Mortality, immigration and emigration
Lindeque (1988) saw the Etosha population as part of a larger
regional population with movements taking place both into
and out of the park.
Martin (2005) modeled the Etosha
elephant population to gauge the status of the population
from one year to the next. The model regards the status
of the population from year to the next as a function of:
- The intrinsic rate of growth of the population;
- The mortality during the year concerned;
- The immigration into or emigration from
the population during the year concerned.
Martin (2005) tested the model predictions
for migrations by examining the effect of confidence intervals
on the population estimates.
Modelling
The model population in any year can be expressed as
Where
At first glance, there would appear to be a tautology in
these relationships. The two terms involving expected and
actual deaths actually cancel out, so that no matter what
finding factor is specified for actual mortality, the net
migration is adjusted to give the correct population estimate
in each year. The apparent tautology is the key to the
model solution.
The model is attempting the extreme. We are not sure to what
extent the mortality record represents the actual deaths (i.e.
we don't know the finding factor), we don't know the population
growth rate (more specifically we don't know the central mortality
for the population) and, finally, the immigration and emigration
are a movable feast. If we set the finding factor high, we
obtain relatively low mortalities and hence lower levels of
immigration for years in which the population increases and
higher levels of emigration for years in which the population
decreases. If we set the finding factor low, we obtain relatively
high mortalities and hence higher levels of immigration for
years in which the population increases and lower levels of
emigration for years in which the population decreases. Either
way we end up with a model population which faithfully tracks
the population changes year by year.
There is one additional factor that has been overlooked -
the difference between the starting population in 1971 (679
animals) and the ending population which would have been obtained
with no immigration or emigration. If we take a population
of 679 animals in 1971 and allow it to increase at an intrinsic
growth rate of 3.3% assuming no immigration or emigration,
its value after 23 years is 895 animals - a net change of
217 animals. When the difference between the actual and expected
mortalities over the same time period is added to this, the
change becomes 362 animals. A key assumption in the model
is that the net difference between the total number of animals
which have immigrated into the population and the total number
of animals which have left the population between 1971 and
2004 must be equal to this change.
Only one value for the central mortality will give this
result, i.e. we have a unique solution. When the central
mortality is set at 1.045% (giving a population growth rate
of 3.296%) the difference between immigration and emigration
is equal to the change which would have taken place in the
population with no migration. This result is independent of
the assumed finding factor for the reason given at the top
of this page.
From the relationship given on the first page of this appendix,
a central mortality of 1.045% implies a total mortality of
3.31%. By adjusting the finding factor, the total mortality
over the period 1971-2003 can be set to this value (see model
table - 2,021 deaths between 1971 and 2003 when the finding
factor is set to 0.5215). This too is a unique solution.
And, having established the 'actual' mortalities in each
year, the profile of immigration and emigration is automatically
set. The Etosha population estimates and the model values
for mortality and migration are shown in Fig.24. The 'growth
history' of the population is discussed in the main body of
this report and will not be duplicated here.
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The intrinsic rate of growth of the population
The model assumes that this is a constant. Lindeque (1988)
found a fecundity of 0.25 for adult female elephants in Etosha
(i.e. one calf every four years). The mortality schedule for
Etosha elephants is one of the unknowns sought from the modelling
process. For a typical elephant population with a fecundity
of 0.25 and an age-specific mortality of 0.5% for animals
between 5-45 years the expected growth rate is 4.56% per year
(see page 7). Martin (2005) began this analysis by assuming
this growth rate and later modified it to achieve correspondence
between the observed population estimates and the model.
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Mortality
Some clarity is needed in defining the processes involved
here. A population with an intrinsic growth rate of 4.56%
per annum and a central mortality of 0.5% (Figure
22) produces an expected number of deaths in maintaining
a net annual growth of 4.56%. For any given size of population
and a fecundity of 0.25 calves per year, the total mortality
(%) is given fairly accurately by the following formula
A population of 1,000 animals with a central mortality of
0.5% produces about 18 deaths in the course of increasing
to 1,046 animals in the following year. In the model table
on page 81, deaths arising in this manner are shown in the
column 'Expected deaths'.
Elephant deaths have been recorded at Etosha since 1971.
The data shown in the model table from 1971-1987 are from
Lindeque (1988) and those for 1988-2003 are from Kilian (2004).
Lindeque gives grouped totals for the periods 1971-1979, 1980-1983
and 1984-1987 and the deaths have been evenly allocated over
the time span involved in the table.
A further unknown enters the analysis here. No mortality
collection ever captures 100% of the deaths which occur in
any year and the finding factor (i.e. the proportion of total
deaths recorded) is a critical value in attempting to make
use of the data. The model assumes that this proportion remains
the same over the period 1971-2004 but this assumption may
not be true. During the period of Lindeque's study, the intensity
of data collection (1983-1987) may have been higher than at
other times. The finding factor is treated as one of the variables
to be modelled and the values which appear in the column 'Deaths'
in the model table are simply the recorded deaths divided
by the assumed finding factor.
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Immigration and emigration
The net migration in any given year is calculated on the
assumption that each population estimate is correct with zero
confidence interval (later we examine the effect of confidence
intervals) and that any difference between two successive
estimates which does not match what would have been expected
from intrinsic growth is due to migration.
- In any given year the population from the year before
is increased by the intrinsic growth rate;
- The difference between the actual deaths and expected
deaths in any given year is deducted from (1);
- the immigration/emigration in any given year is calculated
by subtracting (2) from the population in the year concerned.
i.e. -
where
| Pt |
is the population in year t |
| Pt-1 |
is the population from the previous
year |
| Rg |
is the intrinsic rate of growth of the
population (%) |
| Da |
is the actual number of deaths in the
previous year |
| De |
is the expected number of deaths in
the previous year |
Some of the effects of this relationship may be counterintuitive.
The greater the mortality in a year in which the population
increases, the greater the immigration needed to achieve the
next given population level (Figure 23). If the mortality
is higher than expected in a year in which the population
decreases, the less is the emigration required to match the
new population level. If the assumed finding factor is low,
the true number of deaths is high and immigration must exceed
emigration by a large amount in order to maintain the population.
A high finding factor implies that the recorded number of
deaths is close to the true number and, for the Etosha data,
emigration must exceed immigration to achieve correspondence
between modelled data and the population estimates.
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The effect of confidence intervals on population estimates
So far this modelling has assumed perfect estimates. In practice,
a 95% confidence interval approximately equal to + two standard
deviations of the mean would be associated with every estimate.
Each estimate for immigration or emigration predicted by the
model has been tested to find the level at which it would
be significant. The method used was simply to examine each
estimate in relation to the preceding year and if it fell
outside the value obtained by adding a specified confidence
interval to the estimate for the year before (taking into
account the expected population growth), it was treated as
significant (Figure
25). The process is equivalent to postulating that if
the standard deviations of two successive estimates do not
overlap, the predicted migration probably occurred.
The immigration of some 875 animals in 1978 is significant
at the 100% level and the immigration of 400 animals in 1973
is significant at a level of 45%. Thereafter a further immigration
of 613 animals in 1987 and emigrations of 956 animals in 1985,
449 in 1974 and 383 in 1977 appear significant at the level
of 30%. All other estimates fall below the 30% significance
level and, since few surveys produce confidence intervals
better than this, in sensu strictu there is no basis for accepting
that any population movements actually took place in these
years. However, these tests are by no means exhaustive. Lindeque
(1988, p236) points out that the increase from 1979-1983 exceeded
the normal rate of increase for elephant populations and that
the decline from 1983-1986 could not be explained in terms
of recorded mortalities. A deeper consideration of the model
data might show that any sequence of predicted emigrations,
such as occurred between 1988 and 1995, may be highly significant
over the period concerned although each successive instance
does not appear significant.
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