Noise and Spike Dynamics for the Gierer-Meinhardt Equations

The stability and dynamic properties of spike-type solutions to the Gierer- Meinhart equations are well understood. We examine the effect of adding noise to the equations on the spike-dynamics. We derive a stochastic ordinary differential equation for the motion of a single spike as well as the distribution of spike location from the associated Fokker-Plank equation. With sufficiently large amplitude noise, it is possible for the spike to reach the boundary of the domain and become effectively trapped for some time. In this case, we calculate the expected time to capture.