Global dynamics of a general Lotka-Volterra competition-diffusion system in heterogeneous environments

Previously in [ 14 ], we considered a diffusive logistic equation with two parameters, \begin{document}$ r(x) $\end{document} – intrinsic growth rate and \begin{document}$ K(x) $\end{document} – carrying capacity. We investigated and compared two special cases of the way in which \begin{document}$ r(x) $\end{document} and \begin{document}$ K(x) $\end{document} are related for both the logistic equations and the corresponding Lotka-Volterra competition-diffusion systems. In this paper, we continue to study the Lotka-Volterra competition-diffusion system with general intrinsic growth rates and carrying capacities for two competing species in heterogeneous environments. We establish the main result that determines the global dynamics of the system under a general criterion. Furthermore, when the ratios of the intrinsic growth rate to the carrying capacity for each species are proportional — such ratios can also be interpreted as the competition coefficients — this criterion reduces to what we obtained in [ 18 ]. We also study the detailed dynamics in terms of dispersal rates for such general case. On the other hand, when the two ratios are not proportional, our results in [ 14 ] show that the criterion in [ 18 ] cannot be fully recovered as counterexamples exist. This indicates the importance and subtleties of the roles of heterogeneous competition coefficients in the dynamics of the Lotka-Volterra competition-diffusion systems. Our results apply to competition-diffusion-advection systems as well. (See Corollary 5.1 in the last section.)

intrinsic growth rate r(x) and carrying capacity K(x) has been considered in [11,14]: in Ω × R + , Here, u(x, t) represents the population density of a species at location x ∈ Ω and at time t > 0, which is therefore assumed to be non-negative; Ω, the habitat, is a bounded domain in R N with smooth boundary ∂Ω; d > 0 is the dispersal rate of u; is the usual Laplace operator, and ∂ ν = ν · ∇, where ν denotes the outward unit normal vector on ∂Ω, is the normal derivative on the boundary. The zero Neumann (no-flux) boundary condition is to ensure that no individual crosses the boundary of the habitat. Denoting we can rewrite (1) as In this form, ξ(x) can be interpreted as density-dependent crowding effect, or intraspecific competition coefficient. The above two different forms of reaction or growth term are referred to as the "Pearl-Verhulst growth" and the "original Verhulst growth" respectively, which were originally used in a non-spatial context. Although both these formulations represent the same underlying model, they may lead to different understanding or interpretation of evolution and speciation from the ecological point of view. (See [34] for more discussions.) The main issue is that in a non-spatial context, the quantity K, seen as the carrying capacity, seems to cause some confusion between the population limited by resources and equilibrium density. Recent studies in [41,15] suggest that, the concept "carrying capacity" perhaps is not well-defined as we had realized and it ought to be "dynamic" in nature -that is, it depends not just on the total amount of available resources, but also on how the resources are distributed as well as how the species disperses in the habitat and consumes the resources.
Generally speaking, the two quantities r(x) and K(x) are not completely independent, some correlations may exist between them. It is shown in [11] that the way in which growth rate r(x) and the quantity K(x) in (1) or ξ(x) in (2) are related is important for total equilibrium population to exceed total carrying capacity. Furthermore, it is established in [14] that the two cases r ≡ cK and r ≡ constant in the logistic equation (1) would produce very different total populations at equilibrium compared to the total carrying capacity. For more details and further discussion in this direction, see [11,14,29] and references therein.
Since the logistic equations are the basis for the corresponding Lotka-Volterra competition-diffusion systems, the way in which growth rate r(x) and the quantity K(x) are related leads to different forms of two-species Lotka-Volterra competition systems. Moreover, the aforementioned opposite phenomena of total populations at equilibrium compared to the total carrying capacity have significant consequences on the outcomes of the two competing species-clearly illustrated in [14]. Given the results in [14], it seems necessary to consider the following Lotka-Volterra GLOBAL DYNAMICS OF LOTKA-VOLTERRA COMPETITION-DIFFUSION SYSTEM 6549 competition-diffusion system where more general carrying capacities and intrinsic growth rates are included: in Ω. ( Here, U (x, t) and V (x, t) represent the population densities of two competing species at location x ∈ Ω and at time t > 0; d 1 and d 2 are the dispersal rates of U and V respectively; r 1 (x) and r 2 (x) are the intrinsic growth rates of U and V respectively; K 1 (x) and K 2 (x) are the carrying capacities of U and V respectively; the constants b, c > 0 are interspecific competition strengths, where we have normalized the intraspecific competition strengths to be 1. Denoting we again rewrite (3) as in Ω.
Similarly as in the single species case, ξ i (x) can be interpreted as intraspecific competition coefficients and ξ i (x) together with the constants b, c can be interpreted as interspecific competition coefficients. When r i and K i are proportional for i = 1, 2 onΩ, i.e., both ξ 1 and ξ 2 are constant, the global dynamics of (3) or (4) has been studied extensively in the past few decades. See [5,6,7,8], [13], [16,17,18,19,20], [24,27,29,30] and references therein. For instance, the celebrated fact -"slower diffuser always prevails" [13] for the case r 1 ≡ r 2 ≡ K 1 ≡ K 2 and b = c = 1 with nonnegative and nonzero initial data (U 0 (x), V 0 (x)) says that, regardless of the initial values, U will always wipe out V , as long as d 1 < d 2 . It is easy to see that this phenomenon also holds true for the more general case r 1 ≡ r 2 , K 1 ≡ K 2 and b = c = 1. Another interesting fact, due to Lou [29], says that the combined effects of diffusion and spatial heterogeneity change the nature of weak competition in Lotka-Volterra competition systems with constant coefficients. Finally, He and Ni [18] obtain a complete classification of the global dynamics of (4) when ξ 1 ≡ ξ 2 ≡ 1 for a large range of the parameters b and c including the region bc ≤ 1.
Our goal in this paper is to study a general Lotka-Volterra competition-diffusion model where intrinsic growth rates and inter-/intra-specific competition coefficients are spatially heterogeneous, which certainly includes model (3) or (4) as a special case. To describe our results mathematically, we focus mainly on system (4) in the introduction for simplicity.
Denote the unique positive steady state of (2) by θ d,r,ξ , i.e., θ d,r,ξ satisfies the following equation: (See, e.g., [11] for a proof of existence and uniqueness results of (5).) For simplicity of notation, throughout this paper, we denote It is easy to see that (4) has two semi-trivial steady states (u d1 , 0) and (0, v d2 ). To characterize their linear stability properties in terms of (d 1 , d 2 ) ∈ Q precisely, we define: We also need to consider the situation when one (or both) of the two semitrivial steady states is (or are) neutrally stable, i.e., neither linearly stable nor linearly unstable. For this purpose, we introduce the following elliptic eigenvalue problem. Definition 1.1. Given a constant d > 0 and a function h ∈ L ∞ (Ω), we define µ 1 (d, h) to be the first eigenvalue of Now we are ready to define the following subsets of Q where at least one of the two semitrivial steady states is neutrally stable: Π := Σ U,0 ∩ Σ V,0 .
We now characterize the global dynamics of (4) in Q assuming further that: Theorem 1.2. Assume that (A1) and (A2) hold. Then we have the following mutually disjoint decomposition of Q: In particular, Moreover, the following statements hold for (4): 4) has a unique coexistence steady state that is globally asymptotically stable.
(iv) For all (d 1 , d 2 ) ∈ Π, (4) has a global attractor consisting of a continuum of steady states connecting the two semitrivial steady states.
We remark that the proof of Theorem 1.2 uses the same idea as in [18,Theorem 1.3]. (See also the remark after Theorem 1.3 in [18].) When ξ 1 ∝ ξ 2 , i.e., ξ 1 /ξ 2 ≡ const onΩ, we obtain a full extension of [18, Theorem 1.3] to system (4). However, when ξ 1 ∝ ξ 2 , we have Hence, [ [14] such that when b = c = 1, there exists an unstable coexistence steady state to system (4). Biologically, this means that the complexity of spatially heterogeneous competition coefficients can change the nature of the competition so that a simple classification of global dynamics for all (d 1 , d 2 ) ∈ Q can no longer be obtained for all b and c satisfying bc ≤ 1. Nevertheless, as we can see from Theorem 3.1 below, when all the intra-/interspecific competition coefficients are identical up to scalar multipliers, [18,Theorem 1.3] can still be fully recovered. This indicates that, besides the self-adjointness of the dispersal operator, certain "symmetric" structure of the quadratic terms in the reaction term seems necessary, from a mathematical point of view. (See also the concluding remarks in Section 5.) Although Theorem 1.2 already characterizes all possible long-term dynamical behaviors of (4) for all (d 1 , d 2 ) ∈ Q, next we would like to further classify the global dynamics of (4) in terms of b, c and (d 1 , d 2 ), under the additional condition that ξ 1 ∝ ξ 2 . With this condition, (A2) reduces to bc ≤ 1. Actually, in this case, globally dynamics of (4) can be completely characterized in a larger region including bc ≤ 1 as follows. Motivated by [18], we define sup Ω r 2 ξ 2 u d1 ∈ (0, ∞), and where Theorem 1.3. Assume that (A1) holds and ξ 1 ∝ ξ 2 . If one of r 1 /ξ 1 and r 2 /ξ 2 is nonconstant, then Assume further that (b, c) ∈ Ξ, then the following statements hold for system (4): (i) If b ≥ S U and c ≤ 1/S U , then Q = Σ U ; i.e., for all d 1 , d 2 > 0, (u d1 , 0) is globally asymptotically stable.
(ii) If c ≥ S V and b ≤ 1/S V ,then Q = Σ V ; i.e., for all d 1 , d 2 > 0, (0, v d2 ) is globally asymptotically stable. (iii) If b < L U and c < L V , then Q = Σ − ; i.e., for all d 1 , d 2 > 0, (4) has a unique coexistence steady state that is globally asymptotically stable; if b = L U and if and only if r 1 ≡ cr 2 ξ 1 /ξ 2 and bc = 1, in which case, if L U < b < S U and c = L V , then the same statement holds except for Σ V,0 which may be nonempty.
if L V < c < S V and b = L U , then the same statement holds except for Σ U,0 which may be nonempty.  In the second part of this paper, we assume that the two species have identical competition abilities, i.e., b = c = 1 and ξ 1 ≡ ξ 2 ≡: ξ, and consider the global dynamics of the following system: in Ω.

GLOBAL DYNAMICS OF LOTKA-VOLTERRA COMPETITION-DIFFUSION SYSTEM 6553
Moreover, for comparison purpose we assume further thatr 1 =r 2 , as we have done in previous papers [16,19,20].
Our goal is to illustrate the global dynamics of (13) in terms of (d 1 , d 2 ) and to determine the asymptotic behaviors of the four sets in the decomposition (8) as d 1 and/or d 2 tend to 0 or ∞. Similarly as in [19] and [20], we will distinguish between the following two cases: (i) r 1 /ξ ≡ const and r 2 /ξ ≡ const; (ii) both r 1 /ξ and r 2 /ξ are nonconstant.
We first consider Case (i): We assume in addition thatr Then for system (13), it holds that Σ U , In other words, the species U never loses, while V never wins! More detailed asymptotic behavior ofd 2 (d 1 ) as d 1 → ∞ will be given in Theorem 4.2 in Section 4 below. For an illustration of shapes of the sets Σ U and Σ − in Theorem 1.4, see     For Case (ii), we have the following result: Theorem 1.5. Assume that (A1) holds with ξ 1 ≡ ξ 2 ≡: ξ. We assume in addition thatr Then for system (13) Moreover, there exist two constant B 1 := B 1 (r 1 , r 2 , ξ) and B 2 := B 2 (r 1 , r 2 , ξ) such that whereD =D(r 1 , r 2 , ξ) is a positive constant. Furthermore, as d 1 , d 2 → ∞, Σ − approaches asymptotically a band in Ω with slope C 1 (r 2 , ξ)/C 1 (r 1 , ξ) and width Therefore, in Case (ii), the competition seems more "balanced". For an illustration of shapes of the sets Σ U , Σ V and Σ − in Theorem 1.5, see Figure 4. The rest of this paper is organized as follows: In Section 2 we establish some preliminary results which will be used in later sections. In Section 3, we classify global dynamics of a general Lotka-Volterra competition system which includes model (4) as a special case. Then we establish Theorems 1.2 and 1.3. Section 4 is devoted to proving Theorems 1.4 and 1.5. Some concluding remarks including an important application of our main result to a general Lotka-Volterra competitiondiffusion-advection systems given by Corollary 5.1 are provided in Section 5.

GLOBAL DYNAMICS OF LOTKA-VOLTERRA COMPETITION-DIFFUSION SYSTEM 6555
2. Preliminaries. In this section, we establish some basic facts and preliminary results which will be need latter.
For linear stability of the trivial steady state (0, 0) and the two semi-trivial steady state (u d1 , 0) and (0, v d2 ) of system (4), we have the following relatively simple criterion. The proof follows essentially from the same arguments as in that of [27, Corollary 2.10] and therefore is omitted here.
To further characterize the principal eigenvalue µ 1 (d, h), we need to introduce the following eigenvalue problem with indefinite weight: where h ∈ L ∞ (Ω) is nonconstant and could change sign in Ω. We say that λ is a principal eigenvalue if (21) has a positive solution. (Notice that 0 is always a principal eigenvalue.) The following result is standard. (See, e.g., [4,37].) We will also use the following lemma derived from the theory of monotone dynamical systems. See, e.g., Proposition 9.1 and Theorem 9.2 in [21]. Lemma 2.4. For any d 1 , d 2 > 0, assume that every co-existence steady state of (4), if exists, is asymptotically stable, then one of the following alternatives holds: (i) There exists a unique co-existence steady state of (4) which is globally asymptotically stable. (ii) System (4) has no co-existence steady state and either one of (u d1 , 0) or (0, v d2 ) is globally asymptotically stable, while the other one is unstable.
3. Global dynamics of general Lotka-Volterra competition systems. In this section, we state and prove our main results concerning the global dynamics of the following general system: in Ω, which includes system (4) as a special case. Throughout the paper, we assume that By similar arguments as in the proofs of existence and uniqueness results in [8] (see also [21,Theorem 28.1]), we see that the following equations and have unique positive steady states, which we denote byũ andṽ respectively. Moreover,ũ andṽ are globally asymptotically stable. Hence system (22) has two semitrivial steady states (ũ, 0) and (0,ṽ). Let µ 1 (a(x), q(x)) be the principal eigenvalue of the following eigenvalue problem: where a ∈ C 1,α (Ω) is positive onΩ. Then µ 1 (a(x), q(x)) admits the following variational characterization: By similar arguments to the proof of [27, Corollary 2.10], we can show that the linear stability of (ũ, 0) is determined by the sign of µ 1 (a 2 , r 2 −b 2ũ ). In other words, (ũ, 0) is linearly stable (resp. unstable) if the principal eigenvalue µ 1 (a 2 , r 2 − b 2ũ ) > 0 (resp. µ 1 (a 2 , r 2 − b 2ũ ) < 0). The linear stability of (0,ṽ) can be determined accordingly by the sign of µ 1 (a 1 , r 1 − c 1ṽ ). The global dynamics of system (22) can be characterized as follows: or holds, then µ 1 (a 2 , r 2 − b 2ũ ) and µ 1 (a 1 , r 1 − c 1ṽ ) cannot be both positive, i.e., exactly one of the following four alternatives holds: Moreover, Furthermore, the following hold for system (22): Proof. In the following, we first prove the theorem assuming that (27) holds. At the end of the proof, we point out the slight differences during the proof when , when condition (27) is replaced by (28).
Multiplying the equation forũ byũ and integrating over Ω, we obtain that Choosingũ as a test function in the variational characterization for µ 1 (a 1 , r 1 − c 1ṽ ), by (26) and (30), we obtain that Therefore, by Hölder's inequality, we see that Similarly, choosingṽ as a test function in the variational characterization for µ 1 (a 2 , r 2 − b 2ũ ), we obtain that

QIAN GUO, XIAOQING HE AND WEI-MING NI
Therefore, Denote (27) and it follows from (31) and (32) that Therefore, one of the four alternative cases (A)-(D) must hold, as µ 1 (a 2 , r 2 −b 2ũ ) > 0 and µ 1 (a 1 , r 1 − c 1ṽ ) > 0 simultaneously would lead to a contradiction to (33). Now assume that Case (D) holds, then it follows from (33) that I(b 1 , c 1 , b 2 , c 2 ) = 1 and all the above inequalities involved in the proof of (33) must be equalities. In other words., we obtain that The above relations and the equations forũ andṽ actually further imply that in Case (D), a 1 , a 2 , r 1 and r 2 must satisfy On the other hand, assuming that (34) holds, we see from the equations forũ and v that Hence, Case (D) holds. This finishes the proof for (29). We now claim that: (S1) if one of Cases (A)-(C) holds, then any coexistence steady state of system (22), if it exists, is linearly stable; (S2) if µ 1 (a 2 , r 2 − b 2ũ ) = 0 > µ 1 (a 1 , r 1 − c 1ṽ ) in Case (A) or µ 1 (a 2 , r 2 − b 2ũ ) < 0 = µ 1 (a 1 , r 1 − c 1ṽ ) in Case (B), system (22) has no coexistence steady state at all.
Then Theorem 3.1(i)-(iii) follow directly from the above two claims and Lemma 2.4.

QIAN GUO, XIAOQING HE AND WEI-MING NI
Moreover, when λ 1 = 0, denote η = b 2 /c 2 andη = U/V , which are positive constants. Then U and V satisfy the following equations: By uniqueness of positive steady state to (23) and (24), we obtain that (1+η/η)U = u and (η η + 1)V =ṽ. Therefore, u/ṽ ≡ c 1 /b 1 = η and U + ηV =ũ, which implies by (29) that Case (D) holds. Therefore λ 1 > 0 and this finishes the proof for Claim (S1). Next we prove Claim (S2). It actually follows from a similar perturbation arguments as in the proof of Theorem 3.4 in [18]. We briefly outline the main ideas here. Assume that µ 1 (a 2 , r 2 − b 2ũ ) = 0 > µ 1 (a 1 , r 1 − c 1ṽ ) in Case (A) and assume for contradiction there exists a coexistence steady state (U * , V * ). We then replace b 2 and c 1 by b 2 and c 1 respectively such that b 2 and c 1 are in a sufficiently small neighborhood of b 2 and c 1 in C α (Ω) satisfying that Then it is easy to show by the implicit function theorem that, the perturbed system of (22) with b 2 and c 1 replaced by b 2 and c 1 respectively, still has a coexistence steady state lying in a small neighborhood of (U * , V * ). On the other hand, we can show that µ 1 (a 2 , r 2 − b 2ũ ) > 0 > µ 1 (a 1 , r 1 − c 1ṽ ). Then it follows from Claim (S1) and Lemma 2.4 that, (ũ, 0) is globally asymptotically stable for the perturbed system of (22). However, this is a contradiction to the existence of the coexistence steady state that is close to (U * , V * ). This finishes the proof for Claim (S2).
We now prove Theorem 1.2.
In the end of this section, we characterize in details how the four disjoint components Σ U ∪ Σ U,0 \ Π, Σ V ∪ Σ V,0 \ Π, Σ − and Π of Q in (8) change in terms of b and c, assuming that ξ 1 ∝ ξ 2 . For this purpose, we first establish the following lemma which determines the relative positions of the points (L U , L V ), (L U , S V ) and (L V , S U ) with respect to the line bc = 1 in the bc-plain, where L U , S V , L V and S U are defined in (9). Lemma 3.2. Assume that (A1) holds, ξ 1 ∝ ξ 2 and that at least one of r 1 /ξ 1 and r 2 /ξ 2 is nonconstant. Then the following hold: Proof. Multiplying the equation for θ d,r,ξ by 1 θ d,r,ξ and integrating over Ω, we have obtain that where the last inequality becomes equality if and only if r/ξ ≡ const. This implies that Ω ξθ d,r,ξ dx > Ω r dx, ∀d > 0 and r/ξ ≡ const .
Hence, (i) follows directly from (41) and the fact ξ 1 ∝ ξ 2 . We now prove (ii). By Lemma 2.1 and ξ 1 ∝ ξ 2 , we see that Here, if r 1 /ξ 1 is nonconstant, the last inequality is strict by Lemma 2.1(iii); if r 2 /ξ 2 is nonconstant, then the first inequality is strict by (41). This finishes the proof of (ii). The proof of (iii) is similar to (ii) and is thus omitted.
We now describe how the sets Σ U and Σ U,0 (resp., Σ V and Σ V,0 ) change in the d 1 d 2 -plane when we vary b (resp., c). To characterize the set Σ U in terms of b > 0, we define for each b > 0, where Note that I b is the union of finitely many open intervals, I 0 b is closed, and Similarly, to characterize the set Σ V in terms of c > 0, we define for each c > 0, where for any s > 0, the set {(d 1 , d 2 ) | r 1 ≡ s ξ 1 v d2 } is either empty or equal to a single straight vertical line segment (10), then the following hold: (v) For Σ − , we have the following characterization: and Σ − = ∅ if and only if either the first case holds or r 1 /ξ 1 ≡ c r 2 /ξ 2 and bc = 1 in the last case.
(vi) For Π, we have the following characterization: Hence, Π = ∅ if and only if bc = 1 and exists (d 1 , The proof for the above theorem follows from simiar arguments as in Theorems 3.3 and 3.5 in [18] and thus is omitted. Theorem 1.3 now follows from Lemma 3.2 and Theorem 3.3.

4.
Proof of Theorems 1.4 and 1.5. We now prove Theorem 1.5. The following asymptotic expansion of θ d,r,ξ as d → ∞ which will be used in this section.
We first establish Theorem 1.5.
Proof of Theorem 1.5. The proof for (17) and (18) follows from similar arguments as in the proof for Theorem 1.3 in [16] and Theorem 1.4 in [20] and hence is omitted. We now prove (19). By Theorem 3.3(i), we obtain that where λ 1 (h) is defined as the unique nonzero principal eigenvalue of (21). By Lemma 2.1 and (16), r 2 − ξu d1 changes sign in Ω for all d 1 large. Then by the definition of d 2 (d 1 ), it suffices to show that there exist two constantsD :=D(r 1 , r 2 , ξ) > 0 and B 1 := B 1 (r 1 , r 2 , ξ) such that For convenience, we denote in the rest of this proof. First we claim that Let ϕ 1 > 0 be the eigenfunction corresponding to λ 1 normalized such that where L =r 1 /ξ =r 2 /ξ. By Lemma 2.1 and (16), Ω (r 2 − ξu d1 ) → 0 as d 1 → ∞. Therefore by Proposition 2.3(i) and similar arguments as in the proof of Lemma 2.4 in [31], we can show that, λ 1 → 0 as d 1 → ∞. By standard elliptic regularity estimates, we deduce that, passing to a subsequence of d 1 if necessary, ϕ 1 converges to ϕ 1 in W 2,p (Ω) ∩ C 1,α (Ω) for some constant ϕ 1 ≥ 0. Therefore, it follows from (58) that ϕ 1 = L. This implies that ϕ 1 = L + o(1) as d 1 → ∞. We now rewrite By direct calculation, ω satisfies the following equation: Multiplying the equation for ϕ 1 by ω and the equation for ω by ϕ 1 , integrating over Ω and subtracting, we obtain that Ω Rϕ 1 = 0.
At the end of this section, we give a more detailed characterization of the asymptotic behavior ofd 2 (d 1 ) in Theorem 1.4. (i) If inf Ω ρ r1,ξ + C 1 (r 1 , ξ) > 0, then for all d 1 >D, (ii) If inf Ω ρ r1,ξ + C 1 (r 1 , ξ) < 0, then for all d 1 >D, r 2 − ξu d1 changes sign in Ω and there exist two numberŝ and Γ r1,ξ ∈ R depending on r 1 and ξ such that ) for all d 1 >D. First, competition systems with variable competition coefficients, spatial and/or temporal, have been considered in many works; see [21, Chapter IV] and references therein. In the recent work by Bai and Li [3], global dynamics of system (22) can be clarified provided that For the case of system (4), i.e., when condition (67) reduces to when ξ 1 or ξ 2 is nonconstant, we see that condition (A2) is much better. For the general system (22), none of the conditions (67), (27) and (28) implies the other two, as examples can be easily constructed. This seems to indicate that those criteria are not sharp, and a more complete understanding of the global dynamics of the system is still open. If we restrict ourselves to conditions (67), (27) and (28) and assume that the four competition coefficients b 1 , c 1 , b 2 and c 2 are nonconstant, then roughly speaking, we have the following observation: (i) if the pair of intraspecific competition coefficients b 1 (x) and c 2 (x) are very close/comparable onΩ up to constant scaling, then condition (27) may be better than the other two; (ii) if the pair of interspecific competition coefficients b 2 (x) and c 1 (x) are very close/comparable onΩ up to constant scaling, then condition (28) may be better than the other two; (iii) if minΩ{b 1 , c 2 } ≥ maxΩ{b 2 , c 1 }, then condition (67) may be better than the other two. Along with the study on global dynamics of Lotka-Volterra competition-diffusion systems, there have been a lot of work devoted to understanding the dynamics of Lotka-Volterra competition-diffusion-advection systems in heterogeneous environments, such as the following: in Ω.
As was pointed out in [33], the key ideas developed in [18] for Lotka-Volterra competition-diffusion systems do not work directly when advection terms are involved, since the diffusion-advection operator is usually not self-adjoint. However, through a standard transformation, the above form of Lotka-Volterra competitiondiffusion-advection systems can be rewritten into general Lotka-Volterra competitiondiffusion systems in the form of (22), in which a i (x) now depends on d i , α i and P i (x). To be more specific, let (U, V ) be a coexistence steady state to (69), if it exists. Denote Then (W, Z) satisfies the following system Therefore, Theorem 3.1 can be applied to system (69) to clarify its global dynamics in terms of the local stability of the two semitrial steady states of (69). Denote u * and v * the unique positive steady states to the following equations and respectively. Then we have the following result:
(iii) If Case (C) holds, then there exists a unique co-existence steady state that is globally asymptotically stable. (iv) If Case (D) holds, then there is a compact global attractor consisting of a continuum of steady states {(ζu * , (1 − ζ)u * /c) | ζ ∈ [0, 1]} connecting the two semi-trivial steady states.
It seems that our current understanding of this general heterogeneous Lotka-Volterra competition is fairly good already -at least for system (3) or (4). Given the recent empirical evidence as well as the theoretical models presented in [41] and [40], it seems that a much more realistic model would involve resource dynamics as well, and perhaps that could become our next direction. In that case, our major tasks should consist of: First, to continue the work started in [15] and obtain a complete understanding of systems for single species with resource dynamics, and then move onto consumer-resource competition systems.