EXPANDING SPEED OF THE HABITAT FOR A SPECIES IN AN ADVECTIVE ENVIRONMENT

. Recently, Gu et al. [7, 8] studied a reaction-diﬀusion-advection equation u t = u xx − βu x + f ( u ) in ( g ( t ) ,h ( t )), where g ( t ) and h ( t ) are two free boundaries satisfying Stefan conditions, f ( u ) is a Fisher-KPP type of nonlinearity. When β ∈ [0 ,c 0 ), where c 0 := 2 (cid:112) f (cid:48) (0), they found that for a spreading solution ( u,g,h ), h ( t ) /t → c ∗ r ( β ) and g ( t ) /t → c ∗ l ( β ) as t → ∞ , and c ∗ r ( β ) > c ∗ r (0) = − c ∗ l (0) > − c ∗ l ( β ) > 0. In this paper we study the ex- panding speed C ∗ ( β ) := c ∗ r ( β ) − c ∗ l ( β ) of the habitat ( g ( t ) ,h ( t )), and show that C ∗ ( β ) is strictly increasing in β ∈ [0 ,c 0 ). When β ∈ [ c 0 ,β ∗ ) for some β ∗ > c 0 , [8] also found a virtual spreading phenomena: h ( t ) /t → c ∗ r ( β ) as t → ∞ , and a back forms in the solution which moves rightward with a speed β − c 0 . Hence the expanding speed of the main habitat for such a solution is C ∗ ( β ) := c ∗ r ( β ) − [ β − c 0 ]. In this paper we show that C ∗ ( β ) is strictly decreasing in β ∈ [ c 0 ,β ∗ ) with C ∗ ( β ∗ − 0) = 0, and so there exists a unique β 0 ∈ ( c 0 ,β ∗ ) such that the advection is favorable to the expanding speed of the habitat if and only if β ∈ (0 ,β 0 ).


1.
Introduction. Consider a reaction-diffusion-advection equation with free boundaries: where µ and h 0 are positive constants, β ≥ 0 is a parameter representing the strength of the advection, u 0 is a C 2 function with support [−h 0 , h 0 ], and f ∈ C 1 ([0, ∞)) is a Fisher-KPP type of nonlinearity which satisfies Typical examples for such functions include the logistic nonlinearity f (u) = u(1−u). When β = 0 (that is, there is no advection in the environment) and f (u) = u(1 − u), the problem (P) was studied by Du and Lin [2]. They used such a problem to model the spreading of a new or invasive species with population density u(t, x) over a one dimensional habitat, with two free boundaries x = g(t), h(t) representing the expanding fronts. Among others, they obtained a dichotomy result: either vanishing happens (that is, u(t, ·) → 0 as t → ∞ uniformly in [g(t), h(t)], and the limiting interval (g ∞ , h ∞ ) is a bounded one, where g ∞ := lim t→∞ g(t) and h ∞ := lim t→∞ h(t)), or spreading happens (that is, u(t, ·) → 1 as t → ∞, locally uniformly in R, and the limiting interval (g ∞ , h ∞ ) = R). Furthermore, when spreading happens, they obtained the existence of the asymptotic spreading speed ([2, Theorem 4.2]): In the last few years, the paper [2] has brought a small boom on the study of reaction diffusion equations with free boundaries. Among others, [3] also obtained the formula (1) for general equations and [4] improved it into the form h(t) = −g(t) = c t + O(1).
In the field of ecology, organisms can often sense and respond to local environmental cues by moving towards favorable habitats, and these movements usually depend upon a combination of local biotic and abiotic factors such as stream, climate, food and predators. For example, in studying the propagation of West Nile virus in North America, it was observed in [9] that West Nile virus appeared for the first time in New York city in the summer of 1999. In the second year the wave front travels 187 km to the North and 1100 km to the South, till 2002, it has been spread across almost the whole America continent. Therefore, the propagation of WNv from New York city to California state is a consequence of the diffusion and advection movements of birds. Especially, bird advection becomes an important factor for lower mosquito biting rates. Another example is that Averill [1] considered the effect of intermediate advection on the dynamics of two-species competition system, and provided a concrete range of advection strength for the coexistence of two competing species. Moreover, three different kinds of transitions from small advection to large advection were illustrated theoretically and numerically. Many other examples involving advection were also found in the field of ecology.
From a mathematical point of view, to involve the influence of advection, one of the simplest but probably still realistic approaches is to assume that species can move up along the gradient of the density. The equation is such an example. Recently, this equation (more precisely, the problem (P) with β > 0) was studied by [6,7,8,10], where the problem is used to model the spreading of a new species in an environment with advection. In [5], the authors investigated a SIS model like (P) (with advections and free boundaries). The impact of the spatial heterogeneity and the advection on the persistence and eradication of an infectious disease was studied.
In this paper, we focus on the impact of the advection strength β on the expanding speed of the habitat, which is the further investigation to the details of [8]. Gu, Lou and Zhou [8] gave a rather complete description for the long time behavior of the solutions. More precisely, there exists β * > c 0 (where c 0 := 2 f (0) is the minimal speed of the traveling waves of the Fisher-KPP equation [8, Theorem 2.2]): either vanishing happens, or virtual spreading happens (which means that −∞ < g ∞ < h ∞ = +∞, u(t, ·) → 0 locally uniformly in (g ∞ , ∞), u(t, ·+ct) → 1 locally uniformly in R for some c > 0), or the solution is a transition one in the sense that neither (virtual) spreading nor vanishing happens. (iii) in large advection case β ≥ β * , vanishing happens for all the solutions of (P) (cf. [8,Theorem 2.4]). In addition, [6,8] also studied the asymptotic spreading speeds and asymptotic profiles for (virtual) spreading solutions, which are of special importance from the ecological point of view. More precisely, when β ∈ (0, c 0 ) and when spreading happens for a solution u, [8, Theorem 2.
Therefore, the main part of the habitat domain is [(β − c 0 )t + o(t), h(t)], which expands with an asymptotic speed C * 2 (β) : A natural question is: Q: How does C * (β) depend on β ? Namely, how does the advection strength β affect the expanding speed of the habitat ?
In [8,Lemma 3.4], the authors proved that C * 2 (β) > 0 (resp. = 0, or < 0) if and only if β < β * (resp. β = β * , or β > β * ). As a consequence of such a result we see that the advection is not always favorable to the spreading. Especially, a large advection may cause vanishing for the species. The following result gives a complete answer to the question Q. Theorem 1.1. Assume β ∈ [0, β * ) and C * (β) is defined as in (5). Then From this theorem we immediately obtain the following corollary (see Figure 1).
This corollary indicates that when the advection strength is not strong (that is, when β ∈ (0, β 0 )), the advection environment is more favorable to the spreading of the species than an environment without advection. and µ = 1.

2.
Proof of the main theorem. In this section, we give the proof for Theorem 1.1. First, we use difference method to prove the following lemma.