Global dynamics of a Lotka-Volterra competition-di ﬀ usion-advection system for small di ﬀ usion rates in heterogenous environment

: We investigate the global dynamics of a Lotka-Volterra competition-di ﬀ usion-advection system for small di ﬀ usion rates in heterogenous environment. Our result suggests that the sign of (cid:82) L 0 ( m 1 − m 2 ) e kx dx plays a signiﬁcant role in understanding the global dynamics. In addition, the limiting behavior of coexistence steady state is obtained when di ﬀ usion rates of two species tend to zero meanwhile.


Introduction
Many ecologists and evolutionary biologists are attracted by the issue of why individuals disperse for a number of years. So far, a great deal of researches are addressed to understand the mechanism of dispersal [1,2]. Over the last few decades, researchers from both biology and mathematics have used the reaction-diffusion equations to model population dynamics in spatial ecology and evolution. Among these models, the two species Lotka-Volterra competition-diffusion system perhaps is the most salient example; refer to, e.g., the books [1,3] and some recent works in [4][5][6].
With further development, researchers become more interested in the study of spatial population dynamics in advective environments which forces organisms to move in certain directions(biased movements) modeled by reaction-diffusion-advection equations. For instance, a part of researchers pay attention to an active research area concerning the population dynamics in which the individuals are very intelligent so that they can sense the surroundings and move upward along the gradients of a resource distribution. Belgacem and Consner [7] was the first to raise the single species model and then Cantrell et al. [2] presented the two-species model. From the direction of different research, a portion of researchers investigate population dynamics of some aquatic ecosystem modeled by reaction-diffusion-advection equations. The aquatic ecosystems are environments which are featured by a predominantly unidirectional flow, such as rivers, water columns or streams [8][9][10][11][12][13][14]. The organisms will move passively toward the downstream end due to unidirectional water flow if species lives in rivers and streams, or move upward(downward) due to buoyancy(gravity) if species is in water columns. Specially, environmental conditions shift can produce biased motion, such as, the movement of temperature isoclines, which is caused by global climate change [15]. Speirs and Gurney [16] proposed the following single compartment model with diffusion, advection and a logistic growth from river ecosystems.
where u(x, t) denotes the population density at location x and time t, d is the diffusion rate, L is the size of the habitat, and in the sequel, we call x = 0 the upstream end and x = L the downstream end, α measures the tendency of the biased movement by water flow (sometimes we call α the advection speed/rate) and we point out here that α should be positive since it is defined that x = L is the downstream end. At the upstream end x = 0, the no-flux type condition is imposed, which indicates that there is no individuals movement in or out through the upstream end. However, at the downstream end x = L, the hostile condition is assumed which means that once individuals pass through the downstream end they do not return back(e.g., stream to ocean [16]). The constant m > 0 accounts for intrinsic growth rate, which indicates that the environment is spatially homogeneous. However, as we know, spatial characteristics of the environment play a vital role in ecology and evolution, and the uneven distribution of resources caused by the effect of geological and environmental heterogeneity could create very interesting phenomena in population dynamics. Moreover, the phenomenon of spatial heterogeneity of resources will create more complexity to investigate the global dynamics.
Recently, Tang and Zhou [17] considered the situation where a new or exotic species invades such advective environment. They investigated the competitive consequence of two species model in nonhomogeneous environment, to be more specific, the following two-species Lotka-Volterra competition-diffusion-advection system: where u and v stand for the population densities of two competing aquatic species with their interspecific and intra-specific competition intensities all equal to 1. d 1 , d 2 and α 1 , α 2 are the diffusion and advection rates respectively. The functions m 1 (x) and m 2 (x) represent the intrinsic growth rates at location x of these two competitors, which also reflect the distribution of resources. The parameter L > 0 measures the length of the one-dimensional habitat. And at the boundary of the habitat, it is assumed that there is no net flux across for either of the two species which implies that the environment is closed. In their study, they imposed that m 2 = M where M is a constant, that is, the distribution of resources for the species u is spatially uneven while for the species v is homogeneous. By employing principal spectral theory, they investigated the global dynamics of system (1.2) on the condition that: (H) α 1 /d 1 = α 2 /d 2 =: k > 0 and found that the competitive result was depended on d 1 , d 2 and m 1 . However, for the general situation where m 1 m 2 and m 1 together with m 2 is the function of spatial variation x, the global dynamics of system (1.2) is far from being completely understood. It is common to ask if the outcome of competition when m 2 is a function with respect to x is more complex than when m 2 is a constant. We will pursue further in this direction.
For the spatially homogeneous case m 1 = m 2 ≡ m 0 with m 0 being a positive constant, a lot of researchers are interested in it and have investigated it qualitatively [18][19][20][21][22]. In the case that m 1 = m 2 =: m(x), non-constant, that is, spatially nonhomogeneous, system (1.2) is more difficult to handle and has been explored in many works. Lam et al. [23] seemed the first to try to discuss the case d 1 d 2 and α 1 = α 2 , which directed at the existence and diversity of evolutionarily stable strategies applying some limiting arguments (in the sense of both diffusion and advection rates are sufficiently small and comparable). Zhao and Zhou [24], considering on the special case d 1 d 2 and α 1 = 0 < α 2 , which meant that one species merely suffered random while the other one underwent both random and advective movements, attempted to uncover some various phenomena. For the general case d 1 d 2 and α 1 α 2 , recently, Lou et al. [25], by developing new techniques to surmount the difficult caused by non-self-adjoint operators, obtained a profound understanding on the global dynamics. The more general case m 1 m 2 now is far from being understood completely. Zhou and Xiao [26] established a classification of all possible long time behaviors for a more general competitive system in the condition of (H). Indeed, they discussed it in higher spatial dimensions.
Motivated by Tang and Zhou [17], in this paper, we mainly investigate the dynamical behaviors of system (1.2). Firstly, we need to make the following basic assumptions: Condition (H 1 ), biologically, means that the movement strategies(random diffusion and advection rates) of two competitors are proportional, and it is a mathematically technical condition in the main body of this paper. In assumption (H 2 ), it means the two competing populations have the same amount of total resources and the distributions of resources are spatially heterogeneous as well as nonidentical. Moreover, we exclude the ideal free distribution introduced in [27].
The definitions of the linear stability/instability and the neutrally stability of a steady state of system (1.3) will be given precisely in Section 2.
Due to assumption (H 1 ), the following complete classification on the global dynamics of system (1.3) can be obtained directly from (Theorem 1.2 [26]). It is a special case of (Theorem 1.2 [26]), where the advective direction P(x) = x, inter-specific competition ability b = c = 1, and the habitat Ω = [0, L].
Assume that (H 1 ) and (H 2 ) hold. Then for system (1.3), we have the following mutually disjoint decomposition of Γ: k,m 2 in (0, L) and system (1.3) has a compact global attractor consisting of a continuum of steady states connecting the two semi-trivial steady states; where g.a.s means that the steady state is globally asymptotically stable among all non-negative and nontrivial initial conditions. A basic classification on all possible long time behaviors of system (1.3) is exhibited by the above statements. However, to obtain a transparent picture of the global dynamics of system (1.3), it remains to know explicitly when Σ u ,Σ u , Σ v ,Σ v and Σ o will happen. Equivalently speaking, is it possible to provide a sharp division of these sets by using certain variable parameters? As mentioned in [26], each component of these sets could be empty, and more challengingly, it is hard to give a criteria guaranteeing the dynamics in these sets. Even for the non-advective case, it is not yet completely solved [4].
In this paper, we study the global dynamics of system (1.3) which contains two competing species. We assume that the two species both are in heterogeneous environment and denote their intrinsic growth rates by the functions m 1 (x) and m 2 (x), respectively. As we known, the distribution of resources is uneven in the natural environment, hence, this case is of more realistic significance. In the condition that m 1 m 2 and total resources of the two species are fixed at the same level( The rest of this paper is organized as follows. Section 2 contains some preliminaries which are useful in later analysis and our main results. In Section 3, we will give the proof of Theorem 1.1. We prove Theorem 1.2 in Section 4. Finally, we give a short discussion.

Preliminaries and main results
This section is aim to display our main results and exhibit some fundamental results which will be utilized in later sections.
We obtain the following result when one of d 1 and d 2 tends to zero. Theorem 1.1. Assume that (H 1 ) and (H 2 ) hold. The following statements are valid: where µ(h) is the unique nonzero principal eigenvalue of problem (2.4). Remark 1.1. In [17], the authors found that the outcome of competition in general heterogeneous distribution is very complicate: either u wins, or v wins, or u − v coexists, depending on the size of diffusion rates d 1 , d 2 and m 1 . Similar to [17], Theorem 1.1 indicates that the outcome of competition depends on the size of diffusion rates d 1 and d 2 ; however, in this paper, because the resource functions of the two competing species u and v are both spatially nonhomogeneous, then the outcome of competition depends not only on the distribution of m 1 but also the distribution of m 2 . Moreover, the competition outcome in this situation will be more abundant. Remark 1.2. According to Theorem 1.1, we find that the sign of the quantity L 0 (m 1 − m 2 )e kx dx plays a significant role in understanding the global dynamics of system (1.3) which is similarly to the description mentioned in [16]. Then we give some sufficient conditions about determining the sign of the quantity L 0 (m 1 − m 2 )e kx dx on the condition that m 1 and m 2 are monotonic with respect to spatial variable x. In view of the monotonicity of m 1 and m 2 , we can divide into six situations to talk about and the graphs of m 1 and m 2 are as follows Figure 1. From Figure 1, the following statements are true: Then
The desired result follows by letting s = k.
When d 1 and d 2 tend to zero meanwhile, the following result is true.
Now we are in a position to prove Theorem 1.1. We first prove statement (i) by fixing d 2 > 0. If  From the first inequality, (θ d 1 ,k,m 1 , 0) is linearly unstable and (0, θ d 2 ,k,m 2 ) is linearly unstable by the second inequality. Due to the statement (c), it is easy to deduce that system 1.3 has a coexistence steady state that is g.a.s. Next we prove the limiting behavior of coexistence steady state of system (1.3). Following the ideas in [35], we obtain the proof of this statement.
Then, for small d 1 , d 2 > 0, denote the coexistence state of system (1.3) by (u, v), the following hold: (ii) Analogously for sequences {u n } n=∞ n=1 and {v n } n=∞ n=1 defined by We have Proof. It is enough to prove Eq (4.1) as the proof of Eq (4.2) follows by the symmetry. The proof is obtained by induction argument. From the equation of u, we obtain that is to say, u is a sub-solution of .

Now, substitute this inequality into the equation of
Back to the equations of u and v, one can obtain For small d 1 and d 2 > 0, we have v 1 > 0 in (0, L), so This completes the proof of Eq (4.1) for n = 1. Now suppose that Eq (4.1) is true for some N > 1. Then it suffices to show that Thus it is sufficient to show that Since v N+1 ≥ v N in (0, L), we get This completes the proof. We now analyze the behavior of the scheme introduced by Lemma 4.1, Eqs (4.1) and (4.2) as d 1 and d 2 tend to 0.
In consideration of Lemma 4.1, for each n ≥ 1, one can define the following limits in the topology of C([0, L]): As a matter of fact, these limits can be described detailedly. Lemma 4.2. The following identities hold: Due to assumption (H 2 ), Eq (4.3) is hold when n = 1. Suppose that Eq (4.3) is valid for every 1 ≤ n ≤ N. Then, by definition, u N+2 := θ d 1 ,k,m 1 −v N+1 . Using the argument above again, one obtains We can demonstrate the rest ones in the similar manner. Hence, the proof is finished.
By the above result, it is allowable to get explicit formulas for each of the sequences U n , U n , V n and V n . The following lemma makes these precisely. Lemma 4.3. The following formulas for U n , U n , V n and V n are true: where n ≥ 1.
Proof. We only prove Eq (4.4) by the induction argument and Eqs (4.5), (4.6) and (4.7) can be verified similarly. From Lemma 4.2, it can be inferred that Hence for n = 1, Eq (4.4) is hold. Now suppose that Eq (4.4) is true for every 1 ≤ n ≤ N, then one has Due to the above two equalities, the following can be obtained that This completes the proof. Now it is in a position to prove

Discussions
In this paper, motivated by Tang and Zhou [17], we studied a classical two-species Lotka-Volterra competition-diffusion-advection system in which the diffusion rates, advection rates and intrinsic growth rates are allowed to take on different values in the space(heterogeneity).
In the condition that total resources for two populations are fixed at the same level, we consider both species u and v are both in heterogeneous environment. We assume that the two competing species have different resource functions and the distributions of resources are uneven, which is different from the literature [17], where the authors supposed that one spatial distribution is even across space while the other one not. We find that the outcome of competition in this situation is very abundant: either one of the two competitors becomes the final single winner or both populations coexist eventually, which is dependent on the diffusion rates of both species and the specific shapes of m 1 and m 2 ; see Theorem 1.1. By limiting arguments, we investigate further the population dynamics when d 1 and d 2 tend to zero and give the asymptotic behaviour of coexistence steady state for small diffusion; see Theorem 1.2. These results partially generalizes Tang and Zhou [17]. In comparison with Tang and Zhou [17], we study a more general case. In their paper, they assume that the resource function of v expressed as m 2 is a constant M. In our research, m 2 is a function of the spatial variable x. This case is more realistic. Moreover, our results indicate that the values of m 1 and m 2 have a significant impact on the global dynamics of (1.3) and the limiting behaviour of the coexistence steady state. Therefore a change in the value of m 2 will produce more complicated spatial population dynamics.
From the above research, we know that both m 1 and m 2 have effects on global dynamics of system (1.3). It seems that the spatial population dynamics will appear to be more abundant. Moveover, assumption (H 1 ) plays an important role in the proof process. When (H 1 ) fails, it is far away from a complete understanding and extremely challenging to deal with. We will continue to explore these problems in the future.