  It has been suggested recently that growth and division of a protocell could
be modeled by a chemically active droplet with simple chemical reactions driven
by an external fuel supply. This model is called the continuum model. Indeed
it's numerical simulation reveals a shape instability which results in droplet
division into two smaller droplets of equal size resembling cell division [1].
In this paper, we investigate the reduced version of the continuum model, which
is called the effective model. This model is studied both in the linear and
nonlinear regime. First, we perform a linear stability analysis for the flat
interface, and then we develop a nonlinear theory using the longwave approach.
We find that the interface at the leading order is governed by the modified
Kuramoto-Sivashinsky equation. Therefore the interface is subject to a
logarithmic blow up after a finite time. In addition, an expression for the
interface local velocity is obtained.
