As with molecular evolutionary models, relaxation of these assumptions has gone in the direction of allowing for rates to vary over time and across lineages, the so-called time-heterogeneous CTMC models. In the case of BIB, Bjelec et al. (2014) extended the DTA model to allow for the overall dispersal rate to vary across time slices in a stratified phylogeny; they used a piecewise-constant stochastic process in which rates of migration are constant within a given time slice but change between time slices. The temporal boundary (breakpoint) between two time slices may be estimated from the phylogenetic and distribution data alongside the biogeographic parameters.
A similar approach was implemented in the time-stratified, “epoch” DEC model (Ree and Smith 2008; Landis 2017): the phylogeny is divided into time intervals, and each interval is assigned a different set of values that scale the baseline dispersal rate according to paleogeographic information; for example, the availability of temporal land bridges facilitating migration between continents (Buerki et al. 2011). Time-stratified DEC models can also be used in biogeographic dating (Landis 2017). Rather than assuming a single CTMC process over time, DEC is allowed to shift between different Q matrices at discrete time points, based on paleogeographic evidence. Phylogeny, molecular dating and biogeographic parameters are jointly estimated using hierarchical BI. Paleogeographic data, that is, the formation of dispersal corridors and barriers over time, is used to inform the rates of a piecewise-constant epoch DEC model, and these time-dependent CTMC probabilities are used in turn to inform estimates of species divergence times in the phylogeny; for example, species can only diverge in allopatry if a paleogeographic barrier is present (Landis 2017).
Another exciting approach is the modeling of non-stationary CTMC models, where the equilibrium frequencies are allowed to change at discrete time points between time slices (Sanmartín 2020). Changes in area carrying capacities could result from a global extinction event that wipes out the biota of an island, decreasing its standing carrying capacity, and thus changing the stationary properties of the CTMC dispersal process. The point in time when there is a change in equilibrium frequencies and also the intensity of the extinction event (which might vary between areas) can be estimated by BI (Sanmartín 2020). Alternatively, the CTMC process may never attain equilibrium, or start with different values at root, such as in a directional CTMC process (Klofstein et al. 2015).
2.5.2. Diversification-dependent models
The latest exciting developments in parametric biogeography have been in the direction of implementing “state-dependent speciation and extinction (SSE) models”, in which there is a causal relationship between range evolution and lineage diversification (Maddison et al. 2007, Goldberg et al. 2011; FitzJohn 2012). As explained above, BIB and DEC do not include a speciation parameter in the stochastic CTMC process that governs geographic evolution. This is unrealistic since diversification and range evolution clearly interact: for example, the dispersal of a species into a new region may result in increased speciation rates due to lower competition or access to novel environmental resources (Moore and Donoghue 2007). Moreover, unlike the DEC model, SSE models provide a complete parametric description of biogeographic evolution, since speciation is a rate parameter in the CTMC process. In the geological state-dependent speciation and extinction model (GeoSSE; Goldberg et al. 2011), the Q matrix includes parameters for anagenetic range expansion and range contraction or extinction, as well as parameters for lineage speciation within single areas (SA, SB) or within a widespread range (SAB). There is also a parameter for lineage extinction within single areas (EA, EB): for widespread ranges, this is modeled as the sum of extinction events in single areas. All these parameters are time-dependent. The SSE counterpart of DEC+J is the ClaSSE model (Goldberg and Igic 2012), which allows changes in states to occur not only along branches (anagenetic) but also at speciation nodes (cladogenetic): this “founder-speciation” event is governed by its own time-dependent rate parameter in the Q matrix.
Coupling diversification with range evolution, as in GeoSSE and ClaSSE, allows statistical testing of classical hypotheses, such as whether widespread ranges lead to higher speciation rates (Goldberg et al. 2011) or whether extinction rates are dependent on area size or environmental heterogeneity (Meseguer et al. 2015). A shortcoming of SSE models is their computational complexity. The stationary distributions and parameter probabilities in SSE models are estimated through numerical integration, rather than analytically by matrix exponentiation as in DEC. One attractive avenue forward to tackle these computationally intractable models is the probabilistic programming language (PPL) framework (Ronquist et al. 2020).
2.5.3. Ecology-integrative models
Another exciting development in recent years is the long-sought-after integration of the ecological and historical (phylogenetic) sides of biogeography. Ecological biogeography is defined as dealing with environmental factors and evolutionary processes that act at short time scales and individual or population levels, such as biotic interaction (facilitation, competition), environmental filtering or random genetic drift. Historical biogeography is concerned with deep-time geological events and species-level evolutionary processes, such as dispersal, extinction or speciation (Sanmartín 2012). The distinction between these two approaches has become blurred. For example, a vicariance barrier can be geological (e.g. a mountain) or environmental (e.g. climatically inhospitable land where a species cannot maintain viable populations).
Similarly, for overland dispersal, both the physical bridge and the right environmental conditions along the corridor are a requisite (Donoghue 2008). Ecological niche models can be used to find areas that are within the environmental tolerances of a species, and this information can be used in a biogeographic analysis for modeling the probability of dispersal along corridors or across barriers (Smith and Donoghue 2010). The ecological preferences of ancestors can also be incorporated through the inclusion of fossil, extinct taxa in the analysis; this offers great potential for reconstructing species distributions over the distant past (Meseguer et al. 2015). Ecological processes such as competition and environmental filtering can be modeled in Quintero and Landis’s (2019) composite biogeographic-trait evolutionary model: the rates of range expansion and range contraction depend on the trait values of other co-distributed species (effect of competition on biogeography), while the rate of divergence and convergence of trait values in a species depends on its sympatry with other species, gained or lost via colonization and extinction rates (effect of biogeography on traits).
2.6. Population-level and individual-based models
All models described above were designed to deal with phylogenies in which the terminal tips represent individual species (though BIB-DTA has been used in a phylogeographic context). CTMC processes are less appropriate to model the geographic evolution of individuals within a population, or between closely related populations, because they require the a priori definition of discrete geographic ranges and assume that movement between states is rare, that is, the chain remains in the same state and rarely jumps among states. When dealing with within-species biogeography or phylogeography, it is often difficult to define geographic ranges because boundaries are blurred by the frequent movement of individuals within populations and by gene flow. A Brownian Motion (BM) process, also termed “random walk” or diffusion model, is typically used for modeling the geographic evolution of populations and individuals (Lemey et al. 2011). This is a stochastic process with one parameter governing range evolution: there is a central value from where individuals move away with speed equal to this parameter. Unlike in biogeographic Markov models, tips in the phylogeny are individuals with associated geographical coordinates. Finally, models based on electric circuit-resistance theory (McRae et al. 2008) have been used in phylogeography to model the rate and path of movement or gene flow on heterogeneous landscapes. A special attraction of this model is the possibility to define connectivity maps based on 2D landscapes with barriers: low resistances are assigned to landscape feature types that are most permeable to movement or best promote gene flow, and high resistances assigned
