Populations with individual variation in dispersal in heterogeneous environments: dynamics and competition with simply diffusing populations

We consider a model for a population in a heterogeneous environment, with logistic type local population dynamics, under the assumption that individuals can switch between two different nonzero rates of diffusion. Such switching behavior has been observed in some natural systems. We study how environmental heterogeneity and the rates of switching and diffusion affect the persistence of the population. The reaction diffusion systems in the models can be cooperative at some population densities and competitive at others. The results extend our previous work on similar models in homogeneous environments. We also consider competition between two populations that are ecologically identical, but where one population diffuses at a fixed rate and the other switches between two different diffusion rates. The motivation for that is to gain insight into when switching might be advantageous versus diffusing at a fixed rate. This is a variation on the classical results for ecologically identical competitors with differing fixed diffusion rates, where it is well known that the slower diffuser wins.Comment: To be published in SCIENCE CHINA Mathematic

Stability Analysis of Phase-Locked Bursting in Inhibitory Neuron Networks

Networks of neurons, which form central pattern generators (CPGs), are important for controlling animal behaviors. Of special interest are configurations or CPG motifs composed of reciprocally inhibited neurons, such as half-center oscillators (HCOs). Bursting rhythms of HCOs are shown to include stable synchrony or in-phase bursting. This in-phase bursting can co-exist with anti-phase bursting, commonly expected as the single stable state in HCOs that are connected with fast non-delayed synapses. The finding contrasts with the classical view that reciprocal inhibition has to be slow or time-delayed to synchronize such bursting neurons. Phase-locked rhythms are analyzed via Lyapunov exponents estimated with variational equations, and through the convergence rates estimated with Poincar\\u27e return maps. A new mechanism underlying multistability is proposed that is based on the spike interactions, which confer a dual property on the fast non-delayed reciprocal inhibition; this reveals the role of spikes in generating multiple co-existing phase-locked rhythms. In particular, it demonstrates that the number and temporal characteristics of spikes determine the number and stability of the multiple phase-locked states in weakly coupled HCOs. The generality of the multistability phenomenon is demonstrated by analyzing diverse models of bursting networks with various inhibitory synapses; the individual cell models include the reduced leech heart interneuron, the Sherman model for pancreatic beta cells, the Purkinje neuron model and Fitzhugh-Rinzel phenomenological model. Finally, hypothetical and experiment-based CPGs composed of HCOs are investigated. This study is relevant for various applications that use CPGs such as robotics, prosthetics, and artificial intelligence

Partial Differential Equations in Ecology

Partial differential equations (PDEs) have been used in theoretical ecology research for more than eighty years. Nowadays, along with a variety of different mathematical techniques, they remain as an efficient, widely used modelling framework; as a matter of fact, the range of PDE applications has even become broader. This volume presents a collection of case studies where applications range from bacterial systems to population dynamics of human riots

Differential Equations arising from Organising Principles in Biology

This workshop brought together experts in modeling and analysis of organising principles of multiscale biological systems such as cell assemblies, tissues and populations. We focused on questions arising in systems biology and medicine which are related to emergence, function and control of spatial and inter-individual heterogeneity in population dynamics. There were three main areas represented of differential equation models in mathematical biology. The first area involved the mathematical description of structured populations. The second area concerned invasion, pattern formation and collective dynamics. The third area treated the evolution and adaptation of populations, following the Darwinian paradigm. These problems led to differential equations, which frequently are non-trivial extensions of classical problems. The examples included but were not limited to transport-type equations with nonlocal boundary conditions, mixed ODE-reaction-diffusion models, nonlocal diffusion and cross-diffusion problems or kinetic equations

A physics approach to ecosystem dynamics

The mathematical modeling of ecosystems began in the 70s, and brought initially to conclusions which were completely at odds with empirical observations. In fact, while field ecologists were basically sure that biodiversity brings stability in an ecosystem, Robert May showed, using random matrix theory, that this was not the case: a randomly constructed ecosystem (i.e. characterized only by its diversity) becomes unstable if it is populated by a large enough number of species. This gave birth to the so-called “diversity-stability debate”, which still continues to date and still hasn’t brought to a final answer to the question “what is the relationship between the diversity of an ecosystem and its stability?”. Furthermore, the so-called “competitive exclusion principle” (which in turn gave rise to another intense debate) predicts that in a single trophic level the number of coexisting species cannot be greater than the number of resources. There are however many cases where this principle is clearly violated, the most famous one being the so-called “paradox of the plankton”: while the available nutrients for phytoplankton in the oceans are less than a dozen, the number of coexisting phytoplankton species in a single environment can be of the order of several hundreds, even in the periods of the year when nutrients are less abundant. Numerous ecological mechanisms and models have been proposed in order to solve these paradoxes, but none of them is flawless. Recently a possible solution has been proposed: Posfai et al. Introduced a model inspired by the paradox of the plankton, which consists of a system of different species competing for a common pool of nutrients, supplied constantly to the system. The main hypothesis of the model is the “metabolic trade-off” condition: every species has a fixed amount of energy budget to use in order to assimilate the resources. With this assumption it is possible to show that under some simple conditions the system can reach an equilibrium where an arbitrary number of species can coexist. In this thesis, after re-deriving the already known properties of this model, we have obtained many original results, like the study of the rank-abundance curves, a more thorough study of the stability of the equilibrium of the system, and the comparison between this equilibrium and May’s stability criterion. The most exciting result pertains an extension of the model where the metabolic strategies of the species are allowed to evolve in time in order to maximize the fitness of their relative species; in other words we have promoted the species-resource “interactions” to become dynamical variables themselves and whose evolution satisfy a variational principle. We have found that the metabolic strategies evolve cooperatively in order to allow all species to survive even though the initial conditions would have not allowed for their coexistence. This result should open new perspectives in ecosystem modeling and at the same time to new paradigms in statistical mechanics itself.ope

Nature’s Optics and Our Understanding of Light

Optical phenomena visible to everyone abundantly illustrate important ideas in science and mathematics. The phenomena considered include rainbows, sparkling reflections on water, green flashes, earthlight on the moon, glories, daylight, crystals, and the squint moon. The concepts include refraction, wave interference, numerical experiments, asymptotics, Regge poles, polarisation singularities, conical intersections, and visual illusions

The Eocene Falkland fossil flora, Okanagan Highlands, British Columbia : paleoclimate and plant community dynamics during the Early Eocene Climatic Optimum

The fossil flora and depositional setting of the early Eocene Falkland site in the southern interior of British Columbia, Canada is reported here in detail for the first time. Falkland is part of a series of fossil localities that occur in a region known as the Okanagan Highlands. These sites represent relatively cool upland environments in the greenhouse world of the early Eocene. Macrofossil collections were obtained from Falkland using an unbiased census approach with systematic sampling through three informal units in the exposed outcrop. A stratigraphic log reveals a lacustrine sequence dominated by finely laminated mudstone or shale with periodic influx of coarser material, punctuated by thin volcanic ash layers. Paleoelevation of the site is estimated based on paleobotanical evidence to have been similar to or slightly higher than modern levels (¡Ý1.3 km) during the early Eocene. Paleoclimate is assessed using both physiognomic and floristic approaches as applied to the Falkland flora. Physiognomic approaches correlate aspects of leaf morphology with climate, while floristic approaches use the tolerances of modern nearest living relatives to infer a climate envelope for the fossil flora. Overall, the different methods give broadly consistent results, with an identifiable zone of overlap in the estimates for mean annual temperature at ~10.5¡ãC, cold month mean temperature at 2.3¨C6.3¡ãC, warm month mean temperature at 20.2¨C23.7¡ãC, and a minimum mean annual precipitation of 82¨C120 cm/yr. Assessment of paleoclimate for the three individual units indicates a cooling trend over time, consistent with a radiometric date of 50.61¡À0.16 Ma that places the site in the waning phase of the Early Eocene Climatic Optimum (EECO). The stomatal frequency of fossil Ginkgo adiantoides from Falkland is used to estimate paleoatmospheric carbon dioxide (pCO2). Results from Falkland indicate that pCO2 was significantly higher than modern (>2x) in the early Eocene, although the upper limit of the estimate is unconstrained due to limitations with modern calibration datasets. Analysis of specimens from the three units indicates that climate and pCO2 were coupled during the EECO. Examination of modern Ginkgo biloba leaves suggests that stomatal density is more likely to be accurately measured than stomatal index. In addition, there is a significant difference between stomatal frequencies of long- and short-shoot leaves, suggesting that this factor needs to be taken into account in modern calibration datasets. The Falkland flora was described in two phases. In the first phase, specimens were assigned to morphotypes, informal categories that ideally correspond to species-level organization. In total, 1561 specimens were assigned to 138 morphotypes encompassing foliage and reproductive structures. The taxonomic literature was then investigated and morphotypes were assigned to formal taxa wherever possible. Gymnosperms are dominated by taxa in Cupressaceae, Pinaceae, and Ginkgoaceae, and there is a diverse angiosperm flora particularly rich in taxa belonging to Rosaceae, Betulaceae, and Sapindaceae. Rarefaction analysis shows Falkland as having diversity comparable to that of the hyper-diverse Laguna del Hunco site in Argentina. These data are consistent with an emerging understanding of high diversity in early Eocene forest communities associated with mild but equable climates. The Falkland flora retains a foundation of common taxa through all three units, including Metasequoia, Ginkgo, and Alnus; however, there is a distinct plant community in the upper unit as angiosperms become more abundant and the assemblage more diverse. Patterns in plant diversity are assessed within a context of changing climate and an active disturbance regime at the Falkland site

Perspectives of Nuclear Physics in Europe: NuPECC Long Range Plan 2010

The goal of this European Science Foundation Forward Look into the future of Nuclear Physics is to bring together the entire Nuclear Physics community in Europe to formulate a coherent plan of the best way to develop the field in the coming decade and beyond.<p></p> The primary aim of Nuclear Physics is to understand the origin, evolution, structure and phases of strongly interacting matter, which constitutes nearly 100% of the visible matter in the universe. This is an immensely important and challenging task that requires the concerted effort of scientists working in both theory and experiment, funding agencies, politicians and the public.<p></p> Nuclear Physics projects are often “big science”, which implies large investments and long lead times. They need careful forward planning and strong support from policy makers. This Forward Look provides an excellent tool to achieve this. It represents the outcome of detailed scrutiny by Europe’s leading experts and will help focus the views of the scientific community on the most promising directions in the field and create the basis for funding agencies to provide adequate support.<p></p> The current NuPECC Long Range Plan 2010 “Perspectives of Nuclear Physics in Europe” resulted from consultation with close to 6 000 scientists and engineers over a period of approximately one year. Its detailed recommendations are presented on the following pages. For the interested public, a short summary brochure has been produced to accompany the Forward Look.<p></p&gt

Dynamical system model of decision making and propagation

International audienceIndividual decision making is described as a bistable dynamical system. It can be influenced by the environment represented by other individuals, public opinion, all kinds of visual, oral and other information. We will study how the interaction of the individual decision making with the environment results in various patterns of decision making in the society