Uma abordagem geométrica da estabilidade local em dinâmicas discretas com aplicação em um modelo de cooperação
Abstract
In the search for understanding natural phenomena with complex dynamics, researchers often resort to mathematical tools. Individual-based models (IBMs) are often described by probabilistic rules, which hampers the analytical description of their stability. However, it is possible to write a Master Equation that describes the average behavior of the variables of interest, enabling such analysis. In this paper, we revisit Dynamical Systems Theory through geometric arguments in order to evaluate the local stability of fixed points in discrete-time equations. After that, we present an IBM, inspired by the cooperation phenomenon observed in some male neotropical birds that form groups to attract females. From this example, we show how to construct the Master Equations that accurately describe the mean population size of cooperators and non-cooperators in the system. Finally, we analyze the local stability of these equations and discuss possible maintenance mechanisms of cooperation. Thus, this work assists students from different backgrounds in their first encounters with Dynamical Systems Theory.
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BibTeX
@article{brito2026abordagem,
title = {{Uma abordagem geométrica da estabilidade local em dinâmicas discretas com aplicação em um modelo de cooperação}},
author = {Gustavo De Brito and Natale Cavezzale Dias and Vitor Marquioni and Sabrina B.L. Araujo},
journal = {Revista Brasileira de Ensino de Física},
year = {2026},
month = nov,
doi = {10.1590/1806-9126-rbef-2025-0449},
url = {https://amu.hal.science/hal-05850125v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Uma abordagem geométrica da estabilidade local em dinâmicas discretas com aplicação em um modelo de cooperação
AU - Gustavo De Brito
AU - Natale Cavezzale Dias
AU - Vitor Marquioni
AU - Sabrina B.L. Araujo
PY - 2026
DA - 2026/11/03
JO - Revista Brasileira de Ensino de Física
DO - 10.1590/1806-9126-rbef-2025-0449
UR - https://amu.hal.science/hal-05850125v1
AB - In the search for understanding natural phenomena with complex dynamics, researchers often resort to mathematical tools. Individual-based models (IBMs) are often described by probabilistic rules, which hampers the analytical description of their stability. However, it is possible to write a Master Equation that describes the average behavior of the variables of interest, enabling such analysis. In this paper, we revisit Dynamical Systems Theory through geometric arguments in order to evaluate the local stability of fixed points in discrete-time equations. After that, we present an IBM, inspired by the cooperation phenomenon observed in some male neotropical birds that form groups to attract females. From this example, we show how to construct the Master Equations that accurately describe the mean population size of cooperators and non-cooperators in the system. Finally, we analyze the local stability of these equations and discuss possible maintenance mechanisms of cooperation. Thus, this work assists students from different backgrounds in their first encounters with Dynamical Systems Theory.
SN - 0102-4744
ER -