Finding an equilibrium state of the traffic assignment plays a significant role in the design of transportation networks. We adapt the path finding mathematical model of slime mold Physarum polycephalum to solve the traffic equilibrium assignment problem. We make three contributions in this paper. First, we propose a generalized Physarum model to solve the shortest path problem in directed and asymmetric graphs. Second, we extend it further to resolve the network design problem with multiple source nodes and sink nodes. At last, we demonstrate that the Physarum solver converges to the user-optimized (Wardrop) equilibrium by dynamically updating the costs of links in the network. In addition, convergence of the developed algorithm is proved. Numerical examples are used to demonstrate the efficiency of the proposed algorithm. The superiority of the proposed algorithm is demonstrated in comparison with several other algorithms, including the Frank-Wolfe algorithm, conjugate Frank-Wolfe algorithm, biconjugate Frank-Wolfe algorithm, and gradient projection algorithm.