\\mathbb{Z}^n$. We show under fairly general conditions that the non-uniform

Physarum dynamics \\[ \\dot{x}_e = a_e(x,t) \\left(|q_e| -- x_e\\right) \\] converges

to the optimum solution $x^*$ of the weighted basis pursuit problem minimize

$c^T x$ subject to $A f = b$ and $|f| \\le x$. Here, $f$ and $x$ are $m$-vectors

of real variables, $q$ minimizes the energy $\\sum_e (c_e/x_e) q_e^2$ subject to

the constraints $A q = b$ and $\\mathrm{supp}(q) \\subseteq \\mathrm{supp}(x)$,

and $a_e(x,t) > 0$ is the reactivity of edge $e$ to the difference $|q_e| -

x_e$ at time $t$ and in state $x$. Previously convergence was only shown for

the uniform case $a_e(x,t) = 1$ for all $e$, $x$, and $t$. We also show

convergence for the dynamics \\[ \\dot{x}_e = x_e \\cdot \\left( g_e

\\left(\\frac{|q_e|}{x_e}\\right) -- 1\\right),\\] where $g_e$ is an increasing

differentiable function with $g_e(1) = 1$. Previously convergence was only

shown for the special case of the shortest path problem on a graph consisting

of two nodes connected by parallel edges.

},\n}\n'