A nonlinear free-boundary model with variable diffusion and advection coefficients for pollutant-population dynamics in rivers

Abstract

In natural aquatic environments, both the diffusion coefficient-characterising the rate of pollutant dispersion-and the advection coefficient-describing transport due to water flow-exhibit significant spatio-temporal variability. These variations stem from changes in river geometry, flow velocity, temperature, and seasonal dynamics. To better capture these complexities, this study presents an enhanced modelling framework that incorporates spatio-temporally variable diffusion and advection coefficients. These coefficients are further assumed to depend on both the population density and the concentration of environmental toxicants, enabling a more realistic representation of contaminant transport processes. This study developed a system of nonlinear partial differential equations (PDEs) with a free boundary to represent the dynamic aspect of toxic substance dispersion. The model characterises the interaction between a riverine biological population and a toxicant, accounting for ecological and hydrodynamic influences. To ensure the regularity of the solution, a priori calculations were established, including the population density  the toxicant concentration  the free boundary position  and the Hölder continuity estimates. The global existence and uniqueness of classical solutions are rigorously proven via the Leray-Schauder fixed-point theorem and energy-based methods. Parameter regimes were identified where the toxicant could not spread throughout the entire river area, thereby allowing the population to survive in unaffected areas. Due to the analytical difficulty of the nonlinear free boundary problem, implicit numerical schemes were used for the simulation. Numerical experiments, implemented in Python with graphical visualisations, validate the theoretical results and illustrate the interplay between ecological parameters and pollutant dynamics. The results obtained show how different environmental conditions affect the stability of biological populations and the spatiotemporal evolution of toxic substance concentrations