SOME GLOBAL DYNAMICS OF A LOTKA-VOLTERRA COMPETITION-DIFFUSION-ADVECTION SYSTEM

. This paper studies some population dynamics of a diﬀusive Lotka-Volterra competition advection model under no-ﬂux boundary condition. We establish the main results that determine the stability of semi-trivial steady states.

1. Introduction and statement of the main results. The classical Lotka-Volterra competition-diffusion system models two competing species. Here u(x, t) and v(x, t) denote respectively the population densities of two competing species at location x ∈ Ω and time t > 0, and µ, ν > 0 are random diffusion rates of species u and v respectively. The habitat Ω is a bounded domain in R N , with smooth boundary ∂Ω, n denotes the unit outer normal vector on ∂Ω, and the no flux boundary condition means that no individuals cross over the boundary. The function m(x) ∈ C 2 (Ω) represents their common intrinsic growth rate or local carrying capacity, which is non-constant and m(x) > 0 in Ω. b > 0 and c > 0 are interspecific competition coefficients. Then the maximum principle [18] yields that u(x, t) > 0, v(x, t) > 0 for every x ∈ Ω and every t > 0. By both mathematicians and ecologists, particular interests in two-species Lotka-Volterra competition models with spatially homogeneous or heterogeneous interactions are the dynamics of (1.1). See [2,7,8,9,10,11,14] and the references therein. We say that a steady state (u s , v s ) of (1.1) is a coexistence state if both components are positive, and it is a semi-trivial state if one component is positive and the other is identically zero. We make the following basic assumption on b, c.
Besides random dispersal, it seems reasonable to suppose that species could move upward along the resource gradient (see e.g. [2,19] and the references therein). In this paper we deal with a diffusive Lotka-Volterra competition-advection model in spatially heterogeneous environment in Ω. (1. 3) The functions m 1 (x) and m 2 (x) stand for the local carrying capacity or intrinsic growth rates of species u and v, respectively. b, c > 0 are inter-specific competition coefficients. The advection rates of two species are denoted by α, β ≥ 0, respectively. Here the movement strategies, growth rates and competition abilities of two species are taken into account and allowed to be different. Throughout this paper, besides Assumption 1, we define and also give the following assumption.
in Ω. (1.5) Recently (1.5) has been frequently used as a standard model to study the evolution of conditional dispersal, see, e.g., [1,4,6] for α, β > 0, [3,15,16] for α > 0 = β. Basically speaking, system (1.5) models the competition between two species with the same population dynamics but different movement strategies as reflected by their diffusion and/or advection rates. Motivated by the above work, we shall consider more global dynamics of (1.3). The rest of this paper is organized as follows. In Section 1.2, we present some preliminary results, which are helpful to verify our results. In Section 1.3 we establish our main results (Theorem 1.6, Theorem 1.7). The proof will be given in Section 2.

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QI WANG The following result is due to [1].
has a unique positive solution θ µ,η,m . Furthermore the following statements hold: The following result can be obtained in [ In order to analyze the principal eigenvalue of problem (1.8) and (1.9), it is more convenient to consider the following more general form of eigenvalue problem: (1.14) The principal eigenvalue of problem (1.13), denoted by σ 1 (µ, η, h), is expressed by the following variational formula (see, e.g. [2]) (1.15) The following lemmas collects some useful properties of σ 1 (µ, η, h).
Main results. Based on the above preparations, we are now ready to state our main results concerning the steady state of (1.3).

2.
Proof of the main results. In this section, we verify our main results. Proof of Theorem 1.6. In fact it is sufficient to prove case (i), and case (ii) can be verified similarly. The proof is split into four steps.
Proof of Theorem 1.7.