(\square \)
2.2 Reduction to a fixed-point problem
Take \(N>-z_2.\) Define
For any given \(({\tilde{\phi }},{\tilde{\psi }})\in \Gamma _N,\) define maps
and
where \(\alpha _1\) and \(\alpha _2\) are positive constants and satisfy \(\alpha _1>\max _{t\in [0,T]}\beta (t)K(t)e^{\lambda _1N}\) and \(\alpha _2>\max _{t\in [0,T]}\)\(\gamma (t),\) respectively. Let \({\mathcal {A}}_iu=d_i\partial _{zz}u-c\partial _zu-\alpha _iu, i=1,2.\) Fix a \(({\tilde{\phi }},{\tilde{\psi }})\in \Gamma _N.\) Consider the following linear parabolic initial boundary value problem:
where \(G_1(t,z)=\frac{1}{2}\phi ^-(t,-N)-\frac{z}{2N}\phi ^-(t,-N)\) and \(G_2(t,z)=\frac{1}{2}\psi ^-(t,-N)-\frac{z}{2N}\psi ^-(t,-N)\) for all \(t\in [0,T]\) and \(z\in [-N,N].\) It is easy to see that \(G_1(t,\pm N)=\phi ^-(t,\pm N), G_2(t,\pm N)=\psi ^-(t,\pm N)\) for \(t\in {\mathbb {R}},\) and the function \(G_i\) is T-periodic and belongs to \(C^{1,2}({\mathbb {R}}\times [-N,N])\) for \(i=1,2.\) Let \(V_1(t,z)=\phi (t,z)-G_1(t,z),V_2(t,z)=\psi (t,z)-G_2(t,z)\) and \({\tilde{G}}_i={\mathcal {A}}_iG_i(t,z)-\partial _tG_i(t,z).\) Then, the problem (2.12) reduces to the following system on \((V_1,V_2):\)
Define the realization of \({\mathcal {A}}_i\) in \(C([-N,N])\) with homogeneous Dirichlet boundary condition,
Let \(T_i(t)_{t\ge 0}\) be the strongly continuous analytic semigroup generated by \(A_i^0:D(A_i^0)\subset C([-N,N])\rightarrow C([-N,N])\) (see, e.g., [7, 31]). It is easy to see that
for \(t>0,x\in [-N,N],\) where \(\Gamma _i, i=1,2\) is the Green function associated with \(d_i\partial _{xx}-c\partial _x, i=1,2\) and Dirichlet boundary condition. Then, system (2.13) can be rewritten as the following integral system
for all \(t\ge 0\) and \(z\in [-N,N].\) Then, \(\left( \phi (t,z),\psi (t,z)\right) \) satisfies that
for all \(t\ge 0\) and \(z\in [-N,N].\) We call a solution of (2.16) as a mild solution of (2.12). Since \(f_i[{\tilde{\phi }},{\tilde{\psi }}]\in C({\mathbb {R}}\times [-N,N])\) and \(f_i[{\tilde{\phi }},{\tilde{\psi }}](t,\cdot )\in C([-N,N]),\) it follows from [31, Theorem 5.1.17] that the functions \(\phi \) and \(\psi \) defined by (2.16) belong to \(C([0,2T]\times [-N,N])\cap C^{\theta ,2\theta }([\epsilon ,2T]\times [-N,N])\) for every \(\epsilon \in (0,2T)\) and \(\theta \in (0,1)\).
Define a set
with the usual supreme norm. Obviously, \(\Gamma '_N\) is a closed and convex set.
Lemma 2.4
For any \((\phi _0,\psi _0)\in \Gamma '_N,\) let \(\left( \phi _N(t,z;\phi _0,\psi _0),\psi _N(t,z;\phi _0,\psi _0)\right) \) be the solutions of the system (2.16) with the initial value \((\phi _0,\psi _0).\) Then,
for \((t,z)\in [0,+\infty )\times [-N,N].\)
Proof
Let us first recall that for the given \(({\tilde{\phi }},{\tilde{\psi }})\in \Gamma _N,\) there hold
while every \((\phi _0,\psi _0)\in \Gamma '_N\) satisfies
We are ready to prove that \(\phi _N(t,z;\phi _0,\psi _0)\le \phi ^+(t,z)\) for all \(t\ge 0\) and \(z\in [-N,N].\) Let \({\overline{\phi }}\) be the solution of the following equation
Since \(f_1[{\tilde{\phi }},{\tilde{\psi }}]\le f_1[\phi ^+,\psi ^-],\) we have
In addition, since \(f_1[\phi ^+,\psi ^-]\in C^{\theta /2,\theta }({\mathbb {R}}\times [-N,N])\) for some \(\theta \in (0,1),\) it follows from [31, Theorems 5.1.18 and 5.1.19] that \({\overline{\phi }}\in C([0,+\infty )\times [-N,N])\) is differentiable with respect to t in \((0,+\infty )\times [-N,N], {\overline{\phi }}(t,\cdot )\) belongs to \(W_{\mathrm{loc}}^{2,p}((-N,N))\) for every \(p\ge 1,\) and \(\partial _t{\overline{\phi }}, {\mathcal {A}}_1{\overline{\phi }}\in C^{\theta /2,\theta }([\delta ,+\infty )\times [-N,N])\) for any \(\delta >0.\) As a consequence, we see that \({\overline{\phi }}\in C([0,+\infty )\times [-N,N])\cap C^{1,2}((0,+\infty )\times [-N,N])\) and satisfies that
On the other hand, it is easy to see that \(\phi ^+\) satisfies
Thus, the parabolic comparison principle indicates that
In view of (2.17) and (2.18), we have that
Let \({\underline{\phi }}\) be the solution of the following equation
Thus, we have
because of \(f_1[{\tilde{\phi }},{\tilde{\psi }}]\ge f_1[\phi ^-,\psi ^+].\) Additionally, since \(f_1[\phi ^-,\psi ^+]\in C^{\theta /2,\theta }({\mathbb {R}}\times [-N,N])\) for some \(\theta \in (0,1),\) we conclude from [31, Theorems 5.1.18 and 5.1.19] that \({{\underline{\phi }}}\in C([0,+\infty )\times [-N,N])\cap C^{1,2}((0,+\infty )\times [-N,N])\) satisfies that
Let \({{\underline{\phi }}}^*\equiv 0.\) Then, \({{\underline{\phi }}}^*\) satisfies
and hence, the parabolic comparison principle implies that \({{\underline{\phi }}}(t,z)\ge 0\) for all \(t\in [0,+\infty )\) and \(z\in [-N,N].\) When \((t,z)\in {\mathbb {R}}\times (-\infty ,z_1),\) it follows from Lemma 2.2 that \(\phi ^-(t,z)=S_0(1-M_1e^{\epsilon _1z})\) satisfies (2.8). Thus,
Hence, it follows from the maximum principle [15, Chapter 2, Theorem 1] that
Note that \(\phi ^-(t,z)=\max \{S_0(1-M_1e^{\epsilon _1z}),0\}.\) Therefore, we further have that
In the following, we consider \(\psi _N(t,z;\phi _0,\psi _0)\) for \(t\in [0,+\infty )\) and \(z\in [-N,N].\) Let \({\overline{\psi }}\) be the solution of the following equation
Clearly,
On the other hand, since \(f_2[\phi ^-,\psi ^+]\in C^{\theta /2,\theta }([0,T]\times [-N,N])\) for some \(\theta \in (0,1),\) it follows from [31, Theorems 5.1.18 and 5.1.19] that \({\overline{\psi }}\in C([0,+\infty )\times [-N,N])\cap C^{1,2}((0,+\infty )\times [-N,N])\) satisfies that
In view of Lemma 2.1, (2.7) can be rewritten as
where \(P[\psi ^+](t,z)=\alpha _2\psi ^++\beta (t)S_0\psi ^+- \gamma (t)\psi ^+, (t,z)\in {\mathbb {R}}\times [-N,N].\) Since \(P[\psi ^+](t,z)\ge f_2[\phi ^+,\psi ^+](t,z)\) for \(t\in (0,+\infty )\) and \(z\in [-N,N], \psi ^+(0,\cdot )\ge \psi _0(\cdot )\) and \(\psi ^+(\cdot ,\pm N)\ge G_2(\cdot ,\pm N),\) we can conclude from the parabolic comparison principle that
Thus, we further have that
Finally, we show that \(\psi _N(t,z;\phi _0,\psi _0)\ge \psi ^-(t,z)\) for all \( t\in [0,+\infty )\) and \(z\in [-N,N].\) Let \({{\underline{\psi }}}\) be the solution of the following equation
It is obvious that \({{\underline{\psi }}}(t,\cdot ;\phi _0,\psi _0)\le \psi _N(t,\cdot ;\phi _0,\psi _0)\) for all \(t\ge 0.\) In addition, since \(f_2[\phi ^+,\psi ^-]\in C^{\theta /2,\theta }([0,T]\times [-N,N])\) for some \(\theta \in (0,1),\) it follows from [31, Theorems 5.1.18 and 5.1.19] that \({{\underline{\psi }}}\in C([0,+\infty )\times [-N,N])\cap C^{1,2}((0,+\infty )\times [-N,N])\) satisfies that
Let \({{{\underline{\psi }}}}^*(t,z)\equiv 0.\) Then, \({{{\underline{\psi }}}}^*(t,z)\) satisfies
and hence, the parabolic comparison principle implies that \({{\underline{\psi }}}(t,z)\ge 0\) for all \(t\in [0,+\infty )\) and \(z\in [-N,N].\) When \((t,z)\in {\mathbb {R}}\times (-\infty ,z_2),\) we see that \(\psi ^-(t,z)=K(t)e^{\lambda _1 z}(1-M_2e^{\epsilon _2z})\). Thus, by Lemma 2.3, we have
Consequently, the maximum principle [15, Chapter 2, Theorem 1] yields that
Therefore, we further have that
This completes the proof. \(\square \)
For a given \(({\tilde{\phi }},{\tilde{\psi }})\in \Gamma _N\), we define a map \(F_{({\tilde{\phi }},{\tilde{\psi }})}:\Gamma '_N\rightarrow C([-N,N],{\mathbb {R}}^2)\) by
where \(\left( \phi _N(t,z;\phi _0,\psi _0), \psi _N(t,z;\phi _0,\psi _0)\right) \) is the solution of (2.12). With the aid of Lemma 2.4 and the periodicity of \(\phi ^-,\psi ^-,\phi ^+\) and \(\psi ^+\), we have \(F_{({\tilde{\phi }},{\tilde{\psi }})}\left( \Gamma '_N\right) \subset \Gamma '_N.\) Clearly, \(\Gamma '_N\) is a complete metric space with a distance induced by the supreme norm. For any \(\left( \phi ^1_0,\psi ^1_0\right) ,\left( \phi ^2_0,\psi ^2_0\right) \in \Gamma '_N,\) it follows from (2.14) and (2.16) that
Similarly, we have
Since \(e^{-\alpha T}<1,\) we see that \(F_{({\tilde{\phi }},{\tilde{\psi }})}:\Gamma '_N\rightarrow \Gamma '_N\) is a contraction map. It then follows from the Banach fixed-point theorem that \(F_{({\tilde{\phi }},{\tilde{\psi }})}\) admits a unique fixed point \((\phi _0^*,\psi _0^*)\in \Gamma '_N.\) Let \(({\hat{\phi }}_N^*(t,z),{\hat{\psi }}_N^*(t,z)) = \left( \phi _N(t,z;\phi _0^*,\psi _0^*),\psi _N(t,z;\phi _0^*,\psi _0^*)\right) \) for all \(t\in [0,+\infty )\) and \(z\in [-N,N],\) where \((\phi _N(t,z;\phi _0^*,\psi _0^*),\psi _N(t,z;\phi _0^*,\psi _0^*))\) is the solution of (2.16) with initial value \((\phi _0^*,\psi _0^*)\). In view of \((\phi _0^*(z),\psi _0^*(z))=(\phi _N(T,z;\phi _0^*,\psi _0^*),\psi _N(T,z;\phi _0^*,\psi _0^*))\), we get \(({\hat{\phi }}_N^*(t+T,z),{\hat{\psi }}_N^*(t+T,z))=({\hat{\phi }}_N^*(t,z),{\hat{\psi }}_N^*(t,z))\) for all \(t\in [0,+\infty )\) and \(z\in [-N,N].\) Define \((\phi _N^*(t,z),\psi _N^*(t,z))=({\hat{\phi }}_N^*(t-kT,z),{\hat{\psi }}_N^*(t-kT,z))\) for \(t\in {\mathbb {R}}\) and \(z\in [-N,N]\), where \(k\in {\mathbb {Z}}\) satisfies \(kT\le t\le (k+1)T.\) Then, \((\phi _N^*(t+T,z),\psi _N^*(t+T,z))=(\phi _N^*(t,z),\psi _N^*(t,z))\) for all \(t\in {\mathbb {R}}\) and \(z\in [-N,N]\). According to Lemma 2.4, we see that \((\phi _N^*,\psi _N^*)\in \Gamma _N.\) Moreover, \((\phi _N^*,\psi _N^*)\) satisfies
for all \(t\ge s\). On the basis of the above discussion, we obtain the following theorem.
Theorem 2.5
For any given \(({\tilde{\phi }},{\tilde{\psi }})\in \Gamma _N,\) there exists a unique \((\phi _N^*,\psi _N^*)\in \Gamma _N\) such that (2.20) holds.
Following Theorem 2.5, we can define an operator \({\mathcal {F}}:\Gamma _N\rightarrow \Gamma _N\) by \({\mathcal {F}}({\tilde{\phi }},{\tilde{\psi }})=(\phi _N^*,\psi _N^*)\). We further show the properties of the operator \({\mathcal {F}}.\)
Lemma 2.6
The operator \({\mathcal {F}}:\Gamma _N\rightarrow \Gamma _N\) is completely continuous.
Proof
For any \(({\tilde{\phi }},{\tilde{\psi }})\in \Gamma _N\), there holds \(f_i[{\tilde{\phi }},{\tilde{\psi }}](\cdot ,\cdot )\in C({\mathbb {R}}\times [-N,N])\) and \(f_i[{\tilde{\phi }},{\tilde{\psi }}](t+T,z)=f_i[{\tilde{\phi }},{\tilde{\psi }}](t,z)\) for \(i=1,2, (t,z)\in {\mathbb {R}}\times [-N,N].\) Note that \(f_i[{\tilde{\phi }},{\tilde{\psi }}], i=1,2\) are uniformly bounded with respect to \(({\tilde{\phi }},{\tilde{\psi }})\in \Gamma _N.\) For any given \(({\tilde{\phi }},{\tilde{\psi }})\in \Gamma _N\), let \((\phi _N^*,\psi _N^*)={\mathcal {F}}({\tilde{\phi }},{\tilde{\psi }})\in \Gamma _N.\) By [31, Theorem 5.1.17], it follows from (2.20) with \(s=0\) that \(\phi _N^*,\psi _N^*\in C^{\theta /2,\theta }([T,2T]\times [-N,N])\) for every \(\theta \in (0,1)\) and there exists \(C_i(\theta )>0, i=1,2\) such that
and
Since \(\phi _N^*,\psi _N^*\) are T-periodic, we have that \(\phi _N^*,\psi _N^*\in C^{\theta /2,\theta }({\mathbb {R}}\times [-N,N])\), and there exists \(K_0^i(\theta )>0, i=1,2\) such that
which implies that \({\mathcal {F}}\) is compact on \(\Gamma _N.\)
We further prove the continuity of \({\mathcal {F}}\). For any \(({\tilde{\phi }}_i,{\tilde{\psi }}_i)\in \Gamma _N, i=1,2,\) there exists a positive constant M such that \(\vert {\tilde{\phi }}_i(t,z)\vert \le M\) and \(\vert {\tilde{\psi }}_i(t,z)\vert \le M\) for \(i=1,2, t\in {\mathbb {R}}\) and \(z\in [-N,N],\) and let \((\phi _{i,N}^*(t,z;{\tilde{\phi }}_i,{\tilde{\psi }}_i),\)\(\psi _{i,N}^*(t,z;{\tilde{\phi }}_i,{\tilde{\psi }}_i)) ={\mathcal {F}}({\tilde{\phi }}_i,{\tilde{\psi }}_i), i=1,2.\) By virtue of (2.14) and (2.20), we have
and
Then, there holds
where \({\tilde{\beta }}:=\max _{t\in [0,T]}\beta (t).\) Since \(\phi _{i,N}^*(t+T,z;{\tilde{\phi }}_i,{\tilde{\psi }}_i)=\phi _{i,N}^*(t,z;{\tilde{\phi }}_i,{\tilde{\psi }}_i)\) for all \(t\in {\mathbb {R}}\) and \(z\in [-N,N],\) we can get from the above inequality that
Additionally, \(\phi _{i,N}^*(t,z;{\tilde{\phi }}_i,{\tilde{\psi }}_i)\) satisfies
Thus, by similar arguments to above, it is not difficult to show that \(\phi ^*_{N}(t,z;{\tilde{\phi }},{\tilde{\psi }})\) is continuous in \(({\tilde{\phi }},{\tilde{\psi }})\). Similarly, we can prove that \(\psi ^*_{N}(t,z;{\tilde{\phi }},{\tilde{\psi }})\) is continuous in \(({\tilde{\phi }},{\tilde{\psi }})\). The proof is complete. \(\square \)
With the aid of Lemma 2.6, we can conclude from the Shauder’s fixed-point theorem that \({\mathcal {F}}\) admits a fixed point \((\phi _N^*,\psi _N^*)\in \Gamma _N.\) In particular, \((\phi _N^*(t+T,\cdot ),\psi _N^*(t+T,\cdot ))=(\phi _N^*(t,\cdot ),\psi _N^*(t,\cdot ))\) for all \(t\in {\mathbb {R}}.\) Note that \(\phi _N^*,\psi _N^*\in C^{\theta /2,\theta }({\mathbb {R}}\times [-N,N])\) for some \(\theta \in (0,1).\) By [31, Theorem 5.1.18 and 5.1.19], we have that \(\phi _N^*,\psi _N^*\in C^{1,2}({\mathbb {R}}\times [-N,N])\) satisfy
The following theorem lists some local uniform estimates on \(\phi _N^*\) and \(\psi _N^*.\)
Theorem 2.7
Let \(p\ge 2.\) For any given \(Z>0,\) there exists a constant \(C(p,Z)>0\) such that for sufficiently large \(N>\max \{Z,-z_2\},\) there hold
Furthermore, there exists a constant \(C'(Z)>0\) such that for any \(z_0\in {\mathbb {R}},\) there hold
for sufficiently large \(N>\max \{Z+\vert z_0\vert ,-z_2\},\) where \(\theta \in (0,1).\)
Proof
Fix \(Z>0\) and \(z_0\in {\mathbb {R}}.\) Let \(N>\max \{Z+\vert z_0\vert ,-z_2\}.\) In view of the above discussion, we see that
for all \((t,z)\in {\mathbb {R}}\times (-N,N).\) Since \((\phi _N^*,\psi _N^*)\in \Gamma _N,\) there exists a \(M>0\) independent of N such that
Let \(W_N^1(t,z):=e^{-\frac{c(z-z_0)}{2d_1}}\phi _N^*(t,z),\)\( W_N^2(t,z):=e^{-\frac{c(z-z_0)}{2d_2}}\psi _N^*(t,z)\) for any \(t\in {\mathbb {R}}\) and \(z\in [-N,N].\) It then follows that
for any \((t,z)\in {\mathbb {R}}\times (-N,N).\) For \((t',z')\in {\mathbb {R}}^2\) and \(r>0,\) define
For the given \(Z>0,\) take \(R=\max \{2Z,\sqrt{3T}\}\). Define
According to [29, Proposition 7.14], for \(N>72R+\vert z_0\vert ,\) there exists a constant \(C_1(p,R)\) independent of N, such that
This implies that there exists a constant \(C_2(p,R),\) which is independent of N, such that
In view of the equations for \(\phi _N^*\) and \(\psi _N^*,\) we further conclude from [29, Proposition 7.18] that there exists a constant \(C_3(p,R)\) independent of N, such that
As a consequence, there exists a constant C(p, R), which is independent of N, such that
On account of \([0,T]\times [-Z,Z]\subset Q((2T,0),R),\) we have
Here, R merely depends on Z, and then, C only relies on Z and p.
Take \(p>3.\) Then, the embedding theorem indicates that
and
where \(C'>0\) is a constant depending upon p and Z. \(\square \)
Let \(\left( \phi _N^*,\psi _N^*\right) \) be the solution of the system (2.21), and we further have the following estimations.
Proposition 2.8
There exists a constant \(C_0\) such that
for any \(N>-z_2.\) In particular, \(\int _0^T\partial _z\phi _N^*(t,z)dt\le 0\) for \(z\in [-N,N]\) and \(N>-z_2.\)
Proof
For \(z\in [-N,N],\) define
Clearly,
In view of (2.21), we have
where the subscripts \(_z\) and \(_{zz}\) represent the first derivative and the second derivative for one function on z, respectively. It follows from (2.23) that
Integrating two sides of the last equality from \(z\in [-N,N)\) to N yields
Since \(\Phi ^*(z)\ge 0=\Phi ^*(N)=\Phi ^-(N)\) for \(z\in [-N,N],\) we have that \(\Phi _z^*(N)\le 0,\) and hence, \(\Phi _z^*(z)\le 0\) for \(z\in [-N,N].\) In particular, \(\Phi _z^*(z)\not \equiv 0.\) Making an integration from \(-N\) to N for Eq. (2.23), we obtain
due to \(\Phi ^*(-N)\le S_0\) and
Let \({\hat{\gamma }}:=\min _{t\in [0,T]}\gamma (t)\) and \({\tilde{\gamma }}:=\max _{t\in [0,T]}\gamma (t)\). Then, \(\Psi ^*\) satisfies
Integrating the two sides of the last equality on \([-N,N],\) we have
Since \(\Psi ^*_z(N)\le 0, \Psi _z^*(-N)\ge \Psi _z^-(-N)>0, \Psi ^*(-N)=\Psi ^-(-N)\) and the inequality (2.24) holds, we can conclude from the last equality that
Thus, there exists a constant \(C_0>0\) independent of \(N>-z_2\) such that
This completes the proof. \(\square \)
2.3 Existence of periodic traveling waves
This subsection is concerned with the existence of periodic traveling waves.
Theorem 2.9
Assume that \(R_0>1.\) For any \(c>c^*,\) the system (2.1) admits a time periodic traveling wave solution \((\phi ^*,\psi ^*)\) satisfying (2.4) and (2.5). Furthermore, there hold \(0<\frac{1}{T}\int _0^T\psi ^*(t,z)dt\le S_0-S^\infty \) for any \(z\in {\mathbb {R}},\) and
Proof
The proof is divided into four parts.
I. Existence of periodic traveling waves Let \(\{N_m\}\) be an increasing sequence such that \(N_m\ge -z_2\) and \(\lim _{m\rightarrow +\infty }N_m\)\(=+\infty .\) It then follows that the solutions \((\phi _{N_m}^*,\psi _{N_m}^*)\in \Gamma _{N_m}\) satisfy Theorem 2.7 and (2.21). In light of the periodicity of \((\phi _{N_m}^*,\psi _{N_m}^*)\) in \(t\in {\mathbb {R}},\) we can extract a subsequence of \((\phi _{N_m}^*,\psi _{N_m}^*)\), still denoted by \((\phi _{N_m}^*,\psi _{N_m}^*),\) tending toward functions \((\phi ^*,\psi ^*)\in C({\mathbb {R}}^2)\) in the following topologies
where \(\beta \in (0,\theta )\) and \(\theta \in (0,1)\) is given in (2.22). It is obvious that \((\phi ^*,\psi ^*)\in C^{\frac{1+\beta }{2},1+\beta }({\mathbb {R}}^2)\cap H_{\mathrm{loc}}^1({\mathbb {R}}^2)\cap L_{\mathrm{loc}}^2({\mathbb {R}},H_{\mathrm{loc}}^2({\mathbb {R}})).\) Since \((\phi _{N_m}^*,\psi _{N_m}^*)\) is T-periodic in t, we have \((\phi ^*(t+T,z),\psi ^*(t+T,z))=(\phi ^*(t,z),\psi ^*(t,z))\) for all \(t\in {\mathbb {R}}\) and \(z\in {\mathbb {R}}\), and hence, the estimation (2.22) implies that for any \(N>0\), there exists a constant \(C_3>0\) such that
Let \(u,v\in C_0^\infty ({\mathbb {R}}^2)\) be given. Then, for sufficiently large \(m\in {\mathbb {N}}\) satisfying \(\mathrm{supp}(u)\times \mathrm{supp}(v)\subset {\mathbb {R}}\times (-N_m,N_m),\) we have that \((\phi _{N_m}^*,\psi _{N_m}^*)\) satisfy the equalities
and
On the basis of (2.25), we have that \((\phi ^*,\psi ^*)\) satisfy
and
for any \(u,v\in C_0^\infty ({\mathbb {R}}^2).\) Then, we conclude that \((\phi ^*,\psi ^*)\) satisfy
almost everywhere in \((t,z)\in {\mathbb {R}}^2.\) Consider the following Cauchy problem
Clearly, \((\phi ^*(t,z),\psi ^*(t,z))\) is a strong solution of (2.27). Moreover, [31, Theorem 5.1.3 and 5.1.4] imply that \((\phi ^*,\psi ^*)\) is the unique strong solution of (2.27), and hence, \(\phi ^*,\psi ^*\in C^{1+\frac{\nu }{2},2+\nu }({\mathbb {R}}^2)\) for some \(\nu \in (0,1)\) and satisfy (2.4), that is,
for \((t,z)\in {\mathbb {R}}^2.\) Furthermore, it follows from Proposition 2.8 that there exists a constant \(C_0>0\) such that
Note that \((\phi ^*,\psi ^*)\) satisfies that
and hence, there hold \(\phi ^*(t,z)\rightarrow S_0\) and \(\psi ^*(t,z)\rightarrow 0\) uniformly for \(t\in {\mathbb {R}}\), as \(z\rightarrow -\infty .\)
II. The asymptotic behavior of \(\psi ^*\)as \(z\rightarrow +\infty \) Define \(\Psi (z)=\frac{1}{T}\int _0^T\psi ^*(t,z)dt\). Then, \(\Psi (z)\) satisfies
where \({\hat{\gamma }}\) is defined as in the proof of Proposition 2.8. Denote by
the two roots of the characteristic equation
In addition, denote
Clearly, \({\hat{\lambda }}^-<0<{\hat{\lambda }}^+\). It follows from (2.30) and (2.29) that
and
Since \({\hat{\lambda }}^-<0<{\hat{\lambda }}^+\) and \(\hat{\rho }:=d_2\left( {\hat{\lambda }}^+-{\hat{\lambda }}^-\right) \), we have
It then follows from the integrability of \(\int _0^T\beta (t)\phi ^*(t,\cdot )\psi ^*(t,\cdot )dt\) on \({\mathbb {R}}\) that \(\Psi _z\) is uniformly bounded. Consequently, following \(\int _{-\infty }^\infty \Psi (z)dz<C_0\), we must have \(\Psi (z)\rightarrow 0\) as \(z\rightarrow \infty .\) We further apply Harnack inequalities ([35, Lemma 2.9] (see also [14]) 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 \(z\in {\mathbb {R}}\)) for the second equation of system (2.28), we have
where \(C'_0\) is a positive constant independent of D. Since \(\psi ^*\) is periodic in time t, \(\psi ^*(t,z)\rightarrow 0\) uniformly for \(t\in {\mathbb {R}},\) as \(z\rightarrow \infty .\) As a consequence, there holds \(\psi ^*(t,z)\le C_0\) for \((t,z)\in {\mathbb {R}}^2.\)
III. The asymptotic behavior of \(\phi ^*\)as \(z\rightarrow \infty \) By virtue of the estimate (2.26) and Laudau type inequalities (see, e.g., [4, 27]), we have
Consequently,
Define \(\Phi (z)=\frac{1}{T}\int _0^T\phi ^*(t,z)dt\). It is obvious that \(\Phi _z(z)\rightarrow 0\) as \(z\rightarrow -\infty .\) It then follows from the first equation of system (2.28) that
It is easy to see from the last equation
Since \(\frac{1}{T}\int _0^T \beta (t)\phi ^*(t,z)\psi ^*(t,z)dt\) is integrable on \({\mathbb {R}}\), an integration from z to \(\infty \) for the last equality yields
which implies that \(\Phi _z(z)<0\) for \(z\in {\mathbb {R}},\) and hence, \(\Phi (\infty )\) exists and \(\Phi (\infty )<\Phi (-\infty )=S_0.\) It follows from the Barbălat’s lemma (see, e.g., [3, 12]) that \(\Phi _z(z)\rightarrow 0\) as \(z\rightarrow \infty .\) Integrating two sides of (2.31) from \(-\infty \) to \(\infty \) on z leads to
where \(S^\infty :=\Phi (\infty )<S_0.\)
By similar arguments to [35, Theorem 2.10], we prove that \(\phi ^*(t,z)\rightarrow S^\infty \) uniformly for \(t\in {\mathbb {R}}\), as \(z\rightarrow \infty \). In the light of T-periodicity of \(\phi ^*,\) it is sufficient to show
Clearly, 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
Let \(\phi _n(t,z)=\phi ^*(t+t_n,z+z_n), \psi _n(t,z)=\psi ^*(t+t_n,z+z_n), \forall n\in {\mathbb {N}}, t\in {\mathbb {R}}, z\in {\mathbb {R}}.\) Based on the estimation (2.26) and the uniform boundedness of \(\Phi ,\Phi _z,\Psi \) and \(\Psi _z\), 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 \(C_{\mathrm{loc}}^{\nu /2,\nu }({\mathbb {R}}\times {\mathbb {R}})\) for some \(\nu \in (0,1)\), as \(n\rightarrow \infty .\) Particularly, we have \(\phi _*(0,0)=S_+^\infty \) and
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
Accordingly, \(\phi _*^+(t,z)\) satisfies
As a result of \(\phi _*^+(t^*,0)=S_+^\infty \) and \(\phi _*^+(t,z)\le S_+^\infty \), the maximum principle indicates that \(\phi _*^+(t,z)\equiv S_+^\infty \) for \(t<t^*.\) Since \(\phi _*^+\) is T-periodic in t, 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,
which implies \(S_+^\infty =S^\infty .\) Thus, \(\limsup _{z\rightarrow \infty }\max _{t\in [0,T]}\phi ^*(t,z)=S^\infty .\) 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 \(z\rightarrow \infty \).
IV. The properties for \(\psi ^*\) We use the similar arguments to [35, Theorem 2.10](see also [33]) check on the properties for \(\psi ^*.\) Since \(\Psi (z)\) satisfies
an integration of (2.32) on \({\mathbb {R}}\) yields
Similar to the aforementioned proof on the asymptotic behavior of \(\phi ^*_z(t,z)\) as \(z\rightarrow -\infty \), we can show that
uniformly for \(t\in {\mathbb {R}}\). For any \(z\in {\mathbb {R}},\) define a function
It is not difficult to see that \(\Psi ^{**}(z)\) satisfies the following equation:
By means of (2.33) and L’Hôpital’s rule, it follows that
and
Define a new function
where \(\Psi (z)=\frac{1}{T}\int _0^T\psi ^*(t,z)dt.\) On the basis of (2.33) and (2.34) that
Multiplying two sides of the above equation by \(e^{-c/d_2z}\) and integrating from z to \(\infty ,\) we have
Then, it is obvious 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 \(z\in {\mathbb {R}}.\) The proof is complete. \(\square \)
3 Nonexistence of periodic traveling waves
In this section, our task is to investigate the nonexistence of time periodic traveling waves for two cases. Firstly, we prove that there is no time periodic traveling wave in the case where \(R_0\le 1\).
Theorem 3.1
Assume that \(R_0=\frac{S_0\int _0^T\beta (t)\mathrm{d}t}{\int _0^T\gamma (t)\mathrm{d}t}\le 1.\) Then, for any \(c\ge 0,\) there is no time periodic traveling wave solutions \((\phi ,\psi )\) satisfying the asymptotic boundary conditions (2.5) uniformly for \(t\in {\mathbb {R}}\).
Proof
By contradiction, we assume that there exists a time periodic, non-trivial and nonnegative solution \(\left( \phi (t,z),\psi (t,z)\right) \) of (2.4) satisfying (2.5) uniformly for \(t\in {\mathbb {R}}\), that is,
and
Due to the T-periodicity of \(\psi (t,z)\) and the parabolic maximum principle, it follows that \(\psi (t,z)>0\) for \(t\in {\mathbb {R}}, z\in {\mathbb {R}}.\) In addition, it is not difficult to show that \(\phi (t,z)\le S_0\) for \(t\in {\mathbb {R}}, z\in {\mathbb {R}}.\) In fact, suppose that there exists \((t_0,x_0)\) such that \(S(t_0,x_0)>S_0\). Thus,
which is a contradiction. Let \({\overline{\psi }}(t)=\int _{-\infty }^\infty \psi (t,z)dz\). Then, by the asymptotical boundary conditions (2.5) and (2.33), we have
where
It is easy to see that \({\overline{\psi }}(t+T)={\overline{\psi }}(t), f(t+T)=f(t), \forall t\in {\mathbb {R}}.\) According to the positivity of \({\overline{\psi }}(t)\), we see that
Integrating both two sides of the above equality from 0 to T , we obtain
due to the periodicity and positivity of \({\overline{\psi }}(t)\) and \(R_0=\frac{S_0\int _0^T\beta (t)\mathrm{d}t}{\int _0^T\gamma (t)\mathrm{d}t}\le 1\). This is a contradiction. \(\square \)
Next, we prove the nonexistence of periodic traveling waves for the case where \(R_0>1\) and \(c<c^*.\)
Theorem 3.2
Assume that \(R_0>1\) and \(0<c<c^*=2\sqrt{d_2\varrho }=2\sqrt{\frac{d_2\int _0^T(S_0\beta (t)-\gamma (t))}{T}}.\) System (2.4) does not have a time periodic traveling waves \((\phi ,\psi )\) satisfying (2.5) uniformly for \(t\in {\mathbb {R}}\).
Proof
Suppose, by contradiction, that there exists such a traveling wave solution \((\phi (t,x+ct),\psi (t,x+ct)\) satisfying (2.5) for some \(c<c^*=2\sqrt{\frac{d_2\int _0^T(S_0\beta (t)-\gamma (t))}{T}}.\) Since \(R_0=\frac{S_0\int _0^T\beta (t)dt}{\int _0^T\gamma (t)dt}\), we have \(\int _0^T[\beta (t)S_0-\gamma (t)]dt>0,\) and hence, there exists a sufficiently small \(\delta _0>0\) such that \(\int _0^T[\beta (t)(S_0-\delta _0)-\gamma (t)]\mathrm{d}t>0\). For each \(\delta \in (0,\delta _0)\), define \(\varrho ^\delta \) by
We fix a \(\delta \in (0,1)\) such that \(c<2\sqrt{d_2\varrho ^\delta }\). Since \(\lim _{z\rightarrow -\infty }\phi (t,z)=S_0\),\(\forall t\in {\mathbb {R}}\), we can choose a \(M_\delta >0\) such that \(S_0-\delta \le \phi (t,z)\le S_0+\delta \), \(\forall z<-M_\delta \) uniformly for \(t\in {\mathbb {R}}\). Fix a \(c_0\in (c,2\sqrt{d_2\varrho ^\delta })\) and let \(M_{c_0}=\frac{\sqrt{4d_2\varrho ^\delta -c_0^2}}{2d_2}\). Define
Clearly,
We consider a function \(w_{c_0}(t,z):=e^{\frac{c_0z}{2d_2}}\sin (M_{c_0}z)Q^\delta (t)\). It is easy to verify that \(w_{c_0}(t,z)\) satisfy \(w_{c_0}(t+T,z)=w_{c_0}(t,z)\) for \(z\in {\mathbb {R}}\). Further, some direct manipulation yields
Let \(k_0\in {\mathbb {N}}^+\) such that \(\frac{(2k_0-1)\pi }{M_{c_0}}> M_\delta \). Then, let \(y_1=-\frac{2k_0\pi }{M_{c_0}}\), \(y_2=-\frac{(2k_0-1)\pi }{M_{c_0}}\). Clearly, \(\sin (M_{c_0}y_1)=\sin (M_{c_0}y_2)=0\), \(\sin (M_{c_0}z)>0\), \(\forall z\in (y_1,y_2)\). Since \(\psi (0,z)\) is strictly positive on \([y_1,y_2]\), then there exists an \(\epsilon >0\) such that \(\epsilon w_0(0,z)\le \psi (0,z),~\forall z\in [y_1,y_2]\). Consider the function \(\phi (t,x+(c-c_0)t)\) and \(\psi (t,x+(c-c_0)t)\), \(\forall t>0\), \(x\in [y_1,y_2]\). Denote \(\hat{\psi }(t,x):=\psi (t,x+(c-c_0)t)\). Since \((\phi (t,z),\psi (t,z))\) is a solution of system (2.4), we have
Since \(\phi (t,z)\ge S_0-\delta , \forall z<-M_\delta \) uniformly for \(t\in {\mathbb {R}},\) it follows from above equality that \(\hat{\psi }\) satisfies
for all \(t>0\) and \(x\in [y_1,y_2]\). In view of \(c-c_0<0\) and \(y_1< y_2< -M_{\delta }\), we have \(x+(c-c_0)t\le -M_\delta \), \(\forall t\ge 0\), \(x\in [y_1,y_2]\). Let \({\check{\psi }}(t,x):=\psi (t,x+(c-c_0)t)-\epsilon w_{c_0}(t,x)={\hat{\psi }}(t,x)-\epsilon w_{c_0}(t,x)\) for all \(t\ge 0\) and \(x\in [y_1,y_2]\). Then, we can derive that
In view of the maximum principle of the parabolic equations, we are led to the conclusion that \({\check{\psi }}\ge 0\) for all \(t>0\) and \(x\in [y_1,y_2]\), which implies that \(\psi (t,x+(c-c_0)t)\ge \epsilon w_{c_0}(t,x)\) for all \(t>0\) and \(x\in [y_1,y_2]\). Since \(c-c_0<0\), there is a contradiction that \(\psi (t,x+(c-c_0)t)\rightarrow 0\) as \(t\rightarrow +\infty \). \(\square \)
4 Discussion
In this paper, we investigated time periodic traveling waves for system (1.1) with bilinear incidence in a seasonal forcing environment. To overcome the unboundedness of mass action (bilinear incidence) function, we considered a truncated problem on a large but finite interval and applied the limiting arguments to obtain the existence of periodic traveling waves for each \(c>c^*\) when \(R_0>1\). In addition, we also proved the nonexistence of periodic traveling waves for either \(R_0\le 1\) or \(c<c^*\) and \(R_0>1\). The idea and method of this paper also apply to other periodic and non-monotone evolution systems provided that some new techniques are developed for the verification of the asymptotic boundary condition. Unfortunately, we cannot prove the existence of time periodic traveling waves with critical wave speed \(c=c^*\), which remains an open problem for future investigation. The substantial difficulty is again due to the unboundedness of bilinear incidence, which makes the construction of proper sub- and super-solutions much more challenging (if not impossible). At the same time, since system (1.1) is non-autonomous and non-monotone, and the I-component of the periodic traveling wave with wave speed \(c>c^*\) is a time periodic pulse wave, it is also difficult to get the existence of critical periodic traveling wave by taking the limit of a sequence of periodic traveling wave with wave speeds \(c_n\), where \(c_n>c^*\) and \(c_n\rightarrow c^*\), see [36, 39]. Nevertheless, when the bilinear incidence is replaced by the standard incidence in (1.1) [i.e., system (1.5)], Zhang and Wang [38] recently proved the existence of time periodic traveling wave with the minimal wave speed \(c^*\) by constructing sub- and super-solutions similar to those for some autonomous systems, see [16, 43] and the references therein.
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Acknowledgements
The authors are grateful to the anonymous referees for their insightful comments and suggestions which contributed to greatly improve the original version of the manuscript. Both Zhang and Wang’s research was supported by NNSF of China (11371179, 11731005, 11701242) and the Fundamental Research Funds for the Central Universities (lzujbky-2017-ot09, lzujbky-2017-27, lzujbky-2019-79), and Zhao’s research was supported in part by the NSERC of Canada.
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Zhang, L., Wang, ZC. & Zhao, XQ. Time periodic traveling wave solutions for a Kermack–McKendrick epidemic model with diffusion and seasonality. J. Evol. Equ. 20, 1029–1059 (2020). https://doi.org/10.1007/s00028-019-00544-2
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DOI: https://doi.org/10.1007/s00028-019-00544-2