leslie.matrix: Leslie Matrix
In demogR: Analysis of Age-Structured Demographic Models

Description Usage Arguments Details Value References See Also Examples

Generates a Leslie matrix for demographic projection from vectors of age-specific cumulative survival and fertility.

leslie.matrix(lx, mx, 
		    L = TRUE, 
		    peryear = 5, 
		    one.sex = TRUE, 
		    SRB = 1.05, 
		    infant.class = TRUE)

`lx`	vector of either age-specific cumulative survival or person-years lived in the interval
`mx`	age-specific fertility rates
`L`	logical; if 'TRUE', `lx` is taken to be person-years lived in the interval nLx, while if 'FALSE', `lx` is taken to be cumulative survival to exact age x+n. Defaults to 'TRUE'.
`peryear`	Multiplier for fertility. Defaults to `peryear=5`.
`one.sex`	logical; If 'TRUE', fertility rates will be divided by (1+SRB).
`SRB`	sex ratio at birth. Defaults to `SRB=1.05`.
`infant.class`	logical; 'TRUE' if `lx` contains a value for the infant age-class.

Constructs a k x k age-classified demographic projection matrix with age-specific survival probabilities along the sub-diagonal and age-specific fertilities along the first row of the matrix.

lx and mx are assumed to be of the same length. The resulting matrix is truncated to insure that there are no post-reproductive classes. This is important for ensuring irreducibility of the resulting matrix.

If mx is longer than lx, mx is trucated to be the same length as lx. If lx is longer than mx, a warning is issed and lx is truncated to be the same length as mx.

Fertility is assumed to be birth-flow (Caswell 2001). That is, breeding is assumed to be continuous and the individual elements of the first row of the Leslie matrix are averaged over successive age-classes. Fertility rates are typically given in annualized form. If this is the case and the age-classes are wider than one year, then peryear can be used to appropriately scale up the annual values.

The default behavior is to use person-years lived in the interval as the survival measure. If infant.class=TRUE, lx is taken to have a value for the infant age class (i.e., a shorter class width than the other elements of lx. What is done when there is an infant class depends on what the values in lx represent. If L=TRUE, then the first two values of lx are combined to form the total person-years for the first age-class in the Leslie matrix. Human demographic data from abridged life tables typically come with age classes x = 0, 1, 5, 10, ... Thus, combining the person-years for the first two age classes gives an initial age class of the correct width. If infant.class=TRUE and L=FALSE, the second element of lx is deleted. Creating a Leslie matrix from other forms of non-standard early age-classes can be accomplished by pre-processing lx and using the option infant.class=FALSE.

The human sex ratio at birth (male births/female births) is remarkably close to SRB=1.05 across a wide range of populations and this is the default value for SRB.

The resulting matrix has class "leslie.matrix". This class is not used extensively but will be in future development.

A k x k age-classified demographic projection matrix with class "leslie.matrix".

Keyfitz, N. 1977. Introduction to the mathematics of populations. 2nd ed. Menlo Park: Addison-Wesley.

Preston, S.H., P. Heuveline, and F. Guillot. 2001. Demography: Measuring and modeling population processes. Oxford: Blackwell.

Caswell, H. 2001. Matrix population models: Construction, analysis, and interpretation. 2nd ed. Sunderland, MA: Sinauer.

odiag, leslie.row1

## Construct lx and mx values
data(goodman)
mlt <- with(goodman, life.table(x=age, nDx=mad.nDx, nKx=mad.nKx))
mx <- goodman$mad.bx/goodman$mad.nKx

## Now make the Leslie matrix
mad <- leslie.matrix(lx=mlt$nLx, mx=mx)

## Might as well do something with it too!
eigen.analysis(mad)