Failure to the shortest path decision of an adaptive transport network with double edges in Plasmodium system (Q960316)

Summary: We study a mathematical model describing behaviour of Physarum polycephalum. 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 [cf. \textit{T. Miyaji, I. Ohnishi}, Int. J. Pure Appl. Math. 47, No. 3, 353--369 (2008; Zbl 1235.92004)]. 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.