  Self-sustained dynamical phases of living matter can exhibit remarkable
similarities over a wide range of scales, from mesoscopic vortex structures in
microbial suspensions and motility assays of biopolymers to turbulent
large-scale instabilities in flocks of birds or schools of fish. Here, we argue
that, in many cases, the phenomenology of such active states can be efficiently
described in terms of fourth- and higher-order partial differential equations.
Structural transitions in these models can be interpreted as Landau-type
kinematic transitions in Fourier (wavenumber) space, suggesting that
microscopically different biological systems can share universal
long-wavelength features. This general idea is illustrated through numerical
simulations for two classes of continuum models for incompressible active
fluids: a Swift-Hohenberg-type scalar field theory, and a minimal vector model
that extends the classical Toner-Tu theory and appears to be a promising
candidate for the quantitive description of dense bacterial suspensions. We
also discuss briefly how microscopic symmetry-breaking mechanisms can enter
macroscopic continuum descriptions of collective microbial motion near surfaces
and conclude by outlining future applications.
