Models of intuitionistic linear logic that reflect the resource interpretation

I think the Petri net semantics for linear logic probably best captures this intuition that it is a logic about resource manipulation. The idea is that Petri nets model the movement of tokens (i.e., resources) through a network. Here's the money quote from Lokhorst 1997.

Petri nets are models of dynamic processes in terms of types of resources, represent ed by places which can hold to arbitrary nonnegative multiplicity, and how these resources are consumed or produced by actions, represented by transitions. They are usually described in terms of multisets.

Lokhorst, Gert-Jan C. 1997. Deontic linear logic with Petri net semantics. Technical report, FICT (Center for the Philosophy of Information and Communication Technology). Rotterdam. http://homepages.ipact.nl/~lokhorst/deopetri.pdf.