An integro-differential model for evolutionary dynamics with mutations is investigated by improving the understanding of its behaviour using numerical simulations. The proposed numerical approach can handle also density dependent fitness, and gives new insights about the role of mutation in the preservation of cooperation.

An elementary model of (1 + 1)-dimensional general relativity, known as "R = T " and mainly developed by Mann and coworkers in the early 1990s, is set up in various contexts. Its formulation, mostly in isothermal coordinates, is derived and a relativistic Euler system of selfgravitating gas coupled to a Liouville equation for the metric's conformal factor is deduced.

Well-balanced schemes were introduced to numerically enforce consistency with longtime behavior of the underlying continuous PDE. When applied to linear kinetic models, like the Goldstein-Taylor system, this construction generates discretizations which are inconsistent with the hydrodynamic stiff limit (despite it captures diffusive limits quite well).

Understanding the conditions ensuring the persistence of a population is an issue of primary importance in population biology. The first theoretical approach to the problem dates back to the 1950s with the Kierstead, Slobodkin, and Skellam (KiSS) model, namely a continuous reaction-diffusion equation for a population growing on a patch of finite size L surrounded by a deadly environment with infinite mortality, i.e., an oasis in a desert. The main outcome of the model is that only patches above a critical size allow for population persistence.