Realistic examples of reaction-diffusion phenomena governing spatial and spatiotemporal pattern formation are rarely isolated systems, either chemically or thermodynamically. However, even formulations of 'open' reaction-diffusion systems often neglect the role of domain boundaries. Most idealizations of closed reaction-diffusion systems employ no-flux boundary conditions, and often patterns will form up to, or along, these boundaries. Motivated by boundaries of patterning fields related to the emergence of spatial form in embryonic development, we propose a set of mixed boundary conditions for a two-species reaction-diffusion system which forms inhomogeneous solutions away from the boundary of the domain for a variety of different reaction kinetics, with a prescribed uniform state near the boundary. We show that these boundary conditions can be derived from a larger heterogeneous field, indicating that these conditions can arise naturally if cell signalling or other properties of the medium vary in space. We explain the basic mechanisms behind this pattern localization and demonstrate that it can capture a large range of localized patterning in one, two, and three dimensions and that this framework can be applied to systems involving more than two species. Furthermore, the boundary conditions proposed lead to more symmetrical patterns on the interior of the domain and plausibly capture more realistic boundaries in developmental systems. Finally, we show that these isolated patterns are more robust to fluctuations in initial conditions and that they allow intriguing possibilities of pattern selection via geometry, distinct from known selection mechanisms.

Department of Mathematics FNSPE Czech Technical University Prague Trojanova 13 120 00 Praha Czech Republic

The Roslin Institute and Royal School of Veterinary Studies University of Edinburgh Easter Bush Campus Midlothian EH25 9RG UK

Wolfson Centre for Mathematical Biology Mathematical Institute University of Oxford Andrew Wiles Building Radcliffe Observatory Quarter Woodstock Road Oxford OX2 6GG UK

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Modern perspectives on near-equilibrium analysis of Turing systems