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

Abstract: This paper presents an energetic derivation of a class of multi-species reaction-diffusion systems incorporating various functional responses, with a focus on and application to ecological models. Starting point is a closed reaction network for which we apply the Energetic Variational Approach (EnVarA) to derive the corresponding reaction-diffusion system. This framework captures both diffusion and nonlinear reaction kinetics, and recovers Lotka-Volterra-type dynamics with species-dependent interactions. Moreover, we find an open subsystem embedded into the larger closed system, that contains the physical and ecological important quantities. By analyzing different asymptotic regimes in the reaction parameters of the resulting system, we formally derive classical Holling type I, II, and III functional responses. To rigorously justify these reductions and their dynamical properties, we apply the generalized geometric singular perturbation theory (GSPT) for PDEs and prove the existence of an attracting and invariant slow manifolds near the critical manifold of the reduced system. Our analysis not only bridges thermodynamic consistency with ecological modeling but also provides a robust framework to study the dynamics of slow manifolds in multi-scale reaction-diffusion systems.