  We deal with a mass-conserved three-component reaction-diffusion system which
is proposed by a model describing the dynamics of wavelike actin polymerization
in the macropinocytosis and numerically exhibits dynamical patterns such as
annihilation, crossover, and nucleation of pulses (Yochelis-Beta-Giv 2020). In
this article we first establish the condition for the diffusion driven
instability in the system. Then we rigorously prove the existence of spiky
stationary solutions to the system in a bounded interval with the Neumann
condition. By numerics these solutions play a crucial role in the nucleation of
pulses. Reducing the stationary problem to a scalar second order nonlinear
equation with a nonlocal term, we construct the desired solution by converting
the equation to an integral equation.
