ICTS

In this class, we will cover basic ecological models on population growth dynamics and saturation, including logistic growth and the Allee effect (cooperative growth). We will see how these models can capture the dynamics of microbial populations, which in turn are convenient experimental models to test hypotheses from ecological theory.

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This session will be mostly practical. Within the framework of the 2-species LV model, we will use nullcline analysis to predict the outcome of competition in different situations. These include cases in which either or both competitors are subject to cooperative growth, which can lead to alternative stable states that emerge in (2-species) co-culture, but that are absent in any of the (1-species) monocultures.

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1. basics on RV (probability, moments, mgf, continuous RVs)
2. summing RVs and central limit theorem
3. sampling (binomial, poisson)
4. intro on stochastic processes
5. random walk
6. birth-death process (basics and exctintiion)
7. birth-death-migration process (stationary probability)

One important interdisciplinary approach of ecological systems is the "complexity" perspective, which is represented by many lectures in this school. This lecture covers complementary ideas about networks of species interactions, their structures, and the use of randomness to understand their general features.

1. Phenomenology of community variability (intro, SAD)
2. Phenomenology in microbial communities (minimal patterns)
3. demographic stochasicity and migrations (birth-death-migration)
4. flavours of neutral theory (assumption, predictions)
5. continuous limit of birth-death-migration process (stationary and dynamic solution)
6. when NT works and fails

Another important interdisciplinary approach in ecology is the search for large-scale or even "universal" empirical patterns with simple explanations, such as distribution of energy and biomass across the world and across various types of organisms (plants, herbivores, carnivores...). This lecture will present some insights and limits of simple deterministic patterns constrained by biological mechanisms.

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Ecology is a very fragmented field with a variety of scientific questions and cultures. Toward the end of the school, I will present a personal perspective on what red threads run through the whole field and when the connection can benefit or not from being made in a mathematical language.

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Microbial communities assemble and thrive in strongly fluctuating boom-and-bust environments by adopting distinct metabolic strategies to consume resources. While much attention has been given to understanding these strategies in isolated species, their ecological implications in complex communities remain poorly understood. Here, we combine theoretical and computational frameworks to investigate the assembly and ecological properties of microbial communities with diauxic (sequential) and co-utilization strategies.
We show that diauxic microbial communities, where species sequentially utilize resources, spontaneously develop complementary resource preferences during assembly. This complementarity arises because sequential utilization disproportionately relies on the top-choice resource for growth, leading to intuitive ecological partitioning. A geometric approach to analyzing these serially diluted communities further explains emergent patterns, such as the absence of species preferring suboptimal resources for growth.
Comparing sequential and co-utilization strategies, we find that sequential utilizers dominate in species-rich, high-competition communities, leveraging their resilience to fluctuating resource ratios. Their ecological advantage lies in growth rate distributions characterized by wider upper tails, despite lower averages, enabling efficient niche packing and structural stability. Conversely, co-utilizers thrive in low-diversity communities, benefiting from consistently higher average growth rates.
Our work provides a unified explanation for the coexistence of sequential and co-utilizing strategies in natural ecosystems and predicts patterns of community assembly shaped by metabolic strategies. These findings offer testable hypotheses for understanding the dynamics of microbial communities in natural and synthetic environments.

I discuss how the interplay between non-linearity and temporal variation allows species coexistence.

Achieving precise control over microbial community composition is critical for applications ranging from bioremediation to human health but remains challenging due to the complexity of microbial interactions and resource variability. This work presents a framework for the bottom-up engineering of microbial communities by leveraging resource dynamics and temporal niches in fluctuating environments.
This approach assembles and maintains microbial "dream teams" - small, defined communities with desired properties—using serial dilution experiments. By treating resource concentrations as dynamic "control knobs," it enables stable coexistence and precise tuning of species abundances. Theoretical models, informed by experimental data from natural and synthetic microcosms, incorporate ecological and metabolic parameters, including species-specific time lags and dilution factors, to identify resource combinations that maximize community stability and diversity.
I will also describe how to engineer a multi-cycle resource strategy to overcome resource limitations and dramatically increase microbial diversity. By systematically varying resource ratios across growth-dilution cycles, this strategy creates additional temporal niches that allow for the coexistence of a larger number of species than traditional methods. Numerical simulations demonstrate that multi-cycle strategies significantly enhance species diversity, approaching the theoretical upper bound of 2^n-1 species coexisting on n resources.