Corrigendum: Dynamics of a Reaction-diffusion-advection Model for Two Competing Species

We provide a corrected proof of [4, Theorem 2.2], which preserves the validity of the theorem exactly under those assumptions as stated in the original paper. 1. Corrigendum. This Corrigendum concerns the proof of [4, Theorem 2.2]. In the original proof we used [1, Theorem 2.4] and [3, Proposition 3.2], which require more restrictive conditions than necessary. We provide here an elementary maximum principle argument which preserves the validity of Theorem 2.2, exactly under the assumptions as appeared in [4]. For the reader’s convenience we recall the statement of Theorem 2.2 and give its complete proof. The result concerns the unique positive solution θμ,α (μ > 0, α ≥ 0) of (See [2] for existence and uniqueness results) { ∇ · (μ∇θ − αθ∇m) + θ(m− θ) = 0 in Ω, μ ∂θ ∂n − αθ ∂m ∂n = 0 on ∂Ω, (1) where Ω is a bounded domain in R with smooth boundary ∂Ω, and ∂ ∂n denotes the outward normal derivative. Denote the set of local maximum points of m by M and Σ0 = {x ∈ Ω : ∇m = 0 and x 6∈M}, M+ = {x ∈M : m(x) > 0}. We recall the following non-degeneracy assumption on m(x) contained in [4]: 2010 Mathematics Subject Classification. Primary: 35J57, 35B40; Secondary: 92D40.