# The genetic Allee effect: a unified framework for the genetics and demography of small populations

Gloria M. Luque, Chloé Vayssade, Benoit Façon, Thomas Guillemaud, Franck Courchamp, Xavier Fauvergue

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Gloria M. Luque, Chloé Vayssade, Benoit Façon, Thomas Guillemaud, Franck Courchamp, et al. The genetic Allee effect: a unified framework for the genetics and demography of small populations. Ecosphere, 2016, 7 (7), pp.e01413. 10.1002/ecs2.1413. hal-01594603

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<figure><img src="image_1.png" /></figure>
# The genetic Allee effect: a unified framework for the genetics and demography of small populations

GLORIA M. LUQUE,<fnref n="1" /> CHLOÉ VAYSSADE,<fnref n="2" /> BENOÎT FACON,<fnref n="3" /> THOMAS GUILLEMAUD,<fnref n="2" /> FRANCK COURCHAMP,<fnref n="1" />,<fnref n="4" /> AND XAVIER FAUVERGUE<fnref n="2" />,†

**Citation:** Luque, G. M., C. Vayssade, B. Facon, T. Guillemaud, F. Courchamp, and X. Fauvergue. 2016. The genetic Allee effect: a unified framework for the genetics and demography of small populations. Ecosphere 7(7):e01413. 10.1002/ecs2.1413

**Abstract.** The Allee effect is a theoretical model predicting low growth rates and the possible extinction of small populations. Historically, studies of the Allee effect have focused on demography. As a result, underlying processes other than the direct effect of population density on fitness components are not generally taken into account. There has been heated debate about the potential of genetic processes to drive small populations to extinction, but recent studies have shown that such processes clearly impact small populations over short time scales, and some may generate Allee effects. However, as opposed to the ecological Allee effect, which is underpinned by cooperative interactions between individuals, genetically driven Allee effects require a change in genetic structure to link the decline in population size with a decrease in fitness components. We therefore define the genetic Allee effect as a two-step process whereby a decrease in population size leads to a change in population genetic structure and, in turn, to a decrease in individual fitness. We describe potential underlying mechanisms and review the evidence for this original type of component Allee effect, using published examples from both plants and animals. The possibility of considering demogenetic feedback in light of genetic Allee effects clarifies the analysis and interpretation of demographic and genetic processes, and the interplay between them, in small populations.

**Key words:** Allee effect; drift load; eco-evolutionary feedback; extinction vortex; inbreeding depression; migration load; small populations.

**Received** 3 May 2016; accepted 11 May 2016. Corresponding Editor: D. P. C. Peters.

**Copyright:** © 2016 Luque et al. This is an open access article under the terms of the Creative Commons Attribution
License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

† **E-mail:** xavier.fauvergue@sophia.inra.fr

## INTRODUCTION

The Allee effect can be defined as a decrease in
fitness caused by a decrease in population size
(Stephens et al. 1999). The Allee effect jeopardizes
the persistence of small populations, whether
declining or bottlenecked (threatened species;
Angulo et al. 2007, Bonsall et al. 2014, Kuparinen
et al. 2014; introduced/invasive species; Davis

et al. 2004, Taylor and Hastings 2005, Johnson
et al. 2006). This concept, first introduced by the
American ecologist Warder Clyde Allee in the
1930s, has been the focus of growing interest from
population ecologists over the last two decades.
In 1999, Stephens et al. formalized the defini-
tion of Allee effects, making a crucial distinction
between component and demographic Allee
effects. A component Allee effect is a decrease

<footnote n="1">¹Ecologie Systématique Evolution, Univ. Paris-Sud, CNRS, AgroParisTech, Université Paris-Saclay, 91400, Orsay, France</footnote>

<footnote n="2">²ISA, UMR INRA CNRS, Université Nicé Côte d'Azur, 06903 Sophia-Antipolis, France</footnote>

<footnote n="3">³CBGP, UMR INRA, CIRAD, IRD, SupAgro, 34988, Montferrier sur Lez, France</footnote>

<footnote n="4">⁴Department of Ecology and Evolutionary Biology, Center for Tropical Research, Institute of the Environment and Sustainability, University of California Los Angeles, Los Angeles, California 90095 USA</footnote>
in the value of any component of individual fit-
ness caused by a decrease in population size.
A demographic Allee effect is the consequent
decrease in the per capita growth rate of the pop-
ulation caused by a decrease in population size
and resulting from one or several component
Allee effects. Allee effects have been discovered
in a wide range of taxa, with various mechanisms
underpinned by a shortage of cooperative inter-
actions at low density (Courchamp et al. 1999,
2008).

Population dynamicists working on the Allee
effect may have overlooked the fact that genetic
processes can also impact populations at an
ecological time scale of only a few generations
(Glémin 2003, Spielman et al. 2004). This is par-
ticularly relevant when genetic processes are
combined with demographic processes, such
as demographic and environmental stochas-
ticity (Tanaka 2000, Hanski and Saccheri 2006).
Some of these genetic mechanisms can gener-
ate component Allee effects (Courchamp et al.
1999) because they yield a positive relationship
between population size and fitness (Fischer
et al. 2000, Willi et al. 2005, Berec et al. 2007).
These mechanisms are inbreeding depression
(Frankham 1995, Spielman et al. 2004), loss of
genetic variation (Gomulkiewicz and Holt 1995,
Amos and Balmford 2001), and the accumula-
tion of deleterious mutations (Lande 1994, Lynch
et al. 1995b).

Contrasting with the pervasive evidence that
genetic mechanisms affect the dynamics of small
populations, even in the short term, fewer than
ten publications have made specific reference to
a genetic Allee effect. There are two possible rea-
sons for the lack of studies considering genetic
Allee effects. First, although mentioned in a hand-
ful of studies, the genetic Allee effect has never
been formally defined, so the concept may still
be too vague to stimulate new studies. Second,
the genetic mechanisms occurring in small pop-
ulations are the classic territory of population
genetics, whereas the Allee effect is a concept
derived principally from population dynamics.
These two communities view populations differ-
ently and usually work on different entities (e.g.,
individual numbers vs. genotype frequencies),
concepts and time scales. Most studies in popu-
lation genetics assume populations of constant
size, unaffected by individual fitness (even when

modeling inbreeding depression; e.g., Glémin
2003) and dynamicists usually view genetic pro-
cesses as affecting populations only in the long
term. This would make them likely to think that
small populations do not generally stay small for
long enough (because they grow or disappear
due to demographic processes) to suffer from
genetic Allee effects. There may have been too
little dialog between the two disciplines as yet
(Kokko and Lopez-Sepulcre 2007, Metcalf and
Pavard 2007, Pelletier et al. 2009) for the genetic
Allee effect to have emerged as a robust unifying
paradigm.

The aim of this work was to propose the genetic
Allee effect as a heuristic framework for studying
the interplay between genetics and demography
in small populations. We first propose a formal
definition for the genetic Allee effect and describe
the processes potentially underlying this effect.
We review the evidence for genetic Allee effects,
including population genetic studies, which,
although not referring to the Allee effect, pro-
vide numerous examples of genetic Allee effects.
Based on these examples, we then propose meth-
ods for detecting genetic Allee effects and distin-
guishing them from ecological Allee effects. In
the last section of the study, we discuss the spec-
ificity, limits, and demographic consequences of
genetic Allee effects.

DEFINITION

One fundamental principle of evolutionary
biology is that a population is a collection of dif-
ferent genotypes vehicled by a number of differ-
ent individuals. Changes in genotype frequencies
correspond to changes in the number, frequency,
and association of different alleles within and
between loci. These genotype frequencies are
referred to hereafter as the (within-population)
genetic structure. Changes in genetic structure
yield changes in the fitness components that
these genotypes influence (Soulé 1980). This rela-
tionship underlies the genetic Allee effect because
population size is one of the determinants of
genetic structure (Nielsen and Slatkin 2013).
Here (following the evolutionary paradigm
described by Waples and Gaggiotti 2006), the
term “population” refers to a “a group of indi-
viduals of the same species living in close enough
proximity that any member of the group can
<figure><img src="image_2.png" /></figure>

Fig. 1. Two successive steps are necessary for a genetic Allee effect to occur. Step 1: A decrease in population size causes a change in (a parameter of) genetic structure (heterozygosity, frequency of detrimental and beneficial alleles, allelic richness). Step 2: The change in genetic structure causes a decrease in the value of a component of fitness through inbreeding depression, drift load, or migration load. When both steps occur in a population, a component genetic Allee effect occurs (left panel).

potentially mate with any other member." Unlike population dynamicists, we use the term "population size" to mean the number of individuals that effectively reproduce. We prefer to use this term, rather than effective population size, because the latter is difficult to measure in natural populations and has an ambiguous meaning resulting from its various definitions (the "inbreeding effective population size," the "variability effective population size," and the "eigenvalue effective population size"; see details in Sjödin et al. 2005). Population size, genetic structure, and fitness components are the three cornerstones of the genetic Allee effect.

The term “genetic Allee effect” is not new (first appearance in Fischer et al. 2000), but it has seldom been used, and still lacks a proper definition. We define the genetic Allee effect as a two-step process characterized by (1) a change in population genetic structure due to a decrease in population size and (2) a consequent decrease in individual fitness. This decrease in individual fitness may subsequently lead to a further decrease in population size. These two steps act sequentially to generate a genetically driven component Allee effect. The first step of the genetic Allee effect occurs when population size affects at least one of the following aspects of genetic structure: heterozygosity, frequency of beneficial or detrimental alleles, or allelic richness (Fig. 1). During the second step, any of these changes in population genetic structure may decrease a component of individual fitness through inbreeding depression, drift load, or migration load. If, and only if, both steps occur, a genetic Allee effect occurs. The genetic Allee effect therefore contrasts with what could be referred to as an ecological Allee

effect, in which individual fitness decreases as a straightforward, one-step consequence of a decrease in population size or population density.

## MECHANISMS UNDERLYING GENETIC ALLEE EFFECTS

Our literature search identified 15 studies showing strong evidence for one or several genetic Allee effects in natural or experimental populations (Table 1). Some of these studies did not actually use the term genetic Allee effect. We also identified about another 40 studies, suggesting the occurrence of genetic Allee effects (see for instance the literature cited in Leimu et al. 2006). Following a thorough analysis of these published studies, we defined three types of genetic mechanisms generating Allee effects, each involving a major evolutionary force: inbreeding, drift, or migration.

### Inbreeding depression

As population size declines, inbreeding becomes more frequent (Fig. 2; Malécot 1969). An inbred individual results from a cross between two genetically related individuals and is characterized by a high inbreeding coefficient. Inbred individuals have fewer heterozygous loci than outbred individuals, resulting in a decrease in heterozygosity in declining populations (Frankham 1996, 1998). A pervasive consequence of low heterozygosity is inbreeding depression, defined as the lower fitness of inbred than of outbred individuals. Hence, inbreeding followed by inbreeding depression fulfills the two conditions defining the genetic Allee effect: (1) a decrease in population size causing a change in population genetic structure (a decrease in heterozygosity) and (2) a decrease in one or
<table><caption>Table 1. Published evidence for genetic Allee effects in natural and experimental populations. Genetic Allee effects in natural populations are classified according to their underlying mechanism.</caption><thead><tr><th>Taxon</th><th>Evidence for Step 1</th><th>Parameter of genetic structure</th><th>Evidence for Step 2</th><th>Fitness component</th><th>Evidence for Allee effect</th><th>Reference</th></tr></thead><tbody><tr><td colspan="7">(A) Inbreeding depression</td></tr><tr><td>Plant <i>Cochealaria bavarica</i></td><td>Yes</td><td>Number of polymorphic loci and observed He for 8 allozymes</td><td>Untested</td><td>Fruit set</td><td>Yes</td><td>Fischer et al. (2003)</td></tr><tr><td>Plant <i>Ranunculus reptans</i></td><td>Yes</td><td>Mean kinship coefficient†</td><td>Yes</td><td>Clonal performance;‡ seed production</td><td>No</td><td>Willi et al. (2005)</td></tr><tr><td>Insect <i>Melitaea cinxia</i></td><td>Yes</td><td>Observed He at 7 enzymes and 1 microsatellite</td><td>Yes</td><td>Larval survival; adult longevity; egg hatching rate</td><td>Untested</td><td>Saccheri et al. (1998)</td></tr><tr><td colspan="7">(B) Drift load</td></tr><tr><td>Plant <i>Ranunculus reptans</i></td><td>Yes</td><td>Allelic diversity at 7 allozymes</td><td>Yes</td><td>Seed production</td><td>No</td><td>Willi et al. (2005)</td></tr><tr><td>Plant <i>Hypericum cumulicola</i></td><td>Yes</td><td>Effective population size estimated from 10 microsatellites</td><td>Untested</td><td>Cumulative fitness§</td><td>Yes</td><td>Oakley and Winn (2012)</td></tr><tr><td>Mollusk <i>Physa acuta</i></td><td>Yes</td><td>Gene diversity at 10 microsatellites</td><td>Untested</td><td>Reproductive life span</td><td>Yes</td><td>Escobar et al. (2008)</td></tr><tr><td>Insect <i>Melitaea cinxia</i></td><td>Yes</td><td>Observed and expected He and allelic richness at 9 microsatellites</td><td>Untested</td><td>Mating rate; egg clutch size; hatching rate; larval survival; larval weight; larval group size at diapause; lifetime larval production</td><td>Yes</td><td>Mattila et al. (2012)</td></tr><tr><td>Amphibian <i>Rana temporaria</i></td><td>Yes</td><td>Difference F<sub>st</sub>-Q<sub>st</sub> for larval body size; allelic richness at 7 microsatellites</td><td>Yes</td><td>Body size; larval survival</td><td>Untested</td><td>Johansson et al. (2007)</td></tr><tr><td>Plant <i>Cochealaria bavarica</i></td><td>Yes</td><td>Number of alleles per locus, number of polymorphic loci and observed He for isoenzymes</td><td>Untested</td><td>Compatibility of crosses</td><td>Yes</td><td>Fischer et al. (2003)</td></tr><tr><td>Plant <i>Ranunculus reptans</i></td><td>Yes</td><td>Allelic diversity at 7 allozymes</td><td>Yes</td><td>Compatibility of crosses</td><td>No</td><td>Willi et al. (2005)</td></tr><tr><td>Plant <i>Brassica insularis</i></td><td>Yes</td><td>Number and frequencies of self-incompatibility alleles</td><td>Untested</td><td>Proportion of flowers pollinated with compatible pollen; fruit set</td><td>Yes</td><td>Glémin et al. (2008)</td></tr><tr><td>Plant <i>Rutidosis leptorrhynchoides</i></td><td>Yes</td><td>Number and frequencies of self-incompatibility alleles</td><td>Untested</td><td>Seed set</td><td>Yes</td><td>Young and Pickup (2010)</td></tr><tr><td>Plant <i>Primula vulgaris</i></td><td>Yes</td><td>Morph frequencies</td><td>Untested</td><td>Number of fruits per flower</td><td>Yes</td><td>Brys et al. (2007)</td></tr><tr><td colspan="7">(C) Migration load</td></tr><tr><td>Plant <i>Eucalyptus aggregata</i></td><td>Yes</td><td>Hybridization rate</td><td>Yes</td><td>Germination rate; survivorship</td><td>No</td><td>Field et al. (2008)</td></tr><tr><td colspan="7">(D) Experimental populations (genetic structure manipulated independently of population size)</td></tr><tr><td>Plant <i>Clarkia pulchella</i></td><td>–</td><td>Number of founding families</td><td>–</td><td>Germination rate; survival rate from flower to fruit</td><td>Yes</td><td>Newman and Pilson (1997)</td></tr><tr><td>Plant <i>Raphanus sativus</i></td><td>–</td><td>Relatedness of founders</td><td>–</td><td>Fruit set; number of seeds per fruit</td><td>Yes</td><td>Elam et al. (2007)</td></tr><tr><td>Plant <i>Lolium multiflorum</i></td><td>–</td><td>Number and relatedness of founders</td><td>–</td><td>Seed set; proportion of florets producing seeds</td><td>Yes</td><td>Firestone and Jasieniuk (2013)</td></tr><tr><td>Insect <i>Bemisia tabaci</i></td><td>–</td><td>Relatedness of founders</td><td>–</td><td>Number of offspring</td><td>Yes</td><td>Hufbauer et al. (2013)</td></tr><tr><td>Insect <i>Tribolium castaneum</i></td><td>–</td><td>Relatedness of founders</td><td>–</td><td>Number of offspring</td><td>Yes</td><td>Szücs et al. (2014)</td></tr></tbody></table>
*Notes*: Evidence for Step 1 indicates whether a relationship between population size and genetic structure was tested and found in the expected direction. Evidence for Step 2 indicates whether a relationship between genetic structure and a component of fitness was tested and found in the expected direction. Strong evidence for a genetic Allee effect implies evidence for both Steps 1 and 2. Evidence for Allee effect indicates whether a positive relationship between population size and a fitness component was observed; this corroborates but does not prove that a genetic Allee effect is occurring. Experimental populations are populations in which genetic structure was manipulated independently of population size. For the four examples concerning experimental populations, the mechanism causing the genetic Allee effect was not identified. He: heterozygosity.

†Smaller populations have a lower allelic diversity at allozyme loci, and populations with lower allelic diversity have a higher kinship coefficient, but the relationship between population size and kinship coefficient was not tested.

‡Proportion of ovules producing seedlings × number of rooted rosettes.

§Proportion fruit set × seed number per fruit × proportion germinating × proportion surviving and reproducing × fecundity.

several components of fitness caused by this change (Fig. 2; Frankham 1998, Reed and Frankham 2003, Spielman et al. 2004). The mechanisms causing inbreeding depression (increased expression of deleterious recessive alleles, overdominance, and epistasis; Li et al. 2008, Charlesworth and Willis 2009) underlie the second step of a genetic Allee effect. For instance, the buttercup *Ranunculus rep-tans* suffers from a genetic Allee effect due to inbreeding depression (Table 1A; Willi et al. 2005): small populations have a higher mean inbreeding coefficient (Step 1), and populations with higher inbreeding coefficients have lower levels of seed production (Step 2).

## Drift load

Here, we define drift as a change in allele frequencies due to the random sampling of alleles from one generation to the next (Wright 1969) and drift load as the decrease in the mean fitness of a population due to drift (Charlesworth et al. 1993, Whitlock 2000). Genetic drift is a consequence of finite population size, and, like other random processes, such as demographic stochasticity, the manifestations of genetic drift are more evident in small populations (Fig. 2; Gabriel et al. 1991, Gabriel and Bürger 1994, Lynch et al. 1995a, Oostermeijer et al. 2003, Frankham 2005, Grueber et al. 2013). The balance between drift and selection determines allelic and genotypic frequencies. Selection is a deterministic process whereby the frequency of an allele (or a genotype) changes across generations according to its effect on fitness: Beneficial alleles increase in frequency, whereas deleterious alleles decrease in frequency. Unlike drift, the intensity of which increases with decreasing population size, selection does not vary in intensity as a function of population size (Fig. 2), that is, we do not consider density-dependent selection here. Hence, in small populations, drift intensity often overwhelms selection, so that any changes in the

number and frequencies of alleles result principally from drift rather than selection (Kimura 1983, Whitlock 2000).

The change in the selection/drift balance at small population size modifies the frequency of beneficial alleles, allelic richness, and the heterozygosity of populations, driving the first step of the genetic Allee effect (Fig. 2). The exact nature of the change in genetic structure depends on the type of selection at work. When drift overwhelms negative selection, mildly deleterious alleles increase in frequency (Lanfear et al. 2014). Mildly deleterious alleles sometimes become fixed and constitute the drift load (Whitlock 2000, Glémin 2003). The corollary is that some beneficial alleles, despite being under positive selection, decrease in frequency or even disappear (Lanfear et al. 2014). The value of fitness components decreases if these components are influenced by loci that fix detrimental alleles and lose beneficial ones (Step 2). Variations in larval body size in the European common frog *Rana temporaria* provide an interesting example, with drift stronger than selection in small populations and weaker than selection in large populations (Step 1). Consequently, small populations have a higher drift load on larval body size and display lower values for this fitness component (Table 1B; Johansson et al. 2007). Balancing selection also results in drift load. Balancing selection maintains several alleles at a locus under selection. Balancing selection can be due to overdominance and negative frequency-dependent selection. In small populations, in which drift overwhelms balancing selection, rare alleles can be lost (Zayed and Packer 2005, Levin et al. 2009, Eimes et al. 2011) and allele frequencies move away from the optimum value (Step 1), decreasing the values of the fitness components they influence (Step 2). For instance, plant self-incompatibility loci undergo balancing selection, triggering a genetic Allee effect. This has been shown in the rare plant *Brassica insularis*,
<figure><img src="image_3.png" /></figure>

Fig. 2. Scenarios underlying a genetic Allee effect. During Step 1, a decrease in population size causes an increase in inbreeding, a change in the drift/selection balance in favor of drift, and/or a higher proportion of immigrants in the population (assuming that both selection and migration are density independent). Various population genetic structure variables are affected: heterozygosity, allelic richness, and the frequencies of detrimental and beneficial alleles. During Step 2, these changes in genetic variation cause a decrease in the value of fitness components through inbreeding depression, drift load, or migration load.

in which smaller populations have fewer alleles at the self-incompatibility locus and, thus, lower rates of fruit set (Glémin et al. 2008).

### Migration load

Migration load is defined as the decrease in the mean fitness of a population due to the immigration of maladapted alleles and outbreeding depression (e.g. Bolnick and Nosil 2007). With effective dispersal, immigrants from a source population can bring new alleles into a sink population. Assuming a constant number of dispersers, the proportion of new alleles in the gene pool is higher in small sink populations than in large ones (Fig. 2), so the change in genetic structure depends on population size, fulfilling the conditions for the first step of the genetic Allee effect (Fig. 1). A migration load can occur as a result of local maladaptation and/or outbreeding depression (Ronce and Kirkpatrick 2001, Lenormand 2002) due to underdominance or deleterious epistatic interactions (Edmands 2002). The overrepresentation of new alleles in small populations results in a higher migration load in small populations, satisfying the conditions for the second step of the genetic Allee effect (Fig. 2). We were able to identify a single published example of a genetic Allee effect due to migration load (Table 1C). Populations of *Eucalyptus aggregata* can hybridize with closely related species of eucalypts; the hybridization rate increases with

decreasing population size, and germination rates and seedling survival are lower in populations with a higher proportion of hybrids (Field et al. 2008).

### Combined mechanisms

As already reported for ecological component Allee effects (Berec et al. 2007), several genetic Allee effects may occur simultaneously in the same population. A good example is provided by the concomitant influences inbreeding depression, drift load, and loss of self-incompatibility alleles on small populations of the buttercup *Ranunculus reptans* (Willi et al. 2005). Genetic Allee effects can also act in conjunction with ecological Allee effects, as shown in experimental populations of the self-incompatible plants *Raphanus sativus* and *Lolium multiflorum* (Elam et al. 2007, Firestone and Jasieniuk 2013).

### HOW CAN A GENETIC ALLEE EFFECT BE DETECTED?

The detection of a genetic Allee effect requires investigations of each of the two successive steps: (1) a decrease in population size causing a change in the within-population genetic structure and (2) a decrease in a fitness component caused by this change. For demonstration of the causal relationship underlying Step 2, two confounding effects must be avoided: environmental effects
and ecological Allee effects. It is possible to con-
trol for environmental effects using experimental
designs involving constant environments (e.g.,
Charlesworth 2006, Oakley and Winn 2012) or
extracting fitness responses to environmental
variability before analyzing the effect of popula-
tion genetic structure (e.g., Saccheri et al. 1998,
Hanski and Saccheri 2006). However, controlling
for environmental variability may nevertheless
fail to separate genetic and ecological Allee
effects, and some previously described ecological
Allee effects may actually turn out to be genetic
Allee effects. For instance, mating failures in
small populations may result from a shortage of
genetically compatible mates rather than a
decrease in the frequency of encounters with
conspecifics (Amos et al. 2001, Moller and
Legendre 2001, Kokko and Rankin 2006). Only
by experimental manipulations of population
size and genetic variation with factorial designs
can the genetic and ecological underpinnings of
Allee effects be correctly disentangled. For exam-
ple, genetic variation can be manipulated by
modifying the number of founding families,
while keeping census size constant (e.g., Elam
et al. 2007, Hufbauer et al. 2013, Szücs et al. 2014).

Three main mechanisms underlie the second
step of genetic Allee effects: inbreeding depres-
sion, drift load, and migration load. Inbreeding
depression occurs when individuals with higher
inbreeding coefficients have lower fitness. It can
be assessed by studying individuals from pop-
ulations with known inbreeding coefficients or
offspring from controlled inbred and outbred
crosses (e.g., Elam et al. 2007, Hufbauer et al.
2013). Drift load is detected when between-
population crosses result in heterosis and if
this effect is stronger in populations with lower
heterozygosity or allelic richness (Escobar et al.
2008, Oakley and Winn 2012). The respective
strengths of drift and selection can be estimated
by comparing F<sub>st</sub> and Q<sub>st</sub> between pairs of popu-
lations, under certain conditions: (1) comparison
of several groups of populations, with similar
population sizes within groups and different
population sizes between groups (F<sub>st</sub> and Q<sub>st</sub>
are calculated between all pairs of populations
within groups); (2) the component of fitness
measured is a quantitative trait, with a continu-
ous distribution, influenced by multiple loci with
small effects. If estimates of F<sub>st</sub> and Q<sub>st</sub> are similar

for a quantitative fitness component, then drift is
probably stronger than selection for this compo-
nent. A positive relationship between the value
of this fitness component and the difference
between Qst and Fst may indicate a relationship
between genetic variation and fitness (Oakley
and Winn 2012). In the particular case of drift
load involving plant self-incompatibility and bal-
ancing selection, the role of self-incompatibility
in the relationship between genetic structure and
fitness can be investigated by determining the
proportion of incompatible crosses for which
plants received pollen but produced no seeds
(Fischer et al. 2003, Willi et al. 2005, Glémin et al.
2008). Finally, the demonstration that immigrants
and/or hybrids have lower fitness components
than residents is indicative of the presence of a
migration load.

AT THE CORE OF THE EXTINCTION VORTEX

A central concept in the biology of small popula-
tions is the extinction vortex, defined as a positive
feedback between environmental, demographic,
and genetic factors, reinforcing one another in a
downward spiral until the last individual has dis-
appeared (the F-vortex; Gilpin and Soulé 1986).
Meta-analyses of small populations suggest that
extinction vortices do occur in declining popula-
tions (Fagan and Holmes 2006), and a body of
models and data support the significant roles of
genetic factors (Frankham and Ralls 1998, Tanaka
2000, Spielman et al. 2004, Frankham 2005).
Nonetheless, despite its importance for the man-
agement of threatened species, the extinction vor-
tex suffers from the rarity of clear demonstrations.

Formalizing the causal relations between
population size, genetic structure, and fitness,
via the genetic Allee effect, should strengthen
the approach of extinction vortices (Fig. 3). The
rationale is that thinking in terms of Allee effects
makes it possible to capitalize on the known dual-
ity of component and demographic Allee effects,
thereby bridging the gap between individual fit-
ness and population growth (Fauvergue 2013).
Here, we have defined the genetic Allee effect
as a component Allee effect, that is an effect of
population size on fitness, triggered by genetic
factors. Like any other component Allee effect,
the genetic Allee effect can lower mean individ-
ual fitness in small populations, thereby causing
<figure><img src="image_4.png" /></figure>

Fig. 3. Extinction vortices driven by Allee effects. In ecological Allee effects, a decrease in population size directly causes a decrease in a component of individual fitness, which may in turn yield a decrease in population growth rate. A genetic Allee effect is a two-step process. A decrease in population size induces a change in the genetic variation of the population via inbreeding, drift/selection balance or migration (Step 1). This change yields inbreeding depression, drift load, or migration load, causing a decrease in the value of fitness components (Step 2). Both ecological and genetic component Allee effects can produce a demographic Allee effect, which, if strong, can generate an extinction vortex.

a decrease in the population growth rate, that is a demographic Allee effect (Lennartsson 2002; Fig. 3; Angulo et al. 2007). In the case of strong demographic Allee effect with negative growth rate, or if combined with detrimental environmental factors, the genetic Allee effect can drive small populations to extinction and is thus at the core of an extinction vortex.

A canonical example is provided by research on the Glanville fritillary butterfly, *Melitaea cinxia*. In this species, small subpopulations are less heterozygous than large subpopulations, and less heterozygous females produce larvae with lower survival rates (Saccheri et al. 1998). The Glanville fritillary is therefore subject to a genetic Allee effect. Controlled crosses in the laboratory and in the field have confirmed the existence of inbreeding depression in this species (Nieminen et al. 2001). The combination of inbreeding depression with poor resource availability causes the extinction of small subpopulations of *M. cinxia*, as shown by the higher probability of extinction for smaller and less heterozygous populations living in less favorable environmental conditions (Saccheri et al. 1998). A genetic Allee effect also impacts demography in the plant *Clarkia pulchella*.

Using an experimental approach, Newman and Pilson (1997) showed that populations founded by genetically related seedlings were subject to inbreeding depression and/or drift load and therefore had a lower growth rate than populations founded by unrelated seedlings. In several other species, genetically eroded populations have been shown to have a lower growth rate (Fauvergue and Hopper 2009, Markert et al. 2010, Wennersten et al. 2012, Turcotte et al. 2013, Vercken et al. 2013). Fitness components were not measured in these studies, but the dynamics observed may reflect genetic Allee effects.

## GENETIC VERSUS ECOLOGICAL ALLEE EFFECTS

### *Temporal issues*

Genetic and ecological Allee effects are both component Allee effects capable of generating demographic Allee effects, but they differ in several ways, including the time scale. Genetic Allee effects are characterized by a time lag between the change in population size and its consequences for mean individual fitness, whereas an ecological Allee effect may occur as soon as population size decreases (e.g., mating success decreases together
with population density). The two steps of a genetic Allee effect may not occur over similar time scales: a decrease in population size may modify the genetic structure of the population over tens or thousands of generations, but fitness consequences may appear in the generation following the change in genetic structure (Amos and Balmford 2001, but see Hufbauer et al. 2013). Conversely, if population size increases, ecological Allee effects should die down almost immediately, whereas genetic Allee effects will not. For instance, after several generations of genetic drift, a population cannot recover its initial level of genetic variation unless new alleles arise by mutation or migration. This implies that a population can be rescued from a demographic threat, but, if the underlying genetic variation is not also restored, it may still suffer from a genetic Allee effect and be prone to extinction (what has been referred to as an extinction debt; Vercken et al. 2013).

## Two steps for the genetic Allee effect

The two steps of genetic Allee effects are idiosyncratic: each step can occur independently of the other, but the occurrence of the two steps in succession is necessary to create (and demonstrate) a genetic Allee effect. The first step, a change in genetic structure caused by a decrease in population size, can occur with no decrease in mean fitness (Step 1, but not Step 2). For instance, inbred individuals do not necessarily suffer from inbreeding depression. Indeed, although counterintuitive at first glance, inbreeding can even result in the purging of deleterious alleles, leading to an increase in mean fitness (Glémin 2003, Facon et al. 2011). Similarly, drift can randomly eliminate detrimental alleles and fix beneficial ones, and immigration in small populations can lower the negative impact of drift and inbreeding by bringing new adapted alleles and increasing heterozygosity. The second step, in which a change in the genetic structure of a population triggers a decrease in a component of fitness, is not necessarily induced by the first step, because genetic structure can change independently of population size, under the influences of mutation, migration, or the mating system.

## Evolutionary consequences

The effects on individual fitness of ecological Allee effects could act as a selective force, driving

the evolution of adaptations mitigating sensitivity to population size (Courchamp et al. 2008). For instance, long-range volatile sex pheromones may prevent mating failures (Fauvergue et al. 2007). The same reasoning applies to genetic Allee effects too. Inbreeding depression may have shaped the evolution of dispersal, inbreeding avoidance and self-incompatibility systems (Penn and Potts 1999, Perrin and Mazalov 2000).

## CONCLUSION: WHY IS ALL THIS IMPORTANT?

The focus on genetic Allee effects does not simply add a new semantic layer to processes that have been thoroughly discussed in the last few decades. On the contrary, it is essential, for several reasons.

First, although a genetic Allee effect has been mentioned in a few articles (starting with Fischer et al. 2000), no formal definition has ever been provided. We show here that a genetic Allee effect is not merely an Allee effect underpinned by genetic mechanisms. The formal definition we provide here includes two successive steps: a decrease in population size modifying the within-population genetic structure, in turn causing a decrease in the mean value of a component of individual fitness. The requirement of these two steps differentiates genetic Allee effects from ecological Allee effects. Genetic Allee effects are unique in terms of their potentially long time scale (particularly for the change in genetic structure in response to a decrease in population size) and their entropy (by contrast to ecological Allee effects, a population cannot recover instantly from a genetic Allee effect). According to the formal definition proposed here, some reported findings refer to a genetic Allee effect (including from our own research; Fauvergue and Hopper 2009, Vercken et al. 2013), but without providing explicit evidence for such an effect. Alternatively, some specific types of Allee effect, such as the so-called S-Allee effect (Wagenius et al. 2007, Hoebee et al. 2012, Busch et al. 2014), may be seen as genetic Allee effects. The provision of a clear definition of the genetic Allee effect should prompt a better examination of the genetic mechanisms affecting fitness and demography in small populations. We identified only 15 demonstrations of this particular type of component Allee effect in the literature, but we predict
that widespread evidence of its existence will be
obtained if the methods we propose are used to
identify genetic Allee effects and to distinguish
them from ecological Allee effects occurring in
the same populations.

Second, a definition of the genetic Allee effect
should promote a more comprehensive approach
to the biology of small populations. The integra-
tion of demography and genetics has long been
recognized as an important endeavor, but tangi-
ble progress is still required, particularly for the
management of declining and/or bottlenecked
populations (e.g., Robert et al. 2007, Fauvergue
et al. 2012). Indeed, early developments in popu-
lation viability analyses clearly assumed a feed-
back between demographic and genetic processes
at the core of extinction vortices (Gilpin and
Soulé 1986, Caughley 1994), and, more recently,
the concept of evolutionary rescue has emerged
as a race between demographic and evolutionary
processes (Bell and Gonzalez 2009, Gonzalez et al.
2013). However, these concepts have generally
been restricted to the analysis of demographic
and environmental stochasticity, resulting in the
neglect of the Allee effect as an important mecha-
nism for small populations. Conversely, the Allee
effect is attracting the attention of an increasing
number of scientists with a general interest in the
biology of small populations and extinction vor-
tices. However, despite widespread evidence for
the significant role of genetic processes, it is only
very recently that theoretical works have started
to combine Allee effects with genetic processes
(Kanarek and Webb 2010, Roques et al. 2012,
Shaw and Kokko 2014, Wittmann et al. 2014a, b,
Kanarek et al. 2015). The genetic Allee effect, as
defined here, is at the very intersection of demog-
raphy and genetics and should serve as a unify-
ing paradigm. In the long term, the genetic Allee
effect should contribute to improving the dialog
between genetics and demography.

ACKNOWLEDGMENTS

This work was supported by the *Agence Nationale de la Recherche* (Sextinction project ANR-2010-BLAN-1717 and RARE project ANR-2009-PEXT-01001) and the *Fondation de Recherche sur la Biodiversité* (VORTEX project APP-IN-2009-052). We thank four anonymous reviewers for valuable insights on previous versions of this manuscript and Nathan Fauvergue for fresh thoughts.

LITERATURE CITED

Amos, W., and A. Balmford. 2001. When does conser-
vation genetics matter? Heredity 87:257–265.

Amos, W., J. W. Wilmer, and H. Kokko. 2001. Do female grey seals select genetically diverse mates? Animal Behaviour 62:157–164.

Angulo, E., G. W. Roemer, L. Berec, J. Gascoigne, and F. Courchamp. 2007. Double allee effects and extinction in the island fox. Conservation Biology 21:1082–1091.

Bell, G., and A. Gonzalez. 2009. Evolutionary rescue can prevent extinction following environmental change. Ecology Letters 12:942–948.

Berec, L., E. Angulo, and F. Courchamp. 2007. Multiple Allee effects and population management. Trends in Ecology and Evolution 22:185–191.

Bolnick, D. I., and P. Nosil. 2007. Natural selection in populations subject to a migration load. Evolution 61:2229–2243.

Bonsall, M. B., C. A. Dooley, A. Kasparson, T. Brereton, D. B. Roy, and J. A. Thomas. 2014. Allee effects and the spatial dynamics of a locally endangered butterfly, the high brown fritillary (Argynnis adippe). Ecological Applications 24:108–120.

Brys, R., H. Jacquemyn, L. De Bruyn, and M. Hermy. 2007. Pollination success and reproductive output in experimental populations of the self-incompatible *Primula vulgaris*. International Journal of Plant Sciences 168:571–578.

Busch, J. W., T. Witthuhn, and M. Joseph. 2014. Fewer S-alleles are maintained in plant populations with sporophytic as opposed to gametophytic self-incompatibility. Plant Species Biology 29:34–46.

Caughley, G. 1994. Directions in conservation biology. Journal of Animal Ecology 63:215–244.

Charlesworth, D. 2006. Balancing selection and its effects on sequences in nearby genome regions. Plos Genetics 2:379–384.

Charlesworth, D., and J. H. Willis. 2009. The genetics of inbreeding depression. Nature Reviews Genetics 10:783–796.

Charlesworth, D., M. T. Morgan, and B. Charlesworth. 1993. Mutation accumulation in finite outbreeding and inbreeding populations. Genetical Research 61:39–56.

Courchamp, F., T. Clutton-Brock, and B. Grenfell. 1999. Inverse density dependence and the Allee effect. Trends in Ecology and Evolution 14:405–410.

Courchamp, F., L. Berec, and J. Gascoigne. 2008. Allee effects in ecology and conservation. Oxford University Press, Oxford, UK.

Davis, H. G., C. M. Taylor, J. G. Lambrinos, and D. R. Strong. 2004. Pollen limitation causes an Allee effect in a wind-pollinated invasive grass (*Spartina*
*alterniflora*). Proceedings of the National Academy of Sciences USA 101:13804–13807.

Edmands, S. 2002. Does parental divergence predict reproductive compatibility? *Trends in Ecology & Evolution* 17:520–527.

Eimes, J. A., J. L. Bollmer, L. A. Whittingham, J. A. Johnson, C. Van Oosterhout, and P. O. Dunn. 2011. Rapid loss of MHC class II variation in a bottle-necked population is explained by drift and loss of copy number variation. *Journal of Evolutionary Biology* 24:1847–1856.

Elam, D. R., C. E. Ridley, K. Goodell, and N. C. Ellstrandt. 2007. Population size and relatedness affect fitness of a self-incompatible invasive plant. *Proceedings of the National Academy of Sciences USA* 104:549–552.

Escobar, J. S., A. Nicot, and P. David. 2008. The different sources of variation in inbreeding depression, heterosis and outbreeding depression in a meta-population of *Physa acuta*. *Genetics* 180:1593–1608.

Facon, B., R. A. Hufbauer, A. Tayeh, A. Loiseau, E. Lombaert, R. Vitalis, T. Guillemaud, J. G. Lundgren, and A. Estoup. 2011. Inbreeding depression is purged in the invasive insect *Harmonia axyridis*. *Current Biology* 21:424–427.

Fagan, W. F., and E. E. Holmes. 2006. Quantifying the extinction vortex. *Ecology Letters* 9:51–60.

Fauvergue, X. 2013. A review of mate-finding Allee effects in insects: from individual behavior to population management. *Entomologia Experimentalis et Applicata* 146:79–92.

Fauvergue, X., and K. R. Hopper. 2009. French wasps in the New World: experimental biological control introductions reveal a demographic Allee effect. *Population Ecology* 51:385–397.

Fauvergue, X., J. C. Malausa, L. Giuge, and F. Courchamp. 2007. Invading parasitoids suffer no Allee effect: a manipulative field experiment. *Ecology* 88:2392–2403.

Fauvergue, X., E. Vercken, T. Malausa, and R. A. Hufbauer. 2012. The biology of small, introduced populations, with special reference to biological control. *Evolutionary Applications* 5:424–443.

Field, D. L., D. J. Ayre, R. J. Whelan, and A. G. Young. 2008. Relative frequency of sympatric species influences rates of interspecific hybridization, seed production and seedling performance in the uncommon *Eucalyptus aggregata*. *Journal of Ecology* 96:1198–1210.

Firestone, J. L., and M. Jasieniuk. 2013. Small population size limits reproduction in an invasive grass through both demography and genetics. *Oecologia* 172:109–117.

Fischer, M., M. van Kleunen, and B. Schmid. 2000. Genetic Allee effects on performance, plasticity and developmental stability in a clonal plant. *Ecology Letters* 3:530–539.

Fischer, M., M. Hock, and M. Paschke. 2003. Low genetic variation reduces cross-compatibility and offspring fitness in populations of a narrow endemic plant with a self-incompatibility system. *Conservation Genetics* 4:325–336.

Frankham, R. 1995. Inbreeding and extinction: a threshold effect. *Conservation Biology* 9:792–799.

Frankham, R. 1996. Relationship of genetic variation to population size in wildlife. *Conservation Biology* 10:1500–1508.

Frankham, R. 1998. Inbreeding and extinction: island populations. *Conservation Biology* 12:665–675.

Frankham, R. 2005. Genetics and extinction. *Biological Conservation* 126:131–140.

Frankham, R., and K. Ralls. 1998. Inbreeding leads to extinction. *Nature* 392:441–442.

Gabriel, W., and R. Bürger. 1994. Extinction risk by mutational meltdown: synergistic effects between population regulation and genetic drift. *Pages 69–84 in* V. Loeschcke, J. Tomiuk, and S. K. Jain, editors. *Conservation genetics. Birkhäuser, Basel, Switzerland*.

Gabriel, W., R. Bürger, and M. Lynch. 1991. Population extinction by mutational load and demographic stochasticity. *Pages 49–59 in* A. Seitz, and V. Loeschcke, editors. *Species conservation: a population-biological approach. Birkhäuser, Basel, Switzerland*.

Gilpin, M. E., and M. E. Soulé. 1986. Minimum viable populations: processes of extinction. *Pages 19–34 in* M. E. Soulé, editor. *Conservation biology: the science of scarcity and diversity. Sinauer Associates, Sunderland, Massachusetts, USA*.

Glémin, S. 2003. How are deleterious mutations purged? *Drift versus nonrandom mating. Evolution* 57:2678–2687.

Glémin, S., C. Petit, S. Maurice, and A. Mignot. 2008. Consequences of low mate availability in the rare self-incompatible species *Brassica insularis*. *Conservation Biology* 22:216–221.

Gomulkiewicz, R., and R. D. Holt. 1995. When does evolution by natural selection prevent extinction? *Evolution* 49:201–207.

Gonzalez, A., O. Ronce, R. Ferriere, and M. E. Hochberg. 2013. Evolutionary rescue: an emerging focus at the intersection between ecology and evolution. *Philosophical Transactions of the Royal Society B* 368:20120404. http://dx.doi.org/10.1098/rstb.2012.0404

Grueber, C. E., G. P. Wallis, and I. G. Jamieson. 2013. Genetic drift outweighs natural selection at toll-like receptor (TLR) immunity loci in a re-introduced population of a threatened species. *Molecular Ecology* 22:4470–4482.

Hanski, I., and I. Saccheri. 2006. Molecular-level variation affects population growth in a butterfly meta-population. *PLoS Biology* 4:e129.
Hoebee, S. E., S. Angelone, D. Csencsics, K. Maattanen, and R. Holderegger. 2012. Diversity of S-alleles and mate availability in 3 populations of self-incompatible wild pear (Pyrus pyraster). Journal of Heredity 103:260–267.

Hufbauer, R. A., A. Rutschmann, B. Serrate, H. V. de Conchard, and B. Facon. 2013. Role of propagule pressure in colonization success: disentangling the relative importance of demographic, genetic and habitat effects. Journal of Evolutionary Biology 26:1691–1699.

Johansson, M., C. R. Primmer, and J. Merilae. 2007. Does habitat fragmentation reduce fitness and adaptability? A case study of the common frog (Rana temporaria). Molecular Ecology 16:2693–2700.

Johnson, D. M., A. M. Liebhold, P. C. Tobin, and O. N. Bjornstad. 2006. Allee effects and pulsed invasion by the gypsy moth. Nature 444:361–363.

Kanarek, A. R., and C. T. Webb. 2010. Allee effects, adaptive evolution, and invasion success. Evolutionary Applications 3:122–135.

Kanarek, A. R., C. T. Webb, M. Barfield, and R. D. Holt. 2015. Overcoming Allee effects through evolutionary, genetic, and demographic rescue. Journal of Biological Dynamics 9:15–33.

Kimura, M. 1983. Rare variant alleles in the light of the neutral theory. Molecular Biology and Evolution 1:84–93.

Kokko, H., and A. Lopez-Sepulcre. 2007. The ecogenetic link between demography and evolution: Can we bridge the gap between theory and data? Ecology Letters 10:773–782.

Kokko, H., and D. J. Rankin. 2006. Lonely hearts or sex in the city? Density-dependent effects in mating systems. Philosophical Transactions of the Royal Society B 361:319–334.

Kuparinen, A., D. M. Keith, and J. A. Hutchings. 2014. Allee effect and the uncertainty of population recovery. Conservation Biology 28:790–798.

Lande, R. 1994. Risk of population extinction from fixation of new deleterious mutations. Evolution 48:1460–1469.

Lanfear, R., H. Kokko, and A. Eyre-Walker. 2014. Population size and the rate of evolution. Trends in Ecology & Evolution 29:33–41.

Leimu, R., P. Mutikainen, J. Koricheva, and M. Fischer. 2006. How general are positive relationships between plant population size, fitness and genetic variation? Journal of Ecology 94:942–952.

Lennartsson, T. 2002. Extinction thresholds and disrupted plant-pollinator interactions in fragmented plant populations. Ecology 83:3060–3072.

Lenormand, T. 2002. Gene flow and the limits to natural selection. Trends in Ecology & Evolution 17:183–189.

Levin, D. A., C. D. Kelley, and S. Sarkar. 2009. Enhancement of Allee effects in plants due to self-incompatibility alleles. Journal of Ecology 97:518–527.

Li, L., K. Lu, Z. Chen, T. Mu, Z. Hu, and X. Li. 2008. Dominance, overdominance and epistasis condition the heterosis in two heterotic rice hybrids. Genetics 180:1725–1742.

Lynch, M., J. Conery, and R. Bürger. 1995a. Mutation accumulation and the extinction of small populations. American Naturalist 146:489–518.

Lynch, M., J. Conery, and R. Bürger. 1995b. Mutational meltdowns in sexual populations. Evolution 49: 1067–1080.

Malécot, G. 1969. Consanguinité panmictique et consanguinité systématique. Annales de Génétique et Sélection Animale 1:237–242.

Markert, J. A., D. M. Champlin, R. Gutjahr-Gobell, J. S. Grear, A. Kuhn, T. J. McGreevy, A. Roth, M. J. Bagley, and D. E. Nacci. 2010. Population genetic diversity and fitness in multiple environments. BMC Evolutionary Biology 10:205. http://dx.doi.org/10.1186/1471-2148-10-205

Mattila, A. L. K., A. Duplouy, M. Kirjokangas, R. Lehtonen, P. Rastas, and I. Hanski. 2012. High genetic load in an old isolated butterfly population. Proceedings of the National Academy of Sciences USA 109:E2496–E2505.

Metcalf, C. J. E., and S. Pavard. 2007. Why evolutionary biologists should be demographers. Trends in Ecology & Evolution 22:205–212.

Moller, A. P., and S. Legendre. 2001. Allee effect, sexual selection and demographic stochasticity. Oikos 92:27–34.

Newman, D., and D. Pilson. 1997. Increased probability of extinction due to decreased genetic effective population size: experimental populations of Clarkia pulchella. Evolution 51:354–362.

Nielsen, R., and M. Slatkin. 2013. An introduction to population genetics: theory and applications. Sinauer, Sunderland, Massachusetts, USA.

Nieminen, M., M. C. Singer, W. Fortelius, K. Schops, and I. Hanski. 2001. Experimental confirmation that inbreeding depression increases extinction risk in butterfly populations. American Naturalist 157:237–244.

Oakley, C. G., and A. A. Winn. 2012. Effects of population size and isolation on heterosis, mean fitness, and inbreeding depression in a perennial plant. New Phytologist 196:261–270.

Oostermeijer, J. G. B., S. H. Luijten, and J. C. M. den Nijs. 2003. Integrating demographic and genetic approaches in plant conservation. Biological Conservation 113:389–398.

Pelletier, F., D. Garant, and A. P. Hendry. 2009. Eco-evolutionary dynamics. Philosophical Transactions of the Royal Society B 364:1483–1489.
Penn, D. J., and W. K. Potts. 1999. The evolution of mating preferences and major histocompatibility complex genes. American Naturalist 153:145-164.

Perrin, N., and V. Mazalov. 2000. Local competition, inbreeding, and the evolution of sex-biased dispersal. American Naturalist 155:116-127.

Reed, D. H., and R. Frankham. 2003. Correlation between fitness and genetic diversity. Conservation Biology 17:230-237.

Robert, A., D. Couvet, and F. Sarrazin. 2007. Integration of demography and genetics in population restorations. Ecoscience 14:463-471.

Ronce, O., and M. Kirkpatrick. 2001. Migrational meltdown in heterogeneous habitats. Evolution 55:1520-1531.

Roques, L., J. Garnier, F. Hamel, and E. K. Klein. 2012. Allee effect promotes diversity in traveling waves of colonization. Proceedings of the National Academy of Sciences USA 109:8828-8833.

Saccheri, I., M. Kuussaari, M. Kankare, P. Vikman, W. Fortelius, and I. Hanski. 1998. Inbreeding and extinction in a butterfly metapopulation. Nature 392:491-494.

Shaw, A. K., and H. Kokko. 2014. Mate finding, Allee effects and selection for sex-biased dispersal. Journal of Animal Ecology 83:1256-1267.

Sjödin, P., I. Kaj, S. Krone, M. Lascoux, and M. Nordborg. 2005. On the meaning and existence of an effective population size. Genetics 169:1061-1070.

Soulé, M. E. 1980. Thresholds for survival: maintaining fitness and evolutionary potential. Pages 151-169 in M. E. Soulé and B. A. Wilcox, editors. Conservation biology: an evolutionary-ecological perspective. Sinauer, Sunderland, Massachusetts, USA.

Spielman, D., B. W. Brook, and R. Frankham. 2004. Most species are not driven to extinction before genetic factors impact them. Proceedings of the National Academy of Sciences USA 101:15261-15264.

Stephens, P. A., W. J. Sutherland, and R. P. Freckleton. 1999. What is the Allee effect? Oikos 87:185-190.

Szücs, M., B. A. Melbourne, T. Tuff, and R. A. Hufbauer. 2014. The roles of demography and genetics in the early stages of colonization. Proceedings of the Royal Society B 281:20141073. http://dx.doi.org/10.1098/rspb.2014.1073

Tanaka, Y. 2000. Extinction of populations by inbreeding depression under stochastic environments. Population Ecology 42:55-62.

Taylor, C. M., and A. Hastings. 2005. Allee effects in biological invasions. Ecology Letters 8:895-908.

Turcotte, M. M., D. N. Reznick, and J. D. Hare. 2013. Experimental test of an eco-evolutionary dynamic

feedback loop between evolution and population density in the green peach aphid. American Naturalist 181:S46-S57.

Vercken, E., F. Vincent, L. Mailleret, N. Ris, E. Tabone, and X. Fauvergue. 2013. Time-lag in extinction dynamics in experimental populations: Evidence for a genetic Allee effect? Journal of Animal Ecology 82:621-631.

Wagenius, S., E. Lonsdorf, and C. Neuhauser. 2007. Patch aging and the S-allee effect: breeding system effects on the demographic response of plants to habitat fragmentation. American Naturalist 169:383-397.

Waples, R. S., and O. Gaggiotti. 2006. What is a population? An empirical evaluation of some genetic methods for identifying the number of gene pools and their degree of connectivity. Molecular Ecology 15:1419-1439.

Wennersten, L., J. Johansson, E. Karpestam, and A. Forsman. 2012. Higher establishment success in more diverse groups of pygmy grasshoppers under seminatural conditions. Ecology 93:2519-2525.

Whitlock, M. C. 2000. Fixation of new alleles and the extinction of small populations: drift load, beneficial alleles, and sexual selection. Evolution 54:1855-1861.

Willi, Y., J. Van Buskirk, and M. Fischer. 2005. A threefold genetic allee effect: population size affects cross-compatibility, inbreeding depression and drift load in the self-incompatible *Ranunculus rep-tans*. Genetics 169:2255-2265.

Wittmann, M. J., W. Gabriel, and D. Metzler. 2014a. Genetic diversity in introduced populations with an Allee effect. Genetics 198:299-310.

Wittmann, M. J., W. Gabriel, and D. Metzler. 2014b. Population genetic consequences of the Allee effect and the role of offspring-number variation. Genetics 198:311-320.

Wright, S. 1969. Evolution and the genetics of populations. Volume 2: the theory of gene frequencies. University of Chicago Press, Chicago, Illinois, USA.

Young, A. G., and M. Pickup. 2010. Low S-allele numbers limit mate availability, reduce seed set and skew fitness in small populations of a self-incompatible plant. Journal of Applied Ecology 47:541-548.

Zayed, A., and L. Packer. 2005. Complementary sex determination substantially increases extinction proneness of haplodiploid populations. Proceedings of the National Academy of Sciences USA 102:10742-10746.