4.1), l1 = 254/746 = 0.340 for the inland subspecies and 204/754 = 0.271 for the coastal subspecies. That is, 34% and 27.1% survived the first step to become established plants in the two cases: a slightly higher survival rate at this inland site for the inland than for the coastal subspecies.
In the next column, to consider mortality more explicitly, the proportion of the original cohort dying during each stage (dx ) is computed, being simply the difference between successive values of lx ; for example, for the marmots, d3 = l3 – l4 = 0.180 − 0.137 = 0.043. Next, the stage‐specific mortality rate, qx, is computed. This considers dx as a fraction of lx. Hence, q3 for example is 0.24 (= 0.043/0.180 or d3/l3). Values of qx may also be thought of as the average ‘chances’ or probabilities of an individual dying during an interval. qx is therefore equivalent to (1 − px ) where p refers to the probability of survival.
The advantage of the dx values is that they can be summed: thus, the proportion of a cohort of marmots dying in the first four years was d0 + d1 + d2 + d3 (= 0.86). The disadvantage is that the individual values give no real idea of the intensity or importance of mortality during a particular stage. This is because the dx values are larger the more individuals there are, and hence the more there are available to die. The qx values, on the other hand, are an excellent measure of the intensity of mortality. For instance, in the present example it is clear from the qx column that the mortality rate declined after the first two years of life but then rose again to a peak around years 9 and 10; this is not clear from the dx column. The qx values, however, have the disadvantage that, for example, summing the values over the first four years gives no idea of the mortality rate over that period as a whole.
k values
The advantages are combined, however, in the next column of the life table, which contains kx values (Haldane, 1949 ; Varley & Gradwell, 1970). kx is defined simply as the difference between successive values of log10 ax or successive values of log10 lx (they amount to the same thing), and is sometimes referred to as a ‘killing power’. Like qx values, kx values reflect the intensity or rate of mortality (as Tables 4.1 and 4.2 show); but unlike summing the qx values, summing kx values is a legitimate procedure. Thus, the killing power or k value for the first four years in the marmot example is 0.26 + 0.31 + 0.18 + 0.12 = 0.87, which is also the difference between log10 a0 and log10 a4 (allowing for rounding errors). Note too that like lx values, kx values are standardised, and are therefore appropriate for comparing quite separate studies. In this and later chapters, kx values will be used repeatedly.
fecundity schedules
Tables 4.1 and 4.2 also include fecundity schedules for Gilia and for the marmots (the final three columns). The first of these in each case shows Fx, the total number of the youngest age class produced by each subsequent age class. This youngest class is seeds for Gilia, produced only by the flowering plants. For the marmots, these are independent juveniles, fending for themselves outside their burrows, produced when adults were between 2 and 10 years old. The next column is then said to contain mx values, which is fecundity: the mean number of the youngest age class produced per surviving individual of each subsequent class. For the marmots, fecundity was highest for eight‐year‐old females: 1.68, that is, 37 young produced by 22 surviving females. We get a good idea of the range of fecundity schedules in Figure 4.2: some with constant fecundity throughout most of an individual’s life, some in which there is a steady increase with age, some with an early peak followed by an extended postreproductive phase. We try to account for some of this variation in the next chapter.
… combined to give the basic reproductive rate
In the final column of a life table, the lx and mx columns are brought together to express the overall extent to which a population increases or decreases over time – reflecting the dependence of this on both the survival of individuals (the lx column) and the reproduction of those survivors (the mx column). That is, an age class contributes most to the next generation when a large proportion of individuals have survived and they are highly fecund. The sum of all the lx mx values, ∑lx mx, where the symbol ∑ means ‘the sum of’, is therefore a measure of the overall extent by which this population has increased or decreased in a generation. We call this the basic reproductive rate and denote it by R0 (‘R‐nought’). That is:
(4.2)
We can also calculate R0 by dividing the total number of offspring produced during one generation (∑Fx, meaning the sum of the values in the Fx column) by the original number of individuals. That is:
(4.3)
For Gilia (Table 4.1), R0 is calculated very simply (no summation required) since only the flowering class produces seed. Its value is 38.27 for the inland subspecies and 11.47 for the coastal subspecies: a clear indication that the inland subspecies thrived, comparatively, at this inland site. (Though the annual rate of reproduction would not have been this high, since, no doubt, a proportion of these would have died before the start of the 1994 cohort. In other words, another class of individuals, ‘winter seeds’, was ignored in this study.)
For the marmots, R0 = 0.67: the population was declining, each generation, to around two‐thirds its former size. However, whereas for Gilia the length of a generation is obvious, since there is one generation each year, for the marmots the generation length must itself be calculated. We address the question of how to do this in Section 4.7, but for now we can note that its value, 4.5 years, matches what we can observe ourselves in the life table: that a ‘typical’ period from an individual’s birth to giving birth itself (i.e. a generation) is around four and a half years. Thus, Table 4.2 indicates that each generation, every four and a half years, this particular marmot population was declining to around two‐thirds its former size.
4.6.2 Survivorship curves
It is also possible to study the detailed pattern of decline in a cohort. Figure 4.10a, for example, shows the numbers of marmots surviving relative to the original population – the lx values – plotted against the age of the cohort. However, this can be misleading. If
