Energetic Derivation and Geometric Reduction of Reaction–Diffusion Systems With Holling-Type Functional Responses

Abstract

We derive and reduce a thermodynamically consistent class of ecological reaction–diffusion models with Holling-type functional responses. Starting from an open reaction network involving water, two biomass states, and reservoir species, we embed the system into a closed reversible network and use the Energetic Variational Approach (EnVarA) to obtain a reaction–diffusion system with entropy structure, detailed-balance equilibria, and mass-action kinetics generated by a logarithmic reaction dissipation. We then pass formally to an open ecological subsystem via a sustained-reservoir limit and identify fast-reaction scalings that lead to Holling type I, II, and III responses. The type I and type III reductions are presented as formal asymptotic limits, together with the estimates needed for rigorous justification. For the Holling type II regime, we apply geometric singular perturbation theory for fast-reaction PDEs. Under normal hyperbolicity, spectral splitting, and bounded-chart assumptions, the singularly perturbed system admits a locally invariant attracting slow manifold near the critical manifold, and the semiflow on this manifold converges to the reduced Holling type II dynamics. The results connect thermodynamic closure, energetic variational modeling, functional-response reductions, and slow-manifold dynamics in multiscale reaction–diffusion systems.