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Analyzed the data: PMA AT FAR. Wrote the paper: PMA AT FAR. Developed the mathematical model: PMA AT FAR. Performed the computer experiments/simulations: PMA. Performed analytical analysis: PMA FAR.

The authors have declared that no competing interests exist.

In isolated populations underdominance leads to bistable evolutionary dynamics: below a certain mutant allele frequency the wildtype succeeds. Above this point, the potentially underdominant mutant allele fixes. In subdivided populations with gene flow there can be stable states with coexistence of wildtypes and mutants: polymorphism can be maintained because of a migration-selection equilibrium, i.e., selection against rare recent immigrant alleles that tend to be heterozygous. We focus on the stochastic evolutionary dynamics of systems where demographic fluctuations in the coupled populations are the main source of internal noise. We discuss the influence of fitness, migration rate, and the relative sizes of two interacting populations on the mean extinction times of a group of potentially underdominant mutant alleles. We classify realistic initial conditions according to their impact on the stochastic extinction process. Even in small populations, where demographic fluctuations are large, stability properties predicted from deterministic dynamics show remarkable robustness. Fixation of the mutant allele becomes unlikely but the time to its extinction can be long.

Underdominance is a component of natural evolution: homozygotes – of either wildtypes or mutants – are advantageous. This can play a role in speciation and as a method to establish artificial genetic constructs in wild populations. The polymorphic state of wildtype and mutant alleles is unstable. However, in subdivided populations limited gene flow can counterbalance this effect. The maintenance of polymorphism sensitively depends on the amount of gene flow. In populations of finite size, the polymorphism is ultimately lost due to stochastic fluctuations, but there are long intermediate periods of polymorphism persistence. We analyze a simple population genetic model to characterize and explore the polymorphic phases depending on population size and genotypic fitness values. Even for large fluctuations (small population size), long periods of neither extinction nor fixation are possible. Since underdominance has been proposed as a genetic strategy in the pest management of disease vectors, it is important to understand the basic features of this system precisely, especially with a focus on gene flow between ecological patches. We assess different release strategies of potentially underdominant mutants, where one seeks to minimize the probability of fixation of the introduced allele but maximize the time to its extinction.

A population can evolve due to differences in relative reproductive success over a life cycle. Fitness, in an evolutionary genetic sense, is defined as the relative expected number of descendants in the next generation based on an individual's genotype. In diploid organisms, two alleles can result in three genotype combinations, two homozygous genotypes with two copies of the same allele, and one heterozygote type with one copy of each allelic type. Heterozygote disadvantage in reproductive success is termed underdominance: Heterozygous individuals have a lower relative fitness than both homozygotes. The fundamental properties of underdominance in large populations with deterministic dynamics are well known

Under natural conditions underdominance can be caused by chromosomal rearrangements

We focus on the dynamics of a single locus with underdominant alleles of large fitness effects, such as those expected with natural reciprocal translocations. There has also been research into multiple loci of weaker individual effects, which can have interesting self-organizing properties

As an artificial genetic construct, underdominance has been proposed as a method to stably establish linked alleles with desirable properties in the wild; for example, rendering insect populations resistant to diseases that otherwise can be transferred to humans (or other species), such as malaria or Dengue fever

Underdominant polymorphism is eventually lost or completely fixed in single populations. However, it is known that it can become stable at mixed frequencies due to a migration-selection equilibrium in large subdivided populations that exchange a fraction of migrants

Initial testing of genetic pest management systems is likely to take place on more isolated physical or ecological islands

Generations are overlapping in species that do not strictly follow discrete time reproductive patterns. Hence, we concentrate on Moran models describing the stochastic invasion and fixation of transformed or mutant alleles in a system of coupled populations. A Moran process considers a single reproductive event in one time step such that after

The manuscript is organized in the following way. The next part of this section briefly repeats aspects of evolutionary dynamics in two infinitely large populations coupled by migration. Then, we introduce our model based on a Moran process for two populations with migration. The section ends with the introduction of a one-dimensional island continent model, which directly follows as a solvable limit case. In the

With

For two local populations that exchange migrants we introduce the rate of migration

We focus on a Moran model with fixed population sizes. Our main assumption is that mate choice is random. In this case individuals in the population can be thought of as passing through the Hardy-Weinberg expectations at some point in life-cycle before selection. Hence, we can consider the system as if individual alleles (i.e. gametes) reproduce and die. Reproduction is proportional to fitness and death is random. Such discrete stochastic birth-death processes are typically used to describe the (transient) microscopic evolutionary dynamics in single uncoupled populations of finite size

Our two populations are of size

First, with probability

We show a phase portrait of the gradient of selection with

Secondly, in population

Thirdly, in each population, the total number of alleles is held constant. This implies that for each birth event, there is an independent death event: a randomly chosen individual allele is removed from the population. A type

Overall, given the state

Typical trajectories for the loss of the mutant allele (extinction process) in a system of two populations of the same size,

Let us first consider an island and continent situation. On the large continent, migrants from the small island introduce

In the

Another case that leads to one-dimensional evolutionary dynamics is the limit of high migration rate, such that the two populations become genetically indistinguishable. This yields slightly different dynamics in a population of

With migration the number of

The average allelic fitness values in the island population of size

The parameter transition from high to low migration leads to a change of the local gradient of selection

Let

First, we address the ratio of fixation to loss in the system of two coupled sub-populations of equal size. An ideal case for a locally controlled genetic pest management strategy emerges when the resistant allele (

(

The replicator dynamics for two populations shows a maximum of nine fixed points and an associated bifurcation pattern depending on the migration rates, see

The mean extinction time as a function of the migration rate (

The impact of system size in two equally large populations can now be quantified in terms of the average extinction time of type

Analyzing the extinction times as a function of migration rate reveals the transition from one power law to another in the region around the critical migration rate predicted by the replicator system; We can identify two regimes. In the first regime,

The transition of one population approaching infinite size, while the other remains relatively small, leads to stochastic evolutionary dynamics in one dimension. A benefit in using a Moran model is that in one dimensional systems we can obtain exact analytical results for the hierarchy of moments of extinction/fixation times

Assuming migration is low enough such that it can be locally counteracted by selection, how likely is a mutant allele to fix or be removed depending on the initial transformation? Here, we give a characterization of the two population system in terms of complete loss or fixation in both, or temporary reciprocal fixation and loss. In principle, a release strategy is based on two parameters. First, the amount of new mutant alleles added to the system,

The upper left panel illustrates the deterministic basins of attraction for

We have proposed a simple model to analyze the influence of small system size and system size asymmetry on the evolutionary dynamics of an underdominant system in a structured population. The population structure itself is chosen to be as simple as possible: We consider two sub-populations that exchange migrants at a given rate. This allows a direct comparison with findings in infinitely large populations

We review previous findings in infinitely large populations in the introduction and use them as a basis to examine the influence of demographic fluctuations in small populations. We argue that the transient dynamics are important, as the influence of noise may alter the outcome of the evolutionary dynamics in this regime.

Our main results can be summarized as follows. First, for fitness asymmetry, extinction rather than total fixation of the potentially underdominant allele is the most likely outcome, even if this allele is initially at high frequency in one of the populations. In a migration-selection equilibrium the (quasi-stationary) variance in allele frequencies is low. High migration disrupts this dynamic equilibrium such that extinction is facilitated and the variance in frequency increases.

Second, we find that migration rate has a strong impact on the extinction process. We identify a threshold below which the mutant allele can be maintained for a long time, which corresponds to a bifurcation point in the deterministic system. For example, if mutant homozygotes suffer from a

Third, we evaluate the consequences for release strategies. For conservative estimates of a release of potentially underdominant mutants into wildtype populations, we can give a statistical evaluation that can be tested

Fourth, the limit case of one population becoming very large reveals that the potentially underdominant allele can be kept in the small population for long times. A small population with incoming migrants from a large wildtype reservoir is well described by a one-dimensional process if the reservoir is about

Results from infinite population assumptions may, in some cases, be misleading when observing finite allele frequencies. Under demographic fluctuations the stochastic evolutionary dynamics slow down near corners and along edges: In the vicinity of equilibrium points the flow induced by selection can become squeezed between boundary and equilibrium. High flow density means low flow velocity, which also affects the transition rates. Due to this nature we may observe large waiting times near the corners and along edges which happen to be near internal equilibria. However, under neutral evolution, the system also slows down near corners and edges.

If selection is strong, underdominance and sufficiently low migration can maintain a polymorphic state for many generations even in small populations. This bodes well for using underdominance to control initial testing of genetically modified insects in isolated settings so that the natural species remains untransformed in its broader range. The system can be stable for so long that additional factors are likely to be more important in ultimately disrupting the system. Such additional factors can be the occurrence of new mutations and/or behavioral changes

The stability properties of underdominance in small finite populations may have particular value both in initial testing of genetically modified vectors and in species conservation applications. For example,

Genetically modified chromosomes are typically less fit than wildtypes as homozygotes, see

P.M.A acknowledges discussions with Maria Abou Chakra.