Fractional Patlak--Keller--Segel Equations for Chemotactic Superdiffusion

Abstract

The long range movement of certain organisms in the presence of a chemoattractant can be governed by long distance runs, according to an approximate Lévy distribution. This article clarifies the form of biologically relevant model equations: We derive Patlak-Keller-Segel-like equations involving nonlocal, fractional Laplacians from a microscopic model for cell movement. Starting from a power-law distribution of run times, we derive a kinetic equation in which the collision term takes into account the long range behaviour of the individuals. A fractional chemotactic equation is obtained in a biologically relevant regime. Apart from chemotaxis, our work has implications for biological diffusion in numerous processes.

Interacting Particles with Lévy Strategies: Limits of Transport Equations for Swarm Robotic Systems

“…When only alignment is considered, ζ = 0, this reduces to the Chapman-Enskog expansionp 0 = Φ(Λ · θ)û obtained in [14,22], while for run and tumble processes, ζ = 1, one recovers the leading term of the eigenfunction expansionp 0 = Tp 0 = |S| −1 (û + nθ ·ŵ) [15].…”

“…A series of studies dating to Patlak [34] has generated solid understanding on how microscopic detail translates into a diffusion-advection type equation [4,48] for random walks subject to an external bias and interactions. Recent work has made progress towards nonlocal PDE descriptions of Lévy movement [15,35,44]. Emergence of superdiffusion without Lévy movement is discussed in [17].…”

“…While diffusive behavior dominates for long times, swarming on shorter hyperbolic time scales is not affected by the Lévy movement, so that a rich body of work such as [5] on swarming applies to Lévy robotics. Combining long range dispersal and alignment as in the kinetic equation for the macroscopic density (31), allows us to obtain either a space fractional diffusion equation for a pure nonlocal movement of the individuals (see [15]), or a Vicsek-type equation for the case of pure alignment [10,47].…”

Mathematical models for chemotaxis and their applications in self-organisation phenomena

“…In bacteria with well understood signalling and motion closely approximated by a velocity-jump process, such as E. coli, one can even link parameters and functions characterising molecular signalling and motor control to the parameters and functions that define diffusive/chemotactic sensitivity terms in a PKS model . Moreover, novel and interesting variations can emerge, such as the "perpendicular gradient following" that arises from swimming biases (Xue and Othmer, 2009), fractional operator terms due to non-Poisson type turning rate distributions (Estrada-Rodriguez et al, 2017) or "flux-limited" forms (Perthame et al, 2018). Noteworthy, the above derivations rely on ignoring interactions and Stevens (2000) is noted for providing the first rigorous derivation of a PKS equation for a population of stochastic (weakly) interacting particles.…”

Fast and accurate algorithms for the computation of spherically symmetric nonlocal diffusion operators on lattices

“…Nonlocal models are important for their fidelity and versatility in handling a broad range of applications in material science, thermodynamics, fluid dynamics, and image processing (Silling (2000); Gilboa & Osher (2009);Estrada-Rodriguez et al (2018); Du & Tian (2018)). Mathematically, nonlocal diffusion is usually formulated through weakly singular integral operators, as in Du et al (2012).…”