01861nas a2200157   4500008004100000245013500041210006900176300001300245490000800258520128900266100002101555700002101576700001701597700001901614856007001633       2018                        eng d00aRevealing new dynamical patterns in a reaction&\#x2013;diffusion model with cyclic competition via a novel computational framework0 aRevealing new dynamical patterns in a reactionx2013diffusion mod  a201706080 v4743 aUnderstanding how patterns and travelling waves form in chemical and biological reaction–diffusion models is an area which has been widely researched, yet is still experiencing fast development. Surprisingly enough, we still do not have a clear understanding about all possible types of dynamical regimes in classical reaction–diffusion models, such as Lotka–Volterra competition models with spatial dependence. In this study, we demonstrate some new types of wave propagation and pattern formation in a classical three species cyclic competition model with spatial diffusion, which have been so far missed in the literature. These new patterns are characterized by a high regularity in space, but are different from patterns previously known to exist in reaction–diffusion models, and may have important applications in improving our understanding of biological pattern formation and invasion theory. Finding these new patterns is made technically possible by using an automatic adaptive finite element method driven by a novel a posteriori error estimate which is proved to provide a reliable bound for the error of the numerical method. We demonstrate how this numerical framework allows us to easily explore the dynamical patterns in both two and three spatial dimensions.1 aCangiani, Andrea1 aGeorgoulis, E.H.1 aMorozov, Yu.1 aSutton, O., J.  uhttps://royalsocietypublishing.org/doi/abs/10.1098/rspa.2017.0608