The fractional Keller-Segel model

Abstract

The Keller-Segel model is a system of partial differential equations modelling chemotactic aggregation in cellular systems. This model has blowing up solutions for large enough initial conditions in dimensions d >= 2, but all the solutions are regular in one dimension; a mathematical fact that crucially affects the patterns that can form in the biological system. One of the strongest assumptions of the Keller-Segel model is the diffusive character of the cellular motion, known to be false in many situations. We extend this model to such situations in which the cellular dispersal is better modelled by a fractional operator. We analyze this fractional Keller-Segel model and find that all solutions are again globally bounded in time in one dimension. This fact shows the robustness of the main biological conclusions obtained from the Keller-Segel model.

References in corpus (3)

• Extinction in population dynamics
• Collapsing Bacterial Cylinders
• Chemotactic Collapse and Mesenchymal Morphogenesis

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• Blowup of solutions to a diffusive aggregation model
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• Critical Keller-Segel meets Burgers on ${\mathbb{S}}^1$: large-time smooth solutions
• Spikes and diffusion waves in one-dimensional model of chemotaxis
• A Finite-Volume Scheme for Fractional Diffusion on Bounded Domains
• The Optimal Temporal Decay Estimates for the Fractional Power Dissipative Equation in Negative Besov Spaces
• Mathematical modeling and analysis for the chemotactic diffusion in porous media with incompressible Navier-Stokes equations over bounded domain