The sterile insect technique is protected from evolution of mate discrimination

SIT basics: population collapse accelerates over time

To set the stage for our study, we provide a brief introduction to some fundamental properties of the basic SIT. The essence of suppression under the SIT is described by a fertility function (P, following Barclay, 2005), which gives the probability of a female mating with a wild-type male when sterile males are present: (1)${\displaystyle P=\pi M/\left(\pi M+S\right).}$

S is the density of sterile males, M is the density of fertile males, and π is female preference for wild-type over sterile males (π ≥ 1). Except for the preference, this formula is the same as the numerical model proposed by Knipling (1955) and mathematically equivalent to a fertility function in which males are inferior to wild-type at mating (Barclay, 2005; Barclay, 2021), but using π in this way allows the emphasis to be placed on female discrimination.

Whether the population grows or declines not only depends on P but also on the birth rate, b (the number of daughters born to a female who survive to reproduce). The population will decline if b P <1. Birth rates of wild populations are usually considered to obey density dependence, whereby more young are born than will survive to reproduce. Thus b itself is not necessarily constant across all conditions. Our numerical analyses (based on the continuous-time model in the appendix) incorporate density dependence, but some of our derivations using discrete-time models omit density dependence (justified on the grounds that, as the population starts declining, density dependence will then become negligible).

The fertility function P has several interesting properties. First, for a fixed sterile release rate (S), P is monotonically increasing with M, but the increase is decelerating (Fig. 1). Second, P approaches 0 as M approaches 0. Together, both properties imply that, once the population starts to decline, it continues to decline ever faster. Equally, there is a threshold value of M below which any fixed level of preference π ceases to rescue the population (at which b P <1).

In any dynamic process, the fertility function changes with time because the population density of wild males (and females) is changing. A more accurate formulation of Eq. 1 is thus (2)${\displaystyle P_t=\pi M_t/\left(\pi M_t+S\right),}$

with a subscript t indicating time or generation number. (We will assume the release rate S remains constant unless indicated otherwise.) This formula is not a recursion, but by assuming the number of adult females equals the number of adult males (F_t = M_t, hence a sex ratio of 1:1), a fixed birth rate of b reproductive daughters (and b sons) per female such that M_t+1 = b P_t, F_t = bP_tM_t, it follows that, for π = 1, (3a)${\displaystyle P_{t+1}=M_{t+1}/\left(M_{t+1}+S\right)=b{P}_t{M}_t/\left(b{P}_t{M}_t+S\right)=P_t/\left[P_t+S/\left(b{M}_t\right)\right]}$

and from S / M_t = (1-P_t)/P_t,

(3b)${\displaystyle P_{t+1}=P_t/\left[P_t+\left(1-P_t\right)/\left(b{P}_t\right)\right].}$

If b P_t <1, then P_t+1 will be less than P_t. If this inequality is met, the proportional decline in P_t will accelerate with increasing t, so there is no stopping. (The formula is trivially modified for an unequal adult sex ratio, but the main points below are unaffected by introducing this complication).

When rare, discrimination need not rescue a collapsing population

The properties of P_t provide insight to the evolution of female preference, and as will be shown, evolution is unlikely to rescue a population that is collapsing under the SIT when discrimination is initially rare (Fig. 2). Suppose that most of the females lack preference (π = 1) but a small fraction (denoted by *) has a potentially strong preference for fertile males. At the start (t =0), the fertility function for each type of female is (4)${\displaystyle P_0=M_0/\left(M_0+S\right)\text{(no-preference females)}.P_0^{\ast }=\pi M_0/\left(\pi M_0+S\right)\text{(discriminating females)}.}$

We will consider the favorable case for discrimination that b P₀<1 but b P ₀^* >1, i.e., that the SIT will cause an initial decline in a population of mostly non-discriminating females, but preference is strong enough to rescue a population of the current size if all females possess it.

The key to this process, and the reason that a low frequency of discriminating females will not escape extinction, is that the same pool of wild-type males is common to both fertility functions—M_t is the same for both types of females (assuming the genetic basis of the female preference does not affect male fitness). With initiation of the SIT, the total population declines because most females have no preference and, for them, b P₀<1. Under a fixed progeny sex ratio (e.g., 50% sons), this leads to a progressive decrease in the population size/density of fertile males: M ₀ >M ₁ >M ₂ >M ₃ and so on, with concomitant effects on the fertility function P_t.

Even when discrimination increases initially, its long term persistence is not assured (Fig. 2). Indeed, the total population will decline even though discriminating females increase in density at first. But as M_t drops, it can eventually descend to a level at which b P_t^* <1; henceforth, discriminating females are also declining in numbers but not frequency. If discriminating females comprise enough of the population, or if the release rate S is small enough, then the early increase of discriminating females may halt the population decline. But when discriminating females start out rare, their density increases too slowly to reverse the decline.

The preceding addresses population sizes, not evolution. As long as sterile males are released, the discrimination allele (A) always increases in frequency (although the universality of this outcome requires an intrinsic death rate greater than 0 in the model). Despite the increase in the frequency of A, the discrimination allele, the absolute abundance of discriminating females avoids decline only if the total fertile male population is sufficient to maintain adequate fertility. The process is a race between adaptive evolution (with discrimination continuing to evolve) and population decline, much as in other evolutionary rescue problems (elaborated below). One difference from standard evolutionary rescue is that the decline in numbers of discriminator females is affected by the background abundance of non-discriminators. Persistence is enhanced by fertile males of either type, allowing females to avoid sterile matings. Thus rescue in this case depends on the entire population, not just the subset of individuals with the best genotypes. However, the effect is not strictly frequency-dependent selection (Svensson & Connallon, 2019).

Figure 2 offers a minimal sample of parameter values and initial conditions. One outcome not necessarily evident from the figure is a dependence of evolution and dynamics on initial densities. When the population is near carrying capacity, evolution is slow (and sterile releases are less effective) because there is little opportunity for growth: the model multiplies log fecundity by (1 − N/K), and as this density-dependent term approaches zero, there is little addition to the population. Thus, fitness differences due to discrimination vanish and even sterile releases have little effect on evolution or population density. This works against preference evolving fast enough to sustain a declining population, but it is offset by the intrinsic death rate (artificial population suppression, as with an insecticide, would have a similar effect). Our trials thus started away from carrying capacity and assumed a modest intrinsic death rate. A different model of density dependence might lack this property.

The criterion b P₀^* >1 requires that discrimination (π) must be sufficiently strong for the abundance of discriminating females to increase initially. How strong? Inserting the expression for P₀^* and rearranging shows that π ≥ π_minM₀) [S/(b − 1) ], where π_min is the minimum discrimination required for preference to increase initially. (π_min is equally the minimum population-wide preference required to prevent population decline at the onset of release rate S.) π_min increases with S and decreases with b and M ₀. Since the absence of preference in non-discriminating females is represented in the model by π = 1, the minimum effect of a mutation that would enable rescue is π_min –1. (Note that this result merely implies that a preference level of π_min would begin to increase at the onset of sterile release; it does not ensure that the increase will continue as the population declines—see above). This suggests that rescue will require relatively large-effect mutations when employing large sterile releases (S) or when non-discriminator birth rates b are close to 1 (i.e., just above replacement). And, to the extent that mutations of large effect are rare, the conditions for SIT resistance to arise de novo should also be rare.

The joint population and evolutionary dynamics of resistance to SIT are similar to those of “black-hole sink” adaptation (Holt & Gaines, 1992). A black-hole sink is a population that is a demographic sink but receives a recurrent stream of maladaptive immigrants that maintain a local population at low abundance. In the original concept, cutting off immigrants would doom the population because, as a demographic sink, it will by definition go extinct when isolated (Pulliam, 1988). In the SIT version, however, it is the introduced sterile males that doom the population; extinction would be avoided were the steady flow of steriles interrupted. In black-hole sink models, the evolution of genotypes that would save the population from extinction with or without the immigrant stream is predicted to be rare (Holt & Gomulkiewicz, 1997; Gomulkiewicz, Holt & Barfield, 1999; Holt, Gomulkiewicz & Barfield, 2003). This is used to help explain “niche conservatism”—which is the observation that species only rarely adapt to conditions outside their fundamental niche (Holt & Gaines, 1992). Our argument here for the rarity of resistance evolution to SIT is similar and presumably should extend to systems that assume more complex genetic bases for resistance.

Proportional releases are prone to escape extinction

An alternative to constant-S release is a proportional release, as was considered in models for the evolution of resistance to a version of the SIT using a dominant embryonic lethal (Alphey, Bonsall & Alphey, 2011; Watkinson-Powell & Alphey, 2017). A proportional release means that S is now varied throughout the intervention, with S_t being chosen each generation to maintain the same value of the fertility function across time:

P_t = M_t/(M_t + S_t) = 1/(1 + S_t/M_t) = 1/(1 + d), 

where M_t is the combined density of all males. (The notation S_t/M_t = d, where d is a constant, is from Alphey, Bonsall & Alphey (2011)). The constancy of the ratio S_t/ M_t means that not only the discriminator fertility function P is constant throughout the intervention, but so is the fertility function for discriminators (P^∗ in Eq. (4)).

Intuitively, this strategy seems as if it must be less effective than a constant-S release of the same initial impact, as it lacks the accelerating efficacy of steriles as the population declines. But any intuition that it is less effective leaves open many possibilities –that constant-S release may still be an effective intervention, and of course, it is an intervention that demands less effort. In any actual implementation, maintaining a constant proportion of steriles would be at best approximate, but for the sake of modeling, we assume it is perfect.

To illustrate the contrast between a proportional-S protocol and a constant-S protocol, we again consider the case that b P₀<1 but b P₀^* >1. One difference between the two protocols is immediately evident: as the values of P and P^∗ do not change throughout treatment, the fact that b P₀^* >1 should mean that discrimination is never lost no matter what its initial frequency –there should be no effect of a high abundance of non-discriminators overwhelming discriminators. If true, that would be a major difference from constant-S releases: the long term outcome of a proportional-S release is not affected by the initial abundance of discriminators. This intuition proves to be correct in some respects and false in others.

Two proportional-S trials are shown in Fig. 3. Both satisfy b P <1 and b P^* >1. Figures 3A and 3B use the same initial conditions and parameter values as in Figs. 2A, 2B; for those values, b P is just under 1 and b P* exceeds 3. With constant-S, the population was extinguished; under proportional-S, the population never falls below 10⁵, and the A allele gradually displaces allele a. Figures 3C and 3D use the same inputs except that the initial S is raised from 3 ×10⁶ to 11.7 ×10⁶. Now, the initial population falls rapidly, but discriminators are ultimately maintained at a low (but gradually increasing) level. Thus, the expectation is met in both cases that b P^* >1 ensures maintenance of discriminators. An obvious concern with proportional releases is suggested by Fig. 3C: population suppression works initially against all genotypes, but it ceases to become effective when non-discriminators are purged. There would likely be no way at the outset of an intervention to anticipate the presence of rare discriminators that were capable of avoiding eradication. Of course, and as before, increasing the release rate sufficiently would work to eradicate because a sufficiently higher release rate will drive b P*<1.

The dynamics in (3C, 3D) reveal the anomalous result that females of genotype A decrease in absolute density until genotype a is largely purged, with allele A henceforth maintaining an almost constant density (as its b P* barely exceeds 1.0, growth is expected to be minimal). If b P* exceeds 1, why does the density of discriminators decrease initially? This behavior appears to reflect a ‘poisoning’ from non-discriminators. Inspection of the equations indicates that the initial drop in the absolute abundance of A (despite its monotonic increase in frequency) stems from sexual reproduction. Mating of A females with a males (which is essentially a loss in fecundity for A) is not returned in magnitude when A males mate with the lower-fecundity a females to produce A progeny. Derivations (appendix, assuming discrete time) show the interesting effect that the density of discriminators increases or decreases according to their frequency –an interaction between genetic evolution and population growth. Figure 4 provides numerical examples: for different ratios of S/M (= d, in the figure), the finite growth rate of discriminators (R_A) starts above or below 1 at low frequencies of discriminators and rises above 1 at high frequencies. The lines lie entirely below R_A =1 if b P*<1.

A similar process must occur with constant-S releases but is not easily derived in such a clear fashion as for proportional-S releases. For one case, Fig. 4 therefore also plots the dynamics of constant-S release for comparison to the same initial conditions under proportional-S. The difference is profound and shows how vastly more effective a constant-S release can be. We return to constant-S releases for the remainder of the paper.

Imperfect SIT and the potential for evolution of discrimination

The previous results showed that extinction in the presence of low initial levels of discrimination is sensitive to the sterile release rate, magnitude of discrimination, and initial abundances. These models assumed a single population with constant release rate, S. In practice, the SIT will be applied in a spatial context and perhaps be discontinued when it appears that the population has been eradicated. What is expected when we violate the model to include these more realistic properties? We consider two cases below.

Periodic cessation of sterile release without eradication allows resistance to evolve

If releases are relaxed before eradication, then re-implemented as the population rebounds, discrimination will evolve (Fig. 5). Assuming that the population never goes extinct, this evolution of preference has the potential to allow eventual population recovery even during times of sterile release. Suppression would then require increasing the release rate. Even if the release rate, when applied, is sufficient to suppress the population, the evolved preference may spread or seed the evolution of further levels of preference. This result is consistent with analyses of black-hole sink models which show that temporal variation in the sink facilitates adaptative evolution due to the periods of mild demographic conditions (Holt, Barfield & Gomulkiewicz, 2004).

Simple cases of migration block evolution

Migration and dispersal present possible reasons that the SIT applied to a specific geographic area will fail to eradicate –despite total suppression of a local population, the target species will be continually re-introduced. The question here is whether those reintroductions facilitate the evolution of female preference. There are countless migration models to consider when varying initial population sizes, compositions, migration rates, and sterile release rates (e.g., Watkinson-Powell & Alphey, 2017 offer one of many alternatives). Generalities may exist in the form of multi-dimensional sets of parameter values that lead to common outcomes, but even those generalities will likely be specific to the nature of the model (e.g., the number of islands, relative island sizes, generations discrete or continuous, which sexes migrate, and migration rates varying through time). Our goal here is merely to consider a few simple cases to illustrate a range of possible outcomes.

A common type of migration model assumes an island population with recurring migration from a mainland population of fixed allele frequency (Crow & Kimura, 1970). Here we assume that the mainland and island initially have the same low frequency of discriminating males and females (0.0091), and migration continually reseeds the island. What evolution is expected in the island population?

We consider two examples with low migration rates. In both, the SIT implementation suppresses the population in the absence of migration. (Aside from migration and reducing π to 2, the initial conditions and parameter values are the same as used in Fig. 2A). The two examples differ only in the level of migrants. (A, B): With migrants at 1,031 per generation and a starting population of 3.3 ×10⁵, the population is suppressed (Fig. 6A). There is a brief and shallow rise in the frequency of discriminators that is squelched as the population crashes and becomes dominated by migrants (Fig. 6B). After collapse, the island population is effectively one in which sterile males are released at a high enough rate to prevent ongoing reproduction by the migrants. Even discriminators cannot reproduce at a high enough level to offset their decline, so they cannot evolve fast enough to offset the influx of migrants dominated by non-discriminators. (C, D): An approximate tripling of migrants radically changes the outcome (Figs. 6C, 6D). The population avoids a crash. Even so, the frequency of discriminators never attains a high level and appears to equilibrate only modestly above its frequency in migrants, even though discriminators are essential to avoidance of population suppression. (If π is set to 1 for the trial, the population is suppressed by the SIT).

These two examples merely serve to foreshadow the complexity of the SIT with migration. Initial suppression can be followed by continued suppression despite ongoing migration with discriminators. More surprisingly even when the initial population is not suppressed due to a combination of migration and discriminators, ongoing migration can maintain discriminators at low frequency in the large population. The sensitivity of these outcomes to model structure and parameter values is unknown. Considerable effort may be required to identify generalities that can be translated into robust empirical outcomes.