Towards a stationary Monge-Kantorovich dynamics: the Physarum Polycephalum experience

In this work we propose an extension to the continuous setting of a model describing the dynamics of 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. This model has been shown to be equivalent to a problem of “optimal transportation” on graphs. We propose an extension that abandons the graph structure and moves to a continuous domain. The new model couples an elliptic diffusion equation enforcing PP density balance with an ordinary differential equation governing the flow dynamics. We conjecture that the new system of equations presents a time-asymptotic equilibrium and that such an equilibrium point is precisely the solution of Monge–Kantorovich partial differential equations governing optimal transportation problems. To support this conjecture, we analyze the proposed model by recasting it into an infinite-dimensional dynamical system. We are then able to show well-posedness of the proposed model for sufficiently small times under the hypotheses of H ̈older continuous diffusion coefficients and essentially bounded forcing functions. Numerical results obtained with a simple fixed- point iteration combining P1/P0 finite elements with backward Euler time stepping show that the approximate solution of our formulation of the transportation problem converges at large times to an equilibrium configuration that well compares with the numerical solution of the Monge–Kantorovich equations