On the structure of the set of bifurcation points of periodic solutions for multiparameter Hamiltonian systems

This paper deals with periodic solutions of the Hamilton equation with many parameters. Theorems on global bifurcation of solutions with periods $2\pi/j,$ $j\in\mathbb{N},$ from a stationary point are proved. The Hessian matrix of the Hamiltonian at the stationary point can be singular. However, it is assumed that the local topological degree of the gradient of the Hamiltonian at the stationary point is nonzero. It is shown that (global) bifurcation points of solutions with given periods can be identified with zeros of appropriate continuous functions on the space of parameters. Explicit formulae for such functions are given in the case when the Hessian matrix of the Hamiltonian at the stationary point is block-diagonal. Symmetry breaking results concerning bifurcation of solutions with different minimal periods are obtained. A geometric description of the set of bifurcation points is given. Examples of constructive application of the theorems proved to analytical and numerical investigation and visualization of the set of all bifurcation points in given domain are provided. This paper is based on a part of the author's thesis [W. Radzki, ``Branching points of periodic solutions of autonomous Hamiltonian systems'' (Polish), PhD thesis, Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, Toru\'{n}, 2005].Comment: 35 pages, 4 figures, PDFLaTe

Multiplicity of Positive Solutions for an Obstacle Problem in R

In this paper we establish the existence of two positive solutions for the obstacle problem \displaystyle \int_{\Re}\left[u'(v-u)'+(1+\lambda V(x))u(v-u)\right] \geq \displaystyle \int_{\Re} f(u)(v-u), \forall v\in \Ka where $f$ is a continuous function verifying some technical conditions and \Ka is the convex set given by \Ka =\left\{v\in H^{1}(\Re); v \geq \varphi \right\}, with $\varphi \in H^{1}(\Re)$ having nontrivial positive part with compact support in $\Re$. \vspace{0.2cm} \noindent \emph{2000 Mathematics Subject Classification} : 34B18, 35A15, 46E39. \noindent \emph{Key words}: Obstacle problem, Variational methods, Positive solutions.Comment: To appear in Progress in Nonlinear Differential Equations and their Application

A supercritical elliptic problem in a cylindrical shell

We consider the problem $$-\Delta u=|u|^{p-2}u in \Omega, u=0 on \partial\Omega,$$ where $\Omega:=\{(y,z)\in\mathbb{R}^{m+1}\times\mathbb{R}^{N-m-1}: 0<a<|y|<b<\infty\}$, $0\leq m\leq N-1$ and $N\geq2$. Let $2_{N,m}^{\ast}:=2(N-m)/(N-m-2)$ if $m<N-2$ and $2_{N,m}^{\ast}:=\infty$ if $m=N-2$ or $N-1$. We show that $2_{N,m}^{\ast}$ is the true critical exponent for this problem, and that there exist nontrivial solutions if $2<p<2_{N,m}^{\ast}$ but there are no such solutions if $p\geq2_{N,m}^{\ast}$

Direct and indirect causal effects of heterozygosity on fitness-related traits in Alpine ibex

Heterozygosity–fitness correlations (HFCs) are a useful tool to investigate the effects of inbreeding in wild populations, but are not informative in distinguishing between direct and indirect effects of heterozygosity on fitness-related traits. We tested HFCs in male Alpine ibex (Capra ibex) in a free-ranging population (which suffered a severe bottleneck at the end of the eighteenth century) and used confirmatory path analysis to disentangle the causal relationships between heterozygosity and fitness-related traits. We tested HFCs in 149 male individuals born between 1985 and 2009. We found that standardized multi-locus heterozygosity (MLH), calculated from 37 microsatellite loci, was related to body mass and horn growth, which are known to be important fitness-related traits, and to faecal egg counts (FECs) of nematode eggs, a proxy of parasite resistance. Then, using confirmatory path analysis, we were able to show that the effect of MLH on horn growth was not direct but mediated by body mass and FEC. HFCs do not necessarily imply direct genetic effects on fitness-related traits, which instead can be mediated by other traits in complex and unexpected ways

The Environmental Dependence of Inbreeding Depression in a Wild Bird Population

BACKGROUND: Inbreeding depression occurs when the offspring produced as a result of matings between relatives show reduced fitness, and is generally understood as a consequence of the elevated expression of deleterious recessive alleles. How inbreeding depression varies across environments is of importance for the evolution of inbreeding avoidance behaviour, and for understanding extinction risks in small populations. However, inbreeding-by-environment (IxE) interactions have rarely been investigated in wild populations. METHODOLOGY/PRINCIPAL FINDINGS: We analysed 41 years of breeding events from a wild great tit (Parus major) population and used 11 measures of the environment to categorise environments as relatively good or poor, testing whether these measures influenced inbreeding depression. Although inbreeding always, and environmental quality often, significantly affected reproductive success, there was little evidence for statistically significant I x E interactions at the level of individual analyses. However, point estimates of the effect of the environment on inbreeding depression were sometimes considerable, and we show that variation in the magnitude of the I x E interaction across environments is consistent with the expectation that this interaction is more marked across environmental axes with a closer link to overall fitness, with the environmental dependence of inbreeding depression being elevated under such conditions. Hence, our analyses provide evidence for an environmental dependence of the inbreeding x environment interaction: effectively an I x E x E. CONCLUSIONS/SIGNIFICANCE: Overall, our analyses suggest that I x E interactions may be substantial in wild populations, when measured across relevant environmental contrasts, although their detection for single traits may require very large samples, or high rates of inbreeding

Inbreeding depression across the lifespan in a wild mammal population

Inbreeding depression is of major concern for the conservation of threatened species, and inbreeding avoidance is thought to be a key driver in the evolution of mating systems. However, the estimation of individual inbreeding coefficients in natural populations has been challenging, and, consequently, the full effect of inbreeding on fitness remains unclear. Genomic inbreeding coefficients may resolve the long-standing paucity of data on inbreeding depression in adult traits and total fitness. Here we investigate inbreeding depression in a range of life history traits and fitness in a wild population of red deer (Cervus elaphus) in Scotland using individual inbreeding coefficients derived from dense Single-Nucleotide Polymorphism (SNP) data ([Formula: see text]). We find associations between [Formula: see text] and annual breeding success in both sexes, and between maternal inbreeding coefficient and offspring survival. We also confirm previous findings of inbreeding depression in birth weight and juvenile survival. In contrast, inbreeding coefficients calculated from a deep and comparatively complete pedigree detected inbreeding depression in juvenile survival, but not in any adult fitness component. The total effect of inbreeding on lifetime breeding success (LBS) was substantial in both sexes: for [Formula: see text] [Formula: see text] , a value resulting from a half-sib mating, LBS declined by 72% for females and 95% for males. Our results demonstrate that SNP-based estimates of inbreeding provide a powerful tool for evaluating inbreeding depression in natural populations, and suggest that, to date, the prevalence of inbreeding depression in adult traits may have been underestimated

Fewer invited talks by women in evolutionary biology symposia.

Lower visibility of female scientists, compared to male scientists, is a potential reason for the under-representation of women among senior academic ranks. Visibility in the scientific community stems partly from presenting research as an invited speaker at organized meetings. We analysed the sex ratio of presenters at the European Society for Evolutionary Biology (ESEB) Congress 2011, where all abstract submissions were accepted for presentation. Women were under-represented among invited speakers at symposia (15% women) compared to all presenters (46%), regular oral presenters (41%) and plenary speakers (25%). At the ESEB congresses in 2001-2011, 9-23% of invited speakers were women. This under-representation of women is partly attributable to a larger proportion of women, than men, declining invitations: in 2011, 50% of women declined an invitation to speak compared to 26% of men. We expect invited speakers to be scientists from top ranked institutions or authors of recent papers in high-impact journals. Considering all invited speakers (including declined invitations), 23% were women. This was lower than the baseline sex ratios of early-mid career stage scientists, but was similar to senior scientists and authors that have published in high-impact journals. High-quality science by women therefore has low exposure at international meetings, which will constrain Evolutionary Biology from reaching its full potential. We wish to highlight the wider implications of turning down invitations to speak, and encourage conference organizers to implement steps to increase acceptance rates of invited talks

Prediction of individual genetic risk to prostate cancer using a polygenic score

BACKGROUND Polygenic risk scores comprising established susceptibility variants have shown to be informative classifiers for several complex diseases including prostate cancer. For prostate cancer it is unknown if inclusion of genetic markers that have so far not been associated with prostate cancer risk at a genome-wide significant level will improve disease prediction. METHODS We built polygenic risk scores in a large training set comprising over 25,000 individuals. Initially 65 established prostate cancer susceptibility variants were selected. After LD pruning additional variants were prioritized based on their association with prostate cancer. Six-fold cross validation was performed to assess genetic risk scores and optimize the number of additional variants to be included. The final model was evaluated in an independent study population including 1,370 cases and 1,239 controls. RESULTS The polygenic risk score with 65 established susceptibility variants provided an area under the curve (AUC) of 0.67. Adding an additional 68 novel variants significantly increased the AUC to 0.68 (P-=-0.0012) and the net reclassification index with 0.21 (P-=-8.5E-08). All novel variants were located in genomic regions established as associated with prostate cancer risk. CONCLUSIONS Inclusion of additional genetic variants from established prostate cancer susceptibility regions improves disease prediction

No signs of inbreeding despite long-term isolation and habitat fragmentation in the critically endangered Montseny brook newt (Calotriton arnoldi)

Endemic species with restricted geographic ranges potentially suffer the highest risk of extinction. If these species are further fragmented into genetically isolated subpopulations, the risk of extinction is elevated. Habitat fragmentation is generally considered to have negative effects on species survival, despite some evidence for neutral or even positive effects. Typically, non-negative effects are ignored by conservation biology. The Montseny brook newt (Calotriton arnoldi) has one of the smallest distribution ranges of any European amphibian (8 km2) and is considered critically endangered by the International Union for Conservation of Nature. Here we apply molecular markers to analyze its population structure and find that habitat fragmentation owing to a natural barrier has resulted in strong genetic division of populations into two sectors, with no detectable migration between sites. Although effective population size estimates suggest low values for all populations, we found low levels of inbreeding and relatedness between individuals within populations. Moreover, C. arnoldi displays similar levels of genetic diversity to its sister species Calotriton asper, from which it separated around 1.5 million years ago and which has a much larger distribution range. Our extensive study shows that natural habitat fragmentation does not result in negative genetic effects, such as the loss of genetic diversity and inbreeding on an evolutionary timescale. We hypothesize that species in such conditions may evolve strategies (for example, special mating preferences) to mitigate the effects of small population sizes. However, it should be stressed that the influence of natural habitat fragmentation on an evolutionary timescale should not be conflated with anthropogenic habitat loss or degradation when considering conservation strategies