Genetic Allee Effects for Controlling Invasive Populations

ABSTRACT

Invasive pests threaten food security and devastate ecosystems. A universal problem in their management is that small populations can easily evade detection. This makes identifying new incursions challenging and complicates efforts to eradicate or contain established populations. If newly founded populations exhibited a strong Allee effect, small populations would tend towards extinction and most new incursions would go extinct without the need for detection or intervention. Of course, invasive species rarely exhibit strong Allee effects, but new genetic technologies make it conceivable to impose one. Here we consider how introduction of genetic load can cause a genetic Allee effect that reduces the establishment probability of small founder populations. Using numerical and individual‐based modelling, we examine the fate of populations sampled from a larger invasive source population carrying deleterious recessive alleles. Our analysis reveals that the genetic load unmasked by founding can dramatically reduce the establishment probability of small populations across a wide range of parameter space. A sterile mutation effect is more effective than a lethal mutation effect, but X‐linkage offers minimal benefit over autosomal inheritance. Although extinction of newly founded populations is a common outcome, it may be challenging to achieve in species with very high reproductive outputs. Distributing deleterious recessive alleles across a large number of loci at low frequencies was more effective than distributing them across fewer loci at higher frequencies. Our findings suggest that driving deleterious recessives into a source population may render it less prone to establish in new areas.

1. Introduction

Invasive species are a worldwide issue with ecological and economic costs. A recent UN report shows that 60% of global extinctions from 1970 to 2019 can be attributed to invasive species, and that invasive species impose an annual cost of 423 billion USD on the global economy; a cost that has been quadrupling every decade (Roy et al. 2023).

To mitigate the damage of invasive species, control strategies aim to contain, suppress, or eradicate populations. However, when these populations are small and at low density, they are hard to detect; therefore, it can be difficult to apply these strategies effectively (Rout et al. 2014). The ‘detecting small populations’ problem not only applies when an invasive species makes an incursion into a completely new location (e.g., through a breach in quarantine), but it applies when delimiting an invasion front and also after a control attempt where the rare survivors have the potential to rekindle an invasion.

The reason that these low‐density populations are so important is that they often have a strongly positive growth rate, meaning small populations can rapidly become large populations. But a positive growth rate at low density is not a given. There are many examples in which small populations actually have a weakly positive or even negative growth rate, with the highest per capita growth instead occurring at some intermediate density. This phenomenon, of per capita growth being highest at some intermediate density, is called a demographic Allee effect (Somers et al. 2008; Courchamp et al. 2006). Demographic Allee effects arise through a range of mechanisms that affect fitness components. These include but are not limited to mate limitation, cooperation, and genetic effects (Drake and Kramer 2011; Berec et al. 2007). Strong Allee effects manifest as negative population growth at low density, whereas weak Allee effects manifest as relatively low (but still positive) growth at low density (Drake and Kramer 2011; Berec et al. 2007). Allee effects can result in slower population growth or, in the case of strong Allee effects, a deterministic population decline at low population density (Drake and Kramer 2011; Berec et al. 2007).

If the invasive pests we sought to manage exhibited strong Allee effects, the problem of detecting low‐density populations would be obviated: low‐density populations would simply go extinct on their own, with no additional intervention required. Indeed, naturally occurring Allee effects are known to have important implications for invasion dynamics (Phillips 2025). Invasions exhibiting strong Allee effects are much slower to spread and much easier to contain than equivalent invasions in which per capita growth is highest at the leading edge of the invasion (Tobin et al. 2011). While it may be possible to exploit naturally occurring Allee effects for the management of invasions, many problematic invasions are unlikely to exhibit strong Allee effects (if they did, they probably would not be as problematic). Thus, the question arises: how might we impose an Allee effect on a population?

One possibility is that we impose a genetic Allee effect (Luque et al. 2016). A genetic Allee effect arises when a population experiences a change in genetic structure caused by a decrease in population size and that change in genetic structure affects a component of the fitness (Luque et al. 2016). In this instance, if an invasive population carried an excess of deleterious alleles, a genetic Allee effect could arise. If these deleterious alleles are recessive and at a low frequency across many loci, then when the population is large, there will be negligible genetic load because homozygotes are rare. In small populations, however, genetic bottlenecks and drift can cause deleterious recessive alleles to reach high frequency (Mitton 2013; Slatkin 2004). This, in turn, causes homozygotes to become common, and fitness effects (i.e., genetic load) to manifest (Lynch, Conery, and Burger 1995); an effect compounded by inevitable inbreeding in small populations (Wittmann et al. 2018). This change in genetic structure creates an inbreeding depression and is a primary player in the “extinction vortex” of conservation biology (wherein small populations tend to go extinct as a consequence of demographic and evolutionary stochasticity; (Blomqvist et al. 2010; Frankham 2005; Crnokrak and Roff 1999)). In terms of genetic load, if a population has deleterious recessive alleles at a low frequency across many loci, then we say it has high masked load and low realised load. In small populations, inbreeding will cause this masked load to become realised as inbreeding load, and genetic drift will also contribute to a loss of fitness, i.e., drift load (Bertorelle et al. 2022; Dussex et al. 2023). Thus, a population with sufficient masked load will manifest lower fitness after experiencing low population sizes, displaying a form of Allee effect. Because of its genetic basis, this reduced fitness may persist even if the population grows to a larger size, making genetic Allee effects quite distinct from a typical Allee effect (where fitness is directly linked to current population density; (Luque et al. 2016)). In principle, by creating high masked load in a population through the introduction of deleterious recessive alleles, we impose a genetic Allee effect on that population.

Invasive populations rarely carry enough deleterious recessive alleles to reliably induce a genetic Allee effect throughout their range (Peischl et al. 2015). However, it may soon be possible to drive deleterious recessive alleles into an invasive population using CRISPR/Cas9 and an appropriate gene drive system (Champer et al. 2016; Akbari et al. 2013). To avoid these alleles being purged in the large invasive source population, we propose spreading the recessive mutations between many independent loci, each with a low frequency, creating high masked load (Nanjundiah 1993). In small populations originating from this modified invasive source population, the population bottleneck, drift, and inbreeding can bring some of these deleterious alleles to higher frequency and so cause genetic load to manifest.

Although there is a considerable body of work looking to understand extinction risk caused by genetic load in nature (particularly in a conservation context (Wittmann et al. 2018; Frankham et al. 2002)), the idea that we might design and impose masked load is new. As such, the questions we might ask become subtly different, being less about understanding patterns in nature and more about the emergent consequences of different masked load “designs”. In this paper we look to understand the size and nature of the masked load we would need to impose on an invasive source population in order to reduce the establishment probability of newly founded populations originating from it. We use a combination of numerical and individual‐based modelling to examine newly founded population extinction probabilities given an invasive source population's deleterious recessive allele frequencies, population size, and growth rate. We also use a female‐only mutation effect to assess various load designs by comparing X‐linked mutations to autosomal mutations, varying the frequency of deleterious alleles at a variable number of loci, and switching between lethal and sterile mutation effects.

2. Methods

2.1. Definitions and Metrics

The primary goal of this paper is to investigate the conditions under which masked load imposed onto an invasive source population (e.g., through biotechnological methods) would cause extinction of populations that would otherwise spread or re‐establish an invasion. Therefore, throughout this paper we compare the extinction of newly founded populations, measured as the proportion of extinct populations over a given number of replicates (the ‘extinction proportion’; 1000 replicates in Figures 1 and S1, otherwise 400 replicates). Wittmann et al. (2018) defined a strong genetic Allee effect as a sigmoidal relationship between population survival probability and initial population size (Wittmann et al. 2018). While we show a similar comparison in this paper, we additionally compare mean population fitness against initial founder population size. The latter is closer to the typical per capita growth rate versus current population size used when comparing non‐genetic Allee effects.

FIGURE 1.

Effect of invasive source population deleterious recessive allele frequency on extinction proportions generated by various increasingly realistic models. The red triangles show an initial numerical model. Green triangles show an intermediate model that accounts for variance in genotype frequencies. Both these models consider only the founding generation. Blue triangles indicate a final numerical model which recurses over generations with genetic drift, stochastic mortality, and reproductive output variation. Black circles indicate an analogous multigenerational individual‐based model created in SLiM 4.0 (the “hermaphrodite SLiM model”) with all loci segregating independently. Parameters here were held at $l=200$, $N\left(0\right)=25$, and $b=1.5$.

Our modelling does not include a natural mutation rate or natural genetic variance. As such, any fitness effects and subsequent population extinction cannot be attributed to a loss of genetic variation or an accumulation of de novo deleterious mutations. The deleterious alleles discussed in this study are explicitly introduced and are assumed to always result in a fitness of 0 when homozygous. They may therefore contribute to population extinction through drift and inbreeding depression. In nature, we expect a higher overall extinction probability than reported in this study due to the presence of mutational mechanisms and well‐known ecological drivers of extinction in small populations (Wittmann et al. 2018; Lynch, Conery, and Bürger 1995).

There are numerous ways that we might distribute deleterious recessive alleles within a diploid genome and within an invasive source population. We first clarify that there are at least two metrics for describing these alleles. The first metric is the carried load, $L_c$, which is the expected number of deleterious alleles per diploid individual in the population and is given by

$$L_c=2\sum _{i=1}^l{q}_i,$$

where $q_i$ is the frequency of deleterious recessive alleles at locus $i$ (Wittmann et al. 2018). If we assume a constant frequency, $q$, of deleterious alleles across $l$ loci, then

$$L_c=2\mathit{lq}.$$

The second metric is the realised load, $L_r$, which is the expected proportion of the population that will be homozygous for deleterious recessive alleles before mutation effects occur (under the assumption that mutations at each locus are distributed randomly and independently to other loci):

$$L_r=1-\prod _{i=1}^l\left(1-q_i^2\right)=1-{\left(1-q^2\right)}^l.$$

$L_c$ is a count, whereas $L_r$ is a strict proportion. $L_r$ is a measure of the population‐level fitness impact of the load and determines the rate it will be purged. $L_c$ is a measure of the overall number of deleterious alleles and scales with the persistence of fitness effects over time and the probability that one or more deleterious alleles will be sampled to a high frequency through drift or founder event.

For simplicity, our modelling assumes that there is no population structure in the invasive source population, meaning the described deleterious recessive alleles are randomly distributed within that population.

2.2. Numerical Modelling

We imagine a hermaphroditic population with non‐overlapping generations in which adults die following breeding. Under this model the number of individuals in the population, $N\left(t\right)$, follows a Beverton–Holt growth model (Rohlfs and Weir 2008) in which

$$N\left(t+1\right)=\frac{1}{u+1/B\left(t\right)}.$$ (1)

A full derivation is provided in Text S1, but initially we set $B\left(t\right)=\mathit{bS}\left(t\right)N\left(t\right)$, where $b$ is the average number of births per individual per generation in the absence of genetic load and density dependence (i.e., it is the maximum average growth rate), and $S\left(t\right)$ defines the proportion of individuals that survive to reach breeding age unaffected by deleterious recessive alleles. The remaining term, $u$, is the density‐dependent mortality, set by $u=\left(b-1\right)/\mathit{bK}$, where $K$ is the carrying capacity of the population (set to 1000 throughout). Under this model, the growth rate of the population, $\lambda \left(t\right)$, is

$$\lambda \left(t\right)=\frac{N\left(t+1\right)}{N\left(t\right)}.$$

The effective number of births in our model is reduced by the genetic load, according to the function $S\left(t\right)$. Under the simplest situation, where we assume all deleterious alleles have only a homozygous lethal effect (with complete penetrance), and loci segregate independently, the proportion of individuals whose fitness is unaffected by deleterious recessive alleles is $S\left(t\right)=\prod _{i=1}^l\left(1-H_i\left(t\right)\right)$, where $l$ is the number of loci carrying deleterious recessive alleles, and $H_i$ is the proportion of homozygotes for deleterious recessive alleles at locus $i$. For now, we assume an exact Hardy–Weinberg equilibrium (HWE), i.e., $H_i\left(t\right)=q_i{\left(t\right)}^2$, where q _i is the frequency of deleterious recessive alleles at locus $i$ at a given time $t$. This assumption also results in a direct link between $S\left(t\right)$ and our measure of realised load as $S\left(t\right)=1-L_r\left(t\right)$. We also note that we can use $b$ and $L_r$ to define biologically realistic sets of $l$ and $q$. That is, assuming no purging of deleterious alleles, populations where $b\left(1-L_r\left(t\right)\right)<1$ have a negative growth rate at all population sizes and are unsustainable without a decrease in $L_r$ and/or repeated immigration. Although this metric ignores density‐dependent mortality and therefore overestimates viability, it acts as a useful reference for source population viability. We refer hereon to the $b$, $l$, $q$ set resulting in positive growth at any positive population size as the “positive growth set”.

Our first iteration of numerical modelling simply looked at the chance that the number of independent deleterious mutations sampled in the first generation would lead to a negative growth rate under the assumption of Hardy–Weinberg equilibrium. We look for the conditions where $\lambda \left(0\right)<1$. The process we investigate (founder events) is inherently stochastic, so rather than looking for exact conditions under which $\lambda <1$, we instead look for the probability: $\mathrm{Pr}\left(\lambda <1\right)$.

2.2.1. Basic Numerical Model

To examine the distribution of outcomes arising from founder events, we then assume that $q_i\left(0\right)=\frac{n_{a_i}}{2*N\left(0\right)}$ and that

$$n_{a_i}~\text{Binomial}\left(2N\left(0\right),q\right),$$ (2)

where $N\left(0\right)$ is the founder population size, $n_{a_i}$ is the number of deleterious recessive alleles at locus $i$ and $q$ is the frequency of deleterious recessive alleles for each locus in the invasive source population (held constant across loci). This definition of $n_{a_i}$ results in $S\left(t\right)$ being the product of $l$ identical distributions. Our attempts to derive an analytical solution of the first model (Equation 1) quickly proved intractable due to this product, so a numerical approach was taken to generate model predictions instead. Repeating numerical simulations for a given fixed parameter set of ($q$, $l$, $b$, $N\left(0\right)$) builds an estimate for $\mathrm{Pr}\left(\lambda <1\right)$. This defines our basic numerical model.

2.2.2. Random Genotype Numerical Model

In our next most complex model, we still focus only on the first generation, but we take into account the natural variation that arises in genotype counts within each locus under the assumptions of Hardy–Weinberg equilibrium (Equation 3) as well as the natural variation of associations between genotypes at different independent loci (Equation 4). That is firstly, the process of gametes coming together to form genotypes is actually stochastic and $q_i^2$ is just an expectation value for $H_i\left(t\right)$. The probability mass function for the distribution of homozygous deleterious genotypes ($n_{\mathit{aa}}$) at a locus under HWE is given by (dropping the locus index and time‐dependent notation for brevity; (Lynch, Conery, and Bürger 1995))

$$\mathrm{Pr}\left(n_{\mathit{aa}}|n_a,n_A,\mathrm{HWE}\right)=\frac{N!}{n_{\mathit{AA}}!n_{\mathit{Aa}}!n_{\mathit{aa}}!}\times \frac{2^{n_{\mathit{Aa}}}{n}_A!n_a!}{2N!},$$ (3)

where counts for individuals of genotype ‘aa’, ‘aA’ and ‘AA’ are $n_{\mathit{aa}}$, $n_{\mathit{Aa}}$, $n_{\mathit{AA}}$ respectively and allele counts are $n_a$ and $n_A$. Note that, as with the previous model, $n_a$ is defined binomially and the other terms used in Equation 3 can be calculated as.

$n_{\mathit{Aa}}=n_a-2n_{\mathit{aa}}$ and $n_{\mathit{AA}}=N-n_a+n_{\mathit{aa}}.$

Our proportion of homozygous deleterious genotypes at a given locus $i$ is thus a random variable (with $n_{\mathit{aa}}$ being drawn from the probability mass function, Equation 3),

$$H_i=\frac{n_{\mathit{aa}}}{N}.$$

Secondly, we account for the variation between locus associations which is the stochastic variation of genotype and allele frequencies for a locus between individuals with different genotypes at any other locus. So, for loci 1 and 2, for example, the proportion of double homozygotes was previously the expected value $H_1\times H_2$. But of course, there is stochasticity around this expectation also, resulting in different death counts in a generation. Purging of deleterious recessive alleles is also affected if more or fewer heterozygotes die due to homozygosity at other loci. Allowing stochasticity in these associations means allele frequencies now also vary based on homozygous lethality at other loci. We take this into account iteratively as homozygotes at each locus die. Draws from a multivariate hypergeometric distribution (represented by $\text{MVHG}\left(\right)$) for each locus will give the three remaining genotype counts after the population size is reduced due to lethality at other loci. This distribution exactly represents the random elimination process given independent effects of all loci. So,

$$\left(n_{\mathit{AA}}^{-},n_{\mathit{Aa}}^{-},n_{\mathit{aa}}^{-}\right)~\text{MVHG}\left(N^{-},\left(n_{\mathit{AA}},n_{\mathit{Aa}},n_{\mathit{aa}}\right)\right)$$ (4)

where $N^{-}$ is the lowered population size after lethality at other loci, and $n_{\mathit{AA}}^{-}$, $n_{\mathit{Aa}}^{-}$, $n_{\mathit{aa}}^{-}$ are the resulting genotype counts among the survivors. $\text{MVHG}\left(\right)$ represents a multivariate hypergeometric distribution. Altogether, this results in a more stochastic and more biologically accurate definition of $S\left(t\right)$.

2.2.3. Multigenerational Numerical Model

In the final model, we also included the stochasticity arising from variation in reproductive output, density‐dependent mortality, and genetic drift over generations (Equations (5), (6), (7)). These additional sources of stochasticity require us to iterate the model over generations. We run these simulations until the population reaches carrying capacity, goes extinct, or runs for 2000 generations (although this last limit was never reached). Here we no longer look at the $\mathrm{Pr}\left(\lambda <1\right)$ in the first generation, but instead score whether populations go extinct or not to get an estimate for the extinction probability. Given that the strength of the genetic Allee effect can increase with more time at low population sizes and depends on mutations that are gradually cleared each generation, we expect the multigenerational approach to have greater accuracy.

To include these additional sources of stochasticity, we added stochastic processes for both demographic and evolutionary steps in the model. First, reproductive output in each generation was made stochastic by redefining $B\left(t\right)$ as

$$B\left(t\right)~S\left(\mathrm{t}\right)\times \text{Poisson}\left(\mathrm{bN}\left(t\right)\right)$$ (5)

These offspring need to survive to maturity, and in the Beverton–Holt framework, this survival is density dependent (Beverton and Holt 1993). We make this survival process stochastic also, allowing density‐dependent mortality to occur within a single continuous interval of time after the effects of genetic load are considered. To do this, we follow Beverton–Holt dynamics and define an instantaneous mortality rate, $\varphi =u{n}^2,$ where $n$ is the surviving number of offspring at that instant (initialised at $n=B\left(t\right)$). We then use this rate in repeated calls from an exponential distribution to get the time between deaths during the mortality period, $T$, given by

$$T~\text{Exponential}\left(u{n}^2\right)$$ (6)

We repeat draws from this distribution, iteratively decreasing $n$ by one each time, until the sum of the draws of $T$ exceeds 1 (the end of the mortality period). The number of iterations during the mortality period is taken as the number of individuals dying before maturity, $M\left(t\right)$. We then define the final number of individuals in the population as $N\left(t+1\right)=B\left(t\right)-M\left(t\right)$.

Finally, we incorporate genetic drift at each locus by drawing its deleterious recessive allele count, $n_a$, for the next generation of $N\left(t+1\right)$ individuals, using the binomial probability mass function given by

$$\mathrm{Pr}\left(n_a|N\left(t+1\right),q_a\right)=\left(\genfrac{}{}{0pt}{}{2N\left(t+1\right)}{n_a}\right){q_a}^{n_a}{\left(1-q_a\right)}^{2N\left(t+1\right)-n_a}\mathit{,}$$ (7)

where $q_a$ is the frequency of deleterious recessive alleles at the locus in generation $t$ (again dropping the notation for locus index and time dependence; (Messer 2016)).

2.3. Individual‐Based Modelling

To move beyond the multigenerational model, we would need to replace the Wright‐Fisher model of drift to consider gametic phase disequilibrium caused by rare parental genotypes (Haller and Messer 2019; Slatkin 2008; Excoffier and Slatkin 1998), that is, how alleles are grouped by an individual during breeding. This can be easily done by moving to an individual‐based model. SLiM is software designed for such work and is what we use for further extensions to the model. Individual‐based modelling also allows us to extend our model to include other features which would be much harder to develop numerically (sexes, sterile mutation, linkage effects; (Haller and Messer 2019; Haller and Messer 2023)).

2.3.1. Hermaphrodite SLiM Model

We used discrete time, aspatial, individual‐based simulations in SLiM 4.0 (Haller and Messer 2023) for all further modelling. We first sought to make a SLiM model closely comparable to our numerical work, before adding additional complexities. To specify the initial SLiM model, we make the modelled species a hermaphrodite with all loci segregating independently: a scenario we call the “hermaphrodite SLiM model”. A founder population was created with a given number of migrants, and mutations were randomly introduced across all individuals according to the same binomial distribution used for the numerical simulations (Equation 2). Each generation, individuals mated randomly, and the population grew according to the same Beverton–Holt model (Equation 1). However, deleterious fitness effects were realised by discarding offspring homozygous for deleterious mutations at a locus, rather than by reducing the number of individuals born, with the term $S\left(t\right)$ (Equation 5). The simulation was terminated once the simulated population reached carrying capacity, fell to zero individuals or reached 2000 generations.

To compare models, simulations were conducted on a high‐performance computer varying each parameter individually to obtain an estimate of the probability of population extinction for each parameter set under each model. Parameters were varied from a reference point {$l=200$, $N\left(0\right)=25$, $q=0.04$, and $b=1.5$} and each point was summarised by 1000 replicates (Figures 1 and S1). Prior testing informed the choice of reference point and parameter ranges, with the reference point chosen because it showed an intermediate extinction proportion and the ranges chosen to showcase the transition from extinction to survival across all models (Figures 1 and S1). The final numerical model and the hermaphrodite SLiM model were further compared across a larger subset of parameter space (Table 1) with 400 replicates per point.

TABLE 1.

The subset of parameter space over which the multigenerational numerical model and hermaphrodite SLiM model were compared, with 400 replicate runs at each combination of these parameter values.

Parameter	Range
Founder population size ($N\left(0\right)$)	10, 20, 30 … 100
Deleterious recessive allele frequency in source population ($q$)	0.01, 0.02, 0.03 … 0.15
Deleterious loci count in source population ($l$)	10, 50, 90 … 250
Reproductive output ($b$)	1.0, 1.25, 1.5 … 3.0

2.3.2. Dioecious SLiM Model

We then ran further simulations with an updated dioecious model that allowed sex‐dependent growth. Population growth in this model followed an updated Beverton–Holt model (Beverton and Holt 1993), again with a carrying capacity of 1000 ($K=1000$). To alter the Beverton–Holt model to a dioecious system, we change $u$ in Equation 1 to $u=\left(b_f-2\right)/b_{f}K$ (see Text S1 for full derivation). We draw the number of offspring for individual $j$ (prior to density dependence and genetic load effects) from a Poisson distribution, $o_j~\text{Poisson}\left(b_f\right)$, leading us to

$$B_j\left(t\right)=S\left(t\right)o_j$$

to replace the previous $B\left(t+1\right)$ in Equation 1 where $B_j$ is the number of offspring for female $j$. Note that $b_f$ represents a female's maximum reproductive output which needs to be twice a hermaphrodite individual's output to achieve the same low‐density population growth (that is, $b_f=2b$). The number of offspring in each generation is then $B\left(t\right)=\sum _{j=1}^{N_f}{B}_j\left(t\right)$, each offspring is considered to arise from an independent mating event.

Mutations in the dioecious model were revised to only affect females, which allowed for comparison of X‐linked or autosomal mutation inheritance while still having a recessive mutation effect. The updated model additionally allowed for comparison between a sterile and lethal mutation effect. Sterile individuals in our simulation still mate but do not produce offspring; they will compete with fertile individuals for matings. Given our updated growth equation and female‐only mutation effect, the sex ratio was not fixed. Each generation of the founder population consisted of a random binomial draw of females, where each individual had a 50% chance of being female.

Unless specified otherwise, loci were unlinked. Where we simulated linked loci, they were evenly distributed along a single 100 Mbp chromosome. Loci were assumed to be single points along this chromosome. Following the emergence of pseudo‐overdominance (see results), we also post hoc explored a range of recombination rates for a small subset of our parameter space. As before, in all cases, simulations were stopped if the population size reached 0 or K, or at 2000 generations otherwise.

2.4. Simulations to Identify Informative Parameter Space Subsets

A set of simulations was run to broadly examine the parameter space. Examined parameters and their ranges are summarised in Table 2. Four hundred replicates were used for each parameter set, leading to a total of 134.4 million simulations across parameter space. Informative parts of the parameter space (places where extinction proportions were intermediate, rather than 0% or 100%) were further examined with smaller steps in the parameters of interest to show finer details. Further analyses were focused on a reference point analogous to the previous point used for model comparison, {$l=200$, $N\left(0\right)=25$, $q=0.04$, and $b_f=3$} with autosomal inherited mutations causing a lethal effect acting before density‐dependent mortality.

TABLE 2.

The subset of parameter space explored in the dioecious simulation model, with 400 replicate runs at each combination of these parameter values.

Parameter	Range
Founder population size ($N\left(0\right)$)	5, 10, 15 … 30 and 40, 50, 60 … 120
Deleterious recessive allele frequency in source population ($q$)	0.01, 0.02, 0.03, 0.04 and 0.05, 0.07, 0.09 … 0.15
Deleterious loci count in source population ($l$)	10, 20, 30 … 100 and 120, 140, 160 … 300
Inheritance	X‐linked or autosomal
Mutation effect	Sterile or lethal
Mutation timing a	Before or after density‐dependent mortality
Female reproductive output ($b_f$)	2.5, 3, 3.5 … 6 and 10, 15, 20, 30, 40, 50

^a For all analyses aside from those explicitly looking at the difference between these timings (Figure S9), this parameter was set to “Before density‐dependent mortality”. For more information as to why, see the ‘Mutation inheritance, effect, and timing comparisons’ section of the methods.

2.5. Confirmation of Allee Effect

Increased extinction of smaller populations can result from demographic stochasticity or other processes unrelated to Allee effects. To confirm the presence of an Allee effect as the expected cause of population extinction, we started with the reference point and then varied the founder population size while tracking mean population fitness (Figure 3). By doing this, we can visualise the Allee effect. Specifically, in the case of a lethal mutation effect, relative fitness ($\omega$) was calculated as the average expected number of offspring from a mating relative to a mating with no deleterious recessive alleles. For each possible mating pair in a generation of a given population, we calculated each $\omega$ and then averaged to get the population fitness, $\overline{\omega }$.

$$\omega =\prod _{i=1}^l\left(1-\frac{1}{8}{n}_i^f{n}_i^m\right)$$

where $n_i^m$ and $n_i^f$ are the number of deleterious alleles at locus $i$ in the male (0, 1 or 2) and female (0 or 1) respectively. Considering the 50% chance of passing on a single allele for each parent and the 50% chance for a female offspring, who will be affected, gives the $\frac{1}{8}$ used in the above equation. These calculations were repeated across all generations, 1–50, in each simulation. 400 replicate simulations were performed at each point and the population fitness ($\overline{\omega }$) in each generation was averaged across the replicate populations giving a mean population fitness. Note that for generations beyond the first, extinct and single sex populations were not included in the fitness calculation. In the case of a sterile mutation effect, a simpler fitness calculation was used, the fitness of mating pairs involving a sterile individual was instead 0 and in all other matings the fitness was 1.

FIGURE 3.

Genetic Allee effect over generations: The average of population fitnesses within a given generation across 400 populations is compared against their founder population size ($N\left(0\right)$). Population fitness was calculated as the average of the relative fitness for all individuals in the given generation of the population. As in Figure 2, other parameters were held at $l=200$, $q=0.04$, $b_f=3$ with autosomal lethal mutations acting before density‐dependent mortality. Note that populations that went extinct before reaching a given generation could not be included in that generation's average population fitness calculation, and in later generations, some founder population sizes contained no surviving populations (meaning there is no point on the graph for them).

2.6. Predicting Extinction Probability From Load Metrics

We have two metrics, $L_c$ and $L_r$, describing the introduced deleterious recessive alleles. To test how well these metrics predict extinction probability, we fixed founder size and female reproductive output to 25 and 3 respectively, but systematically varied $l$ and $q$ in the range 80, 90, 100… 400 and 0.0, 0.0025, 0.005… 0.10 respectively. This resulted in a dataset of 541,200 simulations each of which was scored for whether extinction occurred or the population reached carrying capacity. To prepare the data for logistic regression, datapoints were filtered to keep only those with an extinction proportion between 2.5% and 97.5%. We then used our two metrics of load, $L_c$ and $L_r$ as explanatory variables in a logistic regression predicting extinction proportion. To see if the logistic model was generalisable to other parts of the parameter space we then examined seven additional subsets of parameter space, giving us an orthogonal set with founder size {12, 25, 50, 100} and growth rate {3, 5}.

2.7. Mutation Inheritance, Effect, and Timing Comparisons

Using our dataset described by Table 2, we compared X‐linked and autosomal inheritance by looking at pairs of datapoints where all parameter values, apart from mutation inheritance, were the same. For each pair, we firstly recorded, $D_{\text{observed}}$, the difference in extinction proportion between the datapoint with X‐linked mutation inheritance and between the datapoint with autosomal mutation inheritance. Explicitly, we write this as $D_{\text{observed}}=\frac{x_{x-\text{linked}}}{n}-\frac{x_{\text{autosomal}}}{n}$ where $\frac{x_{x-\text{linked}}}{n}$ and $\frac{x_{\text{autosomal}}}{n}$ are the aforementioned extinction proportions and $n$ is the number of replicates for a given parameter set (400 throughout). Note that both $x_{x-\text{linked}}$ and $x_{\text{autosomal}}$ are counts and are observations of binomial distributed random variables (the number of extinctions occurring within a set of 400 replicates). For each pair, we secondly generated the expected distribution under the null hypothesis (that there is no difference in extinction probability between X‐linked mutation inheritance and autosomal mutation inheritance). Under the null hypothesis, $x_{x-\text{linked}}$ and $x_{\text{autosomal}}$ would be observations of the same binomial distributed random variable. We assume that the true probability of extinction for this binomial variable would be the average of our pair of observed extinction proportions, $y_{\mathrm{avg}}$. In this analysis, $y_{\mathrm{avg}}=\frac{x_{x-\text{linked}}+x_{\text{autosomal}}}{2n}$. This means our distribution of the expected difference under the null hypothesis, $D_{\text{expected}}$, is expressed as $D_{\text{expected}}=\frac{X_1}{n}-\frac{X_2}{n}$ given $X_1~\text{Binomial}\left(n,y_{\mathrm{avg}}\right)$, $X_2~\text{Binomial}\left(n,y_{\mathrm{avg}}\right)$. For each pair, we finally used Fisher's exact test to test for a significant difference between the members of a pair in the counts of extinction and survival results from simulations. The Šidák correction for multiple testing was used to adjust the significance level from $\alpha =0.05$ to $\alpha =3.20\times {10}^{-6}$.

Filtering was done to remove parameter set pairs with near‐complete extinction or survival across both members of the pair ($y_{\mathrm{avg}}\le 2.5\%$, $y_{\mathrm{avg}}\ge 97.5\%$), as these pairs made up a majority of the examined parameter‐space subset (Table 2) and were not informative for comparing differences. This filtering removed 78.7% of parameter set pairs in this comparison. Summing the distributions of $D_{\text{expected}}$ that were obtained across all pairs gave us an expected distribution under the null hypothesis which could be compared graphically to the histogram of $D_{\text{observed}}$ values across all the pairs, see Figure 5. Chi‐squared testing found these distributions were significantly different with $p<2.2\times {10}^{-16}$.

FIGURE 5.

The differences in population extinction proportions (in parameter set pairs with average extinction proportions ($y_{\mathrm{avg}}$) between 2.5% and 97.5%) between members of (A) X‐linked and autosomal parameter set pairs and (B) sterile and lethal parameter set pairs. Members in each pair share all other parameter values, differing only in the inheritance or effect (X‐linked versus autosomal; and sterile versus lethal). For a given difference, the grey bar shows the overall frequency and red bars indicate the number of individual pairs that showed significance under chi‐squared analysis ((A) $p<3.20e^{-06}$, (B) $p<2.86e^{-06}$ ($p<0.05$ before applying the Šidák correction for multiple testing)). The expected difference distribution under a null hypothesis (no difference between the extinction proportions of members of each datapoint pair) is shown as black circles. In both cases, chi‐squared testing found the data were significantly different to the null ($p<2.2e^{-16}$). The data was generated from the subset of parameter space given in Table 2.

We compared a sterile mutation effect to a lethal mutation effect using a very similar methodology, with the difference, of course, being that pairing was done by a difference in mutation effect, instead of a difference in mutation inheritance. Filtering in this case removed 80.9% of parameter set pairs, and the final distributions were also found to be significantly different with $p<2.2\times {10}^{-16}$.

In both of these analyses, we only used mutations that took effect before density‐dependent mortality.

We then considered the timing of when lethal and sterile mutation effects manifest (hereafter “mutation timing”). For a lethal mutation, the timings are straightforward; the lethality could occur before or after density‐dependent mortality (both before mating). For a sterile mutation examining the same timing change was more complex. We considered that, in our simulations, sterility of a mother would be equivalent to lethality of the resulting offspring if that offspring died before density‐dependent mortality. Therefore, we compared this to a maternal effect lethal mutation where the lethality occurred after density dependence but before mating. The same methodology for the inheritance and effect comparisons was used for lethal timing and sterile timing comparisons. Filtering for points with intermediate extinction (as before) removed 78.7% of parameter set pairs for the lethal timing analysis, and 82.5% for the sterile timing analysis. Both analyses also showed a significant difference from their null hypotheses with $p<2.2\times {10}^{-16}$.

Sterile mutations that result in individuals not being included in the mating process altogether are also covered by this analysis. In our methodology, these would be equivalent to a lethal effect occurring after density‐dependent mortality. To simplify the discussion in this paper, all other analyses used a mutation timing that acts before the density‐dependent mortality period.

2.8. Data Analysis

Simulation output was examined through R scripts. All code used for simulation and data analysis can be found at https://doi.org/10.5281/zenodo.17020718. Examination of simulations was undertaken through graphical analysis as well as through the use of boosted regression trees (BRT) (Elith et al. 2008). BRT analysis performs additive regression of simple trees to fit complex nonlinear relationships within data and can identify important explanatory variables by measuring how effectively each variable reduces prediction error. We use the BRT approach primarily to understand the relative importance of each axis of our parameter space in affecting extinction probability.

3. Results

We used both numerical and individual‐based approaches to assess the parameters that impact a genetic Allee effect on an invading population. Our first aim was to establish that the two approaches were consistent with each other.

In comparing the three numerical models to our hermaphrodite SLiM model, we found that the more complex the numerical model, the better it matched the hermaphrodite SLiM model (e.g., Figures 1 and S1). The multigenerational numerical model, in particular, was typically very close to the analogous hermaphrodite SLiM model (Figure S2). The difference between models was not consistent across the parameter space. In some parts, non‐multigenerational models gave higher extinction proportions, and in other parts they returned lower extinction proportions than equivalent multigenerational models (Figures S1 and S2). The largest difference in extinction proportion between the hermaphrodite SLiM model and multigenerational numerical model across the parameter‐space subset in Table 1 was −0.4225 and after filtering the data to focus on intermediate extinction proportions, the first and third quartiles were −0.106875 and −0.0075 respectively (Figure S2). The small discrepancy between these two models is likely due to gametic phase disequilibrium caused by rare parental genotypes being accounted for in SLiM but not in our numerical modelling (Slatkin 2008; Excoffier and Slatkin 1998).

Our second aim was to explore how each parameter affected the extinction probability. The individual‐based models (Figures 1, 2 and S1), like all the numerical models (Figures 1 and S1), found population extinction proportion was affected by reproductive output ($b$, $b_f$), deleterious recessive allele frequency ($q$), deleterious loci count ($l$) and founder population size ($N\left(0\right)$); all in expected directions, with sigmoidal curves.

FIGURE 2.

Effect of numerical parameters—female reproductive output (A), deleterious recessive allele frequency (B), founder population size (C), and deleterious loci count (D) — on the proportion of extinct founded populations from the dioecious SLiM model. Unless varying along the x‐axis, parameters were held at $b_f=\left\{3,3.5,4\right\}$, $q=0.04$, $N\left(0\right)=25$, and $l=200$. In A these shared points are indicated with colour, in B–D; a dashed vertical line shows the shared point. Varying each parameter also moves the curves left and right as illustrated by the Female reproductive output curve series (B–D). Hollow points indicate levels of genetic load outside the positive growth set, i.e., genetic load that could not be sustained in a source population even ignoring density‐dependent mortality, i.e., $b_f\left(1-L_r\right)<2$. The mutations were autosomal inherited with a recessive lethal effect.

The fitness decrease causing extinction in these populations took the form of a demographic Allee effect (Figure 3). Typically, a demographic Allee effect is illustrated by comparing a population's growth rate with its current size, whereas a genetic Allee effect is delayed, with low population size affecting allele frequencies in future generations, which then affects population fitness and growth rate (Luque et al. 2016). The effect of a low population size on allele frequencies acts in the next generation and then persists over multiple generations even if the population has grown in the meantime. Therefore, we confirmed the Allee effect by comparing founder population size ($N\left(0\right)$) to mean population fitness (relative to a population without any deleterious alleles) after a number of generations. In the first generation, before any breeding within the founder population, there was no correlation between mean population fitness and initial population size, however in subsequent generations a clear positive correlation emerges. This confirms the demographic Allee effect to be caused by a genetic component Allee effect rather than a non‐genetic “single step” component Allee effect (Luque et al. 2016). After the Allee effect is established, fitness gradually drops in extant populations until around generation 10 and begins to rise again after generation 15 (as the worst affected populations go extinct and/or genetic load is purged). In later generations, a negative relation between fitness and founder population size is observed in extant populations starting with fewer than around 30 founders (Figure 3). We repeated this analysis for each other combination of mutation effects and mutation timings and similar results were seen in all cases (Figure S3).

We next investigated sets of load metrics that would cause extinction of newly founded populations without causing extinction of the invasive source population (Figure 4). To do this, we first identify our positive growth set, which, under the dioecious SLiM model, is the set of $b_f$, $l$, $q$ where $b_f\left(1-L_r\right)>2$, representing the conditions under which our source population is expected to be viable. We found that among viable source populations, for a given $L_r$, higher deleterious loci number ($l$) caused higher extinction proportions. This can be seen graphically as a larger area of high extinction in the positive growth set as deleterious loci count increases (In Figure 4 a larger red area is seen to the left of the black line as loci count rises). This result persisted despite variations to founder population size and female reproductive output, although its strength varied with those parameters.

FIGURE 4.

Proportions of extinct founder populations across a range of loci counts and frequencies. The black line indicates where $b_f\left(1-L_r\right)=2$, meaning where maximum growth rate can exactly sustain the invasive source population despite its genetic load. The region to the right of this line indicates load parameters outside of our positive growth set, i.e., conditions where the genetic load is sufficiently strong that it precludes source population viability. The red area to the left of this line shows where founder population extinction can be ensured without extinction of the source population. The box on each graph describes a logistic model predicting the log‐odds of extinction proportion fitted to the subset of the graph data where extinction proportion was between 0.025 and 0.975. The mutations comprising this load were autosomally inherited with a recessive lethal effect. Details on model fits can be found in Table S1.

Logistic regression was undertaken on each combination of founder population size ($N\left(0\right)$) and female reproductive output ($b_f$) to predict the extinct proportion from the simulation parameters and load metrics. We restricted this analysis to datapoints with an extinction proportion between 0.025 and 0.975. Comparing the Akaike Information Criterion of models using our input parameters ($b_f$, $N\left(0\right)$, $q$, $l$) and metrics of load ($L_c$, $L_r$), we found that $L_c$ and $L_r$ were the best predictors of extinction proportion. Models fitted by the logistic regression are shown in Figure 4 for each set of founder population parameters ($b_f$, $N\left(0\right)$), showing that both metrics predict increased extinction probability for higher values of $L_c$, and $L_r$. Depending on the founder population parameters, $L_r$'s effect on the proportion of extinct populations was 1–2 orders of magnitude greater than $L_c$ and the intercept of the model inversely followed $L_r$'s coefficient. See Table S1 for a complete set of information on the model fit.

These logistic models emphasise the previous result of increased extinction at higher loci counts within the context of source population viability (the positive growth set). Source population viability varies with only $L_r$, but extinction probability in the founder population additionally increases with $L_c$. Increasing loci number ($l$) while holding $L_r$ fixed by decreasing deleterious recessive allele frequency ($q$) accordingly makes $L_c$ rise, and this means a higher extinction probability. In other words, maintaining the same genetic load (i.e., fitness effect) in the invasive source population but spreading deleterious recessive alleles across more loci at a lower frequency can increase extinction probability in the newly founded population. This dynamic is revealed again by these models, where $L_c$'s contribution to extinction probability relative to $L_r$ is found to be higher at lower population sizes, which is in line with previous research by Wittmann et al. (2018).

Using the simulation outputs generated from the subset of parameter space in Table 2, we analysed the differences between X‐linked and autosomal mutation inheritance. We recorded the difference in population extinction proportion between these cases while holding all other parameters equal for each parameter combination. After filtering to keep only parameter set pairs where the extinction proportion was intermediate ($2.5\%<y_{\mathrm{avg}}<97.5\%$), the differences were then compared to the expected differences under the null hypothesis of no difference between X‐linked and autosomal mutation inheritance (Figure 5). These showed the data had a higher spread than expected under the null hypothesis, meaning there were both areas where X‐linked mutations had a significantly higher and significantly lower extinction proportion than an autosomal load (Figure 5). Subsequent analysis revealed that autosomal mutation inheritance resulted in more extinction when female reproductive output was low, and X‐linked mutation inheritance resulted in more extinction when female reproductive output was high (Figures S4 and S5). Overall, there was a trend towards higher extinction with autosomal mutation inheritance.

The same analysis was done for mutation effect datapoint pairs (lethal versus sterile), showing a much bigger difference when compared to the null expectation that they would have equal extinction probabilities (Figure 5B). In the examined subset of parameter space, sterile effects had up to 94.3% more extinction than lethal. Expressing these differences as a risk ratio rather than a difference ($\frac{x_{\text{sterile}}}{x_{\text{lethal}}}$ rather than $x_{\text{sterile}}-x_{\text{lethal}}$) showed sterile effects had on average an eightfold greater risk of extinction, although the first, second, and third quartiles of the risk ratio were 1.10, 1.43, and 2.96 respectively (Figure S6). This indicates a strongly right skewed distribution which was confirmed graphically (Figures 5 and S6).

A boosted regression tree analysis (Figure S7) found female reproductive output and founder population size to have the greatest individual effect in the comparison between sterile and lethal extinction proportions. When looking at interaction effects, the two greatest interaction pairs were deleterious loci count and deleterious recessive allele frequency followed by female reproductive output and founder population size (Figure S8). Overall, this analysis found that switching from lethal mutations to sterile mutations increased the range where we see extinction of founder populations.

Finally, we compared mutation timings for lethal and sterile effects. In both cases, mutation effects occurring after density‐dependent mortality significantly increased the extinction proportion in a subset of the parameter space (Figure S9). This effect was stronger with a sterile mutation effect although neither comparison showed a stronger effect than the difference between sterile and lethal mutation effects (Figure 5).

Our next set of simulations moved away from unlinked mutations and considered a single chromosome with a variable recombination rate between loci. During this experiment, we noticed that not all simulated populations reached carrying capacity or went extinct within the 2000 generation limit we had set. Looking into the haplotypes of these extant populations showed that wildtype haplotypes were completely absent, and instead there was a balance between several haplotypes, each containing different sets of deleterious recessive mutations. Due to the frequency dependence of recessive effects, if any single haplotype were to increase in frequency due to drift, homozygotes of that haplotype would become more common, resulting in a stronger selection against it than against other haplotypes. This resulted in a balance between the frequencies of each haplotype (i.e., balancing selection). The phenomenon behind this has been described previously and is called pseudo‐overdominance (POD). Formally, this is where the presence of several haplotypes which each contain different deleterious recessive mutations cause homozygotes of these haplotypes to have a lower fitness which leads to a heterozygote advantage (Abu‐Awad and Waller 2023; Waller 2021). POD is maintained when these mutations are located in a region with low or no recombination called a POD zone (Abu‐Awad and Waller 2023; Gilbert et al. 2020). In simulations with a low enough recombination rate, the entire genome acted like a POD zone. In contrast, simulations where carrying capacity was reached (Figure 6), showed cases where recombination was able to break up these deleterious haplotypes and form a wildtype haplotype which would be selected for, allowing the carrying capacity to be reached as the deleterious haplotypes were purged.

FIGURE 6.

Effect of recombination rate on the outcome proportions of newly founded populations. Red indicates a drop to 0 individuals (Extinction), blue indicates when the carrying capacity is reached and green represents when neither outcome is reached after 2000 generations and is a result of pseudo‐overdominance. Parameter values were as follows: Deleterious loci count $l=200$, deleterious recessive allele frequency $q=0.04$, founder population size $N\left(0\right)=25$, female reproductive output $b_f=3$, and mutations were inherited autosomally with a lethal effect. Mutations were evenly distributed across a genome of 100 Mbp.

Because of this dependence on recombination, we looked further into the effect of recombination rate on the sampled population (Figure 6). As expected, reducing recombination resulted in a greater proportion of surviving populations maintaining POD and this quickly became all of the surviving populations. Initially, we found a decreased extinction proportion when all mutations were absolutely linked (i.e., a recombination rate of 0) when compared to unlinked mutations (Figures 6 and S10: $q=0.083$, $l=100$), however, further investigation of the parameter space revealed that this difference in extinction proportion depended greatly on the deleterious loci count (l) (Figure S10). Decreasing the number of deleterious loci, while increasing deleterious recessive allele frequency to maintain a similar $L_r$ value, showed that at $l=10$ the relationship inverts so that unlinked mutations cause less extinction than those in absolute linkage with each other (Figure S10).

4. Discussion

Our results demonstrate a wide array of circumstances in which the masked load introduced by deleterious recessive alleles will lead to a high rate of extinction of newly founded populations. Given the close match between our numerical model and the simplest of our simulations, our simulation results appear to have a robust theoretical underpinning. Our simulations show the genetic Allee effect caused by deleterious recessive alleles is often strong enough to cause low‐density populations, those of most concern to biosecurity, to go extinct on their own, without further intervention.

There are also many circumstances where this genetic Allee effect is not sufficiently strong to cause 100% extinction of founded populations. It is worth noting that, from a biosecurity perspective, any increased level of extinction of founded populations is potentially beneficial and we do not necessarily need a 100% extinction probability, though this is an outcome to aim for. Modifying any of the examined parameters affects extinction probability, but this probability was shown to be very sensitive to the birth rate, in particular (Figure 2). When birth rate is high, extinction probability is low unless a much greater number of deleterious recessive alleles are applied. Maintaining this many mutations in the invasive source population without significant purging would likely be challenging, requiring either high rates of mutation introduction or some other mechanism for maintaining the deleterious recessive allele frequencies (e.g., a gene drive or balancing selection due to pseudo‐overdominance). Maintaining these mutations without causing extinction of the invasive source population will also be a tight balance as our logistic models indicate a lower relative contribution of $L_c$ with increasing birth rate (Figure 4).

Generally, our results point to the importance of maximising $L_c$ while keeping $L_r$ constrained. For different sets of introduced mutations with the same $L_r$, those with a higher $L_c$ will create a greater extinction probability in founder populations without affecting the invasive source population's viability (i.e., the strength of the Allee effect increases). Distributing the mutations across more loci (at lower frequency) achieves this aim (Figure 4). In a real system, of course, delivering deleterious recessive mutations to a larger number of loci (i.e., increasing $l$) increases the technical challenge, and physical linkage will increasingly cause non‐independent segregation as the density of modified loci increases. Thus, there are both technical and intrinsic limits to how high $l$ can go in a real system.

In order to relax the constraint placed on $L_r$ by the requirement for a viable source population, load parameters in the source population could be allowed to fluctuate (e.g., allowing natural purging, with occasional reintroduction of deleterious recessive alleles) or might only be held high going into an eradication attempt. A negative growth rate in the source population will cause extinction eventually, but if it is transient, then the population could recover through natural purging. Of course, depending upon the situation, a negative growth rate in the source population could also be a benefit, although this needs to be balanced with the increased death of mutated individuals that could halt the spreading of masked load throughout the source population. Similarly, if this strategy were to be combined with eradication attempts then the eradication attempt would be aiming to cause a population size decrease, which the genetic load would help achieve.

Our work also revealed a potentially strong difference between sterile and lethal mutation effects. Sterile mutation effects caused extinction across a broader swathe of parameter space. This was expected given that sterile individuals can compete for resources and mates, whereas dead individuals do neither. Such effects have been seen in other simulation and empirical studies (Flores 2003; Maïga et al. 2014; Gentile et al. 2015; Alphey 2002). It is worth noting, however, that we observed substantial variation in the lethal/sterile difference meaning the strong advantage of sterile effects is not consistent throughout the parameter space (Figures 5B and S6–S8). Indeed, for much of parameter space, the outcome was not affected by mutation effect (i.e., all populations either went extinct or survived), but there is a substantial set of edge cases where the mutation effect matters. The choice of whether to target sterile genes when using introduced Allee effects should therefore consider not only the constraint of how many recessively sterile mutations have been characterised in the target species, but also where in the parameter space the target species is likely to sit (Figures S7, S8).

A similar result was seen when delaying the mutation effect (Figure S9). By allowing affected individuals to contribute to density‐dependent mortality, they are competing and causing increased mortality of unaffected individuals. Although we do not consider mutation timings that would occur during the density‐dependent mortality period or timings in between a set of mortality periods, we expect these results to be an intermediate of the two timings investigated and in general a later timing would cause more extinction. However, if the mutation effect were to occur much later, i.e., after the mating period, the mutations would not impede the growth rate of the population at all (unless progeny survival relied on parental care). It is unclear how mutation timings during the mating period would vary extinction probability. Would a small increase in successful matings be worth it for a decrease in the rate of purging? Or is this simply a monotonic transition? It may be analogous to a reduction in effect penetrance. Answering this question generally would help guide decisions about which loci to target in actual species, though this is beyond the scope of this paper.

Initially, we expected that X‐linked mutations would also result in an overall greater extinction probability due to the decreased effective population size (and so greater stochasticity during founding) of an X chromosome when compared to an autosome. However, this expectation was not clearly met. This is likely because, on average, for a given mutation frequency, there will be the same number of female homozygotes whether mutations are X‐linked or autosomal, and therefore the same reduction in population growth occurs in both cases. This is true given the mutations we have defined cause female‐only effects, and that definition is why the effective population size argument fails. In fact, since there are fewer X chromosomes in the population, genetic load is on average purged slightly faster when caused by X‐linked mutations instead of autosomal ones (Text S2), which may explain the small bias towards decreased extinction seen in the simulation outputs for datapoints with an X‐linked mutation inheritance (Figure 5A). This result does mean that, under our assumptions, there is overall no disadvantage to using autosomal mutations, and since such mutations can be spread across more chromosomes, they will be less linked and will likely replicate the higher extinction probabilities seen when recombination rate is higher (Figures 6 and S10).

The pseudo‐overdominance (POD) that emerged in our simulations with low or no recombination confirms pre‐existing theory on the topic (Abu‐Awad and Waller 2023; Waller 2021) and further verifies POD's reduction of population fitness and its predicted disappearance under high recombination (Figures 6 and S10). Although many linked loci are less able to cause extinction of founder populations through an Allee effect than the same number of unlinked loci (Figure S10), the genetic load caused by linked mutations could still find some use with another population control method that takes advantage of pseudo‐overdominance. One possibility here is to use the mortality caused by the balancing selection that results from pseudo‐overdominance to reduce the equilibrium population size of the invading species (Figure 5; (Waller 2021)). Such an approach might be useful to mitigate the harmful effects of a species or to increase the feasibility of eradication. We also note that, under competition with native species, a lower equilibrium population size for the invasive species may alone be enough to cause failure of the invasion (Phillips 2025). In principle, such effects could be delivered by modifying only a few loci, as fewer deleterious haplotypes mean higher frequency of each, and so the lower the average fitness and equilibrium population size. Establishing pseudo‐overdominance within an invading species would, however, be difficult. The first major challenge would be identifying, or creating, a low recombination region with viable target genes. Recombination will erode pseudo‐overdominance by breaking up deleterious haplotypes and thereby generating a neutral haplotype (Abu‐Awad and Waller 2023). A neutral haplotype will be selectively favoured because it doesn't carry the same fitness costs as any of the other haplotypes. Methods to reduce recombination could be employed to stop this erosion, for instance using synthetic inversion breakpoints to locally suppress recombination (Dymond et al. 2011; Li et al. 2023). The second challenge would be removing all completely neutral or wild‐type haplotypes within the chosen POD zone (Gilbert et al. 2020). Since this objective is opposed by selection, haplotypes with recessive deleterious alleles would need to be driven into the population, which means resistance to both the gene drive and to the generation of deleterious alleles will need to be overcome. Population structure provides a final challenge here, because even if neutral haplotypes of a POD zone are completely removed in one population or subpopulation, any migration could reintroduce a neutral haplotype, which would readily outcompete the deleterious alleles and remove the heterozygote advantage.

A less general application that may be more feasible is to take advantage of a locus with pre‐existing overdominance by decreasing the number of alleles in the population at that locus. For instance, the complementary sex‐determination locus in Apis cerana shows homozygous lethality in females for all of its alleles (Cho et al. 2006). Over its natural range 22 alleles are present; however, when it spread to Queensland, Australia, it founded a population with only seven alleles which were gradually subject to balancing selection and reached the same frequency (Gloag et al. 2016; Hagan et al. 2024). If we could reduce the number of alleles down to two with biotechnology, then the invasive population would be much less fit and therefore easier to manage. However, Apis cerana populations show extreme polyandry which has been shown to mitigate the fitness costs of overdominance on their colonies. It is likely that other invasive species that exhibit overdominance will have evolved mechanisms to avoid their fitness effects and so the use of overdominance for population control seems limited (Hagan et al. 2024; Ding et al. 2017). At least for the genetic Allee effect method we propose, the best course of action will be to distribute deleterious alleles across more loci that are less linked to have the greatest chance of replicating the higher extinction probabilities we see in our simulations (Figures 5, 6 and S10).

Some further motivation towards using unlinked mutations is that, unlike in our simulations, deleterious mutations in reality are unevenly distributed across chromosomes and can be tightly linked. Tightly linked loci are a disadvantage for manifesting a genetic Allee effect on founding because, when an individual has the mutations in cis, they could be considered a single potentially deleterious linkage unit, effectively reducing the number of loci available for sampling in the founder event. This issue further motivates moving away from a single chromosome that carries all the mutations. Additionally, unlinked mutations are easier to spread through the invasive source population because deleterious alleles will segregate more readily in the gametes of individuals with a high concentration of the initial mutations, thereby affecting their descendants more evenly. Whether the masked load is introduced through releases of male carriers or through a gene drive mechanism, future release strategies would benefit from these mutations having reduced linkage as they will become widely distributed over fewer generations (Harvey‐Samuel et al. 2017).

To spread the large amount of deleterious recessive alleles required for a genetic Allee effect, a gene drive could be an important tool. Equipping this gene drive with a Cas9‐like gene and guide RNAs targeting each locus will allow for the generation of deleterious alleles as the gene drive is spread (Hay et al. 2021; Kandul et al. 2020; Faber et al. 2024). But how does this method compare to a gene drive designed for direct population suppression (Hay et al. 2021; Bier 2022; Hammond et al. 2016)?

When comparing our method to a typical suppression drive (Hay et al. 2021; Bier 2022; Hammond et al. 2016), one key benefit is that it does not matter if resistance to the gene drive emerges (Unckless et al. 2017; Champer et al. 2021), as we have still delivered masked load. If the gene drive were to reach even a proportionally low frequency (e.g., 10%) before resistance becomes widespread enough to select against it, over the generations it had spread and the generations it was being selected against, it still generates mutations that segregate independently to the gene drive and remain in the population after the gene drive is gone. Resistance to a particular load‐generating guide RNA may also occur, but with the number of loci we are targeting, the selection for that resistance will be inhibited, and even if that resistance is fixed, only a small fraction of extinction probability is lost (Kandul et al. 2020; Faber et al. 2024). The cutting efficiency of any particular guide RNA is also less significant in the scheme of things, so long as some mutations are being generated; we do not need to drive introduced alleles to fixation (Kandul et al. 2020; Faber et al. 2024).

Another advantage of using a drive to generate recessive mutations at a low frequency is that we avoid spatial ‘chasing’ that will make suppression drives challenging to implement. Chasing is the phenomenon that can occur after a suppression gene drive has cleared an area and that area is recolonised by wildtypes (Champer et al. 2021; Birand et al. 2022). This mechanism can lead to the failure of otherwise highly efficient suppression drives (Paril and Phillips 2022; Liu and Champer 2022). Since our method does not aim to eradicate the source population, there is no refuge for the wildtype to escape into. Furthermore, by using masked load, we specifically try to target the newly founded population(s) that escape into habitable areas as seen in both gene drive chasing and during the spread of invasive species. That is, masked load delivered into a population may improve the efficacy of a subsequent suppression drive by crippling the founder events that lead to chasing. While a suppression drive and a masked‐load drive have different goals in the management of invasive species, in future it may be that an approach using a combination of methods (introducing a masked load before using a suppression drive) will be best to manage some invasive species where neither method is effective enough by itself.

Overall, our results show that introducing deleterious recessive alleles may be a useful tool for preventing the establishment of small founded populations of invasive species. With enough alleles, drift load and inbreeding load create a genetic Allee effect that pushes small founded populations towards extinction. Our results show that these alleles are most useful when they can be engineered to have lower overall linkage and contain a high proportion of loci causing sterile (rather than lethal) effects. Our results also show that organisms with high realised intrinsic growth rates will require substantially more mutations in order for this idea to be effective beyond all but the smallest of founder population sizes. Although gene drives for delivering such alleles across tens or hundreds of loci into populations have yet to be developed, our initial results suggest such drives may have useful applications, not only in preventing the establishment of invasive populations but also potentially as an aid to suppression and eradication efforts generally.

Author Contributions

Louis Nowell Nicolle performed the simulations, data analysis, wrote the manuscript, and prepared the figures. Alex Fournier‐Level motivated the comparison between individual‐based, numerical, and analytical approaches. He also helped write and review the methodology of the manuscript. Charles Robin supervised the project, helped interpret the data, and write the paper. Ben L. Phillips conceived the idea, supervised the project, performed the boosted regression tree analysis, and helped write the paper.

Funding

This work is funded by an Australian Research Council Discovery Grant (DP230101111). Phillips is supported by the Western Australian Government through the Premier's Science Fellowship Program.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Data S1: mec70228‐sup‐0001‐Supinfo01.pdf.

Nowell Nicolle, L. , Fournier‐Level A., Robin C., and Phillips B. L.. 2026. “Genetic Allee Effects for Controlling Invasive Populations.” Molecular Ecology 35, no. 1: e70228. 10.1111/mec.70228.

Data Availability Statement

Data can be found at https://doi.org/10.5281/zenodo.17021164. Additionally, all code used for simulation and data analysis can be found at https://doi.org/10.5281/zenodo.17020718.