Chapter 5: Rates of Genetic Change
Draft Nov 2001: do not quote or distribute
Bill Sherwin and Craig Moritz

Summary
•	Genetic change underpins evolution and is central to conservation. The likelihood that threatening processes may alter these rates of change, highlights the importance of estimating the expected rates under different conditions.  Each population, with its particular life-history, population size and other characteristics, will have a particular rate of change, which can be expressed as 1/2Ne.  Ne is sometimes called the “effective population size”, but this name can be misleading since Ne is a function of the  population’s entire biology, not just census size.  As a rough approximation, Ne may be thought of as the number of individuals that are passing their alleles on to the next generation.  Ne is often an order of magnitude smaller than adult population size, especially if periods of small population size (“bottlenecks”) have occurred.

•	Estimates of Ne are also important to gauge that rate of change at genes affecting survival and reproduction. This is because the efficiency of selection depends on its strength, s,  relative to that of drift [ i.e s > 1/Ne].

•	In the same population, different genetic processes will not necessarily occur at the same rate, so there wiil be different Ne values for each estimate of genetic variation, such as inbreeding (NeI), gene diversity (NeV), and association between alleles at different loci (genotypic disequilibrium, NeD).  The existence of multiple types of Ne means that conservation biologists must choose carefully which is appropriate to the particular management concerns.

•	There are many ways of determining Ne and the estimate obtained often relate to different time-scales and stages of population history. Several methods can be selected to provide complementary information relevant to management.

•	Current Ne can be estimated from appropriate demographic data, but often these data are difficult to obtain. There have been attempts to define Ne as a function of adult population size given for particular life history and ecology, but these methods are debated still. Estimating Ne over a deeper time-period requires more data, especially fluctuations in population size. Deeper still, Ne over thousands of generations can be estimated from gene trees; such estimates are valuable for comparison to those of current Ne, especially in recently perturbed systems.

•	Ne can also be calculated from data on hypervariable genetic markers.  The most obvious method is somewhat circular: to repeatedly measure  genetic change, and from the rate of this change, back-calculate Ne.  Although an important component of ongoing monitoring, this lacks immediate predictive power, unless molecular analyses of historical collections are possible. Therefore much attention has been given to molecular methods for Ne estimation and bottleneck detection. Especially when combined with information on demography and history from monitoring, these data can be used to estimate simultaneously both current and historical Ne, providing for more direct comparisons.

•	The scope of the Ne estimation is important.  All rates of change are per-generation, which requires careful definition in species with overlapping generations.  Threatened populations with the same Ne but different generation lengths are likely to accumulate genetic problems at a different rate.  Estimates of long–term Ne  such as analyses of allele genealogies within populations, provide a perspective on changes in Ne through time.  These may provide a yardstick against which current Ne and population fluctuations can be compared, with large deviations signaling a need for attention.

•	The geographic scope and structure of the population(s) being managed will affect Ne also, and this will be discussed further in the next chapter.

Genetic changes underpin evolution and is central to conservation. It is known that threatening processes may alter the rate of these changes, and this highlights the importance of estimating the expected rates under different conditions, in different species. It is a major challenge to make general theories of evolutionary biology applicable to individual species, whose biology ranges from a harvested wild plant which produces thousands of offspring, to a large mammal which has small, low-density populations and whose females produce very few offspring in their lifetime, as well as a huge variety of other possibilities (Chapter 4).

Examples of small population size causing reduced genetic variation are now commonplace.  In Elephant seals Mirounga sp, after the northern species had recovered from being hunted to perhaps 10-100 individuals, its genetic variation was about half that of the southern species ((Hoelzel, Halley et al. 1993; Hoelzel 1999) (Hedrick 1995)).  Likewise, in koalas, populations which had been forced to small size or re-established from small numbers showed reduced number of alleles and levels of heterozygosity (Figure 5.1).  Other managed introductions are also often of small size, and can lead to reduced genetic variation, as seen in the Tiburon Island population of Bighorn sheep Ovis canadensis mexicana (Hedrick, Gutierrez-Espeleta et al. 2001), and the introduced pest in New Zealand, Bennett’s Wallaby Macropus rufogriseus rufogriseus (Le Page and al 2000).  Natural recolonisations are often of only a few individuals, and in a metapopulation of the rare butterfly Plebejus argus, it was found that colonisations resulted in significant losses of rare alleles ((Brookes, Graneau et al. 1997)).  

Of course, none of these studies shows that the loss of genetic variation is anything but a consequence of the small population size.  Whether this loss will lead to further problems is the subject of chapters 9 and 10, but here it is worth pointing out that in experimental studies of plants (Clarkia pulchella (Newman and Pilson 1997)) and animals (Drosophila melanogaster (Bijlsma, Bundgaard et al. 2000)), populations with a narrower genetic base did indeed go extinct much more frequently.  In small populations of dipteran fly species, Frankham showed that extinctions due to genetic erosion were considerably in excess of extinctions caused by other effects of small population size, such as demography and behaviour (Frankham 1995).  The rate of genetic change is not determined only by the population size measured by a single census.  Two examples demonstrate this nicely: in the studies of Bijlsma et al, and Newman and Pilson, the populations differed in their genetic variability, but not population size. In another study, populations of Drosophila that were periodically allowed to expand to a larger size showed much lower loss of genetic variation than those held at constant small size (73% compared to 40% (Woodworth, Montgomery et al. 1994)).  There are many other factors that can affect the rate of loss of genetic variation, which will be discussed in this chapter.

This chapter presents the tools and concepts which form a major part of the background for chapters 9 and 10, on processes in small populations.  For any population, the priority given to genetic management depends on the speed of change, which is controlled by the rate of change per generation, and the generation length; both of these are species-specific, or even population-specific.  We are concerned not only with the loss of variation, but also with the effect on adaptability.  For example, in small populations, random genetic changes can overpower other aspects of evolution.  On the other hand, we sometimes try to encourage certain changes, such as selection against harmful recessive alleles in captivity, or adaptation to changed conditions in wild populations.  In the study of Biljsma et al, the extinction rate was significantly worse in the presence of a stress agent (ethanol) which occurs in the species’ natural habitat.  In this chapter, we will see what factors, aside from the availability of genetic variation, affect the ability to adapt to changes in the environment.  We will examine various summaries of the capacity  to change, and the timescales to which they apply.  

Figure 5.1 (a) Numbers of alleles per microsatellite locus in koalas.  The “SE” populations have known histories of population crashes and/or establishment from a small number of founders (From (Houlden 1996))

Figure 5.1 (b) Gene diversity (right bar) and observed heterozygosity (left bar) at microsatellite loci in koalas.  The “SE” populations have known histories of population crashes and/or establishment from a small number of founders (From (Houlden 1996))

5.1 Rate of random genetic change in transmission between generations.

Types of variation, and processes which affect them
Table 5.1 summarises the main types of genetic variation that we have already encountered.  Each of the measures in the table is created, maintained or lost as a result of interacting evolutionary processes which may be altered by threats:
- selection regimes 
- dispersal
- mating patterns, including inbreeding frequency
- genetic architecture
- mutation, recombination
- “Genetic Drift” or random losses in transmission from generation to generation (Wright 1946).  
Basic detail of all theory in this chapter is given in population genetics textbooks such as (Hartl and Clark 1997; Hedrick 2000).

Table 5.1: Types of genetic variation.  Some of these types of variation can be characterised either in diploids, or in haploid organisms and organelles such as mitochondria and chloroplasts, while other measures are particular to diploids.  The range of values which these measures  might take under various circumstances is shown;  in some cases the extreme values are rarely achieved.  For further description of each type of variation, and the sampling requirements for their measurement, see chapters 2 and 3.  

	Description	Symbol	Range of possible values	
Haploid OR Diploid				
	Allelic diversity - number of alleles at a locus 	K, 	> 1	
	Number of polymorphic loci (P) or nucleotide sites (S) 	P, S	0 to all loci or sites	
	Gene diversity – for a locus, the chance that two randomly chosen alleles are different.

When measured in a diploid, gene diversity is also the average expected Hardy-Weinberg heterozygosity under random mating  (He).
  
When measured on the scale of a single nucleotide instead of a whole locus, gene diversity  is called the Nucleotide diversity, P	GD 

or He 

or P	0 to 1	
	Genotypic disequilibrium – a measure of non-random association between alleles at different loci (eg, because of one or more of:  physical linkage, chance, selection etc). Zero if association is random	D	-0.25 to 
+ 0.25	
	Additive  genetic variation	Va	> 0	
	Heritability	h2	0 to 1	
	Epistatic genetic variation (between locus interactions)	 Vi	> 0	
Diploid only				
	Observed heterozygosity	Ho	0 to 1	
	Reduction in heterozygosity (Ho)  relative to HW expectations (He),  due to inbreeding, or other causes.  If Ho< He, then Fis is positive.  On the (debatable) assumption that Fis is only  determined by inbreeding, it is sometimes called the inbreeding coefficient.	Fis	-1 to +1, but range depends on 
measurement method	
	Dominance genetic variation (interactions between alleles at the same locus)	Vd	> 0	

The rate of change of each measure of variation could respond differently to each evolutionary process or its alteration in a threatened species.  Figure 5.2 shows three populations between which there is independent variation of Gene diversity (He), Observed heterozygosity (Ho), Allelic diversity (K ), and the inbreeding coefficient (Fis).  If we know the demographic characterisitics of the managed population, we can predict the rate of change for a particular population.  Rather than creating a bewildering array of possibilities, we will first concentrate on two important processes which may accelerate in threatened populations: drift and inbreeding.  

Figure 5.2  The difference between inbreeding and lowered genetic variation among individuals.
(a) A population in Hardy-Weinberg equilibrium (HWE); Gene Diversity He = 0.67, Observed heterozygosity Ho = 0.67, Allelic Diversity K = 3, Inbreeding coefficient (calculated from the depression of heterozygosity relative to HWE) Fis = 0.
(b) A population with lower genetic variation than "a", but no evidence of inbreeding; this population is also in HWE; He = 0.5, Ho = 0.5, K = 2, Fis = 0.
(c) A population with the same genetic variation (He, K) among individuals as "a", but also a history of recent inbreeding which has resulted in deviation from HWE - deficit of heterozygotes.  H = 0.67, Ho = 0.67, K = 3, Fis= 1.                               After (Sherwin and Moritz 2000).

(a) GD(He) = 0.67, Ho = 0.67   K = 3,  Fis = 0

A1/A1   A1/A1   A1/A1   A1/A1
    
A1/A2   A1/A2   A1/A2   A1/A2   A1/A2   A1/A2   A1/A2   A1/A2
   
A2/A2   A2/A2   A2/A2   A2/A2 
 
A1/A3   A1/A3   A1/A3   A1/A3   A1/A3   A1/A3   A1/A3   A1/A3  
  
A2/A3   A2/A3   A2/A3   A2/A3   A2/A3   A2/A3   A2/A3   A2/A3

A3/A3   A3/A3   A3/A3   A3/A3

(b) GD(He) = 0.5, Ho= 0.5, K = 2, Fis = 0
 
A1/A1   A1/A1   A1/A1   A1/A1   A1/A1   A1/A1   A1/A1   A1/A1   A1/A1

A1/A2   A1/A2   A1/A2   A1/A2   A1/A2   A1/A2   A1/A2   A1/A2   A1/A2   
A1/A2   A1/A2   A1/A2   A1/A2   A1/A2   A1/A2   A1/A2   A1/A2   A1/A2

A2/A2   A2/A2   A2/A2   A2/A2   A2/A2   A2/A2   A2/A2   A2/A2   A2/A2

(c) GD(He)= 0.67, Ho = 0, K = 3, Fis = 1
 
A1/A1   A1/A1   A1/A1   A1/A1   A1/A1   A1/A1   
A1/A1   A1/A1   A1/A1   A1/A1   A1/A1   A1/A1    

A2/A2   A2/A2   A2/A2   A2/A2   A2/A2   A2/A2   
A2/A2   A2/A2   A2/A2   A2/A2   A2/A2   A2/A2    

A3/A3   A3/A3   A3/A3   A3/A3   A3/A3   A3/A3   
A3/A3   A3/A3   A3/A3   A3/A3   A3/A3   A3/A3
Drift (Rate of change of gene diversity, and additive genetic variation) and NeV , the variance effective population size).

Each population will have a rate at which chances in transmission of alleles between generations, as well as a rate at which allele proportions wander up and down due to the same chances of transmission.  This is shown in Figure 5.3 (a and b), where three general points are apparent:  alleles can be lost by chance, allele proportions can change by chance, and identical populations can reach different allele proportions by chance alone (Figure 5.3 b).  Also, rare alleles are much more likely to be completely lost that common ones (Figure 5.3 b, white bars).  Figure 5.3 shows that because of the greater effects of chance in a small sample of gametes, the effects of genetic drift are greater in small populations.  But population size is not the only thing that can affect the rate of drift.  Figure 5.4 shows the lifetime reproductive success of females of an endangered bandicoot species, and indicates that although there were many females, very few of them actually passed on their alleles to the next generation, so any unique alleles they possessed would be lost, and the rate of drift would depend on the smaller number of females which actually reproduced.  Other aspects of the biology can affect the rate of drift also – see Box 5.1, and drift can directly affect any of the types of variation, especially the allelic and gene diversity, and genotypic disequilibrium of unlinked loci.

Figure 5.3 (a) Chance loss of variation in one population of size two, with equal number of B and b alleles. Two of the possible outcomes in generation 2 are shown. The chance is calculated for one generation of mating and reproduction, using Mendelian ratios.

Figure 5.3 (b)  Black bars show the result of a single generation of drift in multiple populations, each starting with two individuals of genotypes Bb and Bb; the chance that a population will have each possible number of B alleles is calculated as in (a).  Note that the average population still has an equal number of the two alleles, but that any other number is possible, from loss of B to loss of b (either of which is called “fixation”).  White bars show the result if the first generation did not have equal allele proportions:  BB and Bb, p=0.75, q=0.25; note that there is a greatly increased chance of losing the rare allele, b.

Figure 5.3 (c)  Drift in a smaller population.  Black bars as in 5.3(b), p = q = 0.5.  White bars show the result of a single generation of drift in multiple populations, each starting with only one individual of genotype Bb, p = q = 0.5 (which self-fertilises).   Note that the chance of fixation of either allele is greatly increased in this small population.

Figure 5.4  Lifetime reproductive success in females of endangered population of eastern barred bandicoots Perameles gunnii ((Sherwin and Prown 1990)).  There were very unequal contributions to the next generation – some females did not reproduce at all, so any allelic variants that were unique to them would be lost, while one flooded the population with her alleles.

# Mothers

•	Eg Eastern Barred Bandicoot

# Young

To predict the rate of change, we must know the demographic characteristics of the managed population, such as its total number, numerical fluctuations, overlap of generations, sex ratio, mating system, mean and variance of lifetime reproductive output, etc.  Most of these characteristics may be altered in managed or threatened populations, so we are keen to investigate their effects on rates of genetic change.    Often, a population's demographic data can be summarised as the expected rate of change 1/2NeV.  This slightly awkward expression can then be used to predict the rate of change in that population; for example, the genetic diversity at generation t (Het) is predicted to be:
                                                               Het = He0 (1-1/2NeV)t

NeV is sometimes called the “variance effective population size" (Wright, 194*), although this name is somewhat misleading since the population size is only a part of the process of calculating the rate of change, and the “effective” does not imply that the population is effective at remaining viable, or anything else.  We will use the term effective population size, but it should be remembered that it is simply the inverse of the rate of genetic change, and can be larger or smaller than the actual population size!  Frequently, Ne is an order of magnitude smaller than adult population size, especially if periods of small population size (“bottlenecks”) have occurred  (Frankham 1995).  Two other ways of thinking about the effective population size are presented in Box 5.1

Box 5.1:   The "Ideal" Population, and a real, endangered, population.

For mathematical convenience, population geneticists have defined an "Ideal" population (Wright 194*), whose census size (N) is equal to its effective population size (Ne).

The ideal population has:
- a constant population size
- an equal probability of reproduction for each individual (A”Poisson distribution, strictly speaking)
- 1:1 sex ratio
- no overlap of generations
- random mating
 
In a real population that departs from these conditions, genetic variation may be lost more slowly, or faster than the ideal population of the same size, so Ne for the real population can be larger or smaller than actual population size, underlining the fact that Ne is not really the size of a population – it is simply an estimate of the rate of genetic change.

The formal definition of effective population size is that it is the size of an ideal population which would have the same rate of genetic change as the actual population that you are studying.  Note that the "ideal" population is not necessary ideal in any way except for mathematical ease.  

An alternative, and not entirely correct, interpretation of Ne of a real population, is that it is the number of individuals which are really passing their alleles on to the next generation.

As an example of the calculation of NeV, the figure and table below show some of the data, and the results of the calculation, for an endangered population of eastern barred bandicoots Perameles gunnii ((Sherwin 1990) (Sherwin and Prown 1990)).  The histogram in Figure 5.4  shows that there were very unequal contributions to the next generation – some females did not reproduce at all, so any allelic variants that were unique to them would be lost, while one flooded the population with her alleles. The correction for this can be seen by comparing the 340 young per generation in the table to the effective female population size of 54.  Because of the loss of variants that is expected from this pattern of reproduction, the effective size of the population is lower than the actual size.  Likewise, because there are many more females than males, the effective size must be corrected further.  At diploid loci, each offspring has one allele from a male and one from a female, so half the alleles for the next generation are coming from the smaller number of males, while the other half will come from the larger number of females, whose alleles will therefore be under-represented in the next generation.  After correcting for the sex ratio, the variance effective population size for the population is 67 – about one tenth of the estimated population size per generation (juveniles plus adults).

Ne =67

Combine sexes

54

f

24

m

Correct for reproductive success

340

f

153

m

Young/generation

633

Total Population (N)

End Box ideal 
 We have already said that the effect of a given population’s biology on the rate of genetic change will not necessarily be the same for different measures of variation, so a multiplicity of Ne calculations is possible.  For example, the same NeV is often used to predict rate of change of heterozygosity per generation, as above, as well as to predict the rate at which replicate populations will differentiate by chance processes, but Ewens ((Ewens 1979)) has shown that in some circumstances the two forecasts require different Ne values, which he gives separate names (“eigenvalue” and “variance”, respectively). Nevertheless, the two are usually used interchangeably.  It is also usually assumed that the same Ne predicts changes of D, the genotypic disequilibrium coefficient, although this is sometimes called NeD.  Finally, NeV can also be used to predict the rate of change of additive genetic variance.  We will call all of these NeV, but in practice it is important to remember that if two NeV values have been calculated in different ways or from different data, there may be comparability problems, which we shall return to later.

The discussion above is based on diploid loci, but conservation geneticists also aim to conserve variation in haploid species, or in haploid loci such as those in mitochondria and chloroplasts or some sex chromosomes.  The difference in behaviour of haploid and diploid loci is also exploited in detection of sex-differences of dispersal (Chapters 3, 4, 6), so it is important to understand how they behave differently for reasons other than their sex-limitation, such as drift.  There has been some debate over the effects of drift on haploid uniparentally inherited loci, but it is now generally accepted that both mitochondrial and Y-linked loci have an effective population size that is one-quarter of that for diploid loci in the same individuals (Birky, Maruyama et al. 1983; Birky, Fuerst et al. 1989).  This assumes that the breeding population is dioecious with 1:1 sex ratio, that the inheritance is really uniparental, and that if new allele arises in an individual, it will pass on only the new or old variant to each offspring, not both.  All of these assumptions are sometimes violated, and appropriate corrections can be made ((Birky, Maruyama et al. 1983; Birky, Fuerst et al. 1989) (Chesser 1998) (Laporte, Cuguen et al. 2000)).  Under certain condition, Ne for X-linked loci in mammals will be 3/4 of Ne for autosomal loci ((Nagylaki 1995; Nagylaki 1995)).

Inbreeding effective size
Another process which may speed up in threatened populations is inbreeding   Note that we use the word "inbreeding" to refer to the result of mating between relatives, causing an increase of homozygotes for alleles which are not only identical in sequence, but copies of the same original allele in one of the ancestors – so called “identical by descent” alleles ((Templeton 1994)).  Where this type of inbreeding is present, the identity by descent will depress observed homozygosity (Ho) relative to random-mating expectations (He), thus Fis will be positive.  In the previous chapter, we saw that mating patterns could be disrupted, eg by loss of a pollinator, resulting in increased mating between relatives, such as selfing.  This is the meaning we will use when we discuss inbreeding and its changes in threatened population, unless stated otherwise.

We do NOT also use “inbreeding” as a general term for reduced genetic variation in a population, as some conservation geneticists do, since this only creates confusion about the two processes.  The reason that this confusion arose is that in a randomly mating population of limited size, if pedigrees are followed for many generations, then gradually some matings between relatives will occur, because there is only a small pool of potential mates. Thus the level of identity by descent will gradually rise.  However, since the population is randomly mating, the proportion of heterozygotes will not be depressed, relative to random mating, and Fis is expected to be zero unless some other factor affects it.   This is a situation very similar to the loss of variance that we have already described, so the rates of decrease of variation (NeV) and increase of inbreeding (NeI) are often similar or identical.  However, this is not always the case.  Since inbreeding depends upon the relationship between the parents, via their ancestors, the rate at which inbreeding increases depends upon the size and mating patterns of previous generations, as well as the current population of parents.  Thus in populations of changing size, as often found in conservation, NeI may be different to NeV ((Ewens 1979)).   It is our contention that NeV is of the greatest importance, since it predicts He, one estimate of the amount of variation available to be arrayed in particular genotypes and acted upon by natural selection (Chapter 9). Secondly, any additional depression of heterozygosity due to inbreeding (Ho<He, as we defined above) is also important (Chapter 10).  Whether particular alleles or haplotypes are identical because they are copies, or for other reasons, has importance as a tool in attempts to trace the history of populations, 

Allelic variation.
Any allele may be vital for adaptation to new conditions in the future, so it is of serious concern that changes of allele diversity are not well represented by changes of gene diversity.  For example, if a population has three alleles, with the rare allele present at only 1%, then loss of this allele will make very little difference to GD, the gene diversity, but the allelic diversity will go down by one-third, from K=3 to K=2.  The expected rate of loss of rare alleles depends upon both the effective population size and the actual population size (Denniston, (Williams)), but some generalisations can be made (Nei, Maruyama et al. 1975).  In general, a period of small population size decreases both GD and K.  If the recovery from the period of small size is fast, the loss of GD is minimised, but the loss of K is little affected.  In the longer term over thousands of generations, K recovers (by mutation) faster than GD, since the presence of one new allele adds one to the value of K.  Alleles for quantitative traits (QTLs) show genetic drift (Frankham and Weber 1997), but it is not known whether this occurs at the same rate as for other alleles.

The existence of multiple types of Ne means that conservation biologists must choose carefully which estimate is appropriate to the particular management concerns.  This will be discussed in more detail in section 5.6, but first we will investigate the practicalities of measuring Ne in different ways.

5.2 Calculating variance effective size from ecological data

Current Ne can be estimated from appropriate ecological data, but often these data are difficult to obtain.  We will set out two well-known methods, which require different amounts of data.  Later, in section 6, we will consider short-cuts that may make the data collection easier.  The calculation of effective size includes formulas for the effect of various characteristics of the population on the rate of genetic change.  One way of calculating the variance effective population size is as follows:

Number of individuals:  A larger population will have a lower rate of chance loss of alleles.  Since the loss happens each generation, the relevant number of individuals is the number of new individuals in a single generation, N1 (below).  The sexes are counted separately in dioecious species; different methods are available for species with more complex life-cycles such as gynodioecius plants (Laporte, Cuguen et al. 2000).

Generation length is readily understood: it is the average time that it takes a newly-formed zygote to become a parent and produce zygotes of its own.  This requires ageing and reproductive data.  A knowledge of generation length is necessary to estimate the effective population size when more than one generation is alive simultaneously (overlapping generations).  However, there is no perfect way of correcting for this (Vucetich, Waite et al. 1997), and there are alternative definitions of generation length ((Charlesworth 1980)).

Lifetime reproduction:  The mean and variance of lifetime reproduction affect the rate of genetic change.  The variance is especially important because if some individuals make relatively small reproductive contributions their alleles are likely to be lost.  The sexes are treated separately, since their reproductive contributions may have very different distributions.

For each sex, one possible calculation is:
Nm or Nf =  [kN1 – 1]/[ k – 1 + V/k]
Where k is average lifetime reproduction, and V is its variance, for that sex.  Both k and V require extensive ecological data, perhaps aided by molecular methods mentioned in the previous chapter.  An alternative method is available if lifetime reproduction data is not available, but it requires age-specific reproduction and survivorship ((Lande 1987))

Sex ratio: If the sex ratio is very skewed, then variation will be lost faster than if it was 1:1.  For example, if there were 1 male and 9 females in a population, all the offspring would receive half of their genes from the male.  This means that alleles in the other parents are under-represented, and may be lost more readily than in a population of five males and five females each contributing equally to the next generation.

To sum the Nm and Nf values for the two sexes, the following equation is standard:
N3 =  [4 Nm Nf ]/[ Nm + Nf]
A quick calculation shows the importance of mating behaviour for Ne: if only one male fertilises all the females, then Ne can never rise above 4, irrespective of how many females there are in the population. 

Fluctuations of population size are very important in Ne calculation, as mentioned before.  Of course, they require very long term population data, and the omission of this correction is probably the cause of many problems with departure of ecological Ne estimates from others ((Frankham 1995)).  Many annual censuses are required to adequately allow for fluctuations in demographic calculation of Ne (Vucetich and Waite 1998).  To correct for fluctuations of population size beteen years, the following calculation can be used:
1/Ne = 1/t [S [ 1/N3]]
Where t is the number of time periods considered.  This is only an approximation of the real correction (Vucetich, Waite et al. 1997), and  (Iizuka 2001) has suggested that another correction should be used if the fluctuations of population size are correlated positively or negatively.

Mating patterns: It is assumed in the above that mating is random, but if not, then the estimate of Ne must be corrected for the expected effect (ie lower NeI if there is inbreeding).  Deliberate avoidance of inbreeding, or failure of inbred fertilisations, could also affect Ne; a possible case of this occurs in gray seals Halichoerus grypus ((Amos, Wilmer et al. 2001)).

A wealth of other Ne calculations are available, requiring slightly different data sets and thus suiting different species (Caballero 1994) (Frankham 1995).  Nunney has produced one widely used method that requires relatively little data: number of adults, sex ratio, survival rate of adults of each sex, age at first breeding, and variance in reproductive output each breeding season  (Nunney and Elam 1994).  The latter is easier to measure than variance of total lifetime reproductive success, but still must be estimated for each sex, often requiring molecular data.  Like any simplification, Nunney’s method also requires assumptions, such as that survivorship is not dependent upon age.   There have been attempts to use even scantier data to judge the likely rate of genetic change (Section 6). 

Each of the measures above can be altered by a threatening process or managed in a conservation program, so there is considerable scope for intervention (chapter 9, 10).  Whether estimating Ne or one of its components, we must also make sure that our estimate applies to an appropriate time-scale.  Usually we want the Ne to be useful for predictions about future rates of change, but it can also be useful to compare estimates which apply to different time periods, answering questions such as “Do short-term estimates of Ne actually predict the rate of genetic change?” or  “Was the rate of genetic change over the past hundred generations the same as the current rate of change?".  If the answer to the latter question is no, then we may need to find and manage the causes of genetic change. 

5.3 Calculating effective sizes from genetic data

Some of the limitations of ecological data can be overcome by taking the reverse approach – observing the genetics of a population and asking “What Ne could have resulted in that pattern?”.  The steady increase of methods for revealing genetic variation in natural populations offers other methods of approaching this question, and the relative speed of some genetic methods is very appealing.  Making estimates of Ne which apply to long time-periods usually require long-term data, because of the effect of fluctuations and trends in populations size.  But of course, we want the answer quickly, and some molecular approaches can allow this.  However, it should be stated at the outset that there are also drawbacks of genetic estimates; the various genetic and ecological methods will be compared in more detail in section 6.

All genetic methods require data on hypervariable genetic markers, but they diverge sharply in their implementation and their predictive scope.  The type of data determines which Ne can be estimated.    If there are multiple loci studied, then we can use the level of genotypic disequilibrium to estimate NeD.  Alternatively, if more than one generation or population has been genotyped, then we may be able to use the difference in expected heterozygosity over time, or the divergence between populations, to estimate NeV (or the related eigenvalue Ne).   Knowledge of genetic variation at a single locus in a single population at a single time provides little power for estimation of Ne, and each method has its own sampling requirements of numbers of loci and individuals. There are a number of different statistical bases to the calculations, which have considerable importance for the power of the method; these considerations, and many of the available computer programs, are discussed in ((Luikart and England 1999)).  Here we do not dwell on the statistical approaches, but discus the genetic and ecological differences between the methods. Some of the methods detailed below can give immediate results, while others require longer-term data.  The time-period to which the resulting Ne applies, is not always the same as the period over which the data collected.  Also, predictions of genetic rate of change will generally apply best to the situation from which they were derived; for example, the rate of divergence between populations can be markedly different to the rate of change within populations ((Hendry and Kinnison 1999)).

Single-time, single-population estimates

Since Ne estimates a rate of change, it seems quite counter-intuitive that Ne could be estimated from genetic analysis of a single population, at a single time.  However, if it could be done, then the immediacy of the result would make these methods very appealing. In fact, there are two general ways in which we can estimate Ne from such limited data, if certain assumptions are made.  Both methods rely upon expectations of the way that genetic variation will be arranged in a population with a particular history and type of mutation, as well as other assumptions.

If it is assumed that the loci analysed in a population are not genetically linked, then in an infinite population, there should be genotypic equilibrium.  That is to say, if pA is the proportion of allele A1 at locus A, and pB is the proportion of allele B1 at locus B, then the expected proportion of haplotype A1 B1 (gametes that carry alleles A1 and B1) would be pA times pB.  However, in a finite population, the haplotype proportion will depart from this value due to the chance effects that we are concerned with in this chapter.  This departure from genotypic equilibrium has been used to estimate the effective population size of hatchery fish, and other species (Hill 1981) (Bartley, Bagley et al. 1992).  Of course, first we must know that we are studying unlinked loci, which can be checked in samples from wild populations, but with low power ((Black and Krafsur 1985; Black and Krafsur 1985)).  It must also be assumed that the loci are selectively neutral.  Linked loci could be used, but the degree of linkage must be known, which is unusual in threatened species (Hill 1981).

Another class of methods does not estimate the effective population size, but merely identifies whether a population has recently suffered a period of contraction (bottleneck) or expansion.  These tests are based upon the expected relationship, in a stable population, between two different measures of genetic variation (Table 5.1).  As pointed out above, different measures are expected to respond at different rates to changes of population size.  Thus a very high heterozygosity coupled with a very low number of alleles per locus might be a sign of a recent bottleneck.  Of course, we must know what are high and low levels for the relevant measures of genetic variation and this relies upon a number of assumptions.  First, we must know or assume the rate of mutation of the markers, and also the type of mutation – infinite alleles model or stepwise mutation model. By choice of appropriate markers, we can often be reasonably sure of these (chapter 2). Often, too, we must assume that the population was somewhere near equilibrium before the expansion or decline, which is a more difficult matter to validate.  Nevertheless, there are a wealth of such methods available, based on statistics such as the relationship between K and He, the number of rare alleles,  and the relationship of number of microsatellite alleles to the range in allele lengths (Luikart 1998; Luikart 1998.; Luikart and England 1999) (Cornuet and Luikart 1996) (Luikart and Cornuet 1998) (Piry, Luikart et al. 1999) (Garza and Williamson 2001).  Many of these methods have quite high success in simulation trials, and some also perform well in trials on populations with known history.  Figure 5.5 shows an example of agreement between data and theory for the contrast between allele numbers (K) and expected heterozygosity (He).  Figure 5.6 shows that in 20 studies of vertebrates and invertebrates, a test based on the relationship of allele size to allele numbers appeared to work well (Garza and Williamson 2001).  These tests are now beginning to be employed by conservation biologists – Rooney et al (Rooney, Honeycutt et al. 1999) showed that genetic data were not consistent with a recent bottleneck in bowhead whales Balaena mysticetus, despite a decline to possibly only 1000 individuals during unregulated hunting.  As well as being useful to warn the manager that an apparently large population has recently suffered a serious loss of genetic variation, these methods might also be useful as a warning of total population size decline, in populations which can be sampled but not easily censused.  Some methods based on a type of phylogenetic statistics called “coalescence” can also estimate how long it is since the bottleneck occurred, and the rate of expansion ((Beaumont 1999) (Wilson and Balding 1998)).  It has been suggested that coalescent methods might not work well on recent declines ((Garza and Williamson 2001)), but for the northern-hairy-nosed wombat Lasiorhinus krefftii, Beaumont’s method did produce a decline rate and date which are consistent with what is known of its very severe bottleneck of approximately 20-30 individuals in the mid-twentieth century.

Figure 5.5  Examples of a test for population decline.  Microsatellite heterozygosity excess (He) is plotted relative to that expected under long-term mutation-drift equilibrium for two species, a non-bottlenecked population of brown bears, and a bottlenecked population of wombats (From (Luikart and Cornuet 1998))
Figure 5.6  A bottleneck test on twenty studies of vertebrates and invertebrates.  The histogram shows M, the ratio (number of different allele sizes)/(number of steps in size between the largest and smallest allele). The lower mode contains eight populations with a history of reduction in population size, while the upper mode contains populations with no known bottleneck.  In some cases, different populations of the same species fall into each group (Garza and Williamson 2001).  

Spatial estimates
Since Ne can predict the rate at which replicate populations differentiate from one another (Figure 5.3 b), conversely, this rate can be used to estimate Ne.  One hundred and forty-two giant toads Bufo marinus were introduced to the island of Oahu in Hawaii in 1932, and then in the next three years, thousands of individuals were taken to four other Hawaiian islands. There was probably little loss of variation in the transfers, so the differentiation between the populations is probably due to Ne of the established populations.  In the 1980s, allozyme studies were used to derive a measure of differentation between the populations (Fst, chapter 6); Fst was converted to the effective population size of each island, using the equation Ne = -t/[2 ln(1-Fst)],  where t is the number of generations elapsed ((Easteal 1985; Easteal 1985).  The Ne appeared to be 390, with 95% confidence limits of 119 and 812.  This was lower than the estimate obtained by ecological methods, probably because the latter were unable to account for fluctuation in population size or variation in reproductive output between individuals.  Could this method be used in conservation biology?  Microsatellite analysis of four populations of African buffalo Syncerus caffer were used to estimate the effective sizes, which appeared to be approximately 15% of current census size ((O'Ryan, Harley et al. 1998)). Usually, wild populations will not all have been founded at approximately the same time, and there will be some degree of dispersal between them, slowing the differentiation and thus inflating the Ne estimate.  The problem of simultaneous estimation of Ne and dispersal rate is not an easy one, but there has been some theoretical progress, which will be dicussed further in chapter 6 ((Vitalis and Couvet 2001) (Beerli and Felsenstein 2001)). One method has been applied to fish data, but without confirmation from independent estimates of Ne (Kitada, Hayashi et al. 2000). 

Temporal estimates
The change though time of each of the measures of genetic variation in Table 5.1 is of prime concern in this book.  One of the easiest ways of knowing this change is, of course to measure it in the target population, or one with similar characteristics.  This rate would be expected to remain unaltered unless the factors that affect Ne change due to management, threats, or other factors.  If required, the rate of change can be converted to the Ne, using the reverse of the equation in section 1 (  Het = He0 (1-1/2NeV)t), or its equivalent for some other measure of genetic variation.    This temporal method requires a control sample to allow estimation of the genetic variation at the start of the time period, He0.  This method has been used in fish, mammals, and many other species ((Waples 1989; Waples 1990)).  Of course, a longer time-period gives a better estimate ((Kitada, Hayashi et al. 2000))  However, there is some evidence that the method can be applied to samples separated by only a single generation, if enough individuals and loci are used (Luikart, Sherwin et al. 1998).  The temporal method is not restricted to He: a study of butterflies Bicyclus anyana used genotypic disequilibrium estimates over only three generations to estimate Ne ((Saccheri, Wilson et al. 1999)).   The temporal method requires a large number of loci to give good estimates.  Slatkin and Williamson ((Williamson and Slatkin 1999)) suggest that increasing up to 50 loci will improve the estimates, but there will be little improvement above that. 
Whatever sampling is used, it is important to remember that the temporal Ne method is only able to estimate an Ne applicable to the time period that the samples span.  Thus if conditions have changed recently, or change in the future, the estimate may be erroneous.  By genotyping Prairie chicken (Tympanus cupido) samples collected over ten years, a low rate of genetic change and therefore a high Ne was inferred ((Bouzat 2000)).  However, when a different baseline (He0) was used, by genetic analysis of museum specimens taken many decades before, it was apparent that  there had been much more genetic erosion per generation, and thus the Ne estimate was much lower.  This underlines the importance of baseline data in conservation genetics, including estimation of past genetic variation using DNA amplified from subfossil or museum collections ((Taylor, Sherwin et al. 1994) (Bouzat 2000) (Matocq, Patton et al. 2000)).  If historical samples cannot be used, the temporal method is a slow procedure, and it is not usually possible to wait many generations into the future for our Ne estimate, which, after all, we are trying to use to predict the future.  One alternative is to use variation measures from an extant large population to estimate the amount of variation present before a decline of the study population ((Sherwin, Murray et al. 1991) (Taylor, Sherwin et al. 1994)). 
 The estimation of bottleneck size can be confounded with a number of other factors, but methods are being developed to deal with this.  Firstly, a bottleneck of long duration, or repeated incidence, can produce the same loss of genetic variation as a smaller bottleneck of shorter duration.  Also, the temporal method assumes that the population has no immigration, but as expected, a very small number of immigrants can slow genetic loss in a small population and thus give an apparently high Ne ((Keller, Jeffery et al. 2001)).  Where it is necessary to simultaneously estimate bottleneck size, duration, and other parameters, a simple equation such as we have shown above is insufficient, and there must be simulation of the evolutionary process, and/or the use of estimation techniques such as maximum likelihood or Bayesian inference ((Williamson and Slatkin 1999) (Anderson, Williamson et al. 2000)) (Saccheri, Wilson et al. 1999) (Keller, Jeffery et al. 2001) (Halley and Hoelzel 1996).

5.4 Timescale

Because Ne forecasts the rate of genetic change per generation, the generation length sets the scale for all these investigations.  We saw above how to estimate the generation length when generations overlap (section 5.2).  Thus in conservation of a wild population, urgency depends on the speed of change per generation, and the generation length.  Threatened populations with the same Ne but different generation lengths are likely to accumulate genetic problems at a different rate.  In longer –lived species, there is more time to develop and implement conservation actions. In ex-situ situations, managers sometimes extend the generation time, by retaining the oldest individuals and their most recent progeny, until the older individuals cease to breed.  Breeding from individuals of intermediate age would simply use up space in the ex-situ facility, and reduce the generation time and therefore accelerate the rate of genetic change (which is usually erosion of variation, as we have seen above).   

We also need to know whether the rate of change per generation has increased as a result of threatening processes.  There are methods which estimate Ne over millennia - can these be used for comparison with current Ne?  Estimates of long–term Ne  such as analyses of allele genealogies within populations, could provide a yardstick against which current Ne and population fluctuations can be compared.  Large deviations would signal a need for attention: a reduction in current Ne relative to historical Ne would be seen as a warning sign for managers (Crandall, Posada et al. 1999).  These authors point out that because the inbreeding Ne integrates over many generations, it may carry longer-term information that the variance or similar Ne values, in a threatened population which has recently changed size.  However, extracting the necessary information to measure the separate Nes is not easy, as discussed above, and considerable research is needed.  The example of the Prairie chicken in the previous section shows how careful we have to be in making estimates on different timescales within one century.

On a longer time-scale, Ne values can be inferred from very deep phylogenies.  A star-like phylogeny indicates an expansion of the population size or range, for example, but the effects of multiple expansions and changing rates of expansion must be teased apart before useful information is available for comparison with shorter-term estimates.  These methods are discussed in more detail in chapter 6.  A single example sets out the difficulties here: colonies of European Beech Fagus sylvatica in glacial refuges have lower gene diversity (He) but higher allelic diversity (K) than recolonised areas.  This is possibly explained by particular patterns of selection or recolonisation, but the occurrence of discordant results like this make for great difficulties in interpreting long-term data ((Comps, Gomory et al. 2001)).  Efforts to identify population size changes from phylogenetic data continue (Nee et al, 1995).

5.5 Spatial scale

So far, we have mostly considered the effective size of an isolated population.  This may well be appropriate for many natural populations, and others are becoming isolated as a result of threats.  However, there are other possibilities:  the species may exist as multiple populations, each following their own drift trajectory (Figure 5.1(c)), or it may have a continuous distribution which is much greater than dispersal distances, so that local regions can differentiate through drift, despite the continuous population – a situation called “isolation by distance”.  There can also be an intermediate state, where there is some subdivision but also limited gene flow.  The discussion of completely or partially subdivided populations is the topic of the next chapter, but here we will examine the effective population size consequences of isolation by distance in a continuous population.

For a continuous population, the question arises ”What portion of this population can be considered separately in population biology?”.   Wright suggested we consider an area called th “neighbourhood”, which contains an approximately random-mating population, which would be affected by drift in much the same way as an isolated population (Wright 1943). The neighbourhood area is defined by the lifetime dispersal distance, or to be exact V, the variance of gene flow  (Figure 5.7).  If D, the population density is known, the population size within the neighbourhood can be calculated  (Figure 5.7).  The population size in a neighbourhood can then be used as the starting point for Ne calculations, i.e., as the census size which can then be corrected for variance in reproductive output between individuals, etc (section 5.2).  The result of this calculation will tell us what is the likely rate of change within the neighbourhood.  It does not tell us what might happen over the entire population, and in fact a population containing many neighbourhoods is expected to lose different alleles in different neighbourhoods, and therefore global drift will be much slower than neighbourhood drift.  For most values of neighbourhood population size, the global rate of drift is best calculated by considering the entire range as a single population ((Chambers 1995)).  The global rate of drift could theoretically be lower in extreme cases, when VD < 1 (Figure 5.8).  Even at VD=1, individuals are likely to encounter very few potential mates because of the low densities and dispersal rates.  Such populations, or those with lower VD, may occur sometimes in conservation, and although they will have relatively slow loss of variation by drift, they are likely to have other problems such as inhibition of breeding due to low density (Allee effect).  If VD >1, then a conservation geneticist needs only to known global Ne for the managed area to predict total rate of loss of variation.  However, calculation of neighbourhood size will also tell us whether different parts of the range could differentiate by drift, and thus more easily show adaptive differences if the environment changes.  

Of course, we are only going to know whether VD is greater or less than one by estimating them. As described in chapter 4, gene flow and population density may be estimated by ecological or genetic methods, or a combination of them both.  In an experimental population of the plant Mimulus ringens, genetic estimates of dispersal were higher than those from pollinator flight measurements, suggesting that there is pollen carry-over beyond a single flight ((Karron, Tucker et al. 1995)).   However, the neighbourhood size was very low (1.66 to 5.53), suggesting that there could be high levels of local differentiation in this population, and that the global rate of genetic change may be lower than expected if it was treated as a single population.  There is no guarantee that this would occur in a wild plant of this species.  In a wild population of skinks Gnypetoscincus queenslandiae, neighbourhood size was estimated two ways, firstly by using density estimates from mark-recapture data and dispersal distances estimated from those data (Sumner, Rousset et al. 2001).  Secondly, an indirect, genetic method was based on the regression of genetic differentiation between individuals and the geographic distance between them, further described in chapter 6 ((Rousset 2000)).  The resulting direct and indirect estimates of neighbourhood population size were 144 and 69 individuals respectively.  Both methods have practical problems, but the genetic method requires only a single sampling period, when genetic and geographical data must be obtained.  A drawback of the genetic method is that it only estimates the VD, and to discover either dispersal or density separately, the other one must be known from ecological methods.  Of course, the sampling for the genetic work will provide an estimate of D, so that V can then be calculated.  There are limited investigations of the sampling requirements of the genetic method, but Rousset pointed out that it may be better to maximise the number of loci rather than individuals, and that for populations with large dispersal, it is important to increase both the number of individuals and the range of geographic distance that separates them.

Figure 5.7  A neighbourhood embedded in a larger range of a continuously distributed species.  The radius of the circle is twice the standard deviation of lifetime dispersal (or more accurately, gene flow, ie dispersal and reproduction) in one direction.  This means that 95% of individuals dispersing from the central point will remain within the population.  The area of the neighbourhood is 4pV, where V is the variance of dispersal.  If we know the population density (D - number of individuals per unit area) then the population size in the neighbourhood is 4pVD.  Note that neighbourhoods can overlap, since their boundaries must be arbitrary in a continuous population.  If dispersal distances is greater in one direction, then the neighbourhood will not be circular.  In the case of a linear population such as a stream-dwelling fish, with dispersal in only one direction (forwards OR backwards along the axis of the population), neighbourhood population size is (4pV)1/2 D.  Note that whether the population is linear or not, the distance moved should be estimated in one dimension only ((Chambers 1995)).

Figure 5.8   Conditions under which a global population will lose genetic variation more slowly than expected because of isolation by distance within the population.  This only occurs in cases where 95% of the population disperse very little, or the density of individuals is very sparse – the tiny region to the left and below the curve.  In these cases, the global loss of heterozygosity per generation is VD/2Ne, where Ne is the global Ne ((Maruyama 1972))

The case where a managed population consists of a number of semi-isolated sub-populations is the subject of the next chapter, and the rate of genetic change depends upon the rate of dispersal between the populations, and the life-history, mating patterns, etc within each population.  One approach to estimate the global rate of genetic change by drift is to obtain the global effective population size: the harmonic mean of the effective population sizes of the subpopulations ((Wright 1968, 1969, 1977, 1978 1977, 1978 #864 1977, 1978 #864; Hartl and Clark 1997; Hedrick 2000)).  Recent work is aimed at building complex models of metapopulations, to allow simultaneous estimation of a number of characteristics, including global and single-population rates of change ((Basset, Balloux et al. 2001))

5.6 Surrogates & comparability of measures

In many species, population size itself is relatively easy to measure, and so an obvious surrogate for Ne is Nc – adult population size estimated from some kind of census. In general, it appears that Ne is usually lower than Nc, by up to two orders of magnitude, with average Ne/Nc = 0.1 (Frankham 1995).  This stands as a warning to conservation biologists: simply seeing a large number of adults is no guarantee that loss of variation by drift will be only slow.  This will be discussed further in chapters 9 and 10.  Given the amount of data necessary to estimate Ne or even Nc, it is important to consider surrogates that require less data.  In a series of papers, Nunney has attempted to establish whether there is a generally applicable ratio of effective population size to census population size, so that only measurement of Nc is necessary ((Nunney 1993; Nunney and Elam 1994; Nunney 1994; Nunney 1995; Nunney 1996; Nunney 1999; Nunney 2000)).   It appears that a broad relationship may hold in some taxa; for example, invertebrates had lower Ne/Nc ratios than vertebrates ((Vucetich, Waite et al. 1997)).   However, it is also known that within a single species (flour beetles, Tribolium castaneum), a change of Nc can be associated with a change in Ne/Nc, (Pray, Goodnight et al. 1996).  The latter result may not be serious conservation concern, since a change of Nc over one order of magnitude resulted in only a 10% change of Ne/Nc, and smaller populations tended to have larger Ne relative to Nc.  Thus the smaller population would have slower rate of genetic change than expected from their Nc, and management based on extrapolation of Ne/Nc ratios from large populations to smaller threatened populations would be conservative.

There have been a number of attempts to single out one factor which has predominant influence on Ne, then use this to estimate Ne/Nc.  Theoretically, is can be shown that the coefficient of variation of population size may be a predictor of Ne/Nc, since fluctuation of population size is an important factor in determination of Ne (Vucetich, Waite et al. 1997).  Waite (Waite and Parker 1996) has investigated another theoretical expectation, that Ne/Nc will depend upon the ratio of the age at reproductive maturity to the average adult lifespan.  Under certain assumptions, mean and variance of lifetime reproductive success can be estimated from partial data, such as success at only one age, and then this can be converted to Ne/N (Rockwell and Barrowclough 1995).  Frankham showed that the most important factors influencing Ne/Nc were (in decreasing order of importance): size fluctuation, variance in family size, taxonomic group, and adult sex-ratio.  Another technical factor, which ranked at third position in this list, was whether the census included all adults, only breeders, or individuals of all ages.  Frankham also found that ecological estimates were higher than genetic estimates, and that genetic estimates differed from one another as well.  The difference between ecological and genetic estimates disappeared when both measures assessed the same factors (ecological estimates often do not account for fluctuation in population size, since many years of data are required to give an adequate estimate - (Vucetich and Waite 1998)). 

It is apparent that each measure of Ne or Nc has its own limitations and assumptions, yet the patterns described above show that they do generally show a predictable relationship to one another, and errors would usually be no more than an order of magnitude if monitoring of Nc was used as a surrogate for Ne, or vice versa.  When doing this, it will be necessary to examine the assumptions of each of the methods being used, and their applicability to the population being studied.  Points to which particular attention should be paid are:
- temporal and spatial scope for the data collection
- assumptions, such as absence of immigrants, details of life history, mating patterns
- type of genetic change being predicted
- temporal and spatial scope that the method is sensitive to (this can be different to the data collection scope, see eg NeD, above)
- numbers of individuals (and loci, and type of loci, if genetic methods used).
 Ideally, a model would be written that is specific for the population and accessible data.  It is also important to consider the importance of each ne measure to conservation.  NeV is expected to predict rate of change of gene diversity and additive genetic variation, but underestimate the rate of loss of rare alleles.  A contrast between variance effective size and NeI, which is sensitive to a longer time-period, may indicate to the conservation biologist that there has been recent change to the size or mating structure of the population.  Very long-term estimates from phylogenies are very difficult to compare with those of current populations, since they sum the effects of a much wider range of ecological conditions.

Finally, monitoring N or Ne on their own is not ideal.  If threatening processes are also monitored, associations between these and changes to N or Ne can be assessed.  Ideally, there will be data on each of the factors that could affect Ne (Section 5.2), so that we can assess whether the threat is depressing Ne by affecting one of those (eg, by altering sex-ratio, Milner-Gulland et al ).  Also, baseline data on genetic variation, and regular sampling are important, so that if problems are suspected with the genetics of the population, there can be genetic analysis of samples, to allow us to check on the Ne estimation and re-calculate Ne.  Direct associations with changes of fitness are important also, but to limit our concern to cases where these associations occur, would be to deny that genetic variation is the basis for long–term adaptation, and worthy of conservation in its own right (chapter 9).
 
5.7 Mutation and  Recombination

Is there a critical Ne below which a population will not be able to maintain its genetic variation?  When will loss of variation outstrip its re-creation by mutation and recombination?  Like drift, the production of new mutations and recombinants is dependent on the size of the population – a bigger population is more likely to have one individual whose germ-line produces a new variant.  We must also take account of the speedy loss of the large percentage of new variants which are harmful (Lande, 1995).  Using calculations of mutation rates for QTLs from Falconer (Falconer and Mackay 1996), and data from Drosophila, Franklin, (Franklin 1980) suggested that an effective size of 500 is necessary for long-term conservation of genetic variation.  This may translate to a census size of up to 50,000 adults, depending upon the Ne/N ratio!  Clearly, many wild populations have survived with population sizes much lower than this for long periods.  Calculations are more difficult for rare alleles, since although they are lost faster than Va, their numbers also recover faster by mutation, as noted above.  Overall, it is not advisable to set a particular “critical Ne” for conservation, but better to assess the measures mentioned in the last section, and take any departure from baseline levels in any of the variables to be a serious sign, worthy of further investigation.

5.8 Selection and population size
Not done yet	

5.9 Changed architecture and population size
Not done yet

5.9 What if we do nothing? 

If we do not monitor effective population size, or one of its surrogates, such as census size, fluctuations of census size, or levels of genetic variation, then we will have no way of knowing whether any adverse changes in the population are due to genetic problems such as low variation and inbreeding, as is clearly the case in some species (See section 1, and chapters 9 and 10).  We would also have no way of assessing whether we are maintaining the within-population component genetic variation.  At least in the short-term, this is probably the major source of variation upon which future adaptation will be based.  In the longer term, if there is some exchange with other populations, then within-population variation may be replenished, is the metapopulation has been well-managed genetically (chapter 11).

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