Traveling Waves in a Three Species Competition-cooperation System

This paper studies the traveling wave solutions to a three species competition cooperation system. The existence of the traveling waves is investigated via monotone iteration method. The upper and lower solutions come from either the waves of KPP equation or those of certain Lotka Volterra system. We also derive the asymptotics and uniqueness of the wave solutions. The results are then applied to a Lotka Volterra system with spatially averaged and temporally delayed competition.


introduction
We study the traveling wave solutions of the following three species competition cooperation system where u(x, t), v(x, t) and w(x, t) stand for the population densities of the three different species, a i > 0 is interaction constant, i = 1, 2 and r > 0 (− 1 τ < 0, respectively) is the relative intrinsic growth rate of the species v (w, respectively). Aside from the intra-specific competitions, system (1.1) describes the relation that the species w competes with u and u competes with v, while v cooperates with w.
The purpose of our study is of two folds: there are less results (see [17,9]) on the traveling wave solutions to the three species systems even for one with simple form as (1.1); on the other hand we would like to extend the results of [1] from the tempo-spatial delayed KPP (Kolmogorov, Petrovsky and Piscounov) equation to the following Lotka Volterra competition system with spatial temporal delay where the function g * * v =´+ Noting that if g * * v is replaced by v in (1.2), we recover the classical Lotka Volterra competition system, and fruitful results have been devoted in the study of traveling waves [3,5,8,7,13,14,11,12,15,19,20] arising from it. It is interesting to see the long term effect of introducing spatio-temporal delay to the competition. The temporal delay accounts for the time once consumed resource the dominated species needs to wait for its re-growth, and the spatial averaging accounts for the fact that individuals are moving around and have therefore not been at the same point in space at different times in their history (see [1]).
On setting we easily verify that w(x, t) satisfies the following equation System (1.2) is now recasted into (1.1). The existence of the traveling wave solution of (1.1) is equivalent to that of (1.2).
A traveling wave solution for (1.1) has the form (u(x, t), v(x, t), w(x, t)) = (u(x+ ct), v(x + ct), w(x + ct)) = (u(ξ), v(ξ), w(ξ)), ξ = x + ct, and satisfies the system For the convenience of later study, we change (1.10) into monotone ( [19]). Let u = u,v = 1 − v andw = 1 − w, and drop the bars on u, v and w, we arive at Conditions (1.7) and (1.8) stipulate that wave solution is either below or above the plane u = v, and this is reflected in the construction of the upper and lower solutions in section 3.
and the boundary conditions A lower solution of (1.11) is defined in a similar way by reversing the inequalities in (1.12) and (1.13).
Since (1.11) is a monotone system, the monotone iteration method ( [20]) is ready to apply once the orderness of the upper and lower solutions is estabished. The key to the monotone iteration is to identify a pair of ordered upper and lower solutions to (1.11) ( [20,3]). There are two methods to set up the upper and lower solutions. The first one was used in [3,8,20], which consists of a pair non-smooth upper and lower solutions, and the similar idea was later successfully generalized to handle local and nonlocal equations, the second one is based on a pair of smooth upper and lower solutions from known equations, and the method was applied in [15] for a general form of two species Lotka Volterra competition system and in [6] for a model system arising from game theory. We will use the ideas of the second method to set up the upper and lower solutions for (1.11). See section 2 for details.
The other interesting aspects of the traveling front solutions are the minimal wave speed, the uniqueness, the asymptotics and the stability. The minimal wave speed is also referred to as the critical wave speed, below which the there will be no monotonic traveling waves. Also the traveling waves with the critical speed behaves differently at −∞, see [15]. We will use a generalized version of sliding domain method (see [2] ) to show the uniqueness of the front solution corresponding to each speed. This paper is organized as follows: In section 2 we gather the necessary information about the KPP and Lotka Volterra waves, in particular we derive the asymptotics of the Lotka Volterra waves at −∞ which is the key in setting up the upper and lower solutions for (1.11); In section 3 we show the existence of the wave solutions for (1.11) and further derive their properties such as strict monotonicity, the uniqueness and the asymptotics.

properties of waves for kpp and a two species lotka volterra competition equations
In this section, we introduce porperties of the wave solutions to KPP equation and to a two species Lotka Volterra competition system, which will be a key ingredient in the construction of the upper and lower solutions for system (1.11). For the rest of the paper the inequality between two vectors is component-wise.
The construction of the smooth upper and lower solution pairs for system (1.11) is based on the known results on the KPP equations and the recent results on a rescaled Lotka Volterra system. It seems that the asymptotics of the Lotka Volterra waves derived in this section is new.
Consider the following form of the KPP equation: We first recall the following result ( [18]): Lemma 3. Corresponding to every c ≥ 2 √ā , system (2.1) has a unique (up to a translation of the origin) monotonically increasing traveling wave solution ω(ξ) for ξ ∈ R. The traveling wave solution ω(ξ), ξ ∈ R has the following asymptotic behaviors: For the wave solution with non-critical speed c > 2 √ā , we have where a ω and b ω are positive constants. For the wave with critical speed c = 2 √ā , we have where the constant d c is negative, b c is positive, and a c ∈ R.
We also need the existence and asymptotics (at −∞) of the solutions of the following rescaled version of Lotka Volterra system: The asymptotics of the wave solutions will be derived by comparing the asymptotic decay rates of the upper and lower solutions of (2.6). Recalling that and we can define the lower solution for (2.6) by reversing the above inequalities.
We have the following conclusions: Lemma 4. Let the parameters satisfy either H1, H2a or H1, H2b, then for each

(2.6) does not have monotonic solution. At −∞ the solution has the following asymptotical properties:
For c > 2 √ 1 − a 1 , the solution (u(ξ), v(ξ)) satisfies, as ξ → −∞; Proof. The existence of the waves under conditions H1-H2a, or H1, H2b is contained in [16], and also refer to [15] for the existence and the asymptotics of the waves in a more general form of a two species competition system under conditions H1-H2a. From now on we will concentrate on system (2.6) under conditions H1 and H2b. The authors in [16] proved the existence of the monotone solutions for c ≥ 2 √ 1 − a 1 by showing that (2.6) is linearly determinated. However such method does not bring us the crucial information on the aymptotics of the wave solutions that is needed in section 3. We show that the traveling wave as derived in [16] is actually squeezed by the lower and upper solutions of (2.6) constructed below. Noting the upper and lower solutions differ from that in [8].
We first set up the lower solution for (2.6). For a fixed c ≥ 2 √ 1 − a 1 , let u(ξ), ξ ∈ R be a corresponding solution of the following KPP equation It is straightforward to verify the following claim. Claim A. Under conditions H1 and H2b, for each fixed . = (u, u)(ξ), ξ ∈ R defines a lower solution for (2.6). We next set up an super solution for (2.6). Choosing a small number l such that Let c ≥ 2 √ 1 − a 1 be fixed andû(ξ) be a solution of the following modified KPP equation: (1, 1), ifû(ξ) ≥ 1.
For the u component we havê and for the v component we have To ensure the non-positiveness of the last expression in (2.14), we require: and either It is a tedious but straightforward verification that for l satisfy (2.11), we have (2.15) and (2.16), or (2.15) and (2.17). Hence in either case the last expression in (2.14) is less than or equal to 0.
Analogous to Remark 2 we see that (2.13) is an upper solution. This finishes the proof of the Claim B.
Noting the relation there follows the orderness of the upper and lower solution By the monotone iteration scheme ( [20]), for each fixed c ≥ 2 Let c ≥ 2 √ 1 − a 1 be fixed and let (u(ξ), v(ξ)), ξ ∈ R be a corresponding solution. The upper and lower solutions have the exactly asymptotic decay rate at −∞by Lemma 3, the estimates (2.8) and (2.9) then readily follow.

Ordered upper and lower solutions under conditions H1 and H2a.
To construct the upper-solution for the system (1.11) in this case, we begin with the following form of KPP system: a 1 ) > 0 and f ′ (1) = −(1 − a 1 ) < 0. According to Lemma 3, for each fixed c ≥ 2 √ 1 − a 1 , system (3.1) has a unique (up to a translation of the origin) traveling wave solutionū(ξ) satisfying the given boundary conditions. Define we have the following result, Lemma 5. Assume the conditions H1 and H2a, for each fixed c ≥ 2 √ 1 − a 1 , (3.2) is a smooth upper solution for system (1.11).
As for the u component, we havē = 0, and for the v componentv due to the condition H2a.
As for thew component, Thus the conclusion follows.
We next construct the lower solution pair for system (1.11). For any small but fixed number l with we choose a numberl such that We begin with yet another KPP system: Corresponding to the notions in Lemma 3, For each c ≥ 2 √ 1 − a 1 let u(ξ), ξ ∈ R be a solution of (3.5) and let we have

(3.6) is a smooth lower solution of system (1.11).
Proof. On the boundary, one has Furthermore, for the u component, because of condition (3.5).
As for the w component we have due to the choice of l andl. The conclusion of the lemma follows.
2. The construction of lower solution in Lemma 5 also applies to the case H1-H2b, where we simply require the condition 0 < l < ra 2 1 − a 1 + r to be replaced by 0 ≤ l < min{ ra 2 1 − a 1 + r , 1}.

3.2.
Ordered upper and lower solutions under conditions H1, H2b and H3. The following estimates of the solutions of system (2.6) is needed in the construction of the upper and lower solutions. , v(ξ)), ξ ∈ R, be a solution of (2.6) for a fixed c ≥ 2 √ 1 − a 1 , then there exists a constant a * 2 > 0 such that for a 2 ≥ a * 2 , a 2 u(ξ) ≥ v(ξ) for ξ ∈ R. Proof. Noting that u(ξ) and v(ξ) have the exactly same (up to the first order) exponential decay rate at −∞ , and at +∞ we have a 2 u(ξ) > v(ξ). The rest of the proof follows easily.
We now set the upper and lower solution pairs for system (1.11).

Lemma 9. Let the parameters satisfy H1 (1.6) and H2b (1.8), then (3.6) consists of a lower solution for (1.11).
Proof. Similar to the proof of Lemma 6 so we skip it.
In the sequal we still write the shifted upper solution as (ū,v,w)(ξ), ξ ∈ R.

Monotone waves and their asymptotics. With such constructed ordered upper and lower solution pairs, we now have
Theorem 13. Assume either the conditions H1 and H2a or H1, H2b and H3, then for every c ≥ 2 √ 1 − a 1 , system (1.11) has a unique (up to a translation of the origin) traveling wave solution. The traveling wave solution is strictly increasing on R and has the following asymptotic properties: 1. Corresponding to the wave speed c > 2 √ 1 − a 1 , as ξ → −∞; while Corresponding to the wave speed c = 2 √ 1 − a 1 , we have For any speed c ≥ 2 √ 1 − a 1 , we have Proof. Starting from the upper and lower solution pairs obtained in section 3.1 and section 3.2 and using the monotone iteration scheme provided in [20,3], we obtain the existence of the solution (u(ξ), v(ξ), w(ξ)) to (1.11) for every fixed c ≥ 2 Lemma 3 and Lemma 4 imply that the upper-and the lower-solutions as derived in section 3.1 and section 3.2 have the same asymptotic rates at −∞. Then (3.20) and (3.21) then follow from Lemmas 3 and 4.
Introducing transformation Ψ = P Y by we can decouple (3.28) into the following equivalent system: (y 3 ) ξξ − c(y 3 ) ξ − 1 τ y 3 = 0, and find its bounded solutions at +∞ explicitely. In fact, for some nonzero constants d 1 , d 2 , d 3 , we have Transforming back to Ψ we have Hence we have (3.22) on intergrating (3.32). We next show the strict monotonicity of the traveling wave solutions, which will be a key ingredient in locating the eigenvalues of the linearized operator about the traveling wave in a separate study. By the monotone iteration process (see [20]), the traveling wave solution U (ξ) is increasing for ξ ∈ R, it then follows that (w 1 (ξ), w 2 (ξ), w 3 (ξ)) T = U ′ (ξ) ≥ 0 and satisfies (3.25), (3.26) and (3.27). The monotonicity of system (1.11) and the Maximum Principle imply that (w 1 , w 2 , w 3 ) T (ξ) > 0 for ξ ∈ R. This concludes the strict monotonicity of the traveling wave solutions.
2. There exists aθ ≥ 0 and ξ 1 ∈ R, such that one of the components of Uθ and U 2 are equal there; and for all ξ ∈ R, we have Uθ 1 (ξ) ≥ U 2 (ξ). On applying the Maximum Principle on R for that component, we find Uθ 1 and U 2 must be identical on that component. To fix ideas, we suppose that the component is the first component. Then Uθ 1 − U 2 satisfies (3.25), (3.26) and (3.27). Plugging w 1 ≡ 0 into (3.26) we find that there is at least one ξθ such that w 2 (ξθ) = 0. Then by applying maximum principle to (3.26), we have w 2 (ξ) ≡ 0 for ξ ∈ R. Similarly we also find w 3 (ξ) ≡ 0, ξ ∈ R. We have then returned to case 1.
Hence, in either situation, there exists aθ ≥ 0, such that for all ξ ∈ R. The nonexistence of the monotone traveling waves for (1.11) comes from the fact that all its solutions are oscillatory for c ≤ 2 √ 1 − a 1 .
The conclusion of the Corollary says the delay does not change the course of the traveling waves, but it may change the asymptotic behaviors of the wave solutions at +∞.