Optimization hardness constrains ecological transients

Living systems operate far from equilibrium, yet few general frameworks provide global bounds on biological transients. In high-dimensional biological networks like ecosystems, long transients arise from the separate timescales of interactions within versus among subcommunities. Here, we use tools from computational complexity theory to frame equilibration in complex ecosystems as the process of solving an analogue optimization problem. We show that functional redundancies among species in an ecosystem produce difficult, ill-conditioned problems, which physically manifest as transient chaos. We find that the recent success of dimensionality reduction methods in describing ecological dynamics arises due to preconditioning, in which fast relaxation decouples from slow solving timescales. In evolutionary simulations, we show that selection for steady-state species diversity produces ill-conditioning, an effect quantifiable using scaling relations originally derived for numerical analysis of complex optimization problems. Our results demonstrate the physical toll of computational constraints on biological dynamics.Author summary: Distinct species can serve overlapping functions in complex ecosystems. For example, multiple cyanobacteria species within a microbial mat might serve to fix nitrogen. Here, we show mathematically that such functional redundancy can arbitrarily delay an ecosystem’s approach to equilibrium. We draw a mathematical analogy between this difficult equilibration process, and the complexity of computer algorithms like matrix inversion or numerical optimization. We show that this computational complexity manifests as a transient chaos in an ecosystem’s dynamics, allowing us to develop scaling laws for the expected length of transients in complex ecosystems. Transient chaos also produces strong sensitivity on the duration and route that the system takes towards equilibrium, affecting the ecosystem’s response to perturbations. Our results highlight the physical implications of computational complexity for large biological networks.