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. Author SummaryWhy do many ecosystems show the highest biodiversity not where energy is most abundant, but at intermediate levels? This unimodal species-energy relationship has been documented across grasslands, wetlands, and rainforests, yet its origin from first principles has remained unclear. We approached this question by developing a simplified model that treats ecological networks as energy-processing systems. In this model, each species captures energy from the environment and exchanges it with others, and the total energy available to the network is explicitly limited. By measuring how the likelihood of species coexistence changes with energy supply within this framework, we found that while a minimum energy threshold is needed for any community to persist, too much energy can paradoxically reduce the chance of long-term coexistence. This creates a bounded energy window most favorable for community persistence. When we extended the analysis to all possible subsets of species, we found that different-sized communities are most likely to persist at different energy levels, and that overall expected diversity peaks at intermediate supply. These results suggest a possible geometric origin for why more energy does not always support more species, providing a theoretical baseline for connecting the structure of energy flow within networks to observed biodiversity patterns.