A Phase transition in zero count probability for Stationary Gaussian Processes

We study the probability that a real stationary Gaussian process has at least $\eta T$ zeros in $[0,T]$ (overcrowding), or at most this number (undercrowding). We show that if the spectral measure of the process is supported on $\pm[B,A]$, overcrowding probability transitions from exponential decay to Gaussian decay at $\eta=\tfrac{A}{\pi}$, while undercrowding probability undergoes the reverse transition at $\eta=\tfrac{B}{\pi}$.Comment: 17 page

Tiger Sharks Eat Songbirds: Reply

In response to our recent paper (Drymon et al. 2019), Yosef (2019) questions the mechanism proposed to explain interactions between tiger sharks (Galeocerdo cuvier) and migratory songbirds, while offering an alternative mechanism based on a single observation. We appreciate the comments from Yosef and the opportunity to respond

On high moments of strongly diluted large Wigner random matrices

We consider a dilute version of the Wigner ensemble of nxn random matrices $H$ and study the asymptotic behavior of their moments $M_{2s}$ in the limit of infinite $n$, $s$ and $\rho$, where $\rho$ is the dilution parameter. We show that in the asymptotic regime of the strong dilution, the moments $M_{2s}$ with $s=\chi\rho$ depend on the second and the fourth moments of the random entries $H_{ij}$ and do not depend on other even moments of $H_{ij}$. This fact can be regarded as an evidence of a new type of the universal behavior of the local eigenvalue distribution of strongly dilute random matrices at the border of the limiting spectrum. As a by-product of the proof, we describe a new kind of Catalan-type numbers related with the tree-type walks.Comment: 43 pages (version four: misprints corrected, discussion added, other minor modifications

Low Genetic Differentiation across Three Major Ocean Populations of the Whale Shark, Rhincodon typus

BACKGROUND:Whale sharks are a declining species for which little biological data is available. While these animals are protected in many parts of their range, they are fished legally and illegally in some countries. Baseline biological and ecological data are needed to allow the formulation of an effective conservation plan for whale sharks. It is not known, for example, whether the whale shark is represented by a single worldwide panmictic population or by numerous, reproductively isolated populations. Genetic analysis of population structure is one essential component of the baseline data required for whale shark conservation. METHODOLOGY/PRINCIPAL FINDINGS:We have identified 8 polymorphic microsatellites in the whale shark and used these markers to assess genetic variation and population structure in a panel of whale sharks covering a broad geographic region. This is the first record of microsatellite loci in the whale shark, which displayed an average of 9 alleles per locus and mean H(o) = 0.66 and H(e) = 0.69. All but one of the eight loci meet the expectations of Hardy-Weinberg equilibrium. Analysis of these loci in whale sharks representing three major portions of their range, the Pacific (P), Caribbean (C), and Indian (I) Oceans, determined that there is little population differentiation between animals sampled in different geographic regions, indicating historical gene flow between populations. F(ST) values for inter-ocean comparisons were low (PxC = 0.0387, CxI = 0.0296 and PxI = -0.0022), and only CxI approached statistical significance (p = 0.0495). CONCLUSIONS/SIGNIFICANCE:We have shown only low levels of genetic differentiation between geographically distinct whale shark populations. Existing satellite tracking data have revealed both regional and long-range migration of whale sharks throughout their range, which supports the finding of gene flow between populations. Whale sharks traverse geographic and political boundaries during their life history and interbreed with animals from distant populations; conservation efforts must therefore target international protection for this species

Risky business for a juvenile marine predator? Testing the influence of foraging strategies on size and growth rate under natural conditions

Mechanisms driving selection of body size and growth rate in wild marine vertebrates are poorly understood, thus limiting knowledge of their fitness costs at ecological, physiological and genetic scales. Here, we indirectly tested whether selection for size-related traits of juvenile sharks that inhabit a nursery hosting two dichotomous habitats, protected mangroves (low predation risk) and exposed seagrass beds (high predation risk), is influenced by their foraging behaviour. Juvenile sharks displayed a continuum of foraging strategies between mangrove and seagrass areas, with some individuals preferentially feeding in one habitat over another. Foraging habitat was correlated with growth rate, whereby slower growing, smaller individuals fed predominantly in sheltered mangroves, whereas larger, faster growing animals fed over exposed seagrass. Concomitantly, tracked juveniles undertook variable movement behaviours across both the low and high predation risk habitat. These data provide supporting evidence for the hypothesis that directional selection favouring smaller size and slower growth rate, both heritable traits in this shark population, may be driven by variability in foraging behaviour and predation risk. Such evolutionary pathways may be critical to adaptation within predator-driven marine ecosystems

Delocalization and Diffusion Profile for Random Band Matrices

We consider Hermitian and symmetric random band matrices $H = (h_{xy})$ in $d \geq 1$ dimensions. The matrix entries $h_{xy}$, indexed by x,y \in (\bZ/L\bZ)^d, are independent, centred random variables with variances s_{xy} = \E |h_{xy}|^2. We assume that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. In one dimension we prove that the eigenvectors of $H$ are delocalized if $W\gg L^{4/5}$. We also show that the magnitude of the matrix entries \abs{G_{xy}}^2 of the resolvent $G=G(z)=(H-z)^{-1}$ is self-averaging and we compute \E \abs{G_{xy}}^2. We show that, as $L\to\infty$ and $W\gg L^{4/5}$, the behaviour of \E |G_{xy}|^2 is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions

Development and characterization of thirteen microsatellite markers for the Fiscal Flycatcher (Sigelus silens) for use in phylogeographic and landscape genetics research

The Fiscal Flycatcher, Sigelus silens, is the only representative of a monotypic genus, endemic to Southern Africa, and may represent two cryptic species. Here we describe the development of thirteen microsatellite markers, and characterize polymorphism for each one. We found that all but one of our 13 loci were highly variable, each having five or more alleles. This suggests that these markers will have high variability across the species range and will be of utility in understanding the extent of gene flow among populations

Zeroes of Gaussian Analytic Functions with Translation-Invariant Distribution

We study zeroes of Gaussian analytic functions in a strip in the complex plane, with translation-invariant distribution. We prove that the a limiting horizontal mean counting-measure of the zeroes exists almost surely, and that it is non-random if and only if the spectral measure is continuous (or degenerate). In this case, the mean zero-counting measure is computed in terms of the spectral measure. We compare the behavior with Gaussian analytic function with symmetry around the real axis. These results extend a work by Norbert Wiener.Comment: 24 pages, 1 figure. Some corrections were made and presentation was improve

EphA3 Expressed in the Chicken Tectum Stimulates Nasal Retinal Ganglion Cell Axon Growth and Is Required for Retinotectal Topographic Map Formation

BACKGROUND: Retinotopic projection onto the tectum/colliculus constitutes the most studied model of topographic mapping and Eph receptors and their ligands, the ephrins, are the best characterized molecular system involved in this process. Ephrin-As, expressed in an increasing rostro-caudal gradient in the tectum/colliculus, repel temporal retinal ganglion cell (RGC) axons from the caudal tectum and inhibit their branching posterior to their termination zones. However, there are conflicting data regarding the nature of the second force that guides nasal axons to invade and branch only in the caudal tectum/colliculus. The predominant model postulates that this second force is produced by a decreasing rostro-caudal gradient of EphA7 which repels nasal optic fibers and prevents their branching in the rostral tectum/colliculus. However, as optic fibers invade the tectum/colliculus growing throughout this gradient, this model cannot explain how the axons grow throughout this repellent molecule. METHODOLOGY/PRINCIPAL FINDINGS: By using chicken retinal cultures we showed that EphA3 ectodomain stimulates nasal RGC axon growth in a concentration dependent way. Moreover, we showed that nasal axons choose growing on EphA3-expressing cells and that EphA3 diminishes the density of interstitial filopodia in nasal RGC axons. Accordingly, in vivo EphA3 ectodomain misexpression directs nasal optic fibers toward the caudal tectum preventing their branching in the rostral tectum. CONCLUSIONS: We demonstrated in vitro and in vivo that EphA3 ectodomain (which is expressed in a decreasing rostro-caudal gradient in the tectum) is necessary for topographic mapping by stimulating the nasal axon growth toward the caudal tectum and inhibiting their branching in the rostral tectum. Furthermore, the ability of EphA3 of stimulating axon growth allows understanding how optic fibers invade the tectum growing throughout this molecular gradient. Therefore, opposing tectal gradients of repellent ephrin-As and of axon growth stimulating EphA3 complement each other to map optic fibers along the rostro-caudal tectal axis