Stability of traveling wave solutions for a nonlocal Lotka-Volterra model

: In this paper, we studied the stability of traveling wave solutions of a two-species Lotka-Volterra competition model in the form of a coupled system of reaction di ﬀ usion equations with non-local intraspeciﬁc and interspeciﬁc competitions in space at times. First, the uniform upper bounds for the solutions of the model was proved. By using the anti-weighted method and the energy estimates, the asymptotic stability of traveling waves with large wave speeds of the system was established.


Introduction
This paper is motivated by the following biological question: How do diffusion and nonlocal intraspecific and interspecific competitions affect the competition outcomes of two competing species?It is well known that if we introduced the spatial dispersal into the Lotka-Volterra competition model, traveling wave solutions are possible.Such solutions effected a smooth transition between two steady states of the space independent system, [1][2][3][4][5][6][7], but for the models that involve nonlocality, the study of traveling waves is challenging and the properties of the traveling waves becomes more complex.Gourley and Ruan [8] proposed a two-species competition model described by a reaction diffusion system with nonlocal terms.By using linear chain techniques and geometric singular perturbation theory, the existence of traveling waves under some conditions were proved.Some other results about the traveling waves of the Lotka-Volterra system or the similar equations with nonlocal terms can be referred to [9][10][11][12][13].
Here the functions u(x, t) and v(x, t) denote the densities of two competing species with respect to location x and time t, respectively.The positive parameter r is the relative growth rate of species v to species u.We assume that the kernels φ i (i = 1, 2, 3, 4) are bounded functions and satisfy the following properties, for all x ∈ R, (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}); (K3) ess inf (−δ,δ) φ i > 0, for some δ > 0. We propose system (1.1) as an extension of the existing two-species reaction diffusion competition models [1][2][3][4][5].For these two species, the terms −u(φ i * u), i = 1, 3 represent intraspecific competition for resources.These two terms involve a convolution in space that arises because of the fact that the animals are moving (by diffusion) and have, therefore, not been at the same point in space at times.Thus, intraspecific competition for resources depends not simply on population density at one point in space, but on a weighted average involving values at all points in space.The terms a 1 u(φ 2 * v) and a 2 v(φ 4 * u), with a 1 and a 2 positive constants, describe the interspecific competition between these two species for resources, which also involve a convolution in space at times.In this paper, we study the weak competition case with 0 < a 1 , a 2 < 1.It is well known in this case that we have (u, v)(t) → (u * , v * ) as t → ∞ in the region {u, v > 0}.
Han et al. [13] proved the existence of traveling wave solutions of the system (1.1) connecting the origin to some positive steady state with some minimal wave speed.Besides the existence and uniqueness of traveling waves, the stability of traveling waves is also a central question in the study of traveling waves.In contrast to the studies on the existence on the traveling waves of the nonlocal Lotka-Volterra system, the study about the stability is very minor.Lin and Ruan [14] proved the asymptotic behavior of traveling waves about a Lotka-Volterra competition system with distributed delays by using Schauder's fixed point theorem, and in [1,14], the delay does not need to be sufficiently small.In addition, if u = 0 or v = 0, the system (1.1) is the Fisher-KPP equation with a nonlocal term in [7,[15][16][17].Recently, there has been some great progress on traveling waves of the nonlocal Fisher-KPP equation (1.2) Hamel and Ryzhik [16] proved uniform upper bounds for the solutions of the Cauchy problem of (1.2).After that, Tian et al. [17] proved the asymptotic stability of traveling waves for the system (1.2) with large wave speeds.
Inspired by [13,[15][16][17], in this paper we study the stability of traveling wave solutions of system (1.1), which describes the scenario when both intraspecific competition and interspecific competition are nonlocal with respect to space.The main mathematical challenge when studying the traveling waves for system (1.1) is that solutions do not obey the maximum principle and the comparison principle cannot be applied to the system.However, we can consider the stability of the zero solution of a perturbation equation about the traveling wave solution and use the anti-weighted method and the energy estimates to reach the expected one.Mei et al. [18] has applied this method in the Nicholsons blowies equation with diffusion, as did [17] in the Fisher-KPP equation with the nonlocal term.For this method, the key step is to establish priori estimates for solutions.Therefore, before presenting the main theorem in this paper, we first give some important preliminaries for the Cauchy problem of system (1.1).
We organize the paper in the following.In section two, we give a global bound of the solutions and some important properties of traveling waves of the system (1.1).The results on the global existence and uniqueness of the perturbation equations about traveling waves are presented in section three.The uniform boundedness for the perturbation equations is given in section four.In section five, we prove the main theorem about the asymptotic stability of traveling waves for the system (1.1).We conclude with a discussion section containing summarization and implications on our findings.

Global bounds of the solutions for system (1.1)
In this section, we first consider the global bounds of the solutions for system (1.1), then give some auxiliary statements of traveling waves of system (1.1).
Proof.By standard parabolic estimates, the solution (u, v) is classical in (0, +∞) × R and we claim that u(t, x), v(t, x) are nonnegative for every t > 0, x ∈ R. Indeed, if the claim is false, without loss of generality, we assume that for t ∈ (0, T ] where T is some fixed constant, there exist constants K, > 0 such that inf u(T, x) = − e KT and − e Kt < u(t, x) < 0, − e Kt < v(t, x).
From the system (1.1), for t ∈ (0, T ], it gives Since u(0, x) is nonnegative, by the maximum principle, it gives that u(t, x) ≥ 0. This is a contradiction.The claim holds, which gives that u(t, x), v(t, x) satisfy for every t > 0 and x ∈ R. Let δ > 0 be defined as in the assumption (K3) and introduce the local average on the scale δ, for (t, x) ∈ [0, +∞) × R, for every (t, x) ∈ (0, +∞) × R. Since the righthand side of the above equations belong to Owing to the assumption (K3), there exists η > 0 such that and let M be any positive real number such that We now show that ū(t, there exists t 0 > 0 such that ū(t 0 , •) L ∞ (R) = M ū and ū(t, •) L ∞ (R) < M ū for all t ∈ [0, t 0 ).Since ū is nonnegative, there exists a sequence of real numbers (x n ) n∈N such that ū(t 0 , x n ) → M ū as n → +∞.We define the translations for n ∈ N and (t, x) ∈ (0, +∞)×R.From standard parabolic estimates, the sequences (u n ) n∈N and (ū n ) n∈N are bounded in C k loc ((0, +∞) × R) for every k ∈ N; they converge in these spaces, up to extraction of a subsequence, to some nonnegative functions u ∞ and ū∞ of class for every (t, x) ∈ (0, +∞) × R. The passage to the limit in the integral terms is possible due to the local uniform convergence of u n , v n to u ∞ , v ∞ in (0, +∞) × R. Furthermore, we have 0 ≤ u ∞ ≤ M ū, for every 0 < t ≤ t 0 and x ∈ R, and ū∞ (t 0 , 0) = M ū.Therefore, we have Hence, everywhere in [−δ/2, δ/2], then the continuous function Hence ū∞ (t 0 , 0) = 0, which contradicts to the assumption that ū∞ (t 0 , 0) = M ū > 0. Therefore, there is a real number Since both functions φ i , i = 1, 2 and u ∞ , v ∞ are nonnegative, from (2.1), it gives that This contradicts to the definition (2.3).Hence, we obtain that ū(t, •) L ∞ (R) ≤ M ū for all t ≥ 0. Since u is nonnegative, this means that for every t ≥ 0 and x ∈ R. To gain a global bound for u, we fix an arbitrary time s ≥ 1 and then for every x ∈ R, by the maximum principle, it gives that where w is the solution of the equation w t = w xx + w with the initial condition at time s − 1 given by w(s − 1, •) = u(s − 1, •).It then follows from (2.5) that, for every x ∈ R, which implies that u is globally bounded.Using the same method, we also prove that v is global bounded.
where 0 < λ 1 < 1 < λ 2 are roots of λ 2 − cλ + 1 = 0 and 0 < λ 3 < √ r < λ 4 are roots of λ 2 − cλ + r = 0. Hence, we have . Using the same process, we also have Finally, from above results, we can assume and denote which will be used in the next section, λ 0 is defined in (3.2) in the next section.
In the following we show p n+1 , Multiplying the first equation of (3.3) by wp n+1 and the second equation of (3.3) by wq n+1 , we have, and . Integrating (3.13) with respect to ξ over R and using the Young inequality, then integrating over [0, t] with respect to t, we can get where where In order to prove p n+1 , q n+1 ∈ L 2 ([0, t 0 ]; H 2 w (R)), we first differentiate the first equation of the system (3.3) with respect to ξ, then multiply it by wp n+1 ξ ; that is,

The uniform boundedness
In this section, we show the uniform boundedness of the solutions of system (3.1).For the global solution of system (3.1),p, q ∈ X(0, T ) for any fixed T > 0, when the initial perturbation p 0 , q 0 ∈ X 0 , we prove u ∈ X(0, ∞) by deriving the uniform boundedness.As stated before, here we adopt the so-called anti-weighted method [17,18].For this, define the following transform: p(t, ξ) = w(ξ)p(t, ξ), q(t, ξ) = w(ξ)q(t, ξ), and it yields that Theorem 4.1.Suppose that the assumptions of Proposition 3.1 hold, then the solution (p(t, ξ), q(t, ξ)) of system (3.1)belongs to X(0, ∞) and there exists a positive constant C, which is independent of t such that Proof.The proof of this Theorem will be accomplished in the following three steps.
Step 1.We claim that the following inequality holds.
where T > 0 is a given constant.
Multiplying the first equation of (4.1) by p and the second equation by q, then integrating them over
R × [0, t] with respect to ξ and t, we get where we use By the similar arguments, we also have From (4.4)-(4.5),we have Step 2. We show pξ (t) where T > 0 is a given constant and C is a positive constant which is independent of T.
Step 3. We show that where C is a positive constant which is independent of T. Indeed, due to p, q ∈ C uni f [0, T ], we find that lim ξ→+∞ p(t, ξ) = p(t, ∞) =: p 1 (t), lim ξ→+∞ q(t, ξ) = q(t, ∞) =: q 1 (t) exists uniformly for t ∈ [0, T ].Let us take the limit to (3.1) as ξ → ∞, then By the theory of order differential equations, we have Thus we can get, for any given 0 > 0, there exists a large number ξ

Discussion
This paper was motivated by the biological question of how diffusion and nonlocal intraspecific and interspecific competitions affect the competition outcomes of two competing species.This may provide us with insights of how species learn to compete and point out species evolution directions.The model (1.1) is a two-species Lotka-Volterra competition model in the form of a coupled system of reaction diffusion equations with nonlocal intraspecific and interspecific competitions in space at times.Han et al. [13] has proved the existence of traveling wave solutions of the system (1.1) connecting the origin to some positive steady state with some minimal wave speed.Following their steps, we studied the stability of these traveling wave solutions.The main mathematical challenge to study the traveling waves for system (1.1) was that solutions do not obey the maximum principle and the comparison principle cannot be applied to the system.We considered the stability of the zero solution of a perturbation equation about the traveling wave solution and used the anti-weighted method and the energy estimates to reach the expected one.The stability of traveling wave solutions with large enough wave speed of system (1.1) was proved.
The existence, stability, and wave speed of traveling wave solutions could help us to understand for phenomenons such as the movement of the hybrid zone.Hybrid zones are locations where hybrids between species, subspecies, or races are found.Climate change has been implicated as driving shifts of hybridizing species' range limits.However, Hunter et al. [19] found that fitness is also linked to both climatic conditions and movement of hybrid zones.These Lotka-Volterra competition models with advection, diffusion, and nonlocal effects can be used to describe the dynamics of species' range [20] and estimate the movement of the hybrid zone under different assumptions.