# Fractional Patlak-Keller-Segel Equations for Chemotactic Superdiffusion

@article{EstradaRodriguez2017FractionalPE, title={Fractional Patlak-Keller-Segel Equations for Chemotactic Superdiffusion}, author={Gissell Estrada-Rodriguez and Heiko Gimperlein and Kevin J. Painter}, journal={SIAM J. Appl. Math.}, year={2017}, volume={78}, pages={1155-1173} }

The long range movement of certain organisms in the presence of a chemoattractant can be governed by long distance runs, according to an approximate Levy 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…

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## References

SHOWING 1-10 OF 62 REFERENCES

### Biased random walk models for chemotaxis and related diffusion approximations

- MathematicsJournal of mathematical biology
- 1980

Stochastic models of biased random walk, which describe the behavior of chemosensitive cells like bacteria or leukocytes in the gradient of a chemotactic factor, are discussed, which are derived from certain biological hypotheses on the background of related experimental observations.

### ABOUT THE KINETIC DESCRIPTION OF FRACTIONAL DIFFUSION EQUATIONS MODELING CHEMOTAXIS

- Mathematics
- 2016

In this paper, we are interested in the microscopic description of fractional diffusion chemotactic models. We will use the kinetic framework of collisional equations having a heavy-tailed…

### The Diffusion Limit of Transport Equations II: Chemotaxis Equations

- MathematicsSIAM J. Appl. Math.
- 2002

The diffusion-limit expansion of transport equations developed earlier are used to study the limiting equation under a variety of external biases imposed on the motion and it is shown that the classical chemotaxis equation---which is called the Patlak--Keller--Segel--Alt (PKSA) equation---arises only when the bias is sufficiently small.

### From Individual to Collective Behavior of Unicellular Organisms: Recent Results and Open Problems

- Biology
- 2009

Recent progress on how to justify partial differential equations from microscopic descriptions of cells, and how to relate the macroscopic quantities in these PDEs to individual‐level parameters are summarized.

### A user’s guide to PDE models for chemotaxis

- Computer ScienceJournal of mathematical biology
- 2009

This paper explores in detail a number of variations of the original Keller–Segel model of chemotaxis from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form.

### Derivation of the bacterial run-and-tumble kinetic equation from a model with biochemical pathway

- BiologyJournal of mathematical biology
- 2016

This program shows how the first class of equations can be derived from the second class with molecular content after appropriate rescaling, and how Randomness of receptor methylation events can be included and used to compute the tumbling frequency in presence of such a noise.

### Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

- Mathematics
- 2015

This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller–Segel model and its subsequent modifications, which, in several…

### A Theoretical Study of Receptor Mechanisms in Bacterial Chemotaxis

- Biology
- 1977

Equations are formulated that link the probability of a bacterium suddenly changing its direction with temporal change in the number of attractant molecules that are bound to the bacteria’s receptor…

### Models of dispersal in biological systems

- MathematicsJournal of mathematical biology
- 1988

Two stochastic processes that model the major modes of dispersal that are observed in nature are introduced, and explicit expressions for the mean squared displacement and other experimentally observable quantities are derived.

### E. coli superdiffusion and chemotaxis-search strategy, precision, and motility.

- BiologyBiophysical journal
- 2009