We study the spectrum of a new class of nonlocal eigenvalue problems (NLEPs) that characterize the linear stability properties of localized spike solutions to the singularly perturbed two-component Gierer-Meinhardt (GM) reaction-diffusion (RD) system with a fixed time-delay $T$ in only the nonlinear autocatalytic activator kinetics. Our analysis of this model is motivated by the computational study of Seirin Lee et al. [Bull. Math. Bio., 72(8), (2010)] on the effect of gene expression time delays on spatial patterning for both the GM and some related RD models. For various limiting forms of the GM model, we show from a numerical study of the associated NLEP, together with an analytical scaling law analysis valid for large delay $T$, that a time-delay in only the activator kinetics is stabilizing in the sense that there is a wider region of parameter space where the spike solution is linearly stable than when there is no time delay. This enhanced stability behavior with a delayed activator kinetics is in marked contrast to the de-stabilizing effect on spike solutions of having a time-delay in both the activator and inhibitor kinetics. Numerical results computed from the RD system with delayed activator kinetics are used to validate the theory for the 1-D case.
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Figure 1.
Left panel: The positive real eigenvalue
Figure 3.
Plot of the numerically computed function
Figure 4.
HB threshold for (3.32) versus the constant multiplier
Figure 5.
HB threshold
Figure 6.
HB values for the infinite-line problem (2.1), with the same caption as in Fig. 5. In the middle panel, the large-delay asymptotic result (dashed curve) is
Figure 7.
Plot of the HB threshold
Figure 8.
Plot of the amplitude
Figure 9.
HB thresholds for the synchronous mode computed from (5.5) and (5.7). Left panel: The HB threshold
Figure 10.
HB threshold for the asynchronous mode computed from (5.13). Left panel: The minimum value
Figure 11.
Left panel: The minimum value
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