A two-species weak competition system of reaction-diffusion-advection with double free boundaries

In this paper, we investigate a two-species weak competition system of reaction-diffusion-advection with double free boundaries that represent the expanding front in a one-dimensional habitat, where a combination of random movement and advection is adopted by two competing species. The main goal is to understand the effect of small advection environment and dynamics of the two species through double free boundaries. We provide a spreading-vanishing dichotomy, which means that both of the two species either spread to the entire space successfully and survive in the new environment as time goes to infinity, or vanish and become extinct in the long run. Furthermore, if the spreading or vanishing of the two species occurs, some sufficient conditions via the initial data are established. When spreading of the two species happens, the long time behavior of solutions and estimates of spreading speed of both free boundaries are obtained.


1.
Introduction. To study the spreading of a new or invasive species in population ecology, plenty of works have been done. In recent years, it has been attracting researchers' attention to understand the role that free boundary problem plays in the dynamics of species. In 2010, Du and Lin [10] studied a logistic model with free boundary. In their work, they elaborated the spreading of a new or invasive species, whose population density is represented by u(t, x), and they used one free boundary h(t) or double free boundaries g(t) and h(t) to represent the expanding fronts. Moreover, spreading-vanishing dichotomy and spreading speed were gained. Later on, a growing number of researchers investigated further extensions, including the model in higher dimension spaces and in spatial heterogeneous environment [7,8] and so on. Besides the homogeneous and spatial heterogeneous environment, if the environment is heterogeneous time-periodic, plenty of results [9,34] have been studied for the case recently. Taking the place of logistic reaction term, Du and Lou [12] investigated the general nonlinear term f (u), containing general monostable, bistable and combustion types. To get more related results on general models for single species case, we refer to [2,25,32,41] and the references cited therein.
For two species case, a typical model is the Lotka-Volterra type competition system. In [18], the two weak competition species shared the same free boundary.
We will elaborate it later. By removing the restriction of weak competition, Wang and Zhao [37] presented a more complete description for the model. Du and Lin [11] and Wang and Zhang [35] considered the diffusive competition problem, which is described as the invasion of a superior or inferior competitor. In their problem, an invasive species exists in a ball initially, and invades into the environment, while the resident species distributes in the whole space. Two different free boundaries for the two species were considered by Guo and Wu [19] and Wu [39]. By using more specific considerations, they established a spreading-vanishing trichotomy, a spreading-vanishing quartering, the notion of the minimal habitat size and so on. For more results, we refer to [6,38] and references cited therein.
The model describes the competition between two species with population densities u(t, x) and v(t, x) at time t and position x. D is diffusion coefficient of species v, r is intrinsic growth rate of species v, k and h are interspecific competition coefficients, the rest parameters are 1 due to nondimensionalization. From the perspective of biology, this model describes the way of the two competing species invade over a one-dimensional habitat with the initial region [0, s 0 ]. It is assumed that the zero Neumann boundary condition is imposed for x = 0. In addition, they supposed that both species have a expanding front s(t), which represents a trend to emigrate from the right boundary to get their new habitat. Here, s(t) satisfies the well-known Stefan type condition. For more biological background of the Stefan type condition, we can refer to [2,20,21,27].
In their paper, they considered the weak competition case: 0 < h, k < 1, and gave the criteria for spreading or vanishing. Besides, in the case of spreading, they obtained a more precise asymptotic behavior and an upper bound for the lim sup t→∞ s(t)/t, which shows that the asymptotic spreading speed can not be faster than the minimal speed of travelling wavefront solutions of the problem in the whole line without a free boundary.
In addition, as a result of rich resource, appropriate climate and so on, organisms can often sense and respond to local environmental factor by moving towards one direction in the field of population ecology. For instance, some diseases spread along the wind direction. In 2009, the propagation of West Nile virus from New York City to California state was studied by Maidana and Yang in [28]. In the summer of 1999, it was observed that West Nile virus appeared for the first time in New York City. In the second year, the wave front travelled 187km to the north and 1100km to the south. Hence, they gave thought to the advection movement and showed that bird advection becomes an important factor for lower mosquito biting rates. In [1], the effect of intermediate advection on the dynamics of two-species competition system was considered by Averill, and a specific range of advection strength for the coexistence of two competing species was provided. Additionally, they illustrated three different kinds of transitions from small advection to large advection theoretically and numerically. From a mathematical point of view, in order to contain the effect of advection, it is one of the simplest but probably still realistic approaches that supposing species can move up along the gradient of the density.
For the case of fixed boundary, there are also many results involving the effect of the advection term. In [4], Cantrell, Cosner and Lou studied a Lotka-Volterra model for two competing species in a heterogeneous environment, i.e., in Ω × (0, ∞), with no-flux boundary conditions This model describes that in a bounded region Ω in R N , species v disperses by random diffusion only, species u disperses by both random diffusion and advection along an environmental gradient. The boundary ∂Ω is smooth, n denotes the unit normal vector on ∂Ω, and the no-flux boundary condition means that no individuals cross the boundary. The migration rates µ and ν are two positive constants, and intrinsic growth rates m(x) of species is assumed to be twice continuously differentiable inΩ. Based on this model, Chen, Lam and Lou [5] studied a model with both species disperse by random diffusion and advection. More models involving the effect of the advection term can be found in related references.
For the case of free boundary involving the effect of the advection term, some results can be found in [14,15,16,17,22]. Gu et al. [14,15] considered the following single case u(t, g(t)) = 0, g (t) = −µu x (t, g(t)), t > 0, where β and µ are positive constants, and the initial function (u 0 , h 0 ) satisfies This model describes the spreading of a new or invasive species under the influence of dispersal and advection (expressed by βu x ). Especially, they studied the influence of small advection coefficient β ∈ (0, 2) on the long time behavior of the solutions, and got a spreading-vanishing dichotomy. Moreover, they gave a sharp threshold between spreading and vanishing. Furthermore, when spreading happens, the asymptotic spreading speed is derived. i.e., there exist two positive constant c * l and c * r such that −c * l := lim which shows how the advection term βu x influences the spreading speed. When the right term is replaced by a more general nonlinear term f (u), which is monostable, bistable or combustion type, such results were carried out in [16,17,22].
In [22], a much sharper estimate for the different spreading speeds of the fronts was obtained, i.e., when β ∈ (0, c 0 ), where c 0 is the minimal speed of the travelling waves of the problem q − cq + f (q) = 0, q > 0 in R, Apart from the above results, how the solution approaches a semi-wave was also described.
Motivated by the above works, specifically, we consider the following problem (TFB): where x = g(t) and x = h(t) are free boundaries which satisfy the well-known onephase Stefan type condition, and the parameters µ and µρ measure the intention for spreading into new habitats of u and v, respectively. For both x = g(t) and x = h(t), we impose the Dirichlet boundary condition. Moreover, we assume D, r, c, b, µ, ρ > 0, and they represent the same meaning as stated in (1) (where k is replaced by c and h is replaced by b). β 1 and β 2 are positive advection coefficients that measure the tendency of the biased movement of species u and v respectively. Besides, the initial data In what follows, we only consider the nonnegative solutions, and focus on the weak competition case: Besides, we only consider the small advection case: We give some notations which are often used in the sequel: Notation.
We can see that h * < h * easily. We give the following definition: uniformly on any compact subset of (−∞, ∞).
The purposes of our paper are to analyse the model of two species competition with the influence of advection on the criteria for spreading and vanishing, the long time behavior of the solution, and the asymptotic spreading speed when spreading occurs. In this sense, this paper can be regarded as an improvement and extension of [18] with adding the advection effect.
The difference between our problem and [18] lies in the discussion of the following problem

BO DUAN AND ZHENGCE ZHANG
and the case in [18] In [18], Guo and Wu considered the case of β = 0, and they investigated the Neumann boundary condition on the left side x = 0 and the Dirichlet boundary condition on the right side x = s(t). In contrast to the problem (5), we consider the left boundary x = g(t), and boundary condition here is Dirichlet type, so the corresponding problems is (4). Therefore, the conclusion in [18] cannot be used directly in this paper. To overcome the difficulty induced by the advection term (β = 0) and the Dirichlet boundary condition on the left side, we calculate the principal eigenvalue for the new problem, and use it to determine the corresponding sharp threshold related to the spreading or vanishing of the two species, and find that both of the advection coefficients β 1 and β 2 can influence the sharp threshold, and the techniques of dealing with the two advection coefficients β 1 and β 2 are more complicated than the single case with only one advection coefficient β. Besides, we need to deal with the double free boundaries at the same time.
Moreover, we give a simple criterion governing spreading or vanishing of the two species, then by adding some more restrictions on the parameters, we derive a spreading-vanishing dichotomy, which was put forward by Du and Lin [10] for a case of single species initially. Last, by using the comparison principle, we give some sufficient conditions for the spreading and vanishing via the initial data (u 0 , v 0 , −h 0 , h 0 ). When spreading of the two species happens, we give two theorems on the long time behavior of solutions, a natural method is finding a pair of supersolution and subsolution to squeeze the solution. Nevertheless, to find them immediately seems not easy, therefore an iteration scheme with constructing better supersolutions and subsolutions step by step is adopted to derive our goal. For the spreading speeds of both h(t) and g(t), we can draw a conclusion that the advection can influence the spreading speed. For the upper bound of lim sup t→∞ h(t)/t and the lower bound of lim inf t→∞ g(t)/t, we adopt the idea of [18], which indicates that lim sup t→∞ h(t)/t (lim inf t→∞ g(t)/t) can not be faster (slower) than the minimal (maximal) travelling wave speed of the related problems. For the lower bound of lim inf t→∞ h(t)/t and the upper bound of lim sup t→∞ g(t)/t, we refer to [38]. Then we give a second estimate of spreading speed, which is better than the first one. The tool we here use is a modification of the single species case.
From this paper, we can see that when the interspecific competition coefficients are small, i.e., 0 < b, c < 1, the two species can coexist when spreading happens. Furthermore, when the advection coefficients are small, i.e., the advection influence is not strong, the two species share similar dynamic behavior with the case of β = 0, i.e., they will spreading or vanishing, and no third case exists. So natural questions arise such as: (1) When 0 < b < 1 ≤ c or 0 < c < 1 ≤ b, i.e., one species has a more competition ability than the other, can the two species coexist, or one overcome another? (2) When the advection coefficients are large, can spreadingvanishing dichotomy still exist? Or does the third case which between spreading and vanishing happen? These interesting problems will be our future research subjects.
The organization of this paper is as follows. In Section 2, we show the problem (TFB) exists a unique global solution. In Section 3, we show several preliminaries including the comparison principles and so on, which will be used throughout the paper. In Section 4, we prove the long time behavior of solutions when g ∞ = −∞ and h ∞ = ∞. In Section 5, we demonstrate both h ∞ and g ∞ are finite or infinite simultaneously, and give some criteria for spreading and vanishing. In Section 6, we provide estimates of spreading speed of both h(t) and g(t) when spreading happens, and give a more detailed depiction on the long time behavior of solutions. In Appendixes A, B and C, we give the brief proofs of Lemmas 2.2, 2.3 and Theorem 2.1, respectively.
2. Existence and uniqueness. In this section, we give Theorem 2.1, which is the global existence and uniqueness of the solution of the problem (TFB). The proof of Theorem 2.1 is standard by modifying the ideas of [10] and [18], and we put the brief proofs of this section in Appendixes A, B and C.
In order to prove Theorem 2.1, we give the following two lemmas.
In order to prove the global existence of solution, we need the following lemma.

Preliminaries.
In this section, we will give some basic lemmas which will be used in the following part. Consider the problem (P 0 ): where r > 0, L 1 and L 2 are constants. The first lemma will be used frequently in the later sections, which can be considered as a special case of Corollary 3.4 in [3].
, then one of the following happens:  Proof. For any τ ≥ 1, t ∈ [τ, τ + 1], by the standard transformation the problem (TFB) can be replaced by a fixed boundary problem. By the standard L p theory and the Sobolev embedding theorem, we obtain thatû andv have a uniform C 1+α 2 ,1+α bound over {(t, y) : t ∈ [τ, τ + 1], −1 ≤ y ≤ 1}, where α ∈ (0, 1). Notice that this uniform bound is independent of τ , hence by using the Stefan type condition in (3), there exists a positive constant C such that We only prove lim t→∞ g (t) = 0, since another is parallel. Assume there exists a sequence {t n } satisfying t n → ∞, g (t n ) → σ 1 as n → ∞ for some σ 1 < 0. Due to (10), we can find ε > 0 small enough such that g (t) ≤ σ1 2 for all t ∈ [t n − ε, t n + ε] and for all large n. Then we get [10,18] for t ≥ 0, then the solution (u, v, g, h) of the problem (TFB) satisfies: Next we give a variant of Lemma 3.3, whose proof only requires some obvious modifications. If uniformly on any compact subset of (−∞, ∞).
In order to prove Theorem 4.1, we give some results firstly.   Then we obtainū n >ū n+1 > 0, and v n < v n+1 < 1 for all n ∈ N. In addition, (ii) Define two sequences {u n } n∈N and {v n } n∈N as follows: Then we obtain u n < u n+1 < 1, andv n >v n+1 > 0 for all n ∈ N. In addition, Proof. The process is similar to Lemma 4.2 of [18], so we omit the details.
In order to use the comparison principle for u and v, constructing some suitable supersolutionŪ (t, x) and subsolution V (t, x) is what we need. Let a ε : where α, b ε are to be determined. It is easy to obtain that φ(t) = a ε − aε It is easy to seeŪ (t, The region in which (13) holds is to be determined.
Last, we obtain lim sup t→∞ v(t, x) ≤v 2 and lim inf t→∞ u(t, x) ≥ u 2 uniformly on any compact subset of (−∞, ∞) in a similar way.
In order to prove Theorem 4.1, we continue the strategy as above to get the following corollary.
uniformly on any compact subset of (−∞, ∞). Now, we are ready to give the proof of Theorem 4.1.
Proof of Theorem 4.1. Letting n → ∞ in Corollary 1 and using Lemma 4.4, we can get the results.
Besides Theorem 4.1, we will give a more detailed depiction on the long time behavior of solutions when spreading of the two species happens, which is stated in Section 6. 5. The criteria governing spreading and vanishing. In this section, we first give a simple criterion for the spreading or vanishing of the two species, then a spreading-vanishing dichotomy is presented. Finally, based on the previous results, we give a corollary.
By Lemma 2.3, we get −g ∞ , h ∞ ∈ (0, ∞], where g ∞ := lim t→∞ g(t) and h ∞ := lim t→∞ h(t). Therefore, we have four cases: Recall the notation of h * from page 5, we get Lemma 5.1, which indicates that the last two cases do not happen, i.e., both g ∞ and h ∞ are finite or infinite simultaneously.
Proof. We divide the discussion into three cases: Case(i). In this case, we have .
The idea of the proof comes from Lemma 3.1 of [26], whose difficulty is induced by the introducing of advection term and double free boundaries. By a contradiction, there are three cases: Without loss of generality, we only assume cases (1) and (2): g ∞ ≥ −∞, h ∞ < ∞, (Whether g ∞ > −∞ or g ∞ = −∞ does not make contributions to the proof), since the proof of case(3) is parallel.
By a similar argument as in Case(i), the proof of Case(ii) can be obtained, so we omit the details here. For Case(iii), without loss of generality, we assume D r Then by using the same method as in Case(i), we can prove it.
Recall the notation of h * from page 5, we get the following lemma.
, we have Proof. We give an argument by a contradiction. Since the proof of D that v(t, ·) converges to some function in C 2 ((g ∞ , h ∞ )), our goal is to obtain a contradiction. Such idea comes from [7]. Let (u, v, g, h) be the solution of the problem (TFB) andū be the solution of Then, by Lemma 3.1, Comparing (ū, 0) with (u, v), we get On the other hand, letv be the solution of By Lemma 3.1 again, we have where v ∞ > 0 satisfies Combining (22) with (24), we get lim sup Next, we estimate lim inf , h ∞ − g ∞ with g n ↓ g ∞ and h n ↑ h ∞ as n → ∞, and fix h 1 − g 1 which is close enough to h ∞ − g ∞ , then {h n − g n } can hold the following property: for all n ∈ N. Thanks to Lemma 3.1, for each fixed n, there exists a unique v n (x) > 0 satisfying For each j ∈ N, as v n is bounded in C 2+α ([g i , h i ]) for all n ≥ j, therefore, by using the Arzela-Ascoli Theorem and the diagonal process, we get v n → v ∞ in C 2 loc ((g ∞ , h ∞ )) as n → ∞ (up to a subsequence), where v ∞ satisfies (23). For each n ∈ N, recalling (20), we can find T n > 0 such that Besides, g(t) < g n and h(t) > h n for t > T n . Last, let v n (t, x) be the solution of From (26), we also see that Therefore, due to (19) and (21), we shall compare as t → ∞. Thus, for each n, Taking n → ∞, we get where v ∞ satisfies (23). Combining (25) with (27), we obtain lim Finally, following the process of the proof of Lemma 2.2 in [7], we derive Therefore, by using (3), we find β > 0 such that g (t) ≤ −β and h (t) ≥ β for all large t. But this is in contradiction with Lemma 3.2. Then the proof is completed.
Based on the above two lemmas, we have the following two theorems.
be a solution of the problem (TFB). Then we have (i) If h ∞ − g ∞ ≤ 2h * , then vanishing of the two species happens.
(ii) If h ∞ − g ∞ > 2h * , then spreading of the two species happens.
Combining with (28), we finish the proof.
(ii) The proof is a direct result of Lemmas 4.3 and 5.1.
As we can see, when 2h * < h ∞ − g ∞ ≤ 2h * , Theorem 5.3 does not give any information for spreading or vanishing. If some more restrictions on the parameters for the problem (TFB) are added, that is, the following sets are introduced: then we can get a spreading-vanishing dichotomy, and it was introduced by Du and Lin [10] for a single species case for the first time.
Proof. To prove the result, by Theorem 5.3, it is enough to show that g ∞ = −∞ and h ∞ = ∞ when h ∞ − g ∞ > 2h * . When (D, r, β 1 , β 2 , b, c, µ, ρ) ∈ A ∪ B, we get Then by Lemma 5.2, we obtain Thus by Theorem 5.3, we have the result. , then either b or c is small, a spreadingvanishing dichotomy can be obtained.
In the last part of this section, as a corollary, for the spreading or vanishing of the two species happens, we give some sufficient conditions via the initial data. , where δ < min{δ 1 , δ 2 , δ 3 }, and δ 1 , δ 2 , δ 3 satisfy respectively. Then vanishing of the two species happens.
6. Spreading speed. In this section, we provide upper bound and lower bound for spreading speeds of both h(t) and g(t) in the spreading case, and give a more detailed depiction on the long time behavior of solutions.
In order to give Theorem 6.4, we give three lemmas first.