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BackgroundTheWolbachiaincompatible insect technique (IIT) shows promise as a method for eliminating populations of invasive mosquitoes such asAedes aegypti(Linnaeus) (Diptera: Culicidae) and reducing the incidence of vector-borne diseases such as dengue, chikungunya and Zika. Successful implementation of this biological control strategy relies on high-fidelity separation of male from female insects in mass production systems for inundative release into landscapes. Processes for sex-separating mosquitoes are typically error-prone and laborious, and IIT programmes run the risk of releasingWolbachia-infected females and replacing wild mosquito populations.ResultsWe introduce a simple Markov population process model for studying mosquito populations subjected to aWolbachia-IIT programme which exhibit an unstable equilibrium threshold. The model is used to study, in silico, scenarios that are likely to yield a successful elimination result. Our results suggest that elimination is best achieved by releasing males at rates that adapt to the ever-decreasing wild population, thus reducing the risk of releasingWolbachia-infected females while reducing costs.ConclusionsWhile very high-fidelity sex separation is required to avoid establishment, release programmes tend to be robust to the release of a small number ofWolbachia-infected females. These findings will inform and enhance the next generation ofWolbachia-IIT population control strategies that are already showing great promise in field trials.

The range of the mosquito Aedes aegypti continues to expand, putting more than two billion people at risk of arboviral infection. The sterile insect technique (SIT) has been used to successfully combat agricultural pests at large scale, but not mosquitoes, mainly because of challenges with consistent production and distribution of high-quality male mosquitoes. We describe automated processes to rear and release millions of competitive, sterile male Wolbachia-infected mosquitoes, and use of these males in a large-scale suppression trial in Fresno County, California. In 2018, we released 14.4 million males across three replicate neighborhoods encompassing 293 hectares. At peak mosquito season, the number of female mosquitoes was 95.5% lower (95% CI, 93.6-96.9) in release areas compared to non-release areas, with the most geographically isolated neighborhood reaching a 99% reduction. This work demonstrates the high efficacy of mosquito SIT in an area ninefold larger than in previous similar trials, supporting the potential of this approach in public health and nuisance-mosquito eradication programs.

Releasing sterile or incompatible male insects is a proven method of population management in agricultural systems with the potential to revolutionize mosquito control. Through a collaborative venture with the "Debug" Verily Life Sciences team, we assessed the incompatible insect technique (IIT) with the mosquito vector Aedes aegypti in northern Australia in a replicated treatment control field trial. Backcrossing a US strain of Ae. aegypti carrying Wolbachia wAlbB from Aedes albopictus with a local strain, we generated a wAlbB2-F4 strain incompatible with both the wild-type (no Wolbachia) and wMel-Wolbachia Ae. aegypti now extant in North Queensland. The wAlbB2-F4 strain was manually mass reared with males separated from females using Verily sex-sorting technologies to obtain no detectable female contamination in the field. With community consent, we delivered a total of three million IIT males into three isolated landscapes of over 200 houses each, releasing ∼50 males per house three times a week over 20 wk. Detecting initial overflooding ratios of between 5:1 and 10:1, strong population declines well beyond 80% were detected across all treatment landscapes when compared to controls. Monitoring through the following season to observe the ongoing effect saw one treatment landscape devoid of adult Ae. aegypti early in the season. A second landscape showed reduced adults, and the third recovered fully. These encouraging results in suppressing both wild-type and wMel-Ae. aegypti confirms the utility of bidirectional incompatibility in the field setting, show the IIT to be robust, and indicate that the removal of this arbovirus vector from human-occupied landscapes may be achievable.

Significance With over 40% of humans at risk from mosquito-borne diseases such as dengue, yellow fever, chikungunya, and Zika, the development of environmentally friendly mosquito-control tools is critical. The release of reproductively incompatible male mosquitoes carrying a Wolbachia bacterium can drive mating events that kill the eggs. Through replicated treatment and control experiments in northern Australia, regular releases of Aedes aegypti males infected with a Wolbachia from Aedes albopictus was shown to drive strong population suppression in mosaic populations of wild-type (no Wolbachia) and wMel-Wolbachia–carrying Ae. aegypti. In a demonstration of bidirectional incompatibility between different Wolbachia strains in the field, we also demonstrate that one season’s suppression experiment can also show an ongoing effect into the following season., Releasing sterile or incompatible male insects is a proven method of population management in agricultural systems with the potential to revolutionize mosquito control. Through a collaborative venture with the “Debug” Verily Life Sciences team, we assessed the incompatible insect technique (IIT) with the mosquito vector Aedes aegypti in northern Australia in a replicated treatment control field trial. Backcrossing a US strain of Ae. aegypti carrying Wolbachia wAlbB from Aedes albopictus with a local strain, we generated a wAlbB2-F4 strain incompatible with both the wild-type (no Wolbachia) and wMel-Wolbachia Ae. aegypti now extant in North Queensland. The wAlbB2-F4 strain was manually mass reared with males separated from females using Verily sex-sorting technologies to obtain no detectable female contamination in the field. With community consent, we delivered a total of three million IIT males into three isolated landscapes of over 200 houses each, releasing ∼50 males per house three times a week over 20 wk. Detecting initial overflooding ratios of between 5:1 and 10:1, strong population declines well beyond 80% were detected across all treatment landscapes when compared to controls. Monitoring through the following season to observe the ongoing effect saw one treatment landscape devoid of adult Ae. aegypti early in the season. A second landscape showed reduced adults, and the third recovered fully. These encouraging results in suppressing both wild-type and wMel-Ae. aegypti confirms the utility of bidirectional incompatibility in the field setting, show the IIT to be robust, and indicate that the removal of this arbovirus vector from human-occupied landscapes may be achievable.

Aedes aegypti is a vector of many significant arboviruses worldwide, including dengue, Zika, chikungunya, and yellow fever viruses. With vector control methodology pivoting toward rearing and releasing large numbers of insects for either population suppression or virus-blocking, economical remote (sentinel) surveillance methods for release tracking become increasingly necessary. Recent steps in this direction include advances in optical sensors that identify and classify insects based on their wing beat frequency (WBF). As these traps are being developed, there is a strong need to better understand the environmental and biological factors influencing mosquito WBFs. Here, we developed new untethered-subject methodology to detect changes in WBFs of male and female Ae. aegypti. This new methodology involves directing an ultrasonic transducer at a free-flying subject and measuring the Doppler shift of the reflected ultrasonic continuous wave signal. This system's utility was assessed by determining its ability to confirm previous reports on the effect of temperature, body size, and age on the WBFs generated from acoustic or optical-based experiments. The presented ultrasonic method successfully detected expected trends for each factor for both male and female Ae. aegypti without the need for subject manipulation and potential impediment of natural flight dynamics due to tethering. As a result, this ultrasonic methodology provides a new method for understanding the environmental and physiological determinants of male and female WBFs that can inform the design of remote mosquito surveillance systems.

Rapid advances in biological and digital support systems are revolutionizing the population control of invasive disease vectors such as Aedes aegypti. Methods such as the sterile and incompatible insect techniques (SIT/IIT) rely on modified males to seek out and successfully mate with females, and in doing so outcompete the wild male population for mates. Currently, these interventions most frequently infer mating success through area-wide population surveillance and estimates of mating competitiveness are rare. Furthermore, little is known about male Ae. aegypti behaviour and biology in field settings. In preparation for a large, community scale IIT program, we undertook a series of mark- release-recapture experiments using rhodamine B to mark male Ae. aegypti sperm and measure mating interactions with females. We also developed a Spatial and Temporally Evolving Isotropic Kernel (STEIK) framework to assist researchers to estimate the movement of individuals through space and time. Results showed that ~40% of wild females captured daily were unmated, suggesting interventions will need to release males multiple times per week to be effective at suppressing Ae. aegypti populations. Males moved rapidly through the landscape, particularly when released during the night. Although males moved further than what is typically observed in females of the species, survival was considerably lower. These unique insights improve our understanding of mating interactions in wild Ae. aegypti populations and lay the foundation for robust suppression strategies in the future., Author summary Incompatible insect techniques for controlling populations of the dengue vector, Aedes aegypti, utilize the mating biology of adult male mosquitoes to achieve suppression through a sterilization process. As the study of Ae. aegypti control has typically focused on adult female mosquitoes, knowledge on male movement, survival and mating interactions in the field is lacking. Here we undertook several mark-release-recapture experiments on adult male Ae. aegypti in Innisfail, Australia, and measured important biological parameters. For the first time in large field experiments, we employed rhodamine B as a marker that when fed to adult males, identified both marked males and the wild females they mated with. We observed males moving further through the landscape, but surviving for a shorter period, than previous measurements undertaken on females in a field setting. A high proportion (~40%) of unmated females suggests individuals are constantly available for mating. As such, sterile male strategies may need to release at regular intervals to achieve effective population suppression. The unique insights provided by this study will assist in designing future sterile male field interventions.

With global expansion of the two main vectors of dengue, Aedes aegypti (Linnaeus, Diptera: Culicidae) and Aedes albopictus (Skuse, Diptera: Culicidae), there is a need to further develop cost-effective and user-friendly surveillance tools to monitor the population dynamics of these species. The abundance of Ae. aegypti and Ae. Albopictus, and associated bycatch captured by Male Aedes Sound Traps (MASTs) and BG-Sentinel (BGS) traps that were unbaited or baited with BG-Lures were compared in Cairns, Australia and Madang, Papua New Guinea. Mean male Ae. aegypti and Ae. albopictus catch rates in MASTs did not significantly differ when deployed with BG-Lures. Similarly, males of both these species were not sampled at statistically different rates in BGS traps with or without BG-Lures. However, MASTs with BG-Lures caught significantly less male Ae. aegypti than BGS traps baited with BG-Lures in Cairns, and MASTs without BG-Lures caught significantly more male Ae. albopictus than BGS traps without BG-Lures in Madang. Additionally, BG-Lures significantly increased female Ae. aegypti catch rates in BGS traps in Cairns. Lastly, bycatch capture rates in BGS traps were not significantly influenced by the addition of the BG-Lures. While this study provides useful information regarding the surveillance of Ae. aegypti and Ae. albopictus in these locations, further development and investigation is required to successfully integrate an olfactory lure into the MAST system.

Aedes aegypti and Aedes albopictus vector dengue, chikungunya and Zika viruses. With both species expanding their global distributions at alarming rates, developing effective surveillance equipment is a continuing priority for public health researchers. Sound traps have been shown, in limited testing, to be highly species-specific when emitting a frequency corresponding to a female mosquito wingbeat. Determining male mosquito capture rates in sound traps based on lure frequencies in endemic settings is the next step for informed deployment of these surveillance tools. We field-evaluated Male Aedes Sound Traps (MASTs) set to either 450 Hz, 500 Hz, 550 Hz or 600 Hz for sampling Aedes aegypti and/or Aedes albopictus and compared catch rates to BG-Sentinel traps within Pacific (Madang, Papua New Guinea) and Latin American (Molas, Mexico and Orange Walk Town, Belize) locations. MASTs set to 450–550 Hz consistently caught male Ae. aegypti at rates comparable to BG-Sentinel traps in all locations. A peak in male Ae. albopictus captures in MASTs set at 550 Hz was observed, with the lowest mean abundance recorded in MASTs set to 450 Hz. While significantly higher abundances of male Culex were sampled in MASTs emitting lower relative frequencies in Molas, overall male Culex were captured in significantly lower abundances in the MASTs, relative to BG-Sentinel traps within all locations. Finally, significant differences in rates at which male Aedes and Culex were positively detected in trap-types per weekly collections were broadly consistent with trends in abundance data per trap-type. MASTs at 550 Hz effectively captured both male Ae. aegypti and Ae. albopictus while greatly reducing bycatch, especially male Culex, in locations where dengue transmission has occurred. This high species-specificity of the MAST not only reduces staff-time required to sort samples, but can also be exploited to develop an accurate smart-trap system—both outcomes potentially reducing public health program expenses., Author summary The continued global expansion of Aedes aegypti and Aedes albopictus, which transmit viruses causing dengue and Zika, necessitates the further development of effective mosquito traps for monitoring and predicting transmission risk. Mosquito sound traps attract male mosquitoes by producing a frequency which mimics the wingbeat sound of female mosquitoes. As few insects will respond to such sounds, these traps can be highly species-specific, saving significant person-time to sort samples, potentially paving the way for more accurate smart-trap systems and reduced surveillance costs. To date, limited research has occurred within field-settings investigating the capture rates of male mosquitoes in sound traps set to different frequencies. To this point, we performed trials in dengue endemic environments to quantify capture rates of mosquitoes in Male Aedes Sound Traps set to different frequencies in Papua New Guinea, Mexico and Belize. We found that both male Ae. aegypti and Ae. albopictus responded positively to frequencies between 450 and 600 Hz. Additionally, male Culex mosquitoes were also caught, but at lower abundances in traps set to higher sound frequencies. These traps, when set to 550 Hz, effectively capture male Ae. aegypti or Ae. albopictus with limited bycatch.

As society has moved past the initial phase of the COVID-19 crisis that relied on broad-spectrum shutdowns as a stopgap method, industries and institutions have faced the daunting question of how to return to a stabilized state of activities and more fully reopen the economy. A core problem is how to return people to their workplaces and educational institutions in a manner that is safe, ethical, grounded in science, and takes into account the unique factors and needs of each organization and community. In this paper, we introduce an epidemiological model (the “Community-Workplace” model) that accounts for SARS-CoV-2 transmission within the workplace, within the surrounding community, and between them. We use this multi-group deterministic compartmental model to consider various testing strategies that, together with symptom screening, exposure tracking, and nonpharmaceutical interventions (NPI) such as mask wearing and social distancing, aim to reduce disease spread in the workplace. Our framework is designed to be adaptable to a variety of specific workplace environments to support planning efforts as reopenings continue.Using this model, we consider a number of case studies, including an office workplace, a factory floor, and a university campus. Analysis of these cases illustrates that continuous testing can help a workplace avoid an outbreak by reducing undetected infectiousness even in high-contact environments. We find that a university setting, where individuals spend more time on campus and have a higher contact load, requires more testing to remain safe, compared to a factory or office setting. Under the modeling assumptions, we find that maintaining a prevalence below 3% can be achieved in an office setting by testing its workforce every two weeks, whereas achieving this same goal for a university could require as much as fourfold more testing (i.e., testing the entire campus population twice a week). Our model also simulates the dynamics of reduced spread that result from the introduction of mitigation measures when test results reveal the early stages of a workplace outbreak. We use this to show that a vigilant university that has the ability to quickly react to outbreaks can be justified in implementing testing at the same rate as a lower-risk office workplace. Finally, we quantify the devastating impact that an outbreak in a small-town college could have on the surrounding community, which supports the notion that communities can be better protected by supporting their local places of business in preventing onsite spread of disease.

As Aedes aegypti (Linnaeus, Diptera: Culicidae) expands its global distribution and vectors a range of debilitating arboviruses there is an increased need for enhanced mosquito surveillance. Consequently, we developed a Male Aedes Sound Trap (MAST) that requires minimal power and is highly species-specific. Two different versions of the MAST were developed, one that uses synthetic pyrethroid to kill captured mosquitoes (MAST Spray) and another which has an internal divider to create a killing chamber in which a sticky panel can be placed to capture mosquitoes (MAST Sticky). We compared weekly capture rates of male Ae. aegypti and bycatch from the two MAST versions to those from BG-Sentinel (BGS) traps and Sound-producing BG-Gravid Aedes Traps (SGATs) throughout Cairns, northern Australia. Weekly mean male Ae. aegypti catches did not significantly differ between trap types. However, the rate of positive weekly detections of male Ae. aegypti was lower for the MAST Sticky than the other three trap types. The MASTs sampled significantly fewer mosquitoes other than male Ae. aegypti, than either the BGS trap or the SGAT. Also, the MASTs and SGATs all caught significantly less non-Culicidae bycatch than the BGS traps. Consequently, we have developed a versatile male Ae. aegypti trap which is potentially of great benefit to Ae. aegypti surveillance programs.

The Wolbachia Incompatible Insect Technique (IIT) shows promise as a method for eliminating invasive mosquitoes such as Aedes aegypti (Linnaeus)(Diptera: Culicidae) and reducing the incidence of vector-borne diseases such as dengue, chikungunya and Zika. Successful implementation of this biological control strategy relies on high-fidelity separation of male from female insects in mass production systems for inundative release into landscapes. Processes for sex-separating mosquitoes are typically error prone, laborious and IIT programs run the risk of releasing Wolbachia infected females and replacing wild mosquito populations. We introduce a simple Markov Population Process (MPP) model for studying mosquito populations subjected to a Wolbachia-IIT program which exhibit an unstable equilibrium threshold. The model is used to study, in silico, scenarios that are likely to yield a successful elimination result. Our results suggest that elimination is best achieved by releasing males at rates that adapt to the ever-decreasing wild population, thus reducing the risk of releasing Wolbachia-infected females while reducing costs. While very high-fidelity sex-separation is required to avoid establishment, release programs tend to be robust to the release of a small number of Wolbachia-infected females. These findings will inform and enhance the next generation of Wolbachia-IIT control strategies that are already showing great promise in field trials.

As Aedes aegypti continues to expand its global distribution, the diseases it vectors (dengue, Zika, chikungunya and yellow fever) are of increasing concern. Modern efforts to control this species include “rear and release” strategies where lab-reared mosquitoes are distributed throughout the landscape to replace or suppress invasive populations. These programs require intensive surveillance efforts to monitor their success, and the Biogents Sentinel (BGS) trap is one of the most effective tools for sampling adult Ae. aegypti. BGS trap catches can be highly variable throughout landscapes, so we investigated the potential impacts of environmental factors on adult Ae. aegypti capture rates during a “rear and release” program in California to better understand the relative contributions of true variability in population density across a landscape and trap context. We recorded male and female Ae. aegypti catches from BGS traps, with and without CO2, throughout control sites where no mosquitoes were released and in treatment sites where males infected with Wolbachia were released. BGS trap catches were positively influenced by higher proportions of shade or bushes in the front yard of the premises as well as the presence of potential larval habitats such as subterranean vaults. In contrast, an increase in residential habitat within a 100 m radius of trap locations negatively influenced BGS trap catches. For male Ae. aegypti, increased visual complexity of the trap location positively influenced capture rates, and the presence of yard drains negatively affected catch rates in control sites. Lastly, for BGS traps using CO2, higher catch rates were noted from traps placed greater than one meter from walls or fences for both male and female mosquitoes. These results have important implications for surveillance programs of Ae. aegypti throughout the Californian urban environment including adult monitoring during “rear and release” programs., Author summary The efficacy of the Biogents Sentinel traps, the gold standard for catching Aedes aegypti mosquitoes, is impacted by various factors. We investigated environmental influences on BGS trap catches of Ae. aegypti during a “rear and release” program using control sites, not containing manipulated populations, and treatment sites where males infected with Wolbachia were released. We found BGS trap catches to be influenced by premises vegetative characteristics, potential larval habitats, trap placement factors and the amount of residential habitat surrounding trap positions. These effects varied depending on the mosquito sex, whether the trap used CO2 and whether the population was impacted by male releases. Our findings can inform a variety of programs requiring the surveillance of adult Ae. aegypti.

Sterile insect technique (SIT) and incompatible insect technique (IIT) are current methods for biological control of insect populations. Critical to the successful implementation of these biocontrol programs is quantifying the competitiveness of sterile/incompatible male insects for female mates relative to wildtype males. Traditionally, entomologists measure this mating competitiveness through a quantity known as Fried’s Index. We establish that Fried’s Index is mathematically equivalent to the mating competitiveness coefficient that features in many population models used in SIT/IIT programs. Using this insight, we propose a new approach for estimating Fried’s Index from mating competitiveness experiments. We show that this approach offers greater precision and accuracy than the traditional approach that is currently used in many studies. This is demonstrated using both simulation experiments and by analysing real experimental data. To facilitate uptake of the proposed method, we also provide an R package that can be used to easily analyse data from mating competitiveness experiments.

As the incidence of arboviral diseases such as dengue, Zika, chikungunya, and yellow fever increases globally, controlling their primary vector, Aedes aegypti (L.) (Diptera: Culicidae), is of greater importance than ever before. Mosquito control programs rely heavily on effective adult surveillance to ensure methodological efficacy. The Biogents Sentinel (BGS) trap is the gold standard for surveilling adult Aedes mosquitoes and is commonly deployed worldwide, including during modern ‘rear and release’ programs. Despite its extensive use, few studies have directly assessed environmental characteristics that affect BGS trap catches, let alone how these influences change during ‘rear and release’ programs. We assessed male and female Ae. aegypti spatial stability, as well as premises condition and trap location influences on BGS trap catches, as part of Debug Innisfail ‘rear and release’ program in northern Australia. We found similar trends in spatial stability of male and female mosquitoes at both weekly and monthly resolutions. From surveillance in locations where no males were released, reduced catches were found at premises that contained somewhat damaged houses and unscreened properties. In addition, when traps were located in areas that were unsheltered, more than 10 m from commonly used sitting areas or more visually complex catches were also negatively affected. In locations where males were released, we found that traps in treatment sites, relative to control sites, displayed increased catches in heavily shaded premises and were inconsistently influenced by differences in house sets and building materials. Such findings have valuable implications for a range of Ae. aegypti surveillance programs.

Background Recent interest in male-based sterile insect technique (SIT) and incompatible insect technique (IIT) to control Aedes aegypti and Aedes albopictus populations has revealed the need for an economical, rapid diagnostic tool for determining dispersion and mating success of sterilized males in the wild. Previous reports from other insects indicated rhodamine B, a thiol-reactive fluorescent dye, administered via sugar-feeding can be used to stain the body tissue and seminal fluid of insects. Here, we report on the adaptation of this technique for male Ae. aegypti to allow for rapid assessment of competitiveness (mating success) during field releases. Methodology/Principle findings Marking was achieved by feeding males on 0.1, 0.2, 0.4 or 0.8% rhodamine B (w/v) in 50% honey solutions during free flight. All concentrations produced >95% transfer to females and successful body marking after 4 days of feeding, with 0.4 and 0.8% solutions producing the longest-lasting body marking. Importantly, rhodamine B marking had no effect on male mating competitiveness and proof-of-principle field releases demonstrated successful transfer of marked seminal fluid to females under field conditions and recapture of marked males. Conclusions/Significance These results reveal rhodamine B to be a potentially useful evaluation method for male-based SIT/IIT control strategies as well as a viable body marking technique for male-based mark-release-recapture experiments without the negative side-effects of traditional marking methods. As a standalone method for use in mating competitiveness assays, rhodamine B marking is less expensive than PCR (e.g. paternity analysis) and stable isotope semen labelling methods and less time-consuming than female fertility assays used to assess competitiveness of sterilised males., Author summary The recent resurgence in male-based “rear and release” strategies (release of radiation sterilised males or those harbouring Wolbachia) to control diseases transmitted by urban Aedes mosquitoes, including dengue, chikungunya, and Zika, has revealed the need for an economical and rapid diagnostic tool to assess the mating competitiveness of released males against wild counterparts for receptive females. Our report details the optimisation of rhodamine B labelling of male seminal fluid and visible body staining via sugar-feeding with no significant impact on survival or mating competitiveness, as well as proof-of-principle field releases. These results reveal rhodamine B labelling to be an economical and rapid evaluation method for male-based control strategies as well as a viable body marking technique for male focused mark-release-recapture experiments without the negative side-effects of traditional marking methods. The use of this method has the potential to enable operators to determine the efficacy of their control method quickly and easily, allowing them to optimise releases to achieve optimal population suppression.

In this paper, we study the evolution of the mutation rate for simple organisms in dynamic environments. A model based on explicit population dynamics at the gene sequence level, with multiple fitness coding loci tracking a moving fitness peak in a random fitness background, is developed and an analytical expression for the optimal mutation rate is derived. The optimal mutation rate per genome is approximately independent of genome length, something that has been observed in nature. Furthermore, the optimal mutation rate is a function of the absolute, not relative, replication rate of the superior gene sequences. Simulations confirm the theoretical predictions.

In this paper we investigate error thresholds on dynamic fitness landscapes. We show that there exists both a lower and an upper threshold, representing limits to the copying fidelity of simple replicators. The lower bound can be expressed as a correction term to the error threshold present on a static landscape. The upper error threshold is a new limit that only exists on dynamic fitness landscapes. We also show that for long genomes and/or highly dynamic fitness landscapes there exists a lower bound on the selection pressure required for the effective selection of genomes with superior fitness independent of mutation rates, i.e. there are distinct nontrivial limits to evolutionary parameters in dynamic environments.

A quasispecies evolving on a fitness landscape with a single peak of fluctuating height is studied. In the approximation that back mutations can be ignored, the rate equations can be solved analytically. It is shown that the error threshold on this class of dynamic landscapes is defined by the time average of the selection pressure. In the case of a periodically fluctuating fitness peak, we also study the phase shift and response amplitude of the previously documented low-pass filter effect. The special case of a small harmonic fluctuation is treated analytically.