Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka-Volterra systems of three competing species

In reaction-diffusion models describing the interaction between the invading grey squirrel and the established red squirrel in Britain, Okubo et al. ([19]) found that in certain parameter regimes, the profiles of the two species in a wave propagation solution can be determined via a solution of the KPP equation. Motivated by their result, we employ an elementary approach based on the maximum principle for elliptic inequalities coupled with estimates of a total density of the three species to establish the nonexistence of traveling wave solutions for Lotka-Volterra systems of three competing species. Applying our estimates to the May-Leonard model, we obtain upper and lower bounds for the total density of a solution to this system. For the existence of traveling wave solutions to the Lotka-Volterra three-species competing system, we find new semi-exact solutions by virtue of functions other than hyperbolic tangent functions, which are used in constructing one-hump exact traveling wave solutions in [2]. Moreover, new two-hump semi-exact traveling wave solutions different from the ones found in [1] are constructed.


1.
Introduction. In the present paper, we study the following Lotka-Volterra system of three competing species in the entire space R: where u(x, t), v(x, t) and w(x, t) represent the density of the three species u, v and w respectively; d i , λ i , c ii (i = 1, 2, 3), and c ij (i, j = 1, 2, 3, i = j) are the diffusion rates, the intrinsic growth rates, the intra-specific competition rates, and the inter-specific competition rates, respectively. These constants are all assumed to be positive.
The problem as to which species will survive in a competitive system is of importance in ecology. In order to tackle this problem, we use traveling wave solutions, which are solutions of the form (u(x, t), v(x, t), w(x, t)) = (U (z), V (z), W (z)), z = x − θ t, where θ is the propagating speed of the traveling wave. Inserting (2) into (1) yields the following system of ordinary differential equations, where we still use u, v and w to denote U, V and W , We note here that the unknown constant θ needs to be determined while solving the above system. In [17,18], a singular perturbation approach is employed to establish existence of traveling wave solutions for (LV ). For existence of standing wave solutions, we refer to [7,8,9,10].
In addition to the wave solutions mentioned above, a natural question is to study the existence of traveling waves which consist of a forward front v, a backward front u, and a pulse w in the middle (see Figure 1). Du to the hump-like profile of w, we call a wave of this type a one-hump wave. Under certain assumptions on the parameters, existence of solutions of such type was established by finding exact traveling wave solutions as well as numerical simulations in [1,2]. Also, the study in those papers indicates that one-hump waves are of importance to be the blocks for building more complicated waves. In [2], the problem of incorporating w into the simpler system for (u, v) alone was considered. Numerical evidence indicates that the invasion of w could dramatically change the inter-specific competitive behavior between u and v. By using the numerical tracking method AUTO, a global branch of one-hump traveling wave solutions was shown to bifurcate from an exact traveling wave solution when a certain parameter was varied. This indicates that, when some parameter is slightly perturbed around an exact solution, traveling wave solutions of (LV ) persist.
In this paper, we continue the study in [1,2] to find new exact and semi-exact traveling wave solutions. In [2], exact one-hump solutions of (LV ) are represented in terms of polynomials in hyperbolic tangent functions with degree 2. It is shown in this paper, that some new semi-exact one-hump solutions can also be constructed using functions other than hyperbolic tangent functions. Moreover we show, in addition to the ones obtained in [1], that there exist more types of semi-exact twohump solutions.
The long-term goal of related study is to find rather precise conditions for the existence of traveling wave solutions. Therefore, on the other hand, we are also interested in the opposite question: Q1: Under what conditions does there fail to exist traveling wave solutions of (LV) with positive u, v and w?
As far as the authors' knowledge, such nonexistence results are rather few in the literature. Based on the maximum principle for elliptic inequalities coupled with estimates of a total density of the three species, we develop in this paper an approach to establish nonexistence results for traveling wave solutions of (LV ) when the diffusion coefficients of the three species are identical.
In an ecosystem, how to estimate the total biomass is an important issue. To study this problem, it is natural to consider the total density η 1 u + η 2 v + η 3 w, where η 1 , η 2 and η 3 are suitable constants which transfer the population densities u, v and w into the ones with the same unit of mass. Then the total biomass can be expressed as the integration of the total density. The total density is not only important in calculating the total mass, but also crucial in the proof of our nonexistence results. The idea is to obtain estimates for the total density which force one of the three species to be zero. To our knowledge, there are few results in the literature regarding the problem as to how to estimate the total density of the three species in (LV ). It remains unknown how to deal with the general total density η 1 u + η 2 v + η 3 w. However, we are able to estimate the quantity u + v + w for the case of equal diffusion coefficients. The method is to derive some differential inequalities for u + v + w and to use the maximum principle for elliptic inequalities. This suffices to establish nonexistence results in some cases. For the general situation, we are interested in the following question: Q2: Find better optimized estimates for the total density of the three species η 1 u + η 2 v + η 3 w in (LV ), where η 1 , η 2 and η 3 are certain non-negative constants. Moreover, we are able to arrive at a rough estimate for η 1 u + η 2 v + η 3 w through Proposition 1 (see Section 2) together with the following inequalities Throughout this paper, we shall always consider (3) with the following two types of boundary conditions at ±∞: , 0, 0). We remark that the first is studied in [2] while in [1] the second is investigated. When w is absent in (1), the resulting two-species system for u and v can be scaled as where d, λ, a 1 and a 2 are positive parameters. To understand the complicated phenomena exhibited by the three-species, the study of the two-species problem (5) is essential. It is easy to see that (5) has four equilibria: e 1 = (0, 0), e 2 = (1, 0), e 3 = (0, 1) and When the domain is bounded, the asymptotic behavior of solutions (u(x, t), v(x, t)) for (5) with initial conditions u(x, 0), v(x, 0) > 0 can be classified into four cases.
, v(x, t)) be the solution of (5) with the entire space R replaced by a bounded domain in R under the zero Neumann boundary conditions. Then for initial conditions u(x, 0),v(x, 0) > 0, we have (i) a 1 < 1 < a 2 ⇒ lim t→∞ (u(x, t), v(x, t)) = (1, 0); (ii) a 2 < 1 < a 1 ⇒ lim t→∞ (u(x, t), v(x, t)) = (0, 1); (iii) a 1 , a 2 > 1 ⇒ (1, 0) and (0, a) are locally stable equilibria; The traveling wave solution of (5), namely where θ represents the wave velocity of the traveling wave, satisfies where it is assumed that the wave tends to some equilibria at infinities and i, j = 1, 2, 3, 4. We remark that the determination of θ is part of the task in solving system (7). For exact solutions of (7), we refer to [6,20,21]. The solution (u(z), v(z)) of (7) is called an (e i , e j )-wave. In this paper, we only concentrate on the (e 2 , e 3 )-wave and slove the problem Problem (P ) for the special case where d = λ = 1, a 1 + a 2 = 2 and a 1 < 1, a 2 > 1 was considered by Okubo et al. ( [19]). They added the two equations in (P ) together to obtain q zz + θ q z + q (1 − q) = 0, (9) where q = u + v and q(±∞) = 1. Then the maximum principle (see Lemma 2.1 in Section 2) yields q = 1 or u + v = 1 for all z ∈ R. Substituting v = 1 − u into the first equation in P gives which is the Fisher equation. It is well-known that the minimal wave speed for (10) is θ u min = 2 √ 1 − a 1 , i.e. θ ≥ θ min . On the other hand, it is readily seen that v and the minimal wave speed for (11) is θ v min = 2 √ a 2 − 1, which coincides with θ u min since a 1 + a 2 = 2. For other (e i , e j )-waves, the reader is referred to [6]. For cases (i) and (iii) in Theorem 1.1 (i.e, monostable and bistable cases), Kan-on ( [11,12]), Fei and Carr ( [5]), Leung, Hou and Li ( [15]), and Leung and Feng ([14]) proved the existence of (e 2 , e 3 )-waves using different approaches. As Q2, we can ask a similar question: Q3: Suppose that a solution of (P ) exists for some d, a 1 , a 2 and λ. Then is u + v larger than 1, smaller than 1, or equal to 1 when d = λ = 1?
The paper is organized as follows. Q1, Q2 and Q3 are investigated in Section 2. When the diffusion coefficients are all identical, some partial answers are obtained.
As an example, we show our results can be applied to the May-Leonard model ( [16]). Section 3 is devoted to reconstructing the exact solution presented in [2] without the aid of Mathematica. By using the exact solution obtained there, we show that the nonexistence results in Section 2 could fail to hold when some parameters vary slightly. As a continuation of the study in [2] and [1], more semi-exact solutions of (3) are constructed respectively in Section 4 and Section 5. Finally, several remarks, including biological meaning of the results in Section 2, are discussed in Section 6.
2. Nonexistence and a priori estimates of u + v + w. As mentioned in the introduction, we consider in this section solutions of (3) with two types of boundary conditions: Before introducing our main results in this section, we prove the following useful lemma.

Lemma 2.1 (Maximum principle).
For any constant c ∈ R and any positive func- Then φ(z) ≤ 1 (resp. ≥ 1) for all z ∈ R. When the inequalities are replaced by equalities, i.e.
Proof. Suppose that there exists z 0 ∈ R with φ(z 0 ) > 1. Then for some which is a contradiction. The alternative case can be shown in a similar manner. By combining the results of the two cases considered above, we obtain φ(z) = 1 for all z ∈ R for the equality case.
In order to prove the nonexistence result, we need to estimate u + v + w first. ).

Suppose that system (3) with boundary conditions either
Proof. Using Lemma 2.1, the proof is elementary. Suppose that (3), with either (12) or (13), has a positive solution (u(z), v(z), w(z)). Let p(z) = u(z) + v(z) + w(z), then the addition of the three equations in (3) gives (in what follows we write for Thus, p satisfies By means of Lemma 2.1, we see that p ≤ 1 and the proof is completed. The alternative case can be shown in a similar manner. From the biological point of view, the interpretation of Proposition 1 is that, whenever the birth rates λ i (i = 1, 2, 3) are relatively larger (respectively, smaller) than the intra-specific competition rates c ii (i = 1, 2, 3), the total density u + v + w is greater (respectively, smaller) than or equal to 1 (after a natural scaling of the system).
For the scaled total density p = u + v + w, the problem of whether p > 1, p < 1 or p = 1 is of interest and is essential, since under certain conditions on the parameters it follows from Corollary 1 that p = 1. In this case, the number of equations under consideration is reduced from three to two due to the fact that w is related to u and v by w = 1 − u − v. As a consequence, the situation among the three species dramatically changes. Therefore, this case can be regarded as a critical case for the competition among the three species.
We are now in a position to show the main result in this section. hold.
Then system (3) with boundary conditions either (12) or (13), has no positive so- Proof. Following the proof of Proposition 1, we see that Using w ≤ 1 − u − v, the first and second equations in (3) become respectively and Let q(z) = u(z) + v(z). Combining (16) and (17) gives By means of [H4], it follows from Lemma 2.1 that q ≥ 1. We conclude that w ≤ 0 since p ≤ 1 and q ≥ 1. This is a contradiction and the proof is completed. , c 13 < λ 1 and c 31 ≥ λ 3 hold. Ecologically, this means that without the presence of v, u is relatively stronger than w in the inter-specific competition between u and w. Similarly, c 32 ≥ λ 3 is true when [H3] (in Theorem 2.2) and [H4] are satisfied. Then c 32 ≥ λ 3 together with c 23 < λ 2 ecologically implies that, without the presence of u, v is relatively stronger than w in the inter-specific competition between v and w. We remark that it follows from Theorem 2.2 that under certain hypotheses, system (3) admits no solutions with profiles similar to those investigated in [1] and [2]. [H4] c 13 < λ 1 and c 23 < λ 2 or λ 1 − c 13 = λ 2 − c 23 < 0; [H5] λ 1 > c 12 ; [H6] c ii = λ i (i = 1, 2, 3). Suppose that system (3) with boundary condition (12) has a nontrivial nonnegative solution (u(z), v(z), w(z)), i.e., there exist at least one component of (u(z), v(z), w(z)) which is strictly positive for all z ∈ R and u(z),v(z),w(z) ≥ 0 for all z ∈ R.
Proof. First of all, due to [H1] ∼ [H3] and [H6], it follows from the proof of Proposition 1 and Theorem 2.2 that p = u + v + w = 1 and q = u + v satisfies According to [H4], we shall divide the proof into two cases. For the first case where c 13 < λ 1 , c 23 < λ 2 hold, Lemma 2.1 gives q = u + v = 1. This proves (i) and (iii). Substituting v = 1 − u and w = 0 into the first equation in (3) gives and u(−∞) = 1, u(∞) = 0. Therefore, (ii) follows and the minimal speed of the wave is given by 2 . For the second case where λ 1 − c 13 = λ 2 − c 23 < 0 hold, once again we have (20). Using λ 1 − c 13 = λ 2 − c 23 , (20) is rewritten as We show q = 1. To this end, multiplying the equation in (22) by q and integrating with respect to z yield This gives θ ∞ −∞ (q ) 2 dz = 0. As a result, either θ = 0 or ∞ −∞ (q ) 2 dz = 0. The latter case yields q is a constant and hence q = 1. For the case θ = 0, the problem (22) becomes We prove q = 1 in this case, too. Indeed, let q = ρ and ρ = ( which gives the relation between ρ and q as where the constant k can be determined by (ρ, q) = (0, 1) and it is easy to see that k = − 1 6 . Thus, we have ρ 2 2 = 1 6 (λ 1 − c 13 ) (q − 1) 2 (2 q + 1). Since c 13 > λ 1 , the only possibility for the last equality to be true is q = 1. The other desired results can be shown in a similar manner as the first case. This completes the proof.
As an immediate application of the results in this section, we consider the May-Leonard model ( [16]). Suppose that we consider a non-transitive relationship among the three competing species u, v and w. That is, in pairwise competition between u, v and w, we assume that v eliminates u, u eliminates w, but w eliminates v. An example to model such a phenomenon is the May-Leonard model where 0 < α < 1 and 1 < β are constants. When the effect of species immigrating described by diffusion is incorporated into (28), we are led to the diffusive May- Suppose that (u, v, w) = (u, v, w)(z), z = x − θ t, is a traveling wave solution of (29) with boundary conditions (u, v, w)(−∞) = (0, 1, 0) and (u, v, w)(∞) = (1, 0, 0), According to Theorem 1, when α + β ≤ 2, the total density of the three species has a lower bound, that is u+v +w ≥ 1, while when α +β ≥ 2, we have an upper bound for the total density of the three species, that is u + v + w ≤ 1. Due to Theorem 2. To answer Q3 mentioned in the introduction, for simplicity we consider (P ) with d = 1, i.e., ( * ) Proof. The proof of the desired result is similar to that of Theorem 1, and is hence omitted here.
3. Exact traveling wave solutions revisited. In [2], we have shown that under certain conditions on the parameters, (3) admits exact traveling wave solutions with the profiles of u(z) being increasing (respectively, u(z) being decreasing) in z, v(z) being decreasing (respectively, v(z) being increasing) in z, and w(z) being a pulse (i.e., w(±∞) = 0 and w(z) > 0 for z ∈ R). All the calculations there were carried out with the aid of Mathematica. In what follows, we show that the same result can be obtained without using Mathematica.
In this section, we show that exact traveling wave solutions with similar profiles can also be constructed using functions other than hyperbolic tangent functions.
To be more precise, let us recall the key ideas in the previous papers. In [2], the hyperbolic tangent function tanh is used as the building block to construct the one-hump solution for three-species problem. We note that all the exact solutions in [2] are polynomials in tanh of degree 2. Those solutions lead us to the question whether there are wave solutions which can be represented by polynomials with degree more than 2 and whether there are other functions which can be used as the building blocks. The papers [20,21] by Rodrigo and Mimura present several exact solutions for two-species case. Their results indicate that, in addition to tanh, the Weierstrass elliptic function and Jacobi elliptic function can be used to represent solutions. Although the ideas in [20,21] are very inspiring, we need to find other methods to deal with the more complicated three-species problem. In [1], the idea to represent a wave solution in terms of a function T (x) which satisfies a simple ODE with a polynomial nonlinearity was proposed. When T (x) is tanh, the wave solution can be written down explicitly and is called an exact solution. When T (x) can only be solved in an implicit form, the wave solution is called a semi-exact solution. The purpose of [1] is to find a two-hump solution (see Figure 5 below). Since it seems hard to construct such a solution by tanh, other T (x) were used.
In this section, we show that in addition to tanh, a lot of one-hump solutions can also be constructed by other T (x). Moreover, the solutions constructed are polynomials in T (x) with degree great than 2. Now let us consider the solution T (z) of the initial value problem d dz where T 0 ∈ (0, 1) and a ∈ R are constants. Suppose that (3) admits a solution of the form where i, m and n are positive integers; k 1 , k 2 and k 3 are positive constants. As we have done in Section 3, using ansätz (44) in (3) gives a system of algebraic equations. We then use Mathematica to solve this system as in Section 3 to obtain seven types of exact solutions. According to different i, m and n, these solutions are classified into seven types as shown in Table 1. It should be noted that we are not able to apply ansätz (44) to find solutions of (3) in a systematic way. In Table 1, 0.178908 < a < 0.34 or a > 0.491722 a > 3.89898 or a < −5.89898 Table 1. Seven sets of solutions of (3) restrictions on a as well as the parameter dependence of the propagation speed θ for each type of solutions are shown. For simplicity, we only discuss one type of solution in more detail here. For further details on the other six types of solutions, refer to [3]. For (i, m, n) = (2, 4, 1), we have the solution to (3), provided that the following relations hold: λ 3 = (7 + 5a) (−15 + a (32 + a (25 + 6a))) d 1 −13 + 3a(8 + 3a) , (46c) c 31 = (5 + 3a)(7 + 5a)(−9 + a(17 + 12a))d 1 (−13 + 3a(8 + 3a))k 1 , c 32 = 15(−1 + 3a)(7 + 5a)d 1 (−13 + 3a(8 + 3a))k 2 , (46g) where k 1 , k 2 , k 3 > 0 are constants. The necessary and sufficient condition for k i , d i , λ i , c ii (i = 1, 2, 3), c ij (i, j = 1, 2, 3, i = j) > 0, and a / ∈ [−1, 0] in (46) is given by Approximately, 0.178908 < a < 0.34 or a > 0.491722. For the particular case where a = 1 and T (0) = 1 2 , the change of variable T 2 (z) = 1 2 (1 + v(z)) transforms the problem (43) into the following initial value problem for v(z): It is readily seen that v(z) = tanh z − 1 2 . Hence, in this case the semi-exact solution (45) to (3) can be rewritten in terms of tanh z as We remark that this solution is essentially different from that presented in [2], in that V (z) and W (z) are not polynomials in hyperbolic tangent functions, as mentioned in the Introduction. We give an example for illustration. If we choose (a, k 1 , k 2 , k 3 , d 1 ) = (  We observe that the significant differences between the solution in Figure 3 and the exact solution in [2] are: (i) W (z) in Figure 3 is not symmetric with respect to the z-axis, while in [2] W (z) is; (ii) Compared with W (z) in [2], W (z) in Figure 3 is relatively larger; (iii) As mentioned, V (z) and W (z) are not polynomials in hyperbolic tangent functions, but V (z) and W (z) in [2] are. 5. New exact two-hump solutions satisfying boundary conditions (13). Two types of semi-exact two-hump solutions are proposed in [1]. As a continuation of the study in [1], we construct two more types of such solutions in this section.
In particular, we note that all the parameter restrictions in [1] are represented in terms of rational functions, while in Theorem 5.1 and Theorem 5.2 below some of the parameter restrictions are expressed in terms of non-rational functions. For the method to obtain these semi-exact solutions, see [1].
An example is given below to illustrate Theorem 5.1. The profiles of the solution in Theorem 5.2 are similar. When n = 3, Figure 4 and Figure 5 show the solution of (57) and the profiles of (56), respectively.  From Figure 6, we see that there exists a solution such that p(z) = u(z) + v(z) + w(z) takes both values greater than 1 for some z and values less than 1 for some other z. This shows the fact that the estimates in Proposition 1 fail to be true for general parameters.  6. Concluding remarks. Motivated by the work of Okubo et al. ( [19]), we have developed an elementary approach relying on the repeated use of the maximum principle for elliptic inequalities, to establish nonexistence results for Lotka-Volterra systems of three competing species, as well as a priori estimates of total scaled species density, namely u + v + w. Such nonexistence results and a priori estimates have not yet been studied in the literature. We employ this approach to investigate these problems in Section 2.
In addition, exact traveling solutions for Lotka-Volterra systems of three competing species presented in [2] are expressed in terms of polynomials in hyperbolic tangent functions. Using functions other than hyperbolic tangent functions, we show that new semi-exact traveling wave solutions with similar wave profiles as in [2] can be found. Furthermore, in addition to the two types of semi-exact traveling wave solutions given in [1], new semi-exact traveling wave solutions are constructed.
Based on the maximum principle for elliptic inequalities, an elementary approach is developed to establish the estimates on the amount of the scaled density of three species, and the nonexistence result of traveling wave solutions. We believe that this approach can be applied to various problems in related studies.