Modelagem matemática da dinâmica do mercúrio no Rio Madeira por meio de equações de diferenças: análise socioambiental e possibilidades para o ensino de Matemática

Abstract

Mercury contamination associated with gold mining constitutes a relevant socio-environmental issue in the Amazon, particularly in river systems upon which riverside populations depend. This research aimed to analyze the contributions of a mathematical model based on difference equations to the understanding of the dynamics of mercury concentrations in the Madeira River, considering its socio-environmental implications and possibilities for Mathematics Education. The study consists of applied research with a predominantly quantitative approach, complemented by bibliographic, documentary, and socio-environmental interpretation procedures. The stretch of the Madeira River between Porto Velho and Calama, in the state of Rondônia, was represented by four consecutive compartments—Porto Velho, São Carlos, Nazaré, and Calama—connected by effective mechanisms of longitudinal transfer, removal, and external inputs. The dynamics were formulated as a discrete linear system of difference equations, implemented in spreadsheets, and analyzed under a reference configuration, a condition of greater transport, and a scenario of intensified external inputs. The temporal evolution of concentrations, equilibrium states, asymptotic stability, and convergence times were examined. The results demonstrated that all analyzed configurations were asymptotically stable, although they produced substantially distinct equilibria. The analysis evidenced that mathematical stability does not necessarily imply low concentrations, and that changes introduced upstream can reverberate in downstream compartments. As an unfolding of the investigation, a didactic proposal for High School was developed, grounded in Mathematical Modeling, Critical Mathematics Education, and Didactic Transposition. It is concluded that difference equations constitute a consistent tool to investigate the relationships between the transport, removal, and inputs of mercury, articulating mathematical analysis, socio-environmental interpretation, and possibilities for Mathematics Education.