Traveling wave solutions in a plant population model with a seed bank

We propose an integro-difference equation model to predict the spatial spread of a plant population with a seed bank. The formulation of the model consists of a nonmonotone convolution integral operator describing the recruitment and seed dispersal and a linear contraction operator addressing the effect of the seed bank. The recursion operator of the model is noncompact, which poses a challenge to establishing the existence of traveling wave solutions. We show that the model has a spreading speed, and prove that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions by using an asymptotic fixed point theorem. Our numerical simulations show that the seed bank has the stabilizing effect on the spatial patterns of traveling wave solutions.     

Population dynamic models of the spread of Wolbachia

Wolbachia are endosymbionts that are found in many insect species and can spread rapidly when introduced into a naive host population. Most Wolbachia spread when their infection frequency exceeds a threshold normally calculated using purely population genetic models. However, spread may also depend on the population dynamics of the insect host. We develop models to explore interactions between host population dynamics and Wolbachia infection frequency for an age-structured insect population regulated by larval density dependence. We first derive a new expression for the threshold frequency that extends existing theory to incorporate important details of the insect's life history. In the presence of immigration and emigration, the threshold also depends on the form of density-dependent regulation. We show how the type of immigration (constant or pulsed) and the temporal dynamics of the host population can strongly affect the spread of Wolbachia. The results help understand the natural dynamics of Wolbachia infections and aid the design of programs to introduce Wolbachia to control insects that are disease vectors or pests.     

Comparing the plant virus spread patterns of non-persistently transmitted virus and persistently transmitted virus using an individual-based model

To compare the spread patterns between two types of plant viruses, non-persistent virus (NPV) and persistent virus (PV), we developed a spatially-explicit individual-based model. Our probability-based model is driven by the actions of insect vectors that are affected by interactions with host plants and plant viruses, considering both biological and behavioral components of their relationship. As a model system, we used potato virus y and potato leafroll virus, respectively for NPV and PV, potato for host plant, and Myzus persicae for the insect vector; empirical results from previous studies were acquired and adjusted to be used as our parameter values. Our simulation results showed that initial infection of PV in the field resulted in over 1.3 times greater number of insect vectors while causing approximately 7 times greater number of virus-infected plants compared to NPV by the end of simulation. Furthermore, spatial analysis showed that PV-infected plants showed greater aggregation in the field, forming larger patches compared to NPV-infected plants. Our results demonstrated the importance of host plant and insect vector manipulation by plant viruses as well as biological properties such as infectious period in the insect on the difference in overall spread pattern.     

Spatial waves of advance with bistable dynamics: cytoplasmic and genetic analogues of Allee effects

Unlike unconditionally advantageous "Fisherian" variants that tend to spread throughout a species range once introduced anywhere, "bistable" variants, such as chromosome translocations, have two alternative stable frequencies, absence and (near) fixation. Analogous to populations with Allee effects, bistable variants tend to increase locally only once they become sufficiently common, and their spread depends on their rate of increase averaged over all frequencies. Several proposed manipulations of insect populations, such as using Wolbachia or "engineered underdominance" to suppress vector-borne diseases, produce bistable rather than Fisherian dynamics. We synthesize and extend theoretical analyses concerning three features of their spatial behavior: rate of spread, conditions to initiate spread from a localized introduction, and wave stopping caused by variation in population densities or dispersal rates. Unlike Fisherian variants, bistable variants tend to spread spatially only for particular parameter combinations and initial conditions. Wave initiation requires introduction over an extended region, while subsequent spatial spread is slower than for Fisherian waves and can easily be halted by local spatial inhomogeneities. We present several new results, including robust sufficient conditions to initiate (and stop) spread, using a one-parameter cubic approximation applicable to several models. The results have both basic and applied implications.     

Evolution of searching and life history characteristics in individual-based models of host-parasitoid-microbe associations

In this paper we develop a novel discrete, individual-based mathematical model of the evolution of life history, dispersal and other behavioural characteristics in insect host-parasitoid-microbe associations, and use it to investigate their evolutionary dynamics. For any individual characteristic the model begins with an even, rectangular distribution of characteristic values. Selection is then allowed to act, and the change in the distribution of the characteristic values is observed. Evolutionary change in the population variance of the characteristic value is also observed, since we would expect this to decline under selection in most cases. The paper, therefore, introduces a general framework for modeling problems of evolution in stochastic, spatially structured environments, where movement and dispersal are under selection. The model then extends this approach to include the sex-distorting bacterium Wolbachia in order to investigate aspects of its horizontal and vertical transmission under different levels of superparasitism by parasitoids. The model also includes a neutral genetic marker, in order to be able to detect changes in phenotype frequency caused by genetic drift, as well as a simplified simulation of sexual reproduction so as to allow the possibility of recombination between genotypes. Key results from the model simulations show that: (i) the refractory time after oviposition affects the value of superparasitism, with short refractory times favouring high rates of superparasitism; (ii) variable levels of superparasitism do not affect the stable proportion of the population of parasitoids infected with Wolbachia, but this is achieved by different evolutionary pathways under low and high superparasitism, respectively. In the case of low superparasitism Wolbachia spreads mainly by vertical transmission, leading to population replacement, whereas when superparasitism rates are high there is significant horizontal transfer.     

Cytoplasmic Incompatibility in Drosophila Simulans: Dynamics and Parameter Estimates from Natural Populations

In Drosophila simulans, cytoplasmically transmitted Wolbachia microbes cause reduced egg hatch when infected males mate with uninfected females. A Wolbachia infection and an associated mtDNA variant have spread northward through California since 1986. PCR assays show that Wolbachia infection is prevalent throughout the continental US and Central and South America, but some lines from Florida and Ecuador that are PCR-positive for Wolbachia do not cause incompatibility. We estimate from natural populations infection frequencies and the transmission and incompatibility parameter values that affect the spread of the infection. On average, infected females from nature produce 3-4% uninfected ova. Infected females with relatively low fidelity of maternal transmission show partial incompatibility with very young infected laboratory males. Nevertheless, crosses between infected flies in nature produce egg-hatch rates indistinguishable from those produced by crosses between uninfected individuals. Incompatible crosses in nature produce hatch rates 30-70% as high as those from compatible crosses. Wild-caught infected and uninfected females are equally fecund in the laboratory. Incompatibility decreases with male age, and age-specific incompatibility levels suggest that males mating in nature may often be 2 or 3 weeks old. Our parameter estimates accurately predict the frequency of Wolbachia infection in California populations.     

Generation of a novel Wolbachia infection in Aedes albopictus (Asian tiger mosquito) via embryonic microinjection

Genetic strategies that reduce or block pathogen transmission by mosquitoes are being investigated as a means to augment current control measures. Strategies of vector suppression and replacement are based upon intracellular Wolbachia bacteria, which occur naturally in many insect populations. Maternally inherited Wolbachia have evolved diverse mechanisms to manipulate host insect reproduction and promote infection invasion. One mechanism is cytoplasmic incompatibility (CI) through which Wolbachia promotes infection spread by effectively sterilizing uninfected females. In a prior field test, releases of Wolbachia-infected males were used to suppress a field population of Culex pipiens. An additional strategy would employ Wolbachia as a vehicle to drive desired transgenes into vector populations (population replacement). Wolbachia-based population suppression and population replacement strategies require an ability to generate artificial Wolbachia associations in mosquitoes. Here, we demonstrate a technique for transferring Wolbachia (transfection) in a medically important mosquito species: Aedes albopictus (Asian tiger mosquito). Microinjection was used to transfer embryo cytoplasm from a double-infected Ae. albopictus line into an aposymbiotic line. The resulting mosquito line is single-infected with the wAlbB Wolbachia type. The artificially generated infection type is not known to occur naturally and displays a new CI crossing type and the first known example of bidirectional CI in Aedes mosquitoes. We discuss the results in relation to applied mosquito control strategies and the evolution of Wolbachia infections in Ae. albopictus.     

Wolbachia and cytoplasmic incompatibility in the California Culex pipiens mosquito species complex: parameter estimates and infection dynamics in natural populations

Before maternally inherited bacterial symbionts like Wolbachia, which cause cytoplasmic incompatibility (CI; reduced hatch rate) when infected males mate with uninfected females, can be used in a program to control vector-borne diseases it is essential to understand their dynamics of infection in natural arthropod vector populations. Our study had four goals: (1) quantify the number of Wolbachia strains circulating in the California Culex pipiens species complex, (2) investigate Wolbachia infection frequencies and distribution in natural California populations, (3) estimate the parameters that govern Wolbachia spread among Cx. pipiens under laboratory and field conditions, and (4) use these values to estimate equilibrium levels and compare predicted infection prevalence levels to those observed in nature. Strain-specific PCR, wsp gene sequencing, and crossing experiments indicated that a single Wolbachia strain infects Californian Cx. pipiens. Infection frequency was near or at fixation in all populations sampled for 2 years along a >1000-km north-south transect. The combined statewide infection frequency was 99.4%. Incompatible crosses were 100% sterile under laboratory and field conditions. Sterility decreased negligibly with male age in the laboratory. Infection had no significant effect on female fecundity under laboratory or field conditions. Vertical transmission was >99% in the laboratory and approximately 98.6% in the field. Using field data, models predicted that Wolbachia will spread to fixation if infection exceeds an unstable equilibrium point above 1.4%. Our estimates accurately predicted infection frequencies in natural populations. If certain technical hurdles can be overcome, our data indicate that Wolbachia can invade vector populations as part of an applied transgenic strategy for vector-borne disease reduction.     

DNA barcoding cannot reliably identify species of the blowfly genus Protocalliphora (Diptera: Calliphoridae)

In DNA barcoding, a short standardized DNA sequence is used to assign unknown individuals to species and aid in the discovery of new species. A fragment of the mitochondrial gene cytochrome c oxidase subunit 1 is emerging as the standard barcode region for animals. However, patterns of mitochondrial variability can be confounded by the spread of maternally transmitted bacteria that cosegregate with mitochondria. Here, we investigated the performance of barcoding in a sample comprising 12 species of the blow fly genus Protocalliphora, known to be infected with the endosymbiotic bacteria Wolbachia. We found that the barcoding approach showed very limited success: assignment of unknown individuals to species is impossible for 60% of the species, while using the technique to identify new species would underestimate the species number in the genus by 75%. This very low success of the barcoding approach is due to the non-monophyly of many of the species at the mitochondrial level. We even observed individuals from four different species with identical barcodes, which is, to our knowledge, the most extensive case of mtDNA haplotype sharing yet described. The pattern of Wolbachia infection strongly suggests that the lack of within-species monophyly results from introgressive hybridization associated with Wolbachia infection. Given that Wolbachia is known to infect between 15 and 75% of insect species, we conclude that identification at the species level based on mitochondrial sequence might not be possible for many insects. However, given that Wolbachia-associated mtDNA introgression is probably limited to very closely related species, identification at the genus level should remain possible.     

The effect of Wolbachia versus genetic incompatibilities on reinforcement and speciation

Wolbachia is a widespread group of intracellular bacteria commonly found in arthropods. In many insect species, Wolbachia induce a cytoplasmic mating incompatibility (CI). If different Wolbachia infections occur in the same host species, bidirectional CI is often induced. Bidirectional CI acts as a postzygotic isolation mechanism if parapatric host populations are infected with different Wolbachia strains. Therefore, it has been suggested that Wolbachia could promote speciation in their hosts. In this article we investigate theoretically whether Wolbachia-induced bidirectional CI selects for premating isolation and therefore reinforces genetic divergence between parapatric host populations. To achieve this we combined models for Wolbachia dynamics with a well-studied reinforcement model. This new model allows us to compare the effect of bidirectional CI on the evolution of female mating preferences with a situation in which postzygotic isolation is caused by nuclear genetic incompatibilities (NI). We distinguish between nuclear incompatibilities caused by two loci with epistatic interactions, and a single locus with incompatibility among heterozygotes in the diploid phase. Our main findings are: (1) bidirectional CI and single locus NI select for premating isolation with a higher speed and for a wider parameter range than epistatic NI; (2) under certain parameter values, runaway sexual selection leads to the increase of an introduced female preference allele and fixation of its preferred male trait allele in both populations, whereas under others it leads to divergence in the two populations in preference and trait alleles; and (3) bidirectional CI and single locus NI can stably persist up to migration rates that are two times higher than seen for epistatic NI. The latter finding is important because the speed with which mutants at the preference locus spread increases exponentially with the migration rate. In summary, our results show that bidirectional CI selects for rapid premating isolation and so generally support the view that Wolbachia can promote speciation in their hosts.     

Comparing short and long‐distance dispersal: modelling and field case studies

Dispersal is a factor of great importance in determining a species spatial distribution. Short distance dispersal (SDD) and long distance dispersal (LDD) strategies yield very different spatial distributions. In this paper we compare spatial spread patterns from SDD and LDD simulations, contrast them with patterns from field data, and assess the significance of biological and population traits. Simulated SDD spread using an exponential function generates a single circular patch with a well‐defined invasion front showing a travelling‐wave structure. The invasive spread is relatively slow as it is restricted to reproductive individuals occupying the outer zone of the circular patch. As a consequence of this dispersal dynamics, spread is slower than spread generated by LDD. In contrast, the early and fast invasion of the entire habitat mediated by power law LDD not only involves a significantly greater invasion velocity, but also an entirely different habitat occupation. As newly dispersed individuals soon reach very distant portions of the habitat as well as the vicinity of the original dispersal focus, new growing patches are generated while the main patch increases its own growth absorbing the closest patches. As a consequence of both dispersal and lower density dependence, growth of the occupied area is much faster than with SDD. SDD and LDD also differ regarding pattern generation. With SDD, fractal patterns appear only in the border of the invasion front in SDD when competitive interaction with residents is included. In contrast, LDD patterns show fractality both in the spatial arrangements of patches as well as in patch borders. Moreover, values of border fractal dimension inform on the dispersal process in relation with habitat heterogeneity. The distribution of patch size is also scale‐free, showing two power laws characteristic of small and large patch sizes directly arising from the dispersal and reproductive dynamics. Ecological factors like habitat heterogeneity are relevant for dispersal, although its importance is greater for SDD, lowering the invasion velocity. Among the life history traits considered, adult mortality, the juvenile bank and mean dispersal distance are the most relevant for SDD. For LDD, habitat heterogeneity and changes in life history traits are not so relevant, causing minor changes in the values of the scale‐free parameters. Our work on short and long distance dispersal shows novel theoretical differences between SDD and LDD in invasive systems (mechanisms of pattern formation, fractal and scaling properties, relevance of different life history traits and habitat variables) that correspond closely with field examples and were not analyzed, at least in this degree of detail, by the previously existing models.     

Gypsy moth invasion in North America: A simulation study of the spatial pattern and the rate of spread

Gypsy moth is regarded as one of the top most harmful invasive species. Its invasion in the northeastern US has led to widespread forest defoliation, wildlife disruption and even a change in biogeochemical conditions over the area of 10⁶ km². Spread of gypsy moth has a few distinct features such as a patchy spatial distribution of the gypsy moth population, which is largely uncorrelated to the environmental heterogeneity, and a high variability (almost over an order of magnitude) in the spread rates. These features are usually explained by human-assisted dispersal, e.g. when masses of gypsy moth eggs are inadvertently transported by cars and vehicles. This theory, however, somewhat disagrees with the existence of the strong Allee effect that tends to wipe out small new colonies. In this paper, we suggest an alternative explanation that the patchy structure can result from the interplay between two natural factors such as wind dispersal and viral infection. In order to check this hypothesis, we describe the gypsy moth spread with a diffusive SI model and study its properties by means of extensive computer simulations. Interestingly, in a certain parameter range our model shows formation of spatial patterns that are qualitatively similar to those observed in the field. To find out the relevant parameter range, we make a careful review of available literature sources. For biologically meaningful parameter values, we then show that the rates of gypsy moth spread predicted by our model are in good agreement with the lower band of the rates observed in nature.     

The role of model selection in describing stochastic ecological processes

A great deal of variation in ecological processes can be generated through variation in environmental conditions. Models describing the underlying processes should be able to account for this variation. However, models may not be flexible enough, so that different realizations of a process may be better described by different models. This may lead to uncertainty in model selection.
Here we examine the question of whether two empirical models can provide consistent fits to different realizations of a process affected by environmental variation. We further examine the sensitivity of the model predictions to the amount of data available and the selection of the model. To study this, we simulated pollen dispersal patterns under varying wind conditions and then investigated whether the datasets consistently supported the same model. The role of the model selection and the impact of the spatial range over which the dispersal distances were observed were assessed by comparing model predictions at long dispersal distances.
There was no consistent pattern of one model providing a better fit than the other across simulations. The model providing better fit varied depending on the range of distances over which the dispersal patterns were observed, and on the amount of long-distance dispersal. The model predictions were found to be very sensitive to the selection of the model.
The variation between datasets produced with the same underlying mechanisms cannot be easily described using one model, which also limits our ability to reliably predict the underlying process. Therefore, the amount of information about a model choice provided by an individual field study may be rather limited. If we are to understand processes that are affected by environmental variation then we have to observe the range of possible outcomes of the processes under varying spatio-temporal conditions.     

Evolutionarily stable infection by a male-killing endosymbiont in Drosophila innubila: molecular evidence from the host and parasite genomes

Maternally inherited microbes that spread via male-killing are common pathogens of insects, yet very little is known about the evolutionary duration of these associations. The few examples to date indicate very recent, and thus potentially transient, infections. A male-killing strain of Wolbachia has recently been discovered in natural populations of Drosophila innubila. The population-level effects of this infection are significant: approximately 35% of females are infected, infected females produce very strongly female-biased sex ratios, and the resulting population-level sex ratio is significantly female biased. Using data on infection prevalence and Wolbachia transmission rates, infected cytoplasmic lineages are estimated to experience a approximately 5% selective advantage relative to uninfected lineages. The evolutionary history of this infection was explored by surveying patterns of polymorphism in both the host and parasite genomes, comparing the Wolbachia wsp gene and the host mtDNA COI gene to five host nuclear genes. Molecular data suggest that this male-killing infection is evolutionarily old, a conclusion supported with a simple model of parasite and mtDNA transmission dynamics. Despite a large effective population size of the host species and strong selection to evolve resistance, the D. innubila-Wolbachia association is likely at a stable equilibrium that is maintained by imperfect maternal transmission of the bacteria rather than partial resistance in the host species.     

Intraspecific specialization of the generalist parasitoid Cotesia sesamiae revealed by polyDNAvirus polymorphism and associated with different Wolbachia infection

As a result of an intense host-parasite evolutionary arms race, parasitic wasps frequently display high levels of specialization on very few host species. For instance, in braconid wasps very few generalist species have been described. However, within this family, Cotesia sesamiae is a generalist species that is widespread in sub-Saharan Africa and develops on several lepidopteran hosts. In this study, we tested the hypothesis that C.?sesamiae may be a cryptic specialist when examined at the intraspecific level. We sequenced exon 2 of CrV1, a gene of the symbiotic polyDNAvirus that is integrated into the wasp genome and is associated with host immune suppression. We found that CrV1 genotype was more closely associated with the host in which the parasitoid developed than any abiotic environmental factor tested. We also tested a correlation between CrV1 genotype and an infection with Wolbachia bacteria, which are known for their ability to induce reproductive isolation. The Wolbachia bacteria infection polymorphism was also found as a major factor explaining the genetic structure of CrV1, and, in addition, the best model explaining CrV1 genetic structure involved an interaction between Wolbachia infection and host species. We suggest that Wolbachia could act as an agent capable of maintaining advantageous alleles for host specialization in different populations of C.?sesamiae. This mechanism could be applicable to other insect models because of the high prevalence of Wolbachia in insects.     

Efficient parametric analysis of the chemical master equation through model order reduction

ABSTRACT: BACKGROUND: Stochastic biochemical reaction networks are commonly modelled by the chemical master equation, and can be simulated as first order linear differential equations through a finite state projection. Due to the very high state space dimension of these equations, numerical simulations are computationally expensive. This is a particular problem for analysis tasks requiring repeated simulations for different parameter values. Such tasks are computationally expensive to the point of infeasibility with the chemical master equation. RESULTS: In this article, we apply parametric model order reduction techniques in order to construct accurate low-dimensional parametric models of the chemical master equation. These surrogate models can be used in various parametric analysis task such as identifiability analysis, parameter estimation, or sensitivity analysis. As biological examples, we consider two models for gene regulation networks, a bistable switch and a network displaying stochastic oscillations. CONCLUSIONS: The results show that the parametric model reduction yields efficient models of stochastic biochemical reaction networks, and that these models can be useful for systems biology applications involving parametric analysis problems such as parameter exploration, optimization, estimation or sensitivity analysis.     

Wolbachia in a major African crop pest increases susceptibility to viral disease rather than protects

Wolbachia are common vertically transmitted endosymbiotic bacteria found in < 70% of insect species. They have generated considerable recent interest due to the capacity of some strains to protect their insect hosts against viruses and the potential for this to reduce vector competence of a range of human diseases, including dengue. In contrast, here we provide data from field populations of a major crop pest, African armyworm (Spodoptera exempta), which show that the prevalence and intensity of infection with a nucleopolydrovirus (SpexNPV) is positively associated with infection with three strains of Wolbachia. We also use laboratory bioassays to demonstrate that infection with one of these strains, a male-killer, increases host mortality due to SpexNPV by 6-14 times. These findings suggest that rather than protecting their lepidopteran host from viral infection, Wolbachia instead make them more susceptible. This finding potentially has implications for the biological control of other insect crop pests.     

How many species are infected with Wolbachia? – a statistical analysis of current data

Wolbachia are intracellular bacteria found in many species of arthropods and nematodes. They manipulate the reproduction of their arthropod hosts in various ways, may play a role in host speciation and have potential applications in biological pest control. Estimates suggest that at least 20% of all insect species are infected with Wolbachia . These estimates result from several Wolbachia screenings in which numerous species were tested for infection; however, tests were mostly performed on only one to two individuals per species. The actual percent of species infected will depend on the distribution of infection frequencies among species. We present a meta-analysis that estimates percentage of infected species based on data on the distribution of infection levels among species. We used a beta-binomial model that describes the distribution of infection frequencies of Wolbachia , shedding light on the overall infection rate as well as on the infection frequency within species. Our main findings are that (1) the proportion of Wolbachia -infected species is estimated to be 66%, and that (2) within species the infection frequency follows a 'most-or-few' infection pattern in a sense that the Wolbachia infection frequency within one species is typically either very high (>90%) or very low (<10%).     

Stochastic spread of Wolbachia

Wolbachia are very common, maternally transmitted endosymbionts of insects. They often spread by a mechanism termed cytoplasmic incompatibility (CI) that involves reduced egg hatch when Wolbachia-free ova are fertilized by sperm from Wolbachia-infected males. Because the progeny of Wolbachia-infected females generally do not suffer CI-induced mortality, infected females are often at a reproductive advantage in polymorphic populations. Deterministic models show that Wolbachia that impose no costs on their hosts and have perfect maternal transmission will spread from arbitrarily low frequencies (though initially very slowly); otherwise, there will be a threshold frequency below which Wolbachia frequencies decline to extinction and above which they increase to fixation or a high stable equilibrium. Stochastic theory was used to calculate the probability of fixation in populations of different size for arbitrary current frequencies of Wolbachia, with special attention paid to the case of spread after the arrival of a single infected female. Exact results are given based on a Moran process that assumes a specific demographic model, and approximate results are obtained using the more general Wright-Fisher theory. A new analytical approximation for the probability of fixation is derived, which performs well for small population sizes. The significance of stochastic effects in the natural spread of Wolbachia and their relevance to the use of Wolbachia as a drive mechanism in vector and pest management are discussed.     

Joint evolution of dispersal and inbreeding load

Inbreeding avoidance is often invoked to explain observed patterns of dispersal, and theoretical models indeed point to a possibly important role. However, while inbreeding load is usually assumed constant in these models, it is actually bound to vary dynamically under the combined influences of mutation, drift, and selection and thus to evolve jointly with dispersal. Here we report the results of individual-based stochastic simulations allowing such a joint evolution. We show that strongly deleterious mutations should play no significant role, owing to the low genomic mutation rate for such mutations. Mildly deleterious mutations, by contrast, may create enough heterosis to affect the evolution of dispersal as an inbreeding-avoidance mechanism, but only provided that they are also strongly recessive. If slightly recessive, they will spread among demes and accumulate at the metapopulation level, thus contributing to mutational load, but not to heterosis. The resulting loss of viability may then combine with demographic stochasticity to promote population fluctuations, which foster indirect incentives for dispersal. Our simulations suggest that, under biologically realistic parameter values, deleterious mutations have a limited impact on the evolution of dispersal, which on average exceeds by only one-third the values expected from kin-competition avoidance.