14.3.1 Key factor analysis

For many years, the demographic approach was represented by a technique called key factor analysis. As we shall see, there are shortcomings in the technique and useful modifications have been proposed, but as a means of explaining important general principles, and for historical completeness, we start with key factor analysis. In fact, the technique is poorly named, since it begins, at least, by identifying key phases (rather than factors) in the life of the organism concerned.

For a key factor analysis, data are required in the form of a series of life tables (see Section 4.5) from a number of different cohorts of the population concerned. Thus, since its initial development (Morris, 1959; Varley & Gradwell, 1968) it has been most commonly used for species with discrete generations, or where cohorts can otherwise be readily distinguished. In particular, it is an approach based on the use of k values (see Sections 4.5.1 and 5.6). An example, for a Canadian population of the Colorado potato beetle (Leptinotarsa decemlineata), is shown in Table 14.1 (Harcourt, 1971). In this species, 'spring adults' emerge from hibernation around the middle ofJune, when potato plants are breaking through the ground. Within 3

or 4 days oviposition (egg laying) begins, continuing for about 1 month and reaching its peak in early July. The eggs are laid in clusters on the lower leaf surface, and the larvae crawl to the top of the plant where they feed throughout their development, passing through four instars. When mature, they drop to the ground and pupate in the soil. The 'summer adults' emerge in early August, feed, and then re-enter the soil at the beginning of September to hibernate and become the next season's 'spring adults'.

The sampling program provided estimates of the population at seven stages: eggs, early larvae, late larvae, pupae, summer adults, hibernating adults and spring adults. One further category was included, 'females X 2', to take account of any unequal sex ratios amongst the summer adults. Table 14.1 lists these estimates for a single season. It also gives what were believed to be the main causes of death in each stage of the life cycle. In so doing, what is essentially a demographic technique (dealing with phases) takes on the mantle of a mechanistic approach (by associating each phase with a proposed 'factor').

The mean k values, determined for a single population over 10 seasons, are presented in the third column of Table 14.2. These indicate the relative strengths of the various factors that contribute to the total rate of mortality within a generation. Thus, the emigration of summer adults has by far the greatest proportional effect (k6 = 1.543), whilst the starvation of older larvae, the frost-induced mortality of hibernating adults, the 'nondeposition' of eggs, the effects of rainfall on young larvae and the cannibalization of eggs all play substantial roles as well.

What this column of Table 14.2 does not tell us, however, is the relative importance of these factors as determinants of the year-to-year fluctuations in mortality. For instance, we can easily imagine a factor that repeatedly takes a significant toll from a population, but which, by remaining constant in its effects, plays key factors? or key phases?

the Colorado potato beetle mean k values: typical strengths of factors

Age interval |
Numbers per 96 potato hills |
Numbers 'dying' |
'Mortality factor' |
LogwN |
k value | |

Eggs |
11,799 |
2,531 |
Not deposited |
4.072 |
0.105 |
(kj |

9,268 |
445 |
Infertile |
3.967 |
0.021 |
(kj | |

8,823 |
408 |
Rainfall |
3.946 |
0.021 |
(U | |

8,415 |
1,147 |
Cannibalism |
3.925 |
0.064 |
(kj | |

7,268 |
376 |
Predators |
3.861 |
0.024 |
kJ | |

Early larvae |
6,892 |
0 |
Rainfall |
3.838 |
0 |
(k2) |

Late larvae |
6,892 |
3,722 |
Starvation |
3.838 |
0.337 |
fe) |

Pupal cells |
3,170 |
16 |
D. doryphorae |
3.501 |
0.002 |
(k) |

Summer adults |
3,154 |
126 |
Sex (52% 5) |
3.499 |
- 0.017 |
(k5) |

5 x 2 |
3,280 |
3,264 |
Emigration |
3.516 |
2.312 |
(*6) |

Hibernating adults |
16 |
2 |
Frost |
1.204 |
0.058 |
(k7) |

Spring adults |
14 |
1.146 |

Mortality factor |
k |
Mean k value |
Regression coefficient on ktota! |
b |
r2 |

Eggs not deposited |
k]a |
0.095 |
-0.020 |
-0.05 |
0.27 |

Eggs infertile |
k1b |
0.026 |
-0.005 |
-0.01 |
0.86 |

Rainfall on eggs |
k-ic |
0.006 |
0.000 |
0.00 |
0.00 |

Eggs cannibalized |
k1d |
0.090 |
-0.002 |
-0.01 |
0.02 |

Eggs predation |
kic |
0.036 |
-0.011 |
-0.03 |
0.41 |

Larvae 1 (rainfall) |
k2 |
0.091 |
0.010 |
0.03 |
0.05 |

Larvae 2 (starvation) |
ks |
0.185 |
0.136 |
0.37 |
0.66 |

Pupae (D. doryphorae) |
K |
0.033 |
-0.029 |
-0.11 |
0.83 |

Unequal sex ratio |
k5 |
- 0.012 |
0.004 |
0.01 |
0.04 |

Emigration |
k6 |
1.543 |
0.906 |
2.65 |
0.89 |

Frost |
k? |
0.1?0 |
0.010 |
0.002 |
0.02 |

ktotal |
2.263 |

Table 14.2 Summary of the life table analysis for Canadian Colorado beetle populations. b is the slope of the regression of each k factor on the logarithm of the numbers preceding its action; r2 is the coefficient of determination. See text for further explanation. (After Harcourt, 1971.)

Table 14.2 Summary of the life table analysis for Canadian Colorado beetle populations. b is the slope of the regression of each k factor on the logarithm of the numbers preceding its action; r2 is the coefficient of determination. See text for further explanation. (After Harcourt, 1971.)

little part in determining the particular rate of mortality (and thus, the particular population size) in any 1 year. This can be assessed, however, from the next column of Table 14.2, which gives the regression coefficient of each individual k value on the total generation value, ktotal.

A mortality factor that is important regressions of k on in determining population changes ktotal: key factors will have a regression coefficient close to unity, because its k value will tend to fluctuate in line with ktotal in terms of both size and direction (Podoler & Rogers, 1975). A mortality factor with a k value that varies quite randomly with respect to ktotal, however, will have a regression coefficient close to zero. Moreover, the sum of all the regression coefficients within a generation will always be unity. The values of the regression coefficients will, therefore, indicate the relative strength of the association between different factors and the fluctuations in mortality. The largest regression coefficient will be associated with the key factor causing population change.

In the present example, it is clear that the emigration of summer adults, with a regression coefficient of 0.906, is the key factor. Other factors (with the possible exception of larval starvation) have a negligible effect on the changes in generation mortality, even though some have reasonably high mean k values. A similar conclusion can be drawn by simply examining graphs of the fluctuations in k values with time (Figure 14.4a).

Thus, whilst mean k values indicate the average strengths of various factors as causes of mortality in each generation, key factor analysis indicates their relative contribution to the yearly changes in generation mortality, and thus measures their importance as determinants of population size.

What, though, of population regu-a role for factors in lation? To address this, we examine regulation? the density dependence of each factor by plotting k values against log10 of the numbers present before the factor acted (see Section 5.6). Thus, the last two columns in Table 14.2 contain the slopes (b) and coefficients of determination (r2) of the various regressions of k values on their appropriate 'log10 initial densities'. Three factors seem worthy of close examination. The emigration of summer adults (the key factor) appears to act in an overcompensating density-dependent fashion, since the slope of the regression (2.65) is considerably in excess of unity (see also Figure 14.4b). Thus, the key factor, although density dependent, does not so much regulate the population as lead to violent fluctuations in abundance (because of overcompensation). Indeed, the Colorado potato beetle-potato system would go extinct if potatoes were not continually replanted (Harcourt, 1971).

Also, the rate of larval starvation appears to exhibit under-compensating density dependence (although statistically this is not significant). An examination of Figure 14.4b, however, shows that the relationship would be far better represented not by a linear regression but by a curve. If such a curve is fitted to the data, then the coefficient of determination rises from 0.66 to 0.97, and the slope (b value) achieved at high densities would be 30.95 (although it is, of course, much less than this in the range of densities observed). Hence, it is quite possible that larval starvation plays an important part in regulating the population, prior to the destabilizing effects of pupal parasitism and adult emigration.

Key factor analysis has been applied to a great many insect populations, but wood frogs and an to far fewer vertebrate or plant popu- annual plant lations. Examples of these, though, are shown in Table 14.3 and Figure 14.5. In populations of the wood frog (Rana sylvatica) in three regions of the United States (Table 14.3), the larval period was the key phase determining abundance in each region (second data column), largely as a result of year-to-year variations in rainfall during the larval period. In low rainfall years, the ponds could dry out, reducing larval survival to catastrophic levels, sometimes as a result of a bacterial infection. Such

Figure 14.4 (a) The changes with time of the various k values of Colorado beetle populations at three sites in Canada. (After Harcourt, 1971.) (b) Density-dependent emigration by Colorado beetle 'summer' adults (slope = 2.65) (left) and density-dependent starvation of larvae (slope = 0.37) (right). (After Harcourt, 1971.)

Figure 14.4 (a) The changes with time of the various k values of Colorado beetle populations at three sites in Canada. (After Harcourt, 1971.) (b) Density-dependent emigration by Colorado beetle 'summer' adults (slope = 2.65) (left) and density-dependent starvation of larvae (slope = 0.37) (right). (After Harcourt, 1971.)

mortality, however, was inconsistently related to the size of the larval population (one pond in Maryland, and only approaching significance in Virginia - third data column) and hence played an inconsistent part in regulating the sizes of the populations. Rather, in two of the regions it was during the adult phase that mortality was clearly density dependent and hence regulatory (apparently as a result of competition for food). Indeed, in two of the regions mortality was also most intense in the adult phase (first data column).

The key phase determining abundance in a Polish population of the sand-dune annual plant Androsace septentrionalis (Figure 14.5; see also Figure 14.1a) was found to be the seeds in the soil. Once again, however, mortality did not operate in a density-dependent manner, whereas mortality of seedlings, which were not the key phase, was found to be density dependent. Seedlings that emerge first in the season stand a much greater chance of surviving.

Overall, therefore, key factor analysis (its rather misleading name apart) is useful in identifying important phases in the life cycles of study organisms. It is useful too in distinguishing the variety of ways in which phases may be important: in contributing significantly to the overall sum of mortality; in contributing significantly to variations in mortality, and hence in determining abundance; and in contributing significantly to the regulation of abundance by virtue of the density dependence of the mortality.

Age interval

Maryland

Larval period

Juvenile: up to 1 year Adult: 1-3 years Total

Virginia

Larval period Juvenile: up to 1 year Adult: 1-3 years Total

Mean k value

Coefficient of regression on ktotal

1.94

4.78

2.35

0.85

0.73

Coefficient of regression on log (population size)

Pond 2: 0.39 (P = 0.50) 0.12 (P = 0.50) 0.11 (P = 0.46)

Table 14.3 Key factor (or key phase) analysis for wood frog populations from three areas in the United States: Maryland (two ponds, 1977-82), Virginia (seven ponds, 1976-82) and Michigan (one pond, 1980-93). In each area, the phase with the highest mean k value, the key phase and any phase showing density dependence are highlighted in bold. (After Berven, 1995.)

Michigan

Larval period 1.12

Juvenile: up to 1 year 0.64

Adult: 1-3 years 3.45

Total 5.21

1.40

1.02

Generation mortality

Seeds not produced

Seeds failing to germinate

Seedling mortality

Vegetative mortality

°.5 r Mortality during flowering 0.0 L •-•--•-

Mortality during fruiting

1969 1970 1971 1972 1973 1974 1975 Year

k3 Seedling mortality

Log number of seedlings

Figure 14.5 Key factor analysis of the sand-dune annual plant Androsace septentrionalis. A graph of total generation mortality (ktotal) and of various k factors is presented. The values of the regression coefficients of each individual k value on ktotal are given in brackets. The largest regression coefficient signifies the key phase and is shown as a colored line. Alongside is shown the one k value that varies in a density-dependent manner. (After Symonides, 1979; analysis in Silvertown, 1982.)

14.3.2 Sensitivities, elasticities and l-contribution analysis

Although key factor analysis has been useful and widely used, it has been subject to persistent and valid criticisms, some technical (i.e. statistical) and some conceptual (Sibly & Smith, 1998). Important among these criticisms are: (i) the rather awkward way in which k values deal with fecundity: a value is calculated for 'missing' births, relative to the maximum possible number of births; and (ii) 'importance' may be inappropriately ascribed to different phases, because equal weight is given to all phases of the life history, even though they may differ in their power to influence abundance. This is a particular problem for populations in which the generations overlap, since mortalities (and fecundities) later in the life cycle are bound to have less effect on the overall rate of population growth than those occurring in earlier phases. In fact, key factor analysis was designed for species with discrete generations, but it has been applied to species with overlapping generations, and in any case, restricting it to the former is a limitation on its utility.

Sibly and Smith's (1998) alternative to key factor analysis, ^-contribution analysis, overcomes these problems. X is the population growth rate (er ) that we referred to as R, for example, in Chapter 4, but here we retain Sibly and Smith's notation. Their method, in turn, makes use of a weighting of life cycle phases taken from sensitivity and elasticity analysis (De Kroon et al., 1986; Benton & Grant, 1999; Caswell, 2001; see also 'integral projection models', for example Childs et al. (2003)), which is itself an important aspect of the demographic approach to the study of abundance. Hence, we deal first, briefly, with sensitivity and elasticity analysis before examining X-contribution analysis.

The details of calculating sensitivities and elasticities are beyond our scope, but the principles can best be understood by returning to the population projection matrix, introduced in Section 4.7.3. Remember that the birth and survival processes in a population can be summarized in matrix form as follows:

where, for each time step, mx is the fecundity of stage x (into the first stage), gx is the rate of survival and growth from stage x into the next stage, and px is the rate of persisting within stage x. Remember, too, that X can be computed directly from this matrix. Clearly, the overall value of X reflects the values of the various elements in the matrix, but their contribution to X is not equal. The sensitivity, then, of each element (i.e. each biological process)

is the amount by which X would change for a given absolute change in the value of the matrix element, with the value of all the other elements held constant. Thus, sensitivities are highest for those processes that have the greatest power to influence X.

However, whereas survival elements (gs and ps here) are constrained to lie between 0 and 1, fecundities are not, and X therefore tends to be more sensitive to absolute changes in survival than to absolute changes of the same magnitude in fecundity. Moreover, X can be sensitive to an element in the matrix even if that element takes the value 0 (because sensitivities measure what would happen if there was an absolute change in its value). These shortcomings are overcome, though, by using the elasticity of each element to determine its contribution to X, since this measures the proportional change in X resulting from a proportional change in that element. Conveniently, too, with this matrix formulation the elasticities sum to 1.

Elasticity analysis therefore offers an especially direct route to plans for the management of abundance. If we wish to increase the abundance of a threatened species (ensure X is as high as possible) or decrease the abundance of a pest (ensure X is as low as possible), which phases in the life cycle should be the focus of our efforts? Answer: those with the highest elasticities. For example, an elasticity analysis of the threatened Kemp's ridley sea turtle (Lepidochelys kempi) off the southern United States showed that the survival of older, especially subadult individuals was more critical to the maintenance of abundance than either fecundity or hatchling survival (Figure 14.6a). Therefore, 'headstarting' programs, in which eggs were reared elsewhere (Mexico) and imported, and which had dominated conservation practice through the 1980s, seem doomed to be a low-payback management option (Heppell et al., 1996). Worryingly, headstarting programs have been widespread, and yet this conclusion seems likely to apply to turtles generally.

Elasticity analysis was applied, too, to populations of the nodding thistle (Carduus nutans), a noxious weed in New Zealand. The survival and reproduction of young plants were far more important to the overall population growth rate than those of older individuals (Figure 14.6b), but, discouragingly, although the bio-control program in New Zealand had correctly targetted these phases through the introduction of the seed-eating weevil, Rhinocyllus conicus, the maximum observed levels of seed predation (c. 49%) were lower than those projected to be necessary to bring X below 1 (69%) (Shea & Kelly, 1998). As predicted, the control program has had only limited success.

Thus, elasticity analyses are valuable in identifying phases and processes that are important in determining abundance, but they do so by focusing on typical or average values, and in that overcoming problems in key factor analysis the population projection matrix revisited

sensitivity and elasticity elasticity analysis and the management of abundance elasticity may say little about variations in abundance ...

Figure 14.6 (a) Results of elasticity analyses for Kemp's ridley turtles (Lepidochelys kempi), showing the proportional changes in X resulting from proportional changes in stage-specific annual survival and fecundity, on the assumption of three different ages of maturity. (After Heppell et al., 1996.) (b) Top: diagrammatic representation of the life cycle structure of Carduus nutans in New Zealand, where SB is the seed bank and S, M and L are small, medium and large plants, and s is seed dormancy, g is growth and survival to subsequent stages, and m is the reproductive contribution either to the seed bank or to immediately germinating small plants. Middle: the population projection matrix summarizing this structure. Bottom: the results of an elasticity analysis for one population, in which the percentage changes in X resulting from percentage changes in s, g and r are shown on the life cycle diagram. The most important transitions are shown in bold, and elasticities less than 1% are omitted altogether. (After Shea & Kelly, 1998.)

8 years | |

12 years | |

■ |
16 years |

Hatchling Juvenile Subadult Adult Fecundity

Animal survival

ms | ||||

SB |
S |
M |
L | |

SB |
S1 |
m1 |
m2 |
m3 |

S |
91 |
m4 |
ms |
me |

M |
0 |
92 |
0 |
0 |

L |
0 |
93 |
94 |
0 |

sense they seek to account for the typical size of a population. However, a process with a high elasticity may still play little part, in practice, in accounting for variations in abundance from year to year or site to site, if that process (mortality or fecundity) shows little temporal or spatial variation. There is even evidence from large herbivorous mammals that processes with high elasticity tend to vary little over time (e.g. adult female fecundity), whereas those with low elasticity (e.g. juvenile survival) vary far more (Gaillard et al., 2000). The actual influence of a process on variations in abundance will depend on both elasticity and variation in the process. Gaillard et al. further suggest that the relative absence of variation in the 'important' processes may be a case of 'environmental canalisation': evolution, in the phases most important to fitness, of an ability to maintain relative constancy in the face of environmental perturbations.

In contrast to elasticity analyses, key factor analysis seeks specifically to understand temporal and spatial variations in abundance. The same is true of Sibly and Smith's ^-contribution analysis, to which we now return. We can note first that it deals with the contributions of the different phases not to an overall k value (as in key factor analysis) but to X, a much more obvious determinant of abundance. It makes use of k values to quantify mortality, but can use fecundities directly rather than converting them into 'deaths of unborn offspring'. And crucially, the contributions of all mortalities and fecundities are weighted by their sensitivities. Hence, quite properly, where generations overlap, the chances of later phases being identified with a key factor are correspondingly lower in X-contribution than in key factor

... but l-contribution analysis does analysis. As a result, X-contribution analysis can be used with far more confidence when generations overlap. Subsequent investigation of density dependences proceeds in exactly the same way in X-contribution analysis as in key factor analysis.

Table 14.4 contrasts the results of the two analyses applied to life table data collected on the Scottish island of Rhum between 1971 and 1983 for the red deer, Cervus elaphus (Clutton-Brock et al., 1985). Over the 19-year lifespan of the deer, survival and birth rates were estimated in the following 'blocks': year 0, years 1 and 2, years 3 and 4, years 5-7, years 8-12 and years 13-19. This accounts for the limited number of different values in the kx and mx columns of the table, but the sensitivities of X to these values are of course different for different ages (early influences on X are more powerful), with the exception that X is equally sensitive to mortality in each phase prior to first reproduction (since it is all 'death before reproduction'). The consequences of these differential sensitivities are apparent in the final two columns of the table, which summarize the results of the two analyses by presenting the regression coefficients of each of the phases against ktotal and Xtotal, respectively. Key factor analysis identifies reproduction in the final years of life as the key factor and even identifies reproduction in the preceding years as the next most important phase. In stark contrast, in X-contribution analysis, the low sensitivities of X to birth in these late phases relegate them to relative insignificance - especially the last phase. Instead, survival in the earliest phase of life, where sensitivity is greatest, becomes the key factor, followed by fecundity in the 'middle years' where fecundity itself is highest. Thus, X-contribution analysis combines the virtues of key factor and elasticity analyses: distinguishing the regulation and determination of abundance, identifying key phases or factors, while taking account of the differential sensitivities of growth rate (and hence abundance) to the different phases.

Table 14.4 Columns 1-4 contain life table data for the females of a population of red deer, Cervus elaphus, on the island of Rhum, Scotland, using data collected between 1971 and 1983 (Clutton-Brock et al., 1985): x is age, lx is the proportion surviving at the start of an age class, kx, killing power, has been calculated using natural logarithms, and mx, fecundity, refers to the birth of female calves. These data represent averages calculated over the period, the raw data having been collected both by following individually recognizable animals from birth and aging animals at death. The next two columns contain the sensitivities of X, the population growth rate, to kx and mx in each age class. In the final two columns, the contributions of the various age classes have been grouped as shown. These columns show the contrasting results of a key factor analysis and a X contribution analysis as the regression coefficients of kx and mx on ktotal and Xtotal, respectively, where Xtotal is the deviation each year from the long-term average value of X. (After Sibly & Smith, 1998, where details of the calculations may also be found.)

Age (years) at start Sensitivity Sensitivity Regression coefficients of kx, Regression coefficients of kx, of class, x lx kx mx of l to kx of l to mx left, and mx, right, on ktota! left, and mx, right, on AMa!

0 |
1.00 |
0.45 |
0.00 |
- 0.14 |
0.16 |
0.01, - |
0.32, - |

1 2 |
0.64 0.59 |
0.08 0.08 |
0.00 0.00 |
-0.14 -0.14 |
0.09 0.08 |
0.01, - |
0.14, - |

4 |
0.54 0.53 |
0.03 0.03 |
0.22 0.22 |
-0.13 -0.11 |
0.07 0.06 |
0.00, 0.05 |
0.03, 0.04 |

5 |
0.51 |
0.04 |
0.35 |
-0.10 |
0.05 | ||

6 |
0.49 |
0.04 |
0.35 |
-0.08 |
0.05 |
-0.00, 0.03 |
0.08, 0.16 |

7 |
0.47 |
0.04 |
0.35 |
-0.07 |
0.04 | ||

8 |
0.45 |
0.06 |
0.37 |
-0.05 |
0.04 | ||

9 |
0.42 |
0.06 |
0.37 |
-0.04 |
0.03 | ||

10 |
0.40 |
0.06 |
0.37 |
-0.03 |
0.03 |
0.01, 0.15 |
0.09, 0.12 |

11 |
0.38 |
0.06 |
0.37 |
-0.02 |
0.02 | ||

12 |
0.35 |
0.06 |
0.37 |
-0.02 |
0.02 | ||

13 |
0.33 |
0.30 |
0.30 |
-0.01 |
0.02 | ||

14 |
0.25 |
0.30 |
0.30 |
-0.006 |
0.01 | ||

15 |
0.18 |
0.30 |
0.30 |
-0.004 |
0.008 | ||

16 |
0.14 |
0.30 |
0.30 |
-0.002 |
0.005 |
-0.05, 0.80 |
0.01, -0.00 |

17 |
0.10 |
0.30 |
0.30 |
-0.001 |
0.004 | ||

18 |
0.07 |
0.30 |
0.30 |
-0.001 |
0.002 | ||

19 |
0.06 |
0.30 |
0.30 |
-0.000 |
0.002 |

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