THE SIGN OF TRAVELING WAVE SPEED IN BISTABLE DYNAMICS

. We are concerned with the sign of traveling wave speed in bistable dynamics. This question is related to which species wins the competition in multiple species competition models. It is well-known that the wave speed is unique for traveling wave connecting two stable states. In this paper, we ﬁrst review some known results on the sign of wave speed in bistable two species competition models. Then we derive rigorously the sign of bistable wave speed for a special three species competition model describing the competition in two diﬀerent circumstances: (1) two species are weak competitors and one species is a strong competitor; (2) three species are very strong competitors. It is interesting to observe that, under certain conditions on the parameters, two weaker competitors can wipe out the strongest competitor.

1. Introduction. In this paper, we are concerned with the sign of wave speed of traveling wave solution to a reaction-diffusion system. Traveling wave solution has been an important object in the study of pattern formations over the last ten decades. Typically a traveling wave solution connects two constant steady states at left and right ends of the spatial domain such that it keeps the same shape for all times and moves with a constant speed. There are the so-called front type wave, pulse type wave and the mixture of both front and pulse types. Here we focus mainly on the front type waves.
A wave is of monostable type, if one of these two steady states is stable and the other is unstable in the ODE sense (i.e. without diffusion). It is of bistable type, if both steady states are stable in the ODE sense. For the monostable waves, there exist a continuum of wave speeds. On the other hand, for the bistable case, it is well-known that the admissible wave speed is unique (in most cases).
In the competition model, the sign of wave speed gives us the information on which species wins the competition. The sign of wave speed decides which species becomes dominant and eventually occupies the whole habitat. Therefore, it is an important task to determine the sign of this unique wave speed in bistable dynamics. However, less attention were paid on the sign of wave speed (cf. [1,9,10,19]) in past years. The main purpose of this work is to provide a new result on a special 3-species bistable competition system. Consider the following system of reaction-diffusion equations where N ∈ N. Assume that there are two constant states u ± := (u 1 ± , · · · , u N ± ) of (1) such that they are stable in the ODE sense. To be a front, we meant that u j − = u j + for all j.
Recently, the following special 3-species competition system was studied ( [11,13,14]): where d i , a i , b i , i = 1, 2, 3, are positive constants. The special nonlinearity models that there is no competition between species u and w. One should note that, in general, a 3-species competition system is not a monotone system. However, under this special circumstance, system (3)-(5) is a monotone system which enjoys a comparison principle. There are two special states (1, 0, 1) and (0, 1, 0), the former is the case when v loses and the latter one is when v wins. To determine whether species v wins the competition, we consider the traveling waves connecting these two states. Under the assumption the traveling wave of (3)-(5) connecting (1, 0, 1) and (0, 1, 0) is of monostable type and this case was studied in [11] including the spatial discrete case for a lattice dynamical system. Indeed, the discrete three species competition system associated with system (3)-(5) is: In this case, the species v always wins the competition, since there exists the positive minimal speed to system (3)-(5).
On the other hand, under the assumption both states (1, 0, 1) and (0, 1, 0) are stable. Intuitively, species v should win the competition, since v is a strong competitor and u (w, resp.) is a weak competitor in the absence of w (u, resp.). However, putting u and w together (under the condition b 1 + b 3 > 1), it is possible that v loses the competition. It is one of the questions to be addressed in this paper. For system (6)- (8), the existence of traveling front connecting (1, 0, 1) and (0, 1, 0) is derived in [13], while the stability and uniqueness of traveling fronts were addressed in [14]. Since our main concern here is the sign of wave speed for traveling fronts of bistable type, we shall not address the existence of traveling fronts to (3)-(5) connecting (1, 0, 1) and (0, 1, 0) with (9) here. We only refer the reader to [22,7] for some general theory to the existence of traveling waves. By some numerical simulations on system (3)-(5) with (9), it is found that v wins the competition, if b 2 > b 1 + b 3 , and v loses the competition when b 2 < b 1 + b 3 . One of the purposes of this paper is to give a rigorous proof of this numerical observation. In addition, we also consider the case that b i 1 for each i. In this case, we will investigate how the diffusion rates affect the sign of the wave speed by a singular limit analysis. This is motivated by a recent work on two species case by Girardin and Nadin [9] in which they provide some results on the wave speed sign when both species are very strong competitors.
The rest of this paper is organized as follows. First, in §2 we shall review some existing results for the 1 and 2 species competition models. Then the 3-species case is treated in §3. In particular, we derive the strict monotonicity of wave profile and the uniqueness of wave speed and wave profiles (up to translations) for system (3)-(5). Finally, we give some criteria to determine the sign of wave speed under certain conditions on the parameters.
As one can see, the understanding of wave speed sign in bistable dynamics is far from complete even for 2-species competition case. For example, should the diffusion coefficients and growth rates be taken into account in the determination of wave speed sign? Next, nothing is known about the sign of wave speed in the discrete lattice dynamical systems. In fact, there is a possibility of propagation failure for small diffusion which makes the question more subtle (see, e.g., [12]). Finally, the sign of wave speed for the 3-species case, both discrete and continuous cases are still largely left open.
2. Review of some existing results.
Multiplying (10) by U and integrating it over Therefore, the sign of wave speed s is determined by the sign of the integral of f over [0, 1].
2.2. Two species case. Consider the following Lotka-Volterra competition diffusion system where u = u(x, t) and v = v(x, t) represent population densities of two competing species, and a, h, k, d are positive constants in which 1, a are the intrinsic growth rates, 1, d are the diffusion coefficients, h, k are the inter-specific competition coefficients. Here the carrying capacity is normalized to be 1 (by taking a suitable unit) for each species.
The u-equation in system (11) can be deduced by taking suitable scales of time and space variables. The parameters h and k influence the asymptotic behaviors of (u, v) and it is the bistable case when h, k > 1. Indeed, both constant states (0, 1) and (1, 0) are stable in the ODE sense. For the existence and stability of traveling waves to (11), we refer to [20,8,3,15,16,17,18,19], etc. In particular, in the bistable case, Kan-on [15] derived the existence of traveling fronts such that the speed is unique and the wave profile is monotone and unique up to translations.
It seems from Theorem 2.1 that the sign of wave speed only depends on the sign of (k − h). This fits perfectly with the intuition that the stronger competitor wins the competition.
Next, by the change of the variables (Ũ ,Ṽ ) = (U, aV ), problem (P) is reduced to the following problem (P): where Here, for given a and d, we have the following relations between parameters (h, k) and (b, c): We recall from [15] the following property of monotone dependence on parameters: From (19) and (h, Also, in [15], it is proved that for any d > 0 and for any positive numbers b, Our question is that, for a given (a, b, c, d) (or, (a, h, k, d)), can we determine the sign ofs (or, s)? To this aim, we first recall some information on s = 0 from [10] as follows.
Hence the case when a = d is completely understood. However, for a = d, we only have It is interesting to observe the following scaling property.
Theorem 2.5 indicates that the sign of wave speed depends only on the ratio of intrinsic growth rate a and diffusion coefficient d of v.
Finally, we have the following results for a > d: Since, for h = k, we have One of the key ingredients in the proof of Theorems 2.6 and 2.7 is the following identities where r := a/d and (s, U, V ) is a traveling front with s = 0. Note that sign(s) is inconsistent with sign(k − h) in Theorem 2.7.
3. Three species case. In this section, we first consider the system (3)-(5) for a 3-species competition system. Without loss of generality, we may assume that d 2 = a 2 = 1 by taking suitable scales of time and space variables. Hence system (3)-(5) becomes Then, for a traveling front (s, U, V, W ) of system (3)-(5) connecting (1, 0, 1) and such that From now on, we shall assume that Note that (25) implies that Then, following the method used in [14], we can prove the following uniqueness theorem. Proof. Indeed, as in [14], it is more convenient to transform system (21)-(23) to the following cooperative system whereÛ := 1 − U ,V := V andŴ := 1 − W . Then, using (25) and (26), the same super-sub-solutions as in [14] with j + ct replaced by x − st based on a given traveling wave solution (s, U, V, W ) of (3)-(5) can be constructed (see (3.10)-(3.12) in [14]), where the operator D 2 is replaced by the second derivative and the monotone functions Finally, since cooperative systems enjoy the (strong) comparison principle, the conclusion follows by the same proof as that of [14,Theorem 4.2]. We safely omit the details.
than (26) is assumed. The proof is rather standard and we only outline it as follows.
3.1. Case of two weak competitors and one strong competitor. In this subsection, we consider the special case: It follows that Hence I < 0 and so s < 0, if b 2 ≥ 2 and b 1 < 1. The theorem follows.
This theorem gives us the case when species v wins the competition. Next, we prepare a lemma as follows.
Proof. First, recall from Lemma 3.3 that U = W . Next, it is easy to check that (s, V, W ) satisfies Hence the conclusion follows from Theorem 2.3.
Theorem 3.4 tells us that two weak competitors can wipe out a strong competitor. This phenomenon can be observed for other ranges of parameters, by using theorems mentioned in section 2. We leave the details here to the interested reader.

3.2.
Case of three very strong competitors. In this subsection, we consider the sign of the wave speed s determined by the following system: where d 1 , d 3 , a 1 , a 3 , β 12 , β 23 , β 32 are given positive constants and k is an arbitrarily large constant.
When W ≡ 0, system (33) is reduced to the one studied by Girardin and Nadin [9]. They consider the infinite competition limit (as k → ∞) and the corresponding limiting problem has the segregation property, which has been discussed by Dancer et al. [4,5] for competition-diffusion systems in bounded domains. It turns out the solution of the limiting problem corresponds to semi-waves studied by Du and Lin [6] (see also [2,9] for more complete description). Then the sign of the wave speed can be determined in terms of the property of semi-waves. Following this idea, for system (33), if taking k → ∞, species u (resp., w) and species v cannot coexist in the same interval. Therefore, we may expect that in the limiting problem, species u and w will coexist in (−∞, ξ 0 ) (since there is no interaction between u and w); while species v will occupy (ξ 0 , ∞) for some ξ 0 ∈ R.
Our main result of this subsection is as follows.
Theorem 3.5. Given positive constants d 1 , d 3 , a 1 , a 3 , β 12 , β 23 , β 32 , there exists a sufficiently large N 0 (depending on d 1 , d 3 , a 1 , a 3 , β 12 , β 23 , β 32 ) such that Theorem 3.5 reveals the role of the diffusion rates for the special 3-species competition-diffusion system. It shows that under a very strong competition, if species u or species w diffuses faster enough, then species v will lose in the competition; while species v will win the competition if both species u and w diffuse slowly enough.
With the help of Proposition A, we can estimate the wave speed as follows. This result is important to determine the sign of the wave speed. Proposition 1. Suppose that (s, U, V, W ) is a solution of (33) for a given k > 0. Then Proof. Recall from the definition of P ± i (i = 1, 2, 3) in the appendix that P + i (−s/d i ) < 0 for i = 1, 3 and P − 2 (−s) < 0. By the definitions of λ 2 , µ 1 and µ 3 , we see that −s d i > µ i , i = 1, 3; and − s < λ 2 .
We now consider the limiting problem by a singular limit process. For each k ∈ N, let {(s k , U k , V k , W k )} be the solution to (33) with s = s k . Thanks to Proposition 1, we may assume, up to extract a subsequence, that s k → s * as k → ∞ for some Moreover, without loss of generality we may assume that This condition makes sure that the limit functions are not null.
Since we have ω k L ∞ (R) = 1 for ω = U, V, W , it is not hard to see that {U k }, {V k } and {W k } are equicontinuous in [−n, n] for any n ∈ N (see, e.g., [9, Proposition 3.1]). By Arzelá-Ascoli theorem, up to extract a subsequence, there exists uniformly on any compact subset of R. Moreover, since U k < 0, V k > 0 and W k < 0 in R, The following result shows that the limit function (U * , V * , W * ) has the so-called segregation property.
Proof. We can follow the same line as in [9,Lemma 3.2] to show this result. For reader's convenience, we give the details here. Multiplying U k -equation by a test function ϕ ∈ C ∞ 0 (R) and integrating over (−∞, ∞), we have where we have used the integration by parts. By Proposition 1 and the fact that 0 < U k < 1 for all k, we have | ∞ −∞ U k V k ϕ| ≤ C ϕ C 2 /k for some constant C > 0 independent of k. By taking k → ∞, U * V * = 0. Similarly, we have W * V * = 0 and thus the lemma follows.
Next, multiplying U k -equation by −1/(a 1 β 12 ) and W k -equation by −β 23 /(a 3 β 32 ), and then summing the two equations with V k -equation, we obtain It follows that in the weak sense. Define Proof. Since U * , V * and W * are continuous in R, with (39) one can follow the process in [9, Lemma 3.5] to finish the proof. We omit the details here.
We are ready to show Proposition 2.
Proof of Proposition 2. We only consider the case s * ≤ 0, since the proof for the case s * > 0 is parallel. Define If A = +∞, from Lemma 3.6 we see that V * ≡ 0. Using the standard elliptic regularity theory, we see that U * ∈ C 2 (R) satisfies in the classical sense. Using (35) and the monotonicity of U * , (s * , U * ) is exactly a traveling front solution connecting 1 and 0, which gives s * ≥ 2 √ a 1 d 1 . This contradicts with s * ≤ 0. Hence, A ∈ (0, ∞). Similarly, we can show that there exists B ∈ (0, ∞) such that S(W * ) = (−∞, B).
Next, we shall show that A = B. If A = B, without loss of generality we may assume that A > B. In this case, Lemma 3.6 yields that W * = V * = 0 in [B, A]. By Lemma 3.7, U * (A) = 0. Also, recall that U * (A) = 0. By the uniqueness of solutions of ODEs, we obtain U * ≡ 0, which is impossible. Therefore, we obtain A = B. Moreover, we have S(V * ) = (A, ∞) in view of Lemma 3.6.
In fact, s * cannot be an endpoint.
Proposition 2 and Lemma 3.8 imply that U * , V * and W * are exactly semi-waves constructed by in [6,2,9]. Therefore, the wave profile (U * , V * , W * ) of the limiting problem is unique. Let us recall a result in [9]: Proof of Theorem 3.5. Let (s * , U * , V * , W * ) satisfy (37) with (38). Without of loss generality, we may assume that ξ 0 = 0 (ξ 0 is defined in Proposition 2). From Proposition 2, we see that U * , V * and W * can be seen as three semi-wave of the Fisher-KPP equation. It turns out that we can use a similar argument in [9, Theorem 4.1] to finish the proof, which is given as follows. By a suitable scaling, we have
Appendix. In the appendix, we provide the asymptotic behavior of the wave profile of traveling wave solutions (U, V, W ) as ξ → ±∞. We assume that (s, U, V, W ) satisfies where b 2 > 1, b 4 > 1, b 1 + b 3 > 1.
Applying a similar approach as in [21] but with some more tedious calculations, we have the following result. Here we omit the proof. where r = 0, if µ 2 = µ j for j = 1, 3, 1 or 2 if µ 2 = µ j for some j ∈ {1, 3},