  The transport of organelles and vesicles in living cells can be well
described by a kinetic tug-of-war model advanced by M\"uller, Klumpp and
Lipowsky. In which, the cargo is attached by two motor species, kinesin and
dynein, and the direction of motion is determined by the number of motors which
bind to the track. In recent work [Phys. Rev. E 79, 061918 (2009)], this model
was studied by mean field theory, and it was found that, usually the tug-of-war
model has one, two, or three distinct stable stationary points. However, the
results there are mostly obtained by numerical calculations, since it is hard
to do detailed theoretical studies to a two-dimensional nonlinear system. In
this paper, we will carry out further detailed analysis about this model, and
try to find more properties theoretically. Firstly, the tug-of-war model is
simplified to a one-dimensional equation. Then we claim that the stationary
points of the tug-of-war model correspond to the roots of the simplified
equation, and the stable stationary points correspond to the roots with
positive derivative. Bifurcation occurs at the corresponding parameters, under
which the simplified one-dimensional equation exists root with zero derivative.
Using the simplified equation, not only more properties of the tug-of-war model
can be obtained analytically, the related numerical calculations will become
more accurate and more efficient. This simplification will be helpful to future
studies of the tug-of-war model.
