We reconsider the deterministic haploid mutation-selection equation with two types. This is an ordinary differential equation that describes the type distribution (forward in time) in a population of infinite size. This paper establishes ancestral (random) structures inherent in this deterministic model. In a first step, we obtain a representation of the deterministic equation's solution (and, in particular, of its equilibria) in terms of an ancestral process called the killed ancestral selection graph. This representation allows one to understand the bifurcations related to the error threshold phenomenon from a genealogical point of view. Next, we characterise the ancestral type distribution by means of the pruned lookdown ancestral selection graph and study its properties at equilibrium. We also provide an alternative characterisation in terms of a piecewise-deterministic Markov process. Throughout, emphasis is on the underlying dualities as well as on explicit results.

We study the common ancestor type distribution in a 2-type Moran model with population size N , mutation and selection, and in the deterministic limit regime arising in the former when N tends to infinity, without any rescaling of parameters or time. In the finite case, we express the common ancestor type distribution as a weighted sum of combinatorial terms, and we show that the latter converges to an explicit function. Next, we recover the previous results through pruning of the ancestral selection graph (ASG). The notions of relevant ASG, finite and asymptotic pruned lookdown ASG permit to achieve this task.

North African knapweed (Centaurea diluta Aiton) is an annual weed that is widespread in southern Spain and is of increasing concern in dryland cropping systems. Despite its expanding range in Spain, there is limited information on the emergence timing and pattern of this species, knowledge of which is critical for developing more timely and effective management strategies. Therefore, there is a need to develop simple and reliable models to predict the timing and emergence of this annual weed under dryland conditions. A multi-location field experiment was established across Spain in 2016 to 2017 to assess the emergence of C. diluta. At each of 11 locations, seeds were sown in the fall, and emergence was recorded. Overall emergence averaged 39% in the first year across all sites and 11% in the second year. In both years, the main emergence flush occurred at the beginning of the growing season. A three-parameter Weibull function best described seedling emergence of C. diluta. Emergence models were developed based on thermal time (TT) and hydrothermal time (HTT) and showed high predictability, as evidenced by root mean-square error prediction values of 10.8 and 10.7, respectively. Three cardinal points were established for TT and HHT at 0.5, 10, and 35 C for base, optimal, and ceiling temperatures, respectively, while base water potential was estimated at −0.5 MPa.

The ubiquity of genetic mixing in nature has eluded unified explanation since the time of Darwin. Conditions that promote the evolution of genetic mixing (recombination) are fairly well understood: it is favored when genomes tend to contain more selectively mismatched combinations of alleles than can be explained by chance alone. Yet, while a variety of theoretical approaches have been put forth to explain why such conditions would have an overarching tendency to prevail in natural populations, each has turned out to be of limited scope and applicability. In our two-part study, we show that, simply and surprisingly, the action of natural selection acting on standing heritable variation creates conditions favoring the evolution of recombination. In this paper, we focus on the mean selective advantage created by recombination between individuals from the same population. We find that the mean selective advantages of recombinants and recombination are non-negative, in expectation, independently of how genic fitnesses in the standing variation are distributed. We further find that the expected asymptotic frequency of a recombination-competent modifier is effectively equal to the probability that the fittest possible genotype is a virtual recombinant; remarkably, expected asymptotic modifier frequency is independent of the strength of selection. Taken together, our findings indicate that the evolution of recombination should be promoted in expectation wherever natural selection is operating.