Abstract
In this paper, the full information about the existence and nonexistence of a time-periodic traveling wave solution of a reaction–diffusion Zika epidemic model with seasonality, which is non-monotonic, is investigated. More precisely, if the basic reproduction number, denoted by \(R_{0}\), is larger than one, there exists a minimal wave speed \(c^* > 0\) satisfying for each \(c > c^*\), the system admits a nontrivial time-periodic traveling wave solution with wave speed c, and for \(c<c^*\), there exist no nontrivial time-periodic traveling waves such that if \(R_0 \leqslant 1\), the system admits no nontrivial time-periodic traveling waves.
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1 Introduction
In this paper, we focus on the following reaction–diffusion Zika epidemic model with seasonality
where the total group of human can be divided into the susceptible group \(S_H\) and the infected group \(I_H\). Similarly, the total group of mosquitoes can be separated into \(S_V\)-susceptible and \(I_V\)-infected. \(D_i(i=1, 2)\) and \(d_i(i=1, 2)\) are the diffusion rate of the susceptible individuals, the susceptible mosquitoes, the infectious individuals and the infectious mosquitoes, respectively. \(\beta _1(t)\), \(\beta _2(t)\), and \(\beta _3(t)\) are the contact rates among the susceptible humans and the infected humans, the susceptible humans and the infected mosquitoes, and the infected humans and the susceptible mosquitoes, respectively. \(r_1(t)\) and \(r_2(t)\) are the removal rate of the infectious individuals and the infectious mosquitoes, respectively. Moreover, we make the following assumption:
- (A):
-
\(D_i(i= 1, 2)\) and \(d_i(i= 1, 2)\) are all positive constants. In addition, \(\beta _i(t)(i= 1, 2, 3)\) and \(r_i(t)(i= 1, 2)\) are H\(\ddot{o}\)lder continuous and positive nontrivial functions on \(\mathbb {R}^+\) and periodic in time with the same period \(T>0\).
In the paper, we study the existence and non-existence of a time-periodic traveling wave solution of system (1.1). Namely, system (1.1) admits a nontrivial time-periodic traveling wave front with each wave speed \(c > c^*\) if \(R_0 >1\). However, the system admits no nontrivial time-periodic traveling wave fronts with \(0<c<c^*\) and \(R_0 >1\) or \(R_{0} \leqslant 1\).
Model (1.1) describes the spatial transmission of Zika virus in human, which were first confirmed in Nigeria [25]. A first severe Zika outbreak has occurred in Island of Yap in 2007. After that, they have also experienced the subsequent outbreak of Zika, such as French Polynesia, South Pacific, New Caledonia, Easter Island, etc. [6]. In 2015, a large outbreak in Brazil was occurred and provided a large number of infected cases. Since then, it had spread freely to many other countries [35]. WHO called Zika a “Public Health Emergency of International Concern” in 2016 [42]. Up to now, there is still no effective drug used to treat Zika patients. In fact, Zika virus infection can be transmitted mainly by the bite of an infected Aedes species mosquito during the day and night. Then a mosquito can be infected with a virus when it bites an infected person during the period of time when the virus can be found in the person’s blood, typically only through the first week of infection [5]. Similar to other viruses transmission through mosquito bites, such as dengue, fever, rash, headache and muscle pain are the most common symptoms of many infected people with Zika virus. However, unlike these infectious disease, Zika virus can be passed through sex [13]. In order to establish a theoretical framework for mathematical analysis of transmission of Zika virus, many ordinary differential epidemic models have been derived, see [5, 8, 13, 15, 19, 26, 28, 32, 34] and the cited reference therein.
Since the human individuals and the mosquitoes usually move randomly in the spatial space, it is reasonable to take to account the random walk of individuals, which can be described by a reaction–diffusion epidemic model. In the literature, there are many results studying the existence and non-existence of traveling wave solutions of some reaction–diffusion epidemic models, see Murray [27], Rass and Radcliffe [29], Ruan [30], Ruan and Wu [31], Ducrot et al. [9, 10], Wang et al. [38, 39], Li and Zou [21], Zhao et al. [51, 52] and the references cited therein. Recently, Zhang and Zhao [49] studied traveling wave solutions for a nonlocal diffusive Zika transmission model with bilinear incidence. Zhao [54] firstly analyzed spreading speed of a reaction–diffusion Zika model with constant recruitment in terms of the basic reproduction number \(R_0\) and the minimal wave speed \(c^*\). On the basis of it, the full information about the existence and nonexistence of traveling wave solutions of the system is investigated.
It was reported that the transmission dynamics of infectious diseases can be significantly influenced by the seasonal change, see Baca\(\mathrm{\ddot{e}}\)ra and Gomes [2], Buonomo [4], Eikenberry and Gumel [11], Grassly and Fraser [14], Hethcote [16], Hethcote and Levin [17] and Soper [33]. Thus, it is crucial to investigate the influence of the seasonal factor on the geographic transmission of infectious diseases. However, the study for traveling waves solutions of non-autonomous epidemic models is few. Wang et al. [40] studied the existence and nonexistence of a time-periodic traveling wave solution for a reaction–diffusion SIR epidemic model with the standard incidence rate and seasonality. After that, they [48] further investigated a traveling wave solution of a time-periodic reaction–diffusion SIR model with the bilinear incidence rate. Compared with the above system in [40], the infection group of such system, denoted by I(t, x), is unbounded. Zhao et al. [53] took into account the asymptotic speed of spread and traveling wave solutions for a time-periodic reaction–diffusion SIR epidemic model with periodic recruitment and standard incidence rate determined by the basic reproduction number \(R_0\) and the minimal wave speed \(c^*\). Wang et al. [36] analyzed the existence and non-existence of a time-periodic traveling wave solution of a generalization of the classical Kermack–McKendrick model with seasonality and nonlocal delayed transmission derived by mobility of individuals during latent period of the infectious disease. Yang and Lin [47] established the speed of asymptotic spreading and minimal wave speed of traveling wave solutions for a time-periodic and diffusive DS-I-A epidemic model. Ambrosio et al. [1] studied the existence of generalized traveling waves for a time-dependent reaction–diffusion SIR epidemic model with the bilinear incidence rate on \(\mathbb {R}^2\). Huang et al. [18] established the spreading speeds and periodic traveling waves for a class of time-periodic and partially degenerate reaction–diffusion systems with monotone and non-monotone nonlinearities. For other related results on the periodic traveling waves for time-periodic and spatially continuous non-monotone epidemic model, we refer to the literature [7, 44, 46]. Recently, Wu [43] analyzed the spreading speed and periodic traveling waves for a time-periodic epidemic model in discrete media, which is the lack of comparison principle and compactness of solution operators.
We mention that the major difficulty to study (1.1) is that it lacks the classical comparison principle. Thus, the theory on the traveling wave solutions for monotone semiflows, see [12, 22, 23, 41] and the cited references therein, doesn’t directly work for system (1.1). In addition, a reaction–diffusion epidemic model describing Zika virus spreading is more complex. Thus, except for [49, 54], there seem no results on a time-periodic traveling wave solution for such a reaction–diffusion Zika epidemic model with seasonality.
The rest of this paper is organized as follows. In Sect. 2, the basic reproduction number \(R_0\) and the minimal wave speed \(c^*\) of the system are defined. On the basis of it, the full information with the existence and non-existence of a time-periodic traveling wave solution of system (1.1) is established for \((t, x) \in \mathbb {R}^2\) in Sects. 3 and 4.
2 Preliminary
The aim of the preliminary is to find the basic reproduction number and the minimal wave speed of system (1.1), denoted by \(R_0\) and \(c^*\), which is related with the existence and non-existence of a time-periodic traveling wave solution for the system. Firstly, let \(\mathbb {C}_T\) be the Banach space of all T-periodic continuous functions from \(\mathbb {R}\) to \(\mathbb {R}^2\), which is endowed with the usual supremum norm. Its positive cone \(\mathbb {C}_T^+\) consists of all functions in \(\mathbb {C}_T\) with both nonnegative components.
Secondly, consider the following ODE system
It is clear that \((S_H^0, 0, 0, S_V^0, 0)\) is always an equilibrium of (2.1), denoted by \(E_0\), which is called the disease-free equilibrium of (2.1). Let
There is an evolution operator U(t, s) for \(t \geqslant s\) such that the following linear T-periodic system
Precisely speaking, for each \(s \in \mathbb {R}\), the \(2 \times 2\) matrix U(t, s) satisfies
where I is the \(2 \times 2\) identify matrix. Define a linear operator \(\mathcal {L}: \mathbb {C}_T \rightarrow \mathbb {C}_T\) by
According to [37], \(\mathcal {L}\) is called by the next infection operator and define the basic reproduction number of system (2.1) by \(R_0:= r(\mathcal {L})\), where \(r(\mathcal {L})\) is the spectral radius of \(\mathcal {L}\).
Linearizing the second equation and the forth equation of system (1.1) at the disease-free equilibrium \(E_0\) yields
Letting \( {I_{H}\atopwithdelims ()I_{V}}(t,x)=e^{\mu x}{\eta _{H}(t)\atopwithdelims ()\eta _{V}(t)}\) and then plugging it into equation (2.2), we obtain the characteristic equations as below
Denote the solution map of system (2.3) by \((\eta _H, \eta _V)_t(\tilde{\eta }_{H0}, \tilde{\eta }_{V0}):= (\eta _H, \eta _V)(t;\tilde{\eta }_{H0}, \tilde{\eta }_{V0}) \), where \((\eta _H, \eta _V)(t;\) \(\tilde{\eta }_{H0},\tilde{\eta }_{V0})\) is the solution of system (2.3) with initial value \((\tilde{\eta }_{H0}, \tilde{\eta }_{V 0}) \in \mathbb {R}^2_+\). Assume that \(r(\mu )\) denotes the spectral radius of the Poincar\(\acute{e}\) map \(B_c:= (\eta _H, \eta _V)_T\) with system (2.3). By using the similar arguments as those in [45], \((\eta _H^*, \eta _V^*)\) is a eigenvalue vector of \(B_c\) associated with the corresponding principal eigenvalue \(r(\mu )\). Furthermore, according to [37] with \(R_0 > 1\), one has \(r_0:= r(0) > 1\), indicating that \(r(\mu )> r_0 > 1\). Define \(\lambda (\mu ):= \frac{\ln r (\mu )}{T}\) and \(\Phi (\mu ): =\frac{\lambda (\mu )}{\mu },~\forall \mu \in (0, \infty )\). It then follows from Lemma 3.8 in [23] that there exist \(\mu ^*,c^* \in (0, +\infty )\) such that
Choose a small enough constant \(\epsilon >0\), which is determined later. Then consider the following system
On the same way, plugging \( {I_{H}^\epsilon \atopwithdelims ()I_{V}^\epsilon }(t,x)=e^{\mu x}{\eta _{H}^\epsilon (t)\atopwithdelims ()\eta _{V}^\epsilon (t)}\) into the above equations causes to
Similarly, define the spectral radius of the Poincar\(\acute{e}\) map with system (2.5) by \(r^\epsilon (\mu )\). Due to \(R_0 > 1\), there exists a \(\epsilon _0 > 0\) small enough such that for any \(\epsilon \in (0, \epsilon _0)\), one has \(r_0^\epsilon := r^\epsilon (0) > 1\), indicating that \(r^\epsilon (\mu )> r_0^\epsilon > 1\). Let \(\lambda ^\epsilon (\mu ):= \frac{\ln r^\epsilon (\mu )}{T}\) and \(\Phi ^\epsilon (\mu ): =\frac{\lambda ^\epsilon (\mu )}{\mu },~\forall \mu \in (0, \infty )\). Then there exist \(\mu _\epsilon ^*,c_\epsilon ^* \in (0, +\infty )\) such that \( c_\epsilon ^* = \Phi ^\epsilon (\mu _\epsilon ^*)= \inf _{\mu > 0}\Phi ^\epsilon (\mu ) \) and
by using (2.4) and (2.5). In addition, it is obvious that \(\lim _{\epsilon \rightarrow 0^+} c_\epsilon ^* = c^*\).
3 Existence of periodic traveling wave solutions
In the section, we establish the existence of the time-periodic traveling wave solutions of model (1.1). We firstly define a time T-periodic traveling wave solution for system (1.1), namely, it is a special solution with the form as follows
In addition, it can satisfy the following epidemic model
posed for \(\forall (t,z)\in \mathbb {R}\times \mathbb {R}.\) We intend to find a nonnegative solution \((u_{1}(t, z), u_{2}(t, z),v_{1}(t, z), v_{2}(t, z))\) of system (3.2) so that the following boundary conditions
uniformly \(t \in \mathbb {R}\), where \(S_H^0>S_H^\infty \) and \(S_V^0>S_V^\infty \), \(S_H^\infty \) and \(S_V^\infty \) are determined later.
Linearizing the second and the last equation of system (3.2) causes to
Letting \({\bar{v}_1\atopwithdelims ()\bar{v}_2}(t,z)=e^{\mu z}{\mathcal {J}_1(t)\atopwithdelims ()\mathcal {J}_2(t))}\) and then plugging it into (3.4), we get the characteristic equations as below
Next, we show that system (3.5) generates a positive time-periodic solution with the period \(T>0\), still denoted by \((\mathcal {J}_1, \mathcal {J}_2)\). Firstly, consider the following system
Define the solution semiflow of system (3.6) by \((\tilde{\eta }_1, \tilde{\eta }_2)_t(\tilde{\eta }_{10}, \tilde{\eta }_{20}):= (\tilde{\eta }_1, \tilde{\eta }_2)(t;\tilde{\eta }_{10}, \tilde{\eta }_{20})\), where \((\tilde{\eta }_1, \tilde{\eta }_2)(t;\tilde{\eta }_{10},\) \(\tilde{\eta }_{20})\) is the solution of system (3.6) with initial value \((\tilde{\eta }_{10}, \tilde{\eta }_{20}) \in \mathbb {R}^2_+\). In addition, denote the Poincar\(\acute{e}\) map of system (3.6) by \(\mathcal {P}_{c}:= (\tilde{\eta }_1, \tilde{\eta }_2)_T\). It further follows that
where \((\kappa _1, \kappa _2)\) is the initial value of system (3.6) and \((\eta _H, \eta _V)(t; \kappa _1, \kappa _2)\) is the solution of system (2.3) with initial value \((\kappa _1, \kappa _2) \in \mathbb {R}^2_+\). Consequently, one has
where \((\eta _H^*, \eta _V^*)\) is a eigenvalue vector of the Poincar\(\acute{e}\) map \(B_c\) of system (3.6) with the principal eigenvalue \(r(\mu )\). Obviously, if \(\mu = \frac{\lambda (\mu )}{c}\), then \((\eta _H^*, \eta _V^*)\) is a fixed point of the Poincar\(\acute{e}\) map \(\mathcal {P}_{c}\), where \(\lambda (\mu )\) has been defined in (2.3). Consequently, \((\tilde{\eta }_1, \tilde{\eta }_2)_t:= (\tilde{\eta }_1, \tilde{\eta }_2)(t; \eta _H^*, \eta _V^*)\) is a positive time-periodic solution of system (3.6) with \(c \mu = \lambda (\mu )\).
According to [23, Theorem 3.8], we obtain that if \(R_0 > 1\), for each \(c > c^*\), there exist \(\mu _1(c)\) and \(\mu _2(c)\) such that \(0< \mu _1(c)< \mu _2(c) < \infty \), \(\Phi (\mu _1) =c \) and \(\Phi (\mu ) <c,~\mu \in (\mu _1, \mu _2)\). Let \(\epsilon _2 \in (0, \mu _2-\mu _1)\), which is determined later, \(\mu _{\epsilon _2} = \mu _1 + \epsilon _2\), \(\lambda (\mu _{\epsilon _2}):= \frac{\ln \rho (\mu _{\epsilon _2})}{T}\), \(\Phi (\mu _{\epsilon _2}):= \inf _{\mu _{\epsilon _2} >0}\frac{\lambda (\mu _{\epsilon _2})}{\mu _{\epsilon _2}}\) and \(c^*< c_{\epsilon _2}:= \Phi (\mu _{\epsilon _2})<c\), where \(\rho (\mu _{\epsilon _2})\) is the spectral radius of the Poincar\(\acute{e}\) map of the system as follows
On the same way, system (3.7) generates a positive time-periodic solution with the period \(T>0\), denoted by \((\mathcal {P}_1(t), \mathcal {P}_2(t))\).
Based on the above arguments, we can obtain the following lemmas.
Lemma 3.1
The vector function \({v_1^+\atopwithdelims ()v_2^+}(t, z):= {\mathcal {J}_1(t) \atopwithdelims ()\mathcal {J}_2(t)} e^{\mu _1 z}\) satisfies the following equations
Lemma 3.2
Assume that \(\epsilon _1\) is sufficiently small with \(0< \epsilon _1 < \min \{\mu _1, \frac{c}{D_i}\}(i= 1, 2)\) and \(\mathcal {M}:= \frac{1}{\epsilon _1}\) is large enough. Then the functions \(u_1^-(t, z):= \max \{S_H^0(1 - \mathcal {M} e^{\epsilon _1 z}), 0\}\) and \(u_2^-(t, z):= \max \{S_V^0(1 - \mathcal {M} e^{\epsilon _1 z}), 0\}\) satisfy
Proof
Here, we show that \(u_1^-\) satisfies (3.8). If \(z > - \epsilon _1^{-1} \ln \mathcal {M}\), then \(u_1^-(t, z) = 0\), and thus, the first equation of (3.8) is valid.
If \(z < - \epsilon _1^{-1} \ln \mathcal {M}\), then \(u_1^-(t, z) = S_H^0(1 - \mathcal {M} e^{\epsilon _1 z})\). In addition, it is needed only to prove that
Therefore, it is sufficient to verify
It is obvious that the above conclusion holds provided that \(\mathcal {M}: = \frac{1}{\epsilon _1}\) is sufficiently large. In addition, \(u_2^-(t, z)\) is discussed similarly and thus we omit it. The proof is completed. \(\square \)
Lemma 3.3
Suppose that \(\epsilon _2\) with \(\epsilon _2 < \min \{\epsilon _1, \mu _2 - \mu _1\}\) is sufficiently small and \(\mathcal {K}\) is large enough such that
where \(c_{\epsilon _2}\), \(\mu _{\epsilon _2}\) and \(\mathcal {P}_i(t)\) have been defined in (3.7) and \(\mathcal {J}_i(t)(i= 1, 2)\) has been defined in (3.5). Then the function \(v_i^-(t, z):= \max \{(\mathcal {J}_i(t)e^{\mu _1 z} - \mathcal {K} e^{\mu _{\epsilon _2} z}\mathcal {P}_i(t)), 0 \}(i= 1, 2)\) satisfies
for any \(z \ne z_2, z_3\), \(z_2(t):= (\epsilon _2)^{-1} \ln \frac{\mathcal {J}_1(t)}{\mathcal {K} \mathcal {P}_1(t)}\), \(z_3(t):= (\epsilon _2)^{-1} \ln \frac{\mathcal {J}_2(t)}{\mathcal {K} \mathcal {P}_2(t)}\) and \(z_2, z_3 < z_1\).
Proof
There may be the two following cases \(z_3(t) \le z_2(t)\) and \(z_2(t) < z_3(t)\) for some \(t\in \mathbb {R} \). Next we show \(z_3(t) \le z_2(t)\) for some \(t \in \mathbb {R}\) and then we omit the condition of \(z_2(t) > z_3(t)\) for some \(t \in \mathbb {R}\).
If \(z> z_2(t)\), then \(v_i^- = 0\) for \(i= 1, 2\).
If \(z_3(t)< z< z_2(t) < z_1\) for some \(t\in \mathbb {R}\), then \(u_1^-(t, z)= S_H^0(1 - \mathcal {M} e^{\epsilon _1 z})\), \(v_1^-(t, z)= \mathcal {J}_1(t)e^{\mu _1 z} - \mathcal {K} e^{\mu _{\epsilon _2} z}\mathcal {P}_1(t)\) and \(v_2^-(t, z) = 0\). Due to (3.9), one can get
If \( z< z_2(t), z_3(t) < z_1\), then \(u_1^-(t, z)= S_H^0 (1 - \mathcal {M} e^{\epsilon _1 z})\), \(v_1^-(t, z)= \mathcal {J}_1(t)e^{\mu _1 z} - \mathcal {K} e^{\mu _{\epsilon _2} z}\mathcal {P}_1(t)\) and \(v_2^-(t, z) = \mathcal {J}_2(t)e^{\mu _1 z} - \mathcal {K} e^{\mu _{\epsilon _2} z}\mathcal {P}_2(t)\). Furthermore, we need to verify that
According to (3.9), the above inequality holds true. In addition, \(v_2^-\) is proved similarly. It completes the proof. \(\square \)
Let \(N > - \min \{z_2, z_3\}\) and \(C_N:=C(\mathbb {R} \times [-N, N], \mathbb {R}^4)\). Define a convex cone \(\mathcal {D}_N\) by
For any given \( (\bar{u}_1, \bar{u}_2, \bar{v}_1, \bar{v}_2) \in \mathcal {D}_N\), consider the following initial value problem:
where
and
for any \(t \in [0, T]\) and \(z \in [-N, N]\). It is easy to see that \(\bar{G}_{\bar{u}_i}(t, \pm N)=u_i^-(t, \pm N)\) and \(\bar{G}_{\bar{v}_i}(t, \pm N) = v_i^-(t, \pm N)\) for \(t \in \mathbb {R}\) and \(i = 1, 2\). Moreover, the functions \(\bar{G}_{\bar{u}_i}\) and \(\bar{G}_{\bar{v}_i}\) are T-periodic and belong to \(C^{1, 2}(\mathbb {R}\times [-N,N])\). Set \(\tilde{u}_i(t, z) = \bar{u}_i(t, z) - \bar{G}_{\bar{u}_i}(t, z)\), \(\tilde{v}_i(t, z) = \bar{v}_i(t, z) - \bar{G}_{\bar{v}_i}(t, z)\), \(\tilde{F}_{\bar{u}_i} = \mathcal {B}_{i} \bar{G}_{\bar{u}_i}(t, z) - \partial _t \bar{G}_{\bar{u}_i}(t, z)\) and \(\tilde{F}_{\bar{v}_i} = \mathcal {T}_{i} \bar{G}_{\bar{v}_i}(t, z) - \partial _t \bar{G}_{\bar{v}_i}(t, z)\) for \(i= 1, 2\). Then the problem (3.10) reduces to
The realization of \(A_i\) in \(C([-N, N])\) with the homogenous Dirichlet boundary condition can be defined by
In fact, \(D(A_i) =\big \{ u \in C^2([-N, N]), u|_{\pm N} = 0 \big \}\) (see, e.g., [24, Section 5.1.2]). Assume that \(\{H_i(t)\}_{t \geqslant 0}\) is the strongly continuous analytic semigroup generated by \(A_i^0: D(A_i^0) \subset C([-N, N]) \rightarrow C([-N, N])\) for \(i= 1, 2\) (see [24]). Note that
and
for \(t > 0\) and \(x \in [-N,N]\), where \(\Gamma _{i}(i=1, 2)\) and \(\Gamma _j(j= 3, 4)\) are the Green functions associated with \(D_i \partial _{xx}-c \partial _x\) and \(d_i\partial _{xx}-c \partial _x\) and Dirichlet boundary condition, respectively. Then system (3.11) can be treated as the following integral system
where \(t \geqslant 0\) and \(z \in [-N, N]\), indicating that \((\bar{u}_1(t, z), \bar{u}_2(t, z), \bar{v}_1(t, z), \bar{v}_2(t, z))\) satisfies
where \(t \geqslant 0\), \(z \in [-N, N]\) and \(i= 1, 2\). A solution of (3.12) can be called as a mild solution of (3.11). Note that \(p_i[\bar{u}_1, \bar{u}_2, \bar{v}_1, \bar{v}_2], q_i[\bar{u}_1, \bar{u}_2, \bar{v}_1, \bar{v}_2] \in C(\mathbb {R}\times [-N, N])\), then it follows from [24, Theorem 5.1.17] that the functions \(\bar{u}_i\) and \(\bar{v}_i(i= 1, 2)\) defined by (3.12) belong to \(C([0, 2T] \times [-N, N]) \bigcap C^{\theta , 2 \theta }([\epsilon , 2 T]\times [-N,N])\) for every \(\epsilon \in (0, 2T)\) and \(\theta \in (0, 1)\). Define a set
Obviously, \(\mathcal {D}_N^0\) is a closed and convex set.
Lemma 3.4
For any \(U_0:=(u_{10}, u_{20},v_{10},v_{20}) \in \mathcal {D}_N^0\), let \((u_{1N}(t,z;U_0), u_{2N}(t,z;U_0), v_{1N}(t,z;U_0),v_{2N}(t,z;\) \(U_0))\) be the solutions of system (3.12) with the initial value \(U_0\). Then
for any \((t, z) \in [0, \infty ) \times [-N, N]\).
Proof
The argumentations are essentially same as those in [53, Lemma 3.3] and [48, Lemma 2.4], so we omit them. \(\square \)
For a given \(U_0:= (u_{10}, u_{20}, v_{10}, v_{20}) \in \mathcal {D}_N^0\), define a map \(F: \mathcal {D}_N^0 \rightarrow C([-N, N], \mathbb {R}^4)\) by
where \((u_{1N}(t,z;U_0),u_{2N}(t,z;U_0), v_{1N}(t,z;U_0), v_{2N}(t,z;U_0))\) is the solution of system (3.12) with the initial value \(U_0\). In view of Lemma 3.4 and the periodicity of \(u_i^-\), \(v_i^-\) and \(v_i^+\), we have \(F[\mathcal {D}_N^0] \in \mathcal {D}_N^0\). Obviously, \(\mathcal {D}_N^0\) is a complete metric space with a distance induced by the supreme norm. For any \(U_0^1:=(u_{10}^1,u_{20}^1,v_{10}^1,v_{20}^1 )\) and \(U_0^2:=(u_{10}^2,u_{20}^2,v_{10}^2,v_{20}^2) \in \mathcal {D}_N^0\), (3.12) indicates
On the same way,
Since \(e^{- \alpha _i T}, e^{- \chi _i T}<1\) for \(i =1, 2\), one has that \(F: \mathcal {D}_N^0 \rightarrow \mathcal {D}_N^0\) is a contraction map. As a consequence, the Banach fixed point theorem implies that F admits a unique fixed point \(U_0^*:=(u_{10}^*,u_{20}^*, v_{10}^*, v_{20}^*) \in \mathcal {D}_N^0\). Let \((u_{1N}^*(t, z),u_{2N}^*(t, z),v_{1N}^*(t, z),v_{2N}^*(t, z)) = (u_{1N}(t, z; U_0^*),u_{2N}(t, z; U_0^*),v_{1N}(t, z; U_0^*),v_{2N}(t, z; U_0^*))\) for \(t \in (0, +\infty )\) and \(z \in [-N, N]\), where \((u_{1N}(t, z; U_0^*),u_{2N}(t, z; U_0^*),v_{1N}(t, z; U_0^*),v_{2N}(t, z; U_0^*))\) is the solution of system (3.10) with the initial value \(U_0^*\). Furthermore, using the similar arguments to these in [53], one has \((u_{1N}^*(t, z),u_{2N}^*(t, z),v_{1N}^*(t, z),v_{2N}^*(t, z)) = (u_{1N}^*(t+T, z), u_{2N}^*(t+T, z), v_{1N}^*(t+T, z), v_{2N}^*(t+T, z))\) for all \(t \in [0, \infty )\) and \(z \in [-N, N]\). According to Lemma 3.4, we can get \((u_{1N}^*(t, z),u_{2N}^*(t, z),v_{1N}^*(t, z),v_{2N}^*(t, z)) \in \mathcal {D}_N\). Then \((u_{1N}^*(t, z),u_{2N}^*(t, z),v_{1N}^*(t, z),v_{2N}^*(t, z))\) satisfies
for any \(t\geqslant s\) and \(i=1, 2\). On the basis of the above discussion, we obtain the theorem as follows.
Theorem 3.5
For any given \((u_{1N}, u_{2N}, v_{1N}, v_{2N}) \in \mathcal {D}_N\), there exists a unique solution \((u_{1N}^*,u_{2N}^*, v_{1N}^*,\) \(v_{2N}^*) \in \mathcal {D}_N\) satisfying (3.13).
By virtue of Theorem 3.5, we can define an operator \(\mathcal {R}: \mathcal {D}_N \rightarrow \mathcal {D}_N\) by \(\mathcal {R}(u_{1N}, u_{2N}, v_{1N}, v_{2N}) = (u_{1N}^*,u_{2N}^*, v_{1N}^*, v_{2N}^*)\). In what follows, by using the similar arguments to those in [53, Lemma 3.5] and [48, Lemma 2.6], we present the complete continuity of the operator \(\mathcal {R}\) without proof.
Lemma 3.6
The operator \(\mathcal {R}: \mathcal {D}_N \rightarrow \mathcal {D}_N\) is completely continuous.
Based on the above arguments, the Schauder’s fixed point theorem expresses that \(\mathcal {R}\) admits a fixed point \((u_{1N}^*,u_{2N}^*, v_{1N}^*, v_{2N}^*) \in \mathcal {D}_N\). In addition, \((u_{1N}^*(t+T,\cdot ), u_{2N}^*(t+T,\cdot ), v_{1N}^*(t+T,\cdot ), v_{2N}^*(t+T,\cdot )) = (u_{1N}^*(t,\cdot ), u_{2N}^*(t,\cdot ),\) \(v_{1N}^*(t,\cdot ), v_{2N}^*(t,\cdot ))\) for all \(t \in \mathbb {R}\). Note that \(u_{iN}^*, v_{iN}^* \in C^{\frac{\theta }{2}, \theta }(\mathbb {R}\times [-N,N])\) for some \(\theta \in (0, 1)\) and \(i= 1, 2\). By [24, Theorem 5.1.18 and 5.1.19], \(u_{iN}^*, v_{iN}^* \in C^{1,2}(\mathbb {R}\times [-N,N])(i=1, 2)\) satisfy
where \(i= 1, 2\). Similar to [53, Theorem 3.6] and [48, Theorem 2.7], we have the following local uniform estimates on \(u_{i}^*\) and \(v_{i}^*(i= 1, 2)\).
Lemma 3.7
Let \(p \geqslant 2\). For any given \(L > 0\), there exists a constant \(C:=C(p, L) > 0\) such that for any \(N> \max \{L, -\min \{z_2,z_3\}\}\) large enough, there hold
In addition, there exists a constant \(\hat{C}:=\hat{C}(L) > 0\) such that, for any \(z_0 \in \mathbb {R}\),
for any \(N> \max \{L+|z_0|, -\min \{z_2,z_3\}\}\), \(\theta \in (0, 1)\) and \(i= 1, 2\).
Now, we estimate the solution of system (3.14), denoted by \((u_{1N}^*, u_{2N}^*, v_{1N}^*, v_{2N}^*)\).
Proposition 3.8
Let N be large enough satisfying \(N > - \min \{z_2, z_3\}\). There exists a constant \(C_0\) independent upon N such that
for any \(z \in [-N, N]\).
Proof
We firstly define
Obviously,
According to (3.14), we have
where \(\tilde{u}_{1N, z}^*(z):= \frac{\textrm{d} \tilde{u}_{1N}^*(z)}{\textrm{d}z}\) and \(\tilde{u}_{1N, zz}^*(z):= \frac{\textrm{d}^2 \tilde{u}_{1N}^*(z)}{\textrm{d}z^2}\). It follows from (3.15) that
Then integrating two sides of the above equation from \(z \in [-N, N)\) to N yields
Due to \(\tilde{u}_{1N}^*(z) \geqslant 0\) for \(z \in [-N, N]\) and \(\tilde{u}_{1N}^*(N) = \tilde{u}_1^-(N) = 0\), one has \( \tilde{u}_{1N,z}^*(N) \leqslant 0\). According to (3.16), it has \( \tilde{u}_{1N,z}^*(z) \leqslant 0\) and \( \tilde{u}_{1N,z}^*(z) \not \equiv 0\) on \([-N, N]\). By using \( \tilde{u}_{1N,z}^*(-N) \geqslant \tilde{u}_{1,z}^-(-N) = -S_H^0\,M \epsilon _1 e^{- \epsilon _1 N} \geqslant -S_H^0\), integrating from \(-N\) to N for equation (3.15) leads to
In addition, \(\frac{1}{T}\int \limits _{-N}^{N}\int \limits _{0}^{T}\beta _3(t)v_{1N}^*(t, z)u_{2N}^*(t, z) \textrm{d}t\textrm{d}z <C_0\) can be discussed similarly.
Let \(\bar{r}_1:= \max _{t \in [0, T]} r_1(t)\). Then, \(\tilde{v}_{1N}^*(z)\) satisfies
Similarly, one has \(\tilde{v}_{1N, z}^*(N) \leqslant 0\), \(\tilde{v}_{1N, z}^*(- N) \geqslant \tilde{v}_{1, z}^-(- N) \geqslant - \mathcal {K} \mu _{\epsilon _2} e^{- \epsilon _2 N}\tilde{\mathcal {P}}_1\), \(\tilde{v}_{1N}^*(N) = 0\) and \(\tilde{v}_{1N}^*(-N) = \tilde{v}_{1}^-(-N)\), where \(\tilde{\mathcal {P}}_1:= \int \limits _0^T \mathcal {P}_1(t) \textrm{d}t\) and \(\mathcal {P}_1(t)\) has been defined in Lemma 3.3. Then by integrating the two sides of the last equality on \([-N,N]\), one has
Furthermore, \(\frac{1}{T} \int \limits _0^{T}\int \limits _{-N}^{N} v_{2N}^*(z) \textrm{d}t\textrm{d}z \leqslant C_0\) can be proved similarly. It completes the proof. \(\square \)
Theorem 3.9
Assume that \(R_0 > 1\). For any \(c > c^*\), system (3.2) admits a time-periodic solution \((u_1^*, u_2^*, v_1^*, v_2^*)\) satisfying (3.3). In addition, there hold \(0 < \frac{1}{T}\int \limits _{0}^{T}v_1^*(t, z)\textrm{d}t \leqslant (S_H^0 - S_H^\infty )\) and \(0 < \frac{1}{T}\int \limits _{0}^{T}v_2^*(t, z)\textrm{d}t \leqslant (S_V^0 - S_V^\infty )\) for any \(z \in \mathbb {R}\), and
Proof
The proof is divided into four steps.
Firstly, we show existence of a periodic solution for system (3.2). Assume that \(\{n_m\}_{m\geqslant 1}\) is an increasing sequence such that \(n_m \geqslant - \min \{z_2,z_3\}\) for \(m\in \mathbb {N}^+\) and \(\lim _{m \rightarrow \infty } n_m = \infty \). It then follows that the solution sequence \((u_{1,n_m}, u_{2,n_m}, v_{1,n_m}, v_{2,n_m}) \in \mathcal {D}_{n_m}\) satisfies Lemma 3.7 and (3.14). By virtue of the periodicity of the solution sequence \((u_{1,n_m}, u_{2,n_m}, v_{1,n_m}, v_{2,n_m})\) with \(t \in \mathbb {R}\), we can extract a subsequence of it, still denoted by \((u_{1,n_m}, u_{2,n_m}, v_{1,n_m}, v_{2,n_m})\), converging to a function \((u_1^*, u_2^*, v_1^*, v_2^*) \in C_{loc}(\mathbb {R}^4)\) in the following topologies
where \(\beta \in (0, \theta )\) and \(\theta \in (0, 1)\). Clearly,
It follows from Lemma 3.7 that for any \(N >0\), there exists a constant \(C_3\) such that
Then using the similar arguments to those in [48, Theorem 2.9], \((u_1^*, u_2^*, v_1^*, v_2^*)\) satisfies
where \((t, z) \in \mathbb {R}^2\). It further follows from Proposition 3.8 that there exists a constant \(C_0 > 0\) such that
Note that \((u_1^*, u_2^*, v_1^*, v_2^*)\) satisfies that
As a consequence, there holds \(u_i^*(t, z) \rightarrow S_H^0(S_V^0)\) and \(v_i^*(t, z) \rightarrow 0\) uniformly for \(t \in \mathbb {R}\) and \(i = 1, 2\), as \(z \rightarrow -\infty \).
Secondly, we prove the asymptotic behavior of \(v_i^*\) as \(z \rightarrow +\infty \). Define \(\hat{v}_1(z)= \frac{1}{T}\int \limits _{0}^{T}v_1^*(t, z)\textrm{d}t\). Then \(\hat{v}_1(t)\) satisfies
where \(\bar{r}_1:= \max _{t \in [0, T]}r_1(t)\). Denote the two roots of the characteristic equation
by
Furthermore, let \(\rho := d_1 (\eta ^+ - \eta ^-) = \sqrt{c^2 + 4d_1\bar{r}_1}\). Then it is easy to see that \(\eta ^-< 0 < \eta ^+\). It follows from (3.20) that
and
According to \(\rho := d_1 (\eta ^+ - \eta ^-)\) and \(\eta ^-< 0 < \eta ^+\), it has
which implies that \(\hat{v}_{1,z}(z)\) is uniformly bounded. Consequently, following \(\int \limits _{- \infty }^{+ \infty } \hat{v}_1(z) \textrm{d}z < C_0\), we must have \(\hat{v}_1(z) \rightarrow 0\) as \(z \rightarrow + \infty \). Using the similar arguments to those in [48, Theorem 2.9], \(v_1^*(t, z) \rightarrow 0\) as \(z \rightarrow + \infty \) uniformly for each \(t \in \mathbb {R}\). As a consequence, \(v_1^*(t, z) \leqslant C_0\) holds for any \((t, z) \in \mathbb {R}^2\). On the same way, \(v_2^*(t, z) \rightarrow 0\) as \(z \rightarrow + \infty \) uniformly for every \(t \in \mathbb {R}\).
Thirdly, the asymptotic behavior of \(u_i^*(i= 1, 2)\) is shown. By using the estimate of (3.18) and Laudau-type inequality (see, e.g., [3, 20]), one has
As a consequence,
Define \(\hat{u}_1^*(z) = \frac{1}{T}\int \limits _{0}^{T} u_1^*(t,z) \textrm{d}t\). It is easy to see that \(\hat{u}_{1, z}^*(z) \rightarrow 0\) as \(z \rightarrow - \infty \). In addition, \(\hat{u}_1^*(z)\) satisfies
which implies that
Then, an integration from z to \(\infty \) for the above equality yields
indicating that \(\hat{u}_{1,z}^*(z) < 0\) for \(z \in \mathbb {R}\). Furthermore, \(\hat{u}_{1}^*(\infty )\) exists and \(\hat{u}_{1}^*(\infty ) < \hat{u}_{1}^*(- \infty )=S_H^0\). barb\(\breve{a}\)lat’s lemma implies that \(\hat{u}_{1,z}^*(z) \rightarrow 0\) as \(z \rightarrow \infty \). Integrating two sides of (3.21) from \(- \infty \) to \(\infty \) on z leads to
where \( S_H^\infty := \hat{u}_{1}(\infty ) < S_H^0\). Using the similar arguments to those in [40, Theorem 2.10] and [48, Theorem 2.9], we get \(u_1^*(t, z) \rightarrow S_H^\infty \) uniformly for \(t \in \mathbb {R}\), as \(z \rightarrow + {\infty }\). In addition, \(u_2^*(t, z)\) can be discussed similarly.
Finally, we discuss the properties of \(v_1^*\). Since \(\hat{v}_1\) satisfies
An integrating of (3.22) on \(\mathbb {R}\) leads to
By using the above arguments on the asymptotic behavior of \(v_1^*(t, z)\) as \(z \rightarrow -\infty \), it is obvious that
For any \(t \in \mathbb {R}\), consider the following equation
Then the solution of (3.23) satisfies
Based on (3.22) and L’H\(\hat{o}\)pital’s rule, it follows that
and
Define a new function
where \(\hat{v}_1(z) = \frac{1}{T}\int \limits _{0}^{T} v_1^*(t, z)\textrm{d}t\). On the basis of (3.22) and (3.23), \(\check{v}_1(z)\) satisfies
Multiplying two sides of the above equation by \(e^{-\frac{c}{d_1 z}}\) and integrating from z to \(\infty \), one has
Then, it is easy to see that \(\check{v}_{1}(z)\) is non-decreasing in \(\mathbb {R}\) and \(\lim _{z \rightarrow \infty } \check{v}_{1}(z) = S_H^0-S_H^\infty \), indicating that \(\check{v}_{1}(z) \leqslant S_H^0-S_H^\infty \) for all \(z \in \mathbb {R}\). In light of the definition of \(\check{v}_{1}(z)\) and \(\bar{v}_{1}(z)\), we conclude that \(\hat{v}_1(z) \leqslant \check{v}_1(z) \leqslant S_H^0-S_H^\infty \) on \(\mathbb {R}\). That is, \(0 \leqslant \frac{1}{T}\int \limits _{0}^{T} v_1^*(t, z)\textrm{d}t \leqslant S_H^0-S_H^\infty \) for any \(z \in \mathbb {R}\). In addition, \(v_2^*(t, z)\) has the similar conclusion as \(v_1^*(t, z)\). The proof is completed. \(\square \)
Remark 3.10
The existence of critical periodic traveling waves is complex, which will be investigated in our future work.
4 Non-existence of periodic traveling wave solutions
In the section, we establish the non-existence of the time-periodic traveling wave solutions of model (1.1) for these cases as below: \(R_0 \leqslant 1\) or \(R_0 > 1\) and \(0< c < c^*\).
4.1 Case 1: \(R_0 > 1\) and \(0< c < c^*\)
With the aim of it, we need to study the following lemma. Firstly, for some \(c \in (0, c^*)\), fix \(c_0 \in (c, c^*)\). Let \(\upsilon _{c_0}=\frac{c_0}{2}\), \(d_1 = d_2=1\) and \(\epsilon \) be small enough, consider the following system
Denote the solution map of system (4.1) by \((\eta _1^\epsilon , \eta _2^\epsilon )_t(\tilde{\eta }_{10}, \tilde{\eta }_{20}):= (\eta _1^\epsilon , \eta _2^\epsilon )(t;\tilde{\eta }_{10}, \tilde{\eta }_{20}) \), where \((\eta _1^\epsilon , \eta _2^\epsilon )(t;\tilde{\eta }_{10},\) \(\tilde{\eta }_{20})\) is the solution of system (4.1) with initial value \((\tilde{\eta }_{10}, \tilde{\eta }_{20}) \in \mathbb {R}^2_+\). In addition, let \(\lambda _{c_0, \epsilon } = \frac{\ln \rho ^\epsilon (\upsilon _{c_0})}{T}\), where \(\rho ^\epsilon (\upsilon _{c_0})\) is the spectral radius of the Poincar\(\acute{e}\) map \(B_{c_0, \epsilon }:= (\eta _1^\epsilon , \eta _2^\epsilon )_T\) of system (4.1). By using the similar arguments as those in [45], \((\eta _1^*, \eta _2^*)\) is a eigenvalue vector of \(B_{c_0, \epsilon }\) associated with the corresponding principal eigenvalue \(\rho ^\epsilon (\upsilon _{c_0})\).
Based on the above arguments, we can obtain the following conclusion.
Lemma 4.1
Suppose that \(\upsilon _{c_0}=\frac{c_0}{2}\), \(L >0\) is large enough and \(\epsilon >0\) is small enough, consider the principal eigenvalue problem of the cooperative elliptic system as below
Then system (4.2) generates a positive time-periodic solution with the period \(T>0\).
Proof
Consider the following system
Define the semiflow of system (4.3) by \((\tilde{\eta }_1, \tilde{\eta }_2)_t(\tilde{\eta }_{10}, \tilde{\eta }_{20}):= (\tilde{\eta }_1, \tilde{\eta }_2)(t;\tilde{\eta }_{10}, \tilde{\eta }_{20}) \), where \((\tilde{\eta }_1, \tilde{\eta }_2)(t;\tilde{\eta }_{10},\) \(\tilde{\eta }_{20})\) is the solution of system (4.3) with initial value \((\tilde{\eta }_{10}, \tilde{\eta }_{20}) \in \mathbb {R}^2_+\). In addition, denote the Poincar\(\acute{e}\) map of system (4.3) by \(\mathcal {P}_{c_0, \epsilon }:= (\tilde{\eta }_1, \tilde{\eta }_2)_T\). It further follows that
where \((\kappa _1, \kappa _2)\) is the initial value of system (4.3) and \((\eta _1^\epsilon , \eta _2^\epsilon )\) is the solution of system (4.1). Consequently, one has
where \((\eta _1^*, \eta _2^*)\) has been defined in (4.1). If \(\lambda = \lambda _{c_0, \epsilon } = \frac{\ln \rho ^\epsilon (\upsilon _{c_0})}{T}\), then \((\eta _1^*, \eta _2^*)\) is a fixed point of the Poincar\(\acute{e}\) map \(\mathcal {P}_{c_0, \epsilon }\). As a consequence, \((\tilde{\eta }_1, \tilde{\eta }_2)_t:= (\tilde{\eta }_1, \tilde{\eta }_2)(t; \eta _1^*, \eta _2^*)\) is a positive time-periodic solution of system (4.3) with \(\lambda = \lambda _{c_0, \epsilon }\). This completes the proof. \(\square \)
Theorem 4.2
Assume that \(R_0 > 1\), \(0< c < c^*\) and \(d_1=d_2=1\). Then system (1.1) admits no nontrivial T-periodic traveling waves \((u_1,u_2, v_1, v_2)\) satisfying (3.2) and (3.3).
Proof
Suppose, by a contradiction way, that there exists such a solution \((u_1,u_2, v_1, v_2)\) satisfying (3.2) and (3.3) for some \(c< c^*\). Firstly, according to \(\lim _{t \rightarrow -\infty } u_1(t, z) = S_H^0,~\forall t \in \mathbb {R}\), we can choose a \(M_\epsilon >0\) large enough and a \(\epsilon >0\) sufficiently small such that
uniformly for \(t \in \mathbb {R}\). Let \(y_1, y_2 < - M_\epsilon \), we take into account the following system
Furthermore, one has
expressing that \(c \upsilon _{c_0} < \lambda _{c_0, \epsilon }\), where \(\lambda _{c_0, \epsilon }\) has been defined in Lemma 4.1, \(r^\epsilon (\mu )\) and \(c_\epsilon ^*\) have been defined in (2.5) and \(\upsilon _{c_0} = \frac{c_0}{2}\).
Secondly, denote \({\bar{w}_{1} \atopwithdelims ()\bar{w}_{2}}(t, z):= e^{\lambda ^* t}e^{\upsilon _{c_0}z}p(z){k_1(t)\atopwithdelims ()k_2(t)}\), where \(\lambda ^* \in (0, \lambda _{c_0, \epsilon }- c_0 \upsilon _{c_0})\) is a constant, \((k_1(t), k_2(t))\) is a solution of system (4.2) and p(z) is the eigenfunction of the principal eigenvalue problem as below
where \(L:= |y_1-y_2|\). Furthermore, one has \(\lim _{L \rightarrow \infty } \rho _L = 0\), indicating that \(\lambda ^* + c_0\upsilon _{c_0} -\lambda _{c_0, \epsilon } + \rho _L \leqslant 0\). According to Lemma 4.1 and the above arguments, plugging \(\bar{w}_{1}(t, z)\) into the first equation of system (4.5) becomes to
Thirdly, let \(\delta > 0\) be small enough such that \(v_{1}(0, z) \geqslant \delta \bar{w}_{1}(0, z),~\forall z \in (y_1, y_2)\). Consider functions \(u_i(t, z+(c-c_0)t)\) and \(v_i(t, z+(c-c_0)t)\) for any \(t \in \mathbb {R}\) and \(z \in (y_1, y_2)\). Denote \(\hat{v}_i(t, z):= v_i(t, z+(c-c_0)t)(i= 1, 2)\), which satisfies
In view of \(c - c_0 < 0\), \(z \in (y_1, y_2)\) and \(y_1< y_2 < -M_\epsilon \), one has \(z + (c - c_0)t < -M_\epsilon ,~\forall t \geqslant 0, ~z \in [y_1, y_2]\). Due to (4.4), \(\hat{v}_1(t, z)\) satisfies
for any \(t \geqslant 0\) and \(z \in [y_1, y_2]\). Since there are
we infer from the parabolic maximum principle that
Due to \(\lambda ^*> 0\), we obtain \(v_1(t, z+(c-c_0)t) \rightarrow \infty \) as \(t \rightarrow \infty \), which leads to a contradiction. On the same way, \(v_2\) is proved similarly and thus we omit it. The proof is completed. \(\square \)
4.2 Case 2: \(R_0 < 1\)
Theorem 4.3
Assume that \(R_0 < 1\). Then for any \(c\geqslant 0\), system (3.2) admits no nontrivial T-periodic solution \((u_1,u_2, v_1, v_2)\) satisfying (3.3).
Proof
Assume that there exists a nontrivial T-periodic solution \((u_1,u_2, v_1, v_2)\) of system (3.2)–(3.3) by a contradiction way. Let \(\bar{v}_i(t):= \int \limits _{- \infty }^{+ \infty } v_i(t, z)\textrm{d}z\) on \(\mathbb {R}\) for \(i= 1, 2\). Obviously, \(\bar{v}_i(t) = \bar{v}_i(t+ T),~\forall t \in \mathbb {R}\) for \(i= 1, 2\). In light of inequality (3.19), one gets that \(\bar{v}_i(t)\) is bounded on [0, T). In addition, for any given \(t \in [0, T)\), there exists a \(\epsilon _0(t)\) depending upon t such that
Furthermore, it follows from \(u_i(t, z) \leqslant S_H^0(S_V^0)(i= 1, 2)\) that
Integrating both two side of the above equations from \(- \infty \) to \(\infty \), we obtain
Then by using the parabolic maximum principle, one has
where \((\tilde{v}_1(t), \tilde{v}_2(t))\) is the solution of the system as below
Due to [50, Theorem 2.1] associated with \(R_0<1\), one has \( \lim _{t \rightarrow + \infty }\tilde{v}_i(t) = 0(i= 1, 2)\), implying that
which leads to a contradiction with (4.7). This completes the proof. \(\square \)
4.3 Case 3: \(R_0 = 1\)
Theorem 4.4
Assume that \(R_0 = 1\). Then for any \(c\geqslant 0\), system (3.2) admits no nontrivial T-periodic solution \((u_1,u_2, v_1, v_2)\) satisfying (3.3).
Proof
Assume that there exists a nontrivial T-periodic solution \((u_1,u_2, v_1, v_2)\) of system (3.2)–(3.3) by a contradiction way. Let \(\bar{v}_i(t):= \int \limits _{- \infty }^{+ \infty } v_i(t, z)\textrm{d}z\) on \(\mathbb {R}\) for \(i= 1, 2\). Due to (3.19), we can get that \(\bar{v}_i(t)\) bounded on [0, T]. In addition, \(\bar{v}_i(t)\) satisfies
where \(f_1(t)= \beta _1(t) \int \limits _{-\infty }^{+\infty } (u_1(t, z)-S_H^0) v_1(t, z)\textrm{d}z + \beta _2(t) \int \limits _{-\infty }^{+\infty } (u_1(t, z)-S_H^0) v_2(t, z)\textrm{d}z\) and \(f_2(t)= \beta _3(t)\) \( \int \limits _{-\infty }^{+\infty } (u_2(t, z)-S_V^0) v_1(t, z)\textrm{d}z\) and \(f(t) = (f_1(t),f_2(t))^T\). System (4.8) owns a positive T-periodic solution \(\bar{v}(t):= (\bar{v}_1(t), \bar{v}_2(t))^T\). Thus, we get
where U(t, s) and \(\mathcal {F}(t)\) have been defined in Sect. 2. In addition, it is not difficult to show that \(u_1(t, z) \leqslant S_H^0\) for \((t, z) \in \mathbb {R}^2\). In fact, suppose that there exists \((t_0, z_0)\) such that \(\max _{(t, z) \in \mathbb {R}^2}u_1(t, z) = u_1(t_0, z_0)>S_H^0\). Thus,
which is a contradiction. Furthermore, \(u_2(t, z)\) can be proved similarly. As a consequence, it has
Consider the following problem:
Due to [50, Theorem 2.1] associated with \(R_0 = 1\), there exists a positive T-periodic solution \(\tilde{v}(t):= (\tilde{v}_1(t), \tilde{v}_2(t))^T\) satisfying the above problem. A straightforward computation leads to
It further follows from the parabolic maximum principle together with (4.10) that
However, due to the periodicity of \(\bar{v}(t)\) and \(\tilde{v}(t)\), one has \(\tilde{v}(T)= \tilde{v}(0)=\bar{v}(0)=\bar{v}(T)\), that is,
In view of (4.10), one has
implying that there exists a \(t_0 \in [0, T)\) satisfying
As a consequence, it contradicts with (4.12). It completes the proof. \(\square \)
Data Availability
The authors declare that the data are available on request from the authors.
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Researcher was supported by National Natural Science Foundation of China (12161052) and Natural Science Foundation of Gansu, China (21JR7RA240).
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Zhao, L. Time-periodic traveling wave solutions of a reaction–diffusion Zika epidemic model with seasonality. Z. Angew. Math. Phys. 75, 32 (2024). https://doi.org/10.1007/s00033-023-02173-9
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DOI: https://doi.org/10.1007/s00033-023-02173-9
Keywords
- Nontrivial time-periodic traveling wave solutions
- Zika epidemic model
- A reaction–diffusion system
- Seasonality