Combined effects of nonlinear proliferation and logistic damping in a three-component chemotaxis system for alopecia areata. (August 2022) - Tao, Youshan; Xu, Danni
Abstract: This work considers the zero Neumann initial–boundary problem to the chemotaxis system u t = Δ u − χ 1 ∇ ⋅ ( u ∇ w ) + w − μ 1 u 2, v t = Δ v − χ 2 ∇ ⋅ ( v ∇ w ) + w + r u v − μ 2 v 2, 0 = Δ w + u + v − w, in a bounded domain Ω ⊂ R n, n ≥ 1, with smooth boundary, which was initially proposed by Dobreva et al. (2020) to describe the process of alopecia areata lesions that involves the complicated interplay between CD4 + T cells, CD8 + T cells and interferon-gamma. Compared with the well-known Keller–Segel-type systems, the appearance of the nonlinear proliferation term r u v constituted by the product of two immune cell densities along with a new production term w activated by the chemical in the first two equations forms a novel feature of this model. For any given suitably regular initial data, it is firstly shown that if the positive parameters χ 1, χ 2, μ 1, μ 2 and r satisfy the assumptions μ 1 − r 2 > ( n − 2 ) + n ( 2 χ 1 + χ 2 2 ) and μ 2 − r > ( n − 2 ) + n ( 2 χ 2 + χ 1 2 ), then the problem admits a global and bounded classical solution. Moreover, we identify a set of parameter conditions: μ 1 < μ 2 < 3 μ 1, r = μ 2 − μ 1 and χ 1 2 + χ 2 2 < 2 μ 1 ( 3 μ 1 − μ 2 ), under which any global smooth solution will stabilize to the constant coexistence equilibrium as the time goes to infinity. In addition, in the case of μ 1 > μ 2, the nonlinear stability of the unique positive stationary solution to the corresponding ODE system is also discussed.