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

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 {−Δu1=u1(r−u1)−a1u1u2e1+u1+u2inΩ,−Δu2=m1u1u2e1+u1+u2−b1u2−a2u2u3e2+u2+u3inΩ,−Δu3=m2u2u3e2+u2+u3−b2u3inΩ,k1∂u1∂ν+u1=k2∂u2∂ν+u2=k3∂u3∂ν+u3=0on∂Ω, where ΩΩ is a bounded domain of RNRN, N≥1N≥1, with boundary ∂Ω∂Ω of class C2+αC2+α for some α∈(0,1)α∈(0,1), νν is the outward unit vector on ∂Ω∂Ω, the parameters r,ai,bi,ei,mi(i=1,2) are strictly positive, and ki≥0(i=1,2,3), ui(i=1,2,3) are the respective densities of prey, predator, and top predator.