Measure-valued solutions for a hierarchically size-structured population

We present a hierarchically size-structured population model with growth, mortality and reproduction rates which depend on a function of the population density (environment). We present an example to show that if the growth rate is not always a decreasing function of the environment (e.g., a growth which exhibits the Allee effect) the emergence of a singular solution which contains a Dirac delta mass component is possible, even if the vital rates of the individual and the initial data are smooth functions. Therefore, we study the existence of measure-valued solutions. Our approach is based on the vanishing viscosity method.