Mathematical models for the evolution of species

Phylogenetics is the study of the evolutionary history of organisms. These organisms encompass biological entities such as viruses, genes, species, and higher taxa. Their relationships can be visualised using a mathematical graph, known as a phylogenetic tree. In this thesis, our objects of interest are species trees. In particular, we develop various mathematical models for the evolution of species and their corresponding phylogenetic tree. These models can be used to elucidate underlying biological processes that drive past speciation and extinction events. Understanding these evolutionary dynamics is important since it can help with preserving present-day species and reconstructing how biodiversity has evolved through time.
The models introduced in this thesis capture different factors driving speciation and extinction events, from species age to environmental niches. In Chapter 1 we give an overview of the thesis and introduce relevant biological background and terms concerning phylogenetic trees. In Chapter 2 we provide the mathematical prerequisites needed to understand models in the following chapters.
In Chapter 3, we develop a model where times to speciation and extinction are drawn from a continuous distribution, known as the phase-type (PH) distribution. Essentially, this is an age-dependent model where the increasing age of a newly produced lineage affects its subsequent probability of speciating or going extinct. In this work, we explore the shape of species trees evolved under different values of the model parameters, and compute tree balance for sets of trees using one of the tree balance metrics (β statistic). Furthermore, we derive a closed-form likelihood expression for the probability of observing a reconstructed tree using information on extant species. Then, we assess how well our model fits empirical tree data, and how it compares to other models.
Throughout history, species may inherit and change traits at various point in time. To capture the possibility of (discrete) trait change driving evolutionary dynamics, in Chapter 4, we explore trait-dependent models of species evolution. We review Markovian Binary Trees (MBTs) from the theory of Matrix-Analytic Methods (MAMs) and explain how they can be applied in phylogenetics. Then, we describe how some earlier models with trait dependency, the Binary-State Speciation and Extinction (BiSSE) model and its derivatives, can be expressed as MBTs. Furthermore, we emphasise the importance of acknowledging past events on a tree that are unobserved in a reconstructed tree. That is, we consider the probability of a lineage becoming extinct after the end of an internal branch, and so not observable at the present day. Given that acknowledgement, we describe a likelihood expression for observing a reconstructed tree, and provide an example of likelihood computation of trees evolving under various scenarios of the model parameters. Moreover, we derive a conditional probability for computing tree balance under this model, give examples of the computation, and draw a connection with tree balance computed using the model in Chapter 3. Also, we represent the general MBT as a Level-Dependent Quasi-Birth-and-Death (LD-QBD) model, derive some probability results of observing an extinction event, and explain computational advantages from this representation.
Models in Chapter 3 and Chapter 4 only use species information (age and inherited trait) to model the species evolution. In Chapter 5 we develop a model where a process formed by change in environmental variables across a geographical region drives diversification. We define both species speciation and extinction processes under the model, which are linked to the appearance and disappearance of habitable regions. We provide a simulation example for generating species trees under our model and explore tree balance under different model parameters. Also, we discuss potential applications of our model to study other phylogenetic topics, including the loss of phylogenetic diversity.
Finally, in Chapter 6, we discuss further research to extend models developed in this thesis. In conclusion, we have developed various models for species evolution, including models that use spatial and temporal information.