## Macroevolutionary theory

Imagine a monophyletic clade evolving. What parameters are necessary to describe its evolution? At minimum, we must describe the number of species, but we may also wish to include a description of its disparity. Since describing

 Clade Question Fossil Tree shape Historical Extant property record extinctions radiations Disparity When? ✓ ✓ Where? (✓) ✓ Who? (✓) Speciation When? ✓ (✓) Where? ✓ (✓) Who? ✓ ✓ Extinction When? ✓ (✓) Where? ✓ (✓) ✓ Who? ✓ ✓ ✓ Net When? ✓ ✓ ✓ diversification Where? ✓ ✓ Who? ✓ ✓

Notes: Brackets indicate current absence of studies.

Notes: Brackets indicate current absence of studies.

disparity requires at minimum a description of cladogenesis, we will start by simply describing the number of species and then add in disparity.

In essence the description of the species richness of a clade is very similar to describing the number of individuals in a population. A clade starts off with a single species, which speciates into two or more, which can also speciate. If this is the only process acting on the clade, the clade multiplies exponentially at a rate determined by the speciation rate. It is then simple to describe the relationship between three parameters: the species richness of the clade, its speciation rate, and the age of the clade. Knowledge of any two of these parameters allows us to calculate the third. This simple exponential growth model assumes a constant-rate deterministic speciation process with no extinction. Of course, we can assume extinction without explicitly modelling it, by just renaming the speciation parameter the net rate of diversification, equal to the speciation rate minus the extinction rate.

This model has rather uninteresting dynamics but has two useful applications. First, if a clade's growth conforms to it, it is an indication that no further complex processes are at work, and therefore that the rate of diversification is constant through time. By adding in the assumption of sto-chasticity in rate, we can ask if the diversities of two clades imply simply sto-chasticity in the same underlying rate of diversification, or if it implies different underlying rates. Thus, this simple exponential model allows us to answer two questions about the net rate of cladogenesis: is it constant (when) and does it vary between clades (who)? The model was extensively used by Stanley (1979) to compare radiation rates in different taxa, but originates much earlier in work by the mathematician Yule (1924).

TO CD C

Time

Fig. 14.2 How to estimate speciation and extinction rates from a phylogeny of extant species. In a

Time

### Present

Fig. 14.2 How to estimate speciation and extinction rates from a phylogeny of extant species. In a semi-log plot of the number of lineages over time, the log number of lineages rises at a characteristic rate, equal to the speciation minus the extinction rate (b - d). Species that are newly formed, however, have not had a chance to go extinct, so the rate of formation of species near the present is simply the speciation rate (b). The difference between the two slopes is therefore the extinction rate (d). After Harvey et al. (1994), with permission from the Society for the Study of Evolution.

A further simple step is to add in two discrete parameters describing the rate of speciation and extinction separately. A clade whose speciation rate is greater than its extinction rate grows in a roughly, but not exactly, exponential fashion. Over much of its history growth rate is constant. However, early on in its history its growth rate is faster, because extinction can only act on lineages that have come into existence through speciation. This early rate of increase represents the speciation rate unfettered by extinction. For clades whose past history can only be inferred from the phylogeny of extant species, this speciation rate can also be estimated as the rate of increase in branching in the phylogeny near to the present (Figure 14.2), once again these species are newly formed and have not had a chance to go extinct (Harvey et al. 1994). Once the speciation rate is known, the 'normal' rate of increase allows the net rate of diversification to be calculated, and the difference between the two is the extinction rate. Hence, knowledge of the dynamics of clade growth allows the rate of speciation and extinction to be calculated from the phylogenies of extant species. These rates can be calculated more simply from the fossil record because origination and extinction are more directly observable.

While a clade is radiating at constant (exponential) rate, what happens to its morphological diversity? Perhaps the simplest model assumes that anagenesis can occur in a random walk fashion and that extinction and speciation is random with respect to morphology. If morphological traits are assumed to be binary, as is practically the case with most morphological traits, then as the clade radiates exponentially, its morphological diversity increases rapidly at first but then asymptotes (Gavrilets 1999). The asymptote is caused simply by the fact that the binary nature of the traits places strong (geometric) constraints on continued morphological diversification. For continuous traits, however, which are less constrained geometrically, disparity continues to increase in a linear fashion (Foote 1996).

It seems unlikely in a finite environment that clades will grow unhindered for ever at a constant rate. The simplest model that puts some constraint on exponential growth is the so-called logistic model of Verhulst (1845), whereby as the clade approaches a theoretical carrying capacity its net rate of diversification is reduced by a feedback parameter.A wide variety of dynamics are possible in the logistic model depending on the value of the feedback parameter and the net rate of diversification, from stable equilibrium through to chaotic dynamics. In addition, interactions between taxa are possible within a logistic framework if the feedback is the result not just of the clade's own species richness but that of other clades too. This now allows displacement dynamics whereby taxa can replace each other. More complex models are possible but their necessity depends on the data to which we shall now turn.

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