Measuring the Variability of Populations

Many measures of variability have been proposed, most of which are unsuitable for ecological purposes because they do not measure proportional change in density or population number. We generally require a measure of variability which gives the same result for a population which changes from 0.1 to 1, 10 to 100, or 1000 to 10000 as in each case the population has increased tenfold. Such a measure is then not dependent on the size of the sample. It is now recommended to use either the standard deviation of the natural logarithm of the abundances or the coefficient ofvariation ofthe abundances (CV(N)). As many time series of abundance have occasional zero values, CV(N) is the preferred measure as the logarithm of zero is undefined.

The merit of CV(N) over a nonproportional measure such as the variance of the time series is illustrated using as an example the 25-year variability of fish and crustaceans in Bridgwater Bay, England. Figure 6 shows clearly that the variance is highly positively correlated with mean abundance and therefore is a useless measure of population variability as it will inevitably show that the most

log 10 Mean annual abundance

log 10 Mean annual abundance

log 10 Mean annual abundance

Figure 6 (a, b) A comparison of the dependence of variance and coefficient of variation on the mean number caught per annum. The data are for the resident fish and crustaceans captured from monthly sampling over a 25-year period in Bridgwater Bay, England. These plots are based on the number caught per annum which may not be directly proportional to the actual number present. This is particularly the case for sampling methods such as trapping that can become saturated at high densities.

log 10 Mean annual abundance

Figure 6 (a, b) A comparison of the dependence of variance and coefficient of variation on the mean number caught per annum. The data are for the resident fish and crustaceans captured from monthly sampling over a 25-year period in Bridgwater Bay, England. These plots are based on the number caught per annum which may not be directly proportional to the actual number present. This is particularly the case for sampling methods such as trapping that can become saturated at high densities.

abundant species are the most variable. This linear relationship between log variance and log mean has long been recognized and is referred to as Taylor's power law. In comparison, the coefficient of variation of annual abundance does not always increase with the mean. In fact, the CV(N) for the Bridgwater Bay animals indicates that proportional variability tends to decrease with increasing abundance and hints at an increased stability with increasing population size. This observation is important, as the choice of the CV(N) as the preferred measure of variation is not to remove all dependence on population size. Even if this were possible, it would be undesirable as we frequently wish to identify stabilizing density-dependent processes, which change the dynamics as mean abundance changes. The standard deviation of the logarithm of the abundances + 1 has been suggested as a measure of variability which can be used with time series that include zeros. It should be avoided as it is biased.

Having settled on the coefficient of variation CV(N) of the time series as an appropriate measure of variability, it is essential, when comparing population variability, to compare time series over the same time period. This is because the CV(N) tends to increase with the increasing length of the time series. This is illustrated in Figure 7 using a 25-year time series of crustacean and fish data. While the common shrimp time series shows a long-term trend of increasing CV(N) (Figure 7a), the whiting data (Figure 7b) shows a time series for which the CV(N) initially rises with length and then stabilizes. A tendency for the CV(N) to stabilize with time is indicative of population regulation and thus may be a general feature of suitably long-term time series. It is notable that the whiting time series showed no long-term trend in abundance (see Figure 2). However, there is little empirical data on this subject and potentially most time series can suddenly increase in variability. The jump in variability of the common shrimp population after 20 years (Figure 7a) was linked to a sudden increase in water temperature from 1999.

Once a measure of variability is defined and it is also noted that a time-series variability should only be compared between populations over the same time period, it is then possible to examine comparative population stability. Figure 8 shows the frequency histogram of the CV(N) in annual abundance for all the species of fish and macro-crustaceans resident in Bridgwater Bay in the Bristol Channel, England. All the populations were monitored contemporaneously for a 25-year period commencing in 1981. It can be seen that CV(N) is approximately normally distributed around a mean of with a slight skew towards higher CV(N) values. This skew would appear to be related to human impacts and climate change as high CV(N) populations are dominated by species that have been showing a trend in abundance over the entire 25-year period. Some

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10 15 20 25 30 Years of study

10 15 20 25 30 Years of study

Years of study

Figure 7 (a, b) The change in the coefficient of variation with the length of the time series for (a), the brown shrimp, Crangon crangon and (b) whiting, Merlangius merlangus. The data were collected by monthly sampling over a 25-year period in Bridgwater Bay, England.

Years of study

Figure 7 (a, b) The change in the coefficient of variation with the length of the time series for (a), the brown shrimp, Crangon crangon and (b) whiting, Merlangius merlangus. The data were collected by monthly sampling over a 25-year period in Bridgwater Bay, England.

Coefficient of variation

Figure 8 The distribution of the coefficient of variation in annual abundance for the resident fish and crustacean species in Bridgwater Bay, England. The data were collected by monthly sampling over a 25-year period in Bridgwater Bay, England.

Coefficient of variation

Figure 8 The distribution of the coefficient of variation in annual abundance for the resident fish and crustacean species in Bridgwater Bay, England. The data were collected by monthly sampling over a 25-year period in Bridgwater Bay, England.

species such as the herring, Clupea harengus, which has a CV(N) = 1.6, are slowly recovering from a population collapse caused by overfishing. Others, such as the gurnard, Eutrigla gurnardus (CV(N) = 1.4, see Figure 3), have been increasing in abundance as seawater temperatures have increased. In summary, a wide range of vertebrate and invertebrate species, differing greatly in life history characteristics, display a notably small range of CV(N) between 0.5 and 1.5 over a 25-year period, suggesting that they all are tending to be constrained toward a similar level of population stability. Unfortunately, we do not have similar long-term data sets covering all the common vertebrates and invertebrates for a terrestrial ecosystem to test if this is a common feature of other communities.

It is possible to determine if the fish in Bridgwater Bay are showing a similar degree of variability to terrestrial vertebrates. B. Saether and colleagues in 2003 presented data for the CV(N) for 13 solitary birds over a 15-year time period. Values ranged from 0.08 for the sparrowhawk to 0.71 for the cactus finch with a mean of 0.31. In comparison, the 23 nonshoaling fish from Bridgwater Bay had CV(N)'s over a 15-year period between 0.45 and 1.9 with a mean of 0.88. It is clear that North American birds have, on average, considerably more stable populations than estuarine fish. This difference is probably related to differences in reproductive behavior. Most of the fish have high fecundities and limited or negligible parental care and therefore their populations are able to respond rapidly to changes in climatic conditions. In general, the CV(N) in annual abundance varies greatly between habitats, taxonomic groups, and species because of differences in the reproductive and life history strategy and the degree of environmental variability. From an extensive study of bird population dynamics, B. Saether and colleagues in 2004 demonstrated that the stability of bird populations is clearly related to their life history strategy and concluded that ''The demographic stochasticity decreased with adult survival rate, age at maturity, and generation time or the position of the species toward the slow end of the slow-fast life-history gradient.''

It has also been suggested on both theoretical and observational grounds that the species richness of a habitat may influence population stability. At present the effect of species richness is unclear. The exceptional study of a plant community by D. Tilman has indicated that species richness actually increased instability while the opposite conclusion has been reached in other studies. Results from theoretical studies are equally contradictory. At present, the indications are that there is no universal relationship between the number of interacting species and individual species stability. However, it is difficult to imagine how a species rich system could maintain species

number long-term if it held a dominant species that fluctuated greatly in abundance. Such a species would act like a bull in a china shop.

While the great majority of studied populations fluctuate with little obvious pattern and clearly respond to random climatic events, there has been great interest in those populations that show long-term oscillations. The most famous of these cyclic phenomena is the 4-year cycle of microtine rodents in boreal and artic regions first discussed by Elton in 1924. There has been considerable debate as to how these cycles form and are stabilized. One favored view is that they are caused by delayed density dependence as time lags are frequently observed to produce cyclical or quasi-cyclical behavior. It has been argued that in Fennoscandia, the cycle is imposed by time lags in the population response of mustelid predators and the time series of the rodents is actually chaotic. Studies in both Fennoscandia and Hokkaido, Japan, show that rodent cycles vary geographically and that cycles tend to be more apparent toward the north of the range. Studies on the gray-sided vole, Clethrionomys rufocanus, for example show that the dynamics of this species is influenced by density-dependent, delayed density-dependent, and stochastic climatic changes. The appearance of quasi-regular cycles therefore seems to depend on the relative magnitude of the three types of influence: density-dependent, delayed density-dependent, and climatic. It is notable that as knowledge has deepened, the important role of both biotic and physical environmental factors has been recognized.

The geographical extent of species is an important factor in population stability. K. J. Gaston in a study of 263 species of British moth studied over a time period of up to 14 years concluded that there was a strong relationship between local population variability and species abundance and distribution - with the most abundant and widely distributed species showing the greatest local variability. These observations indicate that the spatial extent of a population may have a great influence on local stability. This pattern may be related to the fact that widely distributed, well-dispersed, organisms can be opportunistic species, which take advantage of temporary niches to invade and then abandon when conditions worsen with low probability of global extinction. Spatial effects are also a key feature of pathogen population dynamics. For example, L. Ericson and colleagues report a 13-year study of the rust fungus, Uromyces valerianae, on the herb valerian, Valeriana salina, in an archipelago of islands in central Sweden. They found that the fungus populations frequently went extinct and re-colonized, producing very different and unstable population dynamics within different island populations. It is now well established that metapopulation dynamics must be considered to understand the stability of many populations.

It has been frequently observed that insect pest outbreaks occur at about the same time over large geographical areas. The degree of spatial synchrony between forest insect populations has been used to test the differing hypotheses as to why populations fluctuate. While there is considerable difference in the degree of spatial synchronicity between insects, there is good evidence that synchrony is related to large-scale climatic variables. This in turn indicates that the notorious instability of insect forest pests is determined, at least in part, by their sensitivity to climatic conditions.

In conclusion, it is clear that populations can vary greatly in abundance through time and space. Huge though this variation can be, there is considerable evidence that populations are constrained within limits by biotic and physical factors and they do not show their full potential for explosive growth or sudden collapse. The amount of variability displayed differs between species, taxa, and habitat and is related, in part, to life history characteristics and, in part, to the variability inherent in the habitat. Populations are composed of individuals that are constrained by and respond to their physical and biotic environment. This response to the environment can have the characteristics of negative feedback control and results in sufficient stability to increase the likelihood of long-term population maintenance. However, many local populations repeatedly go extinct and re-colonization from successful populations is essential for long-term existence. Stability of existence for many species may only exist at the meta-population level.

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