How trait distributions evolve in populations with parametric heterogeneity.

We consider the problem of determining the time evolution of a trait distribution in a mathematical model of non-uniform populations with parametric heterogeneity. This means that we consider only heterogeneous populations in which heterogeneity is described by an individual specific parameter that differs in general from individual to individual, but does not change with time for the whole lifespan of this individual. Such a restriction allows obtaining a number of simple and yet important analytical results. In particular we show that initial assumptions on time-dependent behavior of various characteristics, such as the mean, variance, or coefficient of variation, restrict severely possible choices for the exact form of the trait distribution. This fact must be taken into account for both model formulation and, especially, for testing theoretical models against available real world data. We illustrate our findings by in-depth analysis of the variance evolution and specific examples from population ecology and mathematical epidemiology. We also reanalyze a well known mathematical model for gypsy moth population and show that the knowledge of how trait distributions evolve allows producing oscillatory behaviors for highly heterogeneous populations.