LITMIR.BIZ

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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 (r_AB) and the area “carrying capacities” (π_A, π_B). 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 scenarios of range evolution, is not part of the stochastic CTMC process that governs the evolution of geographic ranges as a function of time. In other words, there is no speciation parameter in the Q matrix, even though speciation has an effect on range evolution in the DEC model (Figure 2.5(b)). As a result, root states in DEC cannot be drawn from the stationary frequencies of the CTMC process, as can be done in BIB. In Ree and Smith’s (2008) ML implementation of DEC, root states are inferred by first estimating the likelihood of alternative ancestral ranges and then selecting the one that maximizes the global likelihood. Another consequence of DEC not being a fully parametric model is that DEC-derived models that differ in the type of implemented cladogenetic scenarios cannot be compared statistically. DEC and DEC-derived models such as DIVALIKE or BAYAREALIKE contain the same number of parameters in the CTMC Q matrix that governs range evolution (i.e. the rates of dispersal and extinction), so it is erroneous to use penalty-based likelihood tests such as AIC (Matzke 2014) to statistically distinguish or identify them. Instead, we can choose between these models, which imply different speciation modes of widespread range division, using biological knowledge about the study group (Sanmartín 2020). The same issue arises when comparing time-homogeneous and time-stratified DEC models (below) because these models do not differ in the number of CTMC parameters. On the other hand, within a Bayesian framework, we can statistically compare any two models using the Bayes factor. The latter computes the ratio of the marginal likelihood of two competing models, or, in other words, the posterior against the prior odds for any of the models as the one generating the data (Goodman 1999). Unlike AIC or LRT, Bayes factor comparisons do not depend on any single set of parameters, as they integrate over all parameters in each model, while at the same time applying a penalty to overfitting, that is, a low ratio of data to parameters (Kass and Raftery 1995).

2.4.2. Extending the DEC and BIB models

      Over time, the DEC and BIB models have been expanded to include more complexity and increasing realism. The original DEC model (Ree and Smith 2008) included dispersal or range expansion only as an anagenetic event, which was modeled as a time-dependent rate within the Q instantaneous rate matrix (Figure 2.5(b)). Matzke (2014) extended this model to include “cladogenetic dispersal” or “founder-event speciation”, as an event of dispersal that is coincident with speciation, with one daughter lineage instantaneously “jumping” into a new area that was not part of the ancestral range, for example, from A to A and C in Figure 2.5(b). This new cladogenetic scenario is modeled in the DEC+J model by a separate parameter j (Matzke 2014), which is not part of the CTMC process that governs range evolution along branches. Therefore, this j parameter is not equivalent to the rate of jump dispersal p and q in the BIB model (Figure 2.5(a)), and it is also not dependent on time, unlike the D_AB or E_A parameters in DEC. Ree and Sanmartín (2018) showed that by decoupling “jump dispersal” from time, the DEC+J model can result in highly counterintuitive scenarios and degenerate likelihood inferences, in particular if founder speciation is assigned a higher likelihood (“weight”) relative to other cladogenetic scenarios such as allopatry or peripheral isolate speciation. Moreover, when estimated as its maximum value, the inclusion of j can lead to underestimation of the rates of the anagenetic, time-dependent parameters: range expansion and range contraction. As a result, the DEC+J model can generate reconstructions with rates of anagenetic dispersal and (especially) of extinction close to zero, and distribution patterns that are explained almost exclusively by cladogenetic events. The end result is a diminishing of the relevance of time (branch lengths) in biogeographic inference, considered as the key advance of parametric over parsimony-based approaches (Ree and Sanmartín 2018). Figure 2.7 shows an example of this potential bias. As pointed out by Ronquist and Sanmartín (2011) and Ree and Sanmartín (2018), the proper modeling of cladogenetic events in parametric range evolution requires the use of trait-dependent speciation-extinction models (Maddison et al. 2007), discussed in more detail below. A different

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