Les Houches lectures on Theoretical Ecology: High-dimensional models and extreme events (2503.02792v1)

Abstract: The study of ecological systems is gaining momentum in modern scientific research, driven by an abundance of empirical data and advancements in bioengineering techniques. However, a full understanding of their dynamical and thermodynamical properties, also in light of the ongoing biodiversity crisis, remains a formidable endeavor. From a theoretical standpoint, modeling the interactions within these complex systems -- such as bacteria in microbial communities, plant-pollinator networks in forests, or starling murmurations -- presents a significant challenge. Given the characteristic high dimensionality of the datasets, alternative elegant approaches employ random matrix formalism and techniques from disordered systems. In these lectures, we will explore two cornerstone models in theoretical ecology: the MacArthur/Resource-Consumer model, and the Generalized Lotka-Volterra model, with a special focus on systems composed of a large number of interacting species. In the second part, we will highlight timely directions, particularly to bridge the gap with empirical observations and detect macroecological patterns.

Overview of High-Dimensional Models and Extreme Events in Theoretical Ecology

In the context of theoretical ecology, a comprehensive exploration of high-dimensional models and extreme events provides crucial insights into the dynamical properties and complexities inherent to ecological systems. The work under review offers a detailed examination of these models, focusing specifically on the MacArthur/Resource-Consumer model and the Generalized Lotka-Volterra (GLV) model. These foundational models facilitate understanding of ecosystems composed of a vast number of interacting species.

Theoretical Background and Model Frameworks

The MacArthur/Resource-Consumer model and the Generalized Lotka-Volterra model are utilized extensively in theoretical ecology to elucidate the interactions within complex ecological systems. These models incorporate interactions mediated via resources, allowing the modeling of species dynamics in resource-limited environments. The MacArthur model, for instance, operates under the assumption of resource-mediated interactions, highlighting the role of consumption rates and carrying capacities, while the GLV model extends this framework by focusing directly on species interactions, incorporating a random interaction matrix (often derived from random matrix theory) to account for the heterogeneity in species interactions.

Dynamical Properties and Analytical Approaches

A significant feature of these models is their basis in random matrix theory and statistical physics, which aids in tackling the high dimensionality encountered in ecological datasets. Such methods allow for the exploration of the stability properties and the resilience of ecosystems, especially under the perturbative influence of environmental changes. Moreover, through approaches like the cavity method and replica symmetry analysis, the paper explores the phase spaces of these models, identifying conditions leading to single or multiple equilibria states within ecosystems.

Numerical and Analytical Insights

The analytical work is further bolstered by strong numerical results which showcase the transition between different stability regimes. For instance, through saddle-point calculations and dynamical mean-field theories, the paper reveals how ecosystems navigate between equilibria and chaotic regimes. Furthermore, by leveraging techniques such as the replica method, the authors rigorously define the conditions predicting the emergence of marginally stable or glassy phases within ecological communities.

Implications and Future Directions

The implications of such detailed theoretical work extend profoundly into both empirical and practical domains. By bridging theory with observable macroecological patterns, the paper proposes methodologies for predicting ecosystem responses and resilience under biodiversity loss scenarios, offering insight into ecosystem management and conservation strategies.

Moreover, with advancements in metagenomics and bioinformatics, the models discussed could facilitate a deeper understanding of microbial community dynamics, especially in gauging stability and functionality in disturbed or transitioning ecosystems. Future research is likely to benefit from integrating these high-dimensional frameworks with empirical datasets, to validate the theoretical predictions and refine the models for practical application in ecological monitoring and biodiversity prediction.

In sum, this research not only deepens our understanding of complex ecological networks and the factors governing species coexistence but also sets a solid foundation for future advancements in ecological modeling, with potential applications ranging from ecosystem management to understanding the impact of anthropogenic changes on natural systems.