• branching process
• Markovian binary tree
• maximal Lyapunov exponent

This thesis is focused on the study of the probability that a certain Markovian binary tree becomes extinct. Diefferently from most of the existing literature, this study has been conducted through a matrix approach instead of probabilistic.
A Markovian binary tree (MBT) is a model suitable for processes of evolution involving a population of individuals, wherein each of them evolves independently from the others. At every moment, the whole population can be partitioned into a finite number of typologies. During its life, an individual evolves and various events may happen, according to established probabilistic rules, in particular:
- an individual can change its type and continue its life;
- an individual can generate another individual and continue its life, the newborn individual will evolve independently from its parent;
- an individual can die.
An MBT is said to be extinct whenever there are no more individuals alive. In particular, in our analysis, we are interested in computing the probability that a certain tree will become extinct conditioned that the starting population consists of only one individual. This problem has been tackled with a renewed interest in the last years, since the vector of extinction probabilities was identied as the minimal nonnegative solution of a certain quadratic vector equation. Various algorithms have been proposed for computing such a solution, and a survey of them is reported in the thesis.
In this work we deal with an extension of this model, in fact, beside the process of the population, we introduce a parallel independent process of catastrophes. When a catastrophe happens it influences every individual alive at the moment, who may die or survive with a probability depending on their current type. Also in this case we are interested in computing the extinction probability vector, however the challenge is much harder since the quadratic vector equation employed in the classical MBT case is not true anymore.
In a recent work, Hautphenne et al. pointed out the importance of a parameter, whose positivity or negativity plays the discriminating role between the processes that will survive forever with a certain positive probability and those which will become extinct almost surely. The problem is that even only the computation of this parameter is beyond our possibilities. In fact, it is showed in the thesis how its computation of is equivalent to the computation of the maximal Lyapunov exponent of a random dynamical system, a problem that is well known to be hard apart from special cases.
Hautphenne et al. proposed an upper and a lower bound for such a parameter by using probabilistic arguments. In this thesis, by using matrix properties, we provide different expressions for the value of the parameter and we derive some upper and lower bounds.
We conclude the thesis by showing with numerical experiment when the various bounds are accurate or not. It becomes evident that the upper bound provided by Hautphenne et al. is still the best upper bound available, on the other hand the lower bound we suggested works better than the one available before.