  We consider a mathematical model for a two-particle system driven by the
spatial gradient of a concentration field of chemicals with conservative
attractive interactions in one dimension. This setup corresponds to an
experimental system with floating camphor particles at a water surface.
Repulsive interaction is introduced, as well as self-propelling force, through
the concentration field of camphor molecules at the water surface. Here we
newly adopt the attractive lateral capillary force due to the deformation of
the water surface. The particles experience competing dissipative repulsion and
conservative attraction. We numerically investigated the mathematical model,
and found six different modes of motion. The theoretical approach revealed that
some of such mode transitions can be understood in terms of bifurcation.
