Coevolutionary interactions between a haploid species and a diploid species

We investigate a general model describing coevolutionary interaction between a haploid population and a diploid population, each with two alleles at a single locus. Both species are allowed to evolve, with the fitness of the genotypes of each species assumed to depend linearly on the frequencies of the genotypes of the other species. We explore the resulting outcomes of these interactions, in particular determining the location of equilibria under various conditions. The coevolution here is much more complex than that between two haploid populations and allows for the possibility of two polymorphic equilibria. To allow for further analysis, we construct a semi-symmetric model. The variety of outcomes possible even in this second model provides support for the geographic mosaic theory of coevolution by suggesting the possibility of small local populations coevolving to very different outcomes, leading to a shifting geographic mosaic as neighboring populations interact with each other through migration.     

Protected polymorphism in the two-locus haploid model with unpredictable fitnesses

 In an unpredictable environment, the distributions of alleles from which polymorphism can be maintained forever belong to a certain set, the C-viability kernel. Such a set is calculated in the two-locus haploid model, as well as the corresponding fitnesses at any time which make this maintenance possible. The dependence of the C-viability kernel on the set U of admissible fitnesses and on the recombination rate r is studied. Notably, the C-viability kernel varies rapidly in the neighborhood of equal fitness of AB and ab; it becomes empty when ab has a fitness below a certain function, which is delineated, of the recombination rate. The properties of the two-locus model under constraints, out of equilibrium and with unpredictable selection are thus presented. Received: 20 May 1999     

Two SIS epidemiologic models with delays

 The SIS epidemiologic models have a delay corresponding to the infectious period, and disease-related deaths, so that the population size is variable. The population dynamics structures are either logistic or recruitment with natural deaths. Here the thresholds and equilibria are determined, and stabilities are examined. In a similar SIS model with exponential population dynamics, the delay destabilized the endemic equilibrium and led to periodic solutions. In the model with logistic dynamics, periodic solutions in the infectious fraction can occur as the population approaches extinction for a small set of parameter values. Received: 10 January 1997 / 18 November 1997     

Convergence in almost periodic Fisher and Kolmogorov models

 We study convergence of positive solutions for almost periodic reaction diffusion equations of Fisher or Kolmogorov type. It is proved that under suitable conditions every positive solution is asymptotically almost periodic. Moreover, all positive almost periodic solutions are harmonic and uniformly stable, and if one of them is spatially homogeneous, then so are others. The existence of an almost periodic global attractor is also discussed. Received: 11 November 1996 / Revised version: 8 January 1998     

Exploring phenotype space through neutral evolution

RNA secondary-structure folding algorithms predict the existence of connected networks of RNA sequences with identical secondary structures. Fitness landscapes that are based on the mapping between RNA sequence and RNA secondary structure hence have many neutral paths. A neutral walk on these fitness landscapes gives access to a virtually unlimited number of secondary structures that are a single point mutation from the neutral path. This shows that neutral evolution explores phenotype space and can play a role in adaptation. Received: 23 December 1995 / Accepted: 17 March 1996     

Identification of sequence variations by PCR-RFLP and its application to the evaluation of cpDNA diversity in wild and cultivated soybeans

The diversity and maternal lineage in wild and cultivated soybeans have previously been assayed using restriction fragment length polymorphism (RFLP) and sequencing analyses of chloroplast DNA (cpDNA). Here we describe a method based on PCR-RFLP for the identification of nucleotides at four mutation sites in non-coding regions of cpDNA. Of the four sites, two were located in restriction enzyme sites and two were not. For the latter two sites, new primers were designed to artificially create restriction sites that spanned them. The PCR-RFLP method enabled us to identify nucleotides at each of the four mutation sites easily and reliably. Fifty-seven wild and sixty-seven cultivated soybeans of different origins and different cpDNA types (types I, II, and III) were assayed. All of the samples tested could be classified into four haplotypes. All of the type-I and -II accessions had the same nucleotides at each of the four mutation sites, while all of the type-III accessions, except for 3 wild ones, had nucleotides that were different from those of types I and II. A sequencing analysis revealed that the 3 wild accessions possessed other single-base variations in the non-coding regions of trnH-psbA and trnT-trnL. The results of this study suggest that the type-I and type-II chloroplast genomes form a group that is distinct from the type-III chloroplast genome. Received: 14 April 2000 / Accepted: 11 July 2000     

Analysis of a model for macroparasitic infection with variable aggregation and clumped infections

 A model for macroparasitic infection with variable aggregation is considered. The starting point is an immigration-and-death process for parasites within a host, as in [3]; it is assumed however that infections will normally occur with several larvae at the same time. Starting from here, a four-dimensional, where free-living larvae are explicitly considered, and a three-dimensional model are obtained with same methods used in [26]. The equilibria of these models are found, their stability is discussed, as well as some qualitative features. It has been found that the assumption of “clumped” infections may have dramatic effects on the aggregation exhibited by these models. Infections with several larvae at the same time also increases the stability of the endemic equilibria of these models, and makes the occurrence of subcritical bifurcations (and consequently multiple equilibria) slightly more likely. The results of the low-dimensional model have also been compared to numerical simulations of the infinite system that describes the immigration-and-death process. It appears that the results of the systems are, by and large, in close correspondence, except for a parameter region where the four-dimensional model exhibits unusual properties, such as the occurrence of multiple disease-free equilibria, that do not appear to be shared by the infinite system. Received 28 October 1996; in revised form 11 April 1997     

Optimization models for the force and energy in competitive running

 In [2] the author has developed an optimization model for the force and energy in competitive running. In this paper the energy processes in the muscle were described by a three-compartment hydraulic model. Here this is reviewed briefly and applied to the current world records in order to determine the key parameters, maximal force, energy reserves and oxygen uptake. These values agree well with those given in the literature and those obtained by other means. The velocity profiles for 100 m sprints are described equally well. The model is then applied to older world records to deduce a relation between the force and energy by linear regression. Finally the fully parameterized model is used to compute the effects of adverse wind and altitude. Inasmuch as there are data available, there is a good agreement. Received 19 July 1995; received in revised form 27 February 1996     

Sampling distributions, biases, variances, and confidence intervals for genetic correlations

Genetic correlations are frequently estimated from natural and experimental populations, yet many of the statistical properties of estimators of are not known, and accurate methods have not been described for estimating the precision of estimates of Our objective was to assess the statistical properties of multivariate analysis of variance (MANOVA), restricted maximum likelihood (REML), and maximum likelihood (ML) estimators of by simulating bivariate normal samples for the one-way balanced linear model. We estimated probabilities of non-positive definite MANOVA estimates of genetic variance-covariance matrices and biases and variances of MANOVA, REML, and ML estimators of and assessed the accuracy of parametric, jackknife, and bootstrap variance and confidence interval estimators for MANOVA estimates of were normally distributed. REML and ML estimates were normally distributed for but skewed for and 0.9. All of the estimators were biased. The MANOVA estimator was less biased than REML and ML estimators when heritability (H), the number of genotypes (n), and the number of replications (r) were low. The biases were otherwise nearly equal for different estimators and could not be reduced by jackknifing or bootstrapping. The variance of the MANOVA estimator was greater than the variance of the REML or ML estimator for most H, n, and r. Bootstrapping produced estimates of the variance of close to the known variance, especially for REML and ML. The observed coverages of the REML and ML bootstrap interval estimators were consistently close to stated coverages, whereas the observed coverage of the MANOVA bootstrap interval estimator was unsatisfactory for some H, n, and r. The other interval estimators produced unsatisfactory coverages. REML and ML bootstrap interval estimates were narrower than MANOVA bootstrap interval estimates for most H, and r. Received: 6 July 1995 / Accepted: 8 March 1996     

Mapping of two genes for resistance to clubroot (Plasmodiophora brassicae) in a population of doubled haploid lines of Brassica oleracea by means of RFLP and AFLP markers

A genetic map covering 615 cM in 12 linkage groups was assembled based on 92 RFLP and AFLP markers segregating in a population of 107 doubled haploid lines (DH lines) of Brassica oleracea. The DH-line population was obtained through microspore culture from the of two homozygous parents: DH-line Bi derived from the cabbage landrace Bindsachsener, and DH-line Gr from broccoli cv ‘Greenia’. Sixty-five percent of the loci, and in some cases complete linkage groups, displayed distorted segregation ratios, a frequency much higher than that observed in populations of the same species. DH-line Bi was resistant to clubroot, which is caused by a Dutch field isolate of Plasmodiophora brassicae. Resistance in the DH-line population was determined in two ways: by assigning symptom grades to each plant, and by measuring the fresh weights of the healthy and affected parts of the root system of each plant. Using a multiple QTL mapping approach to analyze the fresh weight data, we found two loci for clubroot resistance; these were designated pb-3 and pb-4. The additive effects of these loci were responsible for 68% of the difference between the parents and for 60% of the genetic variance among DH-line means. Also, indications for the presence of two additional, minor QTLs were found. Analysis of symptom grades revealed the two QTLs pb-3 and pb-4, as well as one of the two minor QTLs indicated by analysis of the fresh weight data. Received: 29 April 1996 / Accepted: 10 May 1996     

Simple models for excitable and oscillatory neural networks

 Chains of coupled oscillators of simple “rotator” type have been used to model the central pattern generator (CPG) for locomotion in lamprey, among numerous applications in biology and elsewhere. In this paper, motivated by experiments on lamprey CPG with brainstem attached, we investigate a simple oscillator model with internal structure which captures both excitable and bursting dynamics. This model, and that for the coupling functions, is inspired by the Hodgkin–Huxley equations and two-variable simplifications thereof. We analyse pairs of coupled oscillators with both excitatory and inhibitory coupling. We also study traveling wave patterns arising from chains of oscillators, including simulations of “body shapes” generated by a double chain of oscillators providing input to a kinematic musculature model of lamprey.. Received: 25 November 1996 / Revised version: 9 December 1997     

Periodic orbits arising from Hopf bifurcations in a Volterra prey-predator model

 In this paper we derive a formula which enables the stability of periodic solutions to a Volterra integro-differential system to be determined. This system which has been studied by Cushing [1], models a predator-prey interaction with distributed delays. The results are obtained by using the algorithm developed by Kazarinoff, Wan and van den Driessche [2] based on the centre manifold formulas of Hassard and Wan [3]. We discuss an example of the formula for the case of weak kernels and show that under certain conditions stable periodic solutions arising from Hopf bifurcations at different critical values of the parameters can exist together. Received 30 December 1994; received in revised form 12 December 1995     

Multilocus dynamics under haploid selection

 A general haploid selection model with arbitrary number of multiallelic loci and arbitrary linkage distribution is considered. The population is supposed to be panmictic. A dynamically equivalent diploid selection model is introduced. There is a position effect in this model if the original haploid selection is not multiplicative. If haploid selection is additive then the fundamental theorem is established even with an estimate for the change in the mean fitness. On this basis exponential convergence to an equilibrium is proved. As rule, the limit states are single-gamete ones. If, moreover, linkage is tight, then the single-gamete state with maximal fitness attracts the population for almost all initial states. Received 27 November 1995; received in revised form 17 January 1996     

Discrete chiasma formation models and their associated high order interference

. We introduce some special chiasma formation processes. First a family of discrete chiasma formation processes is introduced and we determine the nature of higher order interference associated with those processes. Secondly we consider a two-stage chiasma formation process, where the associated recombination frequency between two markers depends not only on their map distance but also on their location along the chromosomes. We characterise under this process, in some cases, the nature of interference between two segments. Received: 22 January 1996 / Revised version: 17 September 1997     

The diffusion model for migration and selection in a plant population

 The diffusion approximation is derived for migration and selection at a multiallelic locus in a partially selfing plant population subdivided into a lattice of colonies. Generations are discrete and nonoverlapping; both pollen and seeds disperse. In the diffusion limit, the genotypic frequencies at each point are those determined at equilibrium by the local rate of selfing and allelic frequencies. If the drift and diffusion coefficients are taken as the appropriate linear combination of the corresponding coefficients for pollen and seeds, then the migration terms in the partial differential equation for the allelic frequencies have the standard form for a monoecious animal population. The selection term describes selection on the local genotypic frequencies. The boundary conditions and the unidimensional transition conditions for a geographical barrier and for coincident discontinuities in the carrying capacity and migration rate have the standard form. In the diallelic case, reparametrization renders the entire theory of clines and of the wave of advance of favorable alleles directly applicable to plant populations. Received 30 August 1995; received in revised form 23 February 1996     

An analysis of the hydraulic conductivity of the extracisternal space of the cochlear outer hair cell

 The cylindrically shaped cochlear outer hair cell (OHC) plays an important role in the transduction of acoustic energy into electrical energy in the cochlea. The extracisternal space (ECiS) of the lateral wall of the OHC is the fluid-filled space between the plasma membrane (PM) and the intracellular subsurface cisterna (SSC). In the ECiS, an array of cylindrical micropillars extends from the SSC to the PM. We obtain equations for the pressure, osmotic concentration and fluid velocity in the ECiS from the Brinkman-Stokes equations for steady incompressible flow in a plane channel that encloses an array of cylinders and whose upper wall, i.e. the plasma membrane, has a hydraulic conductivity of P _PM . From these equations we obtain an estimate for the hydraulic conductivity of the ECiS, P _ECiS . We show that the ECiS geometry accounts for P _ECiS being several orders of magnitude larger than P _PM and that P _ECiS increases with the width of the ECiS and decreases with the length of the ECiS. Received: 6 January 1998 / Revised version: 12 October 1998     

Overcompensatory recruitment and generation delay in discrete age-structured population models

 The effect of overcompensatory recruitment and the combined effect of overcompensatory recruitment and generation delay in discrete nonlinear age-structured population models is studied. Considering overcompensatory recruitment alone, we present formal proofs of the supercritical nature of bifurcations (both flip and Hopf) as well as an extensive analysis of dynamics in unstable parameter regions. One important finding here is that in case of small and moderate year to year survival probabilities there are large regions in parameter space where the qualitative behaviour found in a general n+1 dimensional model is retained already in a one-dimensional model. Another result is that the dynamics at or near the boundary of parameter space may be very complicated. Generally, generation delay is found to act as a destabilizing effect but its effect on dynamics is by no means unique. The most profound effect occurs in the n-generation delay cases. In these cases there is no stable equilibrium X ^* at all, but whenever X ^* small, a stable cycle of period n+1 where the periodic points in the cycle are on a very special form. In other cases generation delay does not alter the dynamics in any substantial way. Received 25 April 1995; received in revised form 21 November 1995     

In vivo haploid induction in maize

Two haploid-inducing lines, MHI and M741H, were used for the production of maternal haploids. Haploids were obtained from all maternal genotypes involved in the experiment, including dent, flint and flint×dent maize. The maternal genotype had a significant influence on the frequency of haploids obtained. The frequency ranged from 2.7% to 8.0%. For chromosome-doubling seedlings were treated with colchicine solution, and 49.4% of the haploid plants produced fertile pollen, 39.0% could be selfed and 27.3% produced seeds after selfing. Synthetic populations, improved by haploid sib recurrent selection, were tested in a field trial. The results show that the utilization of maternal haploid plants has great potential for maize breeding and maize genetics. Received: 14 May 2001 / Accepted: 3 August 2001     

Computing the selection gradient and evolutionary response of an infinite-dimensional trait

 Following the results developed in a previous paper, an equation describing the evolutionary response to selection is extended from finite- to infinite-dimensional traits. The selection gradient and evolutionary response are then computed for a large class of infinite-dimensional traits of broad biological interest. In this framework, traits are modeled as Gaussian processes, and reproducing kernel Hilbert spaces constitute a primary tool. Received 12 September 1996