A diffusion and osmosis system of ordinary differential equation is considered. The system may represent migration of finite particle among finite places. Such system in the linear form yields a solution in the form of matrix exponential. In this paper, a bounded nonlinear forcing to the system is considered. Although it is bounded, the nonlinear affected so much to the solution. Numerical simulation of some two-dimensional examples are presented. Understanding the diffusion and osmosis system will be important for studying migration people among towns, cities or countries. © 2021 Published under licence by IOP Publishing Ltd.