Fitness, predictability and equilibrium in biological dynamics *
Abstract
Understanding eco-evolutionary dynamics presents a number of mathematical challenges, among which the role of predictability and equilibrium in affecting eco-evolutionary dynamics is paramount. However, we are far from achieving an integral mathematical representation of the role of predictability, in part because the fundamental concept of fitness and its dynamics lack a general representation. At present, Open System Dynamics theory provides a suitable methodological approach to address these challenges, starting from Stochastic Analysis, which appears to be one of their best contemporary representations. Here, we focus on the concept of fitness and provide a mathematical theoretical framework to characterize the history of fitness change and the relationship between fitness and predictability. We illustrate the use of this framework and explore the limiting behavior of biomass abundances modeled by stochastic processes in both discrete time and continuous time counterpart after rescaling (diffusion). In doing so, we focus on the consequences of neutrality in the search for equilibrium. In particular, the modeling of diffusions considers two kinds of basic settings: for relative abundances, our main processes take values on a d dimensional simplex, where orthogonal polynomials played a central role in constructing such models; for non proportional abundances, these processes take values in the space R d + of vectors with positive components, in which we focus on two remarkable families of models. The paper concludes with statistical inference, presenting numerical examples for the case of relative abundances where orthogonal polynomials play a special role when estimating the relative abundance parameters under a maximum likelihood method.
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BibTeX
@misc{rebolledo2026fitness,
title = {{Fitness, predictability and equilibrium in biological dynamics *}},
author = {Rolando Rebolledo and Mauricio Tejo and Mara A Freilich and Pablo A Marquet},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22368},
eprint = {2609.22368},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05752615v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Fitness, predictability and equilibrium in biological dynamics *
AU - Rolando Rebolledo
AU - Mauricio Tejo
AU - Mara A Freilich
AU - Pablo A Marquet
PY - 2026
DA - 2026/09/16
PB - HAL
DO - 10.48550/arxiv.2609.22368
UR - https://hal.science/hal-05752615v1
AB - Understanding eco-evolutionary dynamics presents a number of mathematical challenges, among which the role of predictability and equilibrium in affecting eco-evolutionary dynamics is paramount. However, we are far from achieving an integral mathematical representation of the role of predictability, in part because the fundamental concept of fitness and its dynamics lack a general representation. At present, Open System Dynamics theory provides a suitable methodological approach to address these challenges, starting from Stochastic Analysis, which appears to be one of their best contemporary representations. Here, we focus on the concept of fitness and provide a mathematical theoretical framework to characterize the history of fitness change and the relationship between fitness and predictability. We illustrate the use of this framework and explore the limiting behavior of biomass abundances modeled by stochastic processes in both discrete time and continuous time counterpart after rescaling (diffusion). In doing so, we focus on the consequences of neutrality in the search for equilibrium. In particular, the modeling of diffusions considers two kinds of basic settings: for relative abundances, our main processes take values on a d dimensional simplex, where orthogonal polynomials played a central role in constructing such models; for non proportional abundances, these processes take values in the space R d + of vectors with positive components, in which we focus on two remarkable families of models. The paper concludes with statistical inference, presenting numerical examples for the case of relative abundances where orthogonal polynomials play a special role when estimating the relative abundance parameters under a maximum likelihood method.
ER -