Chemotactic collapse for the Keller-Segel model

 This work is concerned with the system (S) {u _t =Δu − χ∇ (u∇v) for x∈Ω, t>0Γ v _t =Δv+(u−1) for x∈Ω, t>0 where Γ, χ are positive constants and Ω is a bounded and smooth open set in ℝ². On the boundary ∂Ω, we impose no-flux conditions: (N) ∂u∂n =∂v∂n =0 for x∈∂ Ω, t>0 Problem (S), (N) is a classical model to describe chemotaxis corresponding to a species of concentration u(x, t) which tends to aggregate towards high concentrations of a chemical that the species releases. When completed with suitable initial values at t=0 for u(x, t), v(x, t), the problem under consideration is known to be well posed, locally in time. By means of matched asymptotic expansions techniques, we show here that there exist radial solutions exhibiting chemotactic collapse. By this we mean that u(r, t) →Aδ(y) as t→T for some T<∞, where A is the total concentration of the species. Received 9 March 1995; received in revised form 25 December 1995     

Impact of magnetic field inhomogeneity on electron cyclotron radiative loss in tokamak reactors

The potential importance of electron cyclotron (EC) emission in the local electron power balance in the steady-state regimes of ITER operation with high temperatures, as well as in the DEMO reactor, requires accurate calculation of the one-dimensional (over magnetic surfaces) distribution of the net radiated power density, P _EC(ρ). When the central electron temperature increases to ∼30 keV, the local EC radiative loss comprises a substantial fraction of the heating power from fusion alphas and is close to the total auxiliary NBI heating power, P _EC(0) ≃ 0.3P _α(0) ≃ P _aux(0). In the present paper, the model of EC radiative transport in an axisymmetric toroidal plasma is extended to the case of an inhomogeneous magnetic field B(R, Z). The impact of such inhomogeneity on local and total power losses is analyzed in the framework of this model by using the CYNEQ code. It is shown that, for the magnetic field B, temperature T _e , density n _e , and wall reflection coefficient R _w expected in ITER and DEMO, accurate simulations of the EC radiative loss require self-consistent 1.5D transport analysis (i.e., one-dimensional simulations of plasma transport and two-dimensional simulations of plasma equilibrium). It is shown that EC radiative transport can be described with good accuracy in the 1D approximation with the surface-averaged magnetic field, B(ρ) = 〈B(R, Z)〉 _ms . This makes it possible to substantially reduce the computational time required for time-dependent self-consistent 1.5D transport analysis. Benchmarking of the CYNEQ results with available results of the RAYTEC, EXACTEC, and CYTRAN codes is performed for various approximations of the magnetic field.     

Oxygen evolution from single- and multiple-turnover light pulses: temporal kinetics of electron transport through PSII in sunflower leaves

Oxygen evolution per single-turnover flash (STF) or multiple-turnover pulse (MTP) was measured with a zirconium O₂ analyzer from sunflower leaves at 22°C. STF were generated by Xe arc lamp, MTP by red LED light of up to 18000 μmol quanta m⁻² s⁻¹. Ambient O₂ concentration was 10–30 ppm, STF and MTP were superimposed on far-red background light in order to oxidize plastoquinone (PQ) and randomize S-states. Electron (e⁻) flow was calculated as 4 times O₂ evolution. Q _A → Q _B electron transport was investigated firing double STF with a delay of 0 to 2 ms between the two. Total O₂ evolution per two flashes equaled to that from a single flash when the delay was zero and doubled when the delay exceeded 2 ms. This trend was fitted with two exponentials with time constants of 0.25 and 0.95 ms, equal amplitudes. Illumination with MTP of increasing length resulted in increasing O₂ evolution per pulse, which was differentiated with an aim to find the time course of O₂ evolution with sub-millisecond resolution. At the highest pulse intensity of 2.9 photons ms⁻¹ per PSII, 3 e⁻ initially accumulated inside PSII and the catalytic rate of PQ reduction was determined from the throughput rate of the fourth and fifth e⁻. A light response curve for the reduction of completely oxidized PQ was a rectangular hyperbola with the initial slope of 1.2 PSII quanta per e⁻ and V _m of 0.6 e⁻ ms⁻¹ per PSII. When PQ was gradually reduced during longer MTP, V _m decreased proportionally with the fraction of oxidized PQ. It is suggested that the linear kinetics with respect to PQ are apparent, caused by strong product inhibition due to about equal binding constants of PQ and PQH₂ to the Q _B site. The strong product inhibition is an appropriate mechanism for down-regulation of PSII electron transport in accordance with rate of PQH₂ oxidation by cytochrome b₆f.     

Modeling nitrogen flux by larval insect herbivores from a temperate hardwood forest

Herbivorous insects flux considerable amounts of nitrogen from the forest canopy to the soil in the form of frass. The amount of nitrogen fluxed varies depending on the characteristics of the herbivores, their food resources, and their physical environment. We used concepts from metabolic ecology and ecological stoichiometry to develop a general model of individual nitrogen flux via frass fall for moth and sawfly larvae from a temperate hardwood forest in northern Wisconsin, USA. We found that individual nitrogen flux (Q _N, mg N/day) was related to larval body mass (M _B, mg dry), short-term variation in environmental temperature (T, K), and larval nitrogen concentration (N _B, proportion dry mass) as Q _N = e^25.75 M _B^0.77 e^−0.83/kT N _B^−1.56, where k is Boltzmann’s constant (8.62 × 10⁻⁵ eV/K). We also found that larval nitrogen flux did not vary with the nitrogen concentration of food, and suggest that this was due to compensatory feeding by larvae living on low-quality leaves. With further work, models of individual N flux could be used to scale individual fluxes to population and community levels, and thus link the characteristics of insect herbivore communities with the flow of nitrogen through forested ecosystems.     

Approximation of the Basic Reproduction Number <Emphasis Type="Italic">R</Emphasis><Subscript>0</Subscript> for Vector-Borne Diseases with a Periodic Vector Population

The main purpose of this paper is to give an approximate formula involving two terms for the basic reproduction number R ₀ of a vector-borne disease when the vector population has small seasonal fluctuations of the form p(t) = p ₀ (1+ε cos (ωt − φ)) with ε ≪ 1. The first term is similar to the case of a constant vector population p but with p replaced by the average vector population p ₀. The maximum correction due to the second term is (ε²/8)% and always tends to decrease R ₀. The basic reproduction number R ₀ is defined through the spectral radius of a linear integral operator. Four numerical methods for the computation of R ₀ are compared using as example a model for the 2005/2006 chikungunya epidemic in La Réunion. The approximate formula and the numerical methods can be used for many other epidemic models with seasonality. MSC 92D30 ⋅ 45C05 ⋅ 47A55     

Cell cycle control at the first restriction point and its effect on tissue growth

Cell cycle is controlled at two restriction points, R ₁ and R ₂. At both points the cell will commit apoptosis if it detects irreparable damage. But at R ₁ an undamaged cell also decides whether to proceed to the S phase or go into a quiescent mode, depending on the environmental conditions (e.g., overpopulation, hypoxia). We consider the effect of this decision at the population level in a spherical tissue {r < R(t)}. We prove that if the cells have full control at R ₁, they can manipulate the size of R(t) to ensure that 0 < c ≤ R(t) ≤ C < ∞; simulations further show that R(t) can be made nearly stationary. In the absence of such control, R(t) will either increase to ∞ or decrease to 0. The mathematical model and analysis involve a system of PDEs in {r < R(t)}.     

Kinetic processes initiated by a nanosecond high-current discharge in hot air

Results from numerical investigations of kinetic processes initiated by a pulsed nanosecond discharge in hot (T ₀ ≥ 1000 K) air at atmospheric pressure are presented. The calculated results on the dynamics of the electron density, the population of the N₂(B³Π _g ) and N₂(C³Π _u ) states, and the atomic oxygen density in the axial discharge region agree with experiment. The method for determining the gas temperature by measuring the rotational structure of the transitions N₂(C³Π _u , ν) → N₂(B³Π _g , ν′) of the 2⁺ nitrogen system is analyzed. It is shown that, in relatively weak reduced electric fields typical of secondary discharge pulses, the electron impact excitation of the N₂(C³Π _u ) state from the ground state N₂(X¹Σ _g ⁺) can be accompanied by its additional step population from the N₂(B³Π _g ), N₂(a′Σ _u ⁻), and other electronic states. This effect substantially influences the rotational distribution of nitrogen molecules in the N₂(C³Π _u , ν) state; moreover, the temperature determined from this distribution can be substantially higher than the true gas temperature.     

Longevity and growth efficiency of two deep-dwelling Arctic zoarcids and comparison with eight other zoarcid species from different climatic regions

Our basic knowledge of the ecology, especially the age and growth of polar deep-sea biota is still scarce. This study provides first data about the age and growth of the two abundant Arctic fish species Lycodes frigidus and Lycodes squamiventer (Zoarcidae). Lycodes frigidus was caught at the deeper parts (1,546–3,576 m depth) of the HAUSGARTEN observatory (HG), west of Svalbard. The congener Lycodes squamiventer was caught at two HG stations (1,273–1,546 m) and at the H?kon Mosby Mud Volcano (HMMV, ~1,250 m), a cold seep in the southwestern Barents Sea. Age was determined by sagittal otolith increment analysis. Growth performance was assessed by fitting age–length data to a von Bertalanffy growth equation. Our data suggest that L. frigidus and L. squamiventer attain maximum ages of 33 and 21 years, respectively. Lycodes squamiventer from the HMMV had significantly higher growth rates and their maximum age and length was slightly lower compared to conspecifics from the shallow HG stations. Von Bertalanffy growth equations were L _t  = 58.9 ∗ (1 − e^(−0.042*t)) for L. frigidus, and L _t  = 25.3 ∗ (1 − e^(−0.074*t)) and L _t  = 24.2 ∗ (1 − e^{(−0.099 * t)}) for L. squamiventer from HG and the HMMV, respectively. A comparison of these data with those of eight other zoarcids indicates that growth performances are correlated with temperature: the higher the annual mean temperatures experienced, the higher the growth rates. However, maximum ages decrease with increasing temperatures.     

Dynamics of an epidemic model with non-local infections for diseases with latency over a patchy environment

In this paper, with the assumptions that an infectious disease in a population has a fixed latent period and the latent individuals of the population may disperse, we formulate an SIR model with a simple demographic structure for the population living in an n-patch environment (cities, towns, or countries, etc.). The model is given by a system of delay differential equations with a fixed delay accounting for the latency and a non-local term caused by the mobility of the individuals during the latent period. Assuming irreducibility of the travel matrices of the infection related classes, an expression for the basic reproduction number R₀{\mathcal{R}_0} is derived, and it is shown that the disease free equilibrium is globally asymptotically stable if R₀ < 1{\mathcal{R}_0 < 1} , and becomes unstable if ${\mathcal{R}_0 > 1}${\mathcal{R}_0 > 1} . In the latter case, there is at least one endemic equilibrium and the disease will be uniformly persistent. When n = 2, two special cases allowing reducible travel matrices are considered to illustrate joint impact of the disease latency and population mobility on the disease dynamics. In addition to the existence of the disease free equilibrium and interior endemic equilibrium, the existence of a boundary equilibrium and its stability are discussed for these two special cases.     

An urn model study of variability within a compartment

Formulas are derived for the mean and variance of the number of radioactive atoms present in a compartment (or urn). Initally,n ₁ radioactive atoms andb stable atoms are placed in the urn; and subsequently,r stable atoms are added and an equal number,r, of a random mixture of stable and radioactive atoms is removed per unit time. The expected number of radioactive atoms,E(t), present at timet is, as expected,n ₁ e^−λt where λ=(r/Δt)/(b+r+n ₁). The relative variance, σ²(t)/n ₁ ² , vanishes to zero forr=1, atoms per unit time and for a large number ofn ₁ radioactive atoms; but for a large number of bothr andn ₁ atoms the relative variance is ∼e ^−λt , equal to the fractional retention, fort>1/λ. Thus in studies where radionuclides are injected into animals and a single compartment represents the data, if a large variance is observed it might be due to the fact that large numbers of atoms are transferred out in unit time. When a small variance is observed, this is probably due to the fact that few atoms are transferred in smaller units of time (such that λ is the same in both cases). Research sponsored by the Energy Research and Development Administration under contract with Union Carbide Corporation.     

Photosynthesis and dark respiration of leaves of terrestrial carnivorous plants

Using CO₂ gasometry, net photosynthetic (P _N) and dark respiration rates (R _D) were measured in leaves or traps of 12 terrestrial carnivorous plant species usually grown in the shade. Generally, mean maximum P _N (60 nmol CO₂ g⁻¹(DM) s⁻¹ or 2.7 μmol m⁻² s⁻¹) was low in comparison with that of vascular non-carnivorous plants but was slightly higher than that reported elsewhere for carnivorous plants. After light saturation, the facultatively heliophytic plants behaved as shade-adapted plants. Mean R _D in leaves and traps of all species reached about 50% of maximum P _N and represents the high photosynthetic (metabolic) cost of carnivory.     

Stability of differential susceptibility and infectivity epidemic models

We introduce classes of differential susceptibility and infectivity epidemic models. These models address the problem of flows between the different susceptible, infectious and infected compartments and differential death rates as well. We prove the global stability of the disease free equilibrium when the basic reproduction ratio R₀ ￡ 1{\mathcal{R}_0 \leq 1} and the existence and uniqueness of an endemic equilibrium when ${\mathcal{R}_0 >1 }${\mathcal{R}_0 >1 } . We also prove the global asymptotic stability of the endemic equilibrium for a differential susceptibility and staged progression infectivity model, when ${\mathcal{R}_0 >1 }${\mathcal{R}_0 >1 } . Our results encompass and generalize those of Hyman and Li (J Math Biol 50:626–644, 2005; Math Biosci Eng 3:89–100, 2006).     

Irradiance influences tea leaf (<Emphasis Type="Italic">Camellia sinensis</Emphasis> L.) photosynthesis and transpiration

Rates of net photosynthesis (P _N) and transpiration (E), and leaf temperature (T_L) of maintenance leaves of tea under plucking were affected by photosynthetic photon flux densities (PPFD) of 200–2 200 μmol m⁻² s⁻¹. P _N gradually increased with the increase of PPFD from 200 to 1 200 μmol m⁻² s⁻¹ and thereafter sharply declined. Maximum P _N was 13.95 μmol m⁻² s⁻¹ at 1 200 μmol m⁻² s⁻¹ PPFD. There was no significant variation of P _N among PPFD at 1 400–1 800 μmol m⁻² s⁻¹. Significant drop of P _N occurred at 2 000 μmol m⁻² s⁻¹. PPFD at 2 200 μmol m⁻² s⁻¹ reduced photosynthesis to 6.92 μmol m⁻² s⁻¹. PPFD had a strong correlation with T_L and E. Both T_L and E linearly increased from 200 to 2 200 μmol m⁻² s⁻¹ PPFD. T_L and E were highly correlated. The optimum T_L for maximum P _N was 26.0 °C after which P _N declined significantly. E had a positive correlation with P _N.     

Plasma free amino acid kinetics in rainbow trout (<Emphasis Type="Italic">Oncorhynchus mykiss</Emphasis>) using a bolus injection of <Superscript>15</Superscript>N-labeled amino acids

To gain insight into the metabolic design of the amino acid carrier systems in fish, we injected a bolus of ¹⁵N amino acids into the dorsal aorta in mature rainbow trout (Oncorhynchus mykiss). The plasma kinetic parameters including concentration, pool size, rate of disappearance (R _d), half-life and turnover rate were determined for 15 amino acids. When corrected for metabolic rate, the R _d values obtained for trout for most amino acids were largely comparable to human values, with the exception of glutamine (which was lower) and threonine (which was higher). R _d values ranged from 0.9 μmol 100 g⁻¹ h⁻¹ (lysine) to 22.1 μmol 100 g⁻¹ h⁻¹ (threonine) with most values falling between 2 and 6 μmol 100 g⁻¹ h⁻¹. There was a significant correlation between R _d and the molar proportion of amino acids in rainbow trout whole body protein hydrolysate. Other kinetic parameters did not correlate significantly with whole body amino acid composition. This indicates that an important design feature of the plasma-free amino acids system involves proportional delivery of amino acids to tissues for protein synthesis.     

Theory of transport in linear biological systems: I. Fundamental integral equation

The conditions under which the output,γ _b (t), of a biological system is related to the input,γ _a (t), by an integral equation of the typeγ _b (t) = ∫ ₀ ^t γ _a (ω)w(t−ω)dω, where ω(t) is a transport functioncharacteristic of the system, are analyzed in detail. Methods of solving this type of integral equation are briefly discussed. The theory is then applied to problems in tracer kinetics in which input and output are sums of exponentials, and explicit formulae, which are applicable whether or not the pool is uniformly mixed, are derived for “turnover time” and “pool” size.     

Photosynthetic and leaf respiration activity of <Emphasis Type="Italic">Malcolmia littorea</Emphasis> (L.) R. Br. in response to air temperature

Plant traits of Malcolmia littorea growing at the Botanic Garden of Rome and transplanted from the wild population developing along the Latium coast (Italy) were analyzed. The highest photosynthetic rates [P _N, 22.5 ± 0.5 μmol(CO₂) m⁻² s⁻¹], associated to the highest chlorophyll content (Chl, 60 ± 5 SPAD units), and respiration rates [R, 11.1 ± 0.2 μmol(CO₂) m⁻² s⁻¹] were reached in spring, when mean air temperature (T _m) was in the range 17°C to 23°C. P _N, Chl, and R decreased by 86, 38, and 59% in summer when mean maximum air temperature (T _max) was 30.3 ± 2.6°C. Leaf water potential decreased by 34% in summer compared to the spring value, and it was associated to a relative water content (RWC) of 74 ± 4%, and to a water-use efficiency (WUE) of 2.15 ± 0.81 μmol(CO₂) mmol⁻¹(H₂O). Moreover, also low air temperatures determined a significant P _N and R decreases (by 52 and 40% compared to the maximum, respectively). Responsiveness of gross photosynthetic rate (P _g) to R was higher than that to P _N as underlined by the slope of the regression line between the two variables. The results underlined a low tolerance to both high- and low air temperatures of M. littorea. The selected key traits (R, WUE, Chl) by the discriminant analysis might be used to monitor the M. littorea wild population in the long time. The ex situ cultivated plants could be propagated and used to increase the individuals number of the wild population.     

Nonquasineutral current equilibria as elementary structures of plasma dynamics

A study is made of the fundamental features of current filaments with a nonzero electron vorticity Ω _e ≡ B − (c/e) ▿ × p _ee ≠ 0 and the corresponding Lagrangian invariant I _e . Such current structures can exist on spatial scales of up to ω _pi ⁻¹. It is shown that the dissipative stage of the plasma evolution and the violation of Thomson’s theorem on vorticity conservation in an electron fluid are of fundamental importance for the onset of electron current structures. A key role of the screening of electric and magnetic fields at distances on the order of the magnetic Debye radius r _B = B/(4πen _e )—the main property of such current structures in a Hall medium with σB/(en _e c) ≫ 1—is stressed. Since the minimum size of a vortex structure is the London length c/ω _pe , the structures under consideration correspond to the condition r _B > c/ω _pe or B ² > 4πn _e m _e c ², which leads to strong charge separation in the filament and relativistic electron drift. It is demonstrated that the specific energy content in current structures is high at a filament current of 10–15 kA: from 100 J/cm³ at a plasma density of 10¹⁴ cm⁻³ (the parameters of a lightning leader) to 10⁷ J/cm³ for a fully ionized atmospheric-pressure air. Estimates are presented showing that the Earth’s ionosphere, circumsolar space, and interstellar space are all Hall media in which current vortex structures can occur. A localized cylindrical equilibrium with a magnetic field reversal is constructed—an equilibrium that correlates with the magnetic structures observed in intergalactic space. It is shown that a magnetized plasma can be studied by using evolutionary equations for the electron and ion Lagrangian invariants I _e and I _i . An investigation is carried out of the evolution of a current-carrying plasma in a cylinder with a strong external magnetic field and with a longitudinal electron current turned on in the initial stage—an object that can serve as the simplest electrodynamic model of a tokamak. In this case, it is assumed that the plasma conductivity is low in the initial stage and then increases substantially with time. Based on the conservation of the integral momentum of the charged particles and electromagnetic field in a plasma cylinder within a perfectly conducting wall impenetrable by particles, arguments are presented in support of the generation of a radial electric field in a plasma cylinder and the production of drift ion fluxes along the cylinder axis. A hypothesis is proposed that the ionized intergalactic gas expands under the action of electromagnetic forces.     

Analysis of a mathematical model for the growth of tumors

 In this paper we study a recently proposed model for the growth of a nonnecrotic, vascularized tumor. The model is in the form of a free-boundary problem whereby the tumor grows (or shrinks) due to cell proliferation or death according to the level of a diffusing nutrient concentration. The tumor is assumed to be spherically symmetric, and its boundary is an unknown function r=s(t). We concentrate on the case where at the boundary of the tumor the birth rate of cells exceeds their death rate, a necessary condition for the existence of a unique stationary solution with radius r=R ₀ (which depends on the various parameters of the problem). Denoting by c the quotient of the diffusion time scale to the tumor doubling time scale, so that c is small, we rigorously prove that (i) lim inf _t→∞ s(t)>0, i.e. once engendered, tumors persist in time. Indeed, we further show that (ii) If c is sufficiently small then s(t)→R ₀ exponentially fast as t→∞, i.e. the steady state solution is globally asymptotically stable. Further, (iii) If c is not “sufficiently small” but is smaller than some constant γ determined explicitly by the parameters of the problem, then lim sup _t→∞ s(t)<∞; if however c is “somewhat” larger than γ then generally s(t) does not remain bounded and, in fact, s(t)→∞ exponentially fast as t→∞. Received: 25 February 1998 / Revised version: 30 April 1998     

Photosynthesis and photoinhibition in two differently coloured varieties of <Emphasis Type="Italic">Oxalis triangularis</Emphasis> — the effect of anthocyanin content

The purpose of this study was to clarify effects of anthocyanins on photosynthesis and photoinhibition in green and red leaves of Oxalis triangularis. Gas analysis indicated that green plants had the highest apparent quantum yield for CO₂ assimilation [0.051 vs. 0.031 μmol(CO₂) μmol⁻¹(photon)] and the highest maximum photosynthesis [10.07 vs. 7.24 μmol(CO₂) m⁻² s⁻¹], while fluorescence measurements indicated that red plants had the highest PSII quantum yield [0.200 vs. 0.143 μmol(e⁻) μmol⁻¹(photon)] and ETR_max [66.27 vs. 44.34 μmol(e⁻) m⁻² s⁻¹]. Red plants had high contents of anthocyanins [20.11 mg g⁻¹(DM)], while green plants had low and undetectable levels of anthocyanin. Red plants also had statistically significantly (0.05>p>0.01) lower contents of xanthophyll cycle components [0.63 vs. 0.76 mg g⁻¹(DM)] and higher activities of the reactive oxygen scavenging enzyme ascorbate peroxidase [41.2 vs. 10.0 nkat g⁻¹(DM)]. Anthocyanins act as a sunscreen, protecting the chloroplasts from high light intensities. This shading effect causes a lower photosynthetic CO₂ assimilation in red plants compared to green plants, but a higher quantum efficiency of photosystem II (PSII). Anthocyanins contribute to photoprotection, compensating for lower xanthophyll content in red plants, and red plants are less photoinhibited than green plants, as illustrated by the F_v/F_m ratio.     

Global Dynamics of an In-host Viral Model with Intracellular Delay

The dynamics of a general in-host model with intracellular delay is studied. The model can describe in vivo infections of HIV-I, HCV, and HBV. It can also be considered as a model for HTLV-I infection. We derive the basic reproduction number R ₀ for the viral infection, and establish that the global dynamics are completely determined by the values of R ₀. If R ₀≤1, the infection-free equilibrium is globally asymptotically stable, and the virus are cleared. If R ₀>1, then the infection persists and the chronic-infection equilibrium is locally asymptotically stable. Furthermore, using the method of Lyapunov functional, we prove that the chronic-infection equilibrium is globally asymptotically stable when R ₀>1. Our results shows that for intercellular delays to generate sustained oscillations in in-host models it is necessary have a logistic mitosis term in target-cell compartments.