MONOTONE TRAVELING WAVES FOR DELAYED LOTKA-VOLTERRA COMPETITION SYSTEMS

. We consider a delayed reaction-diﬀusion Lotka-Volterra competition system which does not generate a monotone semiﬂow with respect to the standard ordering relation for competitive systems. We obtain a necessary and suﬃcient condition for the existence of traveling wave solutions connecting the extinction state to the coexistence state, and prove that such solutions are monotone and unique (up to translation)


Introduction. The classical Lotka-Volterra ODE competition model
has the trivial equilibrium E 0 := (0, 0) and two boundary equilibria E 1 , E 2 . Under the assumption that equilibria are isolated, this Lotka-Volterra ODE competition system may exhibit three different types of global dynamics. If one of the boundary equilibria is a saddle and the other is a sink, then the system has no positive coexistence equilibrium. Otherwise, the system has a coexistence equilibrium E * which can be either a saddle point or a sink depending on whether the boundary equilibria are both sinks or are both saddle points. Such a system generates a monotone dynamical system with respect to the standard ordering for competitive systems, the aforementioned classification of global dynamics is natural and can be obtained by applying the powerful monotone dynamical systems theory, see for example [32]. This flow monotonicity with respect to the standard ordering relation for competitive systems has also made it possible to establish the existence of traveling waves connecting equilibria and to analytically describe the range of wave speeds for the corresponding reaction-diffusion model Consequently, it is well-known that traveling waves in (2) can be either monostable or bistable and this classification is completely (linearly) determined by the stability of the involved equilibria connected by a traveling wave. For example, traveling waves connecting two boundary equilibria E 1 , E 2 are monostable if one of these equilibria is a saddle and the other is a sink, and the traveling waves connecting E 1 , E 2 are bistable if both of the equilibria are sinks. General results about monostable and bistable traveling waves for reaction-diffusion equations admitting comparison principles can be found in the monograph [36], and relevant studies for monotone (i.e., order-preserving) dynamical systems can be found in [38,26,39,20,25]. There are substantial recent developments on the linear or nonlinear determinacy of minimal wave speeds for (2), see [23,14,13] and references therein. Incorporating time delay into interspecific competition does not alter the aforementioned results for the corresponding ODE models since the order-preserving property remains valid. However, incorporating time delay in the intraspecific competition in model (1) may alter these results considerably. Even for a single species population with delayed intraspecific competition (self-limitation) such as the delayed Fisher-KPP equation with τ being a given positive number, the generated semiflow (either the kinetic ordinary delayed differential equation or the PDE analogue) is no longer orderpreserving with respect to any closed cone of phase space when τ is large. This is obvious since a Hopf bifurcation of stable periodic solutions may occur. Similar difficulties arise for the nonlocal Fisher-KPP equation where k is a smooth function with R k(x)dx = 1. For model equation (3), there are some partial answers for the global dynamics and the existence of traveling waves. In particular, when the delay is small, Smith and Thieme [34,35] obtained a general result that shows that the model equation without diffusion generates a monotone semiflow under an exponential ordering, and as such most solutions converge to equilibria and the stability of equilibria is essentially the same as that for the ordinary differential equation model. Additional results on the global dynamics in various delayed competition systems can be found in, for example, [16,17,18,7,9].
When both time delay and diffusion are incorporated, we generally obtain a time-delayed system with nonlocal interaction since populations in a specific spatial location at time t − τ will distribute over all spatial locations at time t due to diffusion (see, e.g., [3]). Friesecke [10] proved that most solutions converge to equilibria in some delayed reaction-diffusion equations similar to (3) in bounded domains subject to Neumann or Dirichlet boundary conditions when delay is sufficiently small. There seems to be no further study to verify whether solutions of such delayed reaction-diffusion equations can generate monotone semiflows even with small delay, except the work of [40] that introduced some PDE analogue of exponential ordering similar to that introduced by [34,35] for delayed ODEs. For general classes of reaction-diffusion equations including (3) in unbounded domains, monotone traveling waves for non-quasi monotone delayed/nonlocal reaction-diffusion equation were established when delay is sufficiently small by Wu and Zou [41] ( using the idea of exponential ordering), and similarly for (4) when the nonlocal response is sufficiently narrow by [12] (using a geometric singular perturbation method). These techniques have been further refined and used in [6,30,31,8] to establish the existence of traveling waves when 1). the delay is small; or 2). the nonlocal interaction is narrow; or 3) the wave speed is large. See [21,22] for another approach based on the cross-iteration scheme. Recently, Kwong and Ou [19] and Gomez and Trofimchuk [11] independently found the critical value of delay for the existence of monotone traveling waves of the delayed Fisher-KPP equation (and the work [11] also obtained the uniqueness and asymptotic behavior of such wave solutions ). Moreover, Fang and Zhao [5] established the critical value of the rate of nonlocal interactions for the existence of monotone traveling waves and proved the uniqueness of such solutions to the nonlocal Fisher-KPP equation, for which Berestycki et al. [2] proved that traveling waves exist for all rates of nonlocal interactions and Nadin et al. [29] numerically showed that there is such a critical value for the existence of monotone traveling waves. Finally, we should mention the work [15,24,4,37] for locally quasi-monotone systems, and the work [28,33] for quasi-monotone delayed systems.
The purpose of this paper is to establish a necessary and sufficient condition for the existence of monotone traveling waves for the reaction-diffusion competition models with delayed intraspecific competition. In particular, we calculate precisely the minimal wave speed and provide the sharpest lower bound of the delay for which monotone traveling waves exist. We describe our results and develop our technical details for the following time-delayed model For the simplicity of notations, we scale system (5) as follows: Under the hypothesis that there exists a unique coexistence equilibrium E * , that is, (a 1 − 1)(a 2 − 1) > 0, we will prove the following results on solutions connecting the trivial equilibrium E 0 := (0, 0) to the positive equilibrium E * := ( 1−a1 1−a1a2 , 1−a2 1−a1a2 ) for system (6): (I) There exist monotone traveling waves with speed c if and only if c ≥ c min := max{2, 2 √ dr} and τ ≤ τ (c); (II) τ (c) is strictly decreasing in c to the positive limit τ (∞), which is the critical value for the existence of monotone heteroclinic orbits of the kinetic system; (III) Such a monotone solution is unique up to translation.
In the above results, τ (c) and τ (∞) are both implicitly determined by some eigenvalue problems. More precisely, c ≥ c min is necessary and sufficient for the existence of a non-negative eigenvector associated with a positive eigenvalue for the linearized problem of the wave profile equation at equilibrium E 0 . τ ≤ τ (c) and τ ≤ τ (∞) are the necessary and sufficient conditions for the existence of a nonpositive eigenvector associated with a negative eigenvalue for the linearized problem of the wave profile equation and the kinetic system at equilibrium E * , respectively. As such, we have the full linear determinacy of the considered model.
The rest of this paper is organized as follows. Section 2 is devoted to the study of the two aforementioned eigenvalue problems. Section 3 describes the properties of all possible monotone wave profiles and reveals their exact convergence rate to E * . These preliminary results are then used in Sections 4 and 5 to derive our main results. We conclude the paper with a relevant system, for which we describe the existence of monotone traveling waves and describe their properties in the critical case. This resolves an unsolved problem in [11,5].
2. Eigenvalue problem. This section is devoted to the study of the eigenvalue problems for the wave profile equation, which is obtained by substituting u(t, x) = U (x + ct) and v(t, x) = V (x + ct) in (6). Here (U, V ) is called the wave profile, ξ := x + ct the wave coordinate and c the speed. For the sake of convenience, we still use u, v, x instead of U, V, ξ, respectively. And hence, we have the following wave profile equation after the scaling u( In what follows, we say u ∈ R 2 is strongly positive if u 0 in the sense that each component of u is positive. We first study the eigenvalue problem at E 0 . Linearizing (7) yields which implies that eigenvalue µ is governed by the following equation Direct computations show that there exists at least one positive eigenvalue if and only if c ≥ c min := max{2, 2 √ dr}. There are nonnegative eigenvectors associated with these eigenvalues. Moreover, for c > c min , there are two eigenvalues µ 1 , µ 3 solving c −2 µ 2 − µ + 1 = 0 and two eigenvalues µ 2 , µ 4 solving c −2 dµ 2 − µ + r = 0.
Next we analyze the eigenvalue problem at E * = (u * , v * ) T . Linearizing (7) yields Plugging u(x) = m 1 e λx , v(x) = m 2 e λx into the above equation, we obtain the following eigenvalue problem: where About the distribution of negative eigenvalues and their associated eigenvectors, we have the following results: Assume that c ≥ c min , then we have the following statements.
Proof. (i) In order to define the value τ (c), we begin with the study of the possible location of a solutionλ to (12). Equation det A(λ, τ ) = 0 requires that h i (λ, τ ), i = 1, 2 has the same sign which, together with A(λ, τ )y = 0 with y 0, requires that Here we shall mention thatλ, λ i1 , λ i2 all depend on τ . Now we can define the set Then it is easy to see S(τ ) is an interval having the form (s 1 (τ ), s 2 (τ )) with s i (τ ) being the zeros of either It then follows that s 1 (τ ) is increasing in τ while s 2 (τ ) is decreasing. These facts imply that quantity max is decreasing in τ . It is not difficult to see that lim τ →0 S(τ ) = (−∞, s 2 (0)), and hence, for small τ equation det A(λ, τ ) = 0 always has a negative solution. On the other hand, h i (λ, τ ) < 0 for all λ < 0 if τ is sufficiently large. Therefore, there exists a critical value τ (c) such that max λ∈S(τ ) h 1 (λ, τ )h 2 (λ, τ ) > ra 1 a 2 u * v * if and only if τ < τ (c). This proves statement (i).
(ii) In order to study the dependence on c, we write h i (λ, τ, c) instead of h i (λ, τ ). Note that h i (λ, τ, c) is decreasing in c, so is max λ∈S(τ ) h 1 (λ, τ, c)h 2 (λ, τ, c). It then follows from the analysis in the proof of statement (i) that τ (c) is decreasing in c and the limit τ (∞) is the maximal value of τ such that system (12) with c = +∞ has a solution.
(iii) Denote the solution set of (λ, τ )-system by K. Hence, there are finitely many elements in any bounded subset of K due to the analyticity of the functions det A(λ, τ ) and (15). It follows that λ 1 (τ ) is bounded for large τ due to the monotonicity of s 1 (τ ). On the other hand, there exists some λ 0 such that det A(λ, τ ) > 0 for all small τ and λ ≤ λ 0 . Therefore, λ 1 (τ ) is bounded in τ .
3. Properties of wave profiles. In this section, we conduct some qualitative analysis for wave profiles. The results will be used in the proof of uniqueness and nonexistence of traveling waves. In what follows, we always assume that (u, v) is a given wave profile between E 0 and E * .
Lemma 3.1. Any non-constant wave profile between E 0 and E * is strictly increasing and connecting E 0 to E * .
Similar results hold for w 2 .
Proof. It is easy to see that w 1 (−∞) = u * , w 1 (+∞) = 0. We can rewrite (19) as the following integral equation for w 1 : where m c (x) = 1 − e c 2 x . For any > 0, there exists x > 0 such that w 1 (x) ≤ , ∀x ≥ x . Since w ≥ 0 by hypothesis, it follows that where Consequently, by arguments similar to those in the proof of [5, Lemma 3.1] we can obtain In view of Lemma 3.2, we may define the Laplace transform Owing to Lemma 3.2, L i (λ) converges at least for λ > −β i1 and diverges for λ < −β i2 . As w i is positive, there exists a singular point λ * i ∈ (−β i1 , −β i2 ) such that L i (λ) < +∞ if λ > λ * i and L i (λ) = +∞ if λ < λ * i . Based on Lemma 3.2, we further give an exact priori estimation of w i (x) as x → +∞. Theorem 3.3. There exists an eigenvector (m 1 , m 2 ) T associated with eigenvalue λ 1 defined in (13) and a polynomial p of order k such that where k + 1 is the multiplicity of eigenvalue λ 1 .
Proof. Note that (w 1 , w 2 ) satisfies the following system of differential equations (29) Using the variation of constants formula for the second order ODE, we have the following integral equation where g 1 , g 2 are defined as in (39). Under the transform (28), equation (30) becomes with Clearly, A(λ, τ ) has the same eigenvalues and eigenvectors with A(λ, τ ), which is defined in (11). Note that if L i (λ) converges for λ >λ, then the right-hand side of (31) converges for λ > 2λ. Such difference in the abscissa of convergence gives rise to the conclusion that L i (λ) has possible singularity only at the zeros of det A(λ, τ ). Further, L 1 (λ) and L 2 (λ) shares the singular point, say λ * < 0. Next, we employ the argument in the proof of [27, Proposition 6.1] (see , where in particular we choose τ = −∞. Hence we see (31) is the similar required version as equation (7.5) with ψ = 0 in [27]. Then following the steps in the proof of [27, Proposition 6.1], we can choose a = λ * − and b = λ * − with > 0 small enough to reach that where p is a polynomial of order k and k + 1 is the multiplicity of λ * as the pole of Next we prove λ * = λ 1 defined in (13) and that (m 1 , m 2 ) T is an associated eigenvector with the assumption that the coefficient of x k in p is the unit. Assume for the sake of contradiction that λ * < λ 1 . Then choosing λ = λ 1 in (31) reads However, two components should have different signs due to h i (λ 1 , τ ) > 0 and det A(λ 1 , τ ) = 0, a contradiction. Assume again for the sake of contradiction that λ * > λ 1 , then h i (λ * , τ ) < 0 due to the definition of λ 1 . Dividing the term x k e λ * x in both sides of the integral equality (30) and taking x → +∞, we arrive at which means that (m 1 , m 2 ) T is a positive eigenvector of λ * , a contradiction with h i (λ * , τ ) < 0. Thus, λ * = λ 1 and (m 1 , m 2 ) T is an associated eigenvector. In what follows, we focus on the existence of monotone traveling waves with speed c ≥ c min and τ ≤ τ (c). We develop our proof in two steps. (i) For the case where c > c min and τ ∈ D c , we employ the results for eigenvalue problems to construct upper and lower fixed points for an appropriate monotone integral operator; (ii) For the case where c = c min or τ ∈ D c , we employ some limiting arguments.
Define the differential operator L : C 2 (R, R 2 ) → C(R, R 2 ) by the left hand side of wave profile equation (7), that is, For fixed c > c min , we have four eigenvalues µ i , i = 1 · · · 4, of the linearized problem at E 0 . Define the integral operator T : C(R, R 2 ) → C(R, R 2 ) by separating the linear and nonlinear parts of the wave profile equation and solving it with the variation of constants formula, that is, with Note that zeros of operator L and fixed points of T are the same, and both are wave profiles with speed c > c min . Next we construct upper and lower solutions to L[φ] = 0. Let λ 1 be defined as in (13) and (m 1 , m 2 ) T its associating eigenvector. Define where l 1 , x − 1 and l 2 , x − 2 are uniquely determined, respectively, by Proof. It suffices to show that the first component ( due to u * − m 1 e λ1x ≤ l 1 e µ1x and v * − m 2 e λ1x ≤ l 2 e µ2x . It then follows that for where equalities 1 − u * − a 1 v * = 0 and A(λ 1 , τ )m = 0 are used. For and for For any τ ∈ D c and small > 0, we see from Lemma 2.1(iv) that there exists p := (p 1 , p 2 ) T 0 such that Moreover, we fix the value of p 1 /p 2 . Then we define φ + p := (φ + 1,p , φ + 2,p ) T with and where δ 1 , x + 1 and δ 2 , x + 2 are uniquely determined, respectively, by and −m 2 λ 1 e λ1x + 1 + p 2 (λ 1 − )e (λ1− )x + 2 = 0, Clearly, and it is increasing in p i to +∞.
Proof. It suffices to show that the first component ( x , ∀x ∈ R due to u * −m 1 e λ1x +p 1 e (λ1− )x ≥ δ 1 and v * −m 2 e λ1x +p 2 e (λ1− )x ≥ δ 2 , respectively, and It then follows that for x > x + 1 , and hence, because A(λ 1 , τ )m = 0 and p i e − x are uniformly bounded in p i and x > x + i , where M > 0 is a constant depending on fixed parameters m, λ 1 , , τ and p 1 /p 2 . This, together with the fact that A(λ 1 − , τ )p 0, implies that ( It then follows that (  Proof. We first consider the case where c > c min and τ ∈ D c . Let φ − and φ + p be defined as in (40) and (48)-(49), respectively. Then φ − and φ + p are C 1 -functions on R. In view of Lemmas 4.1 and 4.2, together with [11,Corollary 16], we see that φ − and φ + p (after necessary translations) are a pair of ordered lower and upper fixed points of the monotone operator T for sufficiently large p. Define the iteration scheme ψ 0 = φ − , ψ n+1 = T [ψ n ], ∀n ≥ 0.
We then obtain a sequence of functions {ψ n } with φ − = v 0 ≤ ψ 1 ≤ · · · ≤ ψ n ≤ · · · ≤ φ + p . It follows that the sequence ψ n converges, as n → ∞, to a continuous and nondecreasing function ψ pointwise on R, and ψ(±∞) both exist. Clearly, ψ is a fixed point of T with φ − ≤ ψ ≤ φ + p . Moreover, it is easy to see from (38) that ψ(±∞) satisfy the algebraic (x, y)-equation which, together with the fact that ψ(−∞) In the case where c = c min or τ ∈ D 1 c , we employ a limiting argument. Without loss of generality, we only consider the case c = c min and τ = τ (c). Choose sequences c n > c min and τ n < τ (c n ) with c n → 2, τ n → τ (c) and τ n ∈ D cn . Then for each n, there is a monotone traveling wave ψ n with speed c n . By appropriate translations, we fix (ψ n ) 1 (0) = 1/2 for all n. Since ψ n is monotone in n, we see from Helly's theorem that there exists a subsequence of ψ n converging to a monotone function ψ pointwise. By Lebesgue's dominated convergence theorem, it follows that ψ is a fixed point of T , and hence, is of C 2 . Furthermore, because (ψ n ) 1 (0) = 1/2, we see that ψ connects E 0 to E * .

5.
Uniqueness. By uniqueness of traveling waves, we mean that any two wave profiles with the same speed must be the same after appropriate translations. The main result of this section is as follows. Proof. Assume, for the sake of contradiction, that there are two wave profiles (u i , v i ), i = 1, 2 with speed c. From Theorem 3.3 we have with (m i 1 , m i 2 ) T being eigenvectors associated with λ 1 . Consequently, m i 1 /m i 2 = a 1 u * /h 1 (λ 1 , τ ), ∀i = 1, 2. Set x 0 = 1 λ1 ln a1u * h1(λ1,τ ) and construct the comparison function (P, Q) as follows: Clearly, P (−∞) = 0 = Q(−∞).
Using the fact 0 ≤ (u i , v i ) ≤ (u * , v * ) and triangular inequality, we obtain from the wave profile equation [P (y) + u * e −λτ P (y − τ ) + a 1 u * Q(y)]dy g 2 (x − y)e −λ(x−y) [Q(y) + v * e −λτ Q(y − τ ) + a 2 v * P (y)]dy . Define and Then (57) can be rewrite as Now we claim that P (+∞) = 0 = Q(+∞). Indeed, direct computations show that R J(λ, y)dy − Id = −A(λ, τ ) which is defined in (32). Assume, for the sake of contradiction, that P (x) and Q(x) have limits greater than 0 (possibly +∞) as x goes to +∞. Then multiplying the matrix in both sides of (60) and letting x → +∞, we obtain However, the components of the right hand side have different signs due to the property of A(λ 1 , τ ), which leads to a contradiction.

6.
Remarks. Based on the proof and results in the previous sections, we also obtain the following result on the heteroclinic orbits of the kinetic system.
The idea of the proof is to pass c → ∞ in Theorems 4.3 and 5.1, we omit the details here.
To conclude this paper, we address an unsolved question on the uniqueness of traveling waves for the delayed Fisher-KPP equation It has been proved in [11,Theorem 4] that a monotone traveling wave exists if and only if delay h is less than some value h 1 and speed c ∈ [0, c * (h)], and such traveling wave is unique if c ∈ [2, c * (h)). The following result fills the gap on the uniqueness when c = c * (h) by employing Lemma 7 in [1]. The similar gap in [5, Thorem 1.2] for the nonlocal Fisher-KPP equation can also be filled in the same way.
Theorem 6.2. Let number h 1 be defined as in [11,Theorem 4] and h ≤ h 1 . Then the traveling wave of (67) with speed c = c * (h) is unique.
Since R N (y)dy = 1, R yN (y) = 0 and R |y|N (y)dy < ∞, we can employ [1,Lemma 7] to conclude that such w can not exist. This completes the proof.