Entropy production in stochastic Riemannian geometries: Jacobi metric observables in chemical ecology

We consider formal stochastic diffusion theories associated to a general class of underlying deterministic theories of chemical ecology, coupled to external forces and satisfying an optimality principal. The external forces (e.g. pollution etc.) are introduced into the diffusion theories in a non- traditional way via the use of Jacobi metrics. Observable properties of the resulting stationary probability densities are considered, particularly in relation to an energy parameter representing the strength of coupling to the external forces, and entropy.