  A system of two Schnakenberg-like reaction-diffusion equations is
investigated analytically and numerically. The system has previously been used
as a minimal model for concentrations of GTPases involved in the process of
cell polarisation. Source and loss terms are added, breaking the mass
conservation, which was shown previously to be responsible for the generation
of stable fronts via a so-called wave-pinning mechanism. The extended model
gives rise to a unique homogeneous equilibrium in the parameter region of
interest, which loses stability via a pattern formation, or Turing bifurcation.
The bistable character of the reaction terms ensures that this bifurcation is
subcrtical for sufficiently small values of the driving parameter multiplying
the nonlinear kinetics. This subcriticality leads to the onset of a multitude
of localised solutions, through the homoclinic snaking mechanism. As the
driving parameter is further decreased, the multitude of solutions transforms
into a single pulse through a Belyakov-Devaney transition in which there is the
loss of a precursive pattern. An asymptotic analysis is used to probe the
conservative limit in which the source and loss terms vanish. Matched
asymptotic analysis shows that on an infinite domain the pulse solution
transitions into a pair of fronts, with an additional weak quadratic core and
exponential tails. On a finite domain, the core and tails disappear, leading to
the mere wave-pinning front and its mirror image.
