The hidden side of the Allee eﬀect: correlated demographic traits and extinction risk in experimental populations

Because Allee eﬀects (i.e., the presence of positive density-dependence at low population size or density) have major impacts on the dynamics of small populations, they are routinely included in demographic models for the evaluation of extinction risks. How-ever, the structure of most common models implies that other demographic parameters (like the maximum growth rate) are modiﬁed by the inclusion of an Allee eﬀect, which also aﬀects in return the extinction risk of the population. Whether such correlations between demographic traits occur in natural populations or merely reﬂect a practical con-straint related to model formalism is of primary importance to understand better the dynamics of small populations. We investigated this question using 30 populations of Trichogramma wasps raised under similar conditions, of which 8 were subject to an Allee eﬀect. We showed that these 8 populations were also characterized by lower maximum growth rate and lower population size, and that their extinction probability was higher than for non-Allee populations. These results provide the ﬁrst empirical demonstration of a correlation between the presence of positive density-dependence and impaired demographic performance, which increases the extinction risk of a population, especially during the establishment phase.


Introduction
Allee effects (or "demographic" Allee effects, Stephens et al. 1999) characterize small populations in which per capita performance increases with population density, a phenomenon also known as positive density-dependence (Dennis, 1989;Courchamp et al., 2008). Consequently, populations subject to Allee effects have lower per capita growth rates when population density decreases. In some cases, referred to as a "strong" Allee effects, there is a critical density threshold below which the population declines and ultimately goes extinct. In contrast, in presence of "weak" Allee effects, population growth is reduced at low density but always remains positive (Courchamp et al., 2008). As the presence of an Allee effect is often considered a major risk factor for population extinction (Lande, 1988;Boukal & Berec, 2002), Allee effects are commonly included in population dynamics models for the management of small populations, either for conservation or management purposes (Tobin et al., 2011;Molnár et al., 2014;Hutchings, 2015;Wittmann et al., 2018;Bajeux et al., 2019). One of the most common heuristic population dynamics model accounting for an Allee effect is a modified version of the logistic model (Lewis & Kareiva 1993;Amarasekare 1998;Boukal & Berec 2002, with analogous models existing in discrete time, Liebhold & Bascompte 2003;Tobin et al. 2007) that reads: where A quantifies the severity of the Allee effect (weak Allee effect for ∈ (− ,0] ; strong Allee effect for ∈ (0, )), r > 0 is the maximum per capita population growth rate in the corresponding logistic model, and K the carrying capacity.
However, this modified structure impacts in return several key population properties, with potential significant consequences beyond the mere presence of positive density-dependence. For instance, when theoretical parameters r and K are held constant, the maximum value reached by the per capita growth rate 1 over is always lower in populations subject to an Allee effect as described by the model above than in populations following a logistic model (Courchamp et al., 2008;Drake & Kramer, 2011;Figure 1). A direct consequence of this lower growth rate is that the establishment or growth phase of an initially small population, in which the population is most vulnerable to stochastic influences, lasts longer in presence of an Allee effect (Veit & Lewis, 1996;Dennis, 2002;Drake, 2004;Sinclair & Arnott, 2016) and increases with the magnitude of the Allee effect (Bajeux et al., 2019). Both these parameters (maximum per capita growth rate and duration of the establishment phase) have been shown theoretically and empirically to influence extinction risk (Lande, 1993;Liebhold & Bascompte, 2003;Cassey et al., 2014). In theoretical analyses of population dynamics, populations with an Allee effect are thus characterized not only by the presence of positive density-dependence, but also by lower maximum growth rate, both of which increase the extinction risk of populations.

Figure 1.
Population growth rate (left) and per capita growth rate 1 (right) for a logistic model (in black), a model with a weak Allee effect (A=-5, in blue) or a model with a strong Allee effect (A=20, in purple) when r=0.1 and K=100. After Drake and Kramer (2011).
Because this non-independence between demographic properties arises from the structure of population dynamics models like model (1), its relevance in natural populations might be questioned. Yet a demographic Allee effect reveals specific processes internal to the population, so it is likely to correlate with other components of population dynamics, like growth rate or carrying capacity. Such characteristics might then add up or even interact to increase the probability of population extinction. Alternatively, if the presence of an Allee effect is independent in practice from other demographic properties, it implies that the extinction risk related to positive density-dependence only might have been overestimated in theoretical studies.
A major difficulty in addressing this issue is to find independent populations whose life histories are close enough to allow quantitative comparisons of demographic properties, but that still present variation regarding the Allee effect. Previous studies that aimed at estimating the frequency of Allee effects by analyzing population time series from several species found very little evidence of Allee effects (Myers et al., 1995;Saether et al., 1996;Gregory et al., 2010). However, as these approaches were based on modelfitting, they were constrained by the structure of the models they used. In these studies, either several alternative models were compared (some including negative density-dependence only, others including also positive density-dependence, see for instance (Gregory et al., 2010), or a generic model allowing for some degree of positive density-dependence was fitted and the presence of an Allee effect was derived from the estimates of model parameters (Myers et al., 1995;Perälä & Kuparinen, 2017). Yet, while such theoretical models are extremely useful to examine the theoretical consequences of Allee effects on population eco-evolutionary dynamics, there is no reason why real populations should comply with their underlying hypotheses (e.g., continuous density-dependent response, negative quadratic relationship). This is especially true for models that don't aim at fitting closely to the ecology of a given population (for which enough prior knowledge might exist to allow for an informed decision about modelling), but at comparing general patterns across populations or species.
An alternative method to detect Allee effects from population time series might be to test for the presence of positive density-dependence in the per capita growth rate at low density, without any a priori on the shape of the general density-dependence function (see for instance (Stenglein & Deelen, 2016). For this purpose, we analyzed the relationship between per capita growth rate and population size in 30 populations of parasitoid wasps of the genus Trichogramma using generalized additive models that allow the fitting of smooth relationships using spline functions. We detected positive density-dependence at low population size, i.e. the presence of an Allee effect, in 8 of these populations. Then we compared the demographic performance -maximum growth rate, population size, and extinction probability -of these populations to those without an Allee effect. This privileged experimental system allowed us to demonstrate that general components of population dynamics correlate with the presence of an Allee effect, which increases the extinction probability of the populations.

Biological model
We used parasitoid wasps of the genus Trichogramma as biological models. Our study populations were provided by the Biological Resource Center EP-Coll (doi.org/10.15454/AY4LMT), specialized in egg parasitoids. 30 populations were chosen from a set of 70 populations collected in the field in 2015 to evaluate the diversity of Trichogramma in France (D. Muru, unpublished data). Detailed information on each population is available in Supplementary Material (doi.org/10.5281/zenodo.4650051, Table S1). A "population" was defined as individuals sampled on the same plant in a given locality, so that several populations could have been collected from the same locality. Seven species were included in the 30 selected populations: T. brassicae (2 populations), T. cacoeciae (16), T. cordubensis (3), T. euproctidis (2), T. evanescens (2), T. principium (1), T. semblidis (4). Trichogramma are haplodiploids, i.e. females are diploid and males are haploid. However, some species are arrhenotokous (sexual reproduction is needed to produce diploid females, and unfertilized eggs develop into males, like T. brassicae, T. euproctidis, T. evanescens, T. principium, T. semblidis), while others are thelytokous (T. cacoeciae and T. cordubensis, where females reproduce clonally).

Monitoring of population dynamics
Trichogramma are minute solitary parasitoids of Lepidopteran eggs that can be reared in small volumes. Experimental population units were housed in plastic tubes (diameter 50 mm, length 100 mm). Food for adults was provided ad libitum as drops of honey placed on the tube walls. Hosts for reproduction were provided as 3 mm-diameter patches of Ephestia kuehniella eggs glued on paper strips (10 egg patches at each generation; average number of eggs on a patch and 95% confidence interval: 458 ). Eggs of E. kuehniella were irradiated previously, which prevented host development while allowing parasitoid development. For this experiment, temperature and light conditions were set on a cycle of 16 h daylight (22 °C)/8 h dark (20 °C) with constant 70% humidity. Under these conditions, generation time was 14 days for all populations.
Each population was replicated 8 times, each replicated corresponding to increasing levels of initial density (each replicate being founded with 10-12-14-16-18-20-25 or 30 parasitized eggs). Because of asynchrony in emergence, the different populations were distributed into 6 experimental blocks, spread over 2 consecutive weeks. At emergence, fresh host eggs were introduced in the tubes and exposed 48h to parasitism, then put aside for parasitoid development until the emergence of the next generation. Population size at the next generation was estimated by counting the total number of parasitized eggs before emergence. Parasitized eggs turn black when the parasitoid reaches the nymphal stage and the number of black eggs directly reflects adult population size at emergence. At the beginning of emergence, fresh host patches were introduced in the tubes and similarly exposed during 48 h to obtain the next parasitoid generation. Parasitoid population dynamics were monitored during three generations after introduction between January and March 2016. Time series of population dynamics are provided as Supplementary Material (doi.org/10.5281/zenodo.3963856, Figure S1). All replicates from one T. cordubensis population went extinct at the first generation (see Figure S1), and it was removed from further analyses. The final dataset thus included 29 populations only.

Detection of Allee effects
The presence of an Allee effect is indicated by a positive correlation between population size and per capita growth rate, but usually present for small population sizes only. To account for the potential nonlinearity in density-dependence of the per capita growth rate, we fitted a thin-plate regression spline on the relationship between N t+1 (population size at generation t+1) and N t (population size at generation t) using general additive models (package 'mgcv', Wood 2018) in R (R Core Team, 2017). The function 'gam' fits penalized regression splines, and the optimal degree of smoothness is determined using generalized cross-validation (Wood 2006). To avoid overfitting, we set an upper limit for the smooth term at 7 degrees of freedom (dimension parameter k=8).
The definition of the per capita growth rate ( ) = +1 transcribed into the following model: +1~( ) + (log ( )) N t+1 followed a Poisson distribution with a log link to the linear predictors, so that the smooth function s(N t ) was directly equal to the log of per capita growth rate  t . We acknowledged the presence of an Allee effect for the population if the fitted function was strictly non-monotonic within the 95% confidence bands, with a maximum s(N t ) for an intermediate value of population size.

Estimation of demographic parameters and statistical analyses
Population dynamics were characterized in each replicate by the maximum per capita growth rate, the maximum and mean population size across all 3 generations for non-extinct populations. These three variables followed Normal distributions and were analyzed using linear mixed models. We also computed the probability of extinction of a replicate, which followed a binomial distribution and was analyzed with a generalized linear mixed models. In order to evaluate potential differences in the amount of demographic stochasticity between the different populations, we estimated the variance between replicates by computing the sum of squares between non-extinct replicates of the same population at the first generation. This variable followed a negative binomial distribution and was analyzed with a generalized linear mixed models. Both linear mixed models and generalized linear mixed models were fitted using the R package 'lme4' (Bates et al., 2015) using Maximum Likelihood estimation following (Zuur et al., 2009). For all variables, a full model, including the presence of an Allee effect as a fixed factor and population nested within species as a random effect, was compared with a null model including the random effect only using a likelihood ratio test. All data and scripts for analyses are available with supplementary materials on Zenodo (doi.org/10.5281/zenodo.3963856).
Population dynamics were found to be markedly different between Allee and non-Allee populations (Figure 3). Allee populations were characterized by lower maximum per capita growth rate (p=1.4x10 -4 ), lower mean population size (p=0.013), and higher extinction probability (p=7.6x10 -3 ). Maximum population size was also marginally smaller (p=0.06). There was no difference in the variability among replicates between Allee and non-Allee populations (p=0.98).

Discussion
Our analysis demonstrated the presence of an Allee effect in some, but not all, Trichogramma populations raised in laboratory conditions. This finding further confirms that Allee effects can operate in small volumes and affect population persistence even in favourable, controlled conditions (Gertzen et al., 2011;Kaul et al., 2016). Unlike most classical experimental microcosm systems (e.g., Tribolium, Daphnia, Tetrahymena), our collection of Trichogramma populations allows us to compare a diversity of populations with similar life-histories and raised under the same conditions, in order to single out the impact of the presence or absence of an Allee effect. In this study, we found that Trichogramma populations for which an Allee effect was detected were characterized by qualitatively different dynamics from populations with no Allee effect, with lower per capita growth rate, lower population size and increased extinction probability. The demographic parameters correlated with the presence of an Allee effect have direct consequences on short-term extinction risk. Indeed, low per capita growth rates and low population sizes interact to maintain the population in a unfavorable demographic zone, where it is most vulnerable to extinction through the influence of genetic, demographic, and environmental stochasticity (Dennis et al., 2015).
This high-risk, low-density phase, might be transitory when a newly founded population struggles to establish, but it might also be encountered more regularly, for instance during recovery phases after a perturbation. Modelling results and empirical data from fish stock populations show that the presence of an Allee effect delays population recovery after a major decline, such that a population could stay at a low level for a long time where its vulnerability to external stressors and therefore its extinction risk were substantially increased (Kuparinen et al., 2014).

Figure 2.
Raw data (circle of the same color belong to the same experimental replicate) and GAM fits (dashed lines: predicted values ; dotted lines: 95% confidence intervals around prediction) for the relationship between population size and per capita growth rate. Experimental data were modified as ( +1 +1 ) so that extinction events could be represented.  Figure 3. Comparison of demographic parameters between populations with or without an Allee effect. All graphs represent mean +/-2 SE.
Expanding populations are also exposed to repeated low-density phases at every new colonization beyond the occupied area. It is already widely acknowledged that populations subject to an Allee effect experiment colonization failures when the number of migrants for a new colonization event falls below the critical threshold population persistence, which reduces their spread rate (Lewis & Kareiva, 1993;Kot et al., 1996;Wang & Kot, 2001). However, if population growth is also reduced even at intermediate densities, it will take longer for a newly colonized population to reach a large enough size so it can send enough migrants to overcome the colonization threshold and proceed with the expansion. Indeed, in a simulation study, (Walter et al., 2017) showed that expanding populations with lower Allee thresholds were characterized by shorter colonization times and higher population densities.
In addition to these direct effects of low population size and reduced population growth, this prolonged demographic bottleneck in presence of an Allee effect might increase genetic drift and inbreeding, so that even populations that manage to reach carrying capacity will suffer from higher long-term extinction risk (Fauvergue et al., 2012). This special case of genetic extinction debt was previously reported in laboratory populations of a different species of Trichogramma (Vercken et al., 2013). Furthermore, even parameters related to equilibrium dynamics (like maximum population size) differed between Allee and non-Allee populations. This suggests that the influence of the Allee effect on population dynamics may persist beyond the early establishment phase and affect the demographic trajectory over wide ranges of densities.
In our case, the mechanism responsible for the presence of positive density-dependence in some populations has not been identified. Out of the eight populations subject to an Allee effect, five belong to the species T. cacoeciae that is strictly parthenogenous. Mate limitation is thus unlikely to be the main factor impairing population performance at small density in our system, although it could still play a role in the other three populations. In sexual populations of Trichogramma, sex-ratio is usually female-biased (between 60-80% of individuals being females, (Martel, 2003). This means that, in small populations, the number of males may be too low to mate all females, especially when mating is allowed for a limited time as was the case in our experimental system. Such populations are arrhenotokous, i.e. unmated females may still produce male offspring, so that mate limitation should not theoretically affect the population growth rate and the number of eggs produced at the next generation. However, several studies on arrhenotokous parasitoids have shown that unmated females lay fewer eggs, especially in the beginning of their adult life (Fauvergue et al., 2008;Abe et al., 2010), which may have played a role in the emergence of positive density-dependence in our populations of T. principium and T. semblidis. In addition to this first mechanism, patch exploitation strategy and parasitism pattern in both sexual and asexual populations of Trichogramma might have responded to conspecific density. Direct observations of oviposition behavior suggest that females may adjust their oviposition rate to the competition level they experience: females reared in isolation will wait longer before they start laying eggs and examine host eggs suitability more thoroughly than females reared in group (G. Groussier, pers. obs). Furthermore, previous results on T. minutum showed that female search rate and residence time on a patch are increased in presence of conspecifics (Mills & Lacan, 2004;Robert et al., 2016), which should increase parasitism rate in a singlepatch context. In contrast, Robert et al. (2016) found no response of T. petoi to intraspecific competition, which confirms that major differences in patterns of density-dependence may emerge between Trichogramma species.

Variance among replicates
Apparent positive density-dependence may also emerge in some cases even in the absence of specific ecological mechanisms. Demographic stochasticity in particular may introduce an unstable equilibrium at low population size below which most population trajectories tend to decline towards extinction (Lande, 1998;Saether & Engen, 2004). However, such a phenomenological Allee-like effect does not seem to be the most likely explanation to our results. First, positive density-dependence was detected on population growth rate and not on extinction probability, which does fit the original definition by (Odum & Allee, 1954) and is considered as a generic evidence for the presence of an Allee effect (Johnson et al., 2006;Kramer et al., 2009;Gregory et al., 2010). Second, the dispersion of data at low density does not support the hypothesis of an extreme variability driving artefactually the signal of positive density-dependence ( Figure  2). While it remains undeniable that population dynamics at low density are subject to complex interactions between the Allee effect and stochastic forces, we are confident that our findings reflect true variation in density-dependence among Trichogramma populations.
Independently of the mechanisms responsible for the emergence of an Allee effect, its weak or strong nature is expected to induce major qualitative differences in dynamics in theoretical models related to the existence of an unstable equilibrium in the case of a strong Allee effect (Dennis, 1989;Lewis & Kareiva, 1993;Wang & Kot, 2001). In practice however, the distinction between weak and strong Allee effect in empirical data relies on the quantitative estimation of the per capita growth rate at small density, where stochastic variance is the highest and therefore the estimation is least precise (Kramer et al., 2009). Among our eight Allee populations, two have an estimated growth rate inferior to 1 at low density (i.e., negative values for the log of the per capita growth rate, see Fig. 2), four have a growth rate strictly superior to 1 at low density, while the remaining two have confidence intervals large enough to include 1, i.e. compatible with both weak and strong Allee effect. Furthermore, estimations of Allee effects in nature have demonstrated that the intensity of the Allee effect is context-dependent and can vary geographically (Tobin et al., 2007), while results from stochastic models have demonstrated that weak Allee effects can be amplified by stochasticity (demographic or environmental) so as to induce a lower unstable equilibrium, similar to an Allee threshold (Roth & Schreiber, 2014;Dennis et al., 2015). This converging evidence suggests that the dichotomous distinction of weak and strong Allee effect, though essential in deterministic theoretical approaches, might be less relevant in stochastic modelling or in nature where the "strength" or "intensity" of an Allee effect could be quantified continuously.
These results provide the first empirical demonstration of a correlation between the presence of positive density-dependence and impaired demographic performance during establishment phase. If the presence of an Allee effect is correlated with other demographic parameters, in particular the maximum per capita growth rate, then it remains a major risk factor for population extinction, even in the case of a weak Allee effect, or if the population introduced is superior to the theoretical Allee threshold for a strong Allee effect. The non-independence between these demographic parameters is thus likely to be of critical importance for predictions of dynamics and extinction risk of any population subject to low-density stages, like expanding populations, periodic systems, and populations subject to environmental perturbations or extinction-colonization events.