THE SPREADING FRONTS IN A MUTUALISTIC MODEL WITH ADVECTION

. This paper is concerned with a system of semilinear parabolic equations with two free boundaries, which describe the spreading fronts of the invasive species in a mutualistic ecological model. The advection term is introduced to model the behavior of the invasive species in one dimension space. The local existence and uniqueness of a classical solution are obtained and the asymptotic behavior of the free boundary problem is studied. Our results indicate that for small advection, two free boundaries tend monotoni-cally to ﬁnite limits or inﬁnities at the same time, and a spreading-vanishing dichotomy holds, namely, either the expanding environment is limited and the invasive species dies out, or the invasive species spreads to all new environ- ment and establishes itself in a long run. Moreover, some rough estimates of the spreading speed are also given when spreading happens.


1.
Introduction. The spreading of species from their native habitats to alien environments is a serious threat to biological diversity [22], and mathematical investigation on the spreading of population has been attracting much attention.
In the pioneering works of Fisher [11] and Kolmogorov et al [17], the diffusive logistic equation over the entire space R: was studied, traveling wave solutions have been found for (1): For any c ≥ c * := 2 √ ad, there exists a solution u(x, t) := W (x − ct) such that W (y) < 0 for y ∈ R 1 , W (−∞) = a/b and W (+∞) = 0; no such solution exists if c < c * . The number c * is then called the minimal speed of the traveling waves. These results have been further developed in [1,2,15,26,27,28] and the references therein.
To describe the spreading process of invasive species and the front of the expanding habitat, Du and Lin [6] studied the following diffusive logistic problem, where the unknown u(x, t) stands for the population density of an invasive species over a one-dimensional habitat, and the unknown x = h(t) is the free boundary and is used to describe the expanding front. The free boundary condition is given by h (t) = −µu x (h(t), t), which means that the spreading front expands at a speed that is proportional to the population gradient at the front, the positive constant µ measures the ability of the invasive species to transmit and diffuse in the new habitat, see [20] for details. A spreading-vanishing dichotomy was first presented in [6] for problem (2), namely, as time t → ∞, the population u(x, t) either successfully establishes itself in the new environment (called spreading), in the sense that h(t) → ∞ and u(x, t) → a/b, or the population fails to establish and vanishes eventually (called vanishing), namely h(t) → h ∞ ≤ π 2 d a and u(x, t) → 0. It was also shown that if spreading occurs, for large time, the spreading speed approaches a positive constant k 0 , i.e., h(t) = [k 0 + o(1)]t as t → ∞. k 0 is then called the asymptotic spreading speed, which is uniquely determined by an auxiliary elliptic problem induced from (2). Furthermore, they found that k 0 < c * , where c * (:= 2 √ ad) is the minimal speed of the traveling waves [7]. Hereafter, Du and Guo [4] extended the free boundary problem (1.2) to a higher dimension domain.
Since the work of Du and Lin [6], there have been many theoretical developments on the free boundary problem in homogeneous environment. For example, Du and Lou [9] considered a two free boundaries problem with a general nonlinear term. In [12,13], Gu, Lin and Lou studied how advection term (βu x ) affects the asymptotic spreading speeds when spreading occurs. See also [16] for a free boundary problem with a general nonlinear term, [18,31] for diffusive logistic model in heterogeneous environment, [5] for diffusive logistic model in time-periodic environment, [23] for diffusive logistic model with seasonal succession, [19] for information diffusion in online social networks, [25,29] for Lotka-Volterra type prey-predator model and [8,14] for Lotka-Volterra type competition model.
In this paper, we consider a two-species mutualistic model, which was proposed by May [21] in 1976, and the model is described by the following coupled O.D.E. system: where r i , K i , α i , ε i (i = 1, 2) are positive constants. Linearization and spectrum analysis show that the unique positive equilibrium is locally asymptotically stable, and moreover, it is globally asymptotically stable in the positive quadrant by constructing Lyapunov functional.
Considering the spatial spreading, we assume that one species is native and the other is invasive. Inspired by the former work, we will study how invasive species are spreading spatially over further to larger area, especially, small advection will be introduced to consider the long time behavior of the native species and the invasive species. For simplicity, assume that the native species lives in the whole habitat (−∞, ∞), and the environment in g(t) < x < h(t) is occupied by the invasive species, whose density is denoted by u(x, t) and density of the native is denoted by v(x, t). Assume that h(t) grows at a rate that is proportional to population gradient of the invasive species at the front [20], then the conditions on the right front is Similarly, the conditions on the left front is In such a case, we have the problem for u(x, t) and v(x, t) with free boundaries x = g(t) and x = h(t) such that where x = g(t) and x = h(t) are the moving left and right boundaries to be determined, h 0 , µ and β are positive constants, and the initial functions u 0 and v 0 are nonnegative and satisfy In this paper, we assume that β < 2 √ r 1 d, it is well known that 2 √ r 1 d is the minimal speed of the traveling waves to the cauchy problem and it is also the maximal asymptotic spreading speed of the free boundary to problem (4) with v ≡ 0 ( [6]). The remainder of this paper is organized as follows. In the next section, the global existence and uniqueness of the solution to problem (4) are proved by using a contraction mapping theorem. Section 3 is devoted to sufficient conditions for the invasive species to vanish or spread, a spreading-vanishing dichotomy will be given. Some rough estimates of the spreading speed are also given in Section 4.

2.
Existence and uniqueness. In this section, we first present the following local existence and uniqueness result by the contraction mapping theorem and then give the global existence by using suitable estimates.

MEI LI AND ZHIGUI LIN
Proof. As in [30], we first straighten the double free boundary fronts by making the following change of variable: Then problem (4) can be deduced to This transformation changes the free boundaries x = h(t) and x = g(t) to the fixed lines y = h 0 and y = −h 0 respectively, and the equations become more complex, since now the coefficients in the first and second equations of (8) contain unknown functions h(t) and g(t).
The rest of the proof is by the contraction mapping argument as in [6,30] with suitable modifications, we omit the details here.
To show the global existence of the solution, we need the following estimates.
be a solution to problem (4) defined for t ∈ (0, T 0 ] for some T 0 ∈ (0, +∞). Then there exist constants C 1 and C 2 independent of T 0 such that Proof. The positivity of u and v are obvious since that the initial values are nontrivial and nonnegative, and the system is quasi-increasing. Considering its upper bounds, it is easy to show that The next lemma shows that the left free boundary for problem (4) is strictly monotone decreasing and the right boundary is increasing.
be a solution to problem (4) defined for t ∈ (0, T 0 ] for some T 0 ∈ (0, +∞). Then there exists a constant C 3 independent of T 0 such that Proof. Using the strong maximum principle to the equation of u gives that Hence h (t) > 0 for t ∈ (0, T 0 ] by using the free boundary condition in (4). Similarly, It remains to show that −g (t), h (t) ≤ C 3 for t ∈ (0, T 0 ] and some C 3 . The proof is similar as that of Lemma 2.2 in [6], but we sketch it here since we found that the advection term have different effects to the left and right free boundaries. Set and constitute an auxiliary function In the following proof, we will choose M such that w(x, t) is the supersolution of u(x, t) in Ω. Straightforward computation show that It follows that . On the other hand, . Making use of the comparison principle yields u( Recollecting the free boundary condition in (4) deduces where C 31 is independent of T 0 . Analogously, we can define over the region Since that u, v and g (t), h (t) are bounded by constants independent of T 0 , the global solution is guaranteed. In what follows, we exhibit the comparison principle, which can be proved similarly as Lemma 3.5 in [6].
then the solution (u, v; g, h) to the free boundary problem (4) satisfies Remark 1. The pair (u, v; g, h) in Lemma 2.5 is usually called an upper solution of problem (4). We can define a lower solution by reversing all the inequalities in the obvious places. Moreover, one can easily prove an analogue of Lemma 2.5 for lower solution.
We next fix v 0 and µ, let u 0 = σφ(x) and examine the dependence of the solution on σ, and we write (u σ , v σ ; g σ , h σ ) to emphasize this dependence. As a corollary of Lemma 2.5, we have the following monotonicity: 3. Spreading-vanishing dichotomy. It follows from Lemma 2.3 that h(t) and g(t) are monotone, and therefore there exists h ∞ , −g ∞ ∈ (0, +∞] such that lim t→+∞ h(t) = h ∞ and lim t→+∞ g(t) = g ∞ . Thus, we have four cases: (I) : The next lemma shows that the last two cases do not happen, both h ∞ and g ∞ are finite or infinite simultaneously. Lemma 3.1. If h ∞ < ∞ or g ∞ > −∞, then both h ∞ and g ∞ are finite and Proof. Without loss of generality, we assume that h ∞ < ∞, and divide the proof in the following steps.
Step 1. We prove that h ∞ − g ∞ ≤ 2πd √ 4r1d−β 2 . This step can be obtained by the similar argument in Lemma 5.2 of [6]. But much difficulty is induced by the introduce of β, we sketch it here to see the effect of β. By contradiction, if then one can choose T > 0 large and l > 0 such that It follows that there exists x 0 ∈ (g(T ), h(T )) such that Now, consider the eigenvalue problem Since l > πd √ 4r1d−β 2 , for small ε 3 > 0, the principal eigenvalue of problem (9), donoted by λ 1 , satisfies λ 1 < 0, and the following logistic-type problem admits a unique positive solution satisfies 0 < w < K1r1 1+K1ε1 in (x 0 − l, x 0 + l). By the maximum principle, one sees w (x 0 + l) < 0 < w (x 0 − l), so w has zero point in (x 0 − l, x 0 + l). Let the one that is closest to x 0 + l be x 1 ∈ (x 0 − l, x 0 + l). Then which is very crucial in constructing a sub-solution of u in [x 1 , x 0 + l] × [T, +∞) later.

It follows that
On the other hand, we have

This contradiction proves that
The comparison principle gives 0 ≤ u(t, x) ≤ u(t, x) for t > 0 and x ∈ [g(t), h(t)].
(1) The limit superior of the solution It follows from the comparison principle that (u( It is well known that the unique positive equilibrium (u * , v * ) is globally stable for the ODE system (14) and lim t→∞ (u(t), v(t)) = (u * , v * ); therefore we deduce lim sup uniformly for x ∈ (−∞, ∞).
(2) The lower bound of the solution for a large time.
Note that β < 2 √ dr 1 by assumption, there is L 0 such that This implies that the principal eigenvalues λ * 1 and µ * satisfy and the corresponding eigenfunctions can be choose as φ(x) = e β 2d x cos π 2L0 x and ψ(x) = cos π 2L0 x. Since h ∞ = −g ∞ = +∞, for any L ≥ L 0 , there exists t L > 0 such that g(t) ≤ −L and h(t) ≥ L for t ≥ t L .
Letting u = δφ and v = δψ, we can choose δ sufficiently small such that (u, v) satisfies , which implies that the solution can not decay to zero, this result will be used in the next part.
Similarly, if we consider µ instead of u 0 as a varying parameter, the following result holds, see also Theorem 4.4 in [8].

4.
Estimates of spreading speed. To derive the estimates of the asymptotic spreading speed, we first recall the known result for (4) with β = 0 and α 1 = 0, see Proposition 1 in [7].  where (c 0 , q(x)) is the unique positive solution of the problem Theorem 4.1 shows that if there is no advection, the asymptotic spreading speed of the left frontier and that of the right frontier are the same when invasive species is spreading.
where ω > 0. Usually, q(z) is called a semi-wave with speed c. We will derive the rightward spreading speed by this semi-wave. Consequently, for the leftward