Table 1.
Simulation runtimes of the Gray-Scott model.
Table 2.
Simulation runtimes for the Oregonator model.
Table 3.
Simulation runtimes for the model.
Table 4.
Simulation parameters for the Gray-Scott model.
Figure 1.
Nullclines in the Gray-Scott model.
We display the nullclines of the activator species (blue curve) and the inhibitor
(brown curve) for the Gray-Scott model (without diffusion) in the deterministic limit for the parameter set given in Table 4. The blue (brown) arrow illustrates the direction of the gradient in phase space of the activator (inhibitor) on either side of the nullcline and the unstable fix point is marked with
. We demonstrate that the system is in the excitable regime by plotting an example trajectory (dashed curve) for a larger perturbation, starting at point
, from the stable homogeneous state (marked by
in the figure). The system relaxes towards
via a long excursion.
Figure 2.
Formation of a spike spiral wave in the Gray-Scott model.
Shown are snap-shots of a spike spiral wave in the Gray-Scott model Eqs. (1)–(4), initialized as shown in the top left panel, at in the deterministic simulation (bottom left) and in stochastic simulations for different scale factors
(rightmost columns).
Table 5.
Identification of symbolic species in the Oregonator model.
Table 6.
Simulation parameters for the Oregonator model.
Figure 3.
Nullclines in the Oregonator model.
We display the nullclines (logarithmic scale) of the activator species (blue curve) and the inhibitor
(brown curve) for the Oregonator model in the deterministic limit for the parameter set given in Table 6 and
. We assume that the intermediary species
is in a steady-state equilibrium with
and
and ignore diffusion. The blue (brown) arrow illustrates the gradient in phase space of the activator (inhibitor) on either side of the nullcline and the unstable fix point is marked with
. The system is in the unstable (oscillatory) regime. We plot an example trajectory (dashed curve) of a larger perturbation from the (linearly stable) trivial homogeneous state. Starting at point
, the system enters a limit cycle in phase space.
Figure 4.
Formation of a spiral wave in the Oregonator model.
Shown are snap-shots of the formation of a spiral wave in the Oregonator model for the BZ reaction Eqs. (7)–(14), initialized as shown in the top left panel, at in the deterministic simulation (bottom left) and in stochastic simulations for different scale factors
(rightmost columns).
Figure 5.
We display the nullclines of the concentration
(blue curve), which can be regarded as the activator, and the fraction of open channels
(brown curve) for the Calcium oscillation model in the deterministic limit [Eqs. (27)–(29)] for the parameter set given in Table 7 and assume that diffusion is switched off. The blue (brown) arrow illustrates the gradient in phase space of the activator (inhibitor) on either side of the nullcline and the fix points are marked with
. We plot an example trajectory (dashed curve) of a larger perturbation from the homogeneous state
. Starting at point
, the system relaxes towards
via a long excursion.
Table 7.
Simulation parameters for the model.
Figure 6.
Formation of a spike spiral wave in the model.
Shown are snap-shots of a spiral wave in the model Eqs. (18)–(26), initialized as shown in the top left panel, at
in the deterministic simulation (bottom left) and in stochastic simulations for different scale factors
(rightmost columns).