Differential equations models for interacting wild and transgenic mosquito populations

We formulate and study continuous-time models, based on systems of ordinary differential equations, for interacting wild and transgenic mosquito populations. We assume that the mosquito mating rate is either constant, proportional to total mosquito population size, or has a Holling-II-type functional form. The focus is on the model with the Holling-II-type functional mating rate that incorporates Allee effects, in order to account for mating difficulty when the size of the total mosquito populations is small. We investigate the existence and stability of both boundary and positive equilibria. We show that the Holling-II-type model is the more realistic and, by means of numerical simulations, that it exhibits richer dynamics.     

Differential equations models for interacting wild and transgenic mosquito populations

We formulate and study continuous-time models, based on systems of ordinary differential equations, for interacting wild and transgenic mosquito populations. We assume that the mosquito mating rate is either constant, proportional to total mosquito population size, or has a Holling-II-type functional form. The focus is on the model with the Holling-II-type functional mating rate that incorporates Allee effects, in order to account for mating difficulty when the size of the total mosquito populations is small. We investigate the existence and stability of both boundary and positive equilibria. We show that the Holling-II-type model is the more realistic and, by means of numerical simulations, that it exhibits richer dynamics.     

Dynamics of malaria transmission model with sterile mosquitoes

In this paper, a malaria transmission model with sterile mosquitoes is considered. We first formulate a simple SEIR malaria transmission model as our baseline model. Then sterile mosquitoes are introduced into the baseline model. We consider the case that the release rate of sterile mosquitoes is proportional to the wild mosquito population size. To investigate the impact of releasing sterile mosquitoes on the malaria transmission, the dynamics of the baseline model and the models with the sterile mosquitoes are discussed. We derive formulas of the reproductive numbers and explore the existence of endemic equilibrium as the reproductive number is more than unity for these models. It is shown that both the baseline model and the models with the sterile mosquitoes undergo backward bifurcations. Based on theoretical analysis and numerical simulation, we investigate the impact of releasing sterile mosquitoes on malaria transmission.     

Asymmetric mating in the brachypterous grasshopper Podisma sapporensis

Crossing of genetically differentiated populations often results in assortative mating within populations. However, asymmetric sexual isolation or negative assortative mating has occasionally been reported. Previous studies suggested that sexual selection or sexual conflicts would lead to asymmetric mating when local populations are crossed. In order to evaluate the extent of assortative or disassortative mating in population crosses, we conducted laboratory crosses using the flightless grasshopper Podisma sapporensis. Crossing was conducted for all pairwise combinations of three populations, 150–240 km from one another – Teine, Shimokawa, and Akan. We found evidence for asymmetric mating for all the pairs of the populations. In particular, when the Teine and Akan populations were crossed, mating in the Teine male–Akan female cross was significantly more frequent than mating in both within‐population crosses, whereas mating in the Teine female–Akan male cross was significantly less frequent than mating in both within‐population crosses. We examined whether these results can be explained by any of the three hypotheses: (1) Kaneshiro's hypothesis, (2) differentiation in attractiveness, or (3) coevolution between male vigor and female receptivity. All the results were consistent with male vigor differing between populations balanced by different female potential to reject males. The available evidence suggests that antagonistic coevolution between the sexes has led local populations to different equilibria and that crossing of populations at different equilibria has resulted in asymmetry in mating frequencies.     

Discrete-time models with mosquitoes carrying genetically-modified bacteria

We formulate a homogeneous model and a stage-structured model for the interactive wild mosquitoes and mosquitoes carrying genetically-modified bacteria. We establish conditions for the existence and stability of fixed points for both models. We show that a unique positive fixed point exists and is asymptotically stable if the two boundary fixed points are both unstable. The unique positive fixed point exists and is unstable if the two boundary fixed points are both locally asymptotically stable. Using numerical examples, we demonstrate the models undergoing a period-doubling bifurcation.     

Optimal control strategy of malaria vector using genetically modified mosquitoes

The development of transgenic mosquitoes that are resistant to diseases may provide a new and effective weapon of diseases control. Such an approach relies on transgenic mosquitoes being able to survive and compete with wild-type populations. These transgenic mosquitoes carry a specific code that inhibits the plasmodium evolution in its organism. It is said that this characteristic is hereditary and consequently the disease fades away after some time. Once transgenic mosquitoes are released, interactions between the two populations and inter-specific mating between the two types of mosquitoes take place. We present a mathematical model that considers the generation overlapping and variable environment factors. Based on this continuous model, the malaria vector control is formulated and solved as an optimal control problem, indicating how genetically modified mosquitoes should be introduced in the environment. Numerical simulations show the effectiveness of the proposed control.     

Cytonuclear Theory for Haplodiploid Species and X-Linked Genes. I. Hardy-Weinberg Dynamics and Continent-Island, Hybrid Zone Models

We develop models that describe the cytonuclear structure for either a cytoplasmic and nuclear marker in a haplodiploid species or a cytoplasmic and X-linked marker in a diploid species. Sex-specific disequilibrium statistics that summarize nonrandom cytonuclear associations in such systems are defined, and their basic Hardy-Weinberg dynamics and admixture formulae are delimited. We focus on the context of hybrid zones and develop continent-island models whereby individuals from two genetically differentiated source populations migrate into and mate within a single zone of admixture. We examine the effects of differential migration of the sexes, assortative mating by pure type females, and census time (relative to mating and migration), as well as special cases of random mating and migration subsumed under the general models. We show that pure type individuals and nonzero cytonuclear disequilibria can be maintained within a hybrid zone if there is continued migration from both source populations, and that females generally have a greater influence over these cytonuclear variables than males. The resulting theoretical framework can be used to estimate the rates of assortative mating and sex-specific gene flow in hybrid zones and other zones of admixture involving haplodiploid or sex-linked cytonuclear data.     

Field performance of engineered male mosquitoes

Dengue is the most medically important arthropod-borne viral disease, with 50-100 million cases reported annually worldwide. As no licensed vaccine or dedicated therapy exists for dengue, the most promising strategies to control the disease involve targeting the predominant mosquito vector, Aedes aegypti. However, the current methods to do this are inadequate. Various approaches involving genetically engineered mosquitoes have been proposed, including the release of transgenic sterile males. However, the ability of laboratory-reared, engineered male mosquitoes to effectively compete with wild males in terms of finding and mating with wild females, which is critical to the success of these strategies, has remained untested. We report data from the first open-field trial involving a strain of engineered mosquito. We demonstrated that genetically modified male mosquitoes, released across 10 hectares for a 4-week period, mated successfully with wild females and fertilized their eggs. These findings suggest the feasibility of this technology to control dengue by suppressing field populations of A. aegypti.     

LABORATORY EVOLUTION OF LONGEVITY AND REPRODUCTIVE FITNESS COMPONENTS IN MALE FRUIT FLIES: MATING ABILITY

Populations of Drosophila melanogaster that had been selected for divergent rates of senescence were compared with respect to age-specific male mating ability. The competitive mating ability of males from populations with delayed senescence was inferior to that of males from populations with higher rates of senescence when males were young. This relationship was reversed when males were older. For noncompetitive mating ability and for recovery of fertility after an exhaustive mating bout, there was no difference between populations with different rates of senescence when males were young. However when males were older, flies from populations selected for delayed senescence again had superior mating ability. Thus, rates of male reproductive senescence can be altered in predictable ways by natural selection. The results for the competitive mating tests are consistent with the hypothesis that antagonistic gene action between early- and late-life fitness components influences the evolution of senescence in these populations.     

Culturally transmitted paternity beliefs and the evolution of human mating behaviour

Recent anthropological findings document how certain lowland South American societies hold beliefs in 'partible paternity', which allow children to have more than one 'biological' father. This contrasts with Western beliefs in 'singular paternity', and biological reality, where children have just one father. Here, mathematical models are used to explore the coevolution of paternity beliefs and the genetic variation underlying human mating behaviour. A gene-culture coevolutionary model found that populations exposed to a range of selection regimes typically converge on one of two simultaneously stable equilibria; one where the population is monogamous and believes in singular paternity, and the other where the population is polygamous and believes in partible paternity. A second agent-based model, with alternative assumptions regarding the formation of mating consortships, broadly replicated this finding in populations with a strongly female-biased sex ratio, consistent with evidence for high adult male mortality in the region. This supports an evolutionary scenario in which ancestral South American populations with differing paternity beliefs were subject to divergent selection on genetically influenced mating behaviour, facilitated by a female-biased sex ratio, leading to the present-day associations of female control, partible paternity and polygamy in some societies, and male control, singular paternity and monogamy in others.     

Modelling Allee effects in a transgenic mosquito population during range expansion

ABSTRACT

Mosquitoes are vectors for many diseases that cause significant mortality and morbidity. As mosquito populations expand their range, they may undergo mate-finding Allee effects such that their ability to successfully reproduce becomes difficult at low population density. With new technology, creating target specific gene modification may be a viable method for mosquito population control. We develop a mathematical model to investigate the effects of releasing transgenic mosquitoes into newly established, low-density mosquito populations. Our model consists of two life stages (aquatic and adults), which are divided into three genetically distinct groups: heterogeneous and homogeneous transgenic that cause female infertility and a homogeneous wild type. We perform analytical and numerical analyses on the equilibria to determine the level of saturation needed to eliminate mosquitoes in a given area. This model demonstrates the potential for a gene drive system to reduce the spread of invading mosquito populations.     

Competitive exclusion and coexistence of pathogens in a homosexually-transmitted disease model

A sexually-transmitted disease model for two strains of pathogen in a one-sex, heterogeneously-mixing population has been studied completely by Jiang and Chai in (J Math Biol 56:373-390, 2008). In this paper, we give a analysis for a SIS STD with two competing strains, where populations are divided into three differential groups based on their susceptibility to two distinct pathogenic strains. We investigate the existence and stability of the boundary equilibria that characterizes competitive exclusion of the two competing strains; we also investigate the existence and stability of the positive coexistence equilibrium, which characterizes the possibility of coexistence of the two strains. We obtain sufficient and necessary conditions for the existence and global stability about these equilibria under some assumptions. We verify that there is a strong connection between the stability of the boundary equilibria and the existence of the coexistence equilibrium, that is, there exists a unique coexistence equilibrium if and only if the boundary equilibria both exist and have the same stability, the coexistence equilibrium is globally stable or unstable if and only if the two boundary equilibria are both unstable or both stable.     

Phenotypic structure and bifurcation behavior of population models

Discrete time models for density-regulated populations have been shown to exhibit periodic and chaotic motion in the absence of any external signal. We show how the genetic structure of a population can initiate bifurcations to periodic and chaotic trajectories. We investigate by simulation the dependence of this phenomenon on the strength of assortative mating, the level of heterozygosity, and the intensity of selection. The implications of internally generated chaos for population modeling are discussed.     

Exposure to parasites increases promiscuity in a freshwater snail

Under the Red Queen hypothesis, outcrossing can produce genetically variable progeny, which may be more resistant, on average, to locally adapted parasites. Mating with multiple partners may enhance this resistance by further increasing the genetic variation among offspring. We exposed Potamopyrgus antipodarum to the eggs of a sterilizing, trematode parasite and tested whether this altered mating behaviour. We found that exposure to parasites increased the number of snail mating pairs and the total number of different mating partners for both males and females. Thus, our results suggest that, in host populations under parasite-mediated selection, exposure to infective propagules increases the rate of mating and the number of mates.     

Equilibrium properties of a multi-locus, haploid-selection, symmetric-viability model

Under haploid selection, a multi-locus, diallelic, two-niche Levene (1953) model is studied. Viability coefficients with symmetrically opposing directional selection in each niche are assumed, and with a further simplification that the most and least favored haplotype in each niche shares no alleles in common, and that the selection coefficients monotonically increase or decrease with the number of alleles shared. This model always admits a fully polymorphic symmetric equilibrium, which may or may not be stable.We show that a stable symmetric equilibrium can become unstable via either a supercritical or subcritical pitchfork bifurcation. In the supercritical bifurcation, the symmetric equilibrium bifurcates to a pair of stable fully polymorphic asymmetric equilibria; in the subcritical bifurcation, the symmetric equilibrium bifurcates to a pair of unstable fully polymorphic asymmetric equilibria, which then connect to either another pair of stable fully polymorphic asymmetric equilibria through saddle-node bifurcations, or to a pair of monomorphic equilibria through transcritical bifurcations. As many as three fully polymorphic stable equilibria can coexist, and jump bifurcations can occur between these equilibria when model parameters are varied.In our Levene model, increasing recombination can act to either increase or decrease the genetic diversity of a population. By generating more hybrid offspring from the mating of purebreds, recombination can act to increase genetic diversity provided the symmetric equilibrium remains stable. But by destabilizing the symmetric equilibrium, recombination can ultimately act to decrease genetic diversity.     

Evolutionary consequences of ecological factors: pollinator reliability predicts mating‐system traits of a perennial plant

The reproductive‐assurance hypothesis predicts that mating‐system traits will evolve towards increased autonomous self‐pollination in plant populations experiencing unreliable pollinator service. We tested this long‐standing hypothesis by assessing geographic covariation among pollinator reliability, outcrossing rates, heterozygosity and relevant floral traits across populations of Dalechampia scandens in Costa Rica. Mean outcrossing rates ranged from 0.16 to 0.49 across four populations, and covaried with the average rates of pollen arrival on stigmas, a measure of pollinator reliability. Across populations, genetically based differences in herkogamy (anther–stigma distance) were associated with variation in stigmatic pollen loads, outcrossing rates and heterozygosity. These observations are consistent with the hypothesis that, when pollinators are unreliable, floral traits promoting autonomous selfing evolve as a mechanism of reproductive assurance. Extensive covariation between floral traits and mating system among closely related populations further suggests that floral traits influencing mating systems track variation in adaptive optima generated by variation in pollinator reliability.     

Modeling and dynamics of Wolbachia-infected male releases and mating competition on mosquito control

Despite centuries of continuous efforts, mosquito-borne diseases (MBDs) remain enormous health threat of human life worldwide. Lately, the USA government has approved an innovative technology of releasing Wolbachia-infected male mosquitoes to suppress the wild mosquito population. In this paper we first introduce a stage-structured model for natural mosquitos, then we establish a new model considering the releasing of Wolbachia-infected male mosquitoes and the mating competition between the natural male mosquitoes and infected males on the suppression of natural mosquitoes. Dynamical analysis of the two models, including the existence and local stability of the equilibria and bifurcation analysis, reveals the existence of a forward bifurcation or a backward bifurcation with multiple attractors. Moreover, globally dynamical properties are further explored by using Lyapunov function and theory of monotone operators, respectively. Our findings suggest that infected male augmentation itself cannot always guarantee the success of population eradication, but leads to three possible levels of population suppression, so we define the corresponding suppression rate and estimate the minimum release ratio for population eradication. Furthermore, we study how the release ratio of infected males and natural ones, mating competition, the rate of cytoplasmic incompatibility and the basic offspring number affect the suppression rate of natural mosquitoes. Our results show that the successful eradication relies on assessing the reproductive capacity of natural mosquitoes, a selection of suitable Wolbachia strains and an appropriate release amount of infected males. This study will be helpful for public health authorities in designing proper strategies to control vector mosquitoes and prevent the epidemics of MBDs.

     

The first releases of transgenic mosquitoes: an argument for the sterile insect technique

Potential applications for reducing transmission of mosquito-borne diseases by releasing genetically modified mosquitoes have been proposed, and mosquitoes are being created with such an application in mind in several laboratories. The use of the sterile insect technique (SIT) provides a safe programme in which production, release and mating competitiveness questions related to mass-reared genetically modified mosquitoes could be answered. It also provides a reversible effect that would be difficult to accomplish with gene introgression approaches. Could new technologies, including recombinant DNA techniques, have improved the success of previous mosquito releases? Criteria for an acceptable transgenic sterile mosquito are described, and the characteristics of radiation-induced sterility are compared with that of current transgenic approaches. We argue that SIT using transgenic material would provide an essentially safe and efficacious foundation for other possible approaches that are more ambitious.     

Evolution of mating preference and sexual dimorphism

A quantitative genetic model of the joint evolution of female mating preferences and sexual dimorphism in homologous characters of the sexes is described for polygamous species with no male parental effort, such that mating preferences are selectively neutral and evolve only by indirect selection on genetically correlated characters. The male character and the homologous female character are each under stabilizing natural selection toward an optimum phenotype. At an evolutionary equilibrium the female character under natural selection is at its optimum, whereas there is a line of possible equilibria between female mating preferences and the male character. The line of equilibria may be stable or unstable, depending on the intensity of natural selection, the type of mating preferences, and the inheritance of the characters. Various mechanisms for maladaptive evolution of mating preferences and sexual dimorphism are discussed.     

Mate‐sampling costs and sexy sons

Costly female mating preferences for purely Fisherian male traits (i.e. sexual ornaments that are genetically uncorrelated with inherent viability) are not expected to persist at equilibrium. The indirect benefit of producing ‘sexy sons’ (Fisher process) disappears: in some models, the male trait becomes fixed; in others, a range of male trait values persist, but a larger trait confers no net fitness advantage because it lowers survival. Insufficient indirect selection to counter the direct cost of producing fewer offspring means that preferences are lost. The only well‐cited exception assumes biased mutation on male traits. The above findings generally assume constant direct selection against female preferences (i.e. fixed costs). We show that if mate‐sampling costs are instead derived based on an explicit account of how females acquire mates, an initially costly mating preference can coevolve with a male trait so that both persist in the presence or absence of biased mutation. Our models predict that empirically detecting selection at equilibrium will be difficult, even if selection was responsible for the location of the current equilibrium. In general, it appears useful to integrate mate sampling theory with models of genetic consequences of mating preferences: being explicit about the process by which individuals select mates can alter equilibria.