is present in both areas (Figure 2.5(b)). This type of dispersal may be appropriate for continental settings in which areas are adjacent, that is, share a physical edge, and we expect gene flow to be maintained for some period of time between the allopatric populations (Ree and Sanmartín 2009). Yet, it comes with the necessity of modeling cladogenetic events or range inheritance scenarios (Figure 2.5(b)).
Figure 2.6. Parametric biogeographic reconstruction of the spatio-temporal evolution of genus Canarina. Canarina is a three-species genus in the angiosperm family Campanulaceae, with a disjunct distribution between the Canary Islands in the west and the Eastern African mountains and Horn of Africa plateaus in the east (Mairal et al. 2015). Both BIB and DEC explain this geographic disjunction as a sequence of migration events from Asia, where sister-genera occur, to Eastern Africa, and to Macaronesia; the latter along the 7 million-year branch separating C. eminii and C. canariensis. DEC infers a similar scenario, but vicariance in a widespread distribution is inferred at some nodes, mostly involving short internal branches and descendants with non-overlapping distributions. Pie charts represent the uncertainty in the estimation of ancestral ranges. For a color version of this figure, see www.iste.co.uk/guilbert/biogeography.zip
The original DEC model (Ree and Smith 2008) includes a “null” state in the Q matrix (∅), equivalent to global extinction. A species can become extinct across its entire range but only if this comprises a single area (A to ∅; Figure 2.5(b)); global extinction across a widespread range is assigned a zero rate in the DEC model (AB to ∅; Figure 2.5(b)). Also, global extinction behaves as an “absorbing state” in the Q matrix because the rate of abandoning this range is zero (e.g. ∅ to A; Figure 2.5(b)). One of the consequences of including a null range in the Q matrix is that extinction rates are typically underestimated in the DEC model, even several orders of magnitude compared with dispersal rates. Because global extinction can only occur within single-area ranges, it is often inferred at terminal branches, where dispersal cannot be countered off by a loss of areas via cladogenesis (vicariance or peripatry). Removing the null range from the Q matrix has the effect that extinction events are forced to occur within widespread ranges in order to counteract dispersal or range expansion along internal branches. In other words, loss of areas would be achieved via extinction in ancestral branches rather than via cladogenesis, and thus extinction rates are increased relative to dispersal if global extinction is disallowed (Massana et al. 2014). Bayesian extensions of DEC (Landis 2017) correct for this bias by conditioning the DEC inference to the survival of all lineages in the extant phylogeny, that is, never entering the null state.
The issues above do not affect the BIB model because it uses a simpler stochastic CTMC model similar to those employed in molecular character evolution. Widespread states (equivalent to “polymorphism” in nucleotide models) are not allowed in the CTMC matrix, and range evolution is limited to the anagenetic component. This is modeled as instantaneous transitions between single-area states, equivalent to “jump dispersal”, but which can vary across area pairs and may also be asymmetric, that is, the rate of moving from A to B, p, is not the same as from B to A, q (Figure 2.5(a)). Modeling dispersal as an instantaneous process, without going through a widespread state, may seem unrealistic but allows the BIB model to “borrow” the sophisticated machinery and statistical algorithms used in molecular models of nucleotide substitution; in fact, initial implementations of BIB used software routinely employed in molecular phylogenetics (Sanmartín et al. 2008; Lemey et al. 2009). In standard molecular models, nucleotide substitutions within a species DNA sequence are considered as instantaneous. In-between demographic-level processes, involving increased allele polymorphism within gene trees, competition among mutations in terms of fitness, and rates of fixation differing between alleles (De Maio et al. 2015), are typically ignored. Similarly, in the BIB model, the species is assumed to instantaneously change the area relative to its current range, ignoring the intermediate population-level processes, such as changes in effective population size due to migration, introgression, etc. The BIB model can thus be appropriate to model scenarios in which areas are discrete entities isolated by dispersal barriers, so that migration to a new area effectively leads to speciation; in other words, the ancestor is not expected to maintain the widespread distribution for long, as in the case of founder effects in oceanic islands isolated by geographic barriers (Sanmartín et al. 2008), or in continental islands isolated by ecological barriers (Sanmartín et al. 2010). However, the assumption of single-state ancestral ranges means that BIB is most useful to explore and test general patterns of geographic movement or dispersal; if the interest lies on inferring speciation modes or possible ways in which ancestral ranges are divided, BIB is not well suited. Notice that constraining ancestors to single areas in the Q matrix does not imply that phylogenies with extant widespread species cannot be analyzed with BIB. As in molecular evolutionary models, these widespread terminals will be treated as sources of “ambiguity” in the BIB analysis: 50% of the time the MCMC chain will sample from one of the discrete states, and 50% from the other. Another solution is to use an expanded, constrained Q matrix in which transitions between widespread states are allowed only between spatially adjacent ranges, as in an ordered “character step matrix” in parsimony-based approaches (Bribiesca-Contreras et al. 2019).
One advantage of the direct analogy between the BIB model and nucleotide substitution models is the possibility to infer the stationary frequencies of the states in the CTMC process (Sanmartín et al. 2008, 2020). The standard CTMC models used in parametric biogeography and molecular evolution are “time-homogeneous” or “stationary” Markov models. They have the property that the rates of transition between states are constant and, over time, tend to reach a stationary equilibrium state. They are also often time-reversible, that is, independent from the flow of time (i.e. this is not the same as symmetric). Over time, the frequencies of the states of a time-homogeneous Markov process converge to the stationary values regardless of the starting point. In a time-reversible stationary CTMC, the state equilibrium frequencies are built into the Q rate matrix, so that the transition rates can be decomposed into two parameters: the relative exchangeability rates and the state stationary frequencies. Similarly, the rate of moving from A to B in the Q matrix of the BIB model (p) can be broken down into two parameters: the relative dispersal rate per migrating lineage (rAB) and the area “carrying capacities” (πA, πB). The latter are the model “stationary” frequencies: the number of lineages at equilibrium conditions, or in other words, the number of lineages expected in each area if the CTMC dispersal process is let to run for a very long time without external disturbances (Sanmartín et al. 2008). Disentangling transition rates into two parameters allows the root states, that is, the states at the start of the process, to be drawn from the stationary frequencies of the CTMC. Also, the two parameters account for different aspects of the dispersal process. Relative dispersal rates can be informed (scaled) by the geographic distance between areas or the strength of wind and ocean currents, while carrying capacities can be partitioned by area size or the degree of environmental heterogeneity versus a species ecology; this allows researchers to measure the role played by abiotic factors and biotic factors in shaping area colonization patterns (Sanmartín et al. 2008; Sanmartín 2020). Finally, though there is no speciation parameter in the BIB model, carrying capacities can be used as a proxy for rates of “within-area diversification”. Stationary frequencies represent the time the CTMC process spends without transitioning between states, or, in a biogeographic context, without migrating between areas. Sanmartín et al. (2010) used this equivalence in a continental-island context, to demonstrate that the southern African component of the Rand Flora was formed through within-area diversification, whereas the Macaronesian component was shaped by immigration events from nearby regions.
The partitioning of CTMC transition rates into stationary frequencies and relative exchange rates is not possible in DEC. The reason was pointed by Ronquist and Sanmartín (2011) and discussed extensively in Ree and Sanmartín (2018). DEC and DEC-derived models are not complete parametric models like BIB because one key component of the biogeographic model, cladogenetic
