Pattern formation in flocking models: A hydrodynamic description

Polar pattern formation: hydrodynamic coupling of driven filaments

How order can emerge spontaneously from a disordered system has always fascinated scientists from numerous disciplines. Especially in active systems like flocks animals, self-propelled microorganisms or the cytoskeleton, a unifying understanding of the pattern formation remains elusive. This is attributed to the inherent complexity of most model systems that prevents a thorough identification o...

Hydrodynamic Limit of the Kinetic Cucker-smale Flocking Model

The hydrodynamic limit of a kinetic Cucker-Smale flocking model is investigated. The starting point is the model considered in [17], which in addition to free-transport of individuals and a standard Cucker-Smale alignment operator, includes Brownian noise and strong local alignment. The latter was derived in [18] as the singular limit of an alignment operator first introduced by Motsch and Tadm...

Models of Liesegang pattern formation

In this article different mathematical models of the Liesegang phenomenon are exhibited. The main principles of modeling are discussed such as supersaturation theory, sol coagulation and phase separation, which describe the phenomenon using different steps and mechanism beyond the simple reaction scheme. We discuss whether the underlying numerical simulations are able to reproduce several empir...

A Critical Review of Flocking Models

From Particle to Kinetic and Hydrodynamic Descriptions of Flocking

We discuss the Cucker-Smale’s (C-S) particle model for flocking, deriving precise conditions for flocking to occur when pairwise interactions are sufficiently strong long range. We then derive a Vlasov-type kinetic model for the C-S particle model and prove it exhibits time-asymptotic flocking behavior for arbitrary compactly supported initial data. Finally, we introduce a hydrodynamic descript...