Darwinian evolution in Malthusian population growth model and Markov chains

The paper is devoted to the study of Darwinian evolution in two mathematical models. The first one is a variation on the Malthusian population growth model with Verhulst's environmental capacity. The second model is grounded in the theory of Markov chains and their stationary distributions. We prove preliminary results regarding both models and pose conjectures, which are supported by computer simulations.