  We present two results on slime mold computations. In wet-lab experiments
(Nature'00) by Nakagaki et al. the slime mold Physarum polycephalum
demonstrated its ability to solve shortest path problems. Biologists proposed a
mathematical model, a system of differential equations, for the slime's
adaption process (J. Theoretical Biology'07). It was shown that the process
convergences to the shortest path (J. Theoretical Biology'12) for all graphs.
We show that the dynamics actually converges for a much wider class of
problems, namely undirected linear programs with a non-negative cost vector.
  Combinatorial optimization researchers took the dynamics describing slime
behavior as an inspiration for an optimization method and showed that its
discretization can $\varepsilon$-approximately solve linear programs with
positive cost vector (ITCS'16). Their analysis requires a feasible starting
point, a step size depending linearly on $\varepsilon$, and a number of steps
with quartic dependence on $\mathrm{opt}/(\varepsilon\Phi)$, where $\Phi$ is
the difference between the smallest cost of a non-optimal basic feasible
solution and the optimal cost ($\mathrm{opt}$).
  We give a refined analysis showing that the dynamics initialized with any
strongly dominating point converges to the set of optimal solutions. Moreover,
we strengthen the convergence rate bounds and prove that the step size is
independent of $\varepsilon$, and the number of steps depends logarithmically
on $1/\varepsilon$ and quadratically on $\mathrm{opt}/\Phi$.
