Route to chaos in multi-species ecosystems

We investigate species-rich mathematical models of ecosystems. While much of the existing literature focuses on the properties of equilibrium fixed-points, persistent dynamics (e.g., limit cycles or chaos) have also been observed, both in natural or laboratory-controlled ecosystems and in mathematical models. Here, we emphasize the emergence of limit cycles following Hopf bifurcations tuned by the variability of interspecies interaction. As this variability increases and owing to the large dimensionality of the system, limit cycles typically acquire a growing spectrum of frequencies. This often leads to the appearance of strange attractors, with a chaotic dynamics of species abundances characterized by a positive Lyapunov exponent. We observe that limit cycles and strange attractors preserve biodiversity to some extent, as they maintain dynamical stability without species extinction. We give numerical evidences that this route to chaos dominates in ecosystems with strong enough interactions and where predator-prey behavior dominates over competition and mutualism. Based on arguments from random matrix theory, we further conjecture that this scenario is generic in ecosystems with a large number of species and identify the key parameters driving it. Overall, we show that the model we consider provides a unifying framework, where a wide range of population dynamics emerge from a simple few-parameter model.