**Sebastián Buedo Fernández**

*Universidade de Santiago de Compostela, Spain*

Delay differential equations are a tool that can be used to model several phenomena naturally involving a certain lag. In particular, when modelling the destruction and delayed production of a certain quantity $x$ over time, e.g., the evolution of the number of individuals in a population, the following differential equation has been extensively considered in the last decades:

$$x'(t)=-a(t)x(t)+f(t,x_t).$$

In the former equation, $a$ represents the destruction rate, $f$ is the production function and $x_t$ is an item that gathers the history of the function $x$ up to time $t$.

Due to the nature of the solutions to any equation of the above-mentioned type, one of the best ways to analyse their behaviour is to consider a qualitative approach. In this talk, we will introduce this kind of equations, show some of their applications and explain several qualitative techniques to handle them, which include the use of related difference equations, integral inequalities and fixed point theorems.

This work has been included in the PhD thesis of the speaker and thus constitutes a product of the collaboration between his supervisors, Eduardo Liz Marzán and Rosana Rodríguez López, and himself.