# Influence of Curvature, Growth, and Anisotropy on the Evolution of Turing Patterns on Growing Manifolds

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## Abstract

We study two-species reaction–diffusion systems on growing manifolds, including situations where the growth is anisotropic yet dilational in nature. In contrast to the literature on linear instabilities in such systems, we study how growth and anisotropy impact the qualitative properties of nonlinear patterned states which have formed before growth is initiated. We produce numerical solutions to numerous reaction–diffusion systems with varying reaction kinetics, manner of growth (both isotropic and anisotropic), and timescales of growth on both planar elliptical and curved ellipsoidal domains. We find that in some parameter regimes, some of these factors have a negligible effect on the long-time patterned state. On the other hand, we find that some of these factors play a role in determining the patterns formed on surfaces and that anisotropic growth can produce qualitatively different patterns to those formed under isotropic growth.

## Keywords

Turing patterns Growing surfaces Anisotropic growth Pattern selection Pattern robustness## 1 Introduction

Since Turing first proposed reaction–diffusion systems as a model for pattern formation (Turing 1952), much work has been performed to understand the theoretical and biological aspects of this phenomenon (Satnoianu et al. 2000; Green and Sharpe 2015; Marcon et al. 2016; Woolley et al. 2017). Murray (2003) discusses the importance of domain size and shape on the formation of patterns, and the impact of geometry on the kinds of admissible patterns that can arise due to a Turing instability. Curved and complex geometries have been of recent interest, as these provide for more realistic regimes within which to test morphogenetic hypotheses (Tse et al. 2010; Trinh and Ward 2016; Núñez-López et al. 2017). Since reaction–diffusion systems are more difficult to analyze on growing domains, pattern formation has been considered on different-sized static domains to simulate very slow growth (Varea et al. 1997). This requires the reaction and diffusion of the chemical species to occur on a much faster timescale than the growth and also be independent of the growth. An alternative simplification would be to assume growth occurs on a much faster timescale than reaction and diffusion and consider the quasi-static approximation of the reaction–diffusion system on the final shape of the domain. Since neither of these approximations are likely to be valid for all systems modeling biological growth, we consider reaction–diffusion systems where growth, reaction, and diffusion occur on comparable timescales.

Crampin et al. (1999) explicitly considered uniform and isotropic domain growth in one-dimensional reaction–diffusion systems in the slow and fast growth regimes, demonstrating frequency doubling of the emergent Turing patterns. This approach was discussed in the context of biological patterning problems in Crampin and Maini (2001), Barrass et al. (2006) and extended in Crampin et al. (2002) to consider the influence of non-uniform domain growth on one-dimensional reaction–diffusion systems, including apical or boundary growth. Castillo et al. (2016) considered Turing and Turing–Hopf instabilities in an exponentially and isotropically growing square and suggested that anisotropy and curvature are important considerations for extending their analysis. Following work on characterizing instabilities in non-autonomous reaction–diffusion equations on slowly growing domains in Madzvamuse et al. (2010), Hetzer et al. (2012) derived conditions for diffusion-driven instability on time-dependent manifolds. Plaza et al. (2004) derived a general formulation of reaction–diffusion theory on isotropically evolving one and two-dimensional manifolds, with motivation from biological settings where growth and curvature both play a role in organism development.

Beyond computing linear instability criteria, some authors have analytically explored how patterns change and evolve under growth (Comanici and Golubitsky 2008) by exploiting the framework of amplitude (or Ginzburg–Landau) equations (Cross and Hohenberg 1993; van Saarloos et al. 1994). While analytical results on mode competition and selection can be valuable, these are often extremely limited as they only apply near the bifurcation boundary in the parameter space, and they become computationally intractable in many cases of interest. Additionally, to our knowledge, there does not exist an analytical framework applicable to curved manifolds. In contrast, many more studies have explored such systems numerically, developing a variety of techniques to capture the transient and the final patterned states in reaction–diffusion systems on manifolds (Madzvamuse et al. 2003; Barreira et al. 2011; Macdonald et al. 2013; Madzvamuse and Maini 2007; Tuncer and Madzvamuse 2017). While these numerical methods and their applications will always be isolated to specific instances of reaction–diffusion systems, one can hope to at least explore the kinds of emergent behaviors in candidate systems as a way to understand realistic pattern formation and pattern evolution in more complex settings.

All of the above models of reaction–diffusion systems on growing domains only analyzed the case of isotropic (or apical) growth, which is unable to recapitulate the full range of complex biological structures found in developing organisms (Ubeda-Tomás et al. 2008; Corson et al. 2009; Peaucelle et al. 2015). Investigating arbitrary anisotropic growth in the context of biological patterning is a natural extension to reaction–diffusion theories of pattern formation and has been considered in biomechanical models of growth across a range of tissues and organisms (Menzel 2005; Saez et al. 2007; Bittig et al. 2008; Amar and Jia 2013). Madzvamuse and Barreira (2014) considered anisotropic and concentration-dependent growth in the context of cross-diffusion models, but their emphasis was on cross-diffusion-driven instabilities in complicated settings, rather than on understanding anisotropic growth in simple geometries, which has been suggested as a useful direction (Castillo et al. 2016). Rossi et al. (2016) also studied concentration-dependent growth of a scalar reaction–diffusion equation on a time-dependent manifold. Krause et al. (2018a) investigated the effects of Turing instabilities in two-species reaction–advection–diffusion models on a sphere for several well-known reaction kinetics. They find that advection and the compact geometry of the sphere allow for complex spatiotemporal patterns. Since growth can induce advection-like terms into the morphogen evolution equations (Plaza et al. 2004), it is of interest to consider the effect of anisotropic growth on compact domains, such as the sphere.

Much of the work in reaction–diffusion theory on growing domains has emphasized differences between quasi-static, slow, and fast growth regimes. Growth can lead to a robustness of patterning processes in some scenarios (Crampin et al. 1999), where the period-doubling of pattern splitting from an existing pattern leads to insensitivity to the initial symmetry-breaking perturbation which initiated patterning. Despite such differences, most reaction–diffusion systems give rise to qualitatively similar patterns, with quantities such as the pattern wavelength determined primarily by the final domain size, independent of growth. Determining when different kinds of growth influence the final pattern, and the effects that growth has on patterns, is an important problem to determine the validity of quasi-static approximations. Indeed, other questions arise such as: *To what extent does the rate of deformation, type of growth or extremity of growth affect the patterns formed on the domain, if there is any effect at all?*

In this paper, we generalize the modeling of reaction–diffusion systems on growing manifolds developed by Plaza et al. (2004) to allow for dilational anisotropic growth. The restriction of dilational growth allows for relatively simple governing equations amenable to simulation, yet still allows us to compare between quasi-static, isotropic, and anisotropic growth regimes. We then systematically explore three kinds of reaction kinetics on planar and curved two-dimensional manifolds. We also vary the kinds of growth, both in terms of average growth rate, functional form of the growth, and exploring growth in different stages. To address the questions of when these factors matter, we restrict attention to the evolution of established patterns far from homogeneous equilibrium states that arise due to Turing instabilities. In other words, all parameters are chosen within the Turing space such that these systems will always develop a heterogeneous solution, and we focus on qualitative features which change in such a solution as the manifold grows. This allows us to determine when each of these different growth scenarios gives rise to qualitatively different patterns. We emphasize qualitative differences and very pronounced quantitative effects, but do not focus on small quantitative differences between simulations, as these can depend sensitively on parameters and the initial data. We find that all of these features (growth rates, anisotropy, curvature) can give rise to qualitatively different patterns in some regimes, but that some of them (e.g., growth rate) are typically more important than others (e.g., the functional form of growth), and that some reaction kinetics, and some parameter regimes, are more or less influenced by growth.

The remainder of this paper is organized as follows. In Sect. 2, we develop our models to study two-species reaction–diffusion systems on a general two-dimensional manifold undergoing isotropic or anisotropic, but dilational growth. In Sect. 3, we discuss a systematic framework by which we study the effects of such growth on a range of reaction kinetics, demonstrating systems where patterning is generally robust (e.g., qualitatively independent of growth). In the following sections we shall employ this systematic approach to study Turing patterns on growing domains in the case where the domain is a flat ellipse (Sect. 4) and in the case where the domain is an ellipsoid surface (Sect. 5), the former demonstrating the role of anisotropy and the latter additionally demonstrating the role played by local curvature of the domain. We then discuss the evolution and numerical stability of stripe and target patterns in the presence of growth in Sect. 6, considering both the ellipse and ellipsoid domains. We finally discuss the implications of our results in Sect. 7.

## 2 Mathematical Model for Anisotropic Dilational Growth

We now introduce the model accounting for reaction–diffusion on a growing manifold. We first define the kinds of stationary patterns we are interested in and recall the conditions for diffusion-driven instability. We then introduce the concept of a reaction–diffusion system on a growing manifold and show how to transform this system onto a stationary manifold. This is a generalization of Plaza et al. (2004) where we allow for anisotropic dilational growth. Finally, we apply this general framework to two example manifolds which we will simulate in the following sections: a planar ellipse and the surface of an ellipsoid.

### 2.1 Turing Instability

*u*and

*v*are the two interacting chemical species, \(\delta _1\) and \(\delta _2\) are the diffusion coefficients of

*u*and

*v*, respectively, \(\nabla ^2\) is the Laplace–Beltrami operator on \(\varOmega \), and

*f*and

*g*are nonlinear kinetic functions (reaction terms) to be specified. We will consider manifolds with and without boundaries. When boundaries are present, we augment Eq. (1) with the conditions

*f*and

*g*with respect to

*u*and

*v*. Such conditions have been extended to time-dependent domains (Madzvamuse et al. 2010; Klika and Gaffney 2017), as well as to isotropically growing manifolds (Plaza et al. 2004). In these cases, the conditions are more complicated, but the basic idea is the same: differences in the diffusion rates of each species allow for the growth of spatial perturbations to a homogeneous steady state and hence allow for patterning. Here we are primarily concerned with the interactions between growth and the kinds of patterns formed and so will always study cases where diffusion-driven instability occurs in order to elucidate important differences in the kinds of growth. We note that while growth may influence the stability criteria, we intend to always let patterns emerge on a static manifold before initiating growth. In this way, we will use the above conditions to choose parameters within the Turing space such that patterns form before considering the influence of growth on such patterns.

### 2.2 Reaction–Diffusion Equations on Growing Two-Dimensional Manifolds

We now consider a growing domain in the form of a smooth two-dimensional manifold \(\varOmega (t)\) which is simply connected and compact. The manifold \(\varOmega (t) \) will either have a smooth boundary \(\partial \varOmega (t) \) for all time \(t \ge 0 \) or will be a manifold without boundary for all time \(t \ge 0\). Let \(\hat{\varOmega }(t)\) be an area element of the manifold, such that \(\hat{\varOmega }(t) \subset \varOmega (t)\). Let \((u,v) : \varOmega (t) \rightarrow \mathbb {R}\) be a concentration function defined on the manifold \(\varOmega (t)\), where, for example, \(u,\ v\) may describe the concentration of a chemical species, or morphogen, on the manifold \(\varOmega (t)\). We shall assume that *u* and *v* are \(C^1([0,\infty ))\) in time and \(C^2(\varOmega (t))\) in spatial coordinates.

*u*and \(\mathrm{d}\varOmega \) is the local area element on the manifold. Using Reynold’s transport theorem on the left-hand side of Eq. (4), we have

*v*. Note that we denote \(\nabla _{\varOmega (t)}\cdot \) as the divergence operator on \(\varOmega (t)\). Similarly, we define \(\nabla _{\varOmega (t)}^2\) to be the Laplace–Beltrami operator on \(\varOmega (t)\).

*u*and

*v*given by:

*u*due to local area changes.

### 2.3 Coordinate Transformation to Stationary Manifolds

In order to solve Eqs. (6a) and (6b), it is often necessary to map the equations to a stationary manifold and solve for *u* and *v* numerically in this stationary domain, before transforming the solved problem back to the original time-dependent manifold. Here, we outline a method to map the evolution equations for *u* and *v* to the stationary manifold \(\varPhi \subset \mathbb {R}^2\). We follow a similar procedure to Section 2 of Plaza et al. (2004), however, in a more general form, since we do not insist upon the parametrization for \(\varOmega (t)\) to be orthogonal and therefore permit anisotropic growth.

### 2.4 Reaction–Diffusion System on the Stationary Manifold

### 2.5 Examples of Flat and Curved Domains

We now define a pair of two-dimensional manifolds which we will use as examples for isotropic and anisotropic growth: a planar ellipse and the surface of an ellipsoid. There are many biologically relevant manifolds one could study, but these examples are sufficient to compare the influence of isotropic and anisotropic growth, and compare effects due to (boundary) curvature of the manifold.

#### 2.5.1 Planar Ellipse

*A*(

*t*) and

*B*(

*t*), respectively. These must satisfy

#### 2.5.2 Growing Tri-Axial Ellipsoid Surface

Another manifold we consider is the surface of a growing tri-axial ellipsoid. Murray (2003) discusses reaction–diffusion models as a mechanism for spiral patterning on amphibian egg shells. Since the shape of these eggs may be approximated by an ellipsoid, studying reaction–diffusion systems on these sorts of domains could help predict the patterns formed due to the calcium waves diffusing over the surface of the shell.

*C*(

*t*) with respect to time as \(\dot{A} = \frac{\mathrm{d}A}{\mathrm{d}t},\, \dot{B} = \frac{\mathrm{d}B}{\mathrm{d}t},\) and \(\dot{C} = \frac{\mathrm{d}C}{\mathrm{d}t}\), respectively.

## 3 Methods for the Study of Growth Rates, Curvature, and Anisotropic Growth

In this section, we will describe the various kinds of reaction kinetics that we have explored, as well as our systematic approach to studying them on growing domains. We are chiefly interested in fully nonlinear and stationary patterns, rather than transient patterning, so we always allow the reaction–diffusion process to settle onto a steady state on a stationary manifold before and after it has undergone growth. In this way, we can compare the kinds of stationary patterns which are arrived at via different growth processes. We will proceed to discuss parameter choices and numerical methods which will be used in our analysis.

### 3.1 Types of Reaction Kinetics

There are a multitude of functions \(f, \ g\) which can represent the interaction between two morphogens. We consider three different reaction schemes: Schnakenberg, Gierer–Meinhardt, and FitzHugh–Nagumo. These are canonical examples of cross- and pure activator–inhibitor kinetics, as well as excitable media, respectively.

#### 3.1.1 Schnakenberg Reaction Kinetics

First, we consider Schnakenberg (also known as activator-depleted) kinetics, a well-studied form of reaction which models activator–inhibitor behavior (Schnakenberg 1979). They were originally developed to explore simple chemical kinetics that gave rise to oscillatory behavior, but they have since been used in a huge amount of literature as a canonical example of activator–inhibitor chemistry (Iron et al. 2004; Liu et al. 2013; Sarfaraz and Madzvamuse 2017).

*a*and

*b*are parameters to be chosen. In order for patterns to form, the parameters \(a, \ b, \ \delta _1, \ \delta _2 \) must satisfy Turing instability condition Eq. (3) given in Sect. 2.1. Following the methodology outlined in Sect. 2.1, we first find the homogeneous steady state of the system by setting \(f= 0 = g\). This gives the steady-state \(\mathbf {u}^*\) and the Jacobian of the linearized system \(\varvec{J}^*\) as

#### 3.1.2 Gierer–Meinhardt Reaction Kinetics

We also consider Gierer–Meinhardt reaction kinetics (Gierer and Meinhardt 1972). These have been used in a variety of theoretical and applied contexts and are particularly used as an example of Turing instabilities which lead to spatially localized patterns, such as spikes in 1-D domains and spots in 2-D domains (Wei and Winter 2013). We note in particular the study by Tse et al. (2010) of these kinetics on the sphere, demonstrating the effect of curvature on the stability and structure of patterns.

*au*and

*bv*correspond to linear degradation of the morphogens, at rates

*a*and

*b*, respectively, and the terms involving \(u^2\) correspond to the activator’s production of both

*u*and

*v*. We note the addition of the constant 1 to the activator kinetics, corresponding to basal production of the activator, which allows for robust spontaneous spot formation (Page et al. 2005; Krause et al. 2018b).

#### 3.1.3 FitzHugh–Nagumo Reaction Kinetics

### 3.2 Numerical Approach

Throughout the rest of the paper, we will demonstrate solutions to equations (25) or (41) for different reaction kinetics *f* and *g*, as well as in different growth scenarios. We will use the commercially available finite element solver COMSOL, version 5.3, which will discretize the manifolds using second-order triangular finite elements. We note that these simulations were checked in various static domain cases using the Matlab package Chebfun (Townsend and Trefethen 2013; Townsend et al. 2016), in addition to convergence checks in spatial and time discretizations. In all simulations, we used a relative tolerance of \(10^{-5}\) and fixed an initial time step of \(10^{-6}\) (but let the solver increase the time step freely as the solution evolved). Convergence in time was checked by restricting the maximum time step, and convergence in space was determined via using different numbers of finite elements and comparing the norm of solutions over time and space. For the growing ellipse, we used 25,970 domain elements, and for the ellipsoid we used 19,704 boundary elements, which we now describe in some detail.

*n*can be constructed from the Laplace operator in the ambient space \(\mathbb {R}^{n+1}\) (Dziuk and Elliott 2007, 2013; Olshanskii and Xu 2017), so that dilation of a particular coordinate in \(\mathbb {R}^{n+1}\) allows a natural construction of the Laplace–Beltrami operator on the surface. We note that the planar ellipse model given by (25) is a flat 2-D manifold, and so the projection in this case is trivial. For Eq. (41), however, we can define both the Laplace–Beltrami operator and the dilution term in the ambient space \(\mathbb {R}^3\) as the operators

*x*,

*y*,

*z*) from the extrinsic coordinate system restricted to the surface. We can then define (41) in the whole space \(\mathbb {R}^3\) using (55) and simply project the equations to the unit sphere in order to obtain the correct dynamics on a growing ellipsoid. We also note that this ambient formulation allows us to see an explicit interaction between curvature and anisotropic growth—namely, the dilution term (55b) contains explicit spatial heterogeneity due to curvature and anisotropy, as these do not appear in the planar ellipse, nor in the isotropically growing sphere described in Plaza et al. (2004).

### 3.3 Robustness of Patterning to Growth

We now outline our systematic study of these three kinds of reaction kinetics on the elliptical and ellipsoidal domains, with dynamics governed by Eqs. (25) and (41), respectively. We vary the functional forms of *A*(*t*), *B*(*t*), and in the ellipsoidal case, *C*(*t*) in order to elucidate effects due to growth rates, anisotropy, and curvature. We consider different forms and rates of growth, in addition to the case of no growth. We note that on many domains, multistability of several non-uniform patterns is generic, especially in manifolds with dimension greater than one (Hunding 1980; Jensen et al. 1993; Borckmans et al. 1995). Because of this, we will emphasize qualitative differences between the kinds of patterns, rather than quantitative differences in the final patterned state. We now describe all of the different growth scenarios, which correspond to different choices in the parameters, manifolds, growth functions, etc. We will always take parameters within the Turing regime, so that the homogeneous steady state is unstable and leads to the formation of stationary patterns. Unless otherwise mentioned, we will always assume initial data of the form \(\mathbf {u}(0) = \mathbf {u}^* + \xi \), where \(\mathbf {u}^*\) is the spatially homogeneous steady state and \(\xi \) is a normally distributed spatial random variable with zero mean and standard deviation \(10^{-3}\). We note that the same realization of the spatial noise was used in all simulations.

For each scenario, we will consider a fixed static domain \(\varOmega ^*\), such as a sphere or an ellipsoid, and compare it to a growing domain \(\varOmega (t)\) such that after a time period *T*, \(\varOmega (T) = \varOmega ^*\). We are interested in the equilibrium states on the final manifold, and we are also interested in the evolution of non-uniform states. For these reasons, for each scenario we identify a kinetic timescale \(T^*\) such that the reaction–diffusion equations have approximately reached their steady-state behavior. We determine \(T^*\) by simulating the equations on the static manifold \(\varOmega ^*\) and observe when the patterning process settles into a stationary state, and we check that no further pattern movement or creation occurs in the time period up to \(10T^*\). With such a kinetic timescale in hand, we always let the pattern develop fully before initiating growth, and we always let the pattern equilibrate after growth. Mathematically this means that we take \(\varOmega (t)=\varOmega (0)\) for \(t \le T^*\) and \(\varOmega (t)=\varOmega (T)\) for \(t \ge T-T^*\), so that growth is confined to the interval of time \(t \in (T^*, T-T^*)\). We will always begin in a symmetric manifold, so that \(\varOmega (0)\) is either a circular disk or a sphere, and we will take the initial radius \(r_0\) to be large enough so that patterning occurs.

*A*(

*t*) and assume that this axis has an initial value of \(A(0)=r_0\) and a final value of \(A(T)=r_T\). Note that these functional forms are applied analogously to all growing axes. We also note that these functions are constant at the beginning and end of the simulation time, as described above. We define linear growth, where the domain is growing at a constant rate, by:

*A*(

*t*) and

*B*(

*t*) (and in the ellipsoidal case

*C*(

*t*)) grow in two (three) different stages subsequently and reach the same final value of \(r_T\), so that the final domain is a larger disk (sphere). We note in this case that we also considered an analogous isotropic growth scenario for the same manifold (e.g., without stages). In the second scenario, we consider a final manifold which is not symmetric, and so only one of the axes grows leading to an elongated ellipse or ellipsoid at the final time \(t=T\). Additionally, we consider the effect of the timescale over which the domain grows on the patterns formed, denoted as \(T_g = T-2T^*\). We consider three different timescales: fast growth given by \(T_g = T^*/2\), growth comparable to the kinetic timescale so \(T_g = T^*\), and growth which is much slower given by \(T_g = 10T^*\). We note that \(T^*\) is somewhat arbitrary, and in our simulations we take it to be sufficiently large to ensure we are in a patterned state, but we believe these growth timescales at least highlight differences in how the rate of growth influences the evolution of patterns. We note that in staged growth, we increase the growth timescale so that growth in each direction occurs over the same period of time \(T_g\).

List of the various parameter values studied in our systematic study

Name | Kinetics | Parameters | \(T^*\) | Static behavior |
---|---|---|---|---|

S1 | Schnakenberg | \(a=0.05, b=1.5, \delta _2=100\) | 5000 | Spots |

S2 | Schnakenberg | \(a=0.05, b=1.6, \delta _2=20\) | 60,000 | Labyrinthine |

GM1 | Gierer–Meinhardt | \(a=0.8, b=5.5, \delta _2=100\) | 13,000 | Large spots |

GM2 | Gierer–Meinhardt | \(a=1.5, b=3, \delta _2=300\) | 13,000 | Small spots |

FHN1 | FitzHugh–Nagumo | \(a=0.8, b=0.4, \delta _2=100\) | 2000 | Spots |

FHN2 | FitzHugh–Nagumo | \(a=0.6, b=0.99, \delta _2=50\) | 8000 | Labyrinthine |

We ran simulations of Eqs. (25) and (41) for each of the three reaction kinetics described in Sect. 3.1. In general, each of these models can give rise to a wide variety of behaviors, but for simplicity we consider the 6 parameter sets shown in Table 1. In each case, we explored all combinations of the 4 growth functions, 3 timescales, and both the final manifold scenarios (including an isotropic growth case), as well as 12 simulations on the static domain given by \(\varOmega ^*=\varOmega (T)\), leading to a total of 444 simulations. In each case, we compared the final pattern (in terms of the activator, *u*, as the inhibitor can be related to it via a phase), and we summarize differences below. We emphasize that many simulations will lead to quantitatively different steady states (e.g., the number of spots may differ slightly), but these quantitative differences will also occur due to different realizations of the initial data, as the precise organization of the final non-uniform state can sensitively depend on the initial data (Maini et al. 2012). Additionally, we ran other simulations with variations in these parameters or growth scenarios to ensure that our results were robust. In particular, we did not systematically vary staged growth at different rates, but did explore some simulations in this case to understand the possible behaviors.

Broadly speaking, we found that for Schnakenberg and Gierer–Meinhardt kinetics, the kinds of patterns observed and their qualitative properties did not change much, except due to the rate of growth, or whether or not the final domain was curved. The functional form of growth is particularly unimportant, only really changing the ‘effective’ rate of growth in some boundary cases which we will describe below. This leads us to conjecture that for the long-time steady states observed, linear growth with varying rates in growing domains is sufficient to capture all of the qualitative dynamics that can be observed, and that static domain simulations capture many qualitative properties of the final patterned state for these kinds of reaction kinetics. FitzHugh–Nagumo kinetics result in a richer structure, in terms of its interaction with growth, curvature, and anisotropy in the domain or growth. We do note, however, that analogous comments about the functional form of growth can be made. As the functional form of growth only influences the rate of growth, we will only show figures using the case of linear growth, but remark that the results with all other growth laws were qualitatively identical throughout. Similarly, we only show cases where noticeable effects occurred, and omit figures with comparable behavior to those shown. Finally, we note that solutions using the Schnakenberg and Gierer–Meinhardt kinetics will always be positive, as these models correspond to morphogen concentrations, whereas the activator *u* in FitzHugh–Nagumo is the voltage with respect to an arbitrary gauge and hence can locally take positive and negative values.

## 4 Reaction–Diffusion Systems on Planar Elliptical Domains

We now describe some of the results obtained in the case of Schnakenberg and Gierer–Meinhardt kinetics in the case of a planar elliptical domain. In Fig. 1, we compare a static ellipse with an ellipse having grown from an initial circle with growth only along the semimajor axis of the final ellipse. We note that growth appears to generate a more symmetric final distribution of spots (and that this result is identical across growth rates and functions in this case). This is consistent with the robustness attributed to growth in 1-D domains (Crampin et al. 1999). We also consider isotropic and staged growth for a circle in Fig. 2, here showing that isotropic growth, staged anisotropic growth, and the static domain all admit a distribution of spots with comparable symmetry. We remark that while some mild robustness under growth is generally independent of growth rates, it is also a small effect that depends on the initial data and the nature of the domain itself, and is in a sense only a weak qualitative effect. Overall, we find that formation of spot patterns in Schnakenberg is qualitatively independent of growth altogether, at least in two spatial dimensions. For all further plots, we omit any simulation which is qualitatively identical to one already presented.

Turning now to simulations of the Gierer–Meinhardt kinetics (47), we plot some example simulations in Fig. 5. As shown in Fig. 2, the case of slow isotropic growth gives rise to some symmetry to the final patterned state, but this was not observed for either staged growth or faster growth regimes, which gave the final spot patterns comparably disordered to the static growth case. We note that in the case of the final domain being elliptical, all growth scenarios gave rise to comparably disordered final patterns. We note the inclusion of the constant term (feed rate) in Eq. (47) leads to generally robust dynamics. If this term was not present, then after the initial production of spots, changes to the manifold do not lead to spot splitting or production of new spots. Growth in the case without this feed rate will, in all cases we explored, lead to transport of spot solutions, but no meaningful interactions or new behaviors.

**GM2**from Table 1) and plot an example in Fig. 6. We note that the static domain gives rise to many more spots than the case of isotropic growth. In fact, staged growth or isotropic growth at all rates exhibits the same number of spots in analogous configurations as shown in the case of isotropic growth. We suspect this is due to a conservation of spots inherent in the system, as discussed above, despite a positive feed rate. This occurs here as the inhibitor is able to suppress the constant production of activator throughout the domain (due to a much larger diffusion rate of the inhibitor and a smaller degradation rate). New spots are formed as the domain grows, but typically on the boundary and far from existing spots (which generate large regions of inhibition), so that the final number of spots in all growth cases is 12–14, compared with the 31 present in the static domain. This quantitative difference was consistent across 5 different realizations of the initial conditions (with different numbers of spots forming, but always less than 50% in the case of any growth). A similar effect was also observed in the elliptical domain (not shown).

*y*direction), but the effect is weaker due to the introduction of more minimal (blue) regions.

*y*axis). All rates of growth changed the final elliptical state in qualitatively similar ways as shown in Fig. 11, in that both maximal and minimal regions align along the semimajor axis of the ellipse, which is also the only axis of growth.

## 5 Reaction–Diffusion Systems on Ellipsoidal Domains

We now demonstrate solutions to Eq. (41) for the various reaction kinetics and growth regimes discussed. We note that now with three axes of possible growth, that there are three final possible domains (an oblate and a prolate spheroid or ellipsoid) where two axes are of equal size, depending on if one axis is smaller or larger than the other two. We did not systematically explore all intermediate domains with all variations in growth or kinetics, but we did include some in order to determine the influence of anisotropic curvature (as opposed to the isotropic curvature of a perfect sphere) on the final pattern. These effects can be seen during the transient staged growth, despite the shown states not being completely stationary, so we omit these results for brevity. Similarly, in all cases the effects of growth on different kinetics persist in these simulations, so we only show simulations where the influence of curvature (as well as new directions of anisotropic growth) can affect the final pattern. In particular, we do not include simulations of Gierer–Meinhardt, although we did observe the same kinds of patterns as shown in Figs. 5 and 6.

We next consider the evolution of labyrinthine Schnakenberg patterns in Fig. 13. Here we compare staged growth of these patterns in fast and slow regimes. The transient behaviors in the fast and slow growth regimes are quite different, but the final steady-state patterns are comparable with the fast growth case exhibiting some minor defects in the observed final labyrinthine pattern. We also note that the final patterns in the static domain and the slow growth domain, as shown in Fig. 13b, h, respectively, are essentially rotations of one another. Similar behavior is observed in the isotropic growth cases, where the final steady state differs from the static domain in the same way that different random initial data would (not shown).

## 6 Stripe Stability in Isotropic and Anisotropic Growth Regimes

Having systematically explored a wide range of reaction kinetics and growth regimes on planar and curved domains, we now briefly mention some results concerning the numerical stability and evolution of target and stripe patterns under growth. These patterns are special solutions that typically require spatial heterogeneity or specific initial data in order to observe (although we note that growth can do this as well, as shown in Figs. 4b, 13f). There is a large literature on the applications of stripe patterns in real organisms and chemical models (Ouyang and Swinney 1991; Nagorcka and Mooney 1992; Kondo and Asai 1995) (and this is something that Turing mentioned in his original 1952 paper), as well as a large literature on their stability and other theoretical properties (Ermentrout 1991; Lyons and Harrison 1991; Moyles et al. 2016).

*k*is a positive integer which determines the number of circular stripes in the target pattern. The vector of opposite signs is used as we will only consider the Schnakenberg kinetics with parameters that give rise to stripe solutions (

**S2**), and the activator and inhibitor are always out of phase as is typical for models with cross-kinetics. Depending on the size of the domain, these kinetics and parameter choices lead to a stable pattern very close to the original one, with \(k-1\) interior stripes, one stripe along the boundary, and a single central spot completing the target pattern. One can readily generate a bifurcation diagram in radius \(r_0\) and

*k*and numerically observe stability of different modes, but for brevity we set \(k=2\) and \(r_0=20\) in the elliptical case, and consider growth up to \(r_T=4r_0=80\).

*y*axis. In the extrinsic and stationary Cartesian frame, this is given by

For staged growth, we show some example simulations in Fig. 18 which all correspond to different stages and directions of anisotropy given the same initial state shown in Fig. 17a. Independent of the initial direction of growth, we see staged growth develop into an approximate target pattern for an elongated ellipsoid, but further growth along other axes leads to more complicated labyrinthine patterning, leading to the final patterns comparable to 17c. Compared with Fig. 16, the target patterns on curved domains appear much less stable to growth, likely due to the influence of curvature.

## 7 Discussion

Extending the work of Plaza et al. (2004), we have generalized reaction–diffusion systems on growing two-dimensional manifolds to include situations where the growth is anisotropic yet dilational in nature. This allowed us to systematically study the evolution of patterns in a variety of growth scenarios, in order to understand how growth affects the robustness of non-equilibrium patterns. We primarily concentrated on patterns which naturally emerged from Turing instabilities, but also gave some examples of stripe patterns, and their sensitivity to growth and curvature. While the restriction to only dilational anisotropy is somewhat restrictive, it allows for an exploration of the role that anisotropy has on the evolution of patterns under growth in a wide variety of growth scenarios.

Our results show several key findings. Firstly, the growth rate and kind of growth (e.g., isotropic or anisotropic) can significantly change the final pattern, even though we can demonstrate that the final pattern is a steady state. This suggests that the high level of multistability in these systems on fixed domains can be heavily influenced by their history due to growth, as suggested by Klika and Gaffney (2017). However, the functional form of growth was never really important; all simulations using different forms of the growth rates were qualitatively the same with linear growth at a suitable rate, as long as the initial and final manifolds were fixed. Secondly, we have used anisotropy in staged growth as compared to both quasi-static growth and isotropic growth to the same final manifold, as a concrete way of demonstrating the importance of both anisotropy and history in determining patterning. A final key finding is that the qualitative nature of patterns heavily depends on both the nonlinear reaction kinetics under consideration and the specific parameter regimes studied. While we only explored three candidate choices of reaction kinetics, we believe these are good exemplars of reaction–diffusion systems sufficient to obtain a qualitative picture of the influence of growth on patterned states.

Spot patterns in Gierer–Meinhardt and Schnakenberg were often not influenced by growth (Figs. 1, 2, 5) and manifold curvature only gave rise to slight local distortions in spots (Fig. 12b–c). The conservation of spots in the Gierer–Meinhardt system is something that has not, to our knowledge, been reported in the literature. Specifically, dropping the constant term in the first of Eq. (47) and letting spot patterns stabilize before growth always led to no new spots forming, independent of the manner of growth. With this feed rate term, however, the behavior of the Gierer–Meinhardt system then became heavily dependent on both the parameters used, as well as the growth rate. ‘Large’ spot solutions behaved as in Schnakenberg, with comparable numbers and distributions of spots in both quasi-static and grown domains. The smaller spot solutions, in Fig. 6, showed a dependence on the growth rate wherein slow enough growth permitted fewer final spots in the final domain. Finally, spot solutions arising from the FitzHugh–Nagumo kinetics exhibited an interesting dependence on both isotropy and curvature (see Figs. 7, 8, 9, 14). Isotropic growth led to labyrinthine-like patterns, whereas staged growth had an obvious effect on the curved regions. For slow growth rates, the total number of such regions was the same between the initial domain and the final one, whereas fast growth could initiate the formation of new regions, akin to spot splitting due to growth. This stark contrast between these reaction kinetics suggests that excitable media might behave quite differently under growth, compared to their dissipative counterparts.

Labyrinthine patterns exhibited a similar richness of behavior. Using Schnakenberg kinetics, stripes could be aligned if the domain was static, but not symmetric, whereas growth would destabilize such alignment (Fig. 3). For some growth rates and kinds of growth, these solutions could spontaneously form target patterns with symmetric stripes (Figs. 4b, 13d). Solutions for FitzHugh–Nagumo kinetics displayed slightly different behaviors, but also showed a strong dependence on anisotropy (Figs. 10, 11, 15). In this case, we did not see the emergence of target patterns or similarly perfectly symmetric structures, but we suspect that for some parameters this might be possible.

Finally, we explored the evolution of striped patterns in Sect. 6. These findings reinforce the key roles of growth rates, anisotropy, and curvature in determining the properties of the final steady-state patterns. It is interesting that in any growing ellipsoid we do not find any stable target or striped pattern that is not a labyrinthine-like pattern. This may be due to the specific parameter choices we have made, and we leave further investigation of stripes and other specific kinds of patterns to future work.

We have demonstrated the complex influence of growth, curvature, and anisotropy on non-uniform patterns arising from Turing instabilities on planar and curved manifolds. There are many possible extensions to our work, including considering manifolds with holes or cusps, or considering bulk-surface reaction–diffusion systems as in Madzvamuse et al. (2015), under the influence of growth. Additionally, further generalizing the reaction–diffusion framework to account for arbitrary non-uniform growth would allow for more accurate modeling of realistic pattern formation in development and elsewhere. Finally, our work also shows behaviors of non-uniform solutions to nonlinear PDE in complicated domains, and the stability and evolution of such solutions are usually confined to illustrative special cases, as in Trinh and Ward (2016). One could pursue similar analyses in the context of anisotropic growth, in order to deepen our understanding of the effects we have demonstrated here.

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