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Turing conditions for pattern forming systems on evolving manifolds

Abstract

The study of pattern-forming instabilities in reaction–diffusion systems on growing or otherwise time-dependent domains arises in a variety of settings, including applications in developmental biology, spatial ecology, and experimental chemistry. Analyzing such instabilities is complicated, as there is a strong dependence of any spatially homogeneous base states on time, and the resulting structure of the linearized perturbations used to determine the onset of instability is inherently non-autonomous. We obtain general conditions for the onset and structure of diffusion driven instabilities in reaction–diffusion systems on domains which evolve in time, in terms of the time-evolution of the Laplace–Beltrami spectrum for the domain and functions which specify the domain evolution. Our results give sufficient conditions for diffusive instabilities phrased in terms of differential inequalities which are both versatile and straightforward to implement, despite the generality of the studied problem. These conditions generalize a large number of results known in the literature, such as the algebraic inequalities commonly used as a sufficient criterion for the Turing instability on static domains, and approximate asymptotic results valid for specific types of growth, or specific domains. We demonstrate our general Turing conditions on a variety of domains with different evolution laws, and in particular show how insight can be gained even when the domain changes rapidly in time, or when the homogeneous state is oscillatory, such as in the case of Turing–Hopf instabilities. Extensions to higher-order spatial systems are also included as a way of demonstrating the generality of the approach.

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Notes

  1. We note that for complex reaction–diffusion systems spatio-temporal base states consisting of plane waves are also possible (Knobloch and Krechetnikov 2014). However, as our concern is with generalising the Turing conditions to account for evolving domains, we only consider spatially uniform base states.

References

  • Amar MB, Jia F (2013) Anisotropic growth shapes intestinal tissues during embryogenesis. Proc Natl Acad Sci 110(26):10525–10530

    Article  Google Scholar 

  • Baker RE, Gaffney E, Maini P (2008) Partial differential equations for self-organization in cellular and developmental biology. Nonlinearity 21(11):R251

    Article  MathSciNet  MATH  Google Scholar 

  • Bánsági T, Vanag VK, Epstein IR (2011) Tomography of reaction–diffusion microemulsions reveals three-dimensional Turing patterns. Science 331(6022):1309–1312

    Article  MathSciNet  MATH  Google Scholar 

  • Barrass I, Crampin EJ, Maini PK (2006) Mode transitions in a model reaction–diffusion system driven by domain growth and noise. Bull Math Biol 68(5):981–995

    Article  MathSciNet  MATH  Google Scholar 

  • Barreira R, Elliott CM, Madzvamuse A (2011) The surface finite element method for pattern formation on evolving biological surfaces. J Math Biol 63(6):1095–1119

    Article  MathSciNet  MATH  Google Scholar 

  • Bellman RE (1949) A survey of the theory of the boundedness, stability, and asymptotic behavior of solutions of linear and non-linear differential and difference equations. Tech. Rep. NAVEXOS P-596, Office of Naval Research, Washington DC (1949)

  • Beloussov L, Grabovsky V (2003) A geometro-mechanical model for pulsatile morphogenesis. Comput Methods Biomech Biomed Eng 6(1):53–63

    Article  Google Scholar 

  • Binder BJ, Landman KA, Simpson MJ, Mariani M, Newgreen DF (2008) Modeling proliferative tissue growth: a general approach and an avian case study. Phys Rev E 78(3):031912

    Article  Google Scholar 

  • Bittig T, Wartlick O, Kicheva A, González-Gaitán M, Jülicher F (2008) Dynamics of anisotropic tissue growth. New J Phys 10(6):063001

    Article  Google Scholar 

  • Cahn JW, Hilliard JE (1958) Free energy of a nonuniform system. I. Interfacial free energy. J Chem Phys 28(2):258–267

    Article  MATH  Google Scholar 

  • Callahan T, Knobloch E (1999) Pattern formation in three-dimensional reaction–diffusion systems. Physica D 132(3):339–362

    Article  MathSciNet  MATH  Google Scholar 

  • Castillo JA, Sánchez-Garduño F, Padilla P (2016) A Turing–Hopf bifurcation scenario for pattern formation on growing domains. Bull Math Biol 78(7):1410–1449

    Article  MathSciNet  MATH  Google Scholar 

  • Chaplain MA, Ganesh M, Graham IG (2001) Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth. J Math Biol 42(5):387–423

    Article  MathSciNet  MATH  Google Scholar 

  • Coen E, Rolland-Lagan AG, Matthews M, Bangham JA, Prusinkiewicz P (2004) The genetics of geometry. Proc Natl Acad Sci 101(14):4728–4735

    Article  Google Scholar 

  • Cohen DS, Murray JD (1981) A generalized diffusion model for growth and dispersal in a population. J Math Biol 12(2):237–249

    Article  MathSciNet  MATH  Google Scholar 

  • Colbois B, Provenzano L (2019) Neumann eigenvalues of the biharmonic operator on domains: geometric bounds and related results. arXiv preprint arXiv:190702252

  • Comanici A, Golubitsky M (2008) Patterns on growing square domains via mode interactions. Dyn Syst 23(2):167–206

    Article  MathSciNet  MATH  Google Scholar 

  • Corson F, Hamant O, Bohn S, Traas J, Boudaoud A, Couder Y (2009) Turning a plant tissue into a living cell froth through isotropic growth. Proc Natl Acad Sci 106(21):8453–8458

    Article  Google Scholar 

  • Crampin E, Maini P (2001) Reaction–diffusion models for biological pattern formation. Methods Appl Anal 8(3):415–428

    Article  MathSciNet  MATH  Google Scholar 

  • Crampin EJ, Gaffney EA, Maini PK (1999) Reaction and diffusion on growing domains: scenarios for robust pattern formation. Bull Math Biol 61(6):1093–1120

    Article  MATH  Google Scholar 

  • Crampin E, Hackborn W, Maini P (2002) Pattern formation in reaction–diffusion models with nonuniform domain growth. Bull Math Biol 64(4):747–769

    Article  MATH  Google Scholar 

  • Cross MC, Hohenberg PC (1993) Pattern formation outside of equilibrium. Rev Mod Phys 65(3):851

    Article  MATH  Google Scholar 

  • De Wit A, Borckmans P, Dewel G (1997) Twist grain boundaries in three-dimensional lamellar Turing structures. Proc Natl Acad Sci 94(24):12765–12768

    Article  Google Scholar 

  • Dhillon DSJ, Milinkovitch MC, Zwicker M (2017) Bifurcation analysis of reaction diffusion systems on arbitrary surfaces. Bull Math Biol 79(4):788–827

    Article  MathSciNet  MATH  Google Scholar 

  • Dolnik M, Zhabotinsky AM, Epstein IR (2001) Resonant suppression of Turing patterns by periodic illumination. Phys Rev E 63(2):026101

    Article  Google Scholar 

  • Drasdo D, Loeffler M (2001) Individual-based models to growth and folding in one-layered tissues: intestinal crypts and early development. Nonlinear Anal Theory Methods Appl 47(1):245–256

    Article  MathSciNet  MATH  Google Scholar 

  • Ermentrout B (1991) Stripes or spots? Nonlinear effects in bifurcation of reaction–diffusion equations on the square. Proc R Soc Lond A 434(1891):413–417

    Article  MathSciNet  MATH  Google Scholar 

  • Facsko S, Bobek T, Stahl A, Kurz H, Dekorsy T (2004) Dissipative continuum model for self-organized pattern formation during ion-beam erosion. Phys Rev B 69(15):153412

    Article  Google Scholar 

  • Feijó JA, Sainhas J, Holdaway-Clarke T, Cordeiro MS, Kunkel JG, Hepler PK (2001) Cellular oscillations and the regulation of growth: the pollen tube paradigm. BioEssays 23(1):86–94

    Article  Google Scholar 

  • FitzHugh R (1955) Mathematical models of threshold phenomena in the nerve membrane. Bull Math Biol 17(4):257–278

    Google Scholar 

  • Gerber M, Hasselblatt B, Keesing D (2003) The Riccati equation: pinching of forcing and solutions. Exp Math 12(2):129–134

    Article  MathSciNet  MATH  Google Scholar 

  • Ghadiri M, Krechetnikov R (2019) Pattern formation on time-dependent domains. J Fluid Mech 880:136–179

    Article  Google Scholar 

  • Gierer A, Meinhardt H (1972) A theory of biological pattern formation. Kybernetik 12(1):30–39

    Article  MATH  Google Scholar 

  • Gjorgjieva J, Jacobsen JT (2007) Turing patterns on growing spheres: the exponential case. In: Discrete and continuous dynamical systems supplement, pp 436–445

  • Grigoryan G (2015) On the stability of systems of two first-order linear ordinary differential equations. Differ Equ 51(3):283–292

    Article  MathSciNet  MATH  Google Scholar 

  • Hetzer G, Madzvamuse A, Shen W (2012) Characterization of Turing diffusion-driven instability on evolving domains. Discrete Contin Dyn Syst Ser A 32(11):3975–4000

    Article  MathSciNet  MATH  Google Scholar 

  • Hilali M, Métens S, Borckmans P, Dewel G (1995) Pattern selection in the generalized Swift–Hohenberg model. Phys Rev E 51(3):2046

    Article  Google Scholar 

  • Horváth AK, Dolnik M, Munuzuri AP, Zhabotinsky AM, Epstein IR (1999) Control of Turing structures by periodic illumination. Phys Rev Lett 83(15):2950

    Article  Google Scholar 

  • Hunding A (1985) Morphogen prepatterns during mitosis and cytokinesis in flattened cells: Three dimensional Turing structures of reaction–diffusion systems in cylindrical coordinates. J Theor Biol 114(4):571–588

    Article  MathSciNet  Google Scholar 

  • Hyman JM, Nicolaenko B (1986) The Kuramoto–Sivashinsky equation: a bridge between PDE’s and dynamical systems. Physica D 18(1–3):113–126

    Article  MathSciNet  MATH  Google Scholar 

  • Ince E (1956) Ordinary differential equations. Dover, New York

    Google Scholar 

  • Josić K, Rosenbaum R (2008) Unstable solutions of nonautonomous linear differential equations. SIAM Rev 50(3):570–584

    Article  MathSciNet  MATH  Google Scholar 

  • Keener JP, Sneyd J (1998) Mathematical physiology, vol 1. Springer, Berlin

    Book  MATH  Google Scholar 

  • Kielhöfer H (1997) Pattern formation of the stationary Cahn–Hilliard model. Proc R Soc Edinb Sect A Math 127(6):1219–1243

    Article  MathSciNet  MATH  Google Scholar 

  • Klika V (2017) Significance of non-normality-induced patterns: transient growth versus asymptotic stability. Chaos Interdiscip J Nonlinear Sci 27(7):073120

    Article  MathSciNet  MATH  Google Scholar 

  • Klika V, Gaffney EA (2017) History dependence and the continuum approximation breakdown: the impact of domain growth on Turing’s instability. Proc R Soc A Math Phys Eng Sci 473(2199):20160744

    MathSciNet  MATH  Google Scholar 

  • Knobloch E, Krechetnikov R (2014) Stability on time-dependent domains. J Nonlinear Sci 24(3):493–523

    Article  MathSciNet  MATH  Google Scholar 

  • Knobloch E, Krechetnikov R (2015) Problems on time-varying domains: formulation, dynamics, and challenges. Acta Appl Math 137(1):123–157

    Article  MathSciNet  MATH  Google Scholar 

  • Kondo S, Miura T (2010) Reaction–diffusion model as a framework for understanding biological pattern formation. Science 329(5999):1616–1620

    Article  MathSciNet  MATH  Google Scholar 

  • Konow C, Somberg NH, Chavez J, Epstein IR, Dolnik M (2019) Turing patterns on radially growing domains: experiments and simulations. Phys Chem Chem Phys 21(12):6718–6724

    Article  Google Scholar 

  • Korzec MD, Evans PL, Münch A, Wagner B (2008) Stationary solutions of driven fourth-and sixth-order Cahn–Hilliard-type equations. SIAM J Appl Math 69(2):348–374

    Article  MathSciNet  MATH  Google Scholar 

  • Krause AL, Burton AM, Fadai NT, Van Gorder RA (2018a) Emergent structures in reaction–advection–diffusion systems on a sphere. Phys Rev E 97(4):042215

    Article  MathSciNet  Google Scholar 

  • Krause AL, Ellis MA, Van Gorder RA (2018b) Influence of curvature, growth, and anisotropy on the evolution of Turing patterns on growing manifolds. Bull Math Biol 81:759–799

    Article  MathSciNet  MATH  Google Scholar 

  • Krause AL, Klika V, Woolley TE, Gaffney EA (2020) From one pattern into another: analysis of Turing patterns in heterogeneous domains via WKB.J Royal Society Interface 17:20190621

  • Krechetnikov R (2017) Stability of a growing cylindrical blob. J Fluid Mech 827:R3

    Article  MathSciNet  MATH  Google Scholar 

  • Krechetnikov R, Knobloch E (2017) Stability on time-dependent domains: convective and dilution effects. Physica D 342:16–23

    Article  MathSciNet  MATH  Google Scholar 

  • Kuramoto Y (1978) Diffusion-induced chaos in reaction systems. Prog Theor Phys Suppl 64:346–367

    Article  Google Scholar 

  • Laptev A (1997) Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces. J Funct Anal 151(2):531–545

    Article  MathSciNet  MATH  Google Scholar 

  • Leppänen T, Karttunen M, Kaski K, Barrio RA, Zhang L (2002) A new dimension to Turing patterns. Physica D 168:35–44

    Article  MathSciNet  MATH  Google Scholar 

  • Levine H, Protter M, Payne L (1985) Unrestricted lower bounds for eigenvalues for classes of elliptic equations and systems of equations with applications to problems in elasticity. Math Methods Appl Sci 7(1):210–222

    Article  MathSciNet  MATH  Google Scholar 

  • Liu R, Liaw S, Maini P (2006) Two-stage Turing model for generating pigment patterns on the leopard and the jaguar. Phys Rev E 74(1):011914

    Article  Google Scholar 

  • Lloyd DJ, Sandstede B, Avitabile D, Champneys AR (2008) Localized hexagon patterns of the planar Swift–Hohenberg equation. SIAM J Appl Dyn Syst 7(3):1049–1100

    Article  MathSciNet  MATH  Google Scholar 

  • Macdonald CB, Merriman B, Ruuth SJ (2013) Simple computation of reaction–diffusion processes on point clouds. Proc Natl Acad Sci 110(23):9209–9214

    Article  MathSciNet  MATH  Google Scholar 

  • Madzvamuse A (2008) Stability analysis of reaction–diffusion systems with constant coefficients on growing domains. Int J Dyn Syst Differ Equ 1(4):250–262

    MathSciNet  MATH  Google Scholar 

  • Madzvamuse A, Gaffney EA, Maini PK (2010) Stability analysis of non-autonomous reaction–diffusion systems: the effects of growing domains. J Math Biol 61(1):133–164

    Article  MathSciNet  MATH  Google Scholar 

  • Maini PK, Crampin EJ, Madzvamuse A, Wathen AJ, Thomas RD (2002) Implications of domain growth in morphogenesis. In: Mathematical modelling & computing in biology and medicine: 5th ESMTB Conf, vol 1, pp 67–73

  • Marcon L, Sharpe J (2012) Turing patterns in development: what about the horse part? Curr Opin Genet Dev 22(6):578–584

    Article  Google Scholar 

  • Matthews P (2003) Pattern formation on a sphere. Phys Rev E 67(3):036206

    Article  Google Scholar 

  • Meinhardt H, Koch AJ, Bernasconi G (1998) Models of pattern formation applied to plant development. In: Symmetry in plants. World Scientific, pp 723–758

  • Mendez V, Fedotov S, Horsthemke W (2010) Reaction-transport systems: mesoscopic foundations, fronts, and spatial instabilities. Springer, Berlin

    Book  Google Scholar 

  • Menzel A (2005) Modelling of anisotropic growth in biological tissues. Biomech Model Mechanobiol 3(3):147–171

    Article  Google Scholar 

  • Mierczynski J (2017) Instability in linear cooperative systems of ordinary differential equations. SIAM Rev 59(3):649–670

    Article  MathSciNet  MATH  Google Scholar 

  • Míguez DG, Pérez-Villar V, Muñuzuri AP (2005) Turing instability controlled by spatiotemporal imposed dynamics. Phys Rev E 71(6):066217

    Article  Google Scholar 

  • Míguez DG, Dolnik M, Munuzuri AP, Kramer L (2006) Effect of axial growth on Turing pattern formation. Phys Rev Lett 96(4):048304

    Article  Google Scholar 

  • Miura T, Shiota K, Morriss-Kay G, Maini PK (2006) Mixed-mode pattern in doublefoot mutant mouse limb-Turing reaction–diffusion model on a growing domain during limb development. J Theor Biol 240(4):562–573

    Article  MathSciNet  MATH  Google Scholar 

  • Murray JD (2003a) Mathematical biology. I An introduction. In: Interdisciplinary applied mathematics, vol 18. Springer, New York

  • Murray JD (2003b) Mathematical biology. II Spatial models and biomedical applications. In: Interdisciplinary applied mathematics, vol 18. Springer, New York

  • Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc IRE 50(10):2061–2070

    Article  Google Scholar 

  • Nechaeva MV, Turpaev TM (2002) Rhythmic contractions in chick amnio-yolk sac and snake amnion during embryogenesis. Comp Biochem Physiol Part A Mol Integr Physiol 131(4):861–870

    Article  Google Scholar 

  • Novick-Cohen A, Segel LA (1984) Nonlinear aspects of the Cahn–Hilliard equation. Physica D 10(3):277–298

    Article  MathSciNet  Google Scholar 

  • Ochoa FL (1984) A generalized reaction diffusion model for spatial structure formed by motile cells. Biosystems 17(1):35–50

    Article  Google Scholar 

  • Page KM, Maini PK, Monk NA (2005) Complex pattern formation in reaction–diffusion systems with spatially varying parameters. Physica D 202(1–2):95–115

    Article  MathSciNet  MATH  Google Scholar 

  • Pawłow I, Zajaczkowski WM (2011) A sixth order Cahn–Hilliard type equation arising in oil–water-surfactant mixtures. Commun Pure Appl Anal 10(6):1823–1847

    Article  MathSciNet  MATH  Google Scholar 

  • Peaucelle A, Wightman R, Höfte H (2015) The control of growth symmetry breaking in the arabidopsis hypocotyl. Curr Biol 25(13):1746–1752

    Article  Google Scholar 

  • Plaza RG, Sanchez-Garduno F, Padilla P, Barrio RA, Maini PK (2004) The effect of growth and curvature on pattern formation. J Dyn Differ Equ 16(4):1093–1121

    Article  MathSciNet  MATH  Google Scholar 

  • Pleijel Å (1950) On the eigenvalues and eigenfunctions of elastic plates. Commun Pure Appl Math 3(1):1–10

    Article  MathSciNet  MATH  Google Scholar 

  • Preska Steinberg A, Epstein IR, Dolnik M (2014) Target Turing patterns and growth dynamics in the chlorine dioxide–iodine–malonic acid reaction. J Phys Chem A 118(13):2393–2400

    Article  Google Scholar 

  • Raspopovic J, Marcon L, Russo L, Sharpe J (2014) Digit patterning is controlled by a Bmp-Sox9-Wnt turing network modulated by morphogen gradients. Science 345(6196):566–570

    Article  Google Scholar 

  • Rossi F, Duteil NP, Yakoby N, Piccoli B (2016) Control of reaction–diffusion equations on time-evolving manifolds. In: 2016 IEEE 55th Conference on Decision and Control (CDC). IEEE, pp 1614–1619

  • Rost M, Krug J (1995) Anisotropic Kuramoto–Sivashinsky equation for surface growth and erosion. Phys Rev Lett 75(21):3894

    Article  Google Scholar 

  • Rozada I, Ruuth SJ, Ward M (2014) The stability of localized spot patterns for the Brusselator on the sphere. SIAM J Appl Dyn Syst 13(1):564–627

    Article  MathSciNet  MATH  Google Scholar 

  • Rüdiger S, Miguez D, Munuzuri A, Sagués F, Casademunt J (2003) Dynamics of Turing patterns under spatiotemporal forcing. Phys Rev Lett 90(12):128301

    Article  Google Scholar 

  • Saez A, Ghibaudo M, Buguin A, Silberzan P, Ladoux B (2007) Rigidity-driven growth and migration of epithelial cells on microstructured anisotropic substrates. Proc Natl Acad Sci 104(20):8281–8286

    Article  Google Scholar 

  • Sánchez-Garduno F, Krause AL, Castillo JA, Padilla P (2019) Turing–Hopf patterns on growing domains: the torus and the sphere. J Theor Biol 481:136–150

    Article  MathSciNet  MATH  Google Scholar 

  • Sarfaraz W, Madzvamuse A (2018) Domain-dependent stability analysis of a reaction–diffusion model on compact circular geometries. Int J Bifurc Chaos 28(08):1830024

    Article  MathSciNet  MATH  Google Scholar 

  • Satnoianu RA, Menzinger M, Maini PK (2000) Turing instabilities in general systems. J Math Biol 41(6):493–512

    Article  MathSciNet  MATH  Google Scholar 

  • Schnakenberg J (1979) Simple chemical reaction systems with limit cycle behaviour. J Theor Biol 81(3):389–400

    Article  MathSciNet  Google Scholar 

  • Seifert U, Berndl K, Lipowsky R (1991) Shape transformations of vesicles: phase diagram for spontaneous-curvature and bilayer-coupling models. Phys Rev A 44(2):1182

    Article  Google Scholar 

  • Seul M, Andelman D (1995) Domain shapes and patterns: the phenomenology of modulated phases. Science 267(5197):476–483

    Article  Google Scholar 

  • Shimoda Y, Kumagai J, Anzai M, Kabashima K, Togashi K, Miura Y, Shirasawa H, Sato W, Kumazawa Y, Terada Y (2016) Time-lapse monitoring reveals that vitrification increases the frequency of contraction during the pre-hatching stage in mouse embryos. J Reprod Dev 62:187–193

    Article  Google Scholar 

  • Shoji H, Iwasa Y, Kondo S (2003) Stripes, spots, or reversed spots in two-dimensional Turing systems. J Theor Biol 224(3):339–350

    Article  MathSciNet  MATH  Google Scholar 

  • Shoji H, Yamada K, Ueyama D, Ohta T (2007) Turing patterns in three dimensions. Phys Rev E 75(4):046212

    Article  MathSciNet  Google Scholar 

  • Sigrist R, Matthews P (2011) Symmetric spiral patterns on spheres. SIAM J Appl Dyn Syst 10(3):1177–1211

    Article  MathSciNet  MATH  Google Scholar 

  • Sivashinsky G (1977) Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations. Acta Astronaut 4:1177–1206

    Article  MathSciNet  MATH  Google Scholar 

  • Stoop N, Lagrange R, Terwagne D, Reis PM, Dunkel J (2015) Curvature-induced symmetry breaking determines elastic surface patterns. Nat Mater 14(3):337

    Article  Google Scholar 

  • Striegel DA, Hurdal MK (2009) Chemically based mathematical model for development of cerebral cortical folding patterns. PLoS Comput Biol 5(9):e1000524

    Article  MathSciNet  Google Scholar 

  • Swift J, Hohenberg PC (1977) Hydrodynamic fluctuations at the convective instability. Phys Rev A 15(1):319

    Article  Google Scholar 

  • Tan Z, Chen S, Peng X, Zhang L, Gao C (2018) Polyamide membranes with nanoscale Turing structures for water purification. Science 360(6388):518–521

    Article  Google Scholar 

  • Thery M, Bornens M (2006) Cell shape and cell division. Curr Opin Cell Biol 18(6):648–657

    Article  Google Scholar 

  • Toole G, Hurdal MK (2013) Turing models of cortical folding on exponentially and logistically growing domains. Comput Math Appl 66(9):1627–1642

    Article  MathSciNet  MATH  Google Scholar 

  • Toole G, Hurdal MK (2014) Pattern formation in Turing systems on domains with exponentially growing structures. J Dyn Differ Equ 26(2):315–332

    Article  MathSciNet  MATH  Google Scholar 

  • Townsend A, Trefethen LN (2013) An extension of Chebfun to two dimensions. SIAM J Sci Comput 35(6):C495–C518

    Article  MathSciNet  MATH  Google Scholar 

  • Toyama Y, Peralta XG, Wells AR, Kiehart DP, Edwards GS (2008) Apoptotic force and tissue dynamics during drosophila embryogenesis. Science 321(5896):1683–1686

    Article  Google Scholar 

  • Turing AM (1952) The chemical basis of morphogenesis. Philos Trans R Soc Lond B Biol Sci 237(641):37–72

    Article  MathSciNet  MATH  Google Scholar 

  • Ubeda-Tomás S, Swarup R, Coates J, Swarup K, Laplaze L, Beemster GT, Hedden P, Bhalerao R, Bennett MJ (2008) Root growth in arabidopsis requires gibberellin/della signalling in the endodermis. Nat Cell Biol 10(5):625

    Article  Google Scholar 

  • Ueda KI, Nishiura Y (2012) A mathematical mechanism for instabilities in stripe formation on growing domains. Physica D 241(1):37–59

    Article  MATH  Google Scholar 

  • Van Gorder RA (2020a) Influence of temperature on Turing pattern formation. Proc R Soc A 476(2240):20200356

    Article  MathSciNet  Google Scholar 

  • Van Gorder RA (2020b) Turing and Benjamin-Feir instability mechanisms in non-autonomous systems. Proc R Soc A 476(2238):20200003

    Article  MathSciNet  Google Scholar 

  • Van Gorder RA, Kim H, Krause AL (2019) Diffusive instabilities and spatial patterning from the coupling of reaction–diffusion processes with stokes flow in complex domains. J Fluid Mech 877:759–823

    Article  MathSciNet  MATH  Google Scholar 

  • Varea C, Aragón J, Barrio R (1997) Confined Turing patterns in growing systems. Phys Rev E 56(1):1250

    Article  Google Scholar 

  • Varea C, Aragon J, Barrio R (1999) Turing patterns on a sphere. Phys Rev E 60(4):4588

    Article  Google Scholar 

  • Vinograd R (1952) On a criterion of instability in the sense of Lyapunov of the solutions of a linear system of ordinary differential equations. Doklady Akad Nauk SSSR (NS) 84:201–204

    MathSciNet  Google Scholar 

  • Wang H, Zhang K, Ouyang Q (2006) Resonant-pattern formation induced by additive noise in periodically forced reaction–diffusion systems. Phys Rev E 74(3):036210

    Article  Google Scholar 

  • Wang Q, Zhao X (2015) A three-dimensional phase diagram of growth-induced surface instabilities. Sci Rep 5:8887

    Article  Google Scholar 

  • Wu M (1974) A note on stability of linear time-varying systems. IEEE Trans Autom Control 19(2):162

    Article  MathSciNet  Google Scholar 

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Van Gorder, R.A., Klika, V. & Krause, A.L. Turing conditions for pattern forming systems on evolving manifolds. J. Math. Biol. 82, 4 (2021). https://doi.org/10.1007/s00285-021-01552-y

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