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From 08-12 August, 2022, 32 individuals participated in a workshop, Stability and Fluctuations in Complex Ecological Systems, at the Lorentz Center, located in Leiden, The Netherlands. An interdisciplinary dialogue between ecologists, mathematicians, and physicists provided a foundation of important problems to consider over the next 5-10 years. This paper outlines eight areas including (1) improving our understanding of the effect of scale, both temporal and spatial, for both deterministic and stochastic problems; (2) clarifying the different terminologies and definitions used in different scientific fields; (3) developing a comprehensive set of data analysis techniques arising from different fields but which can be used together to improve our understanding of existing data sets; (4) having theoreticians/computational scientists collaborate closely with empirical ecologists to determine what new data should be collected; (5) improving our knowledge of how to protect and/or restore ecosystems; (6) incorporating socio-economic effects into models of ecosystems; (7) improving our understanding of the role of deterministic and stochastic fluctuations; (8) studying the current state of biodiversity at the functional level, taxa level and genome level., Comment: 22 pages

The prediction of critical transitions, such as extinction events, is vitally important to preserving vulnerable populations in the face of a rapidly changing climate and continuously increasing human resource usage. Predicting such events in spatially distributed populations is challenging because of the high dimensionality of the system and the complexity of the system dynamics. Here, we reduce the dimensionality of the problem by quantifying spatial patterns via Betti numbers ($\beta_0$ and $\beta_1$), which count particular topological features in a topological space. Spatial patterns representing regions occupied by the population are analyzed in a coupled patch population model with Ricker map growth and nearest-neighbors dispersal on a two-dimensional lattice. We illustrate how Betti numbers can be used to characterize spatial patterns by type, which in turn may be used to track spatiotemporal changes via Betti number time series and characterize asymptotic dynamics of the model parameter space. En route to a global extinction event, we find that the Betti number time series of a population exhibits characteristic changes. We hope these preliminary results will be used to aide in the prediction of critical transitions in spatially extended systems. Additional applications of this technique include analysis of spatial data (e.g., GIS) and model validation., Comment: Published in Contemporary Mathematics: Dynamical Systems and Random Processes, Volume 736, 2019