In Hanski, I. and O. Gaggiotti (eds.)

I consider a series of simple mathematical models describing the dynamics of

speciation and diversification in metapopulations. The system under consideration

is comprised by a number of "patches" arranged on one- or two-dimensional

lattices. Populations are sexual, generations are discrete and

non-overlapping, the number of loci is large, each locus is diallelic. Each

population is described by a sequence of genes in its most common genotype.

Each population is subject to mutation, extinction, and recolonization.

Populations accumulate genetic changes which, as a by-product, lead to

reproductive isolation. Reproductive isolation is described in terms of the

Bateson-Dobzhansky-Muller model and its multi-dimensional generalization

- holey adaptive landscapes. Populations are assigned to different genetic clusters

(e.g., subspecies, species, genera, etc.) using the single-linkage clustering technique.

At time zero all populations in the clade are identical genetically.

I am interested in the following characteristics: the waiting time to the beginning

of radiation, the duration of radiation, the number of clusters (that is, the overall

diversity)

and their spatial ranges at a stochastic equilibrium, the degree of genetic variation

within clusters, the clade's overall disparity and turnover rate, the average genetic

distance to the species-founder. The important

parameters of the models are the dimensionality of the system, the overall number

of patches $n$, the rates of mutation $\nu$ and extinction-recolonization $\gd$, and

the number

of genetic changes $K$ required for reproductive isolation (or for the population to

be assigned to a different cluster). The following results are based on both analytical

approximations and numerical simulations.

The waiting time to the beginning of radiation is on the order of $K/\nu$

generations. The duration of radiation depends mostly on the fixation rate.

The transient dynamics of the diversity and the disparity

are decoupled to a certain degree. At low taxonomic levels, the

diversity increases faster than the disparity, whereas at high taxonomic

levels the diversity increases slower than the disparity.

The clade as a whole keeps changing genetically even after its diversity has

approached an equilibrium level.

The turnover rates do not depend (or weakly depend) on extinction rate $\gd$

and are mostly controlled by parameters $\nu$ and $K$.

Diversification requires that the overall number of patches in (or spatial

area of) the system exceeds a certain minimum value.

In general, the characteristics of two-dimensional systems

(such as describing oceans, continental areas, etc.)

are (much) more sensitive to parameter values than those of one-dimensional

systems (such as describing rivers, shores of lakes and oceans, areas at a

constant elevation in a mountain range, etc.).

The diversity in one-dimensional system is (much) higher than

that in two-dimensional systems.