Traveling Wave Solutions of Competitive Models with Free Boundaries

We study two systems of reaction diffusion equations with monostable or bistable type of nonlinearity and with free boundaries. These systems are used as multi-species competitive model. For two-species models, we prove the existence of a traveling wave solution which consists of two semi-waves intersecting at the free boundary. For three-species models, we also prove the existence of a traveling wave solution which, however, consists of two semi-waves and one compactly supported wave in between, each intersecting with its neighbor at the free boundary.


Introduction
In this paper, we study the following two systems: where α, β, β l , β r , γ are positive constants, f, g, f 1 , f 2 and f 3 are monostable or bistable types of nonlinearities and c is a constant to be determined together with the unknowns φ, ψ, φ 1 , etc.. In what follows, we say that f is a monostable type of nonlinearity (f is of (f M ) type, for short), if f ∈ C 1 ([0, ∞)) and we say that f is a bistable type of nonlinearity (f is of (f B ) type, for short), if . It is known that the equation (1.1) 1 has monotonically decreasing traveling front on R when c = c * f , where c * f > 0 is the minimal traveling speed when f is of (f M ) type, or the unique traveling speed when f is of (f B ) type (c.f. section 2). Similarly, the equation (1.1) 2 has monotonically increasing traveling front when c = c * g , where c * g < 0 is the maximal speed when g is of (f M ) type, or the unique speed when g is of (f B ) type (c.f. section 2).
On the problem (1.1) we have the following main result.
(i) Let α > 0 be a given constant. Then for any c ∈ (c * g ,ĉ f ), whereĉ f > 0 depends only on α and f , there exists a unique β(c) > 0 such that (1.1) has a unique solution (φ, ψ, c). Moreover, β(c) is continuous and strictly decreasing in c ∈ (c * g ,ĉ f ) and Let β > 0 be a given constant. Then for any c ∈ (ĉ g , c * f ), whereĉ g < 0 depends only on β and g, there exists a unique α(c) > 0 such that (1.1) has a unique solution (φ, ψ, c). Moreover, α(c) is continuous and strictly increasing in c ∈ (ĉ g , c * f ) and On the problem (1.2) we have the following result.
Theorem 1.2. Assume that f 1 , f 2 , f 3 are of (f M ) or (f B ) type. Let α, γ > 0 be given constants, σ ∈ (0, 1) (in case f 2 is of (f M ) type), or σ ∈ (θ, 1) (in case f 2 is of (f B ) type) be a given constant. Then there exist c − < 0 < c + depending only on f 1 , f 2 , f 3 , α, γ and σ such that for any c ∈ (c − , c + ), there exists a unique pair (β l (c), β r (c)), β l (c) (resp. β r (c)) is continuous and strictly decreasing (resp. increasing) in c, such that problem Problem (1.1) arises in the study of traveling wave solutions of the following system of reaction diffusion equations: is the free boundary to be determined together with u and v, f, g ∈ C 1 satisfying f (0) = g(0) = 0. In population ecology, the appearance of regional partition of multi-species through strong competition is one interesting phenomena. In [9,10,11], Mimura, Yamada and Yotsutani used problem (1.6) to describe regional partition of two species, which are struggling on a boundary to obtain their own habitats. Among others, they obtained the global existence, uniqueness, regularity and asymptotic behavior of solutions for the problem. Later [4,5,8,12] studied similar strong competitive models. Recently Du and Lin [6] and Du and Lou [7] studied a free boundary problem, which is essentially the problem (1.6) in case v ≡ 0. They constructed some semi-waves to characterize the spreading of u which represents the density of a new species. Motivated by these works, Chang and Chen [3] recently study the traveling wave solution of (1.6) (i.e. problem (1.1)) with logistic type of nonlinearities: They obtain the existence and uniqueness of traveling wave solution, similar as our Theorem 1.1 but for logistic type of f and g. One of our purpose in this paper is to study problem (1.6) for general monostable or bistable type of nonlinearity. In what follows, when f and g are of (f M ) type and (f B ) type, respectively, we call the solution of (1.1) a MB-type traveling wave solution for convenience. MM-type, BM-type and BB-type of traveling wave solutions are defined similarly (see Figure 1). Thus [3] presented a special MM-type traveling wave solution, while our Theorem 1.1 gives all these four types of traveling wave solutions.
When three (or more) species are involved in contesting the habitats, one should consider the following competitive model: Our problem (1.2) is nothing but the problem for the traveling wave solutions of (1.7): As above, if f 1 , f 2 and f 3 are of (f B ), (f M ) and (f B ) types of nonlinearities, respectively, we call the solution of (1.2) a BMB-type traveling wave solution for convenience. Similarly, one can define MMM-type, MBM-type and other types of traveling wave solutions (see Figure 1). Our Theorem 1.2 indeed includes all of these types. We point out that similar conclusions as in Theorems 1.1 and 1.2 remain true for the models including four or more species. In other words, for such a model, one can construct a traveling wave which consists of two semi-waves and several compactly supported waves in between, each intersecting with its neighbor at the free boundary.
In section 2, we give some basic phase plane analysis and prove Theorem 1.1. In section 3 we prove Theorem 1.2.

The Proof of Theorem 1.1
In this section we prove Theorem 1.1 for BB-type traveling wave solutions, that is, for the case where both f and g are of (f B ) type. Other types can be proved similarly.
We also need to consider a similar semi-wave ψ with increasing profile: Denote c * g the unique traveling speed of the following problem Then c * g < 0 and in a similar way as above one can obtain the following result.