Segregation Pattern in a Four-Component Reaction–Diffusion System with Mass Conservation

We deal with a four-component reaction–diffusion system with mass conservation in a bounded domain with the Neumann boundary condition. This system serves as a model describing the segregation pattern which emerges during the maintenance phase of asymmetric cell devision. By utilizing the mass conservation, the stationary problem of the system is reduced to a two-component elliptic system with nonlocal terms, formulated as the Euler–Lagrange equation of an energy functional. We first establish the spectral comparison theorem, relating the stability/instability of equilibrium solutions to the four-component system to that of the two-component system. This comparison follows from examining the eigenvalue problems of the linearized operators around equilibrium solutions. Subsequently, with an appropriate scaling, we prove a \(\Gamma \) $\mathrm{\Gamma }$ -convergence of the energy functional. Furthermore, in a cylindrical domain, we prove the existence of equilibrium solutions with monotone profile representing a segregation pattern. This is achieved by applying the gradient flow and the comparison principle to the reduced two-component system.