Modeling the indirect contamination of a structured population with continuous levels of exposure

We shall be concerned with the mathematical analysis of a deterministic model describing the spread of a contamination which structures a population on different and continuous levels, each level representing a degree of contamination. Our approach is essentially devoted to describe a population when exposed to pollution or affected by any non environmentally-friendly source.
    Mathematically, the problem consists of an advection-reaction partial differential equation with variable speed, coupled by mean of its boundary condition to an ordinary differential equation. Using a method of characteristics, we prove the global existence, uniqueness and nonnegativity of the mild solution to this system, and also the global boundedness of the total population when subjected to controlled growth dynamics such as so-called logistic behaviors.