Aim of the course

Starting with the groundbreaking works of Charles Darwin and Gregor Mendel and passing through all the achievements of genetics, ecology, and modern molecular biology, we now accept the view that evolutionary change results from a mechanism of innovation, i.e. genetic mutation, introducing new variant forms of individuals in the community, and from a process of competition, i.e. the demography of the community. This evolutionary view goes far beyond biology, since culture, behavioral strategies, companies, goods, technology, and many other artificial agents often evolve by means of slight innovations and compete with the other agents of the system.

The typical pattern of evolutionary change of an individual trait, such as body size or shape, can be imagined as a sequence of small innovations which confer an advantage to affected individuals in terms of survival and reproductive success. After each advantageous innovation, the group of innovative individuals has the potential to spread in the community and replace the previous group of similar individuals, thus leading to a small step in the evolution of the trait.

Darwin first realized that those individuals best adapted to survive and reproduce should come to dominate the populations in the absence of further innovations. He called natural selection the demographic process leading to the dominance of the best adapted individuals. As a result, all individual traits affecting demography evolve through innovation-competition processes, by adapting to the local environmental circumstances. The evolutionary hypothesis is that the superposition of all such processes have led to the origin of all species starting form a common ancestral species more than 3 billion years ago.

The aim of the course is to present the mathematical approaches for modeling evolutionary change, with emphasis to Evolutionary game theory and Adaptive dynamics. Several applications in biology and social sciences will be discussed in detail.

The course is addressed to Ph.D. students and scientists interested in the field of evolutionary dynamics. Only undergraduate skill in mathematics is required. All basic notions of linear and nonlinear dynamical systems and bifurcation theory will be introduced.