Yancheng, Jiangsu , P. R. China b Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA.

Transcription

1 *Manuscript On an Advection-reaction-diffusion Competition System with Double Free Boundaries Modeling Invasion and Competition of Aedes Albopictus and Aedes Aegypti Mosquitoes Canrong Tian a and Shigui Ruan b, a School of Mathematics and Physics, Yancheng Institute of Technology, Yancheng, Jiangsu 43, P. R. China b Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA March 31, 18 Abstract. Based on the invasion of the Aedes albopictus mosquitoes and the competition between Ae. albopictus and Ae. aegypti mosquitoes in the United States, we consider an advection-reaction-diffusion competition system with two free boundaries consisting of an invasive species (Ae. albopictus) with density u and a local species (Ae. aegypti) with density v in which u invades the environment with leftward front x = g(t) and rightward front x = h(t). In the case that the competition between the two species is strong and species v wins over species u, the solution (u, v) converges uniformly to the semi-positive equilibrium (, 1), while the two fronts satisfy that lim (g(t), h(t)) = (g, h ) R. In the case that the competition between the two species is weak, we show that when the advection coefficients are less than fixed thresholds there are two scenarios for the long time behavior of solutions: (i) when the initial habitat h < π ( 4 ν1) 1 and the initial value of u is sufficiently small, the solution (u, v) converges uniformly to the semi-positive equilibrium (, 1) with the two fronts (g, h ) R; (ii) when the initial habitat h π ( 1, 4 ν1) the solution (u, v) converges locally uniformly to the interior equilibrium with the two fronts (g, h ) = R. In addition, we propose an upper bound and a lower bound for the asymptotic spreading speeds of the leftward and rightward fronts. Numerical simulations are also provided to confirm our theoretical results. Keywords: Invasion; competition; advection-reaction-diffusion model; free boundary; traveling wave; asymptotic spreading speed. MSC: 35B35; 35K6; 9B5 1 Introduction Aedes aegypti and Aedes albopictus are two prominent mosquito species which transmit the viruses that cause dengue fever, yellow fever, West Nile fever, chikungunya, and Zika, along with many other diseases. Ae. aegypti mosquito is a species with tropical and subtropical worldwide distribution and is an insect closely associated with humans and their dwellings. Ae. albopictus, a mosquito native to the tropical and subtropical areas of Southeast Asia and a most invasive Research was partially supported by NSF grant (DMS ) and CDC Southeastern Regional Center of Excellency in Vector-Borne Diseases - The Gateway Program (1U1-CK51-1). 1

2 Figure 1: Estimated range of Ae. aegypti and Ae. albopictus in the U.S., 16 (CDC [6]). species, has spread recently to many countries (including the U.S.) through the transport of goods and international travel. Inter-specific competition among mosquito larvae on larval, adult, and life-table traits exists between Ae. aegypti and Ae. albopictus and affects primarily larva-to-adult survivorship and the larval development time (Noden et al. [36]). Before the arrival of Ae. albopictus, Ae. aegypti was a common mosquito in artificial containers throughout Florida (Morlan and Tinker [35], Frank [17]). Ae. albopictus was found for the first time in northern counties in Florida in 1986 (Peacock et al. [39]). Over the next six years, Ae. albopictus spread slowly but steadily southward, and by the summer of 1994 it had spread to all 67 counties of the state (O Meara et al. [37]). Meanwhile, major declines in Ae. aegypti abundance were associated with the invasion and expansion of Ae. albopictus populations, not only in Florida, but elsewhere in the southern part of the continental United States (Hobbs et al. [6], O Meara et al. [38]). By 8, Ae. albopictus had spread to 36 states and continued to expand its range (Enserink [16]). In 13, Rochlin et al. [4] predicted that North American land favoring the environmental conditions of the Ae. albopictus mosquito is expected to more than triple in size in the next years, especially in urban areas. By the estimates of CDC [6] in 16, Ae. albopictus not only has spread to all states where Ae. aegypti presents but also has reached habitats beyond Ae. aegypti s boundaries (see Fig. 1). Taking into account the effect of wind on the movement of mosquitoes, Takahashi et al. [44] proposed an advection-reaction-diffusion equation model to investigate the dispersal dynamics of A. aegypti and predicted the existence of stable traveling waves in several situations. Although Takahashi et al. gave an estimation of the speed of traveling waves of A. aegypti, it is the asymptotic wave speed that usually gives an approximation of the progressive spreading speed of A. aegypti, and it does not really show the spread of A. aegypti in the early stage of spatial expanding to larger areas. To describe the spatial spreading of A. aegypti, we (Tian and Ruan [46]) generalized the model of Takahashi et al. [44] to an advection-reaction-diffusion equation model with free boundary, where the population of the vector mosquitoes is described by a system for the two life stages: the winged form (mature female mosquitoes) and an aquatic population (eggs, larvae and pupae), the expanding front is expressed by a free boundary which models the spatial expanding of the source area. The female mosquitoes are initially located at a habitat, then spread to other places owing to their dispersal ability, which includes the long distance dispersal as well as the short distance dispersal. The long distance dispersal of A. aegypti is caused by wind and described

3 by the advection, while the short dispersal is due to the random walk of each individual mosquito and described by the classical Laplacian diffusion. The free boundary theory was used to show the existence and uniqueness of the global solution to an advection-reaction-diffusion equation model and the boundedness of moving speed of the free boundary was also estimated. To model the invasion of Ae. albopictus species and the competition between A. aegypti and Ae. albopictus and take into account the effect of wind, we consider the following advection-reactiondiffusion system with two free boundaries: U t = D 1 U xx ν 1 U x + r 1 U(1 U ) ã 1 UV, t >, g(t) < x < K h(t), 1 U(t, x), t >, x ( g(t), h(t)), V t = D V xx ν V x + r V (1 V ) ã UV, K t >, x R, U =, g (t) = µu x (t, g(t)), t, x = g(t), U =, h (t) = µu x (t, h(t)), t, x = h(t), g() = h, h() = h, U(, x) = U (x), x [ h, h ]; V (, x) = V (x), x R. (1.1) The biological meanings of (1.1) are described as follows: V (t, x) represents the density of the local species (A. aegypti) and U(t, x) represents the density of the invasive species (Ae. albopictus) at time t and space location x, respectively. These two species have a competition relation. The invasive species, initially limited to a specific part of the domain [ h, h ], spreads over the space with the leftward front x = g(t) and rightward front x = h(t) (i.e. free boundaries). The boundaries are assumed to evolve according to the Stefan boundary condition g (t) = µu x (t, g(t)) and h (t) = µu x (t, h(t)) (Hilhorst et al. [4, 5]), which as a kind of free boundary conditions. D 1 and D are the diffusion rates of the two species respectively; ν 1 and ν are the advection speeds of the two species respectively; r 1 and r are the intrinsic growth rates of the two species respectively; K 1 and K are the carrying capacities of the two species respectively; ã 1 and ã are the interspecific competition rates. Free boundary problems have been extensively studied in the literature including realistic biological problems with free boundaries, see for example, Bounting and Du [3], Cao et al. [5], Chen and Friedman [7], Du and Lin [8], Du and Lou [1], Du et al. [11, 13, 14], Lin and Zhu [31], Liu and Lou [3], and the references cited therein. In order to minimize the number of parameters involved in the model, we introduce the dimensionless variables. Set u = 1 K 1 U, v = 1 K V, t = r 1 t, x = Then the free boundaries become r1 D 1 g( t r 1 ) and 1 K 1 D 1 x. (1.) r1 D 1 h( t r 1 ). Denote by g( t) r1 D 1 g( t r 1 ) and h( t) r1 D 1 h( t r 1 ), respectively. For the sake of simplicity, we omit the caps of t and x. Rewrite 3

4 the problem (1.1) as follows: u t = u xx ν 1 u x + u(1 u a 1 v), t >, g(t) < x < h(t), u(t, x), t >, x (g(t), h(t)), v t = Dv xx ν v x + v(r a u rv), t >, x R, u =, g (t) = µu x (t, g(t)), t, x = g(t), u =, h (t) = µu x (t, h(t)), t, x = h(t), g() = h, h() = h, u(, x) = u (x), x [ h, h ]; v(, x) = v (x), x R, (1.3) where ν 1 = ν 1 1 r 1 K 1 D 1, a 1 = K ã 1 r 1, D = D r 1 K 1 D 1, a = K 1ã r 1, µ = K 1 µ 1 r 1 K 1 D 1, h = r1 D 1 h, u (x) = 1 K 1 U ( K 1 D 1 x), and v (x) = 1 K V ( K 1 D 1 x). Moreover, we assume that u and v satisfy u C ([ h, h ]), u (±h ) =, u (x) > in ( h, h ), v C(R) L (R), v (x) > in R. (1.4) Two special cases of problem (1.3) have been studied in the literature. (a) In absence of the local species, named v, problem (1.3) reduces to the following singlespecies advection-reaction-diffusion equation with free boundary: u t = u xx ν 1 u x + u(1 u), t >, g(t) < x < h(t), u =, g (t) = µu x (t, g(t)), t, x = g(t), u =, h (t) = µu x (t, h(t)), t, x = h(t), (1.5) g() = h, h() = h, u(, x) = u (x), x [ h, h ]. Problem (1.5) was first studied by Gu et al. [19] who showed that if < ν 1 <, then the behavior is characterized by a spreading-vanishing dichotomy, namely, one of the following alternatives occurs: Spreading: lim (g(t), h(t)) = R, lim u(t, x) = 1 locally uniformly in R. Vanishing: lim (g(t), h(t)) = (g, h ), lim u(t, ) C([g(t),h(t)]) =, here g and h are finite constants. In [], Gu et al. also showed that the leftward and rightward asymptotic spreading speeds c l and c r exist and satisfy c l < c < c r, here c g(t) l = lim t, c h(t) r = lim t, c is the minimal speed of traveling wave speed connecting and 1. In [1], they further proved that when ν 1 is large the solution of problem (1.5) has an asymptotic behavior more complicated than spreading and vanishing. (b) When ν 1 = ν =, that is there is no advection in the environment, many researchers studied the qualitative properties of problem (1.3) under one or two free boundaries, see for example, Du and Lin [9], Guo and Wu [], Lin [3], Wang and Zhao [49], Wu [51], and Zhao and Wang [53]. These researchers extended the asymptotic stability and traveling wave solutions to the corresponding Cauchy problem of system (1.3), which can be listed in details as follows: (i) If a 1 > 1 and a < r, then the semi-positive equilibrium (, 1) is globally asymptotically stable, and there exists a traveling wave solution connecting the two semi-positive equilibria (1, ) and (, 1) (cf. Kan-On [7]); 4

5 (ii) If a 1 < 1 and a < r, then the interior equilibrium ( r(1 a 1 ) r a r a 1 a, ) r a 1 a is globally asymptotically stable, and there exists a traveling wave solution connecting the trivial equilibrium (, ) and the interior equilibrium ( r(1 a 1 ) ) r a r a 1 a, r a 1 a (cf. Tang and Fife [45]); (iii) If a 1 < 1 and a > r, then the semi-positive equilibrium (1, ) is globally asymptotically stable, and there exists a traveling wave solution connecting the two semi-positive equilibria (, 1) and (1, ) (cf. Kan-On [7]). The competition in cases (i) and (iii) is called weak-strong since one species wins the competition in the long run, whereas the competition in case (ii) is called weak since the two species coexist with no one winning or losing the competition. Recently, Du et al. [15] found the exact spreading speed in the case of spreading. The current studies show that when the size of the initial habitat is big, the solutions of free boundary problem (without advection) have the same asymptotic convergence with the corresponding Cauchy problem, and the asymptotic spreading speed of the front is no more than the minimal speed of the traveling wave solution. However, when the initial habitat is small, the free boundary problem (without advection) cannot spread over the whole space, which is different from the Cauchy problem where the traveling wave solution exists. Our main purpose in this paper is to study the influence of the advection terms on the asymptotic behavior of the competition system (1.3). We will extend the results of asymptotic convergence of Cauchy problem with cases (i) and (ii) to the free boundary problem (with advection). Moreover, we will estimate an upper and lower bounded asymptotic spreading speed of the front to improve the previous results of competition system (1.3). We would like to mention that the advection coefficients in problem (1.3) cause substantial technical difficulties, studying the asymptotic stability and traveling wave solutions for reaction-diffusion systems to advection-reaction-diffusion systems is far from trivial, even for regular boundary value problems (Zhao and Ruan [5]). The rest of the article is organized as follows. In section we prove the global existence and uniqueness of solutions to the free boundary problem (1.3). Moreover, we give a priori estimate of the derivative of boundary. In section 3, we investigate the weak-strong competition case in which u will vanish. In section 4, we investigate the weak competition case that u and v will coexist and the boundary will spread. Section 5 deals with the asymptotic spreading speed when the boundary spreads. In section 6 we carry out numerical simulations to confirm our analytical findings. In section 7 we use our conclusions to interpret the invasion of Ae. albopictus mosquitoes and the competition between A. aegypti and Ae. albopictus as an example of biological applications. Existence and uniqueness In this section, we first present the following local existence and uniqueness result by using the contraction mapping theorem. Theorem.1 For any given (u, v ) satisfying (1.4) and any α (, 1), there is a T > such that problem (1.3) admits a unique bounded solution Moreover, where (u, v, g, h) C 1+α,1+α (D T ) ( C 1+ α,+α loc (D T ) C(D T ) ) [ C 1+ α ([, T ]) ]. u 1+α + v C,1+α (D T ) L (D T ) + g C 1+ α ([,T ]) + h C 1+ α C, (.1) ([,T ]) D T = {(t, x) R : < t T, g(t) < x < h(t)}, 5

6 D T = {(t, x) R : < t T, x R}, here positive constants C and T depend only on h, α, u C ([ h,h ]) and v L (R). Proof. We introduce an auxiliary function ζ(y) satisfying ζ (y) < 6 h for y [, ), and { 1, if y h < h ζ(y) = 4,, if y h > h, it is obvious that such a C 3( [, ) ) function ζ(y) exists. As in Wang and Zhao [5], we perform the following coordinate transformation here ξ(y) = ζ( y). When t is confined to x = y + ζ(y)(h(t) h ) + ξ(y)(g(t) + h ), y R, (.) h(t) h + g(t) + h h 16, the transformation (.) implies that x y is a diffeomorphism from R onto R, meanwhile, the free boundaries x = g(t) and x = h(t) are transformed to the lines y = h and y = h, respectively. The rest of the proof is similar to that of Theorem.1 of Du and Lin [9]. We omit the details. We extend the local solution obtained in Theorem.1 to the maximal time. To do so, we need the following priori estimate. Lemma. Let (u, v, g, h) be a solution to problem (1.3) defined for t [, T ) for some T (, ). Then there exist constants M 1, M, and M independent of T such that < u(t, x) M 1, for t T, g(t) < x < h(t), (.3) < v(t, x) M, for t T, x R, (.4) M 3 g (t) <, < h (t) M 3, for < t T. (.5) Proof. As g(t), h(t) are fixed, we use the strong maximum principle to get u > for (t, x) [, T ] (g(t), h(t)), and v > for (t, x) [, T ] R. (.6) Consequently, since v satisfies { vt Dv xx ν v x + rv(1 v), < t T, x R, v(, x) = v (x), x R, we as well obtain v max{ v L (R), 1} := M. Similarly, u satisfies { ut u xx ν 1 u x + u(1 u), < t T, g(t) < x < h(t), u(, x) = u (x), x [ h, h ]. It is obvious that u max{ u C([ h,h ]), 1} := M 1. It remains to prove (.5). We only prove the boundedness of h (t), since g (t) can be treated in a similar way. Applying the Hopf Lemma (cf. Du and Lin [8]) to problem (1.3), we have u x (t, h(t)) < for t (, T ]. We thereby obtain h (t) > for t (, T ]. In order to prove the upper bound of h (t), we compare u with an auxiliary function. Define Ω =: {(t, x) : < t T, h(t) 1 M < x < h(t)} 6

7 and construct an auxiliary function w(t, x) := M 1 [M(h(t) x) M (h(t) x) ]. We choose M so that w(t, x) u(t, x) in Ω. Owing to h (t) >, for (t, x) Ω we calculate Consequently, As u, v >, we have w t = M 1 Mh (t)(1 M(h(t) x)) >. w t w xx + ν 1 w x > M 1 M ν 1 M 1 M for (t, x) Ω. (.7) u t u xx + ν 1 u x = u(1 u a 1 v) In light of (.7) and (.8), setting z = w u, we obtain provided < u(1 u) < M 1 for (t, x) Ω. (.8) z t z xx + ν 1 z x > M 1 (M ν 1 M 1), for (t, x) Ω, (.9) On the other hand, we have M ν 1 + ν1 +. (.1) z(t, h(t) 1 M ) = M 1 u(t, h(t) 1 ) for t (, T ), M z(t, h(t)) = for t (, T ). (.11) Hence, we expect to find M such that u (x) w(, x) for x [h 1 M, h ]. (.1) In view of (.9), (.11), and (.1), we can apply the parabolic maximum principle to z over Ω to deduce that u(t, x) w(t, x) for (t, x) Ω. Consequently, u x (t, h(t)) w x (t, h(t)) = MM 1, which leads to h (t) = µv x (t, h(t)) MM 1 µ := M 3. (.13) It remains to seek for some M independent of T such that (.1) holds. We compute In view of (.1), choosing we have w x (, x) = M 1 M(1 M(h x)) M 1 M for x [h 1 M, h ]. M := max { ν 1 + ν1 +, 4 u C 1 ([ h,h ]) 3M 1 }, (.14) w x (, x) MM u C 1 ([ h,h ]) u (x) for x [h (M) 1, h ]. 7

8 As w(, h ) = u (h ) =, the above inequality implies that w(, x) u (x) for x [h 1 M, h ]. Moreover, in term of M 4 3M 1 u C 1 ([ h,h ]) for x [h 1 M, h 1 M ], we obtain w(, x) 3 4 M 1, u (x) 1 M u C 1 ([ h,h ]) 3 4 M 1, for x [h 1 M, h 1 M ]. Therefore, u (x) w(, x) for x [h 1 M, h ], which means that (.13) is valid. The proof is complete. Theorem.3 For any given (u, v ) satisfying (1.4), the solution of problem (1.3) exists and is unique for all t [, ). Proof. It follows from the uniqueness of solutions (Theorem.1) that there is a fixed T such that the solution (u, v, g, h) is confined to [, T ). We now fix δ (, T ). By L p estimates, the Sobolev s embedding theorem, and the Hölder estimates for parabolic equations, we can find M 4 > depending on δ, T, M i (i = 1,, 3) such that u(t, ) C ([g(t),h(t)]), v(t, ) C (R) M 4 for t [δ, T ]. Again it follows from the proof of Theorem.1 that there exists a τ > depending on M i (i = 1,, 3) but not on t such that the solution of problem (1.3) with initial time T τ can be extended uniquely to the time T + τ. This means that the solution of (1.3) can be extended as long as u and v remain bounded. Therefore, the solution of problem (1.3) can be extended to the infinite time interval [, ). 3 Weak-strong competition In view of Lemma., we can observe that x = g(t) is monotone decreasing and x = h(t) is monotone increasing. Therefore, there exist g [, ) and h (, ] such that lim g(t) = g and lim h(t) = h. In this section, we try to extend the long time behavior to solutions of Cauchy problem corresponding to (1.3) in the case of weak-strong competition and v winning. Theorem 3.1 Suppose that a 1 > 1 and a < r. (3.1) If δ is an arbitrary small positive constant such that v (x) δ for x R, then lim (u(t, x), v(t, x)) = (, 1) uniformly in any compact subset of R. Proof. Consider the following ordinary differential equation: { U = U(1 U), t >, U() = u L ([ h,h ]). We solve the equation and find U = e t( e t ) 1. U() 8

9 By the comparison principle, we see that u(t, x) U(t) for t > and x [g(t), h(t)]. lim U(t) = 1, we have Since lim sup u(t, x) 1 uniformly in R. (3.) Similarly, we can deduce that lim sup v(t, x) 1 uniformly in R. (3.3) Therefore, for ε 1 = 1 ( r a 1), there exists t 1 > such that u(t, x) 1 + ε 1 for t t 1, x R. Thus, v satisfies { vt Dv xx + ν v x v(ε 1 rv), t > t 1, x R, v(t 1, x) >, x R. Let V (t) be the unique solution to { V = V (ε 1 rv ), t > t 1, V (t 1 ) = inf x R v(t 1, x). Since v (x) δ, we have V (t 1 ) >. By using the comparison principle, we have v(t, x) V (t) for t t 1 and x R. Since lim V (t) = ε 1 r, for any L >, there exists t L > t 1 such that v(t, x) V (t) ε 1 r for t t L, L x L. We obtain that (u, v) satisfies u t = u xx ν 1 u x + u(1 u a 1 v), t > t L, g(t) < x < h(t), v t = Dv xx ν v x + v(r a u rv), t > t L, x R, u(t, x) 1 + ε 1, v(t, x) ε 1 r, t t L, x R. Consider (u, v) as a unique solution to the following equations u t = u xx ν 1 u x + u(1 u a 1 v), t > t L, L < x < L, v t = Dv xx ν v x + v(r a u rv), t > t L, L < x < L, u(t, ±L) = 1 + ε 1, v(t, ±L) = ε 1 r, t > t L, u(t L, x) = 1 + ε 1, v(t L, x) = ε 1 r, L < x < L. Since u for t > t L and x / (g(t), h(t)), no matter whether or not (g(t), h(t)) ( L, L), the comparison principle implies that u u and v v in [t L, ) [ L, L]. Thanks to the fact that system (3.5) is quasimonotone nonincreasing, the theory of monotone dynamical systems (see, e.g. Smith [43, Corollary 7.3.6]) yields that lim u(t, x) = u L(x), (3.4) (3.5) lim v(t, x) = v L (x) uniformly in [ L, L], (3.6) where (u L (x), v L (x)) is the solution to u Lxx ν 1 u Lx + u L (1 u L a 1 v L ) =, L < x < L, Dv Lxx ν v Lx + v L (r a u L rv L ) =, L < x < L, (3.7) u L (±L) = 1 + ε 1, v L (±L) = ε 1 r. Note that if < L 1 < L, by employing the comparison principle to system (3.5) with the corresponding boundary conditions, we discover that u L1 (x) u L (x) and v L1 (x) v L (x) in [ L 1, L 1 ]. Letting L, by classical elliptic regularity theory and a diagonal procedure, we 9

10 obtain that (u L (x), v L (x)) converges uniformly on any compact subset of R to (u (x), v (x)), which satisfies u xx ν 1 u x + u (1 u a 1 v ) =, x R, Dv xx ν v x + v (r a u rv ) =, x R, (3.8) u (x) 1 + ε 1, v (x) ε 1 r, x R. Recalling (3.1), we shall illustrate that (u, v ) = (, 1). Let us consider the following ordinary differential equations: z = z(1 z a 1 w), t >, w = w(r a z rw), t >, z() = 1 + ε 1, w() = ε 1 r. By (3.1), it follows that (z, w) (, 1) as t (see, e.g. Morita and Tachibana [34]). If (Z, W ) is the solution to Z t = Z xx ν 1 Z x + Z(1 Z a 1 W ), t >, x R, W t = DW xx ν W x + W (r a Z rw ), t >, x R, (3.9) Z(, x) = 1 + ε 1, W (, x) = ε 1 r, x R, then (Z, W ) (, 1) as t uniformly in R. By utilizing the comparison principle to (3.8) and (3.9), we have u (x) Z(t, x) and v (x) W (t, x) for t >, which immediately gives that u (x) = and v (x) = 1. Thanks to (3.6), u(t, x) u(t, x) u L (x) and v(t, x) v(t, x) v L (x). Letting L, we obtain that lim sup u(t, x) and lim inf v(t, x) 1 uniformly in any compact subset of R. Taking into account (3.3) and lim inf u(t, x), we can deduce that lim (u(t, x), v(t, x)) = (, 1) uniformly in any compact subset of R. Remark 3. Consider the Cauchy problem corresponding to (1.3): u t = u xx ν 1 u x + u(1 u a 1 v), t >, x R, v t = Dv xx ν v x + v(r a u rv), t >, x R, u(, x) = u (x), v(, x) = v (x), x R, (3.1) where u (x) and v (x) are nonnegative and bounded. Note that the proof of Theorem 3.1 is valid. In addition, the solution (u, v) of (3.1) converges to (, 1) locally uniformly in R as t. In the above theorem, we only give the long time behavior of (u(t, x), v(t, x)). The next theorem addresses the long time behavior of the double boundaries x = g(t) and x = h(t). Theorem 3.3 Suppose that (3.1) holds. If δ is an arbitrary small positive constant such that v (x) δ for x R, then g, h < and lim u(t, x) = and lim v(t, x) = 1 uniformly in R. Proof. In view of (3.), for any given ε 1 = 1 ( r a 1), there exists t 1 > such that u(t, x) 1 + ε 1 for t t 1, x R. In addition, it follows from Lemma. that < u(t, x) < M 1 := max{1, u C([ h,h ])} and < v(t, x) < M := max{1, v L (R)} for t and x R. Hence, v satisfies { vt Dv xx + ν v x v(r a M 1 rm ), t >, x R, v(, x) δ, x R. 1

11 Therefore, we obtain that v(t, x) δe ( a M 1 rm )t for t > and x R. We now consider the following problem: z = z(1 z a 1 w), t > t 1, w = w(r a z rw), t > t 1, z(t 1 ) = 1 + ε 1, w(t 1 ) = δe ( a M 1 rm )t 1. By employing the comparison principle, we obtain that u(t, x) z(t) and v(t, x) w(t) for t t 1 and x R. Owing to the assumption (3.1) that a 1 > 1 and a < r, it follows that (z, w) (, 1) as t. Thus we have lim u(t, x) = uniformly in R. Next we claim that lim v(t, x) = 1 uniformly in R. Since lim u(t, x) = uniformly in R, for any ε > there exists t > such that < u(t, x) ε for t t and x R. Recalling that v(t, x) δe ( a M 1 rm )t, we thereby obtain { vt Dv xx + ν v x v(r a ε rv), t > t, x R, v(t, x) δe ( a M 1 rm )t, x R. Let us consider the following ordinary differential equation: { ṽ = ṽ(r a ε rṽ), t >, ṽ(t ) = δe ( a M 1 rm )t. By the comparison principle, we have v(t, x) ṽ(t) for t > t and x R. Since lim ṽ(t) = 1 a ε r, we deduce that lim inf v(t, x) 1 a ε r uniformly in R. Due to the arbitrariness of ε, it follows that lim v(t, x) 1 uniformly in R. It remains to show that g and h <. Since (u(t, x), v(t, x)) (, 1) as t uniformly in R, for any ε >, there exists T > such that u(t, x) < ε and v(t, x) > 1 ε for t > T and x R. By setting ε = 1 1 a 1, we have u(1 u a 1 v) u(1 a 1 + a 1 ε) for t > T and x R. (3.11) Owing to (3.11), it follows from the boundary condition of (1.3) that d dt h(t) g(t) u(t, x)dx = = h(t) g(t) h(t) g(t) h(t) g(t) u t (t, x)dx + h (t)u(t, h(t)) g (t)u(t, g(t)) (u xx ν 1 u x )dx + (u xx ν 1 u x )dx h(t) g(t) u(1 u a 1 v)dx u x (h(t)) u x (g(t)) ν 1 u(h(t)) + ν 1 u(g(t)) = u x (h(t)) u x (g(t)) = h (t) + g(t). µ (3.1) Integrating (3.1) from T to t gives h(t) g(t) u(t, x)dx h(t ) g(t ) u(t, x)dx 11 h(t) g(t) h(t ) + g(t ). (3.13) µ

12 It follows from (3.13) that h(t) g(t) h(t ) g(t ) + µ h(t ) g(t ) + µ h(t ) g(t ) h(t ) g(t ) u(t, x)dx µ u(t, x)dx. h(t) g(t) u(t, x)dx (3.14) By letting t in (3.14), we have h g h(t ) g(t ) + µ h(t ) g(t ) u(t, x)dx <, which implies that h and g <. 4 Weak competition In the section, we try to extend the long time behavior to solutions of Cauchy problem corresponding to (1.3) in the case of weak competition. In order to do that, we give the following two lemmas. These lemmas without the advection terms were proved in Propositions.1 and. of Wang and Zhao [49] or in Propositions B1 and B of Wang and Zhao [5]. Lemma 4.1 Assume that ν, D, α and β are fixed positive constants. Suppose that ν < Dα. (4.1) For any given ε > and L >, there exist l ε > max { L, Dπ ( 4Dα ν ) 1} and Tε >, such that when the continuous and non-negative function w(t, x) satisfies then w t Dw xx + νw x ( )w(α βw), t >, l ε < x < l ε, w(t, ±l ε ) (=), t, w(, x) >, x ( l ε, l ε ), (4.) w(t, x) > α β ε (w(t, x) < α β + ε), for t T ε, x [ L, L]. (4.3) Moreover, the above inequality implies that lim inf w(t, x) > α β ε (lim sup w(t, x) < α + ε) uniformly in [ L, L]. (4.4) β Proof. Consider the following eigenvalue problem with advection: DΦ + νφ = λφ in( l, l), Φ(±l) =. (4.5) The first eigenvalue λ 1 and eigenfunction Φ 1 of (4.5) are ( ) λ 1 = ν 4D + D π l, Φ 1(x) = e ν D x λ1 sin D ν 4D x. Let l > Dπ ( 4Dα ν ) 1, we obtain λ1 < α. Thus, the problem { Dw xx + νw x = w(α βw), l < x < l, w(±l) = (4.6) 1

13 admits a unique positive solution < w = w l < α β (Proposition 3.3 in Cantrell and Cosner [4]). By the comparison principle, w l (x) is monotone decreasing with respect to l. Letting l, by using the classical elliptic regularity theory and a diagonal procedure, the limit lim l w l (x) = w (x) exists and satisfies the following Cauchy equation: In view of Liouville theorem, we have lim l w l(x) = α β Dw xx + νw x = w(α βw), x R. (4.7) uniformly in any compact subset of R. (4.8) Therefore, for any given L > and ε >, there exists l ε > Dπ ( 4Dα ν ) 1 such that α β ε < w l(x) < α β + ε, for x [ L, L], l l ε. (4.9) Since the parabolic equation w t Dw xx + νw x = w(α βw), t >, l ε < x < l ε, w(t, ±l ε ) =, t, w(, x) = w (x), x [ l ε, l ε ]. (4.1) is a gradient system, the solution w(t, x) of (4.1) converges to a solution w l (x) of the corresponding stationary problem (4.6) as t (see Brunovsky and Chow [], Hale and Massatt [3]). By (4.9), there exists a T ε > such that α β ε < w(t, x) < α β + ε, for t T ε, x [ L, L]. (4.11) Combining (4.) and (4.1), the comparison principle implies that (4.3) and (4.4) hold immediately. Lemma 4. Assume that ν, D, α and β are fixed positive constants and (4.1) holds. For any given ε > and L >, there exist l ε > max { L, Dπ ( 4Dα ν ) 1} and Tε >, such that when the continuous and non-negative function z(t, x) satisfies then z t Dz xx + νz x ( )z(α βz), t >, l ε < x < l ε, z(t, ±l ε ) ( )k, t, z(, x) >, x ( l ε, l ε ), (4.1) z(t, x) > α β ε (z(t, x) < α β + ε), for t T ε, x [ L, L]. (4.13) Moreover, the above inequality implies that lim inf z(t, x) > α β ε (lim sup z(t, x) < α + ε) uniformly in [ L, L]. (4.14) β Proof. Let l > Dπ ( 4Dα ν ) 1. By using a similar argument as in the proof of Lemma 4.1, we can show that the problem { Dz xx + νz x = z(α βz), l < x < l, (4.15) z(±l) = k 13

14 admits a unique positive solution z = z l. We claim that lim l z l(x) = α β uniformly in any compact subset of R. (4.16) For the case k > α β, the strong maximum principle implies that α β z l(x) k for x [ l, l]. Since z l (x) k, the comparison principle yields that z l (x) is monotone decreasing with respect to l. Hence, the limit lim l z l (x) = z (x) exists and satisfies the following Cauchy equation: Dz xx + νz x = z(α βz), x R. (4.17) By Liouville theorem, lim l z l (x) = α β uniformly in any compact subset of R. Thus (4.16) holds. For the case k α β, choose k > α β and let z l (x) be the unique solution of (4.15) with k = k. By the comparison principle, w l (x) z l (x) zl (x) for x [ l, l], where w l(x) is the solution of (4.6). Since w l (x) and zl (x) converge to α β as l, we obtain (4.16) immediately. By (4.16), for any given L > and ε >, there exists l ε > max { L, Dπ ( 4Dα ν ) 1} such that α β ε < z l(x) < α β + ε for x [ L, L], l l ε. (4.18) Let z (x) C([ l ε, l ε ]) be a positive function and z(t, x) be the solution of z t Dz xx + νz x = z(α βz), t >, l ε < x < l ε, z(t, ±l ε ) = k, t, z(, x) = z (x), x [ l ε, l ε ]. (4.19) In view of l ε > max { L, Dπ ( 4Dα ν ) 1}, we show that lim z(t, x) = z l(x) uniformly in any compact subset of ( l ε, l ε ). (4.) Fix a positive constant q and let ψ q (t, x) be the unique solution of ψ t Dψ xx + νψ x = ψ(α βψ), t >, l ε < x < l ε, ψ(t, ±l ε ) = k, t, ψ(, x) = q, x [ l ε, l ε ]. (4.1) We choose M >> 1 and < m << 1 such that M and m are the upper and lower solutions of (4.15). Then ψ M (t, x) is monotone decreasing and ψ m (t, x) is monotone increasing with respect to t. Hence, the limits lim ψ M (t, x) = ψ M (x) and lim ψ m (t, x) = ψ m (x) exist. ψ M (x) and ψ m (x) are both positive solutions of (4.15) with l = l ε. Therefore, ψ M (x) = ψ m (x) = z l (x), where z l (x) is the solution of (4.15) with l = l ε. (4.) By comparing the equations (4.19) and (4.1), the comparison principle leads to ψ m (t, x) z(t, x) ψ M (t, x). Since (4.) holds, lim z(t, x) = z l (x). By using the interior estimate, the limit lim z(t, x) = z l (x) is uniformly in any compact subset of ( l ε, l ε ). Owing to (4.14) and (4.), there exists T ε >> 1 such that α β ε < z(t, x) < α β + ε, for t T ε, x [ L, L]. By comparing the two systems (4.1) and (4.19), we obtain that (4.13) and (4.14) hold immediately. 14

15 4.1 Long term behavior of boundaries and convergence of solutions In fact, whether the size of (g, h ) is finite determines the asymptotic behavior of u and v. In the next two theorems, we give the relationship between the long term behavior of boundaies and the convergence of u and v. Theorem 4.3 Assume that h = and g =. If a 1 < 1, a < r, ν 1 < 1 a 1, ν < D(r a ), (4.3) then the solution (u, v, g, h) to (1.3) satisfies lim (u(t, x), v(t, x)) = ( r(1 a 1 ) r a ), uniformly in any compact subset of R. (4.4) r a 1 a r a 1 a Proof. By Lemma., < u(t, x) M 1 for t > and x (g(t), h(t)), < v(t, x) M for t > and x R. Then we find that v satisfies { vt Dv xx ν v x v(r rv), t >, x R, The comparison principle yields that v(, x) = v (x), x R. lim sup v(t, x) 1 := v 1 uniformly in R. For any < ε 1 << 1, there exists T 1 > such that v(t, x) v 1 + ε 1 for t T 1, x R. (4.5) (i) For any given L >, and < ε << 1, let l ε be given by Lemma 4.1. In view of h = and g =, there exists T T 1 such that g(t) < l ε, h(t) > l ε, for t T. Recalling (4.5), we see that u satisfies u t u xx + ν 1 u x u(1 u a 1 (v 1 + ε 1 )), t > T, x ( l ε, l ε ), u(t, ±l ε ), t T, u(t, x) >, x ( l ε, l ε ). In order to use Lemma 4.1, we need to verify condition (4.1), here it corresponds to ν 1 < 1 a 1 (v 1 + ε 1 ). Thus, we should choose ε 1 < 4(1 a 1v 1 ) ν1 4a 1. Due to ν 1 < 1 a 1 of (4.3), we induce that 4(1 a 1v 1 ) ν1 4a 1 = 4(1 a 1) ν1 4a 1 >. Hence ε 1 satisfying Lemma 4.1 exists. Applying Lemma 4.1, we have lim inf u(t, x) > 1 a 1v 1 a 1 ε 1 ε uniformly in [ L, L]. Since 1 a 1 >, by the arbitrariness of L, ε and ε 1, we have lim inf u(t, x) 1 a 1v 1 := u 1 > uniformly in any compact subset of R. (4.6) (ii) For any given L > and < ε << 1, let l ε be given by Lemma 4. with k = M. Taking into account (4.6), there exists T 3 T such that u(t, x) u 1 + ε for t T 3, x [ l ε, l ε ]. 15

16 Then v satisfies v t Dv xx ν v x v(r a (u 1 + ε ) rv), t > T 3, x ( l ε, l ε ), v(t, ±l ε ) M, t T 3, v(t 3, x) >, x ( l ε, l ε ). In order to use Lemma 4., we need to verify condition (4.1), here it corresponds to ν < D(r a (u 1 + ε )). Thus we should choose ε < 4D(r a u 1 ) ν 4Da. Since u 1 < 1, 4D(r a u 1 ) ν 4Da > 4D(r a ) ν 4Da. Consequently, in order to ensure ε existing, we only need to verify 4D(r a ) ν 4Da >, which can be induced by the condition ν < D(r a ) of (4.3). Applying Lemma 4., we have lim sup v(t, x) < 1 a r (u 1 + ε ) + ε uniformly in [ L, L]. By the arbitrariness of L, ε and ε, we have lim sup v(t, x) 1 a r u 1 := v uniformly in any compact subset of R. (4.7) (iii) For any given L > and < ε << 1, let l ε be given by Lemma 4.1. Taking into account (4.7), there exists T 4 T 3 such that g(t) < l ε, h(t) > l ε, v(t, x) v + ε 1, for t T 4, x [ l ε, l ε ]. We see that u satisfies u t u xx + ν 1 u x u(1 u a 1 (v + ε 1 )), t > T 4, x ( l ε, l ε ), u(t, ±l ε ), t T 4, u(t 4, x) >, x ( l ε, l ε ). Since v < v 1, the above 4(1 a 1v 1 ) ν1 4a 1 > implies 4(1 a 1v ) ν1 4a 1 the condition (4.1). Applying Lemma 4.1, we have >. Thus, our choice ε 1 satisfies lim inf By the arbitrariness of L, ε and ε 1, we have u(t, x) > 1 a 1v a 1 ε 1 ε uniformly in [ L, L]. lim inf u(t, x) 1 a 1v := u uniformly in any compact subset of R. (4.8) (iv) For any given L > and < ε << 1, let l ε be given by Lemma 4. with k = M. Taking into account (4.8), there exists T 5 T 4 such that u(t, x) u + ε for t T 5, x [ l ε, l ε ]. Then v satisfies v t Dv xx ν v x v(r a (u + ε ) rv), t > T 5, x ( l ε, l ε ), v(t, ±l ε ) M, t T 5, v(t 5, x) >, x ( l ε, l ε ). Since u < 1 still holds, the above 4D(r a ) ν 4Da > implies 4D(r a u ) ν 4Da >. Thus the chosen ε satisfies condition (4.1). Applying Lemma 4., we have lim sup v(t, x) < 1 a r (u + ε ) + ε uniformly in [ L, L]. 16

17 By the arbitrariness of L, ε and ε, we have lim sup v(t, x) 1 a r u := v 3 uniformly in any compact subset of R. (4.9) Therefore, as long as the sequence {u n } is monotone increasing and the sequence {v n } is monotone decreasing, the condition (4.1) is naturally satisfied. We can apply Lemmas 4.1 and 4. again. Repeating the above procedure such as (4.6), (4.7), (4.8) and (4.9), we obtain two sequences {u n } and {v n }, which satisfy u 1 = 1 a 1, v 1 = 1, and u n = 1 a 1 v n, v n+1 = 1 a r u n, for n = 1, (4.3) We now claim that {u n } is monotone increasing and {v n } is monotone decreasing. We prove it by using an induction method. For the case n = 1, since a 1 < 1, it is easy to see that v v 1 = a r (1 a 1) <, u u 1 = a 1 (v v 1 ) >. Suppose that v n v n 1 <, u n u n 1 >. By (4.3), we compute v n+1 v n = 1 a r u n (1 a r u n 1) = a r (u n u n 1 ) <, u n+1 u n = a 1 (v n+1 v n ) >. Thus the induction principle implies the claim. Since the sequences {u n } is monotone increasing and {v n } is monotone decreasing, the limits lim n u n and lim n v n exist, which satisfy lim inf u(t, x) u = r(1 a 1) r a 1 a uniformly in any compact subset of R. lim sup v(t, x) v = r a uniformly in any compact subset of R. r a 1 a (4.31) By using a similar argument, we can show lim sup u(t, x) r(1 a 1) uniformly in any compact subset of R. r a 1 a lim inf v(t, x) r a r a 1 a uniformly in any compact subset of R. Combining (4.31) and (4.3), we conclude that (4.4) holds. (4.3) Remark 4.4 For the Cauchy problem (3.1), the proof of Theorem 4.3 is valid. When (4.3) holds, the solution (u, v) of (3.1) converges to the coexistence state ( r(1 a 1) r a r a 1 a, r a 1 a ) locally uniformly in R as t. Theorem 4.5 Let (u, v, g, h) be a solution of (1.3). If h g <, then lim C([g(t),h(t)]) =, (4.33) lim g (t) = lim h (t) =, (4.34) lim v(t, x) = 1 uniformly in any compact subset of R. (4.35) 17

18 Proof. We first show that u C 1+α,(1+α)/ ([1, ) [g(t),h(t)]) + g C 1+α/ ([1, )) + h C 1+α/ ([1, )) C (4.36) for any α (, 1), where the constant C depends on α, h, u C ([ h,h ]), g and h. straighten the free boundaries by the following transformation We y = x h(t) + g(t) h(t) g(t) h(t) g(t) such that x = h(t) and x = g(t) change to y = ±1. Some calculations show that y x = h(t) g(t) := A(g(t), h(t), y), y x = 4 := A(g(t), h(t), y), (h(t) g(t)) y t = (t) g (t)) + (g(t)h (t) + h(t)g (t)) x(h (h(t) g(t)) := B(g(t), g (t), h(t), h (t), y). Our proof is motivated by Theorem.1 in Wang [47, 48] (see also Theorem 4.1 in Lei et al. [9] and Theorem 4.1 in Cao et al. [5]). Let w(t, y) = u(t, x) and z(t, y) = v(t, x), then we obtain that w(t, y) satisfies For any integer n, define then w n (t, y) satisfies w t Aw yy (A B ν 1 A)wy = f 1 (w, z), t >, 1 < y < 1 w(t, ±1) =, t >, w(, y) = u ( y h ), 1 y 1. w n (t, y) = w(t + n, y), z n (t, y) = z(t + n, y), wt n A n wyy n (A n B n ν 1 A n )wy n = f 1 (w n, z n ), t (, 3], 1 < y < 1, w n (t, ±1) =, t (, 3], w n (, y) = u(n, y(h(n) g(n))+h(n)+g(n) ), 1 y 1, where A n = A(t + n) and B n = B(t + n). In view of Lemma., it follows that w n, z n, A n and B n are bounded uniformly on n, and max t 1 <t 3, t 1 t τ An (t 1 ) A n (t ) 8(hn (t) g n (t)) (h n (t) g n (t)) 3 M 3τ h 3 as τ with g n (t) = g(t + n), h n (t) = h(t + n). Moreover, we have A n 4 (h g ) for all n and < t 3 as h g <. If we choose the integer p sufficiently large, applying the interior L p estimate yields that there exists a positive constant C independent of n such that u n W 1, p ([1,3] [ 1,1]) C 1 for all n >. By Sobolev s imbedding theorem, we therefore have u n C (1+α)/,1+α ([1,3] [ 1,1]) C 1 for all n >, which implies that u C (1+α)/,1+α ([n+1,n+3] [ 1,1]) C 1. Since g (t) = µu x (t, g(t)), u x (t, g(t)) = h(t) g(t) u y( 1, t), h (t) = µu x (t, h(t)), u x (t, h(t)) = 18 h(t) g(t) u y(1, t),

19 hold in [n + 1, n + 3] [ 1, 1] and < g (t), h(t) M 3, we have g C 1+α/ ([n+1,n+3]) + h C 1+α/ ([n+1,n+3]) C. Since the intervals [n+1, n+3] [ 1, 1] overlap and the above constants C 1 and C are independent of n, it follows that (4.36) holds. Noting that h(t) is bounded and h C 1+α/ ([, )) M, we have h (t) as t +. Analogously, g (t) as t +. Thus (4.34) is proved. We then show (4.33). Assume that lim sup t + u(t, ) C([g(t),h(t)]) = δ > by contradiction. Then there exists a sequence {(t k, x k )} in (, ) (g(t), h(t)) such that u(t k, x k ) δ/ for all k N, and t k as k. It follows from Theorem.1 that there exists a constant C depending on α, h, (u, v ) and h such that u C (1+α)/,1+α (G) + v C (1+α)/,1+α (G) + h C 1+α/ ([, )) + g C 1+α/ ([, )) C, (4.37) where G = {(t, x) R : t, x [g(t), h(t)]}. Combining u(t, h(t)) = u(t, g(t)) = and (4.37), we have u x (t, h(t)) and u x (t, g(t)) are uniformly bounded for t [, ). There exists σ > such that x k h(t k ) σ for all k 1. Therefore there exists a subsequence of {x k } converging to x (g, h σ]. Without loss of generality, we assume x k x as k. Define with U k (t, x) = u(t k + t, x) and V k (t, x) = v(t k + t, x) for (t, x) G k G k = {(t, x) R : t ( t k, ), x [g(t k + t), h(t k + t)]}. It follows from (4.37) that {(U k, V k )} has a subsequence {(U kn, V kn )} such that and (Û, ˆV ) satisfies with (U kn, V kn ) (Û, ˆV ) C 1, (G kn ) C 1, (G kn ) as n, { Ût = Ûxx ν 1 Û x + Û(1 Û a 1 ˆV ), < t <, g < x < h, ˆV t = D ˆV xx ν ˆVx + ˆV (r a Û r ˆV ), < t <, g < x < h, Û(t, h ) = for t (, ). Since Û(, x ) δ/, we have Û > in (, ) (g, h ) by the strong comparison principle. Let M = 1 Û a 1 ˆV L ([, )), then Ût Ûxx + ν 1 Û x + MÛ. Using the Hopf Lemma at the point (, h ), we get Ûx(, h ) := σ <. It follows that u x (t kn, h(t kn )) = u kn (, h(t kn )) σ / < (4.38) for all large n. Hence, h (t kn ) µσ / > for all large n. On the other hand, since h C 1+α/ ([, )) C, h (t) > and h(t) h, we have h (t) as t. We obtain a contradiction, thus (4.33) is true. 19

20 It remains to show (4.35). Let v(t) solve the following ordinary differential equation: { v = rv(1 v), t >, v() = v L (R), where v = e rt( e rt ) 1. v() Applying the comparison principle yields that v(t, x) v(t) for (t, x) [, T ) R. lim v(t) = 1, we have Since lim sup v(t, x) 1 uniformly for x R. (4.39) On the other hand, we have (4.33) and u(t, x) for x / (g(t), h(t)). Hence, for any given positive σ, there exists T σ > such that u(t, x) < σ for (t, x) [T σ, ) R. Consequently, v satisfies { vt Dv xx + ν v x v(r a σ rv), t > T σ, x R, (4.4) v(t σ, x) >, x R. Considering the corresponding Cauchy problem of (4.4), { wt Dw xx + ν w x = w(r a σ rw), t > T σ, x R, w(, x) >, x R, (4.41) invoking Theorem 5.4 of Du and Ma [1] shows that uniformly in any compact subset of R. deduce that lim inf u(t, x) r a σ r lim w(t, x) = r a σ r By applying the comparison principle to u and w, we uniformly in any compact subset of R. (4.4) Letting σ, (4.4) implies lim inf u(t, x) 1 uniformly in any compact subset of R. Recalling (4.4), (4.33) is proved. The proof is complete. 4. Sufficient conditions for spreading and vanishing By Theorems 4.3 and 4.5, the boundaries determine the asymptotic behavior of u and v. Hence, we only need to study the asymptotic behavior of the free boundaries h and g. In order to do that, we give the spreading-vanishing alternative for the following single free boundary problem: u t = u xx νu x + u(α βu), t >, g(t) < x < h(t), u =, g (t) = µu x (t, g(t)), t, x = g(t), u =, h (t) = µu x (t, h(t)), t, x = h(t), (4.43) g() = h, h() = h, u(, x) = u (x) = σφ(x), x [ h, h ], where φ(x) C ([ h, h ]), φ(±h ) =, φ ( h ) >, φ (h ) <, φ (x) > in ( h, h ). Note that the global existence of solutions and boundedness to problem (4.43) are valid by a similar argument in Theorem.3 and Lemma.. Gu et al. [19, 1] have used the zero number argument method to show that when ν < α the problem (4.43) possesses a spreading-vanishing dichotomy; that is, the solution converges either to 1 locally uniformly in R or to uniformly into a finite domain.

21 Lemma 4.6 (Theorem 1.1 in [19] or Theorem.1 in [1]) Assume that ν < α and (u, g, h) is a global solution of (4.43). Then either (i) Spreading: (g, h ) = R and lim u(t, x) = α β uniformly in any compact subset of R; or (ii) Vanishing: (g, h ) is a finite interval with h g π ( 4α ν ) 1 and lim u(t, ) C([g(t),h(t)]) =. Moreover, in the case h π ( 4α ν ) 1, spreading happens; In the case h < π ( 4α ν ) 1, there exists σ = σ (h, φ) (, ] such that vanishing happens when < σ σ, and spreading happens when σ > σ. Next we give the comparison principle for the single free boundary problem (4.43). Lemma 4.7 Let T (, ), g, h C 1 ([, T ]). Assume that D T := {(t, x) R : < t T, g(t) < x < h(t)} and u C(D T ) C1, (D T ), and u(t, x) > in D T, and (u, g, h) satisfies u t u xx + νu x u(α βu), < t T, g(t) < x < h(t), u(t, g(t)) =, g (t) µu x (t, g(t)), < t T, u(t, h(t)) =, h (t) µu x (t, h(t)), < t T, g() h, h() h, u(, x) u (x), x [ h, h ]. (4.44) Then (u, g, h) is called an upper solution of (4.43). Moreover, the solution (u, g, h) of the free boundary problem (4.43) satisfies g(t) g(t), h(t) h(t) for t [, T ], (4.45) u(t, x) u(t, x) for (t, x) [, T ] [g(t), h(t)]. (4.46) By reversing all the inequalities in (4.44), we can define a lower solution (u, g, h), which satisfies g(t) g(t), h(t) h(t) for t [, T ], (4.47) u(t, x) u(t, x) for (t, x) [, T ] [g(t), h(t)]. (4.48) Proof. We first prove the case of g() < h and h() > h. We claim that g(t) < g(t) and h(t) > h(t) on (, T ]. Clearly, this is true for small t >. If our claim is not true, without loss of generality, there exists a T > such that h(t ) = h(t ), and g(t) < g(t), h(t) > h(t) for all t (, T ). Thus, h (T ) h (T ). (4.49) We now show that u u in [, T ] [g(t), h(t)]. Letting U = (u u)e Kt, taking the difference of the equations for (4.43) and (4.44) implies that < t T, g(t) < x < h(t), U t U xx + νu x KU + αu βu(u + u), U(t, g(t)), U(t, h(t)), < t T, U(, x), x [ h, h ]. 1 (4.5)

22 Since u C(DT ), by setting M = u L (DT ) + u L (DT ), the first inequality of (4.5) implies that U t U xx + νu x KU + αu βmu, < t T, g(t) < x < h(t). Upon choosing K α βm (4.51) and applying the maximum principle directly to (4.5), we obtain U in [, T ] [g(t), h(t)]. Therefore, u u in [, T ] [g(t), h(t)]. Since u(t, h(t )) = u(t, h(t )), utilizing Hopf Lemma to U shows U x (T, h(t )) < ; that is, u x (T, h(t )) < u x (T, h(t )). Thus, we obtain that h (T ) < h (T ), which is a contradiction to (4.49). Hence, we proved our claim that h(t) > h(t) and g(t) < g(t) on all [, T ]. We can obtain the equation (4.5) for t (, T ]. By applying the maximum principle again, we conclude that u u in [, T ] [g(t), h(t)]. Hence, (4.45) and (4.46) hold in the case of g() < h and h() > h. It remains to prove the general case that g() = h and h() = h. We set h ε = (1 ε)h, µ ε = (1 ε)µ for small ε >, and (u ε, g ε, h ε ) solves the following equations: u t = u xx νu x + u(α βu), t >, g(t) < x < h(t), u =, g (t) = µ ε u x (t, g(t)), t, x = g(t), u =, h (t) = µ ε u x (t, h(t)), t, x = h(t), (4.5) g() = h ε, h() = hε, u(, x) = u ε (x), x [ hε, hε ]. here u ε (x) C ([ h ε, hε ]), which also satisfies Moreover, as ε, < u ε (x) u (x) for x [ h ε, h ε ]. u ε (h x ) u (x) in C ([ h, h ]). Thus, (u ε, g ε, h ε ) satisfies the first case. Consequently we have h ε g ε (t) g(t), h ε (t) h(t) for t [, T ], u ε (t, x) u(t, x) for t [, T ], x [ h ε, h ε ]. (4.53) Since the unique solution of (4.43) depends continuously on the parameters in problem (4.5), as ε, (u ε, g ε, h ε ) converges to (u, g, h) the unique solution of (4.43). Then (4.45) and (4.46) are true by taking ε. We now give a sufficient condition for spreading in (1.3). Theorem 4.8 Assume that (4.3) holds and (u, v, g, h) is a global solution of (1.3). If h g <, then h g π ( 4 ν 1) 1. (4.54) If h π ( 4 ν 1) 1, then (g, h ) = R. (4.55) Moreover, when (g, h ) = R, spreading happens, and the solution (u, v, g, h) to (1.3) satisfies lim (u(t, x), v(t, x)) = ( r(1 a 1 ) r a ), uniformly in any compact subset of R. r a 1 a r a 1 a

23 Proof. In order to prove (4.54), we assume h g > π ( 4 ν 1) 1 to get a contradiction. By Theorem 4.5, if h g <, then u(t, ) C([g(t),h(t)]) =, lim v(t, x) = 1 uniformly in any compact subset of R. For any small < ε 1 << 1 and any ε >, there exists T 1 > such that v(t, x) < 1 + ε 1, for t > T 1, x [g, h ]. h(t 1 ) g(t 1 ) > max{h, π ( 4 ν 1 ε 1) 1}. Set l 1 = g(t 1 ) and l = h(t 1 ), then l l 1 > π ( 4 ν 1 ε 1) 1. Consider the following initial boundary value problem: w t w xx + ν 1 w x = w(1 w a 1 (1 + ε )), t > T 1, l 1 < x < l, w(t, l 1 ) = w(t, l ) =, t > T 1, w(t 1, x) = u(t 1, x) >, x (l 1, l ). By the comparison principle, we have (4.56) w(t, x) u(t, x) for t T 1, x [l 1, l ]. (4.57) By choosing ε < ε 1 4a 1, we verify that the first eigenvalue of (4.56) λ 1 = ν π < 1 a l 1 a 1 ε. Since (4.56) is a gradient system (see [, 3]), as t, the solution w(t, x) converges to a solution θ(x) of the following stationary problem: { θxx + ν 1 θ x = θ(1 θ a 1 (1 + ε )), l 1 < x < l, θ(l 1 ) = θ(l ) =. Thus, it follows from (4.57) that lim inf u(t, x) lim t w(t, x) = θ(x) > in (l 1, l ), which is a contradiction to u(t, ) C([g(t),h(t)]) =. Hence (4.54) is true. In addition, if h π ( 4 ν 1) 1, the monotone properties of g(t) and h(t) yield that h g > π ( 4 ν 1) 1, which leads to (4.55) immediately. Since (4.3) holds, applying Theorem 4.3, the proof is completed. In the next, we discuss the case of h < π ( 4 ν 1) 1. We give a sufficient condition for vanishing in (1.3). Theorem 4.9 Assume that (4.3) holds and h < π ( 4 ν 1) 1. The initial condition of (1.3) satisfies u(, x) = u (x) = σφ(x) for x [ h, h ], where φ(x) C ([ h, h ]), φ(±h ) =, φ ( h ) >, φ (h ) <, φ (x) > in ( h, h ). Then there exists σ = σ (h, φ) (, ) such that when < σ σ, h g <. Moreover, when h g <, vanishing happens, and the solution (u, v, g, h) to (1.3) satisfies lim u(t, ) C([g(t),h(t)]) =, v(t, x) = 1 uniformly in any compact subset of R. lim Proof. Since v(t, x) > in (, ) R, the problem (1.3) implies that u t u xx ν 1 u x + u(1 u), t >, g(t) < x < h(t), u =, g (t) = µu x (t, g(t)), t, x = g(t), u =, h (t) = µu x (t, h(t)), t, x = h(t), g() = h, h() = h, u(, x) = u (x) = σφ(x), x [ h, h ]. (4.58) 3

24 Applying Lemma 4.7 directly, we know that (u, g, h) is a lower solution of u t = u xx ν 1 u x + u(1 u), t >, g(t) < x < h(t), u =, g (t) = µu x (t, g(t)), t, x = g(t), u =, h (t) = µu x (t, h(t)), t, x = h(t), g() = h, h() = h, u(, x) = u (x) = σφ(x), x [ h, h ]. Thus, (4.59) g(t) g(t), h(t) h(t), for t >. (4.6) By using Lemma 4.6 to (4.59) with α = 1 and ν = ν 1, there exists σ = σ (h, φ) (, ] such that when < σ σ, h g <. In view of (4.6), by choosing σ = σ (h, φ), we obtain that when < σ σ, h g <. Since (4.3) holds, applying Theorem 4.5, the proof is completed. 5 Asymptotic spreading speed In term of Theorem 4.8, we obtain sufficient conditions for spreading. Our aim is to show that when spreading happens, the leftward front g(t) and the rightward front h(t) move at different speeds for large time. To do so, we recall the asymptotic spreading speed for the scalar free boundary problem (4.43). Gu et al. [] used the phase plane analysis to show that when ν < α the problem (4.43) possesses the fixed leftward and rightward asymptotic spreading speeds. Lemma 5.1 (Theorems 1.1 and 1. in []) Assume that ν < α and (u, g, h) is a global solution of (4.43). Then there exist positive constants c l and c r such that < c g(t) l := lim < c < c h(t) r := lim, t t where c l and c r depend on µ, ν, α and β; c is the asymptotic spreading speed of the logistic free boundary problem without the advection, which is given as k in Proposition 4.1 of Du and Lin [8] or c in Theorem 1.1 of Du and Lou [1]. If µ, ν, and β are fixed, then c l, c and c r are strictly increasing in α. Moreover, (i) If ν (, α) is fixed, then c l, c and c r are strictly increasing in µ, and lim µ c l =, lim µ c l = α ν, lim µ c =, lim µ c = α, lim µ c r =, lim µ c r = α + ν. (ii) If µ is fixed, then c l and c r are strictly increasing in ν, and lim ν c l = lim c r = c, ν lim ν c l =. α Theorem 5. Assume that (4.3) and h π ( 4 ν1) 1. (u, v, g, h) is a global solution of (1.3). Then there exist positive constants c l, c r, c l, and c r such that < c r < c l lim inf lim inf h(t) t g(t) t lim sup lim sup 4 h(t) c t r, g(t) t c l. (5.1)