We study the connections between Physarum Dynamics and Dynamic Monge Kantorovich (DMK) Optimal Transport algorithms for the solution of Basis Pursuit problems. We show the equivalence between these two models and unveil their dynamic character by showing existence and uniqueness of the solution for all times and constructing a Lyapunov functional with negative Lie-derivative that drives the large-time convergence. We propose a discretization of the equation by means of a combination of implicit time-stepping and Newton method yielding an efficient and robust method for the solution of general basis pursuit problems. Finally, we propose a simple modification to the dynamic equation that can be interpreted as a gradient flow system for the Lyapunov functional. Several numerical experiments run on literature benchmark problems are used to show the accuracy, efficiency, and robustness of the proposed method.