## Computation of the First Passage Time From Trajectories

### Description

These functions compute the first passage time using trajectories of class `"ltraj"` of type II (time recorded).

### Usage

```fpt(lt, radii, units = c("seconds", "hours", "days"))
varlogfpt(f, graph = TRUE)
meanfpt(f, graph = TRUE)
## S3 method for class 'fipati'
plot(x, scale, warn = TRUE, ...)
```

### Arguments

 `lt` an object of class `"ltraj"` of type II (time recorded) `radii` a numeric vector giving the radii of the circles `units` The time units of the results `f,x` an object of class `fipati` returned by the function `fpt` `graph` logical. Whether the results should be plotted `scale` the value of the radius to be plotted `warn` logical. Whether the function should warn the user when the given scale does not correspond to possible radii available in the object of class `fipati` `...` additional arguments to be passed to the generic function `plot`

### Details

The first passage time (FPT) is a parameter often used to describe the scale at which patterns occur in a trajectory. For a given scale r, it is defined as the time required by the animals to pass through a circle of radius r. Johnson et al. (1992) indicated that the mean first passage time scales proportionately to the square of the radius of the circle for an uncorrelated random walk. They used this property to differenciate facilitated diffusion and impeded diffusion, according to the value of the coefficient of the linear regression `log(FPT) = a * log(radius) + b`. Under the hypothesis of a random walk, `a` should be equal to 2 (higher for impeded diffusion, and lower for facilitated diffusion). Note however, that the value of a converges to 2 only for large values of radius.

Fauchald & Tveraa (2003) proposed another use of the FPT. Instead of computing the mean of FPT, they propose the use of the variance of the log(FPT). This variance should be high for scales at which patterns occur in the trajectory (e.g. area restricted search). This method is often used to determine the scale at which an animal seaches for food.

### Value

`fpt` computes the FPT for each relocation and each radius, and for each animals. This function returns an object of class `"fipati"`, i.e. a list with one component per animal. Each component is a data frame with each column corresponding to a value of `radii` and each row corresponding to a relocation. An object of class `fipati` has an attribute named `"radii"` corresponding to the argument `radii` of the function `fpt`.

`meanfpt` and `varlogfpt` return a data frame giving respectively the mean FPT and the variance of the log(FPT) for each animal (rows) and rach radius (column). These objects also have an attribute `"radii"`.

### Author(s)

Clement Calenge clement.calenge@ofb.gouv.fr

### References

Johnson, A. R., Milne, B.T., & Wiens, J.A. (1992) Diffusion in fractal landscapes: simulations and experimental studies of tenebrionid beetle movements. Ecology 73: 1968–1983.

Fauchald, P. & Tveraa, T. (2003) Using first passage time in the analysis of area restricted search and habitat selection. Ecology 84: 282–288.

`ltraj` for additional information on objects of class `ltraj`

### Examples

```
data(puechcirc)
i <- fpt(puechcirc, seq(300,1000, length=30))
plot(i, scale = 500, warn = FALSE)

toto <- meanfpt(i)
toto