  We consider a system of nonlinear ordinary differential equations for the
solution of linear programming (LP) problems that was first proposed in the
mathematical biology literature as a model for the foraging behavior of
acellular slime mold Physarum polycephalum, and more recently considered as a
method to solve LPs. We study the convergence time of the continuous Physarum
dynamics in the context of the linear programming problem, and derive a new
time bound to approximate optimality that depends on the relative entropy
between projected versions of the optimal point and of the initial point. The
bound scales logarithmically with the LP cost coefficients and linearly with
the inverse of the relative accuracy, establishing the efficiency of the
dynamics for arbitrary LP instances with positive costs.
