Measure solutions for some models in population dynamics
Jos\'e A. Ca\~nizo, Jos\'e A. Carrillo, S\'ilvia Cuadrado

TL;DR
This paper establishes well-posedness of solutions for population dynamics models with measure initial data, using optimal transport techniques to handle measure-valued stationary states more naturally.
Contribution
It provides a simplified, unified proof of well-posedness for structured population models with measures, applicable to a broad class of models.
Findings
Well-posedness proven for measure-valued solutions
Techniques based on optimal transport distances
Applicable to various population dynamics models
Abstract
We give a direct proof of well-posedness of solutions to general selection-mutation and structured population models with measures as initial data. This is motivated by the fact that some stationary states of these models are measures and not functions, so the measures are a more natural space to study their dynamics. Our techniques are based on distances between measures appearing in optimal transport and common arguments involving Picard iterations. These tools provide a simplification of previous approaches and are applicable or adaptable to a wide variety of models in population dynamics.
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