Stage-dependent life-tables

   Stage-dependent life tables are built in the cases when:
     * The life-cycle is partitioned into distinct stages (e.g., eggs, larvae, pupae
       and adults in insects)
     * Survival and reproduction depend more on organism stage rather than on
       calendar age
     * Age distribution at particular time does not matter (e.g., there is only one
       generation per year)

   Stage-dependent life tables are used mainly for insects and other terrestrial
   invertebrates.

   Example. Gypsy moth (Lymantria dispar L.) life table in New England (modified
   from Campbell 1981)

   Stage Mortality factor Initial no. of insects No. of deaths Mortality (d)
   Survival (s) k-value [-ln(s)]
   Egg Predation, etc. 450.0 67.5 0.150 0.850 0.1625
   Egg Parasites 382.5 67.5 0.176 0.824 0.1942
   Larvae I-III Dispersion, etc. 315.0 157.5 0.500 0.500 0.6932
   Larvae IV-VI Predation, etc. 157.5 118.1 0.750 0.250 1.3857
   Larvae IV-VI Disease 39.4 7.9 0.201 0.799 0.2238
   Larvae IV-VI Parasites 31.5 7.9 0.251 0.749 0.2887
   Prepupae Desiccation, etc. 23.6 0.7 0.030 0.970 0.0301
   Pupae Predation 22.9 4.6 0.201 0.799 0.2242
   Pupae Other 18.3 2.3 0.126 0.874 0.1343
   Adults Sex ratio 16.0 5.6 0.350 0.650 0.4308
   Adult females   10.4
   TOTAL     439.6 97.69 0.0231 3.7674

   Specific features of stage-dependent life tables:
     * There is no reference to calendar time. This is very convenient for the
       analysis of poikilothermous organisms.
     * Gypsy moth development depends on temperature but the life table is
       relatively independent from weather.
     * Mortality processes can be recorded individually and thus, this kind of life
       table has more biological information than age-dependent life tables.

K-values

   K-value is just another measure of mortality. The major advantage of k-values as
   compared to percentages of died organisms is that k-values are additive: the
   k-value of a combination of independent mortality processes is equal to the sum
   of k-values for individual processes.

   Mortality percentages are not additive. For example, if predators alone can kill
   50% of the population, and diseases alone can kill 50% of the population, then
   the combined effect of these process will not result in 50+50 = 100% mortality.
   Instead, mortality will be 75%!

   Survival is a probability to survive, and thus we can apply the theory of
   probability. In this theory, events are considered independent if the probability
   of the combination of two events is equal to the product of the probabilities of
   each individual event. In our case event is survival. If two mortality processes
   are present, then organism survives if it survives from each individual process.
   For example, an organism survives if it was simultaneously not infected by
   disease and not captured by a predator.

   Assume that survival from one mortality source is s1 and survival from the second
   mortality source is s2. Then survival from both processes, s12, (if they are
   independent) is equal to the product of s1 and s2:

                                       [eq9.gif]

   This is a "survival multiplication rule". If survival is replaced by 1 minus
   mortality [s=(1-d)], then this equation becomes:

                                      [eq10.gif]

   For example, if mortality due to predation is 60% and mortality due to diseases
   is 30%, then the combination of these two death processes results in mortality of
   d = 1-(1-0.6)(1-0.3)=0.72 (=72%).

   Varley and Gradwell (1960) suggested to measure mortality in k-value which is the
   negative logarithms of survival:

                                      k = -ln(s)

   We use natural logarithms (with base e=2.718) instead of logarithms with base 10
   used by Varley and Gradwell. The advantages of using natural logarithms will be
   shown below.

   It is easy to show that k-values are additive:

                                      [eq11.gif]

   The k-values for the entire life cycle (K) can be estimated as the sum of
   k-values for all mortality processes:

                                      [eq11a.gif]

   In the life table of the gypsy moth (see above), the sum of all k-values (K =
   3.7674) was equal to the k-value of total mortality.

   [gmort2k.gif] This graph shows the relationship between mortality and the
   k-value. When mortality is low, then the k-value is almost equal to mortality.
   This is the reason why the k-value can be considered as another measure of
   mortality. However, at high mortality, the k-value grows much faster than
   mortality. Mortality cannot exceed 1, while the k-value can be infinitely large.

   The following example shows that the k-value represents mortality better than the
   percentage of dead organisms: One insecticide kills 99% of cockroaches and
   another insecticide kills 99.9% of cockroaches. The difference in percentages is
   very small (