Unraveling emergent network indeterminacy in complex ecosystems: A random matrix approach.

Indeterminacy of ecological networks-the unpredictability of ecosystem responses to persistent perturbations-is an emergent property of indirect effects a species has on another through interaction chains. Thus, numerous indirect pathways in large, complex ecological communities could make forecasting the long-term outcomes of environmental changes challenging. However, a comprehensive understanding of ecological structures causing indeterminacy has not yet been reached. Here, using random matrix theory (RMT), we provide mathematical criteria determining whether network indeterminacy emerges across various ecological communities. Our analytical and simulation results show that indeterminacy intricately depends on the characteristics of species interaction. Specifically, contrary to conventional wisdom, network indeterminacy is unlikely to emerge in large competitive and mutualistic communities, while it is a common feature in top-down regulated food webs. Furthermore, we found that predictable and unpredictable perturbations can coexist in the same community and that indeterminate responses to environmental changes arise more frequently in networks where predator-prey relationships predominate than competitive and mutualistic ones. These findings highlight the importance of elucidating direct species relationships and analyzing them with an RMT perspective on two fronts: It aids in 1) determining whether the network's responses to environmental changes are ultimately indeterminate and 2) identifying the types of perturbations causing less predictable outcomes in a complex ecosystem. In addition, our framework should apply to the inverse problem of network identification, i.e., determining whether observed responses to sustained perturbations can reconstruct their proximate causalities, potentially impacting other fields such as microbial and medical sciences.