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.
Figure 2.5. Parametric models in biogeographic inference. Top: a continuous-time Markov chain (CTMC) process is used to describe the probability of range evolution; the states of the CTMC are discrete geographic ranges (A, B, AB), and transitions between states (A to B) are governed by an instantaneous Q matrix with rate parameters (q) that are estimated from the data. Bottom: given a two-species phylogeny with associated distributions and branch lengths measured as time since divergence, we can use maximum likelihood or Bayesian inference to estimate the ancestral geographic range and the sequence of biogeographic events that gave rise to the current distributions. a) The BIB model implements a CTMC process with only single-area ranges as discrete states (A or B); anagenetic evolution along branches is governed by parameters p and q, describing the instantaneous movement between states; ancestral states are identically inherited by the two descendants through speciation (A/A). b) The DEC model implements a CTMC process with widespread distributions formed by two or more discrete areas (AB), and thus requires an additional cladogenetic component to describe the different ways by which a widespread ancestral range is divided between the two descendants (A/B); anagenetic evolution is governed by a CTMC process with two parameters: range expansion (DAB) and range contraction (EB); direct transitions between single areas (A to B) are not allowed in the Q matrix. For a color version of this figure, see www.iste.co.uk/guilbert/biogeography.zip
2.4.1. Ancestral range versus single state models: DEC and BIB
The first two parametric methods developed in biogeography (Figure 2.5) were the Bayesian island biogeography (BIB) model (Sanmartín et al. 2008) and the dispersal-extinction-cladogenesis (DEC) model (Ree et al. 2005; Ree and Smith 2008). The BIB model uses Markov-Chain Monte Carlo (MCMC) simulations and BI to estimate ancestral ranges and rates of biogeographic parameters alongside phylogenetic parameters, such as the tree topology, rates of molecular evolution and branch lengths; the input data are DNA sequences and tip distributions of the study species (Sanmartín et al. 2008). The DEC model was originally implemented in an ML framework (Ree and Smith 2008) and later extended to BI (Landis et al. 2013), but it is typically applied to a phylogeny with fixed topology and branch lengths. Though they both use CTMC processes to model range evolution, BIB and DEC models are slightly different (Figure 2.5). BIB (Figure 2.5(a)) implements a simpler character evolutionary model, in which ancestors can only occupy single areas (A or B) and range evolution along the branches if governed by a CTMC process with only one type of parameter equivalent to range switching or instantaneous dispersal; the Q matrix describes the instantaneous transition from one area as a jump dispersal event (p = A to B). At the speciation events in the phylogeny, the single-area ancestral range is inherited entirely and identically by the two descendants (A/A); in other words, there is no need to include a cladogenetic component in the BIB model because the ancestral range is not altered through speciation (Figure 2.5(a)).
DEC implements a more complex ancestral-state reconstruction model (Figure 2.5(b)), allowing for ranges in the Q matrix to comprise two or more discrete areas (i.e. widespread states, AB). Therefore, the model requires an additional cladogenetic component describing the different ways (range inheritance scenarios) by which a widespread ancestral range is divided between the two descendants. In Figure 2.5(b), this involves vicariance, range division into non-overlapping subsets (A/B), but other possible range inheritance scenarios are “peripheral isolate speciation” (Ree et al. 2005), in which one descendant inherits the whole ancestral range and the other descendant inherits only one area (AB/B), or “widespread sympatry”, in which the two descendants inherit the widespread ancestral range (AB/AB; Landis et al. 2013). In the original ML implementation of DEC (Ree and Smith 2008), all these range inheritance scenarios are assigned equal relative likelihoods or “weights”. In the BI implementation (Landis 2017), different prior probabilities can be assigned to the speciation modes. There are other DEC-derived models (DIVALIKE, BAYAREALIKE, Maztke 2014) which differ on the type of range inheritance scenarios allowed for widespread ranges. Like BIB, DEC implements anagenetic evolution as a CTMC process. Instantaneous transitions between geographic states are governed by two parameters (Figure 2.5(b)): range expansion, where an additional area is added to the current range (DAB = A to AB), and range contraction, with the removal of an area from the ancestral range (EB = AB to B). Notice that instantaneous transitions between single-area states are not allowed in DEC. Moving between single areas requires going through a widespread state in which the ancestor is present in both areas (Figure 2.5(b)). This can be observed in the Q matrix, with zero rate values for direct transitions between A and B (Figure 2.5(b)). A transition from A to B requires two consecutive instantaneous events: range expansion from A to AB and range contraction from AB to B.
Figure 2.6 illustrates, with a real biogeographic problem, some of the differences between BIB and DEC. In sum, the BIB model allows only anagenetic changes along branches in the phylogeny and constrains ancestors to occur in single areas, whereas DEC includes both anagenetic and cladogenetic range evolution; in fact, DEC is the parametric counterpart of DIVA (Ree et al. 2005). However, this additional level of complexity brings some statistical limitations discussed below. Both models implement different sources of uncertainty (phylogenetic and reconstruction in BIB, and reconstruction in DEC).
The DEC model is undoubtedly more realistic than BIB. In biogeography, widespread terminals and ancestors are biologically plausible: an extant or extinct taxon could have occupied more than one area, especially if these areas were connected and there was no dispersal barrier between them. However, such complexity comes with a cost: there are 2N possible ancestral ranges for N areas, so the Q instantaneous rate matrix cannot be analytically calculated with more than 10 areas (1,024 states). This can be reduced by removing certain transitions from the Q matrix, for example, disallowing ancestral ranges that involve discrete areas that are non-adjacent in the physical space (Buerki et al. 2011), or using alternative estimation methods, such as data augmentation (Landis et al. 2013). Dispersal in DEC is equivalent to range expansion – the ancestor moves into a new area but keeps the original distribution for some time; in other words, moving between single
