minimize $c^T x$ subject to $Ax = b$ and $x \\ge 0$ for a positive cost vector

$c$. The directed Physarum dynamics evolves a positive vector $x$ according to

the dynamics $\\dot{x} = q(x) -- x$. Here $q(x)$ is the solution to $Af = b$ that

minimizes the "energy" $\\sum_i c_i f_i^2/x_i$.

In this paper, we study the non-uniform directed dynamics $\\dot{x} = D(q(x) -

x)$, where $D$ is a positive diagonal matrix. The non-uniform dynamics is more

complex than the uniform dynamics (with $D$ being the identity matrix), as it

allows each component of $x$ to react with different speed to the differences

between $q(x)$ and $x$. Our contribution is to show that the non-uniform

directed dynamics solves positive linear programs.

},\n}\n'