Energetic constraints shape the diversity of feasible ecological networks

The relationship between energy supply and biodiversity is a longstanding question in ecology. Although a monotonic increase in diversity with energy availability is often assumed, unimodal species-energy relationships have been widely documented across ecosystems, and their origin from first principles remains unclear. Here, we develop a geometric framework that recasts ecological feasibility in explicitly energetic terms. By treating total energy supply as a system-level constraint on an energy-based network model, we define nested feasibility domains in the space of energy capture rates and quantify feasibility probabilities as their volume ratios. We show that the probability of initializing a feasible network increases monotonically and saturates with energy supply, whereas the probability of sustaining steady-state biomass follows a unimodal relationship-revealing a bounded energetic window within which network maturation is most likely. Extending this analysis to all candidate subcommunities via feasibility partitions, we find that different community sizes are most feasible at different energy levels, and that average diversity itself peaks at intermediate supply. Together, these results suggest that energetic constraints determine the diversity of ecological networks not through energy scarcity alone, but through the geometric interplay between external energy supply and internal energy exchange.