Harmonic trophic potentials and memory effects in ecological dynamics in resource-limited ecosystems through the Generalized Lotka–Volterra model with biomass constraints

This study investigates the thermodynamic and mathematical foundations of trophic dynamics in ecological systems, focusing on the Generalized Lotka-Volterra (GLV) model to analyze prey/predator interactions under resource constraints. We introduce a framework where species' contributions are governed by mean biomass and relative trophic strength, subject to physiological and ecological bounds. The total biomass function and abiotic reservoir are defined to quantify the balance between biotic and abiotic resources, with the functional capture of net energetic flux. Key results demonstrate that dynamic equilibrium is achievable only in trivial cases (extinction) when measures of historical interactions are absolutely continuous. In contrast, memory-free models, where species interactions are instantaneous, permit nontrivial equilibria under harmonic conditions, for the effective trophic potential harmonicity implies local equilibrium, ensuring smooth integration of species into the trophic network without abrupt disruptions. A two-species system (in a bidimensional case) illustrates phase space dynamics: Stability analysis highlights the sensitivity of ecosystems to competitive dynamics and resource influx, with structured ecological inputs that promote equilibrium. The study bridges theoretical ecology and dynamical systems, offering insight into the stability of trophic networks under varying constraints and setting an effective machinery that establishes a clear connection between classical GLV models and mass balance treatments in ecosystems.