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

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.

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

“…It was shown in [12] that the SKS System (1) can be further simplified provided βd 1 /α = ε 1. In this case, one may reduce SKS System (1) to the form:…”

“…Of course, the system derived is still nonlinear; however, one admits infinite-dimensional Lie algebra of invariance generated by the operators [12]:…”

“…In fact, having the specified initial profiles in (10), we find the initial condition for Equation (12) as follows:…”

“…Because the result depends essentially on geometry of the domain Ω, where BVP in question is defined, one needs to examine different cases. In [12], the simplest case, when Ω is a half-plane, was under study. In principle, Ω can be an arbitrary (bounded or unbounded) domain with smooth boundaries.…”

“…In this work, we continue study of a simplified (1+2)-dimensional Keller-Segel model, initiated in the first part of this work [12]. This model is a particular case of the classical Keller-Segel model [13,14] used for modeling a wide range of processes in biology, ecology, medicine, etc.…”