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

  • * Corresponding author: David Iron

    * Corresponding author: David Iron 
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  • 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 $ T $. 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 $ T $ is increased beyond a critical Hopf bifurcation value $ T_H $, 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.

    Mathematics Subject Classification: Primary: 35Q92, 58J55; Secondary: 34E10.

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  • Figure 1.  Left: Plot of the one-spike stable equilibrium solution $ a $ (blue curve) and $ h $ (red curve) for the model in (2) with delay in the inhibitor equation. Right: Plot of the trajectory $ x_0(\tau) $ for the center of the spike. The dotted curve is the full numerical simulation obtained from (2), and the solid curve is the asymptotic result as obtained from (21), with delay in the non-linear term of the inhibitor equation. Parameter values used are $ T = 0.1 $, $ \epsilon = 0.06 $, $ \mu = 1 $, $ L = 1 $ and $ D = 1 $

    Figure 2.  Left: Plot of the one-spike stable equilibrium solution $ a $ (solid curve) and $ h $ (dotted curve) for the model in (22), with delay in the $ h $ term of the activator equation. Right: Plot of the trajectory $ x_0(\tau) $ for the center of the spike. The dotted curve is the full numerical simulation obtained from (22), and the solid curve is the asymptotic result as obtained from (30), with delay in the $ h $ term of the activator equation. Parameter values used are $ T = 0.1 $, $ \epsilon = 0.06 $, $ \mu = 1 $, $ L = 1 $ and $ D = 1 $

    Figure 3.  Left: Plot of the one-spike stable equilibrium solution $ a $ (solid curve) and $ h $ (dotted curve) for the model in (31), with delay in both the activator regulation and inhibitor production. Right: Plot of the trajectory $ x_0(\tau) $ for the center of the spike. The dotted curve is the full numerical simulation obtained from (31), and the solid curve is the asymptotic result as obtained from (35), with delay in the $ h $ term of the activator equation. Parameter values used are $ T = 0.1 $, $ \epsilon = 0.06 $, $ \mu = 1 $, and $ D = 1 $

    Figure 4.  Left: Plot of the one-spike stable equilibrium solution $ a $ (solid curve) and $ h $ (dotted curve) for the model in (36), where the nonlinear term of the activator equation is delayed. Right: Plot of the trajectory $ x_0(\tau) $ for the center of the spike. The dotted curve is the full numerical simulation obtained from (36), and the solid curve is the asymptotic result as obtained from (49). Parameter values used are $ T = 0.1 $, $ \epsilon = 0.06 $, $ \mu = 1 $, $ L = 1 $ and $ D = 1 $

    Figure 5.  Left: Plot of the one-spike stable equilibrium solution $ a $ (solid curve) and $ h $ (dotted curve) for the model in (50), where all the nonlinear terms are delayed. Right: Plot of the trajectory $ x_0(\tau) $ for the center of the spike. The dotted curve is the full numerical simulation obtained from (50), and the solid curve is the asymptotic result as obtained from (51). Parameter values used are $ T = 0.1 $, $ \epsilon = 0.06 $, $ \mu = 1 $, $ L = 1 $ and $ D = 1 $

    Figure 6.  Plot of the motion of the center of the spike as obtained from the asymptotic ODE (21), with delay in the inhibitor equation. Increasing values of the delay $ T $ were used, and initial points as indicated. There are no oscillations in this case, and the equilibrium position $ x_0 = 0.5 $ is always stable. Parameters used are $ \epsilon = 0.06 $, $ \mu = 1 $, and $ D = 1 $

    Figure 7.  Solutions of (64) for various values of $ L $

    Figure 8.  Plot of the asymptotic result $ x_0(t) $ as obtained from (30), with delay in the $ h $ term of the activator equation, for delay $ T = 1.7<T_H $ (left figure) and $ T = 1.9>T_H $ (right figure). Parameters used are $ \epsilon = 0.06 $, $ \mu = 1 $, $ L = 2 $ and $ D = 1 $

    Figure 9.  Solutions of (68) for a range of $ L $

    Figure 10.  Plot of the asymptotic result $ x_0(t) $ as obtained from (35), where delay is in both the activator and inhibitor equations, for delay values $ T_1 = 1.3 $ (feft figure) and $ T_1 = 1.45 $ (right figure). Parameters used are $ \epsilon = 0.06 $, $ \mu = 1 $, $ L = 2 $ and $ D = 1 $

    Figure 11.  Plot of trajectories corresponding to motion of the spike as obtained from the asymptotic ODE (49), with increasing values of delay in the non-linear term of the activator equation, and using various initial points as indicated. No oscillations are observed and all trajectories approach the stable equilibrium position $ x_0 = 0.5 $. Parameters used are $ \epsilon = 0.06 $, $ \mu = 1 $, and $ D = 1 $

    Figure 12.  Solution to (85) for various values of $ L $

    Figure 13.  Amplitude of spike solution to (2) for delay below and above the critical value of $ T_H = 0.519 $. Here $ D = 1 $, $ \mu = 1 $, $ L = 1 $ and $ \varepsilon = 0.06 $

    Figure 14.  Amplitude of spike solution to (31) for delay below and above the critical value of $ T_H\sim0.25 $. Here $ D = 1 $, $ \mu = 1 $, $ L = 1 $ and $ \varepsilon = 0.06 $

    Figure 15.  Plot of $ Re(\lambda_0) $ versus $ Im(\lambda_0) $ for the eigenvalue of matrix $ \mathcal{M} $. In figure (a), the red curve follows $ T = 0 $ to 0.6. Black from 0.6 to 1.4, blue form 1.4 to 1.6 and green 1.6 to 2.2

    Figure 16.  Simulations of (99) with $ D = 1 $, $ \mu = 1 $, $ L = 1 $. In 16(a), $ T = .9 $ and in 16(b), $ T = 1.04 $. Time is plotted on the horizontal axis and $ x_0 $ is plotted on the vertical axis

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