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Uniqueness and exponential stability of traveling wave fronts for a multi-type SIS nonlocal epidemic model
Nonlinear Analysis: Real World Applications 36 (2017) 267–277
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Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa
Uniqueness and exponential stability of traveling wave fronts for a multi-type SIS nonlocal epidemic model Shi-Liang Wu ∗ , Guangsheng Chen School of Mathematics and Statistics, Xidian University, Xi’an, Shaanxi 710071, People’s Republic of China
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Article history: Received 9 December 2016 Accepted 5 February 2017
This paper is concerned with the traveling wave fronts of a multi-type SIS nonlocal epidemic model. From Weng and Zhao (2006), we know that there exists a critical wave speed c∗ > 0 such that a traveling wave front exists if and only if its wave speed is above c∗ . In this paper, we first prove the uniqueness of certain traveling wave fronts with non-critical wave speed. Then, we show that all non-critical traveling wave fronts are asymptotically exponentially stable. The exponential convergent rate is also obtained. © 2017 Elsevier Ltd. All rights reserved.
Keywords: SIS nonlocal epidemic model Traveling wave front Uniqueness Stability
1. Introduction To model the spatial spread of a deterministic epidemic in multi-types of population, Rass and Radcliffe [1] proposed and studied the following nonlocal epidemic model: m ∂ui (x, t) = 1 − ui (x, t) σk βi,k uk (x − y, t)pi,k (y)dy − µi ui (x, t), 1 ≤ i ≤ m, x ∈ R, (1.1) ∂t R k=1
where ui (x, t) represents the proportion of individuals for the ith population σi at position x who are infectious at time t, µi ≥ 0 denotes the combined death/emigration/recovery rate for infectious individuals, βi,k ≥ 0 is the infection rate of a type i susceptible by a type k infectious individual, and pi,k is the corresponding contact distribution. For more details about biological meaning, we refer to [1,2]. In [1, Chapter 8], Rass and Radcliffe made a complete analysis on the global dynamics of the spatially homogeneous m-dimensional system of (1.1). Weng and Zhao [2] further considered the spreading speed and traveling wave fronts of (1.1) and gave an affirmative answer to an open problem presented by Rass and Radcliffe [1]. More precisely, under some reasonable assumptions (see (C1 )–(C3 ) in Section 2), they established ∗ Corresponding author. E-mail address: [email protected] (S.-L. Wu).
http://dx.doi.org/10.1016/j.nonrwa.2017.02.001 1468-1218/© 2017 Elsevier Ltd. All rights reserved.
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the existence of the spreading speed c∗ and showed that it coincides with the critical wave speed for traveling wave fronts. However, to the best of our knowledge, there has been no results on the uniqueness and stability of the traveling wave fronts for such nonlocal epidemic model. This constitutes the purpose of this paper. In the past decades, there have been many significant results on the uniqueness of traveling wave solutions for various evolution equations [3–9]. For example, Chen and Guo [4], Diekmann and Kapper [5], Carr and Chmaj [6] and Huang et al. [7] established the uniqueness theorem of traveling wave solutions for lattice differential equations, integral equations, nonlocal dispersal equations, and delayed reaction–diffusion systems, respectively. Based on the method in Diekmann and Kapper [5] with some nontrivial modifications, we shall prove the uniqueness (up to translation) of certain traveling wave fronts of the nonlocal system (1.1) with non-critical wave speed (see Theorem 3.1). Although the stability of traveling wave solutions for reaction–diffusion equations has been studied intensively in the existing literature (see e.g. [10–21] and references therein), not much is known about traveling wave solutions of monostable nonlocal systems. The main difficulties come from the lack of compactness for solution maps and interaction between different components in a higher dimensional system. Recently, Ouyang and Ou [20] and Wang and Zhao [21] established the asymptotic stability of traveling waves for a nonlocal model in periodic media and a partially degenerate reaction–diffusion system in a periodic habitat, respectively. In this paper, by using the analysis of the principal eigenvalue and establishing a comparison theorem for related linear systems (see Lemma 4.1), we shall show that if the initial function is within a bounded distance from a certain traveling wave front with respect to a weighted maximum norm, then the solution of (1.1) will converge to the traveling wave front exponentially in time. The exponential convergent rate is also obtained (see Theorem 4.2). Our work provides some insights on how to establish the asymptotic stability of the traveling wave fronts for the nonlocal systems. The rest of the paper is organized as follows. In Section 2, we give some preliminaries. Sections 3 and 4 are devoted to the uniqueness and asymptotic stability of the non-critical traveling wave fronts, respectively. 2. Preliminaries In this section, we recall some known results on the existence of the critical wave speed. Throughout this paper, we always use the usual notations for the standard ordering in Rm and denote by ∥ · ∥ the Euclidean norm in Rm . Similar to [2], we make the following assumptions: (C1 ) pi,k (y) ≥ 0, pi,k (y) = pi,k (−y), ∀y ∈ R, R pi,k (y)dy = 1, and for any λ > 0, R eλy pi,k (y)dy < ∞, 1 ≤ i, k ≤ m. (C2 ) The matrix Λ := (σk βi,k )m×m is irreducible in the sense that for every i ̸= k, there exists a distinct sequence i1 , i2 , . . . , ir with i1 = i, ir = k such that σis+1 βis ,is+1 ̸= 0, 1 ≤ s ≤ r − 1. (C3 ) s(Df (0)) > 0, where f = (f1 , . . . , fm ) with fi (u1 , . . . , um ) = (1 − ui )
m
σk βi,k uk − µi ui ,
i = 1, . . . , m.
k=1
From [2, Lemma 2.1], system (1.1) has exactly two spatially homogeneous equilibria 0 := (0, . . . , 0) ∈ Rm and K := (K1 , . . . , Km ) ∈ Rm with 0 ≪ K ≤ 1 := (1, . . . , 1) ∈ Rm . For any λ ∈ R, define a matrix A(λ) = (Ai,k (λ))m×m , where e−λy pi,k (y)dy − µi , i = k, σk βi,k R Ai,k (λ) = (2.1) σk βi,k eλy pi,k (y)dy, i ̸= k. R
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Since A(λ) is cooperative and irreducible, M (λ) := max{ℜα : det(αI − A(λ)) = 0} is a simple eigenvalue of A(λ) with a strongly positive eigenvector ν(λ) = (ν1 (λ), . . . , νm (λ)) (see, e.g. [22, Corollary 4.3.2]). Here and in what follows, ℜα denotes the real part of α. The following result follows from [2, Lemma 3.3 and Theorem 3.2]. Proposition 2.1. Assume that (C1 )–(C3 ) hold. There exists c∗ > 0 and λ∗ > 0 such that c∗ =
M (λ) M (λ∗ ) = inf , λ>0 λ∗ λ
and for any c > c∗ , there exists a unique λ1 (c) ∈ (0, λ∗ ) such that M (λ1 ) = cλ1 , and M (λ) < cλ for any λ ∈ (λ1 , λ∗ ). Moreover, from [2, Theorem 4.2] and the proof of [2, Theorem 4.1], we have the following results on the existence and non-existence of traveling wave fronts. Proposition 2.2. Assume that (C1 )–(C3 ) hold. (i) For any c ∈ (0, c∗ ), system (1.1) has no traveling wave solution connecting 0 and K. (ii) For every c ≥ c∗ , system (1.1) admits a traveling wave front Φ(x + ct) = (φ1 (x + ct), . . . , φm (x + ct)) satisfying Φ(−∞) = 0 and Φ(+∞) = K; and for any c > c∗ , lim Φ(ξ)e−λ1 (c)ξ = ν(λ1 (c)).
ξ→−∞
It is clear that the wave system of the traveling wave solution Φ(ξ) = (φ1 (ξ), . . . , φm (ξ)), ξ = x + ct, of (1.1) has the following form: m σk βi,k cφi (ξ) = 1 − φi (ξ) φk (ξ − y)pi,k (y)dy − µi φi (ξ), 1 ≤ i ≤ m, ξ ∈ R. (2.2) k=1
R
3. Uniqueness of traveling wave fronts In this section, we establish the uniqueness of certain non-critical traveling wave fronts of system (1.1). More precisely, we have the following result. Theorem 3.1. Assume (C1 )–(C3 ) and the following condition: (C4 ) βj,j > 0 and there exists an interval Ij ⊆ R such that pj,j (x) > 0, ∀x ∈ Ij , j = 1, . . . , m. Let Ψ (ξ) = ψ1 (ξ), . . . , ψm (ξ) and Φ(ξ) = φ1 (ξ), . . . , φm (ξ) , ξ = x + ct, be two traveling wave fronts of system (1.1) connecting 0 and K with speed c > c∗ , and satisfy lim Φ(ξ)e−λ1 (c)ξ = h1 ν(λ1 (c)) and
ξ→−∞
lim Ψ (ξ)e−λ1 (c)ξ = h2 ν(λ1 (c)),
ξ→−∞
where h1 , h2 are positive constants. Then, there exists ξ0 ∈ R such that Ψ (ξ) = Φ(ξ + ξ0 ).
(3.1)
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Proof. By a translation if necessary, we may assume lim Φ(ξ)e−λ1 ξ = lim Ψ (ξ)e−λ1 ξ = ν(λ1 ),
ξ→−∞
ξ→−∞
(3.2)
where λ1 := λ1 (c). Set q(ξ) := maxj∈{1,...,m} qj (ξ), ∀ξ ∈ R, where φj (ξ) − ψj (ξ) , eλ1 ξ νj (λ1 )
qj (ξ) :=
∀j = 1, . . . , m, ξ ∈ R.
From (3.2), one can see that qj (±∞) = q(±∞) = 0, ∀j = 1, . . . , m. We first prove that φj (ξ) ≤ ψj (ξ), ∀ξ ∈ R, j = 1, . . . , m. It suffices to show that q(ξ) ≤ 0, ∀ξ ∈ R. Suppose for the contrary that there exists ξ1 ∈ R such that q(ξ1 ) = maxξ∈R q(ξ) > 0. By the definition of q(ξ), there exists j0 ∈ {1, . . . , N } such that q(ξ1 ) = qj0 (ξ1 ). Then, we have qj0 (ξ) ≤ q(ξ) ≤ q(ξ1 ) = qj0 (ξ1 ),
∀ξ ∈ R.
which yields that qj0 (ξ) attains its maximum at ξ1 . According to qj0 (ξ1 ) > 0 and qj0 (±∞) = 0, we can redefine ξ1 such that qj0 (ξ1 ) = maxξ∈R qj0 (ξ) and qj0 (ξ) < qj0 (ξ1 ) for any ξ < ξ1 . Noting that φj0 (ξ1 ) − ψj0 (ξ1 ) = qj0 (ξ1 )eλ1 ξ1 νj0 (λ1 ) > 0 and 0 = qj′ 0 (ξ) =
1 [φ′ (ξ1 ) − ψj′ 0 (ξ1 )]e−λ1 ξ1 − λ1 qj0 (ξ1 ), νj0 (λ1 ) j0
direct computations show that cλ1 νj0 (λ1 )qj0 (ξ1 )eλ1 ξ1 = c[φ′j0 (ξ1 ) − ψj′ 0 (ξ1 )] m σk βj0 ,k pj0 ,k (y)[φk (ξ1 − y) − ψk (ξ1 − y)]dy = 1 − φj0 (ξ1 ) R
k=1
− µj0 [φj0 (ξ1 ) − ψj0 (ξ1 )] − [φj0 (ξ1 ) − ψj0 (ξ1 )]
m k=1
≤ =
m k=1 m
pj0 ,k (y)ψk (ξ1 − y)dy
σk βj0 ,k R
pj0 ,k (y) max{0, φk (ξ1 − y) − ψk (ξ1 − y)}dy − µj0 [φj0 (ξ1 ) − ψj0 (ξ1 )]
σk βj0 ,k R
σk βj0 ,k
pj0 ,k (y)νk (λ1 )eλ1 (ξ1 −y) max{0, qk (ξ1 − y)}dy − µj0 νj0 (λ1 )qj0 (ξ1 )eλ1 ξ1 ,
R
k=1
which implies that cλ1 νj0 (λ1 )qj0 (ξ1 ) ≤
m
σk βj0 ,k νk (λ1 )
pj0 ,k (y)e−λ1 y max{0, qk (ξ1 − y)}dy − µj0 νj0 (λ1 )qj0 (ξ1 ). (3.3)
R
k=1
On the other hand, it follows from A(λ1 )ν(λ1 ) = M (λ1 )ν(λ1 ) = cλ1 ν(λ1 ) that m cλ1 νj0 (λ1 )qj0 (ξ1 ) = σk βj0 ,k νk (λ1 ) pj0 ,k (y)e−λ1 y dyqj0 (ξ1 ) − µj0 νj0 (λ1 )qj0 (ξ1 ).
(3.4)
R
k=1
By (3.3) and (3.4), we see that m
σk βj0 ,k νk (λ1 )
k=1
pj0 ,k (y)e−λ1 y [qj0 (ξ1 ) − max{0, qk (ξ1 − y)}]dy ≤ 0.
R
Note that qj0 (ξ1 ) > 0 and qk (ξ1 − y) ≤ q(ξ1 − y) ≤ q(ξ1 ) = qj0 (ξ1 ),
∀k = 1, . . . , m, y ∈ R.
(3.5)
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It follows from (3.5) that
pj0 ,k (y)e−λ1 y [qj0 (ξ1 ) − max{0, qk (ξ1 − y)}]dy = 0,
σk βj0 ,k νk (λ1 )
∀k = 1, . . . , m.
R
In particular, σj0 βj0 ,j0 νj0 (λ1 )
pj0 ,j0 (y)e−λ1 y [qj0 (ξ1 ) − max{0, qj0 (ξ1 − y)}]dy = 0.
R
Since Λ = (σk βi,k )m×m is irreducible, we see that σj > 0, ∀j = 1, . . . , m. This together with the condition βj,j > 0, ∀j = 1, . . . , m, imply that pj0 ,j0 (y)e−λ1 y [qj0 (ξ1 ) − max{0, qj0 (ξ1 − y)}]dy = 0. R
By the assumptions (C1 ) and (C4 ), we deduce that there exists y0 > 0 such that qj0 (ξ1 ) = max{0, qj0 (ξ1 − y0 )}. It then follows from qj0 (ξ1 ) > 0 that qj0 (ξ1 ) = qj0 (ξ1 − y0 ), which contradicts to the fact qj0 (ξ) < qj0 (ξ1 ) for any ξ < ξ1 . Thus, we conclude that φj (ξ) ≤ ψj (ξ), ∀ξ ∈ R, j = 1, . . . , m. Exchanging the roles of φj (ξ) and ψj (ξ), we also obtain ψj (ξ) ≤ φj (ξ), ∀ξ ∈ R, j = 1, . . . , m. Thus, ψj (ξ) = φj (ξ), ∀ξ ∈ R, j = 1, . . . , m. This completes the proof. Remark 3.2. It should be pointed out that if pj,j (x) ∈ C(R), then it follows the condition that there exists an interval Ij ⊆ R such that pj,j (x) > 0, ∀x ∈ Ij , j = 1, . . . , m.
R
pj,j (y)dy = 1
4. Stability of traveling wave fronts This section is devoted to the asymptotic stability of the non-critical traveling wave fronts of (1.1). Let Φ(x + ct) = φ1 (x + ct), . . . , φm (x + ct) be a traveling wave front of system (1.1) with c > c∗ connecting 0 and K. Given L0 ∈ R ∪ {−∞}. Denote ΩL0 := {(x, t) : x + ct > L0 , t > 0} and ∂ΩL0 := {(x, t) : x + ct > L0 , t = 0} ∪ {(x, t) : x + ct = L0 , t ≥ 0}. The following comparison theorem for related linear system of (1.1) plays a crucial role in proving the stability of the traveling wave fronts. Lemma 4.1. Let Di , di > 0 (i = 1, . . . , m) be any given constants. Assume that W ± (x, t) = W1± (x, t), . . . , ± Wm (x, t) : R × R+ → Rm are two functions which satisfy (i) (ii) (iii) (iv)
W + (x, t) ≥ 0 and W − (x, t) ≤ K for all (x, t) ∈ R × R+ . W + (x, t) ≥ W − (x, t) for all (x, t) ∈ ∂ΩL0 . W + (x, t) ≥ W − (x, t) for all (x, t) ∈ R × R+ with x + ct ≤ L0 . For all (x, t) ∈ ΩL0 , i = 1, . . . , m, there hold m + ∂t Wi (x, t) ≥ Di σk βi,k Wk+ (x − y, t)pi,k (y)dy − di Wi+ (x, t), ∂t Wi− (x, t) ≤ Di
k=1 m k=1
(4.1)
R
σk βi,k R
Then, W + (x, t) ≥ W − (x, t) for all (x, t) ∈ R × R+ .
Wk− (x − y, t)pi,k (y)dy − di Wi− (x, t).
(4.2)
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Proof. Let W (x, t) = (W1 (x, t), . . . , Wm (x, t)) := W + (x, t) − W − (x, t), ∀x ∈ R, t ≥ 0. Then, W (x, t) is bounded from below by −K, W (x, 0) ≥ 0 for x ∈ R, W (x, t) ≥ 0 for all (x, t) ∈ R × R+ with x + ct ≤ L0 , and m ∂t Wi (x, t) ≥ Di σk βi,k Wk (x − y, t)pi,k (y)dy − di Wi (x, t), i = 1, . . . , m, (4.3) R
k=1
for (x, t) ∈ ΩL0 . Choosing K > 3 maxj=1,...,m {Dj σj βj,j }. Suppose that the assertion is false. Then, there exist i0 ∈ {1, . . . , m}, ϖ > 0 and t0 > 0 such that Wi0 (x, t) > −ϖe2Kt and Wj (x, t) ≥ 0
for x ∈ R, t ∈ [0, t0 ),
inf Wi0 (x, t0 ) = −ϖe2Kt0
x∈R
for x ∈ R, t ∈ [0, t0 ], j ̸= i0 .
2Kt0 Thus, there exists a bounded set S ⊂ R with positive Lebesgue measure such that Wi0 (x, t0 ) ≤ − 15 16 ϖe for x ∈ S.
Let ζ(x) be a smooth function satisfying min ζ(x) = 1, x∈R ′
ζ(z) = 1
for z ∈ S,
sup ζ(x) = ζ(±∞) = 3, x∈R
|ζ (·)| ≤ 1
′′
and |ζ (·)| ≤ 1.
For any α ∈ [0, 1], we define the function z(x, t; α) := −ϖ
3 + αζ(x) e2Kt , 4
x ∈ R, t ∈ [0, t0 ].
∂ z(x, t; α) < 0, ∀α ∈ [0, 1], x ∈ R, t ∈ [0, t0 ], It is clear that ∂α 3 1 1 z x, t; = −ϖ + ζ(x) e2Kt ≤ −ϖe2Kt ≤ Wi0 (x, t) for x ∈ R, t ∈ [0, t0 ]; 4 4 4 1 7 3 1 z x, t0 ; + ζ(x) e2Kt0 = − ϖe2Kt0 > Wi0 (x, t0 ) for x ∈ S. = −ϖ 8 4 8 8
Thus, 1 1 , Wi0 (x, t) ≥ z(x, t; α) α∗ := inf α ∈ 8 4
for x ∈ R, t ∈ [0, t0 ]
is well defined. Obviously, Wi0 (x, t) ≥ z(x, t; α∗ ) for x ∈ R, t ∈ [0, t0 ]. Since Wi0 (x, 0) ≥ 0 > z(x, 0; α∗ ) for x ∈ R, 9 z(±∞, t; α∗ ) ≤ − ϖe2Kt < Wi0 (x, t) for x ∈ R, t ∈ [0, t0 ], and 8 Wi0 (x, t) ≥ 0 > z(x, t; α∗ ) for (x, t) ∈ R × [0, t0 ] with x + ct ≤ L0 , we deduce that the function w(x, t) := Wi0 (x, t) − z(x, t; α∗ ) attains its infimum 0 at (x1 , t1 ) ∈ R × (0, t0 ] with x + ct > L0 . Therefore, w(x1 , t1 ) = 0 and wt (x1 , t1 ) ≤ 0. Moreover, w(x1 − y, t1 ) ≥ w(x1 , t1 ) for all y ∈ R, which implies that Wi0 (x1 − y, t1 ) ≥ z(x1 − y, t1 ; α∗ ) + Wi0 (x1 , t1 ) − z(x1 , t1 ; α∗ ) ≥ z(x1 − y, t1 ; α∗ ) + Wi0 (x1 , t1 ),
∀y ∈ R.
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Thus, by (4.3) and the fact that Wj (x, t) ≥ 0 for x ∈ R, t ∈ [0, t0 ], j ̸= i0 , we have 0 ≥ wt (x1 , t1 ) = (Wi0 )t (x, t) − zt (x, t; α∗ ) m ≥ Di0 σk βi0 ,k Wk (x1 − y, t1 )pi0 ,k (y)dy − di0 Wi0 (x1 , t1 ) − zt (x1 , t1 ; α∗ ) R
k=1
≥ Di0 σi0 βi0 ,i0
[z(x1 − y, t1 ; α∗ ) + Wi0 (x1 , t1 )]pi0 ,i0 (y)dy − di0 z(x1 , t1 ; α∗ ) + 2Kϖ R 3 7 3 ≥ − Di0 σi0 βi0 ,i0 ϖe2Kt1 − Di0 σi0 βi0 ,i0 ϖ + α∗ ζ(x1 ) e2Kt1 + Kϖe2Kt1 2 4 4 ≥ K − 3Di0 σi0 βi0 ,i0 ϖe2Kt1 > 0.
3 + α∗ ζ(x1 ) e2Kt1 4
This contradiction implies that W + (x, t) ≥ W − (x, t) for all (x, t) ∈ R × R+ . This completes the proof.
Recall that f (u) = (f1 (u), . . . , fm (u)) with fi (u) = (1 − ui )
m
σk βi,k uk − µi ui ,
i = 1, . . . , m.
k=1
To establish the stability of traveling wave fronts, we need the following additional assumption: (C4 ) Df (K) is irreducible and s(Df (K)) < 0. ¯ := s(Df (K)) < 0 is a simple eigenvalue of Df (K) with an Since Df (K) is cooperative and irreducible, λ ¯ be a fixed number. We eigenvector w = (w1 , . . . , wm ) ≫ 0 (see, e.g. [22, Corollary 4.3.2]). Let ϵ¯ ∈ (0, −λ) can choose ϵ1 ∈ 0, min1≤i≤m {Ki } such that ϵ¯wi ≥ ϵ1
m
σk βi,k wk + ϵ1 wi
k=1
m
σk βi,k ,
∀i = 1, . . . , m.
(4.4)
k=1
Moreover, using limξ→+∞ φj (ξ) = Kj , it is easy to show that lim φj (ξ − y)pi,j (y)dy = Kj , ∀i, j = 1, . . . , m. ξ→+∞
R
Then, we can choose X0 > 0 such that Kj − ϵ1 ≤ φj (z), φj (z − y)pi,j (y)dy ≤ Kj
for any z ≥ X0 , i, j = 1, . . . , m.
R
The stability result of the non-critical traveling wave fronts is stated as follows. Theorem 4.2. Assume that (C1 )–(C4 ) hold. Let Φ(x + ct) = φ1 (x + ct), . . . , φm (x + ct) be a traveling wave front of system (1.1) with c > c∗ connecting 0 and K. For a sufficiently small ϵ > 0, we denote λϵ = λ1 (c)+ϵ and define the weight function W ϵ : R → R+ as follows: e−λϵ (x−X0 ) , x ≤ X0 , ϵ W (x) := (4.5) 1, x > X0 . Then, there exists a number ϵ0 > 0 such that for any given initial value u0 (x) with 0 ≤ u0 (x) ≤ K
and
[u0 (·) − Φ(·)]W ϵ (·) ∈ L∞ (R, Rm ),
the unique solution u(x, t; u0 ) of (1.1) satisfies 0 ≤ u(x, t; u0 ) ≤ K,
∀x ∈ R, t ≥ 0,
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274
and for some C > 0, sup ∥u(x, t; u0 ) − Φ(x + ct)∥ ≤ Ce−ϵ0 t ,
∀ t ≥ 0.
(4.6)
x∈R
Proof. Define two functions U − (x) := min{u0 (x), Φ(x)},
U + (x) := max{u0 (x), Φ(x)},
∀x ∈ R.
Let U ± (x, t) be the solutions of (1.1) with the initial values U + (x) and U − (x), respectively. Then, a simple application of the comparison principle (see e.g. [2, Theorem 2.2]) implies that 0 ≤ U − (x, t) ≤ u(x, t; u0 ), Φ(x + ct) ≤ U + (x, t) ≤ K,
∀x ∈ R, t ≥ 0,
which yields that ∥u(x, t; u0 ) − Φ(x + ct)∥ ≤ max{∥U + (x, t) − Φ(x + ct)∥, ∥U − (x, t) − Φ(x + ct)∥}. Thus, to prove the assertion of this theorem, it is sufficient to show that U ± (x, t) converges to Φ(x + ct) exponentially in time. By symmetry, we only prove that U + (x, t) converges to Φ(x + ct). Set V (x, t) := U + (x, t) − Φ(x + ct). It is clear that V (x, t) ≥ 0 for x ∈ R, t ≥ 0, and 0 ≤ V (x, 0) ≤ |u0 (x) − Φ(x)|,
∀x ∈ R,
ϵ
which implies that V (x, 0)W (x) is uniformly bounded on R. In the sequel, we consider two cases x+ct ≤ X0 and x + ct > X0 , respectively. Case 1. x+ct ≤ X0 . Noting that V (x, t) ≥ 0 and U + (x, t) ≤ K ≤ 1 for x ∈ R and t ≥ 0, a direct calculation implies that m ∂t Vi (x, t) = 1 − Ui+ (x, t) σk βi,k Uk+ (x − y, t)pi,k (y)dy − µi Vi (x, t) R
k=1
m σk βi,k φk (x + ct − y)pi,k (y)dy − 1 − φi (x + ct) R
k=1
= 1 − Ui+ (x, t)
m
− Vi (x, t)
m k=1
≤
m
Vk (x − y, t)pi,k (y)dy − µi Vi (x, t)
σk βi,k R
k=1
φk (x + ct − y)pi,k (y)dy
σk βi,k R
Vk (x − y, t)pi,k (y)dy − µi Vi (x, t),
σk βi,k
i = 1, . . . , m,
R
k=1
for x ∈ R, t > 0. Let νϵ = ν(λϵ ) be the eigenfunction of A(λϵ ) corresponding to the principal eigenvalue Mϵ := M (λϵ ). Since V (x, 0)W ϵ (x) is uniformly bounded on R, we can choose a sufficiently large L1 > 0 such that L1 νϵ eλϵ (x−X0 ) ≥ V (x, 0),
∀x ∈ R.
Now, we define V¯ (x, t) = L1 νϵ eλϵ (x−X0 )+Mϵ t ,
∀x ∈ R, t ≥ 0.
Since A(λϵ )ν(λϵ ) = M (µϵ )ν(λϵ ), one can easily verify that the function V¯ (x, t) satisfies m ∂t V¯i (x, t) = σk βi,k V¯k (x − y, t)pi,k (y)dy − µi V¯i (x, t), ∀x ∈ R, t > 0, i = 1, . . . , m. k=1
R
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275
Using Lemma 4.1 with L0 = −∞, Di = 1 and di = µi (i = 1, . . . , m), we obtain V (x, t) ≤ V¯ (x, t),
∀x ∈ R, t ≥ 0.
Thus, for any x ∈ R and t ≥ 0 with x + ct ≤ X0 , we have V (x, t) ≤ L1 νϵ eλϵ (x−X0 )+Mϵ t ≤ L1 νϵ eλϵ (x+ct−X0 )t e−(cλϵ −Mϵ )t ≤ L1 νϵ e−(cλϵ −Mϵ )t . Case 2. x + ct > X0 . Note that Φ(x + ct) ≤ U + (x, t) ≤ K, ∀x ∈ R, t ≥ 0 and Kj − ϵ1 ≤ φj (z), φj (z − y)pi,j (y)dy ≤ Kj for any z ≥ X0 , j = 1, . . . , m. R
Thus, for any (x, t) in the domain ΩX0 := {(x, t) : x + ct > X0 , t > 0}, we have m + σk βi,k Vk (x − y, t)pi,k (y)dy − µi Vi (x, t) ∂t Vi (x, t) = 1 − Ui (x, t) R
k=1
− Vi (x, t)
m
≤ 1 − Ki + ϵ1
φk (x + ct − y)pi,k (y)dy
σk βi,k R
k=1 m
Vk (x − y, t)pi,k (y)dy
σk βi,k R
k=1
m − µi + σk βi,k (Kk − ϵ1 ) Vi (x, t),
i = 1, . . . , m.
(4.7)
k=1
¯ Take ϵ0 = min{cλϵ −Mϵ , −λ−¯ ¯ ϵ}. Choose L2 > 0 such that L2 w ≥ max{L1 νϵ , K}. Recall that ϵ¯ ∈ (0, −λ). Now, we define Vˆ (x, t) := L2 we−ϵ0 t ,
∀x ∈ R, t ≥ 0.
By the result obtained from Case 1 and the fact ϵ0 ≤ cλϵ − Mϵ , we see that V (x, t) ≤ Vˆ (x, t) for all (x, t) ∈ R × R+ with x + ct ≤ X0 and (x, t) ∈ ∂ΩX0 , where ∂ΩX0 := {(x, t) : x + ct > X0 , t = 0} ∪ {(x, t) : x + ct = X0 , t ≥ 0}. ¯ is a simple eigenvalue of Df (K) with a eigenvector w = (w1 , . . . , wm ) ≫ 0, we have Since λ ¯ i = (1 − Ki ) λw
m
σk βi,k wk − µi wi − wi
k=1
m
σk βi,k Kk ,
i = 1, . . . , m.
(4.8)
k=1
¯ − ϵ¯ ≤ −λ, ¯ it follows from (4.8) that In view of ϵ0 ≤ −λ ∂t Vˆi (x, t) = −ϵ0 Vˆi (x, t) ¯ + ϵ¯)Vˆi (x, t) ≥ (λ = [−µi + ϵ¯]Vˆi (x, t) + (1 − Ki ) = [−µi + ϵ¯]Vˆi (x, t) + (1 − Ki )
m k=1 m k=1
σk βi,k Vˆk (x, t) − Vˆi (x, t)
m
σk βi,k Kk
k=1
σk βi,k
Vˆk (x − y, t)pi,k (y)dy − Vˆi (x, t)
R
k=1
From (4.4), we get ϵ¯Vˆi (x, t) ≥ ϵ1
m k=1
σk βi,k R
Vˆk (x − y, t)pi,k (y)dy + ϵ1 Vˆi (x, t)
m
m k=1
σk βi,k .
σk βi,k Kk .
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Thus, the function Vˆ (x, t) satisfies m ∂t Vˆi (x, t) ≥ 1 − Ki + ϵ1 σk βi,k k=1
− µi +
m
Vˆk (x − y, t)pi,k (y)dy
R
σk βi,k (Kk − ϵ1 ) Vˆi (x, t)
(4.9)
k=1
for x ∈ R, t > 0, i = 1, . . . , m. Then, by (4.7)–(4.9) and using Lemma 4.1 with L0 = X0 , Di = 1 − Ki + ϵ1 m and di = µi + k=1 σk βi,k (Kk − ϵ1 ) (i = 1, . . . , m), we obtain V (x, t) ≤ Vˆ (x, t) = L2 we−ϵ0 t
for all x ∈ R and t ≥ 0 with x + ct > X0 .
Combining the above two cases, we conclude that 0 ≤ ∥V (x, t)∥ = ∥U + (x, t) − Φ(x + ct)∥ ≤ M e−ϵ0 t
for all x ∈ R and t ≥ 0,
where M := max{L1 maxj=1,...,m (νϵ )j , L2 maxj=1,...,m wj }, that is U + (x, t) converges to Φ(x + ct) exponentially in time. This completes the proof. Acknowledgments The first author is partially supported by the NSF of China (11671315) and the Fundamental Research Funds for the Central Universities (JB160714, JBG160706). References [1] L. Rass, J. Radcliffe, Spatial Deterministic Epidemics, in: Math. Surveys Monogr., vol. 102, Amer. Math. Soc., Providence, RI, 2003. [2] P. Weng, X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations 229 (2006) 270–296. [3] M. Aguerrea, C. Gomez, S. Trofimchuk, On uniqueness of semi-wavefronts: Diekmann–Kapper theory of a nonlinear convolution equation re-visited, Math. Ann. 354 (2012) 73–109. [4] X. Chen, J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann. 326 (2003) 123–146. [5] O. Diekmann, H.G. Kapper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal. 2 (1978) 721–737. [6] J. Carr, A. Chmaj, Uniqueness of traveling waves for nonlocal monostable equations, Proc. Amer. Math. Soc. 132 (2004) 2433–2439. [7] W. Huang, M. Han, M. Puckett, Uniqueness of monotone mono-stable waves for reaction–diffusion equations with time delay, Math. Model. Nat. Phenom. 4 (2009) 48–67. [8] J. Fang, J. Wei, X.-Q. Zhao, Uniqueness of traveling waves for nonlocal lattice equations, Proc. Amer. Math. Soc. 139 (2011) 1361–1373. [9] Z.-X. Yu, Uniqueness of critical traveling waves for nonlocal lattice equations with delays, Proc. Amer. Math. Soc. 140 (2012) 3853–3859. [10] X. Chen, J.-S. Guo, Existence and asymptotic stability of travelling waves of discrete quasilinear monostable equations, J. Differential Equations 184 (2002) 549–569. [11] S. Ma, X.-Q. Zhao, Global asymptotic stability of minimal fronts in monostable lattice equations, Discrete Contin. Dyn. Syst. 21 (2008) 259–275. [12] S. Ma, X. Zou, Existence, uniqueness and stability of traveling waves in a discrete reaction–diffusion monostable equation with delay, J. Differential Equations 217 (2005) 54–87. [13] M. Mei, C.-K. Lin, C.-T. Lin, J.W.H. So, Traveling wavefronts for time-delayed reaction–diffusion equation: II nonlocal nonlinearity, J. Differential Equations 247 (2009) 511–529. [14] M. Mei, C. Ou, X.-Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction–diffusion equations, SIAM J. Math. Anal. 42 (2010) 2762–2790. [15] Y. Wu, X. Xing, Stability of traveling waves with critical speeds for p-degree Fisher-type equations, Discrete Contin. Dyn. Syst. 20 (2008) 1123–1139. [16] Z.-C. Wang, W.-T. Li, S. Ruan, Travelling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations 20 (2008) 563–607.
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[17] Z.-C. Wang, W.-T. Li, S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations 238 (2007) 153–200. [18] S.-L. Wu, W.-T. Li, S.-Y. Liu, Exponential stability of traveling fronts in monostable reaction–advection–diffusion equations with non-local delay, Discrete Contin. Dyn. Syst. Ser. B 17 (2012) 347–366. [19] S.-L. Wu, H.-Q. Zhao, S.-Y. Liu, Asymptotic stability of traveling waves for delayed reaction–diffusion equations with crossing-monostability, Z. Angew. Math. Phys. 62 (2011) 377–397. [20] Z. Ouyang, C. Ou, Global stability and convergence rate of traveling waves for a nonlocal model in periodic media, Discrete Contin. Dyn. Syst. Ser. B 17 (2012) 993–1007. [21] X.-S. Wang, X.-Q. Zhao, Pulsating waves of a partially degenerate reaction–diffusion system in a periodic habitat, J. Differential Equations 259 (2015) 7238–7259. [22] H.L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, in: Math. Surveys Monogr., vol. 41, Amer. Math. Soc., Providence, RI, 1995.
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa
Uniqueness and exponential stability of traveling wave fronts for a multi-type SIS nonlocal epidemic model Shi-Liang Wu ∗ , Guangsheng Chen School of Mathematics and Statistics, Xidian University, Xi’an, Shaanxi 710071, People’s Republic of China
article
abstract
info
Article history: Received 9 December 2016 Accepted 5 February 2017
This paper is concerned with the traveling wave fronts of a multi-type SIS nonlocal epidemic model. From Weng and Zhao (2006), we know that there exists a critical wave speed c∗ > 0 such that a traveling wave front exists if and only if its wave speed is above c∗ . In this paper, we first prove the uniqueness of certain traveling wave fronts with non-critical wave speed. Then, we show that all non-critical traveling wave fronts are asymptotically exponentially stable. The exponential convergent rate is also obtained. © 2017 Elsevier Ltd. All rights reserved.
Keywords: SIS nonlocal epidemic model Traveling wave front Uniqueness Stability
1. Introduction To model the spatial spread of a deterministic epidemic in multi-types of population, Rass and Radcliffe [1] proposed and studied the following nonlocal epidemic model: m ∂ui (x, t) = 1 − ui (x, t) σk βi,k uk (x − y, t)pi,k (y)dy − µi ui (x, t), 1 ≤ i ≤ m, x ∈ R, (1.1) ∂t R k=1
where ui (x, t) represents the proportion of individuals for the ith population σi at position x who are infectious at time t, µi ≥ 0 denotes the combined death/emigration/recovery rate for infectious individuals, βi,k ≥ 0 is the infection rate of a type i susceptible by a type k infectious individual, and pi,k is the corresponding contact distribution. For more details about biological meaning, we refer to [1,2]. In [1, Chapter 8], Rass and Radcliffe made a complete analysis on the global dynamics of the spatially homogeneous m-dimensional system of (1.1). Weng and Zhao [2] further considered the spreading speed and traveling wave fronts of (1.1) and gave an affirmative answer to an open problem presented by Rass and Radcliffe [1]. More precisely, under some reasonable assumptions (see (C1 )–(C3 ) in Section 2), they established ∗ Corresponding author. E-mail address: [email protected] (S.-L. Wu).
http://dx.doi.org/10.1016/j.nonrwa.2017.02.001 1468-1218/© 2017 Elsevier Ltd. All rights reserved.
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the existence of the spreading speed c∗ and showed that it coincides with the critical wave speed for traveling wave fronts. However, to the best of our knowledge, there has been no results on the uniqueness and stability of the traveling wave fronts for such nonlocal epidemic model. This constitutes the purpose of this paper. In the past decades, there have been many significant results on the uniqueness of traveling wave solutions for various evolution equations [3–9]. For example, Chen and Guo [4], Diekmann and Kapper [5], Carr and Chmaj [6] and Huang et al. [7] established the uniqueness theorem of traveling wave solutions for lattice differential equations, integral equations, nonlocal dispersal equations, and delayed reaction–diffusion systems, respectively. Based on the method in Diekmann and Kapper [5] with some nontrivial modifications, we shall prove the uniqueness (up to translation) of certain traveling wave fronts of the nonlocal system (1.1) with non-critical wave speed (see Theorem 3.1). Although the stability of traveling wave solutions for reaction–diffusion equations has been studied intensively in the existing literature (see e.g. [10–21] and references therein), not much is known about traveling wave solutions of monostable nonlocal systems. The main difficulties come from the lack of compactness for solution maps and interaction between different components in a higher dimensional system. Recently, Ouyang and Ou [20] and Wang and Zhao [21] established the asymptotic stability of traveling waves for a nonlocal model in periodic media and a partially degenerate reaction–diffusion system in a periodic habitat, respectively. In this paper, by using the analysis of the principal eigenvalue and establishing a comparison theorem for related linear systems (see Lemma 4.1), we shall show that if the initial function is within a bounded distance from a certain traveling wave front with respect to a weighted maximum norm, then the solution of (1.1) will converge to the traveling wave front exponentially in time. The exponential convergent rate is also obtained (see Theorem 4.2). Our work provides some insights on how to establish the asymptotic stability of the traveling wave fronts for the nonlocal systems. The rest of the paper is organized as follows. In Section 2, we give some preliminaries. Sections 3 and 4 are devoted to the uniqueness and asymptotic stability of the non-critical traveling wave fronts, respectively. 2. Preliminaries In this section, we recall some known results on the existence of the critical wave speed. Throughout this paper, we always use the usual notations for the standard ordering in Rm and denote by ∥ · ∥ the Euclidean norm in Rm . Similar to [2], we make the following assumptions: (C1 ) pi,k (y) ≥ 0, pi,k (y) = pi,k (−y), ∀y ∈ R, R pi,k (y)dy = 1, and for any λ > 0, R eλy pi,k (y)dy < ∞, 1 ≤ i, k ≤ m. (C2 ) The matrix Λ := (σk βi,k )m×m is irreducible in the sense that for every i ̸= k, there exists a distinct sequence i1 , i2 , . . . , ir with i1 = i, ir = k such that σis+1 βis ,is+1 ̸= 0, 1 ≤ s ≤ r − 1. (C3 ) s(Df (0)) > 0, where f = (f1 , . . . , fm ) with fi (u1 , . . . , um ) = (1 − ui )
m
σk βi,k uk − µi ui ,
i = 1, . . . , m.
k=1
From [2, Lemma 2.1], system (1.1) has exactly two spatially homogeneous equilibria 0 := (0, . . . , 0) ∈ Rm and K := (K1 , . . . , Km ) ∈ Rm with 0 ≪ K ≤ 1 := (1, . . . , 1) ∈ Rm . For any λ ∈ R, define a matrix A(λ) = (Ai,k (λ))m×m , where e−λy pi,k (y)dy − µi , i = k, σk βi,k R Ai,k (λ) = (2.1) σk βi,k eλy pi,k (y)dy, i ̸= k. R
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Since A(λ) is cooperative and irreducible, M (λ) := max{ℜα : det(αI − A(λ)) = 0} is a simple eigenvalue of A(λ) with a strongly positive eigenvector ν(λ) = (ν1 (λ), . . . , νm (λ)) (see, e.g. [22, Corollary 4.3.2]). Here and in what follows, ℜα denotes the real part of α. The following result follows from [2, Lemma 3.3 and Theorem 3.2]. Proposition 2.1. Assume that (C1 )–(C3 ) hold. There exists c∗ > 0 and λ∗ > 0 such that c∗ =
M (λ) M (λ∗ ) = inf , λ>0 λ∗ λ
and for any c > c∗ , there exists a unique λ1 (c) ∈ (0, λ∗ ) such that M (λ1 ) = cλ1 , and M (λ) < cλ for any λ ∈ (λ1 , λ∗ ). Moreover, from [2, Theorem 4.2] and the proof of [2, Theorem 4.1], we have the following results on the existence and non-existence of traveling wave fronts. Proposition 2.2. Assume that (C1 )–(C3 ) hold. (i) For any c ∈ (0, c∗ ), system (1.1) has no traveling wave solution connecting 0 and K. (ii) For every c ≥ c∗ , system (1.1) admits a traveling wave front Φ(x + ct) = (φ1 (x + ct), . . . , φm (x + ct)) satisfying Φ(−∞) = 0 and Φ(+∞) = K; and for any c > c∗ , lim Φ(ξ)e−λ1 (c)ξ = ν(λ1 (c)).
ξ→−∞
It is clear that the wave system of the traveling wave solution Φ(ξ) = (φ1 (ξ), . . . , φm (ξ)), ξ = x + ct, of (1.1) has the following form: m σk βi,k cφi (ξ) = 1 − φi (ξ) φk (ξ − y)pi,k (y)dy − µi φi (ξ), 1 ≤ i ≤ m, ξ ∈ R. (2.2) k=1
R
3. Uniqueness of traveling wave fronts In this section, we establish the uniqueness of certain non-critical traveling wave fronts of system (1.1). More precisely, we have the following result. Theorem 3.1. Assume (C1 )–(C3 ) and the following condition: (C4 ) βj,j > 0 and there exists an interval Ij ⊆ R such that pj,j (x) > 0, ∀x ∈ Ij , j = 1, . . . , m. Let Ψ (ξ) = ψ1 (ξ), . . . , ψm (ξ) and Φ(ξ) = φ1 (ξ), . . . , φm (ξ) , ξ = x + ct, be two traveling wave fronts of system (1.1) connecting 0 and K with speed c > c∗ , and satisfy lim Φ(ξ)e−λ1 (c)ξ = h1 ν(λ1 (c)) and
ξ→−∞
lim Ψ (ξ)e−λ1 (c)ξ = h2 ν(λ1 (c)),
ξ→−∞
where h1 , h2 are positive constants. Then, there exists ξ0 ∈ R such that Ψ (ξ) = Φ(ξ + ξ0 ).
(3.1)
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Proof. By a translation if necessary, we may assume lim Φ(ξ)e−λ1 ξ = lim Ψ (ξ)e−λ1 ξ = ν(λ1 ),
ξ→−∞
ξ→−∞
(3.2)
where λ1 := λ1 (c). Set q(ξ) := maxj∈{1,...,m} qj (ξ), ∀ξ ∈ R, where φj (ξ) − ψj (ξ) , eλ1 ξ νj (λ1 )
qj (ξ) :=
∀j = 1, . . . , m, ξ ∈ R.
From (3.2), one can see that qj (±∞) = q(±∞) = 0, ∀j = 1, . . . , m. We first prove that φj (ξ) ≤ ψj (ξ), ∀ξ ∈ R, j = 1, . . . , m. It suffices to show that q(ξ) ≤ 0, ∀ξ ∈ R. Suppose for the contrary that there exists ξ1 ∈ R such that q(ξ1 ) = maxξ∈R q(ξ) > 0. By the definition of q(ξ), there exists j0 ∈ {1, . . . , N } such that q(ξ1 ) = qj0 (ξ1 ). Then, we have qj0 (ξ) ≤ q(ξ) ≤ q(ξ1 ) = qj0 (ξ1 ),
∀ξ ∈ R.
which yields that qj0 (ξ) attains its maximum at ξ1 . According to qj0 (ξ1 ) > 0 and qj0 (±∞) = 0, we can redefine ξ1 such that qj0 (ξ1 ) = maxξ∈R qj0 (ξ) and qj0 (ξ) < qj0 (ξ1 ) for any ξ < ξ1 . Noting that φj0 (ξ1 ) − ψj0 (ξ1 ) = qj0 (ξ1 )eλ1 ξ1 νj0 (λ1 ) > 0 and 0 = qj′ 0 (ξ) =
1 [φ′ (ξ1 ) − ψj′ 0 (ξ1 )]e−λ1 ξ1 − λ1 qj0 (ξ1 ), νj0 (λ1 ) j0
direct computations show that cλ1 νj0 (λ1 )qj0 (ξ1 )eλ1 ξ1 = c[φ′j0 (ξ1 ) − ψj′ 0 (ξ1 )] m σk βj0 ,k pj0 ,k (y)[φk (ξ1 − y) − ψk (ξ1 − y)]dy = 1 − φj0 (ξ1 ) R
k=1
− µj0 [φj0 (ξ1 ) − ψj0 (ξ1 )] − [φj0 (ξ1 ) − ψj0 (ξ1 )]
m k=1
≤ =
m k=1 m
pj0 ,k (y)ψk (ξ1 − y)dy
σk βj0 ,k R
pj0 ,k (y) max{0, φk (ξ1 − y) − ψk (ξ1 − y)}dy − µj0 [φj0 (ξ1 ) − ψj0 (ξ1 )]
σk βj0 ,k R
σk βj0 ,k
pj0 ,k (y)νk (λ1 )eλ1 (ξ1 −y) max{0, qk (ξ1 − y)}dy − µj0 νj0 (λ1 )qj0 (ξ1 )eλ1 ξ1 ,
R
k=1
which implies that cλ1 νj0 (λ1 )qj0 (ξ1 ) ≤
m
σk βj0 ,k νk (λ1 )
pj0 ,k (y)e−λ1 y max{0, qk (ξ1 − y)}dy − µj0 νj0 (λ1 )qj0 (ξ1 ). (3.3)
R
k=1
On the other hand, it follows from A(λ1 )ν(λ1 ) = M (λ1 )ν(λ1 ) = cλ1 ν(λ1 ) that m cλ1 νj0 (λ1 )qj0 (ξ1 ) = σk βj0 ,k νk (λ1 ) pj0 ,k (y)e−λ1 y dyqj0 (ξ1 ) − µj0 νj0 (λ1 )qj0 (ξ1 ).
(3.4)
R
k=1
By (3.3) and (3.4), we see that m
σk βj0 ,k νk (λ1 )
k=1
pj0 ,k (y)e−λ1 y [qj0 (ξ1 ) − max{0, qk (ξ1 − y)}]dy ≤ 0.
R
Note that qj0 (ξ1 ) > 0 and qk (ξ1 − y) ≤ q(ξ1 − y) ≤ q(ξ1 ) = qj0 (ξ1 ),
∀k = 1, . . . , m, y ∈ R.
(3.5)
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It follows from (3.5) that
pj0 ,k (y)e−λ1 y [qj0 (ξ1 ) − max{0, qk (ξ1 − y)}]dy = 0,
σk βj0 ,k νk (λ1 )
∀k = 1, . . . , m.
R
In particular, σj0 βj0 ,j0 νj0 (λ1 )
pj0 ,j0 (y)e−λ1 y [qj0 (ξ1 ) − max{0, qj0 (ξ1 − y)}]dy = 0.
R
Since Λ = (σk βi,k )m×m is irreducible, we see that σj > 0, ∀j = 1, . . . , m. This together with the condition βj,j > 0, ∀j = 1, . . . , m, imply that pj0 ,j0 (y)e−λ1 y [qj0 (ξ1 ) − max{0, qj0 (ξ1 − y)}]dy = 0. R
By the assumptions (C1 ) and (C4 ), we deduce that there exists y0 > 0 such that qj0 (ξ1 ) = max{0, qj0 (ξ1 − y0 )}. It then follows from qj0 (ξ1 ) > 0 that qj0 (ξ1 ) = qj0 (ξ1 − y0 ), which contradicts to the fact qj0 (ξ) < qj0 (ξ1 ) for any ξ < ξ1 . Thus, we conclude that φj (ξ) ≤ ψj (ξ), ∀ξ ∈ R, j = 1, . . . , m. Exchanging the roles of φj (ξ) and ψj (ξ), we also obtain ψj (ξ) ≤ φj (ξ), ∀ξ ∈ R, j = 1, . . . , m. Thus, ψj (ξ) = φj (ξ), ∀ξ ∈ R, j = 1, . . . , m. This completes the proof. Remark 3.2. It should be pointed out that if pj,j (x) ∈ C(R), then it follows the condition that there exists an interval Ij ⊆ R such that pj,j (x) > 0, ∀x ∈ Ij , j = 1, . . . , m.
R
pj,j (y)dy = 1
4. Stability of traveling wave fronts This section is devoted to the asymptotic stability of the non-critical traveling wave fronts of (1.1). Let Φ(x + ct) = φ1 (x + ct), . . . , φm (x + ct) be a traveling wave front of system (1.1) with c > c∗ connecting 0 and K. Given L0 ∈ R ∪ {−∞}. Denote ΩL0 := {(x, t) : x + ct > L0 , t > 0} and ∂ΩL0 := {(x, t) : x + ct > L0 , t = 0} ∪ {(x, t) : x + ct = L0 , t ≥ 0}. The following comparison theorem for related linear system of (1.1) plays a crucial role in proving the stability of the traveling wave fronts. Lemma 4.1. Let Di , di > 0 (i = 1, . . . , m) be any given constants. Assume that W ± (x, t) = W1± (x, t), . . . , ± Wm (x, t) : R × R+ → Rm are two functions which satisfy (i) (ii) (iii) (iv)
W + (x, t) ≥ 0 and W − (x, t) ≤ K for all (x, t) ∈ R × R+ . W + (x, t) ≥ W − (x, t) for all (x, t) ∈ ∂ΩL0 . W + (x, t) ≥ W − (x, t) for all (x, t) ∈ R × R+ with x + ct ≤ L0 . For all (x, t) ∈ ΩL0 , i = 1, . . . , m, there hold m + ∂t Wi (x, t) ≥ Di σk βi,k Wk+ (x − y, t)pi,k (y)dy − di Wi+ (x, t), ∂t Wi− (x, t) ≤ Di
k=1 m k=1
(4.1)
R
σk βi,k R
Then, W + (x, t) ≥ W − (x, t) for all (x, t) ∈ R × R+ .
Wk− (x − y, t)pi,k (y)dy − di Wi− (x, t).
(4.2)
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Proof. Let W (x, t) = (W1 (x, t), . . . , Wm (x, t)) := W + (x, t) − W − (x, t), ∀x ∈ R, t ≥ 0. Then, W (x, t) is bounded from below by −K, W (x, 0) ≥ 0 for x ∈ R, W (x, t) ≥ 0 for all (x, t) ∈ R × R+ with x + ct ≤ L0 , and m ∂t Wi (x, t) ≥ Di σk βi,k Wk (x − y, t)pi,k (y)dy − di Wi (x, t), i = 1, . . . , m, (4.3) R
k=1
for (x, t) ∈ ΩL0 . Choosing K > 3 maxj=1,...,m {Dj σj βj,j }. Suppose that the assertion is false. Then, there exist i0 ∈ {1, . . . , m}, ϖ > 0 and t0 > 0 such that Wi0 (x, t) > −ϖe2Kt and Wj (x, t) ≥ 0
for x ∈ R, t ∈ [0, t0 ),
inf Wi0 (x, t0 ) = −ϖe2Kt0
x∈R
for x ∈ R, t ∈ [0, t0 ], j ̸= i0 .
2Kt0 Thus, there exists a bounded set S ⊂ R with positive Lebesgue measure such that Wi0 (x, t0 ) ≤ − 15 16 ϖe for x ∈ S.
Let ζ(x) be a smooth function satisfying min ζ(x) = 1, x∈R ′
ζ(z) = 1
for z ∈ S,
sup ζ(x) = ζ(±∞) = 3, x∈R
|ζ (·)| ≤ 1
′′
and |ζ (·)| ≤ 1.
For any α ∈ [0, 1], we define the function z(x, t; α) := −ϖ
3 + αζ(x) e2Kt , 4
x ∈ R, t ∈ [0, t0 ].
∂ z(x, t; α) < 0, ∀α ∈ [0, 1], x ∈ R, t ∈ [0, t0 ], It is clear that ∂α 3 1 1 z x, t; = −ϖ + ζ(x) e2Kt ≤ −ϖe2Kt ≤ Wi0 (x, t) for x ∈ R, t ∈ [0, t0 ]; 4 4 4 1 7 3 1 z x, t0 ; + ζ(x) e2Kt0 = − ϖe2Kt0 > Wi0 (x, t0 ) for x ∈ S. = −ϖ 8 4 8 8
Thus, 1 1 , Wi0 (x, t) ≥ z(x, t; α) α∗ := inf α ∈ 8 4
for x ∈ R, t ∈ [0, t0 ]
is well defined. Obviously, Wi0 (x, t) ≥ z(x, t; α∗ ) for x ∈ R, t ∈ [0, t0 ]. Since Wi0 (x, 0) ≥ 0 > z(x, 0; α∗ ) for x ∈ R, 9 z(±∞, t; α∗ ) ≤ − ϖe2Kt < Wi0 (x, t) for x ∈ R, t ∈ [0, t0 ], and 8 Wi0 (x, t) ≥ 0 > z(x, t; α∗ ) for (x, t) ∈ R × [0, t0 ] with x + ct ≤ L0 , we deduce that the function w(x, t) := Wi0 (x, t) − z(x, t; α∗ ) attains its infimum 0 at (x1 , t1 ) ∈ R × (0, t0 ] with x + ct > L0 . Therefore, w(x1 , t1 ) = 0 and wt (x1 , t1 ) ≤ 0. Moreover, w(x1 − y, t1 ) ≥ w(x1 , t1 ) for all y ∈ R, which implies that Wi0 (x1 − y, t1 ) ≥ z(x1 − y, t1 ; α∗ ) + Wi0 (x1 , t1 ) − z(x1 , t1 ; α∗ ) ≥ z(x1 − y, t1 ; α∗ ) + Wi0 (x1 , t1 ),
∀y ∈ R.
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273
Thus, by (4.3) and the fact that Wj (x, t) ≥ 0 for x ∈ R, t ∈ [0, t0 ], j ̸= i0 , we have 0 ≥ wt (x1 , t1 ) = (Wi0 )t (x, t) − zt (x, t; α∗ ) m ≥ Di0 σk βi0 ,k Wk (x1 − y, t1 )pi0 ,k (y)dy − di0 Wi0 (x1 , t1 ) − zt (x1 , t1 ; α∗ ) R
k=1
≥ Di0 σi0 βi0 ,i0
[z(x1 − y, t1 ; α∗ ) + Wi0 (x1 , t1 )]pi0 ,i0 (y)dy − di0 z(x1 , t1 ; α∗ ) + 2Kϖ R 3 7 3 ≥ − Di0 σi0 βi0 ,i0 ϖe2Kt1 − Di0 σi0 βi0 ,i0 ϖ + α∗ ζ(x1 ) e2Kt1 + Kϖe2Kt1 2 4 4 ≥ K − 3Di0 σi0 βi0 ,i0 ϖe2Kt1 > 0.
3 + α∗ ζ(x1 ) e2Kt1 4
This contradiction implies that W + (x, t) ≥ W − (x, t) for all (x, t) ∈ R × R+ . This completes the proof.
Recall that f (u) = (f1 (u), . . . , fm (u)) with fi (u) = (1 − ui )
m
σk βi,k uk − µi ui ,
i = 1, . . . , m.
k=1
To establish the stability of traveling wave fronts, we need the following additional assumption: (C4 ) Df (K) is irreducible and s(Df (K)) < 0. ¯ := s(Df (K)) < 0 is a simple eigenvalue of Df (K) with an Since Df (K) is cooperative and irreducible, λ ¯ be a fixed number. We eigenvector w = (w1 , . . . , wm ) ≫ 0 (see, e.g. [22, Corollary 4.3.2]). Let ϵ¯ ∈ (0, −λ) can choose ϵ1 ∈ 0, min1≤i≤m {Ki } such that ϵ¯wi ≥ ϵ1
m
σk βi,k wk + ϵ1 wi
k=1
m
σk βi,k ,
∀i = 1, . . . , m.
(4.4)
k=1
Moreover, using limξ→+∞ φj (ξ) = Kj , it is easy to show that lim φj (ξ − y)pi,j (y)dy = Kj , ∀i, j = 1, . . . , m. ξ→+∞
R
Then, we can choose X0 > 0 such that Kj − ϵ1 ≤ φj (z), φj (z − y)pi,j (y)dy ≤ Kj
for any z ≥ X0 , i, j = 1, . . . , m.
R
The stability result of the non-critical traveling wave fronts is stated as follows. Theorem 4.2. Assume that (C1 )–(C4 ) hold. Let Φ(x + ct) = φ1 (x + ct), . . . , φm (x + ct) be a traveling wave front of system (1.1) with c > c∗ connecting 0 and K. For a sufficiently small ϵ > 0, we denote λϵ = λ1 (c)+ϵ and define the weight function W ϵ : R → R+ as follows: e−λϵ (x−X0 ) , x ≤ X0 , ϵ W (x) := (4.5) 1, x > X0 . Then, there exists a number ϵ0 > 0 such that for any given initial value u0 (x) with 0 ≤ u0 (x) ≤ K
and
[u0 (·) − Φ(·)]W ϵ (·) ∈ L∞ (R, Rm ),
the unique solution u(x, t; u0 ) of (1.1) satisfies 0 ≤ u(x, t; u0 ) ≤ K,
∀x ∈ R, t ≥ 0,
S.-L. Wu, G. Chen / Nonlinear Analysis: Real World Applications 36 (2017) 267–277
274
and for some C > 0, sup ∥u(x, t; u0 ) − Φ(x + ct)∥ ≤ Ce−ϵ0 t ,
∀ t ≥ 0.
(4.6)
x∈R
Proof. Define two functions U − (x) := min{u0 (x), Φ(x)},
U + (x) := max{u0 (x), Φ(x)},
∀x ∈ R.
Let U ± (x, t) be the solutions of (1.1) with the initial values U + (x) and U − (x), respectively. Then, a simple application of the comparison principle (see e.g. [2, Theorem 2.2]) implies that 0 ≤ U − (x, t) ≤ u(x, t; u0 ), Φ(x + ct) ≤ U + (x, t) ≤ K,
∀x ∈ R, t ≥ 0,
which yields that ∥u(x, t; u0 ) − Φ(x + ct)∥ ≤ max{∥U + (x, t) − Φ(x + ct)∥, ∥U − (x, t) − Φ(x + ct)∥}. Thus, to prove the assertion of this theorem, it is sufficient to show that U ± (x, t) converges to Φ(x + ct) exponentially in time. By symmetry, we only prove that U + (x, t) converges to Φ(x + ct). Set V (x, t) := U + (x, t) − Φ(x + ct). It is clear that V (x, t) ≥ 0 for x ∈ R, t ≥ 0, and 0 ≤ V (x, 0) ≤ |u0 (x) − Φ(x)|,
∀x ∈ R,
ϵ
which implies that V (x, 0)W (x) is uniformly bounded on R. In the sequel, we consider two cases x+ct ≤ X0 and x + ct > X0 , respectively. Case 1. x+ct ≤ X0 . Noting that V (x, t) ≥ 0 and U + (x, t) ≤ K ≤ 1 for x ∈ R and t ≥ 0, a direct calculation implies that m ∂t Vi (x, t) = 1 − Ui+ (x, t) σk βi,k Uk+ (x − y, t)pi,k (y)dy − µi Vi (x, t) R
k=1
m σk βi,k φk (x + ct − y)pi,k (y)dy − 1 − φi (x + ct) R
k=1
= 1 − Ui+ (x, t)
m
− Vi (x, t)
m k=1
≤
m
Vk (x − y, t)pi,k (y)dy − µi Vi (x, t)
σk βi,k R
k=1
φk (x + ct − y)pi,k (y)dy
σk βi,k R
Vk (x − y, t)pi,k (y)dy − µi Vi (x, t),
σk βi,k
i = 1, . . . , m,
R
k=1
for x ∈ R, t > 0. Let νϵ = ν(λϵ ) be the eigenfunction of A(λϵ ) corresponding to the principal eigenvalue Mϵ := M (λϵ ). Since V (x, 0)W ϵ (x) is uniformly bounded on R, we can choose a sufficiently large L1 > 0 such that L1 νϵ eλϵ (x−X0 ) ≥ V (x, 0),
∀x ∈ R.
Now, we define V¯ (x, t) = L1 νϵ eλϵ (x−X0 )+Mϵ t ,
∀x ∈ R, t ≥ 0.
Since A(λϵ )ν(λϵ ) = M (µϵ )ν(λϵ ), one can easily verify that the function V¯ (x, t) satisfies m ∂t V¯i (x, t) = σk βi,k V¯k (x − y, t)pi,k (y)dy − µi V¯i (x, t), ∀x ∈ R, t > 0, i = 1, . . . , m. k=1
R
S.-L. Wu, G. Chen / Nonlinear Analysis: Real World Applications 36 (2017) 267–277
275
Using Lemma 4.1 with L0 = −∞, Di = 1 and di = µi (i = 1, . . . , m), we obtain V (x, t) ≤ V¯ (x, t),
∀x ∈ R, t ≥ 0.
Thus, for any x ∈ R and t ≥ 0 with x + ct ≤ X0 , we have V (x, t) ≤ L1 νϵ eλϵ (x−X0 )+Mϵ t ≤ L1 νϵ eλϵ (x+ct−X0 )t e−(cλϵ −Mϵ )t ≤ L1 νϵ e−(cλϵ −Mϵ )t . Case 2. x + ct > X0 . Note that Φ(x + ct) ≤ U + (x, t) ≤ K, ∀x ∈ R, t ≥ 0 and Kj − ϵ1 ≤ φj (z), φj (z − y)pi,j (y)dy ≤ Kj for any z ≥ X0 , j = 1, . . . , m. R
Thus, for any (x, t) in the domain ΩX0 := {(x, t) : x + ct > X0 , t > 0}, we have m + σk βi,k Vk (x − y, t)pi,k (y)dy − µi Vi (x, t) ∂t Vi (x, t) = 1 − Ui (x, t) R
k=1
− Vi (x, t)
m
≤ 1 − Ki + ϵ1
φk (x + ct − y)pi,k (y)dy
σk βi,k R
k=1 m
Vk (x − y, t)pi,k (y)dy
σk βi,k R
k=1
m − µi + σk βi,k (Kk − ϵ1 ) Vi (x, t),
i = 1, . . . , m.
(4.7)
k=1
¯ Take ϵ0 = min{cλϵ −Mϵ , −λ−¯ ¯ ϵ}. Choose L2 > 0 such that L2 w ≥ max{L1 νϵ , K}. Recall that ϵ¯ ∈ (0, −λ). Now, we define Vˆ (x, t) := L2 we−ϵ0 t ,
∀x ∈ R, t ≥ 0.
By the result obtained from Case 1 and the fact ϵ0 ≤ cλϵ − Mϵ , we see that V (x, t) ≤ Vˆ (x, t) for all (x, t) ∈ R × R+ with x + ct ≤ X0 and (x, t) ∈ ∂ΩX0 , where ∂ΩX0 := {(x, t) : x + ct > X0 , t = 0} ∪ {(x, t) : x + ct = X0 , t ≥ 0}. ¯ is a simple eigenvalue of Df (K) with a eigenvector w = (w1 , . . . , wm ) ≫ 0, we have Since λ ¯ i = (1 − Ki ) λw
m
σk βi,k wk − µi wi − wi
k=1
m
σk βi,k Kk ,
i = 1, . . . , m.
(4.8)
k=1
¯ − ϵ¯ ≤ −λ, ¯ it follows from (4.8) that In view of ϵ0 ≤ −λ ∂t Vˆi (x, t) = −ϵ0 Vˆi (x, t) ¯ + ϵ¯)Vˆi (x, t) ≥ (λ = [−µi + ϵ¯]Vˆi (x, t) + (1 − Ki ) = [−µi + ϵ¯]Vˆi (x, t) + (1 − Ki )
m k=1 m k=1
σk βi,k Vˆk (x, t) − Vˆi (x, t)
m
σk βi,k Kk
k=1
σk βi,k
Vˆk (x − y, t)pi,k (y)dy − Vˆi (x, t)
R
k=1
From (4.4), we get ϵ¯Vˆi (x, t) ≥ ϵ1
m k=1
σk βi,k R
Vˆk (x − y, t)pi,k (y)dy + ϵ1 Vˆi (x, t)
m
m k=1
σk βi,k .
σk βi,k Kk .
276
S.-L. Wu, G. Chen / Nonlinear Analysis: Real World Applications 36 (2017) 267–277
Thus, the function Vˆ (x, t) satisfies m ∂t Vˆi (x, t) ≥ 1 − Ki + ϵ1 σk βi,k k=1
− µi +
m
Vˆk (x − y, t)pi,k (y)dy
R
σk βi,k (Kk − ϵ1 ) Vˆi (x, t)
(4.9)
k=1
for x ∈ R, t > 0, i = 1, . . . , m. Then, by (4.7)–(4.9) and using Lemma 4.1 with L0 = X0 , Di = 1 − Ki + ϵ1 m and di = µi + k=1 σk βi,k (Kk − ϵ1 ) (i = 1, . . . , m), we obtain V (x, t) ≤ Vˆ (x, t) = L2 we−ϵ0 t
for all x ∈ R and t ≥ 0 with x + ct > X0 .
Combining the above two cases, we conclude that 0 ≤ ∥V (x, t)∥ = ∥U + (x, t) − Φ(x + ct)∥ ≤ M e−ϵ0 t
for all x ∈ R and t ≥ 0,
where M := max{L1 maxj=1,...,m (νϵ )j , L2 maxj=1,...,m wj }, that is U + (x, t) converges to Φ(x + ct) exponentially in time. This completes the proof. Acknowledgments The first author is partially supported by the NSF of China (11671315) and the Fundamental Research Funds for the Central Universities (JB160714, JBG160706). References [1] L. Rass, J. Radcliffe, Spatial Deterministic Epidemics, in: Math. Surveys Monogr., vol. 102, Amer. Math. Soc., Providence, RI, 2003. [2] P. Weng, X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differential Equations 229 (2006) 270–296. [3] M. Aguerrea, C. Gomez, S. Trofimchuk, On uniqueness of semi-wavefronts: Diekmann–Kapper theory of a nonlinear convolution equation re-visited, Math. Ann. 354 (2012) 73–109. [4] X. Chen, J.-S. Guo, Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann. 326 (2003) 123–146. [5] O. Diekmann, H.G. Kapper, On the bounded solutions of a nonlinear convolution equation, Nonlinear Anal. 2 (1978) 721–737. [6] J. Carr, A. Chmaj, Uniqueness of traveling waves for nonlocal monostable equations, Proc. Amer. Math. Soc. 132 (2004) 2433–2439. [7] W. Huang, M. Han, M. Puckett, Uniqueness of monotone mono-stable waves for reaction–diffusion equations with time delay, Math. Model. Nat. Phenom. 4 (2009) 48–67. [8] J. Fang, J. Wei, X.-Q. Zhao, Uniqueness of traveling waves for nonlocal lattice equations, Proc. Amer. Math. Soc. 139 (2011) 1361–1373. [9] Z.-X. Yu, Uniqueness of critical traveling waves for nonlocal lattice equations with delays, Proc. Amer. Math. Soc. 140 (2012) 3853–3859. [10] X. Chen, J.-S. Guo, Existence and asymptotic stability of travelling waves of discrete quasilinear monostable equations, J. Differential Equations 184 (2002) 549–569. [11] S. Ma, X.-Q. Zhao, Global asymptotic stability of minimal fronts in monostable lattice equations, Discrete Contin. Dyn. Syst. 21 (2008) 259–275. [12] S. Ma, X. Zou, Existence, uniqueness and stability of traveling waves in a discrete reaction–diffusion monostable equation with delay, J. Differential Equations 217 (2005) 54–87. [13] M. Mei, C.-K. Lin, C.-T. Lin, J.W.H. So, Traveling wavefronts for time-delayed reaction–diffusion equation: II nonlocal nonlinearity, J. Differential Equations 247 (2009) 511–529. [14] M. Mei, C. Ou, X.-Q. Zhao, Global stability of monostable traveling waves for nonlocal time-delayed reaction–diffusion equations, SIAM J. Math. Anal. 42 (2010) 2762–2790. [15] Y. Wu, X. Xing, Stability of traveling waves with critical speeds for p-degree Fisher-type equations, Discrete Contin. Dyn. Syst. 20 (2008) 1123–1139. [16] Z.-C. Wang, W.-T. Li, S. Ruan, Travelling fronts in monostable equations with nonlocal delayed effects, J. Dynam. Differential Equations 20 (2008) 563–607.
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