  In this work we study and expand a model describing the dynamics of a
unicellular slime mold, Physarum Polycephalum (PP), which was proposed to
simulate the ability of PP to find the shortest path connecting two food
sources in a maze. The original model describes the dynamics of the slime mold
on a finite dimensional planar graph using a pipe-flow analogy whereby mass
transfer occurs because of pressure differences with a conductivity coefficient
that varies with the flow intensity. We propose an extension of this model that
abandons the graph structure and moves to a continuous domain. Numerical
evidence, shows that the model is capable of describing the slime mold dynamics
also for large times, accurately reproducing the PP behavior.
  A notable result related to the original model is that it is equivalent to an
optimal transportation problem over the graph as time tends to infinity. In our
case, we can only conjecture that our extension presents a time-asymptotic
equilibrium. This equilibrium point is precisely the solution of the
Monge-Kantorovich (MK) equations at the basis of the PDE formulation of optimal
transportation problems. Numerical results obtained with our approach, which
combines P1 Finite Elements with forward Euler time stepping, show that the
approximate solution converges at large times to an equilibrium configuration
that well compares with the numerical solution of the MK-equations.
