Stability and Complexity of Ecological Communities with Energy Landscapes

A fundamental issue in ecology is to understand the interaction among species, population cycles, and their implications for the stability and complexity of ecological communities. This can be explored mathematically through the dynamics of mathematical models, such as the generalized Lotka–Volterra system or the community matrix. However, as the number of species increases, the dynamical behaviors of these models become increasingly challenging to predict. To address possible mechanisms underlying the interaction among species, we first propose the hypothesis that ecological communities can be described by a Hamiltonian energy landscape (HEL) framework. This framework captures energy exchange in food webs, growth, and reproduction, based on the principle of least action. By applying this framework to an ecological community modeled by a [Formula: see text]-dimensional generalized Lotka–Volterra system, we derive the expression [Formula: see text] of the HEL and prove that all species interact in pairs and they have only three types of interrelationships: cooperation, competition, and predator-prey for any [Formula: see text] mathematically. Notably, predator-prey interactions lead to population cycles. Hence the ecological community consisting of [Formula: see text] independent such predator-prey systems can maintain the stability and species diversity. Further, we connect this stable community to real ecological communities by HEL, which is an additional small periodic perturbation of [Formula: see text] due to the natural fluctuations of the climate. Using KAM theory, we prove that populations of [Formula: see text] species in the perturbed Hamiltonian model exhibit quasi-periodic coexistence almost everywhere. When [Formula: see text], Arnold diffusion may occur between two invariant Diophantine tori. This theoretical finding indicates that weak interactions among species in communities are likely to maintain the stability of the community in a probabilistic sense. However, when [Formula: see text], there remains a risk of the community transition from stability to May–Wigner instability.