Abstract
By a pattern we usually mean a spatially nontrivial structure and hence its antonym is spatial homogeneity. Alan Turing found that, in a reaction-diffusion system of two species, different diffusion rates can destabilize a spatially uniform state, leading to spontaneous formation of a pattern. This chapter proposes to generalize the notion of pattern to that of spatially heterogeneous environments and to build a unified theory of spontaneous emergence of patterns against spatially homogeneous or heterogeneous backgrounds.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ai, S., Chen, X., Hastings, S.P.: Layers and spikes in non-homogeneous bistable reaction-diffusion equations. Trans. Am. Math. Soc. 358, 3169–3206 (2006)
Cantrell, R.S., Cosner, C.: Spatial Ecology via Reaction-Diffusion Equations. Wiley, Chichester (2003)
Casten, R.G., Holland, C.J.: Instability results for reaction diffusion equations with Neumann boundary conditions. J. Differ. Equ. 27, 266–273 (1978)
Cosner, C., Lou, Y.: Does movement toward better environments always benefit a population? J. Math. Appl. 277, 489–503 (2003)
Gierer, A., Meinhart, H.: A theory of biological pattern formation. Kybernetik (Berlin) 12, 30–39 (1972)
Gui, C., Wei, J.: Multiple interior peak solutions for some singularly perturbed Neumann problems. J. Differ. Equ. 158, 1–27 (1999)
Hale, J., Sakamoto, K.: Existence and stability of transition layers. Jpn. J. Appl. Math. 5, 367–405 (1988)
Hutson, V., Lou, Y., Mischaikow, K.: Spatial heterogeneity of resources versus Lotka-Volterra dynamics. J. Differ. Equ. 185, 97–136 (2002)
Maini, P., Takagi, I., Yamamoto, H.: A two-stage model of activator-inhibitor type in developmental biology (in preparation)
Matano, H.: Asymptotic behavior and stability of solutions of semilinear diffusion equations. Publ. RIMS, Kyoto Univ. 15, 401–454 (1979)
Mimura, M., Nishiura, Y.: Spatial patterns for an interaction-diffusion equation in morphogenesis. J. Math. Biol. 7, 243–263 (1979)
Mimura, M., Tabata, M., Hosono, Y.: Multiple solutions of two point boundary value problems of Neumann type with a small parameter. SIAM J. Math. Anal. 11, 613–631 (1980)
Murray, J.D.: Mathematical Biology, vol. II, 3rd edn. Springer, New York/Berlin/Heidelberg (2003)
Ni, W.-M., Takagi, I.: On the shape of least-energy solutions to a semilinear Neumann problem. Commun. Pure Appl. Math. 44, 819–851 (1991)
Ni, W.-M., Takagi, I.: Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J. 70, 247–281 (2019)
Ni, W.-M., Takagi, I., Yanagida, E.: Stability of least energy patterns of the shadow system for an activator-inhibitor model. Jpn. J. Indust. Appl. Math. 18, 259–272 (2001)
Ni, W.-M., Takagi, I., Yanagida, E.: Stability analysis of point-condensation solutions to a reaction-diffusion system (in preparation)
Omel’chenko, O., Recke, L.: Existence, local uniqueness and asymptotic approximation of spike solutions to singularly perturbed elliptic problem. Hiroshima Math. J. 45, 35–89 (2015)
Ren, X.: Least-energy solution to a nonautonomous semilinear problem with small diffusion coefficient. Electron. J. Differ. Equ. No. 05, approx. 21 pp. (1993)
Takagi, I., Yamamoto, H.: Locator function for concentration points in a spatially heterogeneous semilinear Neumann problem. Indiana Univ. Math. J. 68, 63–103 (2019)
Turing, A.M.: The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. B 237, 37–72 (1952)
Wang, X., Zeng, B.: On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions. SIAM J. Math. Anal. 28, 633–655 (1997)
Wei, J., Winter, M.: Mathematical Aspects of Pattern Formation in Biological Systems. Springer, London (2003)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Takagi, I. (2021). Patterns Versus Spatial Heterogeneity—From a Variational Viewpoint. In: Zheng, Z. (eds) Proceedings of the First International Forum on Financial Mathematics and Financial Technology. Financial Mathematics and Fintech. Springer, Singapore. https://doi.org/10.1007/978-981-15-8373-5_9
Download citation
DOI: https://doi.org/10.1007/978-981-15-8373-5_9
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-8372-8
Online ISBN: 978-981-15-8373-5
eBook Packages: Economics and FinanceEconomics and Finance (R0)