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Global asymptotic stability of bistable traveling fronts in reaction-diffusion systems and their applications to biological models
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Chaos, Solitons and Fractals 40 (2009) 1229–1239 www.elsevier.com/locate/chaos
Global asymptotic stability of bistable traveling fronts in reaction-diffusion systems and their applications to biological models Shi-Liang Wu a
a,b
, Wan-Tong Li
a,*,1
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China b Department of Applied Mathematics, Xidian University, Xi’an, Shanxi 710071, People’s Republic of China Accepted 31 August 2007
Abstract This paper deals with the global asymptotic stability and uniqueness (up to translation) of bistable traveling fronts in a class of reaction-diffusion systems. The known results do not apply in solving these problems because the reaction terms do not satisfy the required monotone condition. To overcome the difficulty, a weak monotone condition is proposed for the reaction terms, which is called interval monotone condition. Under such a weak monotone condition, the existence and comparison theorem of solutions is first established for reaction-diffusion systems on R by appealing to the theory of abstract differential equations. The global asymptotic stability and uniqueness (up to translation) of bistable traveling fronts are then proved by the elementary super- and sub-solution comparison and squeezing methods for nonlinear evolution equations. Finally, these abstract results are applied to a two species competition-diffusion model and a system modeling man–environment–man epidemics. 2007 Elsevier Ltd. All rights reserved.
1. Introduction The theory of traveling wave solutions of parabolic differential equations is one of the fastest developing areas of modern mathematics and has attracted much attention due to its significant nature in biology, chemistry, epidemiology and physics (see, e.g., [1,4,5,8–10,16–18,20,22,23]). Traveling wave solutions are solutions of special type and can be usually characterized as solutions invariant with respect to transition in space. From the physical point of view, traveling waves describe transition processes. These transition processes (from one equilibrium to another) usually ‘‘forget’’ their initial conditions and the properties of medium itself. Among the basic questions in the theory of traveling waves, the global asymptotic stability of traveling wave solutions is an important one. For scalar reaction-diffusion equations, there are many nice results, see e.g., Chen [3] for nonlocal evolution equations, Schaaf [14], Smith and Zhao [15] and Wang et al. [19] for delayed or nonlocal *
1
Corresponding author. E-mail address: [email protected] (W.-T. Li). Supported by NSFC (No. 10571078) and NSF of Gansu Province of China (No. 3ZS061-A25-001).
0960-0779/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.08.075
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reaction-diffusion equations. For systems of reaction-diffusion equations, see [13,17,21] and cited references. In particular, Volpert et al. [17] considered a reaction-diffusion system of the form oui ðx; tÞ o2 ui ðx; tÞ þ F i ðu1 ðx; tÞ; . . . ; un ðx; tÞÞ; ¼ di ot ox2
ð1:1Þ
where x 2 R, t > 0, D = diag(d1, . . . , dn), di > 0, i = 1, . . . , n, and (A1) F ðuÞ ¼ ðF 1 ðuÞ; . . . ; F n ðuÞÞ 2 C 1 ðRn ; Rn Þ satisfies F(w±) = 0; (A2) all the eigenvalues of the matrixes (ojFi(w±)) lie in the left half-plane. In the order interval (w, w+), the only equilibrium w is such that the matrix (ojFi(w)) has at least one eigenvalue in the right half-plane. If the reaction term F satisfies the so-called monotone condition oF i P 0; 1 6 i–j 6 n; ouj
uj 2 R;
ð1:2Þ
then Volpert et al. [17, Theorem 5.6.1] proved that if the initial datum is monotone in x and satisfies decay conditions at ±1, then the solution u(x, t) converges exponentially to a traveling wave front connecting the stable equilibria w and w+. Furthermore, Roquejoffre et al. [13] removed the monotonicity assumption on the initial datum, by operator semigroup precompactness and quasiconvergence, and showed that any solution u(x, t) of the Cauchy problem for system (1.1) which initially has these two stable equilibria w± as spatial limits will converge to a traveling wave front connecting the stable equilibria w and w+. However, the reaction term in a model system arising from a practical problem may not satisfy the monotone condition (1.2). A typical and important example is the two competitive species model (see Gardner [6], Huang [7] and Li et al. [9]): 8 < ovðx;tÞ ¼ d 1 o2 vðx;tÞ þ vMðv; wÞ; ot ox2 ð1:3Þ : owðx;tÞ ¼ d o2 wðx;tÞ þ wN ðv; wÞ; ot
2
ox2
where x 2 R, t > 0, v(x, t) and w(x, t) are the densities of the two species, d1 > 0 and d2 > 0 are the diffusion constants of the species v and w, respectively, and (B1) the partial derivatives Mw(v, w) < 0 and Nv(v, w) < 0 for v P 0 and w P 0; (B2) (1.3) has four spatially uniform equilibria: (0, 0), (v*, 0), (0, w*) and (v0, w0), where 0 < v0 < v* and 0 < w0 < w*. Furthermore, all four equilibria are hyperbolic with (v*, 0), (0, w*) being stable and (0, 0), (v0, w0) being unstable; (B3) the zero sets of M and N intersect exactly once in the positive quadrant at (v0, w0) and are the graphs of functions w = k(v) and v = l(w), respectively. Thus, it is worthwhile to further explore this topic for non-monotone systems, and this constitutes the purpose of this paper. In order to focus on the mathematical ideas and for the sake of simplicity, we consider a reaction-diffusion system of two equations, that is 8 < ou1 ðx;tÞ ¼ d 1 o2 u1 ðx;tÞ þ f1 ðu1 ðx; tÞ; u2 ðx; tÞÞ; ot ox2 ð1:4Þ : ou2 ðx;tÞ ¼ d o2 u2 ðx;tÞ þ f ðu ðx; tÞ; u ðx; tÞÞ; ot
2
ox2
2
1
2
where x 2 R, t > 0, di > 0, i = 1, 2, and the function f(u, v) = (f1(u, v),f2(u, v)) satisfies the following assumptions: (C1) f(0, 0) = f(K1, K2) = 0, f 2 C 2 ðI 2 ; R2 Þ for an open interval I R with [0, +1) I, where Ki are two positive constants, i = 1, 2; (C2) all the eigenvalues of the matrixes (oj fi (0, 0)) and (oj fi (K1, K2)) lie in the left half-plane. In the order interval (E, E+), where E = (0, 0) and E+ = (K1, K2) the only equilibrium E0 = (a1, a2) with 0 < ai < Ki, i = 1, 2 is such that the matrix (ojfi(a1, a2)) has at least one eigenvalue in the right half-plane. Since the results of Roquejoffre et al. [13], Volpert et al. [17] and Xu and Zhao [21] do not apply to the reactiondiffusion systems (1.3) and (1.4), we must search for new techniques that can be applied to (1.4), at least for (1.3).
S.-L. Wu, W.-T. Li / Chaos, Solitons and Fractals 40 (2009) 1229–1239
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To overcome the difficulty, we propose a weak condition for the reaction terms, which is called interval-monotone condition (C3) ojfi(u, v) > 0 for (u, v) 2 [0, K1] · [0, K2], 1 6 i 5 j 6 2. Under the conditions (C1), (C2) and (C3), we first establish the existence and comparison theorem of solutions for the interval monotone reaction-diffusion systems (1.4) on R by appealing to the theory of abstract differential equations developed in [11] and properties of the analytic semigroup generated by the one-dimensional Laplacian operator on the Banach space of all bounded and uniformly continuous functions on R. In order to prove global asymptotic stability of traveling wave fronts, we shall borrow a squeezing technique due to Chen [3] for nonlocal evolution equations, which is similar in spirit to a ‘‘contracting rectangle’’ approach developed in [12]. By the elementary super- and subsolution comparison method and the established global asymptotic stability of traveling wave fronts, the Liapunov stability and uniqueness of traveling wave fronts connecting the stable equilibria E and E+ are then proved. Finally, these abstract results are applied to the competition-diffusion model (1.3) and a system modelling man–environment–man epidemics. Here a traveling wave front of (1.4) always refers to a pair (U, c), where U = U(n) is a monotone function on R and c > 0 is a constant, such that u(x, t): = U(x ct) is a solution of (1.4) and lim U ðnÞ ¼ E ;
n!1
lim U ðnÞ ¼ Eþ :
n!þ1
ð1:5Þ
The rest of this paper is organized as follows. In Section 2, we establish an existence and comparison theorem for interval-monotone reaction-diffusion systems on R and then prove that the traveling wave front is globally asymptotically stable and unique up to translation. In Section 3, we apply our results to the above mentioned model and a system modelling man–environment–man epidemics, and obtain some new results on global asymptotic stability and uniqueness of traveling wave fronts.
2. Main results In this paper, we will use the usual notations for the standard ordering in R2 . That is, for u = (u1, u2) and v = (v1, v2), we denote u 6 v if ui 6 vi, i = 1, 2, and u < v if ui 6 vi but u 5 v. In particular, we denote u v if ui 6 vi but ui 5 vi, i = 1, 2. If u 6 v, we also denote ðu; v ¼ fw 2 R2 : u < w 6 vg, ½u; vÞ ¼ fw 2 R2 : u 6 w < vg, and ½u; v ¼ fw 2 R2 : u 6 w 6 vg. Let kÆk denote the Euclidean norm in R2 . In order to study the asymptotic stability of traveling wave fronts of (1.4), we first study the initial value problem 8 ou1 o2 u1 > > < ot ¼ d 1 ox2 þ f1 ðu1 ðx; tÞ; u2 ðx; tÞÞ; ou2 o2 u2 ð2:1Þ > ot ¼ d 2 ox2 þ f2 ðu1 ðx; tÞ; u2 ðx; tÞÞ; > : ui ðx; 0Þ ¼ /i ðxÞ; i ¼ 1; 2; where x 2 R, t > 0. Let X ¼ BUCðR; R2 Þ be the Banach space of all bounded and uniformly continuous functions from R into R2 with the supremum norm kÆkX. Let X þ ¼ fu 2 X : uðxÞ P 0; x 2 Rg. It is easy to see that X+ is a closed cone of X and X is a Banach lattice under the partial ordering induced by X+. For any /1, /2 2 X, we write /1 6 X/2 if /1 /2 2 X+, /1 < X/2 if /1 /2 2 X þ n f^0g, /1 X/2 if /1 /2 2 int X+, where ^ a denote a constant vector function on R taking the vector a. For /1, /2 2 X with /1 6 X/2, we denote [/1, /2]X = {/ 2 X:/1 6 X/ 6 X/2}. Consider the uncoupled linear system ( oui ¼ d i Dui ; x 2 R; t > 0; ot ð2:2Þ ui ðx; 0Þ ¼ /i ðxÞ; x 2 R; where i = 1, 2. Define the family of linear operator T(t) = (T1(t),T2(t)) on X by T ðtÞ/ ¼ ðT 1 ðtÞ/1 ; T 2 ðtÞ/2 Þ
for each / ¼ ð/1 ; /2 Þ 2 X ;
ð2:3Þ
where T i ðtÞ/i ¼ ui ð; tÞði ¼ 1; 2Þ, and u is the solution of (2.2). It is easy to see that T(t):X ! X is an analytic semigroup on X with T(t)X+ X+ for all t > 0. In fact, by the explicit expression of solution of the heat Eq. (2.2), we have
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1 T i ðtÞ/i ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi 4pd i t x 2 R;
Z
þ1
exp
1
! ðx yÞ2 /i ðyÞdy; 4d i t
/i ðÞ 2 BUCðR; RÞ;
t > 0;
i ¼ 1; 2:
b ¼ f/ 2 X : 0 6 /ðxÞ 6 K; x 2 Rg, where 0 = (0, 0), K = (K1, K2), define F(/)(x) = For any / ¼ ð/1 ; /2 Þ 2 ½^0; K X (F1(/)(x), F2(/)(x)) by F i ð/ÞðxÞ ¼ fi ð/1 ðxÞ; /2 ðxÞÞ;
i ¼ 1; 2:
b ! X is globally Lipschitz continuous. Then F(/) 2 X and F : / 2 ½^0; K X ^ K b , (2.1) has a unique, bounded and nonnegative solution u(x, t, /) on [0, 1), and the Lemma 2.1. For any / 2 ½0; X b 0; K solution semiform of (2.1) is monotone. Moreover, u(x, t, /1) u(x, t, /2) for t > 0 and x 2 R whenever /1 ; /2 2 ½^ X 1 2 with / < X/ , i.e., the solution semiform of (2.1) is strongly monotone. Proof. Let T(t) = (T1(t), T2(t)), then T(t) is a linear semigroup on X. Under an abstract setting in [11], a mild solution of (2.1) is a solution to its associated integral equation ( Rt uðtÞ ¼ T ðtÞuð0Þ þ 0 T ðt sÞF ðuðsÞÞds; t > 0; ð2:4Þ b : uð0Þ ¼ / 2 ½^0; K X
b ! X is globally Lipschitz continuous. Let As aforementioned, F : ½^0; K X Li ¼ max joi fi ðu; vÞj; ðu;vÞ2½^ 0; b K
i ¼ 1; 2:
b in the sense that We claim that F is also monotone on ½^0; K X 1 þ lim distðw / þ h½F ðwÞ F ð/Þ; X Þ ¼ 0; h!0þ h ^ K b with w P /. Indeed, for all w; / 2 ½0; X
F 1 ðwÞ F 1 ð/Þ ¼ f1 ðw1 ðÞ; w2 ðÞÞ f1 ð/1 ðÞ; /2 ðÞÞ P f1 ðw1 ðÞ; /2 ðÞÞ f1 ð/1 ðÞ; /2 ðÞÞ P L1 ðw1 /1 Þ
in X ;
and hence, for any d1 > 0 such that L1d1 < 1, w1 /1 þ d1 ½F 1 ðwÞ F 1 ð/Þ P ð1 L1 d1 Þðw1 /1 Þ P 0 in X : Similarly, we have for any d2 > 0 such that L2d2 < 1, w2 /2 þ d2 ½F 2 ðwÞ F 2 ð/Þ P ð1 L2 d2 Þðw2 /2 Þ P 0 in X : Thus, the existence and uniqueness of u(x, t, /) follows from [11, Corollary 5](take delay as zero) with S(t, s) = T(t, s), t P s P 0, and B(t, /) = F(/). Moreover, by a semigroup theory argument given in proof of [11, Theorem 1], it follows that u(x, t, /) is a classical solution for t > 0. Note that [11, Corollary 5] also implies that the comparison principle holds for (2.1), and it follows that (2.1) defines b . a monotone solution semigroup on ½^0; K X 1 2 i 2 b Suppose that /1 ; /2 2 ½^0; K X with / < X/ . Then u(x, t, / ) P 0 for x 2 R; t P 0. Let u(x, t) = u(x, t, / ) 1 1 2 u(x, t, / ). Then u(x, t) P 0 for x 2 R; t P 0. We shall show that ui(x, t) > 0 for x 2 R; t > 0, i = 1, 2. Since / < X/ , i.e., /1 6 X/2 and /1 f /2. Without loss of generality, we assume that /21 ðÞX/11 ðÞ and /22 ðÞ /12 ðÞ, i.e., u1(Æ, 0) f 0 and u2(Æ, 0) 0. Note that in the first component u1(x, t) of u(x, t) satisfies u1;t ¼ d 1 u1;xx þ f1 ðu1 ðx; t; /2 Þ; u2 ðx; t; /2 ÞÞ f1 ðu1 ðx; t; /1 Þ; u2 ðx; t; /1 ÞÞ P d 1 u1;xx þ f1 ðu1 ðx; t; /2 Þ; u2 ðx; t; /2 ÞÞ f1 ðu1 ðx; t; /1 Þ; u2 ðx; t; /2 ÞÞ P d 1 u1;xx L1 u1 : By the strict positive theorem [17, Theorem 5.5.4], we have u1(x, t) > 0 for all x 2 R; t > 0. Suppose to the contrary that there exists a point (x*, t*) with x 2 R and t* > 0 such that u2(x*, t*) = 0. Then u2,t(x*, t*) 6 0 and u2,xx(x*, t*) P 0. Since u2(x, t) satisfies u2;t ¼ d 2 u2;xx þ f2 ðu1 ðx; t; /2 Þ; u2 ðx; t; /2 ÞÞ f2 ðu1 ðx; t; /1 Þ; u2 ðx; t; /1 ÞÞ ¼ d 2 u2;xx þ o1 f2 ðgÞu1 ðx; tÞ þ o2 f2 ðgÞu2 ðx; tÞ; where g = su(x, t, /2) + (1 s)u(x, t, /1) 2 [0, K], s 2 (0, 1). Since o1f2(g) > 0, we conclude that 0 P u2;t jðx ;t Þ ¼ d 2 u2;xx þ o1 f2 ðgÞu1 ðx; tÞ þ o2 f2 ðgÞu2 ðx; tÞjðx ;t Þ > 0 which is a contradiction. Thus u2(x, t) > 0 for x 2 R; t > 0. This completes the proof.
h
S.-L. Wu, W.-T. Li / Chaos, Solitons and Fractals 40 (2009) 1229–1239
1233
Suppose that u(x ct) = (u1(x ct),u2(x ct)) is a strictly increasing traveling wave solution of (1.4) connecting E and E+. Rescale (1.4) in a moving coordinate frame, i.e., in terms of variables z = x ct and t, it becomes w1;t ðz; tÞ ¼ cw1;z ðz; tÞ þ d 1 w1;zz ðz; tÞ þ f1 ðw1 ðz; tÞ; w2 ðz; tÞÞ; ð2:5Þ w2;t ðz; tÞ ¼ cw2;z ðz; tÞ þ d 2 w2;zz ðz; tÞ þ f2 ðw1 ðz; tÞ; w2 ðz; tÞÞ: Then u(z) is an equilibrium solution of system (2.5). In what follows, we denote by w(z, t, /) = (w1(z, t, /),w2(z, t, /)) the b . Clearly, the solution u(x, t, /) of (2.1) is given by u(x, t, /) = w(x ct, t, /). As solution of (2.5) with w0 ¼ / 2 ½^0; K X noted before, the comparison principle holds for (2.1) and hence for (2.5). For convenience, we set H 1 ðw1 ; w2 Þ :¼ w1;t ðz; tÞ cw1;z ðz; tÞ d 1 w1;zz ðz; tÞ f1 ðw1 ðz; tÞ; w2 ðz; tÞÞ; H 2 ðw1 ; w2 Þ :¼ w2;t ðz; tÞ cw2;z ðz; tÞ d 2 w2;zz ðz; tÞ f2 ðw1 ðz; tÞ; w2 ðz; tÞÞ: b and Lemma 2.2. Assume that / ¼ ð/1 ; /2 Þ 2 ½^0; K X ð/Þ :¼ max lim sup k/ðzÞ E k; lim sup k/ðzÞ Eþ k : z!1
ð2:6Þ
z!þ1
is small enough. Then for any e ¼ ðe1 ; e2 Þ 2 R2 with 0 e, there exist z ¼ zðe; /Þ > 0 and a large time t ¼ tðe; /Þ such that uðz zÞ e 6 wðz; t; /Þ 6 uðz þ zÞ þ e;
8z 2 R:
Proof. From the assumptions (C1), (C2) and (C3), it is easy to see that E± are asymptotic stable equilibria of the following ordinary differential system
z01 ðtÞ ¼ f1 ðz1 ðtÞ; z2 ðtÞÞ; z02 ðtÞ ¼ f2 ðz1 ðtÞ; z2 ðtÞÞ;
t > 0; t > 0:
ð2:7Þ
e E0 , and the solution z(t, k±) of system (2.7) with Then there exists v0 > 0 is small enough such that E < E þ v0~ e approach E± as t ! 1, where ~ zð0; k Þ ¼ k :¼ E þ v0~ e ¼ ð1; 1Þ. Let v±(t) = z(0, k±). Then there exists l P 0 such that 0 6 v±(t) 6 l. b and (/) is small enough, without loss of generality, we assume that /(z) 6 k for z 6 0, where Since / 2 ½^0; K X e. Take k ¼ E þ v0~ L ¼ supfjoij fj ðuÞj : u 2 ½0; l; 1 6 i; j 6 2g: Let
)
(
2 1 þ 1 ðvþ j ðtÞ vj ðtÞÞ þ ~c > c þ maxfd i g þ L sup : t 2 ½0; 1Þ; 1 6 i–j 6 2 ðtÞÞ þ ðv ðtÞ v ðtÞÞ:þ ðvi ðtÞ v i j j i¼1;2 2 2 vþ i ðtÞ vi ðtÞ
be a fixed number. Define 1 sþT fðsÞ ¼ 1 þ tanh ; 2 2
s2R
and vðz; tÞ ¼ vþ ðtÞfðz þ ~ctÞ þ v ðtÞð1 fðz þ ~ctÞÞ;
z 2 R;
t P 0;
where T > 0 is large enough such that f(0)E+ P E+ v0. It is easy to see that f 0 = f(1 f), f00 = f 0 (1 2f) and v(z, 0) P /(z) for z 2 R. We further show that v(z, t) is a super-solution of (2.5). Indeed, by Taylor’s expansion, we have H 1 ðvðz; tÞÞ ¼ v1;t ðz; tÞ cv1;z ðz; tÞ d 1 v1;zz ðz; tÞ f1 ðvðz; tÞÞ ¼ ff1 ðvþ ðtÞÞ þ ð1 fÞf1 ðv ðtÞÞ f1 ðfvþ ðtÞ þ þ ð1 fÞv ðtÞÞ þ fð1 fÞ½ð~c cÞðvþ 1 ðtÞ v1 ðtÞÞ d 1 ð1 2fÞðv1 ðtÞ v1 ðtÞÞ 1 1 2 2 þ þ ¼ fð1 fÞðvþ 1 ðtÞ v1 ðtÞÞ o11 f1 ðh1 Þ þ fð1 fÞðv2 ðtÞ v2 ðtÞÞ o22 f1 ðh1 Þ þ fð1 fÞðv1 ðtÞ v1 ðtÞÞ 2 2 þ c cÞðvþ
ðvþ 2 ðtÞ v2 ðtÞÞo12 f1 ðh1 Þ þ fð1 fÞ½ð~ 1 ðtÞ v1 ðtÞÞ d 1 ð1 2fÞðv1 ðtÞ v1 ðtÞÞ #) ( " 2 1 þ 1 ðvþ þ þ 2 ðtÞ v2 ðtÞÞ P0 P fð1 fÞðv1 ðtÞ v1 ðtÞÞ ~c c d 1 L ðv1 ðtÞ v1 ðtÞÞ þ ðv2 ðtÞ v2 ðtÞÞ þ 2 2 vþ 1 ðtÞ v1 ðtÞ
and
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H 2 ðvðz; tÞÞ ¼ v2;t ðz; tÞ cv2;z ðz; tÞ d 2 v2;zz ðz; tÞ f2 ðvðz; tÞÞ ( 1 þ ðtÞ v P fð1 fÞðv2 ðtÞ v2 ðtÞÞ ~c c d 2 þ ðvþ 2 ðtÞÞo22 f2 ðh2 Þ: 2 2 ) 2 1 ðvþ þ 1 ðtÞ v1 ðtÞÞ o11 f2 ðh2 Þ P 0; þ ðv1 ðtÞ v1 ðtÞÞo12 f2 ðh2 Þ þ 2 vþ 2 ðtÞ v2 ðtÞ where h1, h2 2 (v-(t),v+(t)) [0, l]. Thus v(z, t) is a super-solution of system (2.5). Therefore, by the comparison principle we have w(z, t, /) 6 v(z, t) for z 2 R, t > 0. Note that limt!1v±(t) = E±. It then follows that for any e ¼ ðe1 ; e2 Þ 2 R2 with 0 e, there exist z ¼ zðe; /Þ > 0 and a large time t ¼ tðe; /Þ such that wðz; t; /Þ 6 uðz þ zÞ þ e; 8z 2 R. Indeed, since limt!1v±(t) = E±, there exists t > 0 large enough such that 1 1 kvþ ðtÞ Eþ k 6 minfei g; kv ðtÞ E k 6 minfei g: 4 i¼1;2 4 i¼1;2 By virtue of limz!±1u(z) = E± and lims!1f(s) = 0, there exists M > 0 such that e Eþ uðzÞ þ for z P M; 4 and e ei for z 6 M: E uðzÞ þ ; fðz þ ~ctÞ < min i¼1;2 4li 4 Moreover, since u(Æ) is monotone and limz!1u(z) = E+, there exists z > 0 such that Eþ uðz þ zÞ þ 4e for M 6 z 6 M. Therefore, it is easy to see that wðz; t; /Þ 6 uðz þ zÞ þ eforallz 2 R: The other estimates on the lower bound of the solution are similar and omitted. This completes the proof.
h
±
Note that all eigenvalues of the Jacobian matrixes (ojfi(E )) lie in the left half-plane. Denote by q ¼ ðq 1 ; q2 Þ the Þ, where l be constants such that positive eigenvectors corresponding the principle eigenvalues of the matrixes ðl ij ij oj fi ðE Þ < l , 1 6 i, j 6 2 and (0, 0) is asymptotic stable in both of the linear system 8 ij < dr1 ðtÞ ¼ l r þ l r ; t > 0; 11 1 12 2 dt ð2:8Þ : dr 2 ðtÞ ¼ l r þ l r ; t > 0: 21 1 22 2 dt
Let q1(n),q2(n) be smooth positive functions such that q(n) = (q1(n),q2(n)) ! q± in C2 -topology as n ! 1. Lemma 2.3. There exist two positive numbers r and 10 such that for any 1 P 10, n0 2 R, and 2 (0, 0(1)), v ðz; tÞ ¼ uðz n0 1ð1 ert ÞÞ qðz n0 Þert ;
z 2 R;
t>0
are super- and sub-solutions of systems (2.5), respectively. Proof. Clearly, there exist d, k > 0 such that for ku E k 6 d; u 2 R2 ; oj fi ðuÞ 6 l ij 2 X
l ij qj < kqi
for q ¼ ðq1 ; q2 Þ 2 R2þ
1 6 i;
ð2:9Þ
j 6 2;
with kq q k 6 d;
i ¼ 1; 2:
ð2:10Þ
j¼1
Take B1 > 0 such that kqðgÞk; kq0 ðgÞk; kq00 ðgÞk 6 B1 ±
forall g 2 R:
±
Since u(n) ! E , q(n) ! q , q 0 (n),q00 (n) ! 0 as n ! ±1, there exist 0 < 1 < 1,M > 0 such that k c1 maxfd i g1 > 0; i¼1;2
jq0i ðgÞj; jq00i ðgÞj þ
6 1 qi ðgÞ
for jgj P M 1;
i ¼ 1; 2;
kqðgÞ q k 6 d for g P M 1; kqðgÞ q k 6 d for g 6 M þ 1; d kuðnÞ þ qðgÞ Eþ k 6 for 2 ð0; 1 ; n P M 1; g P M 1; 2 d kuðnÞ þ qðgÞ E k 6 for 2 ð0; 1 ; n 6 M þ 1; g 6 M þ 1: 2
ð2:11Þ ð2:12Þ
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Define eg B2 ¼ supfjoj fi ðuÞj : u 2 ½E d~ e; Eþ þ B1~
and
B3 ¼ inf ku0 ðnÞk: knk6M
Choosing r > 0 sufficiently small such that r 6 k c1 maxi=1, 2{di}. Take B1 1 1 P 10 ¼ ðr þ c þ maxfd i g þ B2 Þ; 0 ¼ min 1 ; : i¼1;2 B3 r 1 Set n = z + n0 + 1(1 ert), g = z + n0 and q = ert. Then, for t P 0, we have þ þ þ H i ðvþ ðz; tÞÞ ¼ vþ i;t ðz; tÞ cvi;z ðz; tÞ d i vi;zz ðz; tÞ fi ðv ðz; tÞÞ
¼ fi ðuðnÞÞ fi ðuðnÞ þ qðgÞqÞ þ 1rqu0i ðnÞ ðrqi ðgÞ þ cq0i ðgÞ þ d i q00i ðgÞÞq: We distinguish among three cases: Case (i): |n| 6 M. In this case, Z 1 X 2 q qj oj fi ðu þ sqqÞds P qB1 B2 : fi ðuÞ fi ðu þ qqÞ ¼ 0
j¼1
Therefore, H i ðvþ ðz; tÞÞ P qB1 B2 B1 qðr þ c þ d i Þ þ B3 1rq P 0: Case (ii): n P M. Since n g 6 1 6 1, n > g P M 1. Thus, d kuðnÞ þ sqqðgÞ Eþ k 6 ; 2
kqðgÞ qþ k 6 d; 8s 2 ð0; 1Þ:
Therefore, by (2.9)–(2.12), we have
H i ðvþ ðz; tÞÞ ¼ fi ðuÞ fi ðu þ qqÞ þ 1rqu0i rqi þ cq0i þ d i q00i q Z 1 X 2 q qj oj fi ðu þ sqqÞds þ 1rqu0i rqi þ cq0i þ d i q00i q ¼ 0
P q
j¼1 2 X
0 0 00 lþ ij qj þ 1rqui rqi þ cqi þ d i qi q P ðk r c1 d i 1 Þqqi P 0:
j¼1
Case (iii): n 6 M. Clearly, g < n < M + 1. Then d kuðnÞ þ sqqðgÞ E k 6 ; 2
kqðgÞ q k 6 d; 8s 2 ð0; 1Þ:
Therefore, again by (2.9)–(2.12), we have Z 1 X 2 q qj oj fi ðu þ sqqÞds þ 1rqu0i rqi þ cq0i þ d i q00i q H i ðvþ ðz; tÞÞ ¼ 0
P q
j¼1 2 X
0 0 00 l ij qj þ 1rqui rqi þ cqi þ d i qi q P ðk r c1 d i 1 Þqqi P 0:
j¼1
Combining cases (i)–(iii), we have Ni(v+(z, t)) P 0 for all 2 (0, 0) and t P 0, i = 1, 2. Thus v+(z, t) is a supersolution of (2.5). In a similar way, we can show that v(z, t) is a sub-solution of (2.5). This completes the proof. h Lemma 2.4. The wave profile u(z) is a Liapunov stable equilibrium of (2.5). Proof. Let 0 and v±(z, t, ) be as in Lemma 2.3 with 1 = 10 and n0 = 0. Then, for z 2 R, t P 0, 2 (0, 0), kv ðz; t; Þ uðzÞk 6 kv ðz; t; Þ uðz 10 ð1 ert ÞÞk þ kuðz 10 ð1 ert ÞÞ uðzÞk ¼ kqðzÞkert þ 10 ku0 ðzÞ s10 ð1 ert Þkð1 ert Þ 6 K;
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b with k/ ukX 6 d, where K > 0 is independent of . For any 2 (0, 0), take d ¼ inf z2R qðzÞ. Thus, for any / 2 ½^ 0; K X we have v ðz; 0; Þ ¼ uðzÞ qðzÞ 6 uðzÞ d 6 /ðzÞ 6 uðzÞ þ d 6 uðzÞ þ qðzÞ ¼ vþ ðz; 0; Þ; 8z 2 R: Then the comparison principle implies that v ðz; t; Þ 6 wðz; t; /Þ 6 vþ ðz; t; Þ; z 2 R;
t P 0:
Therefore, kw(Æ, t, /) u(Æ)k < K for t P 0. This completes the proof.
h
To prove the asymptotic stability of traveling waves, we need the following convergence result for monotone semiflows (see [24]). Lemma 2.5. Let U be a closed convex subset of an ordered Banach space v, and U(t):U ! U be a monotone semiform. Assume that there exists a monotone homeomorphism h from [0, 1] onto a subset of U such that (i) for each s 2 [0, 1], h(s) is a stable equilibrium for U(t):U ! U; (ii) each orbit of U(t) in [h(0),h(1)]v is precompact; (iii) one of the following two properties holds: (a) if h(s0) < vx(/) for some s0 2 [0, 1) and / 2 [h(0),h(1)]v, then there exists s1 2 (s0,1) such that h(s1) 6 vx(/); (b) if x(/) < vh(r1) for some r1 2 [0, 1) and / 2 [h(0),h(1)]v, then there exists r0 2 (0, r1) such that x(/) 6 vh(r0). Then for any precompact orbit c+(/0) of U(t) in U with x(/0) \ [h(0),h(1)]v 5 B, there exists s* 2 [0, 1] such that x(/0) = h(s*). Now we state the main results of this section as follows. Theorem 2.6. Assume that (C1), (C2) and (C3) hold, u(n) is a traveling wave front of system (1.4) and u(x, t, /) is the b . Then for any / 2 ½^ b such that (/) (defined by (2.6)) is small enough, there solution of (2.1) with u0 ¼ / 2 ½^0; K 0; K X X exists s/ 2 R such that lim sup kuðx; t; /Þ uðx ct þ s/ Þk ¼ 0: t!þ1 x2R
Proof. Using Lemmas 2.1–2.5, the proof is similar to that of Theorem 3.1 in Xu and Zhao [21] and is omitted.
h
~ ðnÞ ¼ u ~ ðx ~ctÞ is a traveling wave solution of (1.4) satisfying (1.5) Theorem 2.7. Assume that (C1), (C2) and (C3) hold, u ~ ðÞ < K. Then u ~ ðÞ ¼ uð þ n0 Þ for some n0 ¼ n0 ð~ and 0 < u uÞ 2 R. ~ ðÞ < K, by Theorem 2.6, there exists n0 ¼ n0 ð~ ~ ðnÞ ¼ E , 0 < u uÞ 2 R such that Proof. Since limn! 1 u uðx ~ctÞ uðx ct þ n0 Þk ¼ 0: lim sup k~ t!þ1 x2R
Let z = x ct, we have lim sup k~ uð þ ðc ~cÞtÞ uð þ n0 Þk ¼ 0: t!þ1
~ ðnÞ ! E as n ! ±1, we must have ~c ¼ c, and thus u ~ ðÞ ¼ uð þ n0 Þ. This Since u(Æ) is strictly increasing on R and u completes the proof. h
3. Applications In this section, we shall apply our abstract results to a competition-diffusion model and a system modelling man– environment–man epidemics. 3.1. A competition model We consider the competition model of two species (1.3), that is, 8 < ovðx;tÞ ¼ d 1 o2 vðx;tÞ þ vMðv; wÞ; ot
: owðx;tÞ ¼ d ot
ox2
o2 wðx;tÞ 2 ox2
þ wN ðv; wÞ;
ð3:1Þ
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1237
where x 2 R, t > 0, di > 0, i = 1, 2. Under the assumptions (B1)-(B3), the existence of a monotone traveling wave solution of (3.1) connecting the two stable equilibria is established in Gardner [6] with the use of topological degree theory. In [7], Huang further proved that this traveling wave front is unique up to translation by using a homotopy approach incorporated with the Lypunov–Schmidt method. By the method developed in this paper we shall show that the traveling wave front in fact is global asymptotic stable. To do so, let us first substitute u1 = v and u2 = w* w into (3.1), then system (3.1) becomes 8 < ou1 ðx;tÞ ¼ d 1 o2 u1 ðx;tÞ þ h1 ðu1 ; u2 Þ; ot ox2 ð3:2Þ : ou2 ðx;tÞ ¼ d o2 u2 ðx;tÞ þ h ðu ; u Þ; ot
2
ox2
2
1
2
where h1 ðu1 ; u2 Þ ¼ u1 Mðu1 ; w u2 Þandh2 ðu1 ; u2 Þ ¼ ðu2 w ÞN ðu1 ; w u2 Þ: Obviously, if (B1) and (B2) hold, then we have ojhi(u) > 0 for u 2 [0,v*] · [0,w*], 1 6 i 5 j 6 2, and (3.2) has four spatially uniform equilibria 0: = (0, 0), (v0, w* w0), (0, w*) and K = (v*, w*). Furthermore, (0, 0), (v*, w*) are stable and (v0, w* w0), (0, w*) are unstable. Although (3.2) has a extra unstable equilibrium (0, w*) in (0, K), with a slight modification of the method used in this paper, we are still able to conclude that the bistable traveling wave solution of (3.2) is globally asymptotically stable, i.e., the following result holds. Theorem 3.1. Assume that (B1) and (B2) hold, u(n) is a traveling wave front of (3.2) and u(x, t, /) is the solution of (3.2) b . Then for any / 2 ½^0; K b such that (/) (defined by (2.6)) is small enough, there exists s/ 2 R such with u0 ¼ / 2 ½^0; K X X that lim sup kuðx; t; /Þ uðx ct þ s/ Þk ¼ 0:
t!þ1 x2R
Consequently, the bistable traveling wave solution of (3.1) connecting (0, w*) and (v*, 0) is also globally asymptotically stable. 3.2. An epidemic model In this subsection, we consider an epidemic model which is a monotone system, and establish the global asymptotic stability and uniqueness (up to translation) of bistable traveling wave front of the model under suitable conditions which are less restrictive than that in Roquejoffre et al. [13]. More precisely, we consider the following reaction-diffusion system (see [2]) 8 < ou1 ðx;tÞ ¼ d 1 o2 u1 ðx;tÞ a11 u1 ðx; tÞ þ a12 u2 ðx; tÞ; ot ox2 ð3:3Þ : ou2 ðx;tÞ ¼ d o2 u2 ðx;tÞ a u ðx; tÞ þ gðu ðx; tÞÞ; ot
2
ox2
22 2
1
where u1(x, t) and u2(x, t), respectively, represent the average concentration of bacteria and the infective human population at a point x in the habitat at time t, 1/a11 is the mean lifetime of the agent in the environment, 1/a22 is the multiplicative factor of the mean infectious agent due to the human populations, g(v) is the infection rate of human under the assumption that total susceptible human population is constant during the evolution of the epidemic, and d1 P 0 and d2 P 0 denote the diffusion constants of the bacteria and the infective human, respectively. System (3.3) has been widely studied. For the bistable case, i.e., the function g is sigma-shaped (defined concisely by properties (H1) and (H2) in the following), the existence of bistable traveling wave fronts of (3.3) is a consequence of Volpert et al. [17, Theorem 3.1.1] when di > 0, i = 1, 2. By using phase space analysis, monotone semiflows approach and spectrum method, Xu and Zhao [21] proved the existence, uniqueness and global asymptotic stability of bistable traveling wave fronts of (3.3) when d1 > 0 and d2 = 0. We will establish the global asymptotic stability and uniqueness of bistable traveling wave fronts of (3.3) when d1 > 0 and d2 > 0. Similar to Xu and Zhao [21], we assume that the function g satisfies (H1) g 2 C2(I), g(0) = 0, g 0 (0) P 0, g 0 (z) > 0, "z > 0, limz!1g(z) = 1, and there is a 1 > 0 such that g00 (z) > 0 for z 2 (0, 1) and g00 (z) < 0 for z > 1. (H2) g 0 (0) < c = a11a22/a12 < ccrit, where c = ccrit such that the equation g(z) = cz has one and only one positive root.
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Under the assumptions (H1) and (H2), (3.3) has three equilibrium: E = (0, 0), E0 = (a, a11a/a12) and E+ = (b, a11b/ a12), where 0 < a < b are two roots of the equation g(x) = (a11a22/a12)x. In this case, E0 is a saddle point, E and E+ are stable nodes (see [21, Proposition 2.1]). The function f(u, v) = (f1(u, v), f2(u, v)), where f1(u, v) = a11u + a12v and f2(u, v) = a22v + g(u), satisfies the monotone condition (1.2). Thus, the following result is a consequence of Theorem 2.1 in Roquejoffre et al. [13]. Theorem 3.2. Assume that (H1) and (H2) hold, u(n) is a traveling wave front of (3.3) and u(x, t, /) is the solution of (3.3) b . Then for any / 2 ½^0; K b such that (/) (defined by (2.6)) is small enough, there exists s/ 2 R such with u0 ¼ / 2 ½^0; K X X that lim sup kuðx; t; /Þ uðx ct þ s/ Þk ¼ 0:
t!þ1 x2R
~ ðnÞ of (3.3) satisfying (1.5) and 0 < u ~ ðÞ < K is a translate of u. Moreover, any traveling wave solution u With a slight modification of the method used in this paper (see Lemma 2.2), we are able to show that the above conclusion holds under suitable weak conditions. In fact, the following result holds. Theorem 3.3. Assume that (H1) and (H2) hold, u(n) is a traveling wave front of system (3.3) and u(x, t, /) is the solution of b . Then for any / 2 ½^0; K b such that (3.3) with u0 ¼ / 2 ½^0; K X X lim sup /ðxÞ E0 lim inf /ðxÞ; x!1
x!þ1
ð3:4Þ
there exists s/ 2 R such that lim sup kuðx; t; /Þ uðx ct þ s/ Þk ¼ 0:
t!þ1 x2R
~ ðnÞ of (3.3) satisfying (1.5) and 0 < u ~ ðÞ < K is a translate of u. Moreover, any traveling wave solution u Remark 3.4. Theorem 3.2 implies that any solution of the Cauchy problem for (3.3) which initially has these two stable equilibria as spatial limits will converge to a traveling wave front. Theorem 3.3 further guarantees that the solution will converge to a traveling wave front when the initial function satisfies the positive and negative spatial limits are larger and smaller than the immediate equilibrium, respectively. That is to say, for some monotone reaction-diffusion systems, by the method developed in this paper, we can establish the global asymptotic stability of bistable traveling wave fronts under suitable weaker conditions than those in Roquejoffre et al. [13]. Remark 3.5. With some additional assumptions on f(Æ, Æ), Volpert et al. [17, Theorem 3.1.1] proved the existence of bistable traveling wave front of (1.1). Our results (Theorems 2.6 and 2.7) further conclude that this traveling wave front is globally asymptotically stable with phase shift, that all traveling waves are unique up to translation, and that every traveling wave solution is Liapunov stable. In particular, our conditions are less restrictive than those in Roquejoffre et al. [13] and Volpert et al. [17].
Acknowledgements The authors would like to thank Professor Shigui Ruan (University of Miami) for his helpful comments on this paper.
References [1] Britton NF. Reaction-diffusion equations and their applications to biology. New York: Academic Press; 1986. [2] Capasso V, Maddalena L. Convergence to equlibrium states for a reaction-diffusion system modeling the spatial speed of a class of bactetial and viral diseases. J Math Biol 1981;13:173–84. [3] Chen X. Existence, uniqueness and asymptotic stability of traveling waves in non-local evolution equations. Adv Differ Eqn 1997;2:125–60. [4] Feng Z. Traveling solitary wave solutions to evolution equations with nonlinear terms of any order. Chaos, Solitons & Fractals 2003;17:861–8. [5] Feng Z. Traveling wave behavior for a generalized fisher equation. Chaos, Solitons & Fractals 2008;38:481–8.
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[6] Gardner RA. Existence and stability of traveling wave solution of competition models: a degree theoretic approach. J Differ Eq 1982;44:343–64. [7] Huang W. Uniqueness of the bistable traveling wave for mutualist species. J Dyn Differ Eqn 2001;13:147–83. [8] Jiang G, Lu Q, Qian L. Complex dynamics of a Holling type II prey–predator system with state feedback control. Chaos, Solitons & Fractals 2007;31:448–61. [9] Li WT, Lin G, Ruan S. Existence of traveling wave solutions in delayed reaction-diffusion systems with applications to diffusioncompetition systems. Nonlinearity 2006;19:1253–73. [10] Li WT, Wu SL. Traveling waves in a diffusive predator–prey model with Holling type-III functional response. Chaos, Solitons & Fractals 2008;37:476–86. [11] Martin RH, Smith HL. Abstract functional differential equations and reaction-diffusion system. Trans Am Math Soc 1990;321:1–44. [12] Martin RH, Smith HL. Reaction-diffusion systems with time delays: Monotonicity, invariance, comparison and convergence. J Reine Angew Math 1991;413:1–35. [13] Roquejoffre JM, Terman D, Volpert VA. Global stability of traveling fronts and convergence towards stacked families of waves in monotone parabolic systems. SIAM J Math Anal 1996;27:1261–9. [14] Schaaf KW. Asymptotic behavior and traveling wave solutions for parabolic functional differential equations. Trans Am Math Soc 1987;302:587–615. [15] Smith HL, Zhao XQ. Global asymptotical stability of traveling waves in delayed reaction-diffusion equations. SIAM J Math Anal 2000;31:514–34. [16] Tan Y, Xu H, Liao SJ. Explicit series solution of traveling waves with a front of Fisher equation. Chaos, Solitons & Fractals 2007;31:462–72. [17] Volpert AI et al. Traveling wave solutions of parabolic systems, translations of mathematical monographs, vol.140, Am Math Soc Providence, RI, 1994. [18] Wang ZC, Li WT, Ruan S. Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays. J Differ Eq 2006;222:185–232. [19] Wang ZC, Li WT, Ruan S. Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay. J Differ Eq 2007;238:153–200. [20] Wu J, Zou X. Traveling wave fronts of reaction-diffusion systems with delay. J Dyn Differ Eqn 2001;13:651–87. [21] Xu D, Zhao XQ. Erratum to Bistable waves in an epidemic model. J Dyn Differ Eqn 2005;17:219–47. [22] Xu R, Chaplain MAJ, Davidson FA. Traveling wave and convergence in stage-structured reaction-diffusion competitive models with nonlocal delays. Chaos, Solitons & Fractals 2006;30:974–92. [23] Zhang S, Chen L. Chaotic behavior of a chemostat model with Beddington–DeAngelis functional response and periodically impulsive invasion. Chaos, Solitons & Fractals 2006;29:474–82. [24] Zhao XQ. Dynamical systems in population biology. Verlag, New York: Springer; 2003.
Chaos, Solitons and Fractals 40 (2009) 1229–1239 www.elsevier.com/locate/chaos
Global asymptotic stability of bistable traveling fronts in reaction-diffusion systems and their applications to biological models Shi-Liang Wu a
a,b
, Wan-Tong Li
a,*,1
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China b Department of Applied Mathematics, Xidian University, Xi’an, Shanxi 710071, People’s Republic of China Accepted 31 August 2007
Abstract This paper deals with the global asymptotic stability and uniqueness (up to translation) of bistable traveling fronts in a class of reaction-diffusion systems. The known results do not apply in solving these problems because the reaction terms do not satisfy the required monotone condition. To overcome the difficulty, a weak monotone condition is proposed for the reaction terms, which is called interval monotone condition. Under such a weak monotone condition, the existence and comparison theorem of solutions is first established for reaction-diffusion systems on R by appealing to the theory of abstract differential equations. The global asymptotic stability and uniqueness (up to translation) of bistable traveling fronts are then proved by the elementary super- and sub-solution comparison and squeezing methods for nonlinear evolution equations. Finally, these abstract results are applied to a two species competition-diffusion model and a system modeling man–environment–man epidemics. 2007 Elsevier Ltd. All rights reserved.
1. Introduction The theory of traveling wave solutions of parabolic differential equations is one of the fastest developing areas of modern mathematics and has attracted much attention due to its significant nature in biology, chemistry, epidemiology and physics (see, e.g., [1,4,5,8–10,16–18,20,22,23]). Traveling wave solutions are solutions of special type and can be usually characterized as solutions invariant with respect to transition in space. From the physical point of view, traveling waves describe transition processes. These transition processes (from one equilibrium to another) usually ‘‘forget’’ their initial conditions and the properties of medium itself. Among the basic questions in the theory of traveling waves, the global asymptotic stability of traveling wave solutions is an important one. For scalar reaction-diffusion equations, there are many nice results, see e.g., Chen [3] for nonlocal evolution equations, Schaaf [14], Smith and Zhao [15] and Wang et al. [19] for delayed or nonlocal *
1
Corresponding author. E-mail address: [email protected] (W.-T. Li). Supported by NSFC (No. 10571078) and NSF of Gansu Province of China (No. 3ZS061-A25-001).
0960-0779/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.08.075
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reaction-diffusion equations. For systems of reaction-diffusion equations, see [13,17,21] and cited references. In particular, Volpert et al. [17] considered a reaction-diffusion system of the form oui ðx; tÞ o2 ui ðx; tÞ þ F i ðu1 ðx; tÞ; . . . ; un ðx; tÞÞ; ¼ di ot ox2
ð1:1Þ
where x 2 R, t > 0, D = diag(d1, . . . , dn), di > 0, i = 1, . . . , n, and (A1) F ðuÞ ¼ ðF 1 ðuÞ; . . . ; F n ðuÞÞ 2 C 1 ðRn ; Rn Þ satisfies F(w±) = 0; (A2) all the eigenvalues of the matrixes (ojFi(w±)) lie in the left half-plane. In the order interval (w, w+), the only equilibrium w is such that the matrix (ojFi(w)) has at least one eigenvalue in the right half-plane. If the reaction term F satisfies the so-called monotone condition oF i P 0; 1 6 i–j 6 n; ouj
uj 2 R;
ð1:2Þ
then Volpert et al. [17, Theorem 5.6.1] proved that if the initial datum is monotone in x and satisfies decay conditions at ±1, then the solution u(x, t) converges exponentially to a traveling wave front connecting the stable equilibria w and w+. Furthermore, Roquejoffre et al. [13] removed the monotonicity assumption on the initial datum, by operator semigroup precompactness and quasiconvergence, and showed that any solution u(x, t) of the Cauchy problem for system (1.1) which initially has these two stable equilibria w± as spatial limits will converge to a traveling wave front connecting the stable equilibria w and w+. However, the reaction term in a model system arising from a practical problem may not satisfy the monotone condition (1.2). A typical and important example is the two competitive species model (see Gardner [6], Huang [7] and Li et al. [9]): 8 < ovðx;tÞ ¼ d 1 o2 vðx;tÞ þ vMðv; wÞ; ot ox2 ð1:3Þ : owðx;tÞ ¼ d o2 wðx;tÞ þ wN ðv; wÞ; ot
2
ox2
where x 2 R, t > 0, v(x, t) and w(x, t) are the densities of the two species, d1 > 0 and d2 > 0 are the diffusion constants of the species v and w, respectively, and (B1) the partial derivatives Mw(v, w) < 0 and Nv(v, w) < 0 for v P 0 and w P 0; (B2) (1.3) has four spatially uniform equilibria: (0, 0), (v*, 0), (0, w*) and (v0, w0), where 0 < v0 < v* and 0 < w0 < w*. Furthermore, all four equilibria are hyperbolic with (v*, 0), (0, w*) being stable and (0, 0), (v0, w0) being unstable; (B3) the zero sets of M and N intersect exactly once in the positive quadrant at (v0, w0) and are the graphs of functions w = k(v) and v = l(w), respectively. Thus, it is worthwhile to further explore this topic for non-monotone systems, and this constitutes the purpose of this paper. In order to focus on the mathematical ideas and for the sake of simplicity, we consider a reaction-diffusion system of two equations, that is 8 < ou1 ðx;tÞ ¼ d 1 o2 u1 ðx;tÞ þ f1 ðu1 ðx; tÞ; u2 ðx; tÞÞ; ot ox2 ð1:4Þ : ou2 ðx;tÞ ¼ d o2 u2 ðx;tÞ þ f ðu ðx; tÞ; u ðx; tÞÞ; ot
2
ox2
2
1
2
where x 2 R, t > 0, di > 0, i = 1, 2, and the function f(u, v) = (f1(u, v),f2(u, v)) satisfies the following assumptions: (C1) f(0, 0) = f(K1, K2) = 0, f 2 C 2 ðI 2 ; R2 Þ for an open interval I R with [0, +1) I, where Ki are two positive constants, i = 1, 2; (C2) all the eigenvalues of the matrixes (oj fi (0, 0)) and (oj fi (K1, K2)) lie in the left half-plane. In the order interval (E, E+), where E = (0, 0) and E+ = (K1, K2) the only equilibrium E0 = (a1, a2) with 0 < ai < Ki, i = 1, 2 is such that the matrix (ojfi(a1, a2)) has at least one eigenvalue in the right half-plane. Since the results of Roquejoffre et al. [13], Volpert et al. [17] and Xu and Zhao [21] do not apply to the reactiondiffusion systems (1.3) and (1.4), we must search for new techniques that can be applied to (1.4), at least for (1.3).
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To overcome the difficulty, we propose a weak condition for the reaction terms, which is called interval-monotone condition (C3) ojfi(u, v) > 0 for (u, v) 2 [0, K1] · [0, K2], 1 6 i 5 j 6 2. Under the conditions (C1), (C2) and (C3), we first establish the existence and comparison theorem of solutions for the interval monotone reaction-diffusion systems (1.4) on R by appealing to the theory of abstract differential equations developed in [11] and properties of the analytic semigroup generated by the one-dimensional Laplacian operator on the Banach space of all bounded and uniformly continuous functions on R. In order to prove global asymptotic stability of traveling wave fronts, we shall borrow a squeezing technique due to Chen [3] for nonlocal evolution equations, which is similar in spirit to a ‘‘contracting rectangle’’ approach developed in [12]. By the elementary super- and subsolution comparison method and the established global asymptotic stability of traveling wave fronts, the Liapunov stability and uniqueness of traveling wave fronts connecting the stable equilibria E and E+ are then proved. Finally, these abstract results are applied to the competition-diffusion model (1.3) and a system modelling man–environment–man epidemics. Here a traveling wave front of (1.4) always refers to a pair (U, c), where U = U(n) is a monotone function on R and c > 0 is a constant, such that u(x, t): = U(x ct) is a solution of (1.4) and lim U ðnÞ ¼ E ;
n!1
lim U ðnÞ ¼ Eþ :
n!þ1
ð1:5Þ
The rest of this paper is organized as follows. In Section 2, we establish an existence and comparison theorem for interval-monotone reaction-diffusion systems on R and then prove that the traveling wave front is globally asymptotically stable and unique up to translation. In Section 3, we apply our results to the above mentioned model and a system modelling man–environment–man epidemics, and obtain some new results on global asymptotic stability and uniqueness of traveling wave fronts.
2. Main results In this paper, we will use the usual notations for the standard ordering in R2 . That is, for u = (u1, u2) and v = (v1, v2), we denote u 6 v if ui 6 vi, i = 1, 2, and u < v if ui 6 vi but u 5 v. In particular, we denote u v if ui 6 vi but ui 5 vi, i = 1, 2. If u 6 v, we also denote ðu; v ¼ fw 2 R2 : u < w 6 vg, ½u; vÞ ¼ fw 2 R2 : u 6 w < vg, and ½u; v ¼ fw 2 R2 : u 6 w 6 vg. Let kÆk denote the Euclidean norm in R2 . In order to study the asymptotic stability of traveling wave fronts of (1.4), we first study the initial value problem 8 ou1 o2 u1 > > < ot ¼ d 1 ox2 þ f1 ðu1 ðx; tÞ; u2 ðx; tÞÞ; ou2 o2 u2 ð2:1Þ > ot ¼ d 2 ox2 þ f2 ðu1 ðx; tÞ; u2 ðx; tÞÞ; > : ui ðx; 0Þ ¼ /i ðxÞ; i ¼ 1; 2; where x 2 R, t > 0. Let X ¼ BUCðR; R2 Þ be the Banach space of all bounded and uniformly continuous functions from R into R2 with the supremum norm kÆkX. Let X þ ¼ fu 2 X : uðxÞ P 0; x 2 Rg. It is easy to see that X+ is a closed cone of X and X is a Banach lattice under the partial ordering induced by X+. For any /1, /2 2 X, we write /1 6 X/2 if /1 /2 2 X+, /1 < X/2 if /1 /2 2 X þ n f^0g, /1 X/2 if /1 /2 2 int X+, where ^ a denote a constant vector function on R taking the vector a. For /1, /2 2 X with /1 6 X/2, we denote [/1, /2]X = {/ 2 X:/1 6 X/ 6 X/2}. Consider the uncoupled linear system ( oui ¼ d i Dui ; x 2 R; t > 0; ot ð2:2Þ ui ðx; 0Þ ¼ /i ðxÞ; x 2 R; where i = 1, 2. Define the family of linear operator T(t) = (T1(t),T2(t)) on X by T ðtÞ/ ¼ ðT 1 ðtÞ/1 ; T 2 ðtÞ/2 Þ
for each / ¼ ð/1 ; /2 Þ 2 X ;
ð2:3Þ
where T i ðtÞ/i ¼ ui ð; tÞði ¼ 1; 2Þ, and u is the solution of (2.2). It is easy to see that T(t):X ! X is an analytic semigroup on X with T(t)X+ X+ for all t > 0. In fact, by the explicit expression of solution of the heat Eq. (2.2), we have
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1 T i ðtÞ/i ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi 4pd i t x 2 R;
Z
þ1
exp
1
! ðx yÞ2 /i ðyÞdy; 4d i t
/i ðÞ 2 BUCðR; RÞ;
t > 0;
i ¼ 1; 2:
b ¼ f/ 2 X : 0 6 /ðxÞ 6 K; x 2 Rg, where 0 = (0, 0), K = (K1, K2), define F(/)(x) = For any / ¼ ð/1 ; /2 Þ 2 ½^0; K X (F1(/)(x), F2(/)(x)) by F i ð/ÞðxÞ ¼ fi ð/1 ðxÞ; /2 ðxÞÞ;
i ¼ 1; 2:
b ! X is globally Lipschitz continuous. Then F(/) 2 X and F : / 2 ½^0; K X ^ K b , (2.1) has a unique, bounded and nonnegative solution u(x, t, /) on [0, 1), and the Lemma 2.1. For any / 2 ½0; X b 0; K solution semiform of (2.1) is monotone. Moreover, u(x, t, /1) u(x, t, /2) for t > 0 and x 2 R whenever /1 ; /2 2 ½^ X 1 2 with / < X/ , i.e., the solution semiform of (2.1) is strongly monotone. Proof. Let T(t) = (T1(t), T2(t)), then T(t) is a linear semigroup on X. Under an abstract setting in [11], a mild solution of (2.1) is a solution to its associated integral equation ( Rt uðtÞ ¼ T ðtÞuð0Þ þ 0 T ðt sÞF ðuðsÞÞds; t > 0; ð2:4Þ b : uð0Þ ¼ / 2 ½^0; K X
b ! X is globally Lipschitz continuous. Let As aforementioned, F : ½^0; K X Li ¼ max joi fi ðu; vÞj; ðu;vÞ2½^ 0; b K
i ¼ 1; 2:
b in the sense that We claim that F is also monotone on ½^0; K X 1 þ lim distðw / þ h½F ðwÞ F ð/Þ; X Þ ¼ 0; h!0þ h ^ K b with w P /. Indeed, for all w; / 2 ½0; X
F 1 ðwÞ F 1 ð/Þ ¼ f1 ðw1 ðÞ; w2 ðÞÞ f1 ð/1 ðÞ; /2 ðÞÞ P f1 ðw1 ðÞ; /2 ðÞÞ f1 ð/1 ðÞ; /2 ðÞÞ P L1 ðw1 /1 Þ
in X ;
and hence, for any d1 > 0 such that L1d1 < 1, w1 /1 þ d1 ½F 1 ðwÞ F 1 ð/Þ P ð1 L1 d1 Þðw1 /1 Þ P 0 in X : Similarly, we have for any d2 > 0 such that L2d2 < 1, w2 /2 þ d2 ½F 2 ðwÞ F 2 ð/Þ P ð1 L2 d2 Þðw2 /2 Þ P 0 in X : Thus, the existence and uniqueness of u(x, t, /) follows from [11, Corollary 5](take delay as zero) with S(t, s) = T(t, s), t P s P 0, and B(t, /) = F(/). Moreover, by a semigroup theory argument given in proof of [11, Theorem 1], it follows that u(x, t, /) is a classical solution for t > 0. Note that [11, Corollary 5] also implies that the comparison principle holds for (2.1), and it follows that (2.1) defines b . a monotone solution semigroup on ½^0; K X 1 2 i 2 b Suppose that /1 ; /2 2 ½^0; K X with / < X/ . Then u(x, t, / ) P 0 for x 2 R; t P 0. Let u(x, t) = u(x, t, / ) 1 1 2 u(x, t, / ). Then u(x, t) P 0 for x 2 R; t P 0. We shall show that ui(x, t) > 0 for x 2 R; t > 0, i = 1, 2. Since / < X/ , i.e., /1 6 X/2 and /1 f /2. Without loss of generality, we assume that /21 ðÞX/11 ðÞ and /22 ðÞ /12 ðÞ, i.e., u1(Æ, 0) f 0 and u2(Æ, 0) 0. Note that in the first component u1(x, t) of u(x, t) satisfies u1;t ¼ d 1 u1;xx þ f1 ðu1 ðx; t; /2 Þ; u2 ðx; t; /2 ÞÞ f1 ðu1 ðx; t; /1 Þ; u2 ðx; t; /1 ÞÞ P d 1 u1;xx þ f1 ðu1 ðx; t; /2 Þ; u2 ðx; t; /2 ÞÞ f1 ðu1 ðx; t; /1 Þ; u2 ðx; t; /2 ÞÞ P d 1 u1;xx L1 u1 : By the strict positive theorem [17, Theorem 5.5.4], we have u1(x, t) > 0 for all x 2 R; t > 0. Suppose to the contrary that there exists a point (x*, t*) with x 2 R and t* > 0 such that u2(x*, t*) = 0. Then u2,t(x*, t*) 6 0 and u2,xx(x*, t*) P 0. Since u2(x, t) satisfies u2;t ¼ d 2 u2;xx þ f2 ðu1 ðx; t; /2 Þ; u2 ðx; t; /2 ÞÞ f2 ðu1 ðx; t; /1 Þ; u2 ðx; t; /1 ÞÞ ¼ d 2 u2;xx þ o1 f2 ðgÞu1 ðx; tÞ þ o2 f2 ðgÞu2 ðx; tÞ; where g = su(x, t, /2) + (1 s)u(x, t, /1) 2 [0, K], s 2 (0, 1). Since o1f2(g) > 0, we conclude that 0 P u2;t jðx ;t Þ ¼ d 2 u2;xx þ o1 f2 ðgÞu1 ðx; tÞ þ o2 f2 ðgÞu2 ðx; tÞjðx ;t Þ > 0 which is a contradiction. Thus u2(x, t) > 0 for x 2 R; t > 0. This completes the proof.
h
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1233
Suppose that u(x ct) = (u1(x ct),u2(x ct)) is a strictly increasing traveling wave solution of (1.4) connecting E and E+. Rescale (1.4) in a moving coordinate frame, i.e., in terms of variables z = x ct and t, it becomes w1;t ðz; tÞ ¼ cw1;z ðz; tÞ þ d 1 w1;zz ðz; tÞ þ f1 ðw1 ðz; tÞ; w2 ðz; tÞÞ; ð2:5Þ w2;t ðz; tÞ ¼ cw2;z ðz; tÞ þ d 2 w2;zz ðz; tÞ þ f2 ðw1 ðz; tÞ; w2 ðz; tÞÞ: Then u(z) is an equilibrium solution of system (2.5). In what follows, we denote by w(z, t, /) = (w1(z, t, /),w2(z, t, /)) the b . Clearly, the solution u(x, t, /) of (2.1) is given by u(x, t, /) = w(x ct, t, /). As solution of (2.5) with w0 ¼ / 2 ½^0; K X noted before, the comparison principle holds for (2.1) and hence for (2.5). For convenience, we set H 1 ðw1 ; w2 Þ :¼ w1;t ðz; tÞ cw1;z ðz; tÞ d 1 w1;zz ðz; tÞ f1 ðw1 ðz; tÞ; w2 ðz; tÞÞ; H 2 ðw1 ; w2 Þ :¼ w2;t ðz; tÞ cw2;z ðz; tÞ d 2 w2;zz ðz; tÞ f2 ðw1 ðz; tÞ; w2 ðz; tÞÞ: b and Lemma 2.2. Assume that / ¼ ð/1 ; /2 Þ 2 ½^0; K X ð/Þ :¼ max lim sup k/ðzÞ E k; lim sup k/ðzÞ Eþ k : z!1
ð2:6Þ
z!þ1
is small enough. Then for any e ¼ ðe1 ; e2 Þ 2 R2 with 0 e, there exist z ¼ zðe; /Þ > 0 and a large time t ¼ tðe; /Þ such that uðz zÞ e 6 wðz; t; /Þ 6 uðz þ zÞ þ e;
8z 2 R:
Proof. From the assumptions (C1), (C2) and (C3), it is easy to see that E± are asymptotic stable equilibria of the following ordinary differential system
z01 ðtÞ ¼ f1 ðz1 ðtÞ; z2 ðtÞÞ; z02 ðtÞ ¼ f2 ðz1 ðtÞ; z2 ðtÞÞ;
t > 0; t > 0:
ð2:7Þ
e E0 , and the solution z(t, k±) of system (2.7) with Then there exists v0 > 0 is small enough such that E < E þ v0~ e approach E± as t ! 1, where ~ zð0; k Þ ¼ k :¼ E þ v0~ e ¼ ð1; 1Þ. Let v±(t) = z(0, k±). Then there exists l P 0 such that 0 6 v±(t) 6 l. b and (/) is small enough, without loss of generality, we assume that /(z) 6 k for z 6 0, where Since / 2 ½^0; K X e. Take k ¼ E þ v0~ L ¼ supfjoij fj ðuÞj : u 2 ½0; l; 1 6 i; j 6 2g: Let
)
(
2 1 þ 1 ðvþ j ðtÞ vj ðtÞÞ þ ~c > c þ maxfd i g þ L sup : t 2 ½0; 1Þ; 1 6 i–j 6 2 ðtÞÞ þ ðv ðtÞ v ðtÞÞ:þ ðvi ðtÞ v i j j i¼1;2 2 2 vþ i ðtÞ vi ðtÞ
be a fixed number. Define 1 sþT fðsÞ ¼ 1 þ tanh ; 2 2
s2R
and vðz; tÞ ¼ vþ ðtÞfðz þ ~ctÞ þ v ðtÞð1 fðz þ ~ctÞÞ;
z 2 R;
t P 0;
where T > 0 is large enough such that f(0)E+ P E+ v0. It is easy to see that f 0 = f(1 f), f00 = f 0 (1 2f) and v(z, 0) P /(z) for z 2 R. We further show that v(z, t) is a super-solution of (2.5). Indeed, by Taylor’s expansion, we have H 1 ðvðz; tÞÞ ¼ v1;t ðz; tÞ cv1;z ðz; tÞ d 1 v1;zz ðz; tÞ f1 ðvðz; tÞÞ ¼ ff1 ðvþ ðtÞÞ þ ð1 fÞf1 ðv ðtÞÞ f1 ðfvþ ðtÞ þ þ ð1 fÞv ðtÞÞ þ fð1 fÞ½ð~c cÞðvþ 1 ðtÞ v1 ðtÞÞ d 1 ð1 2fÞðv1 ðtÞ v1 ðtÞÞ 1 1 2 2 þ þ ¼ fð1 fÞðvþ 1 ðtÞ v1 ðtÞÞ o11 f1 ðh1 Þ þ fð1 fÞðv2 ðtÞ v2 ðtÞÞ o22 f1 ðh1 Þ þ fð1 fÞðv1 ðtÞ v1 ðtÞÞ 2 2 þ c cÞðvþ
ðvþ 2 ðtÞ v2 ðtÞÞo12 f1 ðh1 Þ þ fð1 fÞ½ð~ 1 ðtÞ v1 ðtÞÞ d 1 ð1 2fÞðv1 ðtÞ v1 ðtÞÞ #) ( " 2 1 þ 1 ðvþ þ þ 2 ðtÞ v2 ðtÞÞ P0 P fð1 fÞðv1 ðtÞ v1 ðtÞÞ ~c c d 1 L ðv1 ðtÞ v1 ðtÞÞ þ ðv2 ðtÞ v2 ðtÞÞ þ 2 2 vþ 1 ðtÞ v1 ðtÞ
and
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H 2 ðvðz; tÞÞ ¼ v2;t ðz; tÞ cv2;z ðz; tÞ d 2 v2;zz ðz; tÞ f2 ðvðz; tÞÞ ( 1 þ ðtÞ v P fð1 fÞðv2 ðtÞ v2 ðtÞÞ ~c c d 2 þ ðvþ 2 ðtÞÞo22 f2 ðh2 Þ: 2 2 ) 2 1 ðvþ þ 1 ðtÞ v1 ðtÞÞ o11 f2 ðh2 Þ P 0; þ ðv1 ðtÞ v1 ðtÞÞo12 f2 ðh2 Þ þ 2 vþ 2 ðtÞ v2 ðtÞ where h1, h2 2 (v-(t),v+(t)) [0, l]. Thus v(z, t) is a super-solution of system (2.5). Therefore, by the comparison principle we have w(z, t, /) 6 v(z, t) for z 2 R, t > 0. Note that limt!1v±(t) = E±. It then follows that for any e ¼ ðe1 ; e2 Þ 2 R2 with 0 e, there exist z ¼ zðe; /Þ > 0 and a large time t ¼ tðe; /Þ such that wðz; t; /Þ 6 uðz þ zÞ þ e; 8z 2 R. Indeed, since limt!1v±(t) = E±, there exists t > 0 large enough such that 1 1 kvþ ðtÞ Eþ k 6 minfei g; kv ðtÞ E k 6 minfei g: 4 i¼1;2 4 i¼1;2 By virtue of limz!±1u(z) = E± and lims!1f(s) = 0, there exists M > 0 such that e Eþ uðzÞ þ for z P M; 4 and e ei for z 6 M: E uðzÞ þ ; fðz þ ~ctÞ < min i¼1;2 4li 4 Moreover, since u(Æ) is monotone and limz!1u(z) = E+, there exists z > 0 such that Eþ uðz þ zÞ þ 4e for M 6 z 6 M. Therefore, it is easy to see that wðz; t; /Þ 6 uðz þ zÞ þ eforallz 2 R: The other estimates on the lower bound of the solution are similar and omitted. This completes the proof.
h
±
Note that all eigenvalues of the Jacobian matrixes (ojfi(E )) lie in the left half-plane. Denote by q ¼ ðq 1 ; q2 Þ the Þ, where l be constants such that positive eigenvectors corresponding the principle eigenvalues of the matrixes ðl ij ij oj fi ðE Þ < l , 1 6 i, j 6 2 and (0, 0) is asymptotic stable in both of the linear system 8 ij < dr1 ðtÞ ¼ l r þ l r ; t > 0; 11 1 12 2 dt ð2:8Þ : dr 2 ðtÞ ¼ l r þ l r ; t > 0: 21 1 22 2 dt
Let q1(n),q2(n) be smooth positive functions such that q(n) = (q1(n),q2(n)) ! q± in C2 -topology as n ! 1. Lemma 2.3. There exist two positive numbers r and 10 such that for any 1 P 10, n0 2 R, and 2 (0, 0(1)), v ðz; tÞ ¼ uðz n0 1ð1 ert ÞÞ qðz n0 Þert ;
z 2 R;
t>0
are super- and sub-solutions of systems (2.5), respectively. Proof. Clearly, there exist d, k > 0 such that for ku E k 6 d; u 2 R2 ; oj fi ðuÞ 6 l ij 2 X
l ij qj < kqi
for q ¼ ðq1 ; q2 Þ 2 R2þ
1 6 i;
ð2:9Þ
j 6 2;
with kq q k 6 d;
i ¼ 1; 2:
ð2:10Þ
j¼1
Take B1 > 0 such that kqðgÞk; kq0 ðgÞk; kq00 ðgÞk 6 B1 ±
forall g 2 R:
±
Since u(n) ! E , q(n) ! q , q 0 (n),q00 (n) ! 0 as n ! ±1, there exist 0 < 1 < 1,M > 0 such that k c1 maxfd i g1 > 0; i¼1;2
jq0i ðgÞj; jq00i ðgÞj þ
6 1 qi ðgÞ
for jgj P M 1;
i ¼ 1; 2;
kqðgÞ q k 6 d for g P M 1; kqðgÞ q k 6 d for g 6 M þ 1; d kuðnÞ þ qðgÞ Eþ k 6 for 2 ð0; 1 ; n P M 1; g P M 1; 2 d kuðnÞ þ qðgÞ E k 6 for 2 ð0; 1 ; n 6 M þ 1; g 6 M þ 1: 2
ð2:11Þ ð2:12Þ
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Define eg B2 ¼ supfjoj fi ðuÞj : u 2 ½E d~ e; Eþ þ B1~
and
B3 ¼ inf ku0 ðnÞk: knk6M
Choosing r > 0 sufficiently small such that r 6 k c1 maxi=1, 2{di}. Take B1 1 1 P 10 ¼ ðr þ c þ maxfd i g þ B2 Þ; 0 ¼ min 1 ; : i¼1;2 B3 r 1 Set n = z + n0 + 1(1 ert), g = z + n0 and q = ert. Then, for t P 0, we have þ þ þ H i ðvþ ðz; tÞÞ ¼ vþ i;t ðz; tÞ cvi;z ðz; tÞ d i vi;zz ðz; tÞ fi ðv ðz; tÞÞ
¼ fi ðuðnÞÞ fi ðuðnÞ þ qðgÞqÞ þ 1rqu0i ðnÞ ðrqi ðgÞ þ cq0i ðgÞ þ d i q00i ðgÞÞq: We distinguish among three cases: Case (i): |n| 6 M. In this case, Z 1 X 2 q qj oj fi ðu þ sqqÞds P qB1 B2 : fi ðuÞ fi ðu þ qqÞ ¼ 0
j¼1
Therefore, H i ðvþ ðz; tÞÞ P qB1 B2 B1 qðr þ c þ d i Þ þ B3 1rq P 0: Case (ii): n P M. Since n g 6 1 6 1, n > g P M 1. Thus, d kuðnÞ þ sqqðgÞ Eþ k 6 ; 2
kqðgÞ qþ k 6 d; 8s 2 ð0; 1Þ:
Therefore, by (2.9)–(2.12), we have
H i ðvþ ðz; tÞÞ ¼ fi ðuÞ fi ðu þ qqÞ þ 1rqu0i rqi þ cq0i þ d i q00i q Z 1 X 2 q qj oj fi ðu þ sqqÞds þ 1rqu0i rqi þ cq0i þ d i q00i q ¼ 0
P q
j¼1 2 X
0 0 00 lþ ij qj þ 1rqui rqi þ cqi þ d i qi q P ðk r c1 d i 1 Þqqi P 0:
j¼1
Case (iii): n 6 M. Clearly, g < n < M + 1. Then d kuðnÞ þ sqqðgÞ E k 6 ; 2
kqðgÞ q k 6 d; 8s 2 ð0; 1Þ:
Therefore, again by (2.9)–(2.12), we have Z 1 X 2 q qj oj fi ðu þ sqqÞds þ 1rqu0i rqi þ cq0i þ d i q00i q H i ðvþ ðz; tÞÞ ¼ 0
P q
j¼1 2 X
0 0 00 l ij qj þ 1rqui rqi þ cqi þ d i qi q P ðk r c1 d i 1 Þqqi P 0:
j¼1
Combining cases (i)–(iii), we have Ni(v+(z, t)) P 0 for all 2 (0, 0) and t P 0, i = 1, 2. Thus v+(z, t) is a supersolution of (2.5). In a similar way, we can show that v(z, t) is a sub-solution of (2.5). This completes the proof. h Lemma 2.4. The wave profile u(z) is a Liapunov stable equilibrium of (2.5). Proof. Let 0 and v±(z, t, ) be as in Lemma 2.3 with 1 = 10 and n0 = 0. Then, for z 2 R, t P 0, 2 (0, 0), kv ðz; t; Þ uðzÞk 6 kv ðz; t; Þ uðz 10 ð1 ert ÞÞk þ kuðz 10 ð1 ert ÞÞ uðzÞk ¼ kqðzÞkert þ 10 ku0 ðzÞ s10 ð1 ert Þkð1 ert Þ 6 K;
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b with k/ ukX 6 d, where K > 0 is independent of . For any 2 (0, 0), take d ¼ inf z2R qðzÞ. Thus, for any / 2 ½^ 0; K X we have v ðz; 0; Þ ¼ uðzÞ qðzÞ 6 uðzÞ d 6 /ðzÞ 6 uðzÞ þ d 6 uðzÞ þ qðzÞ ¼ vþ ðz; 0; Þ; 8z 2 R: Then the comparison principle implies that v ðz; t; Þ 6 wðz; t; /Þ 6 vþ ðz; t; Þ; z 2 R;
t P 0:
Therefore, kw(Æ, t, /) u(Æ)k < K for t P 0. This completes the proof.
h
To prove the asymptotic stability of traveling waves, we need the following convergence result for monotone semiflows (see [24]). Lemma 2.5. Let U be a closed convex subset of an ordered Banach space v, and U(t):U ! U be a monotone semiform. Assume that there exists a monotone homeomorphism h from [0, 1] onto a subset of U such that (i) for each s 2 [0, 1], h(s) is a stable equilibrium for U(t):U ! U; (ii) each orbit of U(t) in [h(0),h(1)]v is precompact; (iii) one of the following two properties holds: (a) if h(s0) < vx(/) for some s0 2 [0, 1) and / 2 [h(0),h(1)]v, then there exists s1 2 (s0,1) such that h(s1) 6 vx(/); (b) if x(/) < vh(r1) for some r1 2 [0, 1) and / 2 [h(0),h(1)]v, then there exists r0 2 (0, r1) such that x(/) 6 vh(r0). Then for any precompact orbit c+(/0) of U(t) in U with x(/0) \ [h(0),h(1)]v 5 B, there exists s* 2 [0, 1] such that x(/0) = h(s*). Now we state the main results of this section as follows. Theorem 2.6. Assume that (C1), (C2) and (C3) hold, u(n) is a traveling wave front of system (1.4) and u(x, t, /) is the b . Then for any / 2 ½^ b such that (/) (defined by (2.6)) is small enough, there solution of (2.1) with u0 ¼ / 2 ½^0; K 0; K X X exists s/ 2 R such that lim sup kuðx; t; /Þ uðx ct þ s/ Þk ¼ 0: t!þ1 x2R
Proof. Using Lemmas 2.1–2.5, the proof is similar to that of Theorem 3.1 in Xu and Zhao [21] and is omitted.
h
~ ðnÞ ¼ u ~ ðx ~ctÞ is a traveling wave solution of (1.4) satisfying (1.5) Theorem 2.7. Assume that (C1), (C2) and (C3) hold, u ~ ðÞ < K. Then u ~ ðÞ ¼ uð þ n0 Þ for some n0 ¼ n0 ð~ and 0 < u uÞ 2 R. ~ ðÞ < K, by Theorem 2.6, there exists n0 ¼ n0 ð~ ~ ðnÞ ¼ E , 0 < u uÞ 2 R such that Proof. Since limn! 1 u uðx ~ctÞ uðx ct þ n0 Þk ¼ 0: lim sup k~ t!þ1 x2R
Let z = x ct, we have lim sup k~ uð þ ðc ~cÞtÞ uð þ n0 Þk ¼ 0: t!þ1
~ ðnÞ ! E as n ! ±1, we must have ~c ¼ c, and thus u ~ ðÞ ¼ uð þ n0 Þ. This Since u(Æ) is strictly increasing on R and u completes the proof. h
3. Applications In this section, we shall apply our abstract results to a competition-diffusion model and a system modelling man– environment–man epidemics. 3.1. A competition model We consider the competition model of two species (1.3), that is, 8 < ovðx;tÞ ¼ d 1 o2 vðx;tÞ þ vMðv; wÞ; ot
: owðx;tÞ ¼ d ot
ox2
o2 wðx;tÞ 2 ox2
þ wN ðv; wÞ;
ð3:1Þ
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1237
where x 2 R, t > 0, di > 0, i = 1, 2. Under the assumptions (B1)-(B3), the existence of a monotone traveling wave solution of (3.1) connecting the two stable equilibria is established in Gardner [6] with the use of topological degree theory. In [7], Huang further proved that this traveling wave front is unique up to translation by using a homotopy approach incorporated with the Lypunov–Schmidt method. By the method developed in this paper we shall show that the traveling wave front in fact is global asymptotic stable. To do so, let us first substitute u1 = v and u2 = w* w into (3.1), then system (3.1) becomes 8 < ou1 ðx;tÞ ¼ d 1 o2 u1 ðx;tÞ þ h1 ðu1 ; u2 Þ; ot ox2 ð3:2Þ : ou2 ðx;tÞ ¼ d o2 u2 ðx;tÞ þ h ðu ; u Þ; ot
2
ox2
2
1
2
where h1 ðu1 ; u2 Þ ¼ u1 Mðu1 ; w u2 Þandh2 ðu1 ; u2 Þ ¼ ðu2 w ÞN ðu1 ; w u2 Þ: Obviously, if (B1) and (B2) hold, then we have ojhi(u) > 0 for u 2 [0,v*] · [0,w*], 1 6 i 5 j 6 2, and (3.2) has four spatially uniform equilibria 0: = (0, 0), (v0, w* w0), (0, w*) and K = (v*, w*). Furthermore, (0, 0), (v*, w*) are stable and (v0, w* w0), (0, w*) are unstable. Although (3.2) has a extra unstable equilibrium (0, w*) in (0, K), with a slight modification of the method used in this paper, we are still able to conclude that the bistable traveling wave solution of (3.2) is globally asymptotically stable, i.e., the following result holds. Theorem 3.1. Assume that (B1) and (B2) hold, u(n) is a traveling wave front of (3.2) and u(x, t, /) is the solution of (3.2) b . Then for any / 2 ½^0; K b such that (/) (defined by (2.6)) is small enough, there exists s/ 2 R such with u0 ¼ / 2 ½^0; K X X that lim sup kuðx; t; /Þ uðx ct þ s/ Þk ¼ 0:
t!þ1 x2R
Consequently, the bistable traveling wave solution of (3.1) connecting (0, w*) and (v*, 0) is also globally asymptotically stable. 3.2. An epidemic model In this subsection, we consider an epidemic model which is a monotone system, and establish the global asymptotic stability and uniqueness (up to translation) of bistable traveling wave front of the model under suitable conditions which are less restrictive than that in Roquejoffre et al. [13]. More precisely, we consider the following reaction-diffusion system (see [2]) 8 < ou1 ðx;tÞ ¼ d 1 o2 u1 ðx;tÞ a11 u1 ðx; tÞ þ a12 u2 ðx; tÞ; ot ox2 ð3:3Þ : ou2 ðx;tÞ ¼ d o2 u2 ðx;tÞ a u ðx; tÞ þ gðu ðx; tÞÞ; ot
2
ox2
22 2
1
where u1(x, t) and u2(x, t), respectively, represent the average concentration of bacteria and the infective human population at a point x in the habitat at time t, 1/a11 is the mean lifetime of the agent in the environment, 1/a22 is the multiplicative factor of the mean infectious agent due to the human populations, g(v) is the infection rate of human under the assumption that total susceptible human population is constant during the evolution of the epidemic, and d1 P 0 and d2 P 0 denote the diffusion constants of the bacteria and the infective human, respectively. System (3.3) has been widely studied. For the bistable case, i.e., the function g is sigma-shaped (defined concisely by properties (H1) and (H2) in the following), the existence of bistable traveling wave fronts of (3.3) is a consequence of Volpert et al. [17, Theorem 3.1.1] when di > 0, i = 1, 2. By using phase space analysis, monotone semiflows approach and spectrum method, Xu and Zhao [21] proved the existence, uniqueness and global asymptotic stability of bistable traveling wave fronts of (3.3) when d1 > 0 and d2 = 0. We will establish the global asymptotic stability and uniqueness of bistable traveling wave fronts of (3.3) when d1 > 0 and d2 > 0. Similar to Xu and Zhao [21], we assume that the function g satisfies (H1) g 2 C2(I), g(0) = 0, g 0 (0) P 0, g 0 (z) > 0, "z > 0, limz!1g(z) = 1, and there is a 1 > 0 such that g00 (z) > 0 for z 2 (0, 1) and g00 (z) < 0 for z > 1. (H2) g 0 (0) < c = a11a22/a12 < ccrit, where c = ccrit such that the equation g(z) = cz has one and only one positive root.
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Under the assumptions (H1) and (H2), (3.3) has three equilibrium: E = (0, 0), E0 = (a, a11a/a12) and E+ = (b, a11b/ a12), where 0 < a < b are two roots of the equation g(x) = (a11a22/a12)x. In this case, E0 is a saddle point, E and E+ are stable nodes (see [21, Proposition 2.1]). The function f(u, v) = (f1(u, v), f2(u, v)), where f1(u, v) = a11u + a12v and f2(u, v) = a22v + g(u), satisfies the monotone condition (1.2). Thus, the following result is a consequence of Theorem 2.1 in Roquejoffre et al. [13]. Theorem 3.2. Assume that (H1) and (H2) hold, u(n) is a traveling wave front of (3.3) and u(x, t, /) is the solution of (3.3) b . Then for any / 2 ½^0; K b such that (/) (defined by (2.6)) is small enough, there exists s/ 2 R such with u0 ¼ / 2 ½^0; K X X that lim sup kuðx; t; /Þ uðx ct þ s/ Þk ¼ 0:
t!þ1 x2R
~ ðnÞ of (3.3) satisfying (1.5) and 0 < u ~ ðÞ < K is a translate of u. Moreover, any traveling wave solution u With a slight modification of the method used in this paper (see Lemma 2.2), we are able to show that the above conclusion holds under suitable weak conditions. In fact, the following result holds. Theorem 3.3. Assume that (H1) and (H2) hold, u(n) is a traveling wave front of system (3.3) and u(x, t, /) is the solution of b . Then for any / 2 ½^0; K b such that (3.3) with u0 ¼ / 2 ½^0; K X X lim sup /ðxÞ E0 lim inf /ðxÞ; x!1
x!þ1
ð3:4Þ
there exists s/ 2 R such that lim sup kuðx; t; /Þ uðx ct þ s/ Þk ¼ 0:
t!þ1 x2R
~ ðnÞ of (3.3) satisfying (1.5) and 0 < u ~ ðÞ < K is a translate of u. Moreover, any traveling wave solution u Remark 3.4. Theorem 3.2 implies that any solution of the Cauchy problem for (3.3) which initially has these two stable equilibria as spatial limits will converge to a traveling wave front. Theorem 3.3 further guarantees that the solution will converge to a traveling wave front when the initial function satisfies the positive and negative spatial limits are larger and smaller than the immediate equilibrium, respectively. That is to say, for some monotone reaction-diffusion systems, by the method developed in this paper, we can establish the global asymptotic stability of bistable traveling wave fronts under suitable weaker conditions than those in Roquejoffre et al. [13]. Remark 3.5. With some additional assumptions on f(Æ, Æ), Volpert et al. [17, Theorem 3.1.1] proved the existence of bistable traveling wave front of (1.1). Our results (Theorems 2.6 and 2.7) further conclude that this traveling wave front is globally asymptotically stable with phase shift, that all traveling waves are unique up to translation, and that every traveling wave solution is Liapunov stable. In particular, our conditions are less restrictive than those in Roquejoffre et al. [13] and Volpert et al. [17].
Acknowledgements The authors would like to thank Professor Shigui Ruan (University of Miami) for his helpful comments on this paper.
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