  Understanding 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 work, 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 characterised 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
proven 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 both in two and three spatial dimensions.
