Pattern Formation in Turing Systems on Domains with Exponentially Growing Structures

Toole, Gregory; Hurdal, Monica K.

doi:10.1007/s10884-014-9365-2

Cited by 5 publications

(1 citation statement)

References 13 publications

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“…In addition to biological plausibility, another reason for the popularity of exponential growth is that it allows for the volume expansion term to take the form μ µ = N s (where N is the dimension of the space domain), a constant, which greatly simplifies the dynamics of the spatially uniform system. Exponential isotropic growth of surfaces in R 3 was extensively studied in [109], albeit under the assumption of a time-independent base state for (11).…”

Section: Isotropic Evolution Of a Line Segmentmentioning

confidence: 99%

Turing conditions for pattern forming systems on evolving manifolds

Van Gorder,

Klika,

Krause

2019

Preprint

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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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Section: Isotropic Evolution Of a Line Segmentmentioning

confidence: 99%