BOUNDEDNESS AND PERSISTENCE OF POPULATIONS IN ADVECTIVE LOTKA–VOLTERRA COMPETITION SYSTEM

. We are concerned with a two–component reaction–advection– diﬀusion Lotka–Volterra competition system with constant diﬀusion rates sub- ject to homogeneous Neumann boundary conditions. We ﬁrst prove the global existence and uniform boundedness of positive classical solutions to this sys- tem. This result complements some of the global existence results in [Y. Lou, M. Winkler and Y. Tao , SIAM J. Math. Anal., 46 (2014), 1228–1262.], where one diﬀusion rate is taken to be a linear function of the population density. Our second result proves that the total population of each species admits a positive lower bound, under some conditions of system parameters (e.g., when the intraspeciﬁc competition rates are large). This result of population per- sistence indicates that the two competing species coexist over the habitat in a long time.


1.
Introduction. This paper considers a reaction-advection-diffusion system of (u, v) = (u(x, t), v(x, t)) in the following form x ∈ ∂Ω, t > 0, u(x, 0) = u 0 (x) ≥ 0, v(x, 0) = v 0 (x) ≥ 0, x ∈ Ω, (1.1) where Ω is a bounded domain in R N , N ≥ 2, with piecewise smooth boundary ∂Ω. System parameters D i , a i , b i , c i , i = 1, 2 and χ are all positive constants, and φ(u) is a smooth function. (1.1) was proposed in [20] to study the spatialtemporal behaviors of population distributions of the two competing species u and v. It is assumed that v moves over the habitat randomly and u moves through a combination of random and directed dispersal. In particular, from the viewpoint of mathematical modeling, u retreats from the region of v to avoid competition if χφ(u) > 0 and u invades the region of v to seek competition if χφ(u) < 0. The population kinetics are chosen to be the classical Lotka-Volterra type; ν is unit outer normal to ∂Ω and the Neumann boundary condition means that the domain is enclosed hence there is no population flux across the boundary; the initial data u 0 and v 0 are non-negative functions. See [20] for the derivation and biological justifications and significance of (1.1). We also want to draw the attention of the reader to a parallel work [13].
For Ω = (0, L), global existence and boundedness of classical solutions to (1.1) with φ(u) = u are proved by [20]. For Ω being a bounded domain in R N , N ≥ 2, global existence and boundedness are also obtained for the parabolic-elliptic counterpart of (1.1), provided that b1D2 b2χ is sufficiently large. When the sensitivity function φ only depends on v and satisfies φ(v) ≤ C 0 (v + 1) κ for some C 0 > 0 and κ < −1, global existence and boundedness of (1.1) are proved in [22]. On the other hand, from the viewpoint of mathematical modeling, (1.1) is biologically significant in that it gives rise to the hope of modeling a spatial segregation of these two species by adding even one single advection to the purely diffusive system. For instance, the authors of [20] also show that in 1D (1.1) admits transition layer steady states which model the segregation phenomenon through interspecific competitions. Through bifurcation theorems and singular perturbation methods, a shadow system of (1.1) is further studied by [24] to model the aforementioned segregation in the limit of large advection and small diffusion rates; moreover, the numerics there indicate that this system admits very interesting and complicated spatial-temporal dynamics, even over 1D domains.
One of the goals of this current work is to prove the global existence and boundedness of the fully parabolic system (1.1) over multi-dimensional domain under the following assumption φ(u) ≤ Ku m , ∀u > 0, (1.2) where K and m are positive constants, the former being arbitrary and the latter satisfying some technical conditions to be made precise. Our first main result reads as follows.
2N , then for any nonnegative u 0 ∈ C 0 (Ω) and v 0 ∈ C 1 (Ω), there exists a unique pair (u, v) of nonnegative functions belonging to C 0 (Ω × [0, ∞)) ∩ C 2,1 (Ω × (0, ∞)) which solves (1.1) classically in Ω × (0, ∞). Moreover, the solution is bounded by a positive constant C in the following sense We would like to point out that the condition m < N +2 2N is merely technical and we have to leave the global well-posedness open of (1.1) in this paper if this condition fails-see [21] for the work on (1.1) with nonlinear diffusion for instance.
Before presenting our second main result, we remind the reader that in the absence of advection, i.e., when χ = 0, (1.1) reduces to the following classical diffusive Lotka-Volterra competition system the spatial-temporal dynamics of which have been extensively studied over the past few decades. For example, the global existence and boundedness of (1.4) are proved in [3] using maximum principles. Existence or non-existence of stable nonconstant positive steady states is investigated in [11,12,15,16] etc. Extinction through competition is studied in [7,8,10,17]. Generally speaking, species u and v can coexist over the habitat and both persist if the interspecific competition is weak.
However, if the interspecific competition is strong, the dynamics depend on the initial data and it is possible that initially superior species will dominate over and eventually wipe out their interspecific competitors. This phenomenon is referred to as the extinction through competition. We want to point out that large time behaviors of (1.4) under Dirichlet boundary conditions have been studied by various authors [2,4,5,6].
For a better presentation of our second goal in this paper, and without losing much generality, we choose D 1 = D 2 = 1 and φ(u) = u in (1.1) and have x ∈ Ω. (1.5) We shall show that under the following assumption u and v in (1.5) will coexist over the habitat and both species survive in a long time, hence the phenomenon of extinction through competition does not occur. To be precise, let us denote the spatial mean value of a function bȳ then the second main result of this paper is the following Theorem.

Remark 2.
It is interesting to understand if the condition (1.7) is optimal. Though the latter inequality is technically assumed, it seems that a1 a2 < b1 b2 is necessary. For example, if a1 a2 > b1 b2 , we can easily show that the semi-steady-state ( a1 b1 , 0) is locally stable. Therefore, if we choose (u 0 , v 0 ) to be around this semi-steady-state, (u, v) → ( a1 b1 , 0) as t → ∞ and Theorem 1.2 does not hold any more. Condition (1.8) holds when both u 0 and v 0 are strictly positive.
Ecologically, (1.9) and (1.10) mean that the competing species u and v exist over the habitat and the total population of both species persist in a long time. We observe that (1.6) holds for large χ if the intraspecific competition rates b 1 and c 2 are large, with the rest parameters being fixed. This indicates that weak directed dispersal intensity supports the coexistence of interspecific competing species in model (1.5).
2. Local existence and preliminary results. To establish the global existence and boundedness of classical solutions to (1.1), we start with the following results on existence and extension properties of its local solutions due to the classical theories of Amann [1].
Let Ω be a bounded domain in R N , N ≥ 2, with smooth boundary ∂Ω. Then for any initial data satisfying Proof. It is easy to see that (1.1) is a triangular system. Therefore the existence of maximal classical solutions follows from Theorem 14.4 and Theorem 14.6 in [1]. Moreover, the extensibility property follows from Theorem 15.5 in [1].
Next we collect some basic properties of solutions to (1.1).
Proof. First of all, the nonnegativity of u and (2.1) follow upon direct applications of parabolic maximum principle. From the u-equation, we have From the first inequality in (2.5) we also get and by integrating (2.6) over (t 0 , t 0 + T ) we obtain Similarly, we can show that the inequality holds for v and (2.4) is verified.
According to Proposition 1 and (2.1), in order to prove Theorem 1.1, we only need to prove the L ∞ -boundedness of u . In light of the well-known results on global existence of reaction-advection-diffusion system (e.g. Lemma A.1 in [18]), it is sufficient to prove the boundedness of ∇v L ∞ . To this end we first convert v-equation into the following abstract form where g(u, v) = (a 2 − b 2 u − c 2 v)v, then we can employ the smoothing properties of the operator −∆ + 1 to prove the following Lemma for future reference. Here we include its proof for the completeness.
Proof. After applying the L p -L q estimates between semigroups {e t∆ } t≥0 in Lemma 1.3 of [23] on (2.7), thanks to (2.1), we can find positive constants C 1 , where ν is the first Neumann eigenvalue of −∆. On the other hand, we see that under the conditions in Lemma 2.2 In the sequel, for better organizations of the manuscript and consistency of notations, we shall, in all space integrations, skip the differential dx without confusing the reader. The following results are immediate conclusions from (2.2) and Lemma 2.2. (2.10) For any N ≥ 2 we readily see that N N −1 ≤ 2. Indeed the following lemma states that s = 2 can be achieved in (2.10). Proof.
Using v-equation, we have from the integration by parts that 1 2 . Thanks to the Sobolev interpolation inequality and boundedness of v L ∞ , there exist positive constants C 4 and C 5 such that Multiplying (2.13) by b1 4µ and adding it to the first inequality in (2.5), we can get that for all t ∈ (0, then we can conclude from Grönwall's lemma that ∇v(·, t) L 2 (Ω) is bounded.

Existence and boundedness of global solutions.
This section is devoted to the proof of Theorem 1.1. The main vehicle of our approach is to prove the boundedness of ∇v L ∞ which, thanks to Lemma 2.2, reduces to prove that u L p is bounded for some p > N .
3.1. A priori estimates and global existence. For any p > 1, we test the u-equation in (1.1) to obtain that which, in light of (1.2) and the following fact due to Cauchy-Schwartz gives rise to where C 7 is positive constant.
Lemma 3.1. Let (u, v) be the solution of (1.1), then for any p > 1 there exists a positive constant C(p) such that Assuming that Lemma 3.1 holds, we present the proof of our main result.
Proof of Theorem 1.1. By taking p large but fixed in (3.3), we have from Lemma 2.2 that ∇v(·, t) L ∞ < C for all t ∈ (0, T max ) and this, together with standard Moser-Alikakos L p iteration (cf. Lemma A1 in [18]), gives rise to the boundedness of u(·, t) L ∞ < C for all t ∈ (0, T max ). Therefore we must have that T max = ∞ and the local solution (u(x, t), v(x, t)) is global. Finally, one can apply the classical parabolic theory to prove that the classical solution satisfies the regularity properties. This completes the proof of Theorem 1.1.

3.2.
Proof of Lemma 3.1. We are left to prove Lemma 3.1 to conclude this section. In order to estimate Ω u p through (3.2), we shall work on the combined integrals Ω u p + Ω |∇v| 2q as in [18].
Proof of Lemma 3.1. For any q > 1, we have from the v-equation in (1.1) and integration by parts that 1 2q In view of the pointwise identity upon an integration by parts, the first term in the second line of (3.4) becomes To estimate the boundary integral in (3.5), we have from Lemma 2.2-2.4 in [9] and fractional Gagliardo-Nirenberg inequality that for some positive constants C Ω that vary from line to line below ∂Ω |∇v| 2q−2 ∂|∇v| 2 ∂n ≤C Ω |∇v| q 2 where r ∈ (0, 1 2 ) and h 1 : ∈ (0, 1), and we have used the fact (2.11) in Lemma 2.3. By Young's inequality, for any > 0 we have that On the other hand, since we have in (3.5) that For the second term in the second line of (3.4), we have from the integration by parts that To estimate the last four terms in (3.7), we first apply the fact |∆v| 2 ≤ N |D 2 v| 2 and Cauchy-Schwarz inequality to show that Similarly for the rest terms we can estimate Here we note that |∇v| 2q−4 ∇|∇v| 2 2 = 4 q 2 ∇|∇v| q 2 and for some C 14 > 0 From these facts, (3.12) entails that 1 2q For any fixed q > 1, we invoke Gagliardo-Nirenberg-Sobolev inequality to estimate that with h 2 := where > 0 is any positive constant and the last identity follows from the fact |∇v| q 2 L 2 q (Ω) = ∇v 2q L 2 (Ω) and Lemma 2.3. Therefore (3.13) implies that 1 2q with C 20 = C 12 + C 13 and C 21 = C 9 + C 14 + C 19 . Combining this inequality with (3.2) yields d dt To further estimate (3.15), we use the Young's inequality to obtain (3.17) Therefore, (3.15) implies d dt where we put For each κ i , i = 1, 2, we invoke Gagliardo-Nirenberg-Sobolev inequality to estimate where Since m < N +2 2N , by straightforward calculations we can choose p and q to be sufficiently large satisfying such that θ i ∈ (0, 1) and κi q θ i ∈ (0, 2) for each i = 1, 2. Therefore we can obtain from Cauchy-Schwarz that there exists some C 31 > 0 such that Since Ω u p ≤ b1 4 Ω u p+1 + C 32 , we put and conclude from (3.18) that y (t) + y(t) ≤ C 33 . Solving this ODI using Grönwall's lemma gives rise to (3.3). 4. Population persistence of both species. We proceed to prove Theorem 1.2 which states that both species persist for all the time and extinction through competition does not occur in (1.5) under conditions (1.6)-(1.8). Here one does not require the advection coefficient χ to be positive or negative, which models the repulsion and attraction between the competing species respectively. Moreover (1.7) holds for large χ if both b 2 and c 1 are sufficiently large. Our proof of Theorem 1.2 is motivated by and follows [19], by combining several preliminary results. We start with the following lemma.  (1.5) in Ω × (0, ∞). Assume that condition (1.6) holds, then there exists a positive constant A 1 such that for all t ≥ 0 Proof. We test the u-equation in (1.5) by 1 u and have from the integration by parts that Applying Cauchy-Schwarz inequality gives us By straightforward calculations involving v-equation, we have Multiplying (4.4) by χ 2 4 and then adding it to (4.3), we arrive at the following d dt and A 1 > 0 in light of (1.6), (1.7) and the fact Lemma 4.2. Let (u, v) be a positive classical solution of (1.5) in Ω × (0, ∞). Let L be any given positive constant and assume that there exists t 0 > 0 such that Ω ln u(·, t 0 )dx ≥ −L, then for any T > 0 satisfying

5)
where A 1 , K 0 and M 0 are given in (4.1), (2.1) and (2.2) respectively, we have that Proof. Integrating (4.1) over (t 0 , t 0 + T ), we have that which, in light of the fact that ln ξ < ξ for all ξ > 0, entails that Now for T being sufficiently large as in (4.5), we have from which (4.6) readily follows. and define Then for any T > 0 satisfying (4.5), we have that |S 1 | ≥ ηT M0 . Proof. Denote the complement of set S 1 by which in light of (4.6) and (4.8) implies that

QI WANG, YANG SONG AND LINGJIE SHAO
For fixed positive constants K and M , we define the following sets and with their complements being S c i := (t 0 , t 0 + T )\S i , i = 2, 3. . Then for any T satisfying (4.5) and the following condition we have that Proof. The definitions of S 2 and S 3 imply and t0+T t0 Ω Thanks to (2.4), (4.14) entails that On the other hand, by applying the Cauchy-Schwartz to (4.2) and then using (4.7), we obtain from the definition of A 1 that thanks to the facts (2.2), (2.4), ln ξ < ξ for all ξ > 0 and Ω ln u(x, t 0 )dx > −L. Now we have from (4.17) that . (4.18) Finally in light of the conditions on K and M , as well as (4.5) and (4.12), we conclude from Lemma 4.3, (4.16) and (4.18) that This finishes the proof.
From now one, we assume that the positive constants K and M satisfy the conditions in Lemma 4.4 and T satisfies both (4.5) and (4.12). Therefore according to Lemma 4.4, the set S 1 ∩ S 2 ∩ S 3 ∩ (t 0 + η 2M0 T, t 0 + T ) is always nonempty. We give another important about this set in the following lemma. Here both C η,K and L * 0 are independent of t * 0 .
where η * = max{η, e −L * 0 } and L * 0 is given by (4.19). Proof. Our proof is based on recursive applications of Lemma 4.5. Under the conditions in Lemma 4.4, we readily see that the set S 1 ∩ S 2 ∩ S 3 ∩ (t 0 + η 2M0 T, t 0 + T ) is always nonempty for any t 0 such that Ω u(x, t 0 )dx > −L * 0 . Choosing t 0 = 0 in Lemma 4.5, we can find some where L * 0 is given by (4.19) and it is independent of t * 0 . Now put t 0 as t 1 in Lemma 4.5, then we can find some where again L * 0 is independent of t * 1 . We can repeat this process as follows: for any t k , there always exists t k+1 = t * k ∈ (t k + η 2M0 T, t k + T ) such that Ω ln u(x, t k+1 )dx ≥ −L * 0 .
Obviously the sequence {t k } ∞ k=1 is increasing and t k → ∞ as k → ∞ thanks to η > 3M0 4 and (4.8). Each t k is contained in S 1 hence Ω u(x, t k )dx ≥ η. Moreover applying Jensen's inequality on Ω ln u(x, t k )dx > −L * 0 implies that Ω u(x, t k )dx ≥ e −L * 0 . Together with the fact that t k ∈ S 1 for each k ∈ N, this proves (4.22).
Finally we give the proof of the second main result of our paper.
The proof of Theorem 1.2 completes.