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Fig. 2. Illustration of a Brownian motion (random walk in continuous time) model of
character evolution, as might be implemented in a computer program (e.g.
PDSIMUL of Garland et al.,
1993). The goal is to simulate the evolution of two traits,
beginning at the bottom of the phylogenetic tree and ending at the three tips,
species A, B and C. A computer program begins at the bottom of the tree
(internal node `F') with user-specified starting values, in this example 10
and 10 for Traits 1 and 2, respectively. It then draws a random datum from a
bivariate normal distribution of hypothetical evolutionary changes for the two
traits. This distribution is illustrated by concentric rings proportional to
density of data points in the z axis (projecting out of the page),
with darker indicating a higher density of points; the tails of the
distribution diminish to infinity). In this example, we assume that the means
of this distribution are 0 for both traits, such that no general tendency for
either to increase or decrease will be modeled. We also specify 0 correlation
between them, such that they will `evolve' independently, on average. For the
amount of evolutionary change from node F to tip species C, we happen to draw
values of –4 and –2 (red). Thus, species C has values of 6 and 8.
For the change from node F to G, we draw +3 and +2 (blue). Above this, we draw
two separate sets of changes: +1 and –2 leading to tip species A
(green); –1 and +2 leading to tip species B (purple). Note that the
amount of change tends to be greater for longer branches, reflecting a greater
opportunity for evolutionary change. In practice, a computer program might
achieve this by expanding or contracting the widths of the bivariate normal
distribution for relatively longer or shorter branch lengths, respectively.
Thus, under Brownian motion, for a given character, the variance of this
distribution is set to be proportional to divergence time (along the length of
each branch segment sequentially; Felsenstein,
1985,
1988). Note also that the
distribution from which changes are drawn does not need to be (bivariate)
normal (see Felsenstein, 1985,
1988). Moreover, the means of
the distribution can be set to positive or negative values to impose
directional trends in character evolution
(Garland et al., 1993). These
sorts of changes create models that are no longer simple Brownian motion.
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