Stability and dynamics of spike-type solutions to delayed Gierer-Meinhardt equations

Abstract

<p style='text-indent:20px;'>For a specific set of parameters, we analyze the stability of a one-spike equilibrium solution to the one-dimensional Gierer-Meinhardt reaction-diffusion model with delay in the components of the reaction-kinetics terms. Assuming slow activator diffusivity, we consider instabilities due to Hopf bifurcation in both the spike position and the spike profile for increasing values of the time-delay parameter <inline-formula><tex-math id="M1">\begin{document}$ T $\end{document}</tex-math></inline-formula>. Using method of matched asymptotic expansions it is shown that the model can be reduced to a system of ordinary differential equations representing the position of the slowly evolving spike solution. The reduced evolution equations for the one-spike solution undergoes a Hopf bifurcation in the spike position in two cases: when the negative feedback of the activator equation is delayed, and when delay is in both the negative feedback of the activator equation and the non-linear production term of the inhibitor equation. Instabilities in the spike profile are also considered, and it is shown that the spike solution is unstable as <inline-formula><tex-math id="M2">\begin{document}$ T $\end{document}</tex-math></inline-formula> is increased beyond a critical Hopf bifurcation value <inline-formula><tex-math id="M3">\begin{document}$ T_H $\end{document}</tex-math></inline-formula>, and this occurs for the same cases as in the spike position analysis. In all cases, the instability in the profile is triggered before the positional instability. If however the degradation of activator is delayed, we find stable positional oscillations can occur in this system.</p>