Elizabeth E. Holmes. 2004. Beyond theory to application and evaluation: diffusion approximations for population viability analysis. Ecological Applications 14:1272–1292.

Appendix B. The development of a general method for correcting for age-specific inputs into the censused population.

Consider a population A whose stochastic dynamics due to reproduction and survival can be described by a Leslie matrix, A_t, but that experiences external inputs of individuals into the population.  Such a situation can arise, if the population is supplemented such as for management purposes or if it is adjacent to a source population that continually provides immigrants.  If the input individuals reproduce and their offspring cannot be distinguished, their presence (particularly the presence of their offspring) masks the underlying dynamics of population A.  Here I present a method for estimating population A’s underlying growth rate, _A, if the inputs were halted.

Throughout this discussion, I refer to resident and input individuals at year t.  Residents at year t refers all individuals minus the inputs during year t only; residents may include individuals that were externally input at an earlier year or that were born from parents which were externally input in previous years.  Inputs at year t refer only to those individuals that are externally input into the population at year t.  The method assumes that the following data or estimates thereof are available: 1) a census count, O_t; this could be a total population count or an age- or stage-specific count (such as an egg survey or breeder index).  Since residents and inputs are indistinguishable, this count is assumed to include a mixture of both types of individuals, 2) an estimate of the fraction of individuals in O_t that are year t inputs, and 3) an estimate of the ratio of residents to (residents + inputs) for all ages or stages in the population.  Number 3 is not sufficient to give number 2 since the census count may be a combination of ages, such as a total population count, without age information. 

With external inputs, E_t, at year t into a resident population, N_t, the population process is:

where A_t is the stochastic projection matrix for year t.  The objective is to estimate the mean and variance of _A,t, the dominant eigenvalue of A_t.  N_t is the age-specific vector of resident individuals in the population at year t and E_t is the age-specific vector of individuals input into the population at year t:

,

(B.2)

The maximum age of individuals is m.

The vector E_t can be defined in terms of N_t by using the age-specific fraction of residents relative to inputs at year t: 

We can then solve for E_t in terms of N_t:

.

(B.4)

where I is the identity matrix (all ones along the diagonal; zeros elsewhere) and B_t is the matrix with 1/r_i,t on the diagonal.  Combining Eqs. B1 and B4, we have

where B_t is defined in Eq. B4.  Our censuses, combined with information on the fraction of new inputs in the census, give us an estimate N_i,t or of if the census is a combination of ages.  We can then estimate _C,t, the dominant eigenvalue of C_t using the ratio of N_i,t+1 to N_i,t, or the sums thereof (estimation of _C,t from the census data is discussed below). 

However, our goal is to estimate _A,t, the dominate eigenvalue of A_t, thus we need to understand the relationship between _C,t, which we can estimate from our censuses and _A,t, which we want to estimate.  One strategy is to use net reproductive rates, R₀, defined as the mean number of offspring produced by a female over her lifetime.  There is an approximate relationship between the net reproductive rate and : (Caswell 2000)

(B.6)

where T is the mean generation time.  Using this relationship, we can solve for the relationship between the observed _C,t and the unobserved _A,t:

.

(B.7)

The net reproductive rate of matrix A_t is R_0,t.  This is the true net reproductive rate if inputs were not occurring.  The apparent net reproductive rate from the observed time series is .  This is the net reproductive rate of matrix C_t.  The key is to estimate the ratio without knowing much about A_t, in other words not knowing the survivorships, s_i’s, or fecundities, f_i’s.  Depending on the life history and ages at which inputs occur, there may be a simple  relationship that is independent of A_t; at the minimum, one may be able to make a reasonable guess at . 

As an example, suppose that the form of A_t is

(B.8)

The net reproductive rate of A_t in Eq. B8 is (with t subscripts dropped for simplification)

.

(B.9)

The net reproductive rate of the matrix C_t (= A_tB_t as defined in Eq. B4) would then be

.

(B.10)

If individuals enter the population at a single age k, which is at or before the age of first reproduction, = r_k,t where r_k,t is the fraction of resident individuals at age k when the inputs occur.  If this is not the case, going through the exercise of specifying the form of A, R₀, and , even if the parameters are unknown, .  The reader is cautioned to go through this exercise rather than making a “seat of the pants” guess at .  The ratio  is often larger than one would guess since the product of r_i rather than simply r_i appears in Eq. B10.

The last step is to estimate _C,t from the census data.  At equilibrium, _C,t equals the ratio of any element (or sum of elements) of N_t+1 by the corresponding element (or sum) of N_t.  Thus, _C,t could be estimated a number of ways:

If the census, O_t, is of a single age class, k:

 where io_t is the fraction of year t inputs in O_t.

If the census, O_t, is comprised of multiple age classes, j:

 for j = some set of ages

Another option for both these cases would be to use

 rather than

and assume that .  If the fraction of inputs is changing over time, , and using  means that you will correspondingly under- or overestimate _C,t.  Still this may be a justified approximation if your information on inputs, io_t, is rather poor.

With estimates of  and l_C,t, the estimated mean and variance of _A,t are:

.

(B.11)

These are the maximum likelihood estimates.  The example with Pacific salmon in the main text uses running sum and slope estimates instead to deal with extraneous variability in the census counts.

To summarize, the algorithm for input correction involves the following steps:

1) Estimating the fraction of individuals in each year’s census that came from outside the population that year (this io_t).

2) Estimating for year t, what fraction of age i individuals were external inputs from the current year (r_i,t)

3) Estimating the form of A_t..  Estimating at what ages the inputs occur (E_t).  These two can be used to estimated the form of C_t.

4) Using 1-3 above, to estimate or infer .

5) Estimating _C,tfrom the census data.

6) Estimating the mean and variance of _A,tusing  and _C,t.