BackgroundApricot (Prunus armeniaca L.) exhibits a gametophytic self-incompatibility (GSI) system and it is mostly considered as a self-incompatible species though numerous self-compatible exceptions occur. These are mainly linked to the mutated S C-haplotype carrying an insertion in the S-locus F-box gene that leads to a truncated protein. However, two S-locus unlinked pollen-part mutations (PPMs) termed m and m’ have also been reported to confer self-compatibility (SC) in the apricot cultivars ‘Canino’ and ‘Katy’, respectively. This work was aimed to explore whether other additional mutations might explain SC in apricot as well.ResultsA set of 67 cultivars/accessions with different geographic origins were analyzed by PCR-screening of the S- and M-loci genotypes, contrasting results with the available phenotype data. Up to 20 S-alleles, including 3 new ones, were detected and sequence analysis revealed interesting synonymies and homonymies in particular with S-alleles found in Chinese cultivars. Haplotype analysis performed by genotyping and determining linkage-phases of 7 SSR markers, showed that the m and m’ PPMs are linked to the same m 0−haplotype. Results indicate that m 0-haplotype is tightly associated with SC in apricot germplasm being quite frequent in Europe and North-America. However, its prevalence is lower than that for S C in terms of frequency and geographic distribution. Structures of 34 additional M-haplotypes were inferred and analyzed to depict phylogenetic relationships and M 1–2 was found to be the closest haplotype to m 0. Genotyping results showed that four cultivars classified as self-compatible do not have neither the S C- nor the m 0-haplotype.ConclusionsAccording to apricot germplasm S-genotyping, a loss of genetic diversity affecting the S-locus has been produced probably due to crop dissemination. Genotyping and phenotyping data support that self-(in)compatibility in apricot relies mainly on the S- but also on the M-locus. Regarding this latter, we have shown that the m 0-haplotype associated with SC is shared by ‘Canino’, ‘Katy’ and many other cultivars. Its origin is still unknown but phylogenetic analysis supports that m 0 arose later in time than S C from a widely distributed M-haplotype. Lastly, other mutants putatively carrying new mutations conferring SC have also been identified deserving future research.Electronic supplementary materialThe online version of this article (doi:10.1186/s12870-017-1027-1) contains supplementary material, which is available to authorized users.

Abstract. In the paper observability problems are considered in basic dynamic evolutionary models for sexual and asexual populations. Observability means that from the (partial) knowledge of certain phenotypic characteristics the whole evolutionary process can be uniquely recovered. Sufficient conditions are given to guarantee observability for both sexual and asexual populations near an evolutionarily stable state.

In ecology, if the considered area or space is large, the spatial distribution of individuals of a given plant species is never homogeneous; plants form different patches. The homogeneity change in space or in time (in particular, the related change-point problem) is an important research subject in mathematical statistics. In the paper, for a given data system along a straight line, two areas are considered, where the data of each area come from different discrete distributions, with unknown parameters. In the paper a method is presented for the estimation of the distribution change-point between both areas and an estimate is given for the distributions separated by the obtained change-point. The solution of this problem will be based on the maximum likelihood method. Furthermore, based on an adaptation of the well-known bootstrap resampling, a method for the estimation of the so-called change-interval is also given. The latter approach is very general, since it not only applies in the case of the maximum-likelihood estimation of the change-point, but it can be also used starting from any other change-point estimation known in the ecological literature. The proposed model is validated against typical ecological situations, providing at the same time a verification of the applied algorithms.

Linear verticum-type control and observation systems have been introduced for modelling certain industrial systems, consisting of subsystems, vertically connected by certain state variables. Recently the concept of verticum-type observation systems and the corresponding observability condition have been extended by the authors to the nonlinear case. In the present paper the general concept of a nonlinear verticum-type control system is introduced, and a sufficient condition for local controllability to equilibrium is obtained. In addition to a usual linearization, the basic idea is a decomposition of the control of the whole system into the control of the subsystems. Starting from the integrated pest control model of Rafikov and E.H. Limeira (2012) a nonlinear verticum-type model has been set up an equilibrium control is obtained. Furthermore, a corresponding bioeconomical problem is solved minimizing the total cost of integrated pest control (combining chemical control with a biological one).

The effect of radiation on a cell population is described by a two-dimensional nonlinear system of differential equations. If the radiation rate is not too high, the system is known to have an asymptotically stable equilibrium. First, for the monitoring of this effect, the concept of observability is applied. For the case when the total number of cells is observed, without distinction between healthy and affected cells, a so-called observer system is constructed, which, at least near the equilibrium state, makes it possible to recover the dynamics of both the healthy and the affected cells, from the observation of the total number of cells without distinction.Results of simulations with illustrative data are also presented. If we want to control the system into a required new equilibrium state, and maintain this new equilibrium by a constant control, a technique of theory of optimal control can be applied to construct a feedback control system.

Plum pox virus (PPV) is the most important limiting factor for apricot (Prunus armeniaca L.) production worldwide, and development of resistant cultivars has been proven to be the best solution in the long-term. However, just like in other woody species, apricot breeding is highly time and space demanding, and this is particularly true for PPV resistance phenotyping. Therefore, marker-assisted selection (MAS) may be very helpful to speed up breeding programs. Tightly linked ParPMC1 and ParPMC2, meprin and TRAF-C homology (MATH)-domain-containing genes have been proposed as host susceptibility genes required for PPV infection. Contribution of additional genes to PPV resistance cannot be discarded, but all available studies undoubtedly show a strong correlation between ParPMC2-resistant alleles (ParPMC2res) and PPV resistance. The ParPMC2res allele was shown to carry a 5-bp deletion (ParPMC2-del) within the second exon that has been characterized as a molecular marker suitable for MAS (PMC2). Based on this finding, we propose here a method for PPV resistance selection in apricot by combining high-throughput DNA extraction of 384 samples in 2 working days and the allele-specific genotyping of PMC2 on agarose gel. Moreover, the PMC2 genotype has been determined by PCR or by using whole-genome sequences (WGS) in 175 apricot accessions. These results were complemented with phenotypic and/or genotypic data available in the literature to reach a total of 325 apricot accessions. As a whole, we conclude that this is a time-efficient, cost-effective and straightforward method for PPV resistance screening that can be highly useful for apricot breeding programs.

The problem of the possibility to recover the time-dependent state of a whole population system out of the observation of certain components has been studied in earlier publications, in terms of the observability concept of mathematical systems theory. In the present note a method is proposed to effectively calculate the state process. For an illustration an observer system for a simple food web is numerically constructed.

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