Pattern formation in a reaction-diffusion system of Fitzhugh-Nagumo type before the onset of subcritical Turing bifurcation

We investigate numerically the behavior of a two-component reaction-diffusion system of Fitzhugh-Nagumo type before the onset of subcritical Turing bifurcation in response to local rigid perturbation. In a large region of parameters, the initial perturbation evolves into a localized structure. In a part of that region, closer to the bifurcation line, this structure turns out to be unstable and covers all the available space over the course of time in a process of self-completion. Depending on the parameter values in two-dimensional (2D) space, this process happens either through generation and evolution of new peaks on oscillatory tails of the initial pattern, or through the elongation, deformation, and rupture of initial structure, leading to space-filling nonbranching snakelike patterns. Transient regimes are also possible. Comparison of these results with 1D simulations shows that the prebifurcation region of parameters where the self-completion process is observed is much larger in the 2D case.

Figure 1

Type of response of the model, described by Eqs. (1), to initial rigid perturbation depending on the model parameters: (a) in two dimensions, (b) in one dimension. Minus signs indicate initial perturbation extinction over the course of time, dots indicate the formation of stable localized structures, and plus signs indicate the formation of localized structures that evolve into nonlocalized patterns. The dynamics of the one-dimensional self-completion case designated by an asterisk is shown in Fig. 2. The dynamics of the two-dimensional self-completion cases designated by the letters a–f is shown in Fig. 3.

Figure 2

Self-completion of localized spotlike structures arising in the model, described by Eqs. (1), in the one-dimensional case. Variable $u$ is shown, with white corresponding to negative values, black to maximal positive values, and gray to small positive values. Parameter values are $D=0.0015, \alpha \approx 0.074$, and $\mathrm{\Delta }\alpha =0.002$.

Figure 3

Self-completion of localized spotlike structures arising in the model (1) in the two-dimensional case. Cases (a)–(e) demonstrate system dynamics under different parameters values, and they comprise five figures in a row each, except for (e), which comprises four figures. Case (f) shows static distribution for one specific case. Variable $u$ is shown, with white corresponding to negative values, black to maximal positive values, and gray to small positive values. Parameters and time values (from left to right) are as follows: (a) $D=0.001, \alpha =0.057$ ($\mathrm{\Delta }\alpha \approx 0.005$); $t=700$, 1100, 1600, 2200, and 24 000. (b) $D=0.0015, \alpha =0.04$ ($\mathrm{\Delta }\alpha \approx 0.036)$; $t=4800$, 7300, 10 500, 14 000, and 30 000. (c) $D=0.0022, \alpha =0.077$ ($\mathrm{\Delta }\alpha \approx 0.015$); $t=4300$, 4600, 5100, 6000, and 25 500. (d) $D=0.001, \alpha =-0.06$ ($\mathrm{\Delta }\alpha \approx 0.122$); $t=12000$, 18 000, 23 000, 30 500, and 60 000. (e) $D=0.0022, \alpha =0$ ($\mathrm{\Delta }\alpha \approx 0.092$); $t=42800$, 44 800, 55 000, and 84 500. (f) $D=0.01, \alpha =0.185$ ($\mathrm{\Delta }\alpha \approx 0.005$); $t=3000$. $\mathrm{\Delta }\alpha =2\sqrt{D}-D-\alpha .$