Positive solutions for a three-trophic food chain model with diffusion and Beddington–Deangelis functional response

Abstract

In this paper, we investigate the existence, stability, permanence, and global attractor of positive solutions for the following three species food chain model with diffusion and Beddington–Deangelis functional response { − Δ u 1 = u 1 ( r − u 1 ) − a 1 u 1 u 2 e 1 + u 1 + u 2 in Ω , − Δ u 2 = m 1 u 1 u 2 e 1 + u 1 + u 2 − b 1 u 2 − a 2 u 2 u 3 e 2 + u 2 + u 3 in Ω , − Δ u 3 = m 2 u 2 u 3 e 2 + u 2 + u 3 − b 2 u 3 in Ω , k 1 ∂ u 1 ∂ ν + u 1 = k 2 ∂ u 2 ∂ ν + u 2 = k 3 ∂ u 3 ∂ ν + u 3 = 0 on ∂ Ω , where Ω is a bounded domain of R N , N ≥ 1 , with boundary ∂ Ω of class C 2 + α for some α ∈ ( 0 , 1 ) , ν is the outward unit vector on ∂ Ω , the parameters r , a i , b i , e i , m i ( i = 1 , 2 ) are strictly positive, and k i ≥ 0 ( i = 1 , 2 , 3 ) , u i ( i = 1 , 2 , 3 ) are the respective densities of prey, predator, and top predator.