The self-organization of clusters of particles is a fundamental phenomenon across various physical systems, including hydrodynamic and colloidal systems. One example is that of dense spherical particles submerged in a viscous fluid and subjected to horizontal oscillations. The interaction of the particles with the oscillating flow leads to the formation of one-particle-thick chains or multiple-particle-wide bands, both oriented perpendicular to the oscillation direction. In this study, we model the hydrodynamic interactions between such particles and parallel chains using simplified potentials. We first focus on the hydrodynamic interactions between chains, which we characterize using data from fully resolved numerical simulations. Based on these interactions, we propose a simplified model potential, called the Siren potential, which combines the representative hydrodynamic interactions: short-range attraction, mid-range repulsion, and long-range attraction. Through one-dimensional Monte Carlo simulations, we successfully replicate the characteristic patterns observed in hydrodynamic experiments and draw the phase diagram for the model potential. We further extend our analysis to two-dimensional systems, introducing a dipole-capillary model potential that accounts for both chain formation and Siren-like chain interactions. This potential is based on a system with colloidal particles at an interface, where chain formation is driven by an external electric field that induces a dipole moment parallel to the interface in each particle. The capillary force contributes the long-range attraction. Starting with parallel chains, the patterns in the two-dimensional Monte Carlo simulations of this colloidal system are similar to those observed in the hydrodynamic experiments. However, we identify that nonlinear interactions are important for some distinct steps in the chain formation. Still, the model potentials help clarify the dynamic behavior of the particles and chains due to the complex interactions encountered in both hydrodynamic and colloidal systems, drawing parallels between them.