Pulses in a Gierer--Meinhardt Equation with a Slow Nonlinearity

In this paper, we study in detail the existence and stability of localized pulses in a Gierer--Meinhardt equation with an additional “slow” nonlinearity. This system is an explicit example of a general class of singularly perturbed, two component reaction-diffusion equations that goes significantly beyond well-studied model systems such as Gray--Scott and Gierer--Meinhardt. We investigate the existence of these pulses using the methods of geometric singular perturbation theory. The additional nonlinearity has a profound impact on both the stability analysis of the pulse---compared to Gray--Scott/Gierer--Meinhardt-type models a distinct extension of the Evans function approach has to be developed---and the stability properties of the pulse: several (de)stabilization mechanisms turn out to be possible. Moreover, it is shown by numerical simulations that, unlike the Gray--Scott/Gierer--Meinhardt-type models, the pulse solutions of the model exhibit a rich and complex behavior near the Hopf bifurcations.