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This manuscript presents an algorithmic approach to cooperation in biological systems, drawing on fundamental ideas from statistical mechanics and probability theory. Fisher's geometric model of adaptation suggests that the evolution of organisms well adapted to multiple constraints comes at a significant complexity cost. By utilizing combinatorial models of fitness, we demonstrate that the probability of adapting to all constraints decreases exponentially with the number of constraints, thereby generalizing Fisher's result. Our main focus is understanding how cooperation can overcome this adaptivity barrier. Through these combinatorial models, we demonstrate that when an organism needs to adapt to a multitude of environmental variables, division of labor emerges as the only viable evolutionary strategy., (© 2024. The Author(s).)

Competing Interests: Declaration of Competing Interest Herewith, we confirm that the authors have no conflict of interests.

Extinction of species, and even clades, is a normal part of the macroevolutionary process. However, several times in Earth history the rate of species and clade extinctions increased dramatically compared to the observed "background" extinction rate. Such episodes are global, short-lived, and associated with substantial environmental changes, especially to the carbon cycle. Consequently, these events are dubbed "mass extinctions" (MEs). Investigations surrounding the circumstances causing and/or contributing to mass extinctions are on-going, but consensus has not yet been reached, particularly as to common ME triggers or periodicities. In part this reflects the incomplete nature of the fossil and geologic record, which - although providing significant information about the taxa and paleoenvironmental context of MEs - is spatiotemporally discontinuous and preserved at relatively low resolution. Mathematical models provide an important opportunity to potentially compensate for missing linkages in data availability and resolution. Mathematical models may provide a means to connect ecosystem scale processes (i.e., the extinction of individual organisms) to global scale processes (i.e., extinction of whole species and clades). Such a view would substantially improve our understanding not only of how MEs precipitate, but also how biological and paleobiological sciences may inform each other. Here we provide suggestions for how to integrate mathematical models into ME research, starting with a change of focus from ME triggers to organismal kill mechanisms since these are much more standard across time and spatial scales. We conclude that the advantage of integrating mathematical models with standard geological, geochemical, and ecological methods is great and researchers should work towards better utilization of these methods in ME investigations., Competing Interests: Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2022 Elsevier B.V. All rights reserved.)