Invasion and coexistence of competition-diffusion-advection system with heterogeneous vs homogeneous resources

This paper mainly study the dynamics of a Lotka-Volterra reaction-diffusion-advection model for two competing species which disperse by both random diffusion and advection along environmental gradient. In this model, the species are assumed to be identical except spatial resource distribution: heterogeneity vs homogeneity. It is shown that the species with heterogeneous resources distribution is always in a better position, that is, it can always invade when rare. The ratio of advection strength and diffusion rate of the species with heterogeneous distribution plays a crucial role in the dynamics behavior of the system. Some conditions of invasion, driving extinction, and coexistence are given in term of this ratio and the diffusion rate of its competitor.


1.
Introduction. The question of how the interactions between spatial heterogeneity and the organism's dispersing affect the evolution of the population has fascinated ecologists and evolutionary biologists for many decades. For reactiondiffusion model, Hastings [15] and Dockery et al. [12] showed that, for two competing species with different (random) dispersal rate but otherwise identical in a heterogeneous environments, the slower diffuser always wins. To be more precise, consider the following Lotka-Volterra competition-diffusion system( [12]) where the migration rates d 1 , d 2 are two positive constants, U (x, t), V (x, t) represent the densities of two species at location x and time t, m(x) represents the intrinsic growth rate of species, which also reflects the environmental richness of the resources at location x. The habitat Ω is a bounded region in R N with smooth boundary ∂Ω, ∂ ν = ν · ∇, where ν denotes the unit normal vector on ∂Ω, and the no-flux boundary condition means no individuals cross the boundary. For simplicity, we will assume throughout this paper that the initial data U 0 and V 0 are nonnegative and nontrivial, i.e., not identically zero. Let g(x) ∈ C α (Ω)(α ∈ (0, 1)) with Ω g(x)dx ≥ 0 and g(x) ≡ 0. It is well known that the problem d∆θ + θ(g(x) − θ) = 0 in Ω, ∂ ν θ = 0 on ∂Ω, has a unique positive solution(see, e.g., [4]), which is denoted by θ d,g . Then the following remarkable result is established by Hastings ([15]) and Dockery et al. ( [12]): Theorem A. Suppose that 0 < m(x) ≡ const onΩ and m(x) ∈ C α (Ω) (α ∈ (0, 1)). Then the semitrivial steady state (θ d1,m , 0) of (1) is globally asymptotically stable when d 1 < d 2 ; i.e., every solution (U, V ) of (1) converges to (θ d1,m , 0) as t → ∞ regardless of initial values (U 0 , V 0 ).
An intuitive explanation for this surprising result is that slow diffusion helps species to better track favorable regions whereas fast diffusion will move individuals away from such ideal regions and in so doing lose certain competitive advantages.
Recently, by allowing the species U and V to have different intrinsic growth rates or to have different distributions of resources, in a series of works, He and Ni [16,17,18,19] studied the following Lotka-Voltrra model of competition-diffusion system: where m 1 (x) and m 2 (x) represent the carrying capacities or intrinsic growth rates, which reflect the environmental influence on the species U and V respectively. The positive constants b and c are inter-specific competition coefficients, while both intra-specific competition coefficients are normalized to 1. On the other hand, reaction-diffusion-advection equations nowadays seem more and more popular in spatial population dynamics. Belgacem and Cosner in [2] firstly proposed the single species model in the situation where individuals are very smart so that they can sense and follow gradients in resource distribution, and then Cantrell et al. [5] analyzed the corresponding two-species model. This topic has received considerable research attention; see, e.g., [4,5,6,7,8,9,10,11,14,25,26,27,31] and the references therein, for some latest advances, see [1].
Motivated by the previous works, we introduce the following coupled reactiondiffusion-advection system in Ω, where α, β which are positive constants measure the speed of movement upward along the gradient of resources, respectively. Our main concern in paper is to pursue the dynamics of system (4), especially the effect of advection rates α, β on the dynamics of this system.
When α = β = 0, system (4) become the system (3), which is studied by He and Ni [16,17,18,19] recently. By detailed computation and analysis, He and Ni [16,17,18,19] obtained some dramatic picture of global dynamics of (3) based on diffusion rates d 1 and d 2 . Especially, for the case of heterogeneity vs. homogeneity with equal amount of total resources and b = 1, c = 1, He and Ni [16,19] obtained thoroughly complete global dynamics of (3). More precisely, in [16,19], He and Ni proposed the following system in Ω, where m(x) is nonconstant in Ω and m is the average density of the resources m(x), i.e.
The following notations of subsets of the first quadrant of the d 1 d 2 -plane for (5) is also used in [16,19]  For the precise definition of linear stability/instability of a steady state and their characterization, see e.g., [4]. He and Ni obtained the following remarkable characterization of global dynamics result for system (5) Theorem B. ( [16,19]) Assume that m ∈ C γ (Ω) (γ ∈ (0, 1)), m > 0 inΩ, and m is non-constant, then where Σ U denotes the closure of Σ U in Q. Moreover, Σ U can be characterized by (iii) Σ − = ∅, and Q = Σ U ∪ Σ − . Moreover, (5) has a unique coexistence steady state which is globally asymptotically stable for all Theorem B implies that for two competitive species having identical competition abilities and the same amount of total resources, the species with spatial heterogeneous distribution is always in a superior position to its homogenous counterpart: it is always guaranteed to survive, and it will often wipe out its competitor, so long as the diffusion point (d 1 , d 2 ) is above the critical line Following the idea of [17,19], let m 1 (x) = m(x), m 2 (x) = m = 1 |Ω| Ω m(x)dx, and b = c = 1, then system (4) change to the following coupled reaction-diffusion- in Ω.
Note that since m 2 (x) = m is a constant, the gradient ∇m ≡ 0, and then advection rate β has no effect to the system (6). For simplicity, we assume m(x) satisfies the following assumption in the rest of this paper, unless otherwise specified.
We assume that the initial data U 0 , V 0 of (6) are non-negative and not identically zero, then by maximum principle [34], we can obtain U > 0, V > 0. Under assumption (M), (6) have two semi-trivial steady states for all d 1 , d 2 > 0 and α > 0 (see [4], [11]), denoted by (ũ, 0), (0, m), respectively, whereũ is the unique positive solution (see [2]) of and m is the unique positive solution of Now we state our first result Then there exists an positive constant d * . Theorem 1.2 implies that, when the ratio α/d 1 is small, the species V can invade when rare if and only if it's diffusion rate d 2 is less than the critical value d * 2 . By the theory of monotone dynamical systems, we have the following coexistence result. .
Then there exists a positive constant d * 2 = d * 2 (d 1 , α), such that (6) has at least one stable positive coexistence steady state for d 2 < d * 2 .
Furthermore, when d 2 is large enough, we also have the following global asymptotically stability.
Then (ũ, 0) is globally asymptotically stable if d 2 is sufficiently large.
When the ratio α/d 1 is large, we have then (ũ, 0) is unstable for any d 2 > 0, and (6) has at least one stable positive coexistence steady state.
Theorem 1.5 implies that when the ratio α/d 1 is large, then for any d 2 > 0, species V can invade when rare, and the two species coexist.
The rest of this paper is organized as follows. Section 2 contains some preliminaries. In Section 3, we devote to establish our main results. Some concluding remarks are included in section 4.

2.
Preliminaries. The stability of (ũ, 0) is determined by the principal eigenvalue, denoted by µ 1 (d 2 , m −ũ), of the elliptic eigenvalue problem Similarly, the stability of (0, m) is determined by the principal eigenvalue, denoted by µ 1 (d 1 , α, m − m), of the linear problem as follows: More precisely, we have the following well known criterion Let λ 1 (h) denote the unique nonzero principal eigenvalue of where h ≡ Constant, could change sign. The following results are well known, see e.g., [3,35,33].
e., and h 1 , h 2 both change sign.
In order to analysis the principal eigenvalue of problem (13), it is more convenient to consider the following more general form of eigenvalue problem: The principal eigenvalue of problem (16), denoted by µ 1 (d, h), is expressed by the following variational equation (see, e.g. [4]) The following lemma collects some useful properties of µ 1 (d, h) (see Proposition 4.4 in [33]).
Lemma 2.3. The first eigenvalue µ 1 (d, h) of (16) has the following properties: 3. Proofs of the main results. We first prove Theorem 1.1.
Now we are ready to establish Theorem 1.2 and Theorem 1.3.
This complete the proof of Theorem 1.2.
Proof of Theorem 1.3. It is known that (6) is a strongly monotone system, for a proof, refer to Lemma 2.2 in [6]. Assume that the conditions of Theorem 1.3 is satisfied, then by Theorem 1.2 (ii), for d 2 < d * 2 , the semi-trivial steady state (ũ, 0) is linearly unstable. By Theorem 1.1, the other semi-trivial steady state (0, m) is always linear unstable for all d 1 , d 2 , α > 0. Then by the theory of monotone dynamical system (see, e.g., [21,22,23,36]), (6)has at least one stable positive coexistence steady state. This completes the proof.
To prove Theorem 1.4, we also need the following lemma.
Proof. (Ũ ,Ṽ ) satisfies the following elliptic system By the maximal principle and comparison theorem( [13,34]), we have that Ṽ L ∞ < m, and whereũ is the unique positive solution of the equation This completes the proof.
Now we are ready to prove Theorem 1.4.
Proof of Theorem 1.4. By Theorem 1.1, the semi-trivial steady state solution (0, m) is linearly unstable for all d 1 , d 2 , α > 0, then by the monotone dynamical system theory(see [21,22,23,36]), to show that (ũ, 0) is globally asymptotically stable, it suffices to show that (6) has no co-existence steady state for all d 2 sufficiently large. Suppose this is not true, then there exist some d 1 > 0, a positive sequence d and letting k → ∞, we conclude that the limiting functionṼ ∞ satisfies that ∆Ṽ ∞ = 0 in Ω, ∂ νṼ∞ = 0 on ∂Ω.
Proof of Theorem 1.5. It suffices to show that principal eigenvalue µ 1 of the problem is negative. Dividing (38) by ψ, integrating in Ω, we obtain Since α/d 1 ≥ 1/ maxΩ m, we have Ω (ũ − m) < 0 by Lemma 3.1, Then µ 1 < 0 by (39). The proof of the coexistence part is similar to that in the proof of Theorem 1.3 and is thus omitted.
4. Concluding remarks. We study the dynamics of the Lotka-Volterra reactiondiffusion-advection model, in which the two competing species have equal total resources but different strategy of resource distribution. The two species adopt the dispersal strategy of a combination of random dispersal and biased movement upward along the resource gradient(moving to the location with better resource). In this paper, we mainly study the situation that one species, species V , adopts the homogenous distribution, and its competitor, species U , adopts heterogeneous distribution.
We are interest to know how the resource distribution, the random dispersal rate, and the advection rate affect the dynamics of the system. The species U with heterogeneous distribution is always in a better position than its competitor V with homogenous distribution (Theorem 1.1), that is, the semi-trivial steady state (0, m) is always unstable. Then the dynamics of (6) is mainly depending on the stability of the other semi-trivial steady state ( u, 0). It turns out that the ratio α/d 1 plays a crucial role on the stability of (ũ, 0). Our main results can be interpreted biologically in the following statements: (i) The species U can always invade when rare the species V for any d 1 , d 2 > 0, and α ≥ 0. (ii) For α/d 1 ≤ 1/ maxΩ m and d 2 < d * 2 , the species V can invade when rare the species U , and the two species will coexist. (iii) For α/d 1 ≤ 1/ maxΩ m and d 2 > d * 2 , the species V can not invade when rare the species U near ( u, 0), and species U will drive V to extinction for any initial value if d 2 is sufficiently large.
In case (ii) and (iii), the species U has relatively weak advection, its competitor V can evolve if and only if by adopting slow diffusion strategy. (iv) If m > 0 inΩ, and α/d ≥ 1/ minΩ m, then the two species always coexist for any d 2 > 0. In this case, the species U has relatively strong advection, and then it left sufficient habitat for V evolving. These results provide some new mechanisms for the dynamics of competition system (see, e.g., [6,8,16,17,18,19,20]). For α/d 1 in the interval (1/ maxΩ m, 1/ minΩ m), the dynamics behavior of (6) is more complicated, and it is likely depend upon the the geometry of Ω and the specific distribution of the resources.
The results about the dynamics of the general form of system (4) are known very limited. We hope to explore further in this direction in a future paper.