An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems

By employing the N-barrier method developed in the paper, we establish a new N-barrier maximum principle for diffusive Lotka-Volterra systems of two competing species. As an application of this maximum principle, we show under certain conditions, the existence and nonexistence of traveling waves solutions for systems of three competing species. In addition, new $(1,0,0)$-$(u^{\ast},v^{\ast},0)$ waves are given in terms of the tanh function provided that the parameters satisfy certain conditions.


Introduction
Species diversity refers to the number of different species and abundance of each species that live within an ecological system. To be more specific, species diversity takes into consideration species richness and species evenness; the former is defined as the total number of different species and the latter the variation of abundance in individuals per species. The importance of species diversity to an ecological system lies in the fact that the ecological system with a greater species diversity may function more efficiently and productively since more resources available for other species within the ecological system will be made. Therefore, the study of species diversity has been extensively investigated via both field research and theoretical approaches.
We begin with a two-species system of (1.1) in the absence of W , i.e. on (1.2) with the entire space R is replaced by a bounded and convex domain Ω, we conclude from [15] and [24] that any positive solution (u(x, t), v(x, t)) of such a initialboundary value problem converges to either ( σ 1 c 11 , 0) or (0, σ 2 c 22 ) when U and V are strongly competing, i.e. when the following condition hold: (1. 5) In this case, Gause's principle of competitive exclusion occurs between the two species U and V when U and V competing for the same limited resources cannot stably coexist; one will prevail and the other is excluded. When the influence of diffusion in It is readily seen that (1.6) has four equilibria: e 1 = (0, 0), e 2 = σ 1 c 11 , 0 , e 3 = 0, In the diffusion-free case, we can classify the asymptotic behaviour of solutions (u(t), v(t)) for (1.6) as t → ∞ depending on σ i and c ij (i, j = 1, 2), as described in: As in case (iii) mentioned previously, Gause's principle of competitive exclusion also occurs in cases (i) and (ii): one species always wins and the other species become extinct in the long run. It is easy to see that we do not need to treat one of cases (i) and (ii) since one of the two cases is obtained from the other by exchanging the roles of u and v. For the case of weak competition, the Lotka-Volterra model (1.6) predicts that the stable coexistence state (u * , v * ) exists only when intra-specific competition has a greater effect than inter-specific competition.
We shall assume throughout, unless otherwise stated, that either strong or weak competition occurs between the two species U and V : of (1.2). Here θ is the propagation speed of the traveling wave. We note that u * , v * > 0 if and only if either [S] or [W] holds. Substituting (1.7) into (1.2) yields the following system of ordinary differential equations (1.8) 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 of the form (1.7).
In this paper, we treat the following boundary value problem (1.9) and call a solution (u(x), v(x)) of (1.9) an (e 2 , e 4 )-wave. Under various parameters, monotone (e 2 , e 4 )-waves are investigated via different approaches (for instance, [16], [21], [22]). It is not clear from the assumptions of parameters given in the references above that the monotone (e 2 , e 4 )-waves have the property u * > v * or not. Let us see what happens when the answer is affirmative. If u * > v * holds, then we easily see that, since v is monotonically increasing and u is monotonically decreasing the profile of u lies completely above the profile of v. Accordingly, in this case u dominates the entire habitat R. Indeed, it has been proved that there exist two types of (e 2 , e 4 )-waves, one with u * > v * and the other with u * ≤ v * , by giving exact (e 2 , e 4 )-waves in [19]. We note that in the case u * > v * , the phenomenon exhibited by the dominance of u in the entire habitat R is unique and is yet to be explored. In particular, exact (e 1 , e 2 )-waves, (e 1 , e 4 )-waves, (e 2 , e 3 )-waves and (e 2 , e 4 )-waves are given in [19] under certain conditions on the parameters by applying judicious ansätzes for exact solutions.
When the inhabitant of the two competing species U and V is resource-limited, the investigation of the total mass or the total density of the two species U and V is essential. This gives rises to the problem as to the estimate of the total density u(x) + v(x) in (1.9). In addition, another issue which motivates us to study the estimate of u(x) + v(x) is the measurement of the species evenness index J for (1.9). As J is defined via the Shannon's diversity index H ( [3], [11], [31], [33]), i.e. where s is the total number of species, and p i is the proportion of the i-th species determined by dividing the number of the i-th species species by the total number of all species, the species evenness index J for (1.9) is given by One of our primary goals in this paper is to address the problem of giving a priori estimates of u(x) + v(x), which is involved in the calculation of J in (1.12). On the other hand, we also are concerned with the following question when a priori estimates of u(x) + v(x) are given: Q1: How does the estimate of u(x) + v(x) depend on the parameters in (1.9)?
In [6], upper and lower bounds of u(x) + v(x) are given when the two diffusion rates d 1 and d 2 are equal. However, the approach employed in [6] to obtain upper or lower bounds for u + v cannot be applied directly to the case where the diffusion rates d 1 and d 2 are not equal.
Q2: In (1.9), when d 1 = d 2 , can upper and lower bounds of u + v be obtained?
As for the answer to Q2, it seems as far as we know, not available in the literature. To give an affirmative answer to this question, we develop a new but elementary approach. Moreover, employing this approach leads to a affirmative answer to the following question which is more general than Q2: Q3: In (1.9), when d 1 = d 2 , can upper and lower bounds of α u + β v, where α, β > 0 are arbitrary constants, be given?
By adding the two equations in (1.9), we obtain an equation involving For the sake of clear exposition we shall assume from now on that is a non-zero constant multiple of q(x) has been considered in [6]. A mathematical difficulty arises as a consequence of the fact that the approach used in [6] cannot be directly applied, when d 1 = d 2 in the last equation since p(x) no long can be written as a constant multiple of q(x) and such p(x) and q(x) are involved in a single equation like (1.13). The approach proposed here can be employed to give estimates of p(x) in the case where p(x) and q(x) are involved in the single equation (1.13). One of the main results of this paper is the N-barrier maximum principle (Theorem 2.1), which gives an affirmative answer to Q3. The rest of the paper is organized as follows. In the next section, the main results of this paper, including the N-barrier maximum principle (Theorem 2.1) and two applications (Theorem 2.2 and Theorem 2.3) to the system of three species (1.1), are summarized. We prove in Section 3 the N-barrier maximum principle by means of the construction of N-barriers depending on various conditions. Under certain restrictions on the parameters, the existence of exact traveling waves solutions for (1.1) (Theorem 4.1) is presented in Section 4. Finally, Section 5 is devoted to the proofs of Theorem 2.2 and Theorem 2.3. For any α, β > 0, we have

Statement of main theorems
In particular, (I) (Lower bounds for q = q(x)) when the differential equations in (1.9) are replaced by the differential inequalities (II) (Upper bounds for q = q(x)) when the differential equations in (1.9) are replaced by the differential inequalities We would like to add a few remarks concerning Theorem 2.1.
(i) In addition to the diffusion rates d 1 and d 2 , upper and lower bounds q * and q * depend only on the u-intersection (i.e. σ 1 respectively. We note that e 2 = ( σ 1 c 11 , 0) and e 3 = (0, σ 2 c 22 ) represent the two competitively exclusive states. When α = β = 1, the above observation together with the fact that the estimate of α u(x) + β v(x) given in Theorem 2.1 does not explicitly depend on the propagating speed θ gives a possible answer to Q1.
Let us return to the system of three species (1.1) and consider traveling wave solutions satisfying the following boundary value problem Here again θ is the propagation speed of the traveling wave. As mentioned in the beginning of introduction, we will study the influence of an exotic species W on other native species U and V in terms of (2.16). The first question we shall ask is whether competitormediated coexistence occurs for u, v, and w in the system (2.16). If the three species do coexist under certain conditions, then what will be the profiles of u(x), v(x), and w(x)?
The result in [19] indicates that when w(x) is absent in (2.16), the system of two species (1.9) under certain conditions admits solutions (u(x), v(x)) having the profiles with u(x) being monotonically decreasing and v(x) being monotonically increasing. Moreover, we see from the profiles of u(x) and v(x) that u(x) and v(x) dominate the neighborhood of x = −∞ and the neighborhood of x = ∞, respectively. These facts lead us to the expectation that, the profile of w(x) must be pulse-like (w(x) is a pulse if w(−∞) = w(∞) = 0 and w(x) > 0 for x ∈ R) if it exists since w will prevail only when u and v are not dominant.
To simplify the problem, we restrict ourself to the case of σ 1 = c 11 in Section 4 and denote a solution of (2.16) with σ 1 = c 11 by (1, 0, 0)-(u * , v * , 0) wave for convenience. Although we can find exact (1, 0, 0)-(u * , v * , 0) waves for (2.16) (See Theorem 4.1 and we remark that when σ 1 = c 11 , a similar result remains valid.) under certain restrictions on the parameters, it remains an open problem, however, to establish the existence of solutions for (2.16) under more general conditions. In spite of this fact, when we consider the situation where the influence of the invading species W on the native species U and V is of no significance, i.e. c 13 , c 23 ≈ 0 in (2.16), the limiting case c 13 , c 23 → 0 + leads to the boundary value problem Under the assumption of the existence of solutions (u(x), v(x)) = (ũ(x),ṽ(x)) for the system of two species (1.9) ( [16], [19], [21], [22]), it will be proved in Section 5.1 that under certain conditions, a solution w(x) of the third equation in (2.17), i.e. the nonautonomous Fisher equation for w = w(x) can be found applying the supersolution-subsolution method, thereby establishing the existence of solutions for (2.17). We remark that, as an application of the N-barrier maximum principle, upper and lower bounds of c 31ũ + c 32ṽ are used in constructing supersolutions and subsolutions of (2.18). (2.20) Assume that the following hypotheses hold: [H4]K ≥K > 0; Another consequence of the N-barrier maximum principle concerns the nonexistence of solutions for (2.16). In other words, we look for conditions on the parameters under which there exists no positive solution (u(x), v(x), w(x)) for (2.16). Theorem 2.3 (Nonexistence of traveling wave solutions for three competing species). Let Σ 1 = σ 1 c 33 − σ 3 c 13 and Σ 2 = σ 2 c 33 − σ 3 c 23 . Assume that the following hypotheses hold: Then (2.16) has no positive solution (u(x), v(x), w(x)).
It will be clear from the proof in Section 3 that [A1] and [A2] assure that the Nbarrier maximum principle can be applied in proving Theorem 2.3. We note in particular that when σ 3 is sufficiently small, [A3] in Theorem 2.3 clearly holds. From the viewpoint of ecology, the result of Theorem 2.3 states that as the birth rate σ 3 of the species W decreases below a threshold, the three species U , V and W no longer coexist. Intuitively, due to the weakness of the exotic species W , competitor-mediated coexistence cannot occur for the three species U , V and W in (2.16).

N-barrier maximum principle: proof of Theorem 2.1
In this section, we use the notations p( (1.13). We begin with a useful lemma. Proof of Theorem 2.
Clearly, we see from  , we obtain (3.12). Combining the two inequalities (3.5) and (3.12) yields the lower bound of q(x) given by (3.3). This completes the proof of (I).
4 New exact (1, 0, 0)-(u * , v * , 0) waves In this section, we always assume σ 1 = c 11 , unless otherwise stated. Looking for traveling wave solutions (u(x), v(x), w(x)) with the profiles of u(x) being decreasing in x, v(x) being increasing in x, and w(x) being a pulse (i.e., w(±∞) = 0 and w(x) > 0 for x ∈ R) of (2.16) leads to the following ansätz ( [6,7,8,19]) for solving (2.16) where T (x) = tanh(x), k 1 = v * 4 and k 2 is a positive constant to be determined. It is readily verified that the ansätz (4.1) satisfies the boundary conditions at x = ±∞ in (2.16). Since u(x), v(x) and w(x) in (4.1) are expressed in terms of polynomials in 13 tanh(x) and d dx tanh(x) = 1 − tanh 2 (x), inserting (4.1) into the three equations in (2.16) gives d 1 u xx +θ u x +u (σ 1 −c 11 u−c 12 v−c 13 w) = ζ 10 +ζ 11 T (x)+ζ 12 T 2 (x)+ζ 13 T 3 (x) , (4.2a) Equating the coefficients of powers of T (x) to zero yields a system of ten algebraic equations: It turns out that (4.2) can be solved to give The result obtained is summarized in the following theorem.  4). The resulting exact (1, 0, 0)-(u * , v * , 0) wave is given by (4.5) The profiles of u(x), v(x), and w(x) are shown in Figure 4.1. We conclude this section with the remark that the ansätz (4.1) for solutions of (2.16) is inspired by the one proposed in [19], where the ansätz (4.1) for solutions is In order to find a solution of (5.1), an approach based on the supersolution-subsolution method is employed. To this end, we introduce supersolutions and subsolutions.w = w(x) is said to be a supersolution of (5.1) if it satisfies the differential inequality Similarly, a subsolutionw =w(x) is defined by reversing the inequality in (5.2). The following lemma is helpful in constructing non-trivial solutions of (5.1).
Proof of Theorem 2.2. Due to [H4], we clearly havew(x) ≤w(x) for x ∈ R. First of all, we show thatw(x) andw(x) are a subsolution and a supersolution of (5.1) respectively. Indeed, a straightforward computation gives us q ≤ c 31ũ (x) + c 32ṽ (x) ≤q, x ∈ R. (5.5) The existence of a solution w = w(x) for (5.1) lying between the subsolutionw(x) and the supersolutionw(x) constructed above follows from Theorem 2.8 in [26]. In view of [H1], we finally employ Lemma 5.1 to conclude that the solution w(x) of (5.1) has the asymptotic behavior lim |x|→∞ w(x) = 0. This completes the proof.