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In this paper, we investigate the nonlinear dynamics for an attraction-repulsion chemotaxis Keller-Segel model with logistic source term u1t = d1∆u1 − χ∇(u1∇u2) + ξ∇(u1∇u3) + g(u), x ∈ Td, t > 0, u2t = d2∆u2 + αu1 − βu2, x ∈ Td, t > 0, u3t = d3∆u3 + γu1 − ηu3, x ∈ Td, t > 0, ∂u1 ∂xi = ∂u2 ∂xi = ∂u3 ∂xi = 0, xi = 0, π, 1 ≤ i ≤ d, u1(x, 0) = u10(x), u2(x, 0) = u20(x), u3(x, 0) = u30(x), x ∈ Td(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 Td = (0, π)d(d = 1, 2, 3). It is proved that in the condition of the unique positive constant equilibrium point wc = (u1c, u2c, u3c) of above model is nonlinearly unstable. Moreover, our results provide a quantitative characterization for the early-stage pattern formation in the model.

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