Turing instability for a attraction-repolsion chemotaxis system with logistic growth

In this paper, we investigate the nonlinear dynamics for an attraction-repulsion chemotaxis Keller-Segel model with logistic source term u 1 t = d 1 Δ u 1 − χ ∇ ( u 1 ∇ u 2 ) + ξ ∇ ( u 1 ∇ u 3 ) + g ( u ) , x ∈ T d , t > 0 , u 2 t = d 2 Δ u 2 + α u 1 − β u 2 , x ∈ T d , t > 0 , u 3 t = d 3 Δ u 3 + γ u 1 − η u 3 , x ∈ T d , t > 0 , ∂ u 1 ∂ x i = ∂ u 2 ∂ x i = ∂ u 3 ∂ x i = 0 , x i = 0 , π , 1 ≤ i ≤ d , u 1 ( x , 0 ) = u 10 ( x ) , u 2 ( x , 0 ) = u 20 ( x ) , u 3 ( x , 0 ) = u 30 ( x ) , x ∈ T d ( d = 1 , 2 , 3 ) . Under the assumptions of the unequal diffusion coefficients, the conditions of chemotaxis-driven instability are given in a d -dimensional box T d = ( 0 , π ) d ( d = 1 , 2 , 3 ) . It is proved that in the condition of the unique positive constant equilibrium point w c = ( u 1 c , u 2 c , u 3 c ) of above model is nonlinearly unstable. Moreover, our results provide a quantitative characterization for the early-stage pattern formation in the model.