The shape of patterns to come: from initial formation to long-term evolution

Many studies focus on the emergence and development of rhythmic landscape patterns. In this contribution we explore the different behaviors found as patterns evolve; the trajectories that patterns exhibit as they transit from infinitesimal-amplitude perturbation to a statistically steady state (or in some cases to continued statistical evolution). The variety of behaviors observed, either through field and laboratory experiments or numerical modeling, can be reduced to four classes: (a) simple stabilization where predictions based on the initial growth of small perturbations corresponds with the characteristics of patterns observed in nature; (b) significant pattern coarsening en route to saturated wavelength, where non-linear interactions between finite-amplitude pattern elements change the geometric properties of a pattern as it approaches steady-state; (c) perpetual coarsening where the wavelength associated with the emerging pattern continues to grow over time and is only limited by physical boundaries or external constrains; (d) slow evolution toward a different attractor, a novel behavior observed in numerical modeling that involves profound temporal changes in pattern characteristics. Within these classes we also observe generalizable non-linear behaviors: dependence on initial conditions, the emergence of pattern-scale variables such as pattern defects, and the presence of multiple stable states. Predicting the shape of patterns to come remains a challenge - one that we suggest requires a range of modeling approaches to address both initial instabilities and the emergent properties of evolving patterns, which involve disparate forms of non-linear interactions. Consideration of generic system behaviors at the pattern scale could enhance future pattern formation studies, facilitating appropriate pairings of analysis approaches and pattern-evolution modes. Copyright (c) 2013 John Wiley & Sons, Ltd.