Figures
Abstract
We compare spot patterns generated by Turing mechanisms with those generated by replication cascades, in a model one-dimensional reaction-diffusion system. We determine the stability region of spot solutions in parameter space as a function of a natural control parameter (feed-rate) where degenerate patterns with different numbers of spots coexist for a fixed feed-rate. While it is possible to generate identical patterns via both mechanisms, we show that replication cascades lead to a wider choice of pattern profiles that can be selected through a tuning of the feed-rate, exploiting hysteresis and directionality effects of the different pattern pathways.
Citation: Stich M, Ghoshal G, Pérez-Mercader J (2013) Parametric Pattern Selection in a Reaction-Diffusion Model. PLoS ONE 8(10): e77337. https://doi.org/10.1371/journal.pone.0077337
Editor: Jordi Garcia-Ojalvo, Universitat Politecnica de Catalunya, Spain
Received: July 5, 2013; Accepted: August 26, 2013; Published: October 29, 2013
Copyright: © 2013 Stich et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was carried out with support from Repsol S. A. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The study was supported by Repsol S. A. There are no employment, consultancy, patents, products in development and this does not alter the authors' adherence to all the PLOS ONE policies on sharing data and materials.
Introduction
Reaction-diffusion systems are well known to self-organize into a variety of spatio-temporal patterns including, spots, stripes, spirals, as well as spatio-temporal chaos and uniform oscillations [1]–[3]. Their existence in out-of-equilibrium states, connection to idealized chemical systems, and dependence on dimensional parameters, make them a good testbench for the study of general features of pattern generation and evolution. In particular, the dependence of these final states on the rate at which constituents are fed into the system (feed-rate) is of significant interest, since reaction-diffusion systems represent proxies for high-level biological systems that can exchange matter and energy with the environment [4]. Depending on the value of the feed-rate, the system may asymptote into one of many states and thus the feed-rate can be thought of playing the role of a natural control parameter.
While spatio-temporal patterns in reaction-diffusion systems (like replicating spots [5] and Turing patterns [6], [7]) have been found and discussed extensively in the context of chemical systems [3], [8], their phenomenology is ubiquitous. A well-studied example from physics is related to electrical current filament patterns in planar gas-discharge systems [9], [10]. The system dynamics can be described by activator-inhibitor reaction-diffusion models and different mechanisms of spot array formation have been observed: division and self-completion. The relevant control parameter in this system is the feeding voltage. Another example that have attracted interest recently is found in the realm of fluid dynamics where “spots” of turbulent regions in pipe flow [11] and plane Couette flow [12] have been observed: On a laminar background, patches of localized turbulence, called puffs, emerge via finite-amplitude perturbations and also show splitting behavior. These systems have been recently mapped onto excitable reaction-diffusion systems [13], and subsequently, the Turing mechanism has been proposed to explain the periodic arrangement of puffs in [14], suggesting again a reaction-diffusion framework for the dynamics. The corresponding control parameter in this case is the Reynolds number of the flow.
While these examples show that similar phenomena appear in different systems, an even more intriguing feature is that patterns that look qualitatively similar can be generated by very different mechanisms in the same system. Consider the patterns shown in Figures 1(a,b), which are the result of numerical simulations of a typical bistable reaction-diffusion system in two spatial dimensions. While both figures represent stationary arrays of spots (increased concentrations of one or more chemical species relative to others), their evolutionary pathways are quite different. Figure 1(a) was generated by the Turing mechanism [15], i.e. from a uniform stationary state unstable under spatial perturbations, giving rise to a stationary, spatially periodic pattern. This is illustrated by a space-time diagram for a simulation in one space dimension in Fig. 1(c), where an initially uniform state almost simultaneously develops spots as a result of the small random perturbation.
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).
In contrast to the above, the pattern in Fig. 1(b) was generated by perturbing a different uniform steady state, creating a single spot, that after a slight increase in the feed-rate, undergoes a replication cascade of spots, eventually filling the space (again illustrated in Fig. 1(d) by a space-time diagram for a simulation in one space dimension). Thus, while the asymptotic state of the system looks similar in both cases, the initial conditions, the parameter regimes in which they occur, and the mechanisms by which they are generated are different.
In the face of this, it is of interest to investigate if there is an abrupt transition or a smooth continuation –as a function of the feed-rate– between the patterns, as one traverses from one limit to the other. If there is a coexistence region, we want to investigate whether the asymptotic states of these patterns are identical and only the temporal evolution differ. Finally, since it may be desirable to select particular states of the system we seek to determine if it is possible to use the different mechanisms to smoothly engineer transitions between different states.
In this article, we explore these questions in a model reaction-diffusion system that displays both replication cascades and Turing instabilities. In the spirit of simplicity, tractability and clarity, we focus on a medium with only one spatial dimension and investigate the formation of patterns as a function of the feed-rate . Therefore, we do not consider other pathways for the generation of spot solutions, such as transverse instabilities of stripe solutions (requiring at least two spatial dimensions). We find that, while the mechanisms driving the formation of spot arrays are discernibly separated in different regimes of
, the patterns are essentially indistinguishable in intermediate regimes. Nevertheless, we find degeneracies, hysteresis and directionality effects that can be exploited for the purposes of pattern selection, via the tuning of the feed-rate.
Results
The model and basic instabilities
Our model reaction-diffusion system (first introduced in [16]) is described by the differential equations (1a)
(1b)where
can be interpreted as the concentration of an activator
and
as the concentration of a substrate
. There is an autocatalytic step for
at rate
, and decay reactions for
and
at rates
, while
is fed in to the system at a rate
. The model is closely related to a class of well-studied reaction-diffusion systems such as the Sel'kov-Gray-Scott model [17]–[19] (see also Sec. S1 of file S1 ), the Gierer-Meinhardt model [20] and the Brusselator [21].
We begin our analysis by first determining the uniform absorbing states of the model and then proceed to determine the specific instability associated with each state. Without loss of generality, the concentrations can be rescaled to ; then the stationary uniform states are determined by setting the right-hand side of Eq. (1) to zero. Doing so, we obtain
, which we refer to as state 1. At a critical value of the feed-rate
, we find that two more solutions are generated by a saddle-node bifurcation. The first is an unstable intermediate state 2 and the second is a stable state 3 given by
. In addition to this we find that the system undergoes a Hopf bifurcation at yet another critical value
, whereby in the range
, state 3 is potentially unstable with respect to temporal oscillations (for details see Sec. S2 of file S1).
Thus, the primary absorbing states of interest are 1 and 3. These turn out to display distinct forms of instability. At a critical feed-rate , state 3 is linearly unstable with respect to spatially inhomogeneous perturbations, leading to the formation of Turing patterns in the interval
(see Sec. S3 of file S1). The characteristic wavelength of the pattern
can be determined through a standard linear stability analysis, and this determines the total number of spots
that are present in the system through the simple relation
, where
is the system size and
the wavenumber (see Methods).
On the other hand, while state 1 is stable with respect to infinitesimally small perturbations, it is unstable to localized large-amplitude perturbations, that can induce the formation of a single spot. Using the technique of scale-separation, one can calculate the profiles of the spot solutions, along with the parameter regimes for which they exist. In a particular limit, where , we can define a critical feed-rate for the formation of single-spot solutions, such that spots exist for
(details are shown in Sec. S4 of file S1). As
is further increased, the single spot becomes unstable with respect to a replication cascade (at a numerically determined critical feed-rate
) which eventually fills the system with a spot array, for related work for the Gray-Scott model see [22]–[30].
It is essential to point out that the fundamental difference between the formation of spot arrays via the Turing mechanism or via replication cascades, is that the former results from an instability of state 3 to infinitesimally small perturbations with a characteristic wavelength, while the latter is the result of a localized large-amplitude perturbation to state 1.
Turing patterns and localized spot patterns
We next investigate the differences between these two pattern formation mechanisms through the aid of numerical simulations, where we initialize the system in a variety of different initial states and examine the corresponding asymptotic states. To compare the generated patterns, we need to choose a suitable metric to distinguish them. In principle, there are many quantities one can measure, however, as Fig. 1 suggests, a particularly simple choice would be to simply count the number of spots that are generated in the asymptotic state of the evolution of the system.
Consider the plot in Fig. 2, where we show the number of spots as a function of the feed-rate
in the asymptotic state of the simulation (for the numerical details of the simulations, see Methods). We start with a single spot induced on the background of state 1 in the region
(that supports stable spots) and gradually increase
in small increments of
. Doing so, we eventually reach a critical
where the spot splits into two spots (replicates). The 2-spot solution may again be unstable, and the splitting process is repeated. This is the situation if we start with a single spot as initial condition. However, in the region
, we can also directly create a
-spot array with
by inducing multiple large amplitude perturbations in different spatial locations of the system. The size of each spot of course is finite (being determined by the diffusion coefficients
) and consequently there is a maximum number of spots
that can be supported in a finite medium. Thus in the region
we can initialize a wide range of spot arrays within the bounds
and by the same procedure of incrementing
, determine the values of
at which the spot array replicates. The resulting curve is displayed in Fig. 2 as the lower boundary of the stability area. These values of
for each
represent a generalization of the critical feed-rate
for
. Clearly, this also implies that the curve corresponds to the minimum number of stable spots
that can be supported by the system for fixed
, and we thus label this curve
.
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:
,
,
,
.
We next turn our attention to the Turing regime () and the spot patterns found there. The onset of the Turing instability is of special interest: by inducing a small-amplitude perturbation around state 3 at
, we obtain a native Turing pattern of
spots (denoted in Fig. 2 with a black square) in very good agreement with the theoretical value predicted by linear stability analysis (see Sec. S3 and Eq. (S9) of file S1). Away from
, the analysis provides us with a continuous band of unstable wavelengths. Extensive simulations show that in the entire Turing regime (
), small random perturbations of state 3 lead on average to a spot pattern with
spots (marked by the solid curve extending from
at
in Fig. 2), as predicted by linear stability analysis using the most unstable wavelength. This is in agreement with similar findings for the Gray-Scott model [31], confirming that patterns in this regime and initialized in this way are indeed bonafide Turing patterns.
Comparing the replication mechanism with the Turing mechanism, we recognize that the former provides an elegant way to access a number of spots that are different from (the native Turing pattern) within the Turing regime. This is done by first initializing a
-spot pattern for
(outside the Turing regime), and then gradually increasing
until we are within the Turing regime. In this way we can select a wide range of
within the bounds
that differ from
. We note that Turing patterns with
can also be generated by expanding fronts generated by perturbing state 1 in the Turing regime (see Sec. S5 of file S1), however this is not the focus of this article.
Furthermore, starting from any stable -spot array, we are free to reverse the procedure and decrease
in increments of
. We find that after a particular value of
is reached,
now decreases. By continuing this process and repeating it for all
, we obtain the upper curve in Fig. 2 that gives the maximum number of stable spots
that can be sustained for a given
. The area enclosed by the curves
and
thus marks the stability region of
-spot arrays as a function of the feed-rate
. We immediately see from the figure that degenerate
-spot arrays exist for a large range of
, where the arrays can in principle be generated by different mechanisms.
Taken together, these results allow us to interpret as a disappearance boundary where a
spot solution goes to a new value
, and
as a splitting boundary where
. In general, in an infinite system,
spots split into
spots, however in a finite system this is constrained by its size. Therefore even in the region that supports replication, for large enough
, some of the spots in the array splits while other do not. The specific value of
is sensitive to small perturbations, in particular at the moment the splitting or disappearance takes place.
Clearly, as we can create many different initial conditions, many different splitting or disappearance pathways exist. As an illustrative example we show one where a single spot is initialized on the background of state 1. By increasing , the solution reaches the boundary
and splitting ocurrs. The resulting two spots are also unstable, and finally an 8-spot array is formed. By further increasing
, the array splits into 16 spots. Then it maintains stability for a wide range as
is increased further, well into the Turing regime, until it splits as it encounters
again. This evolution is shown via the red path in Fig. 3(a). If we now decrease
, the boundary
is encountered twice, and finally the number of spots decreases to 1 again (shown as a blue path in Fig. 3(a)). This is an example of a hysteresis curve connected to the degeneracy of the
-spot arrays.
(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.
Another example is shown by the green path in Fig. 3(a), where we cycle the spot-array solution between 10 and 20 spots. To illustrate how this cycle lookes in a real simulation, in Fig. 3(b) we show a space-time diagram for the variable along the green path. We start from a 10-spot solution for
, increase
in small steps until splitting to a 20-spot solution is observed. Then, we decrease
while preserving the 20-spot solution (hysteresis) until finally disappearance of spots takes place and the 10-spot solution is recovered.
The hysteresis effect clearly has the consequence of a preferred directionality in the system for inducing a replication pathway. Replication cascades proceed only via an increase in the feed-rate and for . Conversely, the formation of a Turing pattern appears for a fixed
and for a particular class of initial conditions.
Pattern profiles of 
-spot solutions
While measuring has been fruitful in determining the stability region of the solutions, it does not provide any detailed information about the spatial distribution of the pattern. Do the spot arrays created by Turing instability and spot arrays created by replications show any differences? Clearly, as one changes
smoothly, the distribution of the concentration will vary, even as
remains constant. A simple way of determining this is to measure the profile of the spots, which is the spatial range between its maximum and minimum concentrations. A visual illustration of this definition is provided in Fig. 4(a) inset.
(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
).
To investigate this, we initialize a pattern with spots at the Turing boundary
and examine the change in profile as we decrease
. In Fig. 4(a), we plot the profile of
in function of
. In the same figure we mark the existence region of state 3 by the dashed vertical boundary
as well as the steady-state value
by a solid blue curve (note that state 3 exists only for
). We find that close to
the amplitude of the pattern (marked by the vertical solid lines) is small, and the concentration of
oscillates symmetrically around state 3, in line with what is expected at
(see inset). However, as we move away from
, the amplitude increases and the profile shifts in phase space. At some point the pattern ceases to oscillate around state 3, and eventually decouples from state 3, continuing to persist even below
. This decoupling occurs without any qualitative change as the pattern crosses the boundary. The implication of this is that for
the persistent spot pattern can be interpreted as a continuation of a Turing pattern, although it is independent of state 3 (unlike a near-threshold Turing pattern) and no Turing analysis can be applied. In fact, spot arrays with same
, but created either through the Turing mechanism or replication cascades show no quantitative or qualitative difference, implying that arrays created by the two mechanisms are practically indistinguishable in the intermediate regime.
We next examine the change in profile as we vary between the stability boundaries
and
, for an array with
spots (note that for the 22-spot solution we do not reach the
curve). In Fig. 4(b) we represent the resulting patterns in the space of the concentrations
. Again we see that the patterns change continuously as
is varied, from the blue curve for
to the red curve for
. For the former, a spatial plot of the pattern (inset upper right) reveals that it is sharply peaked and that a spot has a small extension. If one perturbs the system by further decreasing
by a small amount, the number of spots decreases. Turning our attention to the other boundary
, an examination of the profiles there reveals the existence of degenerate values of
for fixed
(marked in red). This implies that within the spot, a small dip in the center is formed, as visualized in the inset (lower left). Now, as one increases
by a small amount, the spot pattern eventually splits along this dip.
Discussion
In conclusion, using a simple reaction-diffusion model, we have identified the stability region for -spot solutions in the parameter space spanned by a natural control parameter (the feed-rate
). In general, for a given
, we find multistability of spot solutions, with a range of spot numbers
, bounded by numerically determined curves
and
.
Spot arrays in the reaction-diffusion system (1) can be created in very different ways, with two distinct limiting behaviors (single-spot solution and native Turing pattern). These arrays are indistinguishable in intermediate regimes (the asymptotic states for fixed and
are identical) where both generative mechanisms coexist. This means that either mechanism can be used to generate the same pattern. Therefore, to discriminate between the pattern formation mechanisms is to some degree artificial, as these can only be distinguished during their transient phases. However, due to the different transients in each case, the initial conditions determine the pattern evolution and the final number of spots in a non-trivial manner: While small random perturbations create typical Turing patterns with
coinciding on average with
, through an appropriate tuning of
, we gain access to a wider range of
via replication cascades. As we have shown, one can make use of the hysteresis feature of the system to generate periodic cycles of spot replication and destruction.
Despite the simple and specific chemical nature of our model, we expect the qualitative result to hold for similar non-chemical systems and in general for those complex scenarios whose dynamics (possibly in reduced form) can by described by reaction-diffusion models such as certain fluid systems [11]–[14]. There, cycles of spot replication and destruction could be used to engineer transitions between out-of-equilibrium states. For example, splitting of turbulent stripes is dominant for large Reynolds numbers in plane Couette flow, while for low Reynolds numbers stripe decay is favored [12]. While the specifics in that system are different from our model, analogous to the role played by the feed-rate, we hypothesize that it could be possible to control the number of stripes through switching of the Reynolds number.
As perspectives for future work we mention the possibility to engineer the system by modulating the feed-rate in time, using a self-generated signal (feedback) that can use the splitting/disappearance pathways [32]. Furthermore, transitions between spot arrays with different can be also induced by application of noise. However, the realization of these ideas goes beyond the scope of this article.
In the spirit of simplicity, tractability and clarity, we have focused on a medium with one spatial dimension. Obviously, the dynamics of localized spots and Turing patterns is much richer in two space dimensions. However, we expect that that the main result of this study holds qualitatively also for two-dimensional spot arrays.
Methods
The numerical simulations of Eq. (1) were conducted in a one-dimensional space of size with periodic boundary conditions which ensures that there no spots attached to the boundary (varying
as well as using no-flux boundary conditions have not shown to produce mayor changes). A spatial grid with
was used along with a Euler routine for time integration and a 3-point stencil for the diffusion operator. In order for increased accuracy for patterns close to instabilities and for validation purposes, a 4th-order Runge-Kutta scheme was employed along with a smaller grid resolution
and a 5-point stencil. The two-dimensional simulations shown in Fig. 1(a,b) are only for the purpose of illustration; they correspond to simulations with
and a 5-point stencil for the diffusion operator.
We are not interested in oscillatory behavior and therefore choose and
in order to be far from the Hopf bifurcation curve (compare Fig. S1 of file S1). In order to observe localized spot and Turing patterns, sufficiently strong substrate diffusion is necessary, and we set
and
accordingly. Although for one-dimensional localized patterns, the notation spike is used in the literature, we apply the more general notation spots.
To obtain the limiting curves in Fig. 2, spot solutions are initialized for different in the region
. The asymptotic state of a simulation is determined at
, although transients usually have died out after
(if
changes within the simulation) or
(if
does not change within the simulation). Following this,
is increased in increments of
, and the simulation is allowed to run again until the asymptotic state is reached. This procedure is repeated until splitting is observed. In the same way,
can be either increased further or decreased until spots split again or disappear. This iterative process has been exhaustively performed for all possible
to determine the stability area.
We note that the numerical results come with inherent imprecisions, in particular for large where the amplitude of the Turing pattern vanishes and for small
where the spot pattern disappears. Finite simulation time may mistake a transient for an asymptotic state. Also, the finite size of the medium (together with the periodic boundary conditions) implies that the range of
(which is a positive integer number) is limited. However, simulations for larger system size and no-flux boundary conditions have not revealed qualitatively new behavior, though of course
increases and the curves in Fig. 2 are extensive in system size.
Author Contributions
Conceived and designed the experiments: MS GG JPM. Performed the experiments: MS GG. Analyzed the data: MS GG JPM. Contributed reagents/materials/analysis tools: MS GG. Wrote the paper: MS GG JPM.
References
- 1.
Cross M, Greenside H (2009) Pattern Formation and Dynamics in Nonequilibrium Systems. Cambridge: Cambridge Univ. Press.
- 2.
Mikhailov AS (1994) Foundations of Synergetics I. Berlin: Springer, 2 edition.
- 3.
Walgraef D (1997) Spatio-Temporal Pattern Formation. New York: Springer.
- 4.
Grzybowski BA (2009) Chemistry in Motion: Reaction-Diffusion Systems for Micro- and Nanotechnology. Chichester: Wiley.
- 5. Lee KJ, McCormick WD, Pearson JE, Swinney HL (1994) Experimental observation of self- replicating spots in a reaction-diffusion system. Nature 369: 215–218.
- 6. Castets V, Dulos E, Boissonade J, De Kepper P (1990) Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. Phys Rev Lett 64: 2953–2956.
- 7. Ouyang Q, Swinney HL (1991) Transition from a uniform state to hexagonal and striped Turing patterns. Nature 352: 610–612.
- 8. de Wit A (1999) Spatial patterns and spatiotemporal dynamics in chemical systems. Adv Chem Phys 109: 435–513.
- 9. Astrov YA, Logvin YA (1997) Formation of clusters of localized states in a gas discharge system via a self-completion scenario. Phys Rev Lett 79: 2983–2986.
- 10. Astrov YA, Purwins HG (2006) Spontaneous division of dissipative solitons in a planar gasdischarge system with high ohmic electrode. Phys Lett A 358: 404–408.
- 11. Avila K, Moxey D, de Lozar A, Avila M, Barkley D, et al. (2011) The onset of turbulence in pipe flow. Science 333: 192–196.
- 12. Shi L, Avila M, Hof B (2013) Scale invariance at the onset of turbulence in Couette flow. Phys Rev Lett 110: 204502.
- 13. Barkley D (2011) Simplifying the complexity of pipe flow. Phys Rev E 84: 016309.
- 14. Manneville P (2012) Turbulent patterns in wall-bounded flows: A Turing instability? Europhys Lett 98: 64001.
- 15. Turing A (1952) The chemical basis of morphogenesis. Phil Trans R Soc Lond B 237: 37–72.
- 16. Lesmes F, Hochberg D, Morán F, Pérez-Mercader J (2003) Noise-controlled self-replicating patterns. Phys Rev Lett 91: 238301.
- 17. Sel'kov EE (1968) Self-oscillations in glycolysis. Europ J Biochem 4: 79–86.
- 18. Gray P, Scott SK (1983) Autocatalytic reactions in the isothermal, continuous stirred tank reactor. Chem Eng Sci 38: 29–43.
- 19. Pearson JE (1993) Complex patterns in a simple system. Science 261: 189–192.
- 20. Gierer A, Meinhardt H (1972) A theory of biological pattern formation. Kybernetik 12: 30–39.
- 21. Prigogine I, Lefever R (1968) Symmetry breaking instabilities in dissipative systems II. J Chem Phys 48: 1695.
- 22. Reynolds WN, Ponce-Dawson S, Pearson JE (1997) Self-replicating spots in reaction-diffusion systems. Phys Rev E 56: 185–198.
- 23. Doelman A, Kaper TJ, Zegeling PA (1997) Pattern formation in the one-dimensional Gray-Scott model. Nonlinearity 10: 523–563.
- 24. Nishiura Y, Ueyama D (1999) A skeleton structure of self-replicating dynamics. Physica D 130: 73–104.
- 25. Morgan DS, Doelman A, Kaper TJ (2000) Stationary periodic patterns in the 1D Gray-Scott model. Meth Appl Anal 7: 105–150.
- 26. Muratov CB, Osipov VV (2000) Static spike autosolitons in the Gray-Scott model. J Phys A Math Gen 33: 8893–8916.
- 27. Ei C, Nishiura Y, Ueda K (2001) 2n-splitting or edge-splitting? Japan J Indust Appl Math 18: 181–205.
- 28. Muratov CB, Osipov VV (2002) Stability of the static spike autosolitons in the Gray-Scott model. SIAM J Appl Math 62: 1463–1487.
- 29. Kolokolnikov T, Ward MJ, Wei J (2005) The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The pulse-splitting regime. Physica D 202: 258–293.
- 30. Wei J, Winter M (2008) Stationary multiple spots for reaction-diffusion systems. J Math Biol 57: 53–89.
- 31. Mazin W, Rasmussen KE, Mosekilde E, Borckmans P, Dewel G (1996) Pattern formation in the bistable Gray-Scott model. Math Comput Simul 40: 371–396.
- 32. Mikhailov AS, Showalter K (2006) Control of waves, patterns and turbulence in chemical systems. Phys Rep 425: 79–194.