Phase transition and uniqueness of levelset percolation

The main purpose of this paper is to introduce and establish basic results of a natural extension of the classical Boolean percolation model (also known as the Gilbert disc model). We replace the balls of that model by a positive non-increasing attenuation function $l:(0,\infty) \to (0,\infty)$ to create the random field $\Psi(y)=\sum_{x\in \eta}l(|x-y|),$ where $\eta$ is a homogeneous Poisson process in ${\mathbb R}^d.$ The field $\Psi$ is then a random potential field with infinite range dependencies whenever the support of the function $l$ is unbounded. In particular, we study the level sets $\Psi_{\geq h}(y)$ containing the points $y\in {\mathbb R}^d$ such that $\Psi(y)\geq h.$ In the case where $l$ has unbounded support, we give, for any $d\geq 2,$ exact conditions on $l$ for $\Psi_{\geq h}(y)$ to have a percolative phase transition as a function of $h.$ We also prove that when $l$ is continuous then so is $\Psi$ almost surely. Moreover, in this case and for $d=2,$ we prove uniqueness of the infinite component of $\Psi_{\geq h}$ when such exists, and we also show that the so-called percolation function is continuous below the critical value $h_c$.Comment: 25 page

Fat fractal percolation and k-fractal percolation

We consider two variations on the Mandelbrot fractal percolation model. In the k-fractal percolation model, the d-dimensional unit cube is divided in N^d equal subcubes, k of which are retained while the others are discarded. The procedure is then iterated inside the retained cubes at all smaller scales. We show that the (properly rescaled) percolation critical value of this model converges to the critical value of ordinary site percolation on a particular d-dimensional lattice as N tends to infinity. This is analogous to the result of Falconer and Grimmett that the critical value for Mandelbrot fractal percolation converges to the critical value of site percolation on the same d-dimensional lattice. In the fat fractal percolation model, subcubes are retained with probability p_n at step n of the construction, where (p_n) is a non-decreasing sequence with \prod p_n > 0. The Lebesgue measure of the limit set is positive a.s. given non-extinction. We prove that either the set of connected components larger than one point has Lebesgue measure zero a.s. or its complement in the limit set has Lebesgue measure zero a.s.Comment: 27 pages, 3 figure

Scaling relation for determining the critical threshold for continuum percolation of overlapping discs of two sizes

We study continuum percolation of overlapping circular discs of two sizes. We propose a phenomenological scaling equation for the increase in the effective size of the larger discs due to the presence of the smaller discs. The critical percolation threshold as a function of the ratio of sizes of discs, for different values of the relative areal densities of two discs, can be described in terms of a scaling function of only one variable. The recent accurate Monte Carlo estimates of critical threshold by Quintanilla and Ziff [Phys. Rev. E, 76 051115 (2007)] are in very good agreement with the proposed scaling relation.Comment: 4 pages, 3 figure

Adaptive and non-adaptive divergence in a common landscape

Species in a common landscape often face similar selective environments. The capacity of organisms to adapt to these environments may be largely species specific. Quantifying shared and unique adaptive responses across species within landscapes may thus improve our understanding of landscape-moderated biodiversity patterns. Here we test to what extent populations of two coexisting and phylogenetically related fishes—three-spined and nine-spined stickleback—differ in the strength and nature of neutral and adaptive divergence along a salinity gradient. Phenotypic differentiation, neutral genetic differentiation and genomic signatures of adaptation are stronger in the three-spined stickleback. Yet, both species show substantial phenotypic parallelism. In contrast, genomic signatures of adaptation involve different genomic regions, and are thus non-parallel. The relative contribution of spatial and environmental drivers of population divergence in each species reflects different strategies for persistence in the same landscape. These results provide insight in the mechanisms underlying variation in evolutionary versatility and ecological success among species within landscapes

The effect of temperature and predation on performance in monoculture and in competition in three Daphniidae differing in body size

Zooplankton body size shows a strong association with temperature, competition, and predation. Global warming affects all three drivers of body size and is thus expected to lead to substantial changes in zooplankton community composition and body size distributions. To disentangle the isolated and joint effect of temperature, competition, and fish predation on species biomass and community composition in zooplankton, we monitored population biomasses of three Daphniidae species that differ in body size (Daphnia magna, Daphnia pulex, and Ceriodaphnia reticulata) for 20 days, manipulating competition (monoculture, pairwise trials, and three-species communities), temperature (20°C, 24°C, and 28°C) and presence or absence of fish predation. In the absence of predation, D. magna dominated in all competition experiments, even at high temperatures. D. magna went extinct, however, in the predation treatments at 24°C and 28°C. D. pulex outcompeted C. reticulata and was negatively affected by predation and high temperature. C. reticulata did not reduce biomass at high temperatures and was negatively affected by all competition trials, but was positively affected by predation. Our results indicate that the two larger-bodied species are more negatively affected by the combination of temperature and predation than the smallest species. While higher temperatures reduced the biomass of the larger-bodied species, it did not fundamentally change their ability to dominate over the smallest species in competition. The combined effect of warming and predation changed community composition more fundamentally, resulting in the dominance of small-bodied species. This can have important ecosystem-wide impacts, such as the transition to turbid, algae-dominated systems

Host–parasite dynamics shaped by temperature and genotype: Quantifying the role of underlying vital rates

1. Global warming challenges the persistence of local populations, not only through heat-induced stress, but also through indirect biotic changes. We study the interactive effects of temperature, competition and parasitism in the water flea Daphnia magna. 2. We carried out a common garden experiment monitoring the dynamics of Daphnia populations along a temperature gradient. Halfway through the experiment, all populations became infected with the ectoparasite Amoebidium parasiticum, enabling us to study the interactive effects of temperature and parasite dynamics. We combined Integral Projection Models with epidemiological models, parameterized using the experimental data on the performance of individuals within dynamic populations. This enabled us to quantify the contribution of different vital rates and epidemiological parameters to population fitness across temperatures and Daphnia clones originating from two latitudes. 3. Interactions between temperature and parasitism shaped competition, where Belgian clones performed better under infection than Norwegian clones. Infected Daphnia populations performed better at higher than at lower temperatures, mainly due to an increased host capability of reducing parasite loads. Temperature strongly affected individual vital rates, but effects largely cancelled out on a population-level. In contrast, parasitism strongly reduced fitness through consistent negative effects on all vital rates. As a result, temperature-mediated parasitism was more important than the direct effects of temperature in shaping population dynamics. Both the outcome of the competition treatments and the observed extinction patterns support our modelling results. 4. Our study highlights that shifts in biotic interactions can be equally or more important for responses to warming than direct physiological effects of warming, emphasizing that we need to include such interactions in our studies to predict the competitive ability of natural populations experiencing global warming

Dimension (in)equalities and H\"older continuous curves in fractal percolation

We relate various concepts of fractal dimension of the limiting set C in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in C (the "dust"). In two dimensions, we also show that the set consisting of connected components larger than one point is a.s. the union of non-trivial H\"older continuous curves, all with the same exponent. Finally, we give a short proof of the fact that in two dimensions, any curve in the limiting set must have Hausdorff dimension strictly larger than 1.Comment: 22 pages, 3 figures, accepted for publication in Journal of Theoretical Probabilit

Electroencephalography in normotensive and hypertensive pregnancies and subsequent quality of life

Objectives: To compare electroencephalography (EEG) findings during pregnancy and postpartum in women with normotensive pregnancies and pregnancies complicated by hypertensive disorders. Also the health related quality of life postpartum was related to these EEG findings. Materials and Methods: An observational case-control study in a university hospital in the Netherlands. Twenty-nine normotensive and 58 hypertensive pregnant women were included. EEG's were recorded on several occasions during pregnancy and 6-8 weeks postpartum. Postpartum, the women filled out health related quality of life questionnaires. Main outcome measures were qualitative and quantitative assessments on EEG, multidimensional fatigue inventory, Short Form (36) Health Survey and EuroQol visual analogue scale. Results: In women with severe preeclampsia significantly lower alpha peak frequency, more delta and theta activity bilaterally and a higher EEG Sum Score were seen. Postpartum, these women showed impaired mental health, mental fatigue and social functioning, which could not be related to the EEG findings. Conclusions: Severe preeclamptic patients show more EEG abnormalities and have impaired mental wellbeing postpartum, but these findings are not correlated