Figure 1.
Stable stationary spot arrays in the reaction-diffusion system (1) generated by (a) Turing instability, (b) replication cascade.
Two space dimensions are considered, with system size and periodic boundary conditions. Typical formation pathways for the Turing case (c) and the replication scenario (d) are shown in the space-time diagrams for simulations in one-dimensional space with
. In (a–d), the variable
is displayed in color code: red, respectively white denote large values. Parameters: (a)
; (b)
; (c)
, displayed time interval
, (d)
, displayed time interval
. Other parameters as in Fig. 2. A pattern profile for both variables
and
will be shown in Fig. 4(b).
Figure 2.
Stability area for -spot arrays as a function of
for a system size
with periodic boundary conditions and
(details of simulation are covered in Methods).
The stability area is enclosed by the curves and
, corresponding to the maximum and minimum number of stable spots for a given
.
is changed in steps of
using the asymptotic state of the previous
as initial condition (ramping). Turing patterns are marked by the curve
. Vertical lines correspond to the values for the instabilities:
,
,
,
.
Figure 3.
(a) The red and blue pathways represent a hysteresis curve for an example -spot array induced in state 1. We observe a sequence
spots. The green path represents a cycle between 10 and 20 spots (more see text). (b) Space-time diagram for
along the green path shown in (a).
is changed about
each
. Simulation starts with a 10-spot solution at
with
, and
increases until
, where splitting is observed. Then
is decreased until
, where 10 spots disappear, and after which it is increased again until
is reached.
Figure 4.
(a) The profile of in a Turing pattern as a function of
for fixed
. The blue curve represents the steady-state value
. The vertical dashed lines
and
mark the Turing regime. A Turing pattern appears supercritically at
(inset) and its amplitude increases as one moves away from threshold. At a certain point, the profile ceases to oscillate around
, and continues to exist beyond the Turing regime without qualitative changes. (b) Multiple
-spot profiles in
space, for
between
at
(blue curve) and
at
(red curve). The insets show the corresponding concentration profiles (black is
, red is
).