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      2.4. A new revolution: parametric approaches in biogeography

      The last decade has witnessed the introduction of parametric approaches in biogeography without the biases inherent in the parsimony framework (Ronquist and Sanmartín 2011). A common feature of these methods is the use of statistical probabilistic models, whose variables or parameters are quantifiable biogeographic processes that are dependent on time. Thus, in addition to the tree topology and tip distributions, parametric models incorporate a third source of information: branch lengths – measured in numbers or units of time – provide direct evidence on the rate or probability of geographic evolution. Longer branches imply a higher probability of change in the geographic range than shorter branches do. As more time elapses since the divergence of the species from its ancestor, there is more opportunity for biogeographic change (by dispersal, extinction or range expansion) along the branch. Branch lengths also inform on the certainty or degree of error in biogeographic inference: a species subtended by a long branch would be associated with a higher uncertainty about its ancestral range than one subtended by a short branch (Sanmartín 2020).

Two parametric models with stochastic variables that are time-dependent are typically used to model range evolution: the Brownian Motion (BM) model and the continuous-time Markov Chain (CTMC) model. BM is used to model the random dispersal of individuals within a population in a continuous landscape. We will return to this model later. CTMC models are typically used to describe range evolution at the species level. A Markov chain is a stochastic, memory-less process that models transitions between discrete states over continuous or discrete-time; the probability of each transition event depends only on the state attained in the previous event. The difference between a discrete-time MC (DTMC) and a continuous-time MC (CTMC) is that in DTMC, the chain moves (transitions or changes state) in discrete intervals (Δt), whereas in continuous-time, the moves are in infinitesimal amounts of time (dt) so that it can effectively measure instantaneous change. CTMC models are used in historical inference disciplines, such as phylogenetics or biogeography, where the states observed in the present are the result of a stochastic process that evolves over time. Stochastic means that, unlike in a deterministic process, the outcome of the process cannot be predicted with certainty, that is, we cannot predict the result of evolution. At the same time, evolutionary processes are not entirely random or unpredictable: they can be described by probability distributions with parameters, whose values we infer from the data.

      Figure 2.5 provides an example of the parametric approach. In biogeography, the states of the stochastic CTMC process that governs range evolution are the set of discrete geographic areas that form the distribution range of a taxon (e.g. A, AB, B). The rates of transition or change between states in the CTMC process (e.g. A to B), within an infinitesimal amount of time (dt), are governed by the so-called instantaneous rate matrix Q, which has as parameters biogeographic processes that determine the probability of range evolution as a function of time, for example, dispersal, extinction, speciation. Given a phylogeny with time-calibrated branch lengths, tip distributions coded as discrete entities (A, B) and a stochastic CTMC model of range evolution (Q matrix, Figure 2.5), we can estimate the probabilities of ancestral ranges (A, B, AB) and the rate of parameters describing the transition between geographical ranges (p, q, DAB, EB, in Figure 2.5), using statistical inference approaches such as maximum likelihood (ML) or BI.

Besides the possibility of integrating time into biogeographic inference, parametric methods offer several advantages over parsimony-based approaches (Ree and Sanmartin 2009). Rather than inferring only the most parsimonious reconstruction, one can integrate over all possible biogeographical scenarios in the estimation of rates of parameters and species ancestral ranges; that is, parametric methods account for the “reconstruction uncertainty” (Ronquist 2004; Ree and Smith 2008). If BI is used, parameter estimates are not conditioned on a given phylogeny but marginalized over the tree topology and branch lengths by simultaneously estimating the parameters governing phylogenetic and biogeographic evolution; that is, BI parametric methods account for the “phylogenetic uncertainty” (Ronquist 2004; Sanmartín et al. 2008). Sources of evidence other than the phylogeny and tip distributions (e.g. the fossil record, geological or paleoclimate information or the species ecology) can be integrated into parametric models, either in the form of new parameters in the Q matrix or through scaling parameters that modify the baseline rate of a different parameter (Buerki et al. 2011; Meseguer et al. 2015; Quintero and Landis 2019; Landis et al. 2021). Finally, model selection, that is, the statistical testing of competing biogeographic scenarios, is straightforward with parametric models because, as in EBMs, assumptions about processes governing range evolution are made explicit and integral to the inference framework. Moreover, because the underlying stochastic models are based on well-known probability distributions, parametric models can make use of statistical tests employed in phylogenetics for model choice, such as likelihood ratio tests (LRT), the Akaike information criterion (AIC) or Bayes factor comparisons.