# LTT

##### 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.

- …
arguments passed to

`plot`

(e.g.,`log="y"`

).

##### 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 an example).

##### References

Hallinan, N. (2012) The generalized time variable reconstructed
birth--death process. *Journal of Theoretical Biology*,
**300**, 265--276.

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

Paradis, E. (2015) Random phylogenies and the distribution of
branching times. *Journal of Theoretical Biology*, **387**,
39--45.

##### See Also

##### Examples

```
# NOT RUN {
### 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)
# }
# NOT RUN {
### 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 5.3, License: GPL (>= 2)*