{'fraction_non_alphanumeric': 0.014191106906338695, 'fraction_numerical': 0.0, 'mean_word_length': 4.845303867403315, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Theory of pinned fronts.

The properties of a front between two different phases in the presence of a smoothly inhomogeneous external field that takes its critical value at the crossing point is analyzed. Two generic scenarios are studied. In the first, the system admits a bistable solution and the external field governs the rate in which one phase invades the other. The second mechanism corresponds to a continuous transition that, in the case of reactive systems, takes the form of a transcritical bifurcation at the crossing point. We solve for the front shape and for the response of competitive fronts to external noise, showing that static properties and also some of the dynamical features cannot discriminate between the two scenarios. A reliable indicator turns out to be the fluctuation statistics. These take a Gaussian form in the bifurcation case and a double-peaked shape in a bistable system. Our results are discussed in the context of biological processes, such as species and communities dynamics in the presence of a resource gradient.