Theory of nonlinear convolution with the Keller-Segel equation

Abstract

ABSTRACT A theory of Keller-Segel equation by using nonlinear convolution is presented. Attention is paid to the case where bacteria contain ions in their composition so that one can derive electric relations in contrast to the Navier-Stokes coupled equations. The mathematical methodology presented in this paper is entirely based on integrals of convolution. In concrete the schemes of Wiener series and the Feynman propagator are used. Finally the resulting distributions of substrate and bacteria density would obey possible scenarios of electrical interactions because the ions in their compositions.