Boundedness of solutions in a predator–prey system with density-dependent motilities and indirect pursuit–evasion interaction. (June 2023) - Xiang, Ailing; Wang, Liangchen
Abstract: This paper is concerned with the predator–prey system with density-dependent motilities and indirect pursuit–evasion interaction u t = Δ ( d 1 ( w ) u ) + u ( λ − u + a v ), x ∈ Ω, t > 0, v t = Δ ( d 2 ( z ) v ) + v ( μ − v − b u ), x ∈ Ω, t > 0, 0 = Δ w − w + v, x ∈ Ω, t > 0, 0 = Δ z − z + u, x ∈ Ω, t > 0 in a smooth bounded domain Ω ⊂ R n ( n ≥ 1 ) with homogeneous Neumann boundary conditions, where the parameters λ, μ, a and b are positive constants. The global existence and the boundedness of the classical solution are established in two dimensions if the motility functions d 1 ( w ) and d 2 ( z ) satisfy the following hypotheses d 1 ( w ) ∈ C 3 [ 0, ∞ ), d 1 ( w ) > 0, d 1 ′ ( w ) < 0 for all w ≥ 0, d 2 ( z ) ∈ C 3 [ 0, ∞ ), d 2 ( z ) > 0, d 2 ′ ( z ) > 0 and d 2 ′ ( z ) is bounded for all z ≥ 0 . However, further assume that the motility function d 2 ( z ) fulfills more rigorous conditions d 2 ( z ) ∈ C 3 [ 0, ∞ ), 0 < d 2 ( z ) < η, d 2 ′ ( z ) > 0 and d 2 ′ ′ ( z ) < 0 for all z ≥ 0 with η > 0 . It was also proved that the global solutions are uniformly bounded with respect to time in higher dimensions.