DISCRETE N-BARRIER MAXIMUM PRINCIPLE FOR A LATTICE DYNAMICAL SYSTEM ARISING IN COMPETITION MODELS

. In the present paper, we show that an analogous N-barrier max- imum principle (see [3, 7, 5]) remains true for lattice systems. This extends the results in [3, 7, 5] from continuous equations to discrete equations. In or- der to overcome the difficulty induced by a discretized version of the classical diffusion in the lattice systems, we propose a more delicate construction of the N-barrier which is appropriate for the proof of the N-barrier maximum principle for lattice systems. As an application of the discrete N-barrier maximum principle, we study a coexistence problem of three species arising from biology, and show that the three species cannot coexist under certain conditions.

1. Introduction and main results. This paper introduces a generalization of the N-barrier maximum principle (NBMP) from second order differential operators, such as [3,6], to the second order difference operators in the following boundary value problem for the two-component lattice dynamical system ( [12,13]) where x ∈ R and the parameters h, d, k, a 1 , a 2 are positive constants. For the limiting case h → 0 + , the NBMP of (BVP) has been established in [3,6]. (BVP) is a discrete version of itself with h → 0, which arises in finding a traveling wave solution of the form (u(y, t), v(y, t)) = (u(x), v(x)), x = y − θ t (1) to the Lotka-Voletrra system of two competing species where u(y, t) and v(y, t) represent the density of the two species u and v, respectively, and R is the habitat of the two species. For the problem arising in ecology as to which species will survive in a competitive system, traveling wave solutions serve an important role in understanding the competition mechanism of species. In (1), θ is the propagation speed of the traveling wave, which is an important index to understand the competition mechanism. When θ > 0 in (BVP), u survives and v dies out eventually; when θ < 0 in (BVP), v survives and u dies out eventually. Clearly, (2) has four constant equilibria: e 1 = (0, 0), e 2 = (1, 0), e 3 = (0, 1) and is the intersection of the two lines 1−u−a 1 v = 0 and 1−a 2 u−v = 0 whenever it exists. The asymptotic behavior of solutions (u(x, t), v(x, t)) for (2) with initial conditions u(x, 0), v(x, 0) > 0 can be classified into the following four cases.
In this paper, we restrict ourselves to the case where one of (i), (ii), (iii) and (iv) in Theorem 1.1 occurs. To establish the discrete NBMP for (BVP), we first observe that, without loss of generality we may assume θ ≥ 0 by lettingx = −x and interchanging the boundary conditions at ±∞. Throughout this paper we shall assume unless otherwise stated, that θ ≥ 0. The sign of θ determines which species is stronger and can survive in the ecological system (BVP).
The main contribution of the discrete NBMP for (BVP) is to provide a priori lower bounds for the linear combination of the components of (u(x), v(x)). More precisely, our discrete NBMP gives an affirmative answer to the following question.

Q:
For any h > 0, can we establish the discrete NBMP for (BVP), i.e. can we find nontrivial lower bounds depending on the parameters in (BVP) , v(x)) solves (BVP) and α, β are arbitrary positive constants?
When h → 0 + and d = 1, upper and lower bounds of u(x) + v(x) can be given by the classical elliptic maximum principle ( [4]). When h → 0 + and d ̸ = 1, an affirmative answer to an even more general problem of estimating α u(x) + β v(x) is given in [3].
From an economic point of view, one motivation for addressing the above question is as follows. Suppose that the two species U and V in (BVP) are commercial farming animals or cash crops which are grown for profit. Let u and v represent the units of U and V, respectively. The price of each U unit is α and each V unit is It turns out from the construction of the N-barrier (see Section 3) and the proof of Theorem 1.2 (see Section 4) that an analogous discrete NBMP remains true for more general nonlinearity or reaction terms in (BVP * ). In addition, it is easy to see that a trivial upper bound of p(x) is where (u(x), v(x)) solves (BVP) without assuming the monotonicity of u(x) and v(x). Employing our N-barrier method, a sharper upper bound of p(x) can be found by using the monotonicity of u(x) and v(x) established in [12]. We remark that the NBMP in [3] is recovered by letting h → 0 + in Theorem 1.

2.
Under certain restrictions on the parameters, we obtain nonexistence of traveling wave solutions for the Lotka-Volterra system of three competing species ( [11]), i.e. nonexistence of solutions of the following problem in R (see Theorem 1.3) as an application of the discrete NBMP. This application makes our discrete NBMP more biologically appealing. Here u(x), v(x) and w(x) represent the density of the three species u, v and w respectively; d i (i = 2, 3), σ 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. Except the propagation speed of the traveling wave θ, these parameters are all assumed to be positive. From the viewpoint of the study of competitive exclusion ( [1,14,15,17,20,22]) or competitor-mediated coexistence ([2, 19, 21]), (N) originates from the investigation of the problem when one exotic species (say, w) invades the ecological system of two native species (say, u and v) that are competing in the absence of w.
For the continuous case when h → 0 and w is absent, (N) becomes a two-species system with the asymptotic behavior at ±∞ Under the condition of strong competition (or the bistable condition) such a system admits a unique monotone solution (u(x), v(x)) with u(x) being monotonically decreasing and v(x) being monotonically increasing in x ( [16,18]). However, the situation changes dramatically for the discrete case h > 0. Under the same condition of strong competition (6) and boundary conditions (5), there exists no traveling solution of (N) with w being absent and θ ̸ = 0 if d 1 and d 2 are sufficiently small ( [12]). On the other hand, under the monostable condition (i.e. u is stronger than v and competitive exclusion occurs) (N) with w being absent admits a solution (u(x), v(x)) satisfying u ′ (x) < 0 and v ′ (x) > 0 if and only if θ ≥ θ min for some constant threshold θ min > 0. A similar conclusion can be drawn for the monostable condition under which v is stronger than u and competitive exclusion occurs ( [13]). Under either (7) or (8), we see that when w is absent in (N), u(x) and v(x) dominate the neighborhoods around x = −∞ and x = ∞ respectively. This fact leads us to consider the situation that when w as an exotic species invades (N), the wave profile of w(x) remains pulse-like, i.e. w(±∞) = 0 and w(x) > 0 for x ∈ R, if the three species coexist since w will prevail over u and v only on the region where u and v are not too dominant. Under certain conditions on the parameters, this conjecture turns out to be true for the continuous case h → 0. As indicated in [8,9,4], existence of (N) has been proved by means of the tanh method as well as numerical experiments. When h → 0, nonexistence of (N) under certain conditions on the parameters has been established by using the NBMP for (N) with h → 0 ( [4,3]). In this paper, we also apply the discrete NBMP (Theorem 1.2) to show the following nonexistence of solutions to (N) for some h > 0. andJ Assume that the following hypotheses hold: Here a 1 , a 2 , and k are the parameters which appear in (BVP), and Λ * (c 31 , c 32 , σ 3 ) is given by

Remark 2 (Theorem 1.3).
• It follows from the assumptions in Theorem 1.3 that M 1 * andM 1 * are quadratic in σ 3 , while M 2 * andM 2 * are linearly dependent on σ 3 . Also,λ * 1 does not depend on σ 3 . Thus it is readily seen that [H3] • According to the assumptions in Theorem 1.3,ū depends on a 2 andv depends on a 1 . Moreover, M 1 and M 2 depend on a 1 , andM 1 andM 2 depends on Biological interpretation of Theorem 1.3: It is readily seen that [H3] is true if σ 3 is sufficiently small when we fix other parameters. Ecologically, this means that when the intrinsic growth rate σ 3 is small enough, the three species u, v and w cannot coexist in the ecological system (N) under certain parameter regimes.
The remainder of this paper is organized as follows. We collect in Section 2 some preliminary results including the L 2 estimates of u ′ and v ′ which turn out to be crucial in proving Theorem 1.2. Section 3 is devoted to the construction of the N-barrier for (BVP * ). In Section 4, we make use of the N-barrier constructed in Section 3 to establish Theorem 1.2. As an application of Theorem 1.2, we show in Section 5 the nonexistence result of three species in Theorem 1.3. Finally, the elementary proofs of certain results in Section 2 are given in the Appendix (Section 5).

Preliminaries.
Throughout this paper, we use the notations collected in the following definition.
(iii) (Gaussian-like kernel with compact support) where (v) (Convolution with ϕ) From Definition 2.1 (i) and (ii), it is readily seen that . For other fundamental properties concerning the functions given in Definition 2.1 that will be used in the subsequent sections, see Lemma 2.2 below and its proof in the Appendix (Section 5). Without causing confusion as to what we refer to, we use ) .
For convenience of notation, we let Suppose that (u(x), v(x)) satisfies the boundary conditions Then On the other hand, we obtain by letting k = 1 and replacing a 2 with a 1 in (17). To establish Proposition 1 below, we make use of (17) and (18). Proposition 1 (i) and (ii) are used to prove the L 2 estimates of u ′ and v ′ given by Proposition 1 (iii). and where M(a, d, k, θ) = 3 16 Proof. It will turn out from the following proof that the estimates of u follows immediately from those of v by letting a 2 = a 1 and d = k = 1 in (23) From Definition 2.1 (i), it is easy to see that when Let Integrating the second equation in (BVP) from z 2 − mh to z 1 and using (27) lead to where we have used (26). Let it follows from (28) that On the other hand, summation of (26) from m = 0 to m = N − 1 gives Dividing the last equation by N , we use (30) and the fact that Then the desired estimate follows from Lemma 2.2 (i). This completes the proof of the second part of Proposition 1 (i).
Proposition 1 (ii) follows from multiplying the second equation in (BVP * ) by v and integrating the resulting equation from z 2 to z 1 : where Proposition 1 (i) has been used. We notice that it is not necessary to assume z 1 − z 2 ≤ 1 for the estimates in (ii) to hold. With the aid of Proposition 1 (i) and (ii), we prove Proposition 1 (iii). Multiplying the first equation in (BVP * ) by v ′ (x) and integrating the resulting equation from z 2 to z 1 yield where Young's inequality a b ≤ a 2 2 ϵ + ϵ b 2 2 with ϵ > 0 has been used. Therefore, Now it suffices to give an estimate for the first term on the right hand side of (38). Integrating by parts yields Lets 1 < z 2 <s 2 < s 1 < z 1 < s 2 withs 2 −s 1 = s 2 − s 1 = 1 and s 1 −s 2 < 1. Double integration of (39) with respect to z 1 from s 1 to s 2 and with respect to z 2 froms 1 tos 2 gives rise to where integration by parts has been used. Rearranging (40) and using the fact that From Definition 2.1 (iii) we have Clearly ψ(x − y) = ϕ(x − y) h 2 is an even function with compact support in y ∈ we further consider in (44).
For convenience of notation, let Performing integration by parts on I gives where we have used the fact that ψ(h) = 0. Using the assumption v ′ (x) < 0, the fact that ψ is even, and Proposition 1 (ii), we are led to On the other hand, (See Figure 3) Where we have used the fact that ψ ′ (x − y) = − 1 h 2 when x − h < y < x from Definition 2.1 (iv) and Ψ(x) is given by (See Figure 4) (49) Figure 3. Domain of the integral s2 s1 s2 s1 I 2 (z 2 ) dz 2 dz 1 in (48).
Remark 3. Some remarks related to Proposition 1.
• Note that the assumption z 1 −z 2 ≤ 1 is not used in deriving Proposition 1 (ii), while it is assumed in deriving Proposition 1 (iii). In addition, Proposition 1 (i) is a point-wise estimate.
• It follows from (35) that Moreover, it is easy to see from Figure 5 that the distance between the two lines is given by  Suppose that for some x 1 , x 2 ∈ R, When we plot (S u (x), S v (x)) or for x ∈ [x 1 , x 2 ] in the first quadrant of the S u S v -plane, we see from Figure 5 that (88) ensures  (87) leads to L = 0. Then the two lines S û u + S v v = 1 and S ū u + S v v = 1 coincide, and therefore this case reduces to that in [3,6]. where Proof. Suppose that we can find some (u 0 , v 0 ) ̸ = (u * , v * ), (1, 0), (0, 1) in the first quadrant of the uv-plane such that (u 0 , v 0 ) ∈ F 0 but (u 0 , v 0 ) / ∈ F * . It follows that either However, both cases lead to a contradiction since F (u 0 , v 0 ) = 0. Therefore, F 0 ⊂ F * .
In other words, Lemma 3.1 asserts that the graph of F (u, v) = 0 in the first quadrant of the uv-plane lies between the two lines 1−u−a 1 v = 0 and 1−a 2 u−v = 0. This fact will be used in proving Theorem 1.2.