# LTT

From ape v3.2
by Emmanuel Paradis

##### Theoretical Lineage-Through Time Plots

This function draws the lineage-through time (LTT) plots predicted under a speciation-extinction model (aka birth-death model) with specified values of speciation and extinction rates (which may vary with time).

A prediction interval is plotted by default which requires to define a sample size (100 by default), and different curves can be combined.

- Keywords
- hplot

##### Usage

```
LTT(birth = 0.1, death = 0, N = 100, Tmax = 50, PI = 95,
scaled = TRUE, eps = 0.1, add = FALSE, backward = TRUE,
ltt.style = list("black", 1, 1), pi.style = list("blue", 1, 2))
```

##### Arguments

- birth
- the speciation rate, this may be either a numeric value
or a funtion of time (named
`t`

in the code of the function). - death
- id. for the extinction rate.
- N
- the size of the tree.
- Tmax
- the age of the root of the tree.
- PI
- the percentage value of the prediction interval; set this value to 0 to not draw this interval.
- scaled
- a logical values specifying whether to scale the $y$-axis between 0 and 1.
- eps
- a numerical value giving the resolution of the time axis.
- add
- a logical values specifying whether to make a new plot (the default).
- backward
- a logical value: should the time axis be traced from the present (the default), or from the root of the tree?
- ltt.style
- a list with three elements giving the style of the
LTT curve with, respectively, the colour (
`"col"`

), the line thickness (`"lwd"`

), and the line type (`"lty"`

). - pi.style
- id. for the prediction interval.

##### Details

For the moment, this works well when `birth`

and `death`

are
constant. Some improvements are under progress for time-dependent
rates (but see below for and example).

##### References

Paradis, E. (2011) Time-dependent speciation and extinction from
phylogenies: a least squares approach. *Evolution*, **65**,
661--672.

##### See Also

##### Examples

```
### predicted LTT plot under a Yule model with lambda = 0.1
### and 50 species after 50 units of time...
LTT(N = 50)
### ... and with a birth-death model with the same rate of
### diversification (try with N = 500):
LTT(0.2, 0.1, N = 50, PI = 0, add = TRUE, ltt.style = list("red", 2, 1))
### predictions under different tree sizes:
layout(matrix(1:4, 2, 2, byrow = TRUE))
for (N in c(50, 100, 500, 1000)) {
LTT(0.2, 0.1, N = N)
title(paste("N =", N))
}
layout(1)
### speciation rate decreasing with time
birth.logis <- function(t) 1/(1 + exp(0.02 * t + 4))
LTT(birth.logis)
LTT(birth.logis, 0.05)
LTT(birth.logis, 0.1)
```

*Documentation reproduced from package ape, version 3.2, License: GPL (>= 2)*

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