EVOLUTION OF DISPERSAL IN ADVECTIVE HOMOGENEOUS ENVIRONMENTS

. The eﬀects of weak and strong advection on the dynamics of reaction-diﬀusion models have long been investigated. In contrast, the role of intermediate advection still remains poorly understood. This paper is devoted to studying a two-species competition model in a one-dimensional ad- vective homogeneous environment, where the two species are identical except their diﬀusion rates and advection rates. Zhou (P. Zhou, On a Lotka-Volterra competition system: diﬀusion vs advection, Calc. Var. Partial Diﬀerential Equations , 55 (2016), Art. 137, 29 pp) considered the system under the no- ﬂux boundary conditions. It is pointed that, in this paper, we focus on the case where the upstream end has the Neumann boundary condition and the down- stream end has the hostile condition. By employing a new approach, we ﬁrstly determine necessary and suﬃcient conditions for the persistence of the corre- sponding single species model, in forms of the critical diﬀusion rate and critical advection rate. Furthermore, for the two-species model, we ﬁnd that (i) the strategy of slower diﬀusion together with faster advection is always favorable; (ii) two species will also coexist when the faster advection with appropriate faster diﬀusion. 92D25.

1. Introduction. All species disperse to some extent, in part because resources become limited locally as population grows. The persistence and population dynamics of a species are strongly influenced by dispersal. Therefore, evolutionary biologists and ecologists have been fascinated by the question of why individuals disperse for many decades. So far, they have proposed many kinds of models to represent dispersal patterns, investigate dispersal processes, elucidate the consequences of dispersal for population and communities, and then explain dispersal evolution.
Among these models are ones described by reaction-diffusion equations, which can describe the uneven distributions of concentrations of individuals across an area. One of the most successful examples is the two-species Lotka-Volterra competitiondiffusion system (see, for example, the books [1,14] and some latest advances in [5,9,11]). One well-known and widely accepted result on the evolution of dispersal is due to Hastings [4]. He found that as long as the organisms disperse by random diffusion only, slow diffusion rate will be selected provided that the environment is spatially heterogeneous but constant in time (slower diffuser wins).
As a further development of traditional reaction-diffusion models, recently there is a growing interest in the study of population dynamics in advective environments, where due to certain external environmental force (such as wind, water flow, gravity, etc), there appears a new advection term in the modeling system. Most obviously, it occurs in rivers where individuals are transported downstream by the water flow. In [9], Lou and Lutscher prescribed various boundary conditions at the upstream and downstream ends motivated by different ecological scenarios. As long as advection terms incorporated in classical Lotka-Volterra competition-diffusion systems, the study of the persistence for the corresponding single equation seems nontrivial, and the global dynamics of the resulting systems is very challenging.
A typical example from river ecosystems (see [17]) is given by the following logistic type reaction-diffusion-advection equation where u(x, t) denotes the population density of an aquatic species at location x and time t > 0, d > 0 is the random movement rate due to water turbulence or self-propelling, α > 0 measures the velocity of advective transport caused by river flow, L > 0 signifies the length of the river, and r > 0 accounts for the intrinsic growth rate. The single species problem (1) with different types of boundary conditions has been extensively studied; see Dirichlet type in [17], Neumann type in [20] and Robin type in [12,22].
In this paper, we try to study the equation (1) with special scenarios, and then investigate complexity of competitive consequence when a new or invasive species is introduced into such advective environments. To be precise, the system to be investigated is as follows.
where u(x, t) and v(x, t) represent the population densities of the two competing species at location x and time t > 0, respectively; d 1 , d 2 > 0 denote the random diffusion rates of the two species; α 1 , α 2 > 0 stand for the effective advection rates; and r > 0 signifies the intrinsic growth rate for both species, which indicates that the environment is spatially homogeneous. The upstream end (x = 0) is the Neumann boundary condition and the downstream end (x = L) is the hostile condition. Without loss of generality, we assume that α 1 < α 2 as the results for the cases where α 1 > α 2 can be stated symmetrically. We note here that Tang and Chen [18] concerned the special case α 1 = α 2 with more general boundary conditions. To our knowledge, currently there seems very few work attempting to understand the population dynamics of system (2) with the special scenarios where the upstream end is the free-flow condition(Neumann type) and the downstream end is the hostile condition(Dirichlet type). Biologically, we consider the special scenarios that the river flows from a lake to a sea. The study of system (2) will enable us to better understand the effect of boundary conditions, diffusion rates and advection rates on the outcome of competition.
1.1. Motivation and related work. In the last few years, various special cases and variants of system (2) have been qualitatively investigated.
We begin with the closed environment, that is, both the upstream end and the downstream end are no-flux boundary conditions which means that no individuals will pass through the boundary. If d 1 = d 2 and α 1 = α 2 , Lou, Xiao and Zhou [10] have confirmed that weak advection is more beneficial for species to exclude its competitor; while if d 1 = d 2 and α 1 = α 2 , it was proved by Lou and Zhou [12] that the competitor with faster diffusion rate would displace the slower one, that is, faster diffusion will evolve, in sharp contrast to the well-known "slower diffuser wins" in non-advective case [2,4]. The general case d 1 = d 2 and α 1 = α 2 was later investigated by Zhou [24], where, among other things, two observations were found: (i) the strategy of faster diffusion together with slower advection is always favorable, which can be seen as a generation of [10] and [12]; (ii) two species will coexist when the faster advection with appropriate faster diffusion.
For the other scenarios, Lou [12] and Zhou [27] introduce a net loss of individuals at the downstream end to describe the different scenarios. Mathematically, as the net loss of individuals varies from zero to infinity, it may yields different types of boundary conditions, including the standard Neumann, Robin and Dirichlet types.
However, it seems few work focusing on the special scenarios where the upstream end is the Neumann boundary condition. We will address on the special scenarios in this paper.

1.2.
Main results. In fact, system (2) generates a monotone dynamical system and the qualitative properties of its steady states almost determine the potential population dynamics. For monotone dynamical systems, there are some well known results. To mention a few, see [6, Proposition 9.1 and Theorem 9.2] and [16]. In particular, we can see that Proposition 1. If system (2) has no coexistence steady state, then one of the steady states (û, 0), (0,v) and (0, 0) is globally asymptotically stable and the others are linearly unstable.
The main result of this paper is stated below.

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LI MA AND DE TANG Theorem 1.1. Assume that 0 < α 1 < α 2 . Given r, L > 0, then the following statements hold: , then there exist two small positive number and δ such that (2) admits a co-existence steady state, where d * 1 and d * 2 are given in Remark 3, and by "g.a.s", we mean that the steady state is globally asymptotically stable among all non-negative and nontrivial initial conditions. Remark 1. The results of Theorem 1.1 are in sharp contrast to those of Theorem 1.2 in [24], where the boundary conditions are no-flux. This suggests us that the boundary conditions play an important role in the outcome of competition.
Throughout this paper, we straighten out some terminologies for convenience. •: It is pointed out that (2) only admits three types of nonnegative steady state solutions as follows: is called a semi-trivial steady state, wherê u > 0 andv > 0; -: u > 0, v > 0, and we call (u, v) a coexistence/positive steady state. The rest of this paper is organized as follows. In Section 2, we establish necessary and sufficient conditions for the persistence of the corresponding single specie model. Section 3 is devoted to show that system (2) has no coexistence steady state under the appropriate conditions and prove Theorem 1.1. In Section 4, we give a short discussion. Though the presentation is kind of parallel to those of [12,24], we have to modify most of the arguments, especially that for the non-existence of positive steady states.

2.
Persistence for the single equation. In this section, we study the following model of single specie where, d, α, r, L > 0 are constants. We focus our attention on the steady state of (3), i.e.
Finally, we concern case (ii): m(0) > 0 and m x (x) ≥ 0 in (0, L). Since the proofs of statements (ii.1) and (ii.2) are similar, we only prove statement (ii.1). Given α > 0 and m ∈ C 1 (0, L), for simplicity, we denote λ 1 (d, α, m) by λ 1 (d) stressing the dependence on d in the following proof of this lemma. It is clear that this lemma comes from the following assertion: x (0) = 0 and the uniqueness of solutions of ODEs, it follows that φ * xx (0) < 0, which together with φ * x (0) = 0 infers that there exists some small 2 > 0, such that We use an indirect argument to verify (17) and suppose that there exists some point x * ∈ (0, L) such that φ * x (x * ) = 0. This together with (18) and φ * x (0) = 0, suggests that w * achieves a negative local minimum at some point x min in (0, Again by the strong maximum principle [3], we know that w * (x) > 0 on (0, x * ), which contradicts w * (x min ) < 0. This completes the proof. Proof. Since the nonlinear reaction term of problem (3) is of the logistic type, it is well known that the existence of a positive steady state for problem (3) is equivalent to that u = 0 is linearly unstable (i.e. λ 1 (d, α, r) < 0) [1]. Moreover, we omit the detail on the proof of the uniqueness of positive steady states as it is standard (for example, we refer to the proof of [10, Lemma 2.1]. We firstly estimate the principal eigenvalue λ 1 (d, 0, r). From (i) in Lemma 2.1 and the definition of k, it follows that Specially, λ 1 ( r k , 0, r) = 0. Now, we consider case (i): 0 < d ≤ r k . Based on (20) and (i) in Lemma 2.1, for any α > 0, one obtains λ 1 (d, α, r) < λ 1 (d, 0, r) ≤ 0, which proves (i). Then, we concern case (ii): d > r k . Recall (i) in Lemma 2.1. We obtain that λ 1 (d, α, r) is strictly decreasing function on parameter α. Since λ 1 (d, 0, r) > 0 for d > r k , it suffices to show that lim By an indirect argument, and together with the monotonicity of λ 1 (d, α, r) with respect to d, we suppose that there exist some constant M and d n > 0 satisfying d n → +∞ as n → ∞, such that λ 1 (d n , α, r) ≤ M as n → ∞. For simplicity, we denote λ 1 (d n , α, r) and φ 1 (d n , α, r) by λ n and φ n , which satisfy d n φ n,xx − αφ n,x + rφ n + λ n φ n = 0, x ∈ (0, L), where ||φ n || L ∞ ([0,L]) = 1. By the elliptic regularity [3] and the standard Sobolev imbedding theorem, one can deduce from equation (21) that φ n , passing to a subsequence if necessary, converges to some function φ * in the topology of C 1 ([0, L]) as n → ∞, where φ * satisfies (in the weak form)  Proof. Similar to the proofs of (14) in Lemma 2.1, one finds Define By the maximum principle [3], one sees z(x) < 0 in (0, L), which infers that Besides, from z(x) < 0 in (0, L) and z(0) = 0, it follows that This together with the uniqueness of solutions of ODEs yields that which combined with the boundary condition suggests that Estimating the value of the first equation of (4) at x = 0, one obtains Thus, (22), (23) and (24) complete the proof.
3. Proof of Theorem 1.1. To prove Theorem 1.1, we firstly establish the nonexistence of coexistence steady state for system (2).

3.1.
Nonexistence of coexistence steady state. To establish the non-existence of coexistence steady state for system (2), we need some preliminary results. For any coexistence steady state (u, v)(if exists) of system (2), similar to the proof of (14) in Lemma 2.1, one can obtain that there exists some 3 > 0, such that A directly computation yields that T andŜ have the following properties (see Lemma 3.5 in [12]).
Proof. By an indirect argument, we suppose that there exists a coexistence steady state for system (2), denoted by (u * , v * ). By strong maximum principle, one obtains u * , v * > 0 on [0, L). Similar to the analysis in Lemma 2.1, one can obtain that there exists some 4 > 0, such that For clarification, we prove this Lemma by several claims. Define x (L) < 0 and g(L) = d 2 v * x (L) < 0 due to the boundary conditions and (28).
Claim 1. f , g, u * x and v * x are real analytic. Moreover, all zero points of f and g in [0, L] must be isolated.
Arguing indirectly, we assume that u * (0) + v * (0) = r. Differentiating the equations of u * and v * , then we see As before, set T := u * x u * and S := v * x v * . Then T and S satisfy the following equation Combining the boundary condition u * x (0) = v * x (0) = 0 and the equations of u * and v * , one obtains that u * xx (0) = v * xx (0) = 0. This together with the definitions of T and S suggests that T x (0) = S x (0) = 0. Then by the uniqueness of solutions of ODEs, one obtains T (x) = S(x) = 0 on [0, L], which contradicts (28).
global dynamics. More precisely, we find that the strategy of slower advection together with faster diffusion is always superior, see Theorem 1.1. Although in the current work we have made some progress in understanding the system (2), there are several important problems that are unsolved and deserve further investigation. The first one concerns the case that two species have different interspecific competition abilities, while both interspecific competition coefficients in (2) are normalized to 1. For the non-advective case, He and Ni [5] made a significant breakthrough on the estimate of linear stability of any coexistence steady state and based on this, they finally classified completely all possible long time dynamical behaviors. The second one is to talk about the same topic but in a spatially heterogenous environment, which clearly is more difficult. See some relevant results in [8,13,19,21,23,25,26] and references therein. We leave these for future work.