Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small

In this paper, we study the global dynamics of a general $2\times 2$ competition models with nonsymmetric nonlocal dispersal operators. Our results indicate that local stability implies global stability provided that one of the diffusion rates is sufficiently small. This paper continues the work in \cite{BaiLi2017}, where competition models with symmetric nonlocal operators are considered.

books [6,23]. Indeed, dispersal strategies of organisms have been a central topic in ecology. However, in ecology, in many situations (e.g. [4,7,8,25]), dispersal is better described as a long range process rather than as a local one, and integral operators appear as a natural choice. A commonly used form that integrates such long range dispersal in ecology is the following nonlocal diffusion operator [1,11,12,14,19,20,21,22,24] Lu := Ω k(x, y)u(y)dy − a(x)u(x), where the dispersal kernel k(x, y) ≥ 0 describes the probability to jump from one location to another.
To be more specific, this paper is motivated by the studies of Lotka-Volterra type weak competition models with spatial inhomogeneity, that is f (x, u, v) = m(x)−u− cv, g(x, u, v) = m(x)−bu−v with 0 < bc < 1 in (1). For this case, u(t, x), v(t, x) are the population densities of two competing species, d, D > 0 are their dispersal rates, which measure the total number of dispersal individuals per unit time, respectively, while m(x) is nonconstant and represents spatial distribution of resources. This type of models reflects the interactions among dispersal strategies, spatial heterogeneity of resources and interspecific competition abilities on the persistence of species and has received extensive studies from both mathematicians and ecologists for the last three decades. For models with random diffusion, see [5,9,13,17,18] and the references therein, while for models with nonlocal dispersals, see [2,3,10,16] and the references therein.
Inspired by the nature of this type of models, in [18], an insightful conjecture was proposed and partially verified: Conjucture A. The locally stable steady state is globally asymptotically stable.
For symmetric PDE case, this conjecture has been completely resolved in [9]. Moreover, if random diffusion is replaced by symmetric nonlocal operators or symmetric mixed dispersal strategies, this conjecture is also verified in [3]. In the proofs of these results, the symmetry property of operators and the particular form of reaction terms are crucial. This naturally leads us to investigate the system (1) with nonsymmetric operators and more general reaction terms.
To better present the main results, let us designate the definitions of nonlocal operators in (1) and assumptions imposed on f, g. Denote For φ ∈ X, define = Ω p(x, y)φ(y)dy − Ω p(y, x)dyφ(x), where the kernels k(x, y), p(x, y) describe the rate at which organisms move from point y to point x. Here the operators defined in (D) and (N) correspond to nonlocal operator with lethal boundary condition and no flux boundary condition respectively. See [11] for the derivation of different types of nonlocal operators.
Throughout this paper, unless designated otherwise, we assume that In (A0), the assumption k(x, x), p(x, x) > 0 corresponds to the strict ellipticity condition for differential operators, which guarantees strong maximum principle. Notice that the reaction terms provided that m ∈ L ∞ (Ω) and the assumption (A5) corresponds to 0 < bc < 1. Moreover, different from PDE case, for models with nonlocal operators, the optimal regularity of solutions is at most the same as the regularity of the reaction terms. Hence, (A1) is imposed to guarantee that the solutions could be continuous in space variable.
For clarity, let (U (x), V (x)) denote a nonnegative steady state of (1), then there are at most three possibilities: is called a semi-trivial steady state, where θ d , η D are the positive solutions to single-species models and respectively. • U > 0, V > 0, and we call (U, V ) a coexistence/positive steady state. The main result in this paper gives a classification of the global dynamics to the competition system (1) under the assumptions (A0)-(A5) provided that one diffusion rate is small. Theorem 1.1. Assume that (A0)-(A5) hold. Also assume that (1) admits two semi-trivial steady states (θ d , 0) and (0, η D ). Then for the global dynamics of the system (1) with nonlocal operators defined in (D) or (N), we have the following statements provided that d is sufficiently small: (i) If µ 0 > 0, ν 0 D > 0 and in addition k(x, y) > 0 for x, y ∈Ω, then the system (1) admits a unique positive steady state in X×X, which is globally asymptotically stable relative to X ++ × X ++ ; (ii) If µ 0 > 0 and ν 0 D < 0, then (0, η D ) is globally asymptotically stable relative to X ++ × X ++ ; (iii) If µ 0 < 0, then (θ d , 0) is globally asymptotically stable relative to X ++ × X ++ . Here µ 0 and ν 0 D are defined in (12). Based on the definitions of µ 0 and ν 0 D defined in (12) and Lemma 2.3, indeed Theorem 1.1 verifies Conjecture A for more general reaction-diffusion systems provided that one diffusion rate is small.
To prove Theorem 1.1, the min-max and max-min characterizations of nonlocal eigenvalue problems obtained in [15] play an important role.
In particular, the proof of Theorem 1.1(i) is much more complicated. It is motivated by the techniques originally developed in the proof of [2, Theorem 1.1], where a simplified version of nonlocal operators is considered. We need first investigate the properties of the limiting system to the system (1) as d → 0 + . To be more specific, notice that due to (A2) and implicit function theorem, there exists Then formally we derive the limiting system of the stationary problem of the system (1) as d → 0 + as follows and characterize the existence and uniqueness of solutions (u, v) with v being positive to this system. Here µ 0 and ν 0 D are defined in (12). As demonstrated in Theorem 2.5, to establish Theorem 1.1(i), the key point is to verify the uniqueness of positive steady states, if exists, to the system (1) when d is sufficiently small. According to Theorem 1.2, it is naturally expected that this property holds for d small. However, it is highly nontrivial to realize it mathematically since it is possible that multiple positive steady states to the system (1) converge to the same solution of the limiting system (5) as d → 0 + . To exclude this possibility, we need more precise characterization of the asymptotic behavior of the positive steady states of the system (1) as d → 0 + .
If µ 0 > 0, there exist C 1 , C 2 > 0, independent of d, such that where (U 0 , V 0 ) is the unique solution to (5) with V 0 being positive.
Obviously Theorem 1.3 indicates that indeed the system (5) is the limiting system of the system (6) as d → 0 + . On the basis of Theorem 1.3, we make use of the proof by contradiction and some thorough estimates to derive the uniqueness of positive steady states, if exists, to the system (1) for d small. During this process, it seems that the extra condition that k(x, y) > 0 for x, y ∈Ω in Theorem 1.1(i) is imposed for technical reasons. However, it remains unknown that whether the complexity of nonsymmetric operators could result in multiple positive steady states under the assumption (A5). We will return to this topic in future work.
In this paper, we only demonstrate the proof of Theorem 1.1 for the nonlocal operators defined in (D), since the proof for nonlocal operators defined in (N) is almost the same.
This paper is organized as follows. In Section 2, some useful properties related to single equations, nonlocal eigenvalue problems and monotone systems are prepared. Then the limiting system and the asymptotic behavior of the positive steady states of the system (1) as d → 0 + is investigated in Section 3. At the end, Section 4 is devoted to the proof of Theorem 1.1.

2.
Preliminary. Throughout this paper, when we discuss the spectrum of linear operators, we always think of them as operators from X to X.
See [3, Theorem 2.1] for details. For further analysis, we need estimate the asymptotic behavior of θ d as d → 0 + .
Proof. First of all, by (A4) and strong maximum principle, it is easy to show that 0 < θ d < M . Then due to (A2), there exists c 0 such that On the one side, provided that c 1 is large enough.
On the other side, set u = (F + (x, 0) − c 2 d) + . Then compute as follows provided that c 2 is large enough. At the end, choose C 3 = max{c 1 , c 2 } and the desired conclusion follows.

2.2.
Properties of nonlocal eigenvalue problems. First of all, the linearized operator of (1) at (θ d , 0) is Also, the linearized operator of (1) at (0, η D ) is It is known that the signs of µ (θ d ,0) and ν (0,η D ) determine the local stability/instability of (θ d , 0) and (0, η D ) respectively. This is explicitly stated as follows and the proof is omitted since it is standard.
Lemma 2.1. Assume that the assumptions (A0), (A1) hold. Then The characterization of µ (θ d ,0) and ν (0,η D ) plays an important role in this paper. We present the related results from [15] as follows for the convenience of readers. 15]). Assume that γ ∈ X and let Then we have In particular, Proof. Indeed, (10) is proved in [15] and we only explain how to derive (11). Note that Then it is easy to see that It is proved.
Now we prepare two simple properties, which will be repeatedly used in future. 0) and lim d→0 + ν (0,η D ) exist and denote Moreover, This lemma follows directly from Proposition 1 and Theorem 2.2 and we omit the details of its proof. Theorem 2.5. Assume that the assumptions (A0)-(A4) hold and the system (1) admits two semi-trivial steady states, denoted by (θ d , 0) and (0, η D ). We have the following three possibilities: (i) If both µ (θ d ,0) > 0 and ν (0,η D ) > 0, the system (1) at least has one positive steady state in L ∞ (Ω) × L ∞ (Ω). If in addition, assume that the system (1) has a unique positive steady state in X × X, then it is globally asymptotically stable relative to X ++ × X ++ . (ii) If µ (θ d ,0) > 0 and no positive steady states of the system (1) exist, then the semi-trivial steady state (0, η D ) is globally asymptotically stable relative to X ++ × X ++ .
Proof. The arguments are almost the same as that of [2, Theorem 2.1], where a simplified nonlocal operator is considered.
The following simple property indicates that any positive steady state of the system (1) in L ∞ (Ω) × L ∞ (Ω) belongs to X × X under the assumption (A5). Lemma 2.6. Assume that (A0) holds and f v g u ≤ f u g v inΩ × R + × R + . Then any positive steady state of the system (1) in L ∞ (Ω) × L ∞ (Ω) belongs to X × X.
This lemma can be verified easily by applying implicit function theorem and we omit the details.

3.
Limiting system and asymptotic behavior as d → 0 + . In this section, we will investigate properties of the limiting system (5) and characterize the asymptotic behavior of the solutions of (6) as d → 0 + .

3.1.
Existence and uniqueness of limiting system. The main purpose in this subsection is to study the existence and uniqueness of solution (u, v) with v being positive to the system (5) and establish Theorem 1.2. Notice that the system (5) can be rewritten as First of all, we need establish a property of g(x, F + (x, v), v), which will be used repeatedly throughout this paper.
Proof. Notice that due to (A2), (A3) and (4), F (x, v) is strictly decreasing in v ≥ 0. Thus we only need discuss the following three situations.
Next, we prepare the comparison principle for (13).
It is clear that * > 0 since v ≡ 0. We will further prove that * ≤ 1. Suppose that * > 1. Let z = * v * − v. It is clear that z ≥ 0 and there exists x 2 ∈Ω such that z(x 2 ) = 0. Thus by Lemma 3.1, direct computation yields that Hence at x = x 2 , we have D Ω p(x 2 , y)z(y)dy < 0, which is impossible. Therefore, 0 < * ≤ 1. At the end, if * < 1, then v * > * v * ≥ v inΩ. If * = 1, then similar to the arguments in the proof of the claim, by using (14), it follows that z = v * − v ≡ 0 is the only possibility. The proof is complete. Now, the existence and uniqueness of positive solutions to (13) will be characterized as follows.
Proof. On the one hand, assume that V 0 (x) is a positive solution to (13). Then On the other hand, assume that µ 0 > 0. We first point out that g(x, F + (x, α), v) is strictly increasing in α > 0 due to (A2), (A3) and (4). This simple property will be used repeatedly for the rest of the proof.
Thus the existence and uniqueness of V k is guaranteed by [3, Theorem 2.1]. Moreover, due to (A2), (A3) and (4) again, it is routine to show that Hence V = lim k→∞ V k is a positive solution of (13) in L ∞ (Ω). Moreover, due to Lemma 3.1, it is standard to verify that V ∈ X by applying implicit function theorem.
Finally, we are ready to demonstrate the main result in this subsection.
Proof of Theorem 1.2. Notice that if V 0 is a positive solution of (13), then (u, v) = (F + (x, V 0 ), V 0 ) is a solution of the limiting system (5). Thus it is clear that the first part is equivalent to Proposition 2. It remains to verify that u ≡ 0 if and only if ν 0 D ≤ 0. This is obvious since u ≡ 0 if and only if v = η D .

3.2.
Asymptotic behavior of the solutions of (6) as d → 0 + . This subsection is devoted to the proof of Theorem 1.3, which indicates that (5) is the limiting system of (6) as d → 0 + by characterizing the asymptotic behavior of the solutions of (6) as d → 0 + .
Proof of Theorem 1.3. For clarity, we divide the proof into several steps. Note that 0 < u d , v d < M due to (A4). Then due to (A2), there exists c 0 > 0 such that Step 1. Regard v d as a given function first and thus u d is unique due to (A2). Set if c 1 is large enough and fix it.
if c 2 is large enough and fix it. Now we have derived that by upper/lower solution method.
Step 2. Let us give a preliminary estimate of v d . Since µ 0 > 0, then similar to the proof of Proposition 2, for d sufficiently small, the following problem admits a unique positive solution, denoted byv. Similarly, for d sufficiently small, the problem 20) admits a unique positive solution, denoted by v. According to (18), it is standard to check that v d is a lower solution of (19). Together with the fact thatv is the unique solution of (19), one sees that v d <v inΩ. Similarly, we have v < v d inΩ. Hence Step 3. Now we could improve the estimate of v d and relate it to V 0 . On the one side, consider (1 + c 3 d)V 0 with c 3 > 0 and compute as follows if c 3 is large enough and fix it, where the second inequality is due to Lemma 3.1 and σ 1 > 0 is some constant. This indicates that (1 + c 3 d)V 0 is a upper solution of (19). Sincev is the unique solution of (19), we havev if c 4 is large enough and fix it, where the inequality is due to Lemma 3.1 and σ 2 > 0 is some constant. Similarly, this implies that (1 − c 4 √ d)V 0 is a lower solution of (20). Thus the uniqueness of the positive solution to (20) yields that At the end, it follows from (21), (23) and (24) that This, together with (18) Thanks to Theorem 1.3, we have Note that by Theorem 1.2, the positivity of V 0 is guaranteed by µ 0 > 0, while ν 0 D < 0 implies that U 0 ≡ 0. Thus, in fact, V 0 = η D . Then due to the first equation in (26) and Lemma 2.4, one sees that Therefore, This is a contradiction.
Proof of Theorem 1.1(iii). First of all, we claim that if µ 0 < 0, then ν 0 where the last inequality is due to Lemma 3.1. This indicates that there exists Hence according to (A2) and the definition of F , it follows that The claim is proved. Now by Lemma 2.3 and Theorem 2.5, to prove Theorem 1.1(iii), it suffices to show that the system (1) admits no positive steady states if d is sufficiently small.
Suppose that there exists a sequence {d i } i≥1 with lim i→∞ d i = 0 + such that for d = d i , the problem (26) admits a positive solution (u i , v i ) ∈ X × X. Then where the last inequality is due to Lemma 3.1. Moreover, notice that regardless of whether µ 0 > 0 or not, the estimate in (18) in the proof of Theorem 1.3 always hold. Hence it follows from (27) that which yields a contradiction by letting d i → 0 + .
The rest of this section is devoted to the proof of Theorem 1.1(i).
Proof of Theorem 1.1(i). According to Lemma 2.3, Lemma 2.6 and Theorem 2.5, it is clear that we only need verify the uniqueness of positive steady states in X × X when d is sufficiently small.
Suppose that there exists a sequence {d i } i≥1 with lim i→∞ d i = 0 + such that the system (1) with d = d i admits two different positive steady states (u i , v i ) and (u * i , v * i ) in X × X. Then similar to the discussion at the beginning of the proof of [3, Theorem 1.1(i)], we assume that w.l.o.g., It is routine to check that This is a contradiction to the definitions of φ i and ψ i .