“…(x, t) ∈ Ω × (0, T ), ∂u ∂ν = ∂υ ∂ν = ∂ω ∂ν = 0, (x, t) ∈ ∂Ω × (0, T ), u(x, 0) = u 0 (x) , ρυ(x, 0) = ρυ 0 (x) , ρω(x, 0) = ρω 0 (x) , x ∈ Ω, (1.1) where Ω ⊂ R N (N ≥ 3) is a bounded domain with smooth boundary, ρ ≥ 0, ∂ ∂ν is the derivative with respect to the outer normal of ∂Ω and nonnegative initial data (u 0 , υ 0 , ω 0 ) satisfying suitable regularity, the parameters χ, ξ, α 1 , α 2 , β 1 , β 2 > 0, where χ and ξ are respectively measure the strength of the attraction and repulsion and u(x, t), υ(x, t), ω(x, t) denote the cell density, the chemoattractant concentration, the chemorepellent concentration, respectively and the logistic source f ∈ C ∞ ([0, ∞)) fulfills f (u) ≤ au − µu 2 with f (0) ≥ 0, a ≥ 0, µ > 0.…”