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To monitor and quantify survival, we may follow the fate of individuals from the same cohort within a population: that is, all individuals born within a particular period. The life table then records the survivorship of the members of the cohort over time (Figure 4.9). The most straightforward life table to construct is a cohort life table for an annual species. Putting to one side the caveats raised above, annual life cycles take approximately 12 months or rather less to complete (Figure 4.6b, c). Usually, every individual in a population breeds during one particular season of the year, but then dies before the same season in the next year. Generations are therefore said to be discrete, and each cohort is distinguishable from every other; the only overlap of generations is between breeding adults and their offspring during and immediately after the breeding season.

an annual life table for a plant

Two very simple life tables, for inland and coastal subspecies of the annual plant Gilia capitata, growing in California, USA are shown in Table 4.1. Initial cohorts of around 750 seeds were followed from seed germination to the death of the last adult.

Table 4.1 Two cohort life tables for the annual plant Gilia capitata. One is for the ‘inland’ subspecies, G. capitata capitata, and one for the ‘coastal’ subspecies, G. capitata chamissonis, growing at an inland site in Napa County, California, USA and being easily distinguishable morphologically, despite being cross‐fertile. Cohorts of seeds were planted at the beginning of the season in 1993 and the life cycle divided simply into seeds, plants that emerged from those seeds, and emerged plants that went on to flower. Other column entries are explained in the text. Source: After Nagy & Rice (1997).

Stage (x)	Number alive at the start of each age class ax	Proportion of original cohort surviving to the start of each age class lx	dx	qx	log ax	log lx	kx	Number of female young produced by each age class Fx	Number of female young produced per surviving individual in each age class mx	Number of female young produced per original individual in each age class lxmx
Inland subspecies:
Seed (0)	746	1.00	0.66	0.66	2.87	0.00	0.47	0	0	0
Emergence (1)	254	0.34	0.25	0.74	2.40	–0.47	0.59	0	0	0
Flowering (2)	66	0.09			1.82	–1.05		28,552	432.61	38.29
Coastal subspecies:
Seed (0)	754	1.00	0.73	0.73	2.88	0.00	0.57	0	0	0
Emergence (1)	204	0.27	0.25	0.91	2.31	–0.57	1.03	0	0	0
Flowering (2)	19	0.03			1.28	–1.60		8645	455.00	11.47

a cohort life table for marmots

Even when generations overlap, if individuals can be marked early in their life so that they can be recognised subsequently, it is feasible to follow the fate of each year’s cohort separately. It may then be possible to merge the cohorts from the different years of a study to derive a single, ‘typical’ cohort life table. An example is shown in Table 4.2 of females from a population of the yellow‐bellied marmot, Marmota flaviventris. The population was live‐trapped and marmots marked individually from 1962 through to 1993 in the East River Valley of Colorado, USA and it was this that allowed each individual to be assigned, whenever it was caught, to its own cohort.

Table 4.2 A cohort life table for female yellow‐bellied marmots, Marmota flaviventris in Colorado, USA. The columns are explained in the text. Source: After Schwartz et al. (1998).

Age class (years) x	Number alive at the start of each age class ax	Proportion of original cohort surviving to the start of each age class lx	dx	qx	log ax	log lx	kx	Number of female young produced by each age class Fx	Number of female young produced per surviving individual in each age class mx	Number of female young produced per original individual in each age class lxmx
0	773	1.000	0.457	0.457	2.888	0.00	0.26	0	0.000	0.000
1	420	0.543	0.274	0.505	2.623	−0.26	0.31	0	0.000	0.000
2	208	0.269	0.089	0.332	2.318	−0.57	0.18	95	0.457	0.123
3	139	0.180	0.043	0.237	2.143	−0.75	0.12	102	0.734	0.132
4	106	0.137	0.050	0.368	2.025	−0.86	0.20	106	1.000	0.137
5	67	0.087	0.030	0.343	1.826	−1.06	0.18	75	1.122	0.098
6	44	0.057	0.017	0.295	1.643	−1.24	0.15	45	1.020	0.058
7	31	0.040	0.012	0.290	1.491	−1.40	0.15	34	1.093	0.044
8	22	0.029	0.013	0.455	1.342	−1.55	0.26	37	1.680	0.049
9	12	0.016	0.006	0.417	1.079	−1.81	0.23	16	1.336	0.021
10	7	0.009	0.005	0.571	0.845	−2.04	0.37	9	1.286	0.012
11	3	0.004	0.001	0.333	0,477	−2.41	0.18	0	0.000	0.000
12	2	0.003	0.000	0.000	0.301	−2.59	0.00	0	0.000	0.000
13	2	0.003	0.000	0.000	0.301	−2.59	0.00	0	0.000	0.000
14	2	0.003	0.001	0.500	0.301	−2.59	0.30	0	0.000	0.000
15	1	0.001			0.000	−2.89		0	0.000	0.000
Total

the columns of a life table

The first column in each life table is a list of the stages or age classes of the organism’s life. For Gilia, these are simply the stages ‘seed’, ‘emerged plants’, and ‘flowering plants’. For the marmots, they are years. The second column is then the raw data from each study, collected in the field. It reports the number of individuals surviving to the beginning of each stage or age class (see Figure 4.9). We refer to these numbers as a_x, where the x in the subscript refers to the stage or age class concerned: a₀ means the numbers in the initial age class, and so on.

Ecologists are typically interested not just in examining populations in isolation but in comparing the dynamics of two or more perhaps rather different populations. This was precisely the case for the Gilia populations in Table 4.1. Hence, it is necessary to standardise the raw data so that comparisons can be made. This is done in the third column of the table, which is said to contain l_x values, where l_x is defined as the proportion of the original cohort surviving to the start of age class. The first value in this column, l₀ (spoken: L‐zero), is therefore the proportion surviving to the beginning of this original age class. Obviously, in Tables 4.1 and 4.2, and in every life table, l₀ is 1.00 (the whole cohort is there at the start). Thereafter, in the marmots for example, there were 773 females observed in this youngest age class. The l_x values for subsequent age classes are therefore expressed as proportions of this number. Only 420 individuals survived to reach their second year (age class 1: between one and two years of age). Thus, in Table 4.2, the second value in the third column, l₁, is the proportion 420/773 = 0.543 (that is, only 0.543 or 54.3% of the original cohort survived this first step). In the next row, l₂ = 208/773 = 0.269, and so on. For Gilia (Table 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 (d_x ) is computed, being simply the difference between successive values of l_x ; for example, for the marmots, d₃ = l₃ – l₄ = 0.180 − 0.137 = 0.043. Next, the stage‐specific mortality rate, q_x, is computed. This considers d_x as a fraction of l_x. Hence, q₃ for example is 0.24 (= 0.043/0.180 or d₃/l₃). Values of q_x may also be thought of as the average ‘chances’ or probabilities of an individual dying during an interval. q_x is therefore equivalent to (1 − p_x ) where p refers to the probability of survival.

The advantage of the d_x 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 d_x values are larger the more individuals there are, and hence the more there are available to die. The q_x 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 q_x 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 d_x column. The q_x 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 k_x values (Haldane, 1949 ; Varley & Gradwell, 1970). k_x is defined simply as the difference between successive values of log₁₀ a_x or successive values of log₁₀ l_x (they amount to the same thing), and is sometimes referred to as a ‘killing power’. Like q_x values, k_x values reflect the intensity or rate of mortality (as Tables 4.1 and 4.2 show); but unlike summing the q_x values, summing k_x 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 l_x values, k_x values are standardised, and are therefore appropriate for comparing quite separate studies. In this and later chapters, k_x 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 F_x, 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 m_x 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 l_x and m_x 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 l_x column) and the reproduction of those survivors (the m_x 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 l_x m_x values, ∑l_x m_x, 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 (∑F_x, meaning the sum of the values in the F_x 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.