TRAVELING WAVES FOR NONLOCAL LOTKA-VOLTERRA COMPETITION SYSTEMS

. In this paper, we study the traveling wave solutions of a Lotka-Volterra diﬀusion competition system with nonlocal terms. We prove that there exists traveling wave solutions of the system connecting equilibrium (0 , 0) to some unknown positive steady state for wave speed c > c ∗ = max (cid:110) 2 , 2 √ dr (cid:111) and there is no such traveling wave solutions for c < c ∗ , where d and r respec-tively corresponds to the diﬀusion coeﬃcients and intrinsic rate of an compe- tition species. Furthermore, we also demonstrate the unknown steady state just is the positive equilibrium of the system when the nonlocal delays only appears in the interspeciﬁc competition term, which implies that the nonlocal delay appearing in the interspeciﬁc competition terms does not aﬀect the exis- tence of traveling wave solutions. Finally, for a speciﬁc kernel function, some numerical simulations are given to show that the traveling wave solutions may connect the zero equilibrium to a periodic steady state.


Introduction.
This paper is concerned with the following Lotka-Volterra diffusion-competition system with nonlocal terms: where d i , r i , a i , b i > 0 (i = 1, 2) are constants and (φ i * u)(x, t) := u and v, respectively. In particular, the convolution terms in (1) describe the nonlocal inter-specific competition (φ 2 * v and φ 4 * u) and the nonlocal intra-specific competition (φ 1 * u and φ 3 * v) for the food sources, etc. For the simplicity of notations, let r 1 t → t, and r = r2 r1 , then the system (1) can be rewritten as follows Here, the kernel φ i (x) (i = 1, 2, 3, 4) are bounded functions and satisfy (K1) φ i (x) ≥ 0 and R φ i (x)dx = 1; (K2) R φ i (y)e λy dy < ∞ for any λ ∈ 0, max 1, r d . It is well known that system (2) (resp.(1)) can further be simplified into the classical Lotka-Volterra competition system if φ i (i = 1, 2, 3, 4) are replaced by the Dirac delta functions. Obviously, system (3) always has three equilibria (0, 0), (1, 0), (0, 1). Especially, if a 1 , a 2 < 1 or a 1 , a 2 > 1, there is the fourth equilibrium (co-existence state) For the traveling wave solutions and the spreading speed of system (3), there were many studies, see [5,9,22,38,42,29,13,23]. For more results about traveling wave solutions of system (3), we refer to [15,43,28] and the references therein (especially the review paper [15]). Due to the realistic models in applications, time delays and nonlocal delays have been widely incorporated into kinds of reaction diffusion systems. The earlier studies about traveling wave solutions of the delayed reaction-diffusion systems mostly considered the cases that the reaction term satisfies quasimonotonicity or nonquasimonotonicity conditions, see [25,26,37,41,48,49]. By using Schauder's fixed point theorem, the cross iteration scheme and the upper and lower solutions method, Li et al. [32] established the existence of traveling wave solutions of a delayed Lotka-Volterra competition systems when the nonlinearity satisfies weak quasimonotonicity or weak non-quasimonotonicity conditions and the delays (occurring in intraspecific competition terms) are sufficiently small. Pan [40] further proved the existence of traveling wave solutions in a Gilpin-Ayala types competition diffusion system under weak quasimonotonicity or weak non-quasimonotonicity conditions. By applying the upper-lower technique and the monotone iteration, Lv and Wang [35] showed that the delayed Lotka-Volterra competitive system admits traveling wave solutions connecting the two semi-trivial equilibriums when the delay is sufficiently small. For more results about traveling wave solutions about the delayed Lotka-Volterra competition system with small delay, we can refer to [31,46,27,24] and the references therein. Recently, Fang and Wu [6] considered the following time-delayed model They proved that system (5) admits monotone traveling wave solutions connecting (0, 0) to the coexistence state if and only if c ≥ max 2, 2 √ dr and τ ≤ τ (c) for ome τ (c) > 0. Lin and Ruan [34] proved the existence and the asymptotic behavior of traveling wave solutions about a Lotka-Volterra competition systems with distributed delays by using Schauder's fixed point theorem, contracting rectangles. Note that in [6,34], the delay does not need to be sufficiently small.
In contrast to the studies on the traveling wave solutions of the delayed Lotka-Volterra system, the study about the nonlocal Lotka-Volterra system is relatively fewer. Gourley and Ruan [12] studied the following system ∂u1 ∂t (x, t) = d 1 where r i , a i , b i , i = 1, 2 are all positive constants and By using linear chain techniques and geometric singular perturbation theory, they established the existence of traveling wave solutions connecting two semitrivial equilibria of system (6) when b 2 < a 1 , a 2 < b 1 and τ 1 , τ 2 are sufficiently small. Later, through transforming system (6) into reaction-diffusion system coupled by four equations without delay, Lin and Li [33] proved that system (6) admits traveling wave solutions connecting (0, 0) to the positive steady state when a 1 < b 2 and a 2 < b 1 . Some other results about the traveling wave solutions of Lotka-Volterra system or the similar equations with nonlocal term can be referred to [14,36,44,45,47]. It is worth noting that the most studies above considered the case that the delays (or nonlocal delays) appearing in intra-specific competition terms are sufficiently small or there are no delays in intra-specific competition terms. In fact, even for the Fisher-KPP equation with (nonlocal) delay, the studies of its traveling wave solutions mainly considered the case that the delay is sufficiently small before. More recently, there have been some great progress on traveling wave solutions of the Fisher-KPP equation with delay or nonlocal delay. Particularly, Berestycki et al. [3] considered the following nonlocal Fisher-KPP equation and proved that (7) admits traveling wave solutions connecting 0 to an unknown positive steady state for c ≥ c * = 2 √ µ and there is no such traveling wave solutions for c < c * . It should be emphasized that the work of Berestycki et al. [3] did not require that the nonlocality is weak( that is, the nonlocal delay is small). After that, many important results about traveling wave solution and the spreading speed of equation (7) have been obtained, see [1,2,11,8,7,20,16,17,18,19,39] and the references therein. In addition, traveling wave soltuions of delayed Fisher-KPP equation was also studied deeply, see [4,11,21,30] and the references therein. Inspired by [3,17,18], in this paper we study traveling wave solutions of system (2) when the nonlocality without any restriction. In addition, throughout this paper, we will assume that a 1 and a 2 satisfies the following hypotheses. (H) 0 < a 1 , a 2 < 1. Now, we state our results as follows.
Theorem 1.1. Assume that (H) holds and the kernel φ i (x)(i = 1, 2, 3, 4) satisfy (K1) and (K2). Then, for any c > c * = max 2, 2 √ dr , there exists a traveling and the boundary conditions where both u(x) and v(x) are positive functions. In particular, the wave profiles u and v decrease on Besides, there is no such traveling wave solution (u(x − ct), v(x − ct)) satisfying (8) and (9) for c < c * .
This paper is organized as follows. In Section 2, we establish the existence of traveling wave solutions of system (1) connecting the equilibrium (0, 0) and an unknown positive steady state, that is Theorem 1.1. Furthermore, we consider a special case of system (1) (that is, φ 1 (x) = φ 3 (x) = δ(x)) and prove Theorem 1.2 in Section 3. In Section 4, by numerical simulations and the stability analysis, we show that the unknown steady state can be a periodic steady state.

2.
Existence of traveling wave solutions of system (2).
In this section, we show that system (2) admits traveling wave solutions connecting (0, 0) to an unknown positive steady state. Let c > max 2, 2 √ dr . Setting ξ = x − ct and looking for solutions of system (2) with the form of (u(x, t), v(x, t)) = (U (ξ), V (ξ)), we can get Our method is to first focus on a two-point boundary value problem on a finite interval and then take the limit of solutions of the problem as the interval passes to the whole line. Specifically, the solutions of two-point boundary value problem are obtained by constructing super-and subsolutions and using Schauder's fixed point theorem.

Supersolution. Take
where λ c > 0 is the smaller root of the equation and ζ c > 0 is the smaller root of the equation Then one has −p c − cp c = p c and − dq c − cq c = rq c in R.

Subsolution.
Take where ε ∈ (0, min{λ c , ζ c }) is small enough such that and Following from the condition (K2), we know that Z c i is well defined. Then for x for any x = ln B ε . A two-point boundary value problem. For c > max 2, 2 √ dr , we consider the following problem in a finite domain (−a, a): In order to obtain the existence of the problem (18), we consider the following two-point boundary value problem where In addition, the convex set M a is defined as Let Ψ a be the solution mapping of the problem (19). Namely, Ψ a (u 0 , v 0 ) = (u, v). It is clear that a solution of the problem (18) is a fixed point of the problem (19). It is easy to know Ψ a is compact and continuous. It suffices to prove that the set M a is invariant for the mapping Ψ a . Given On the other hand, for any . By using the maximum principle again, we know that Now using the Schauder's fixed point theorem, we see that Ψ a has a fixed point (u a , v a ) in M a , which is just the solution of (18). In addition, we have the following lemma.
, v a (a) = q c (a), and Next, we prove the lemma. By evaluating Integrating the previous inequality from x M to x > x M and integrating the last where h(y) = e −y +y−1 Since Take a 0 = 1 2 and let Similarly, take a 0 = d 2r and let y 0 := d 2r , then if x N ∈ (a − y 0 , a), it follows from (23) that Furthermore, it follows from y 0 := d 2r and 1− ry 2 2d ≥ 1− Thus, we always have then we can get the inequality (20). This completes the proof.
Take the limit of (u a , v a ) as a → +∞. It follows from the standard elliptic estimates and Lemma 2.1, we see that there exists M 0 > 0 such that where α ∈ (0, 1) is some constant. Letting a → +∞ (possibly along a subsequence), then we know u a → u and v a → v in C 2 loc (R), and (u( In the following, we divide four steps to complete the proof of Theorem 1.1.
We show that there exists a Z 0 > 0 such that u(x) and v(x) are monotonically decreasing for x > Z 0 .
We use a contradiction argument. Assume that u(x) is not eventually monotonic as x → +∞, then there exists a sequence z n → +∞ such that u(x) achieves a local minimum at z n and u(z then, for any n ∈ N, we have On the other hand, since u(x) and v(x) are bounded in C 2 (R), and lim x→+∞ u(x) = lim x→+∞ v(x) = 0, it is easy to get that which contradicts (24). Therefore, u(x) is eventually monotonic. By using the same method, we can also get that v(x) is eventually monotonic.
Step 2. We show that there exists no traveling wave solutions for speed c < c * . We use a contradiction argument. Assume that for c < c * , there exist a traveling wave solution satisfying (8) and (9). Take a sequence z n satisfying z n → +∞ as where u n (x) = u(x + z n ) and v n (x) = v(x + z n ). It is noting that u n (0) = v n (0) = 1 and u n (x), v n (x) are decreasing in (Z 0 − z n , +∞) for n ∈ N, where Z 0 is defined by Step 1. Since u(x) → 0(x → +∞) and v(x) → 0(x → +∞), we know that ( u n , v n ) → (0, 0) locally uniformly in x as n → +∞. and Clearly, u(·) and v(·) are non-increasing on R and satisfy u(0) = v(0) = 1. In addition, we can show that u(·) and v(·) are positive on R. In fact, if there exists some point x 0 ∈ R such that u(x 0 ) = 0, then u(x) = 0 for any x > x 0 . Now by the uniqueness of solutions of ordinary differential equations, we have u(·) ≡ 0 on R, which contradicts the fact u(0) = 1. Therefore, u(x) > 0 for x ∈ R. Similarly, v(x) > 0 for x ∈ R. By the positivity of u and v, we further have u (x) > 0 and v (x) > 0 for x ∈ R. We now have c ≥ max{2, 2 √ dr} because equation (25)  Step 3. We show that lim inf By a contradiction, we assume that there exists a sequence y n satisfying y n → −∞ as n → +∞ such that u(y n ) → 0 and v(y n ) → 0 as n → +∞. Take u(x) = u(−x), v(x) = v(−x) and c = −c, then ( u(−y n ), v(−y n )) → (0, 0) and ( u( where (φ i ⊗ w) (x) = R φ i (y)w(x + y)dy, i = 1, 2, 3, 4. Similar to Step 2, one has c ≥ max 2, 2 √ dr , which implies c ≤ − max 2, 2 √ dr . This is a contradiction.
Step 4. Finally, we show that there hold lim inf if a 1 < 1 M and a 2 < 1 M , where M is defined by (10). By a contradiction, without loss of generality we assume that lim inf x→−∞ u(x) = 0, then there must hold one of the following two cases: Case 1. There exists a sequence x n → −∞ as n → +∞, such that u(x) attains local minimum at x n and u(x n ) → 0 as n → +∞.
Recall that 0 < u(x) ≤ M and 0 < v(x) ≤ M for any x ∈ R. In addition, u(x) satisfies It follows from lim n→∞ u n (0) = lim n→∞ u(x n ) = 0 that for any δ ∈ 0, 1−a1M 2 , there exists N ∈ N such that u n (x) ≤ δ for any x ∈ (−Z, Z) and n > N , namely, u(x) ≤ δ for any x ∈ (x n − Z, x n + Z) and n > N . Now we show that lim n→∞ (φ 1 * u)(x n ) = 0. Fix any ε > 0. Then there exists Z > 0 such that Then there exists N ∈ N such that u(x n + y) = u n (y) < δ for any y ∈ [−Z, Z] and n > N . Thus, we have for any n > N . This implies that lim n→∞ (φ 1 * u)(x n ) = 0. Hence, for sufficiently large n. On the other hand, since u(x) attains local minimum at x n , then −cu (x n ) − u (x n ) ≤ 0.
Obviously, there is a contradiction.
Since lim x→−∞ u(x) = 0 and lim inf x→−∞ (u(x) + v(x)) > 0, then we know that lim inf It follows that there exists a sequence x n → −∞ (n → +∞) such that Assume that u n (x) converges to u(x) and v n (x) converges to v(x) in C 2 loc (R) as n → ∞. Since lim n→∞ u n (0) = 0, it follows from the Harnack inequality (see (27)) that u n (x) → 0 in C loc (R) as n → ∞. Then applying the L p interior estimates and the imbedding theorem (see Gilbarg and Trudinger [10] for any x > 0. Since u(x) > 0, u (x) ≥ 0 and u(0) = 1, then for sufficiently large x, the inequality (28) is not true, which is a contradiction. Thus, one has lim inf x→−∞ u(x) > 0. Consequently, the proof of Theorem 1.1 is completed.
3. Proof of Theorem 1.2. In this section we prove Theorem 1.2, namely consider the case of φ 1 (x) = φ 3 (x) = δ(x) in system (2). Then system (2) reduces to and the corresponding traveling wave solutions satisfies In order to prove Theorem 1.2, we first establish the existence of solutions of system (30) for any c > c * = max{2, 2 √ dr}. The method is similar to that in Section 2. In the following we fix c > c * = max{2, 2 √ dr}. Supersolution. Take where λ c and ζ c is defined in (14) and (15). Then one has

Subsolution. Take
In addition, A > 1 is large enough such that where Z c 2 and Z c 3 is defined in (16) and (17) respectively. Furthermore, let then similar to those done in Section 2 for p c (x) and q c (x), one has for any x = ln A ε and for any x = ln B ε . A two-point boundary value problem. For c > max 2, 2 √ dr , we consider the following problem in a finite domain (−a, a): where a > max ln A ε , ln B ε . In order to obtain the existence of the problem (31), we consider the following two-point boundary value problem where (u 0 , v 0 ) ∈ M a and x > a, u 0 (x), x ∈ [−a, a], x > a, v 0 (x), x ∈ [−a, a], v 0 (−a), x < −a.
Here the convex set M a is defined as Let Φ a be the solution mapping of the problem (32). Namely, Φ a (u 0 , v 0 ) = (u, v). Similar the arguments in Section 2, we have that Φ a has a fixed point (u a , v a ) in M a , which is just the solution of (31). In addition, it is easy to see that u a (x) ≤ 1 and v a (x) ≤ 1. Let a → +∞ (possibly along a subsequence), then we know u a → u and v a → v in C 2 loc (R) and (u, v) satisfies Moreover, we have that which combining the strong maximum principle implies that 0 < u(x) < 1, 0 < v(x) < 1, and lim x→+∞ u(x) = 0 and lim x→+∞ v(x) = 0.
In particular, we have the following observation. Without loss of generality, we assume that lim inf x→−∞ u(x) = 0, which includes two cases: (i) there exist a sequence x n → −∞ (n → ∞), such that u(x n ) → 0 as n → ∞ and u(x) achieves a local minimum at x n ; (ii) lim x→−∞ u(x) = 0 and there exist a Z > 0, such that u (x) ≥ 0 for any x < −Z.
If the case (i) is true, since u(x) attains local minimum at x n , then Following from (34) and On the other hand, since u(x n ) → 0 and v ≤ 1, then when n → +∞. There is a contradiction.
(ii) Secondly, we show lim sup x→−∞ u(x) < 1 and lim sup x→−∞ v(x) < 1. We also use a contradiction argument. Without loss of generality, we assume that lim sup x→−∞ u(x) = 1. Then there exists By a direct computation, we have that for any s ∈ (0, 1) and u = ( where Then there exists s 0 ∈ (0, 1] such that for any s ∈ [0, s 0 ). To complete the proof, it is sufficient to show and Otherwise, we assume, without loss of generality, that there exists a s 1 ∈ (0, 1) such that u * = a 1 (s 1 ) and for s ∈ (0, s 1 ). Note that 0 < u ≤ 1 and 0 < v ≤ 1. Let w k (x) := u(k + x) and w k (x) := v(k + x) for any x ∈ R and k ∈ Z. Then there holds Since 0 < w k (x) ≤ 1 and 0 <w k (x) ≤ 1 for any x ∈ R and k ∈ Z, it follows from the L p interior estimates (see [10,Theorem 9.11]) that there exists some C 1 > 0 such that w k 2,p;(−2,2) ≤ C 1 w k p;(−3,3) ≤ C 1 6 1 p for any k ∈ Z, where p > 1, · 2,p;(−2,2) and · p;(−3,3) denote the norms of the space W 2,p ((−2, 2)) and L p ((−3, 3)), respectively. Furthermore, by the imbedding theorem, there exists some C 2 > 0 such that |w k (x)| ≤ C 2 for any x ∈ [−2, 2] and k ∈ Z. By the arbitrariness of k ∈ Z, we get Similarly, we can get |v (x)| ≤ C 2 ∀x ∈ R for some constant C 2 > 0. By the equations satisfied by u and v, we further have that there exists a constant C 3 > 0 such that which implies that u , u , v and v are bounded on R. In the following we consider two cases: (i) u * < u * ; (ii) u * = u * .
If u * < u * , then there exists a sequence x n satisfying lim n→+∞ x n = −∞ such that lim inf In addition, it is clear that Using (35), we have which is a contradiction. If u * = u * = a 1 (s 1 ), then we have and Using (35) again, we have which is also a contradiction.
To complete the proof of Theorem 1.2, the remainder is to show the nonexistence of traveling wave solutions for c < c * , which can be done by an argument similar to those in Theorem 1.1. Thus, we have completed the proof of Theorem 1.2.

Numerical simulations.
In Section 2, the existence of traveling wave solutions connecting (0, 0) to an unknown positive steady state for all speeds c > c * is proved. Furthermore, we consider the special case that φ 1 (x) = φ 3 (x) = δ(x) (δ(x) is Dirac function) in Section 3. However, we do not know what the shape of the wave profiles will be. In this section, we will investigate these features of traveling wave solutions by numerical simulations and give some explanations about those results. For convenience of calculations, let φ 2 (x) = φ 4 (x) = δ(x) and φ 1 (x) = φ 3 (x) = φ(x), then (2) can simplify into In the following, we always take a specific kernel function x), then we have Obviously, system (37) have positive equilibrium Next, we give initial value conditions. Take and v(x, 0) = 1−a2 By the definition of u(t, x) and u(t, x), we know that and Similarly, we can obtain and v(x, 0) = Before giving numerical simulation, we also need to know the boundary condition.
Here, the zero-flux boundary condition is considered. Along with (39)-(44), system (37) can be simulated through the pedpe package in Matlab (see Fig 1 and Fig 2). From Figs 1 and 2, it can be seen that the solution (u(x, t), v(x, t)) of system (36) firstly occurs a 'hump' as σ increases. If further increase σ, then the stability of u = u * = 1−a1 1−a1a2 and v = v * = 1−a2 1−a1a2 will change and a periodic steady state will arise around u = u * and v = v * respectively. Now we give a simple explanation of this phenomenon. Linearizing system (37) near the equilibrium (u * , v * , u * , u * , v * , v * ) defined in (38), we can get Take the test function with the form of where λ is the growth rate of perturbations in time t, i is the imaginary unit and k is the wave number. Substituting (46) into (45) gives which is equivalent to when k = 0, the Hopf bifurcation can not arise. However, when k = 0. In addition, there exists k > 0 satisfying E < 0 when σ → +∞. Then it is easy to know that there exist k T > 0 and σ T > 0 such that E (k T ) = 0, E(k T , σ T ) = 0.
Combining (47) and (48), we know that the Turing bifurcation will occur in system (36) at the critical wave number k T when σ = σ T . Therefore, it can be seen that the traveling wave solution connects (0, 0) and a periodic steady state in Fig 1 and  Fig 2.