A (1+2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions

Maksym Didovych

doi:10.3390/sym7031463

Abstract

This research is a natural continuation of the recent paper “Exact solutions of the simplified Keller–Segel model” (Commun Nonlinear Sci Numer Simulat 2013, 18, 2960–2971). It is shown that a (1+2)-dimensional Keller–Segel type system is invariant with respect infinite-dimensional Lie algebra. All possible maximal algebras of invariance of the Neumann boundary value problems based on the Keller–Segel system in question were found. Lie symmetry operators are used for constructing exact solutions of some boundary value problems. Moreover, it is proved that the boundary value problem for the (1+1)-dimensional Keller–Segel system with specific boundary conditions can be linearized and solved in an explicit form.

Highlights

In quite a similar way as it was done for boundary-value problems (BVPs) (6) we have proved that only operators P1 and P3 are the Lie symmetry operators of BVP (20) and (21)
In this work we studied a simplified version of (1+2)-dimensional Keller–Segel model
It is well-known that Keller–Segel model is widely used for modeling a wide range of processes in biology and medicine one is extensively examined by means of different mathematical techniques

Summary

Introduction

Segel published a remarkable papers [1,2], which they constructed the mathematical model for describing the chemotactic interaction of amoebae mediated by the chemical (acrasin) in Nowadays their model is called the Keller–Segel model and used for modeling a wide range of processes in biology and medicine. The corresponding Neumann boundary-value problems admit infinite-dimensional Lie algebras. Using these algebras we find exact solutions for (1+1) and (1+2)-dimensional BVPs. This research is a natural continuation of the recent paper [9].

Lie Symmetry of the Neumann Boundary-Value Problem

Exact Solutions of Neumann Problems

Conclusions