Title: Failure to the shortest path decision of an adaptive transport network with double edges in Plasmodium system

Authors: Tomoyuki Miyaji, Isamu Ohnishi, Atsushi Tero, Toshiyuki Nakagaki

Addresses: Department of Mathematical and Life Sciences, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526 Japan. ' Department of Mathematical and Life Sciences, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526 Japan. ' PRESTO, JST, 4-1-8 Honcho Kawaguchi, Saitama, Japan. ' Research Institute for Electronic Science, Hokkaido University, Sapporo, Hokkaido, 060-0812 Japan

Abstract: In this paper we study a mathematical model describing behaviour of Physarum polycephalum proposed by Tero et al. (2007). In the case of linear adaptive term, it has been proved that the model must solve the shortest path problem mathematically rigorously on a general planar graph in Miyaji and Ohnishi (2007, 2008). However, in a laboratory, P. polycephalum sometimes makes a mistake, for example, when there is a |double-edge| in a graph. We study the case mathematically to show both why and how P. polycephalum makes a mistake.

Keywords: adaptive networks; parallel circuits; shortest path decision; stability analysis; phase plane analysis; connection orbit; transport networks; mathematical modelling; Physarum polycephalum; double edges; slime mould; Plasmodium system.

DOI: 10.1504/IJDSDE.2008.019683

International Journal of Dynamical Systems and Differential Equations, 2008 Vol.1 No.3, pp.210 - 219

Published online: 20 Jul 2008 *

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