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     

Bounds on the total population for species governed by reaction-diffusion equations in arbitrary two-dimensional regions

Analysis based on the integration of differential inequalities is employed to derive upper and lower bounds on the total populationN(t) = ∫ _R θ(x ₁,x ₂,t) dx ₁ dx ₂ of a biological species with an area-density distribution function θ=θ(x ₁,x ₂,t) (≥0) governed by a reaction-diffusion equation of the form ∂θ/∂t =D∇²θ +fθ −gθ ⁿ⁺¹ whereD (>0),n (>0),f andg are constant parameters, θ=0 at all points on the boundary ∂R of an (arbitrary) two-dimensional regionR, and the initial distribution (θ(_{x 1},x ₂, 0) is such thatN(0) is finite. Forg≥0 withR the entire two-dimensional Euclidean space, a lower bound onN(t) is obtained, showing in particular thatN(∞) is bounded below by a finite positive quantity forf≥0 andn>1. An upper bound onN(t) is obtained for arbitrary bounded or unbounded)R withn=1,f andg negative, and ∫ _R θ(x ₁,x ₂, 0)² dx ₁ dx ₂ sufficiently small in magnitude, implying that the population goes to extinction with increasing values of the time,N(∞)=0. Forg≥0 andR of finite area, the analysis yields upper bounds onN(t), predicting eventual extinction of the population if eitherf≤0 or if the area ofR is less than a certain grouping of the parameters in cases for whichf is positive. These results are directly applicable to biological species with distributions satisfying the Fisher equation in two spatial dimensions and to species governed by certain specialized population models.     

Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations

 In this paper we study the existence of one-dimensional travelling wave solutions u(x, t)=φ(x−ct) for the non-linear degenerate (at u=0) reaction-diffusion equation u _t =[D(u)u _x ] _x +g(u) where g is a generalisation of the Nagumo equation arising in nerve conduction theory, as well as describing the Allee effect. We use a dynamical systems approach to prove: 1. the global bifurcation of a heteroclinic cycle (two monotone stationary front solutions), for c=0, 2. The existence of a unique value c ^*>0 of c for which φ(x−c ^* t) is a travelling wave solution of sharp type and 3. A continuum of monotone and oscillatory fronts for c≠c ^*. We present some numerical simulations of the phase portrait in travelling wave coordinates and on the full partial differential equation. Received 15 December 1995; received in revised form 14 May 1996     

Galerkin-Ritz procedures for approximate solutions to systems of reaction-diffusion equations

For any essentially nonlinear system of reaction-diffusion equations of the generic form ∂c_i/∂t=D_i∇²c_i+Q_i(c,x,t) supplemented with Robin type boundary conditions over the surface of a closed bounded three-dimensional region, it is demonstrated that all solutions for the concentration distributionn-tuple function c=(c ₁(x,t),...,c _n (x,t)) satisfy a differential variational condition. Approximate solutions to the reaction-diffusion intial-value boundary-value problem are obtainable by employing this variational condition in conjunction with a Galerkin-Ritz procedure. It is shown that the dynamical evolution from a prescribed initial concentrationn-tuple function to a final steady-state solution can be determined to desired accuracy by such an approximation method. The variational condition also admits a systematic Galerkin-Ritz procedure for obtaining approximate solutions to the multi-equation elliptic boundary-value problem for steady-state distributions c=−c(x). Other systems of phenomenological (non-Lagrangian) field equations can be treated by Galerkin-Ritz procedures based on analogues of the differential variational condition presented here. The method is applied to derive approximate nonconstant steady-state solutions for ann-species symbiosis model.     

Dynamics of production oftrans-zeatin andtrans-zeatin riboside by immobilized cytokinin-autonomous and cytokinin-dependent tobacco cells

Two strains of cultured tobacco cells (Nicotiana tabacum L. cv. Wisconsin 38) differing in their requirement for exogenous cytokinins (cytokinin-dependent and cytokinin-autonomous) were immobilized on polyphenylenoxide (Sorfix) activated with glutaraldehyde. Columns packed with immobilized cells were continually eluted with diluted Murashige and Skoog's medium lacking or supplemented with synthetic cytokinin (6-benzylaminopurine; BA). Purified samples of column eluates were fractionated by HPLC, andtrans-zeatin (t-Z) andtrans-zeatin riboside (t-ZR) content was estimated by enzyme immunoassay. Both cytokinin-autonomous and cytokinin-dependent tobacco cells produced and excretedt-Z and its riboside, and there were significant quantitative differences between the strains. The steady-state excretion rate oft-Z was 19.8 ng · g^–1 dw · h^–1 and 4 ng · g^–1 dw · h^–1, respectively, and that oft-ZR 4 ng · g^–1 dw · h^–1 and 1 ng · g^–1 dw · h^–1, respectively. Exposure of cytokinin-dependent cells to BA after 72 h of starving for this synthetic cytokinin caused temporary increase in excretion of both zeatin and its riboside. After the application of 5 M BA for 24 h, the excretion rate oft-ZR reached 5 ng · g^–1 dw · h^–1 (5-fold increase), and that oft-Z achieved 12 ng · g^–1 dw · h^–1 (3-fold increase). The elevation oft-Z excretion was delayed about 13 h compared witht-ZR excretion, which started increasing almost immediately after BA application. A pulse of BA in lower concentration (1.5 M for 30 h) provoked lower response.     

A note on imitative behavior

In a preceding paper (Bull. Math. Biophysics,27, 175–185) the distribution function ofφ=ɛ ₁-ɛ ₂,—the difference of excitations in the two mutually inhibiting centers, has been derived in terms of the distribution functionsf ₁(ɛ ₁) andf ₂(ɛ ₂) of the two excitations. In the present note some properties of the distribution functionf(ϕ) in terms of the propertiesf ₁(ɛ ₁) andf ₂(ɛ ₂) are derived.     

Characterizing conditions for generalized Verhulst logistic growth of a biological population

It is shown that a biological population or subpopulation composed ofN individuals can be governed by a generalized Verhulst logistic equation with time-dependent rate functions if and only if certain characterizing conditions are satisfied by ∂N/∂N ₀ whereN ₀ is the value ofN att=0.     

Complete ablation of small squamous cell carcinoma xenografts with186Re-labeled monoclonal antibody E48

In previous studies we have shown that in mice bearing head and neck squamous cell carcinoma (HNSCC) xenografts, radioimmunotherapy (RIT) with¹⁸⁶Re-labeled MAb E48 resulted in complete regressions in one-third of the tumors (followup >150 d). MAb E48 was labeled with¹⁸⁶Re following a novel labeling procedure developed at our institute. The injected dose was 600 μCi, which was the maximum tolerated dose (MTD; <15% wt loss) in these studies. The mean size of the tumors was 140±60 mm³. To investigate whether the therapeutic efficacy of RIT in our xenograft model would be improved when treating smaller xenografts, mice bearing 2 HNSCC xenografts with a vol of 75±17 mm³ (number of mice,n=6; number of tumors,t=12) were treated with 600 μCi of¹⁸⁶Re-labeled MAb E48 IgG. All tumors completely regressed and did not regrow during followup (>150 d). In all mice, weight loss did not exceed 10%. to obtain biodistribution data, mice bearing two xenografts with a vol of 58±31 mm³ were injected with 100 μCi of¹⁸⁶Re-labeled MAb E48 IgG. The maximum uptake in blood was 26.4% injected dose/g (%ID·g⁻¹) at 2 h pi and was 53.1%ID·g⁻¹ in the tumor at d 7 pi. In normal tissues, no nonspecific accumulation was observed. Based on these biodistribution data, the absorbed cumulative radiation dose was calculated. The accumulated dose in blood and tumor was 2004 cGy and 8580 cGy, respectively. In other tissues, the dose was less than 8.1% of the dose delivered to tumor. These data implicate that RIT with¹⁸⁶Re-labeled MAb E48 may be especially suited to be used as adjuvant therapy for the treatment of head and neck cancer patients with minimal residual disease.     

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.     

The gas washout determination under a symmetry assumption

In an open circuit gas washout determination the output of test substance is shown to be of the form ∑ _i=0 ^∞ A _i λ _i ^k on thekth expiration whereA _i >0,i=1,2,... and 1 > λ₁ ≥ λ₂...≥ 0 provided the transition from inspiration to expiration has certain symmetry properties with the transition from expiration to inspiration. In general, no direct physical interpretation such as volume for theA _t ’s or fraction of retained gas on expiration for the λ _i is justified.     

Microdistribution of sulfate-reducing bacteria in sediments of a hypertrophic lake and their response to the addition of organic matter

To clarify the ecological significance of the association of sulfate-reducing bacteria (SRB) with sediment particle size, SRB utilizing lactate (l-SRB), propionate (p-SRB) and acetate (a-SRB) were examined with different sizes of sediment particles in a hypertrophic freshwater lake using the anaerobic plate count method. The numbers ofl-SRB anda-SRB were 10⁴–10⁵ colony forming units (CFU) per ml in the 0–3 cm layer and 10²–10³ CFU ml⁻¹ in the 10–13 cm layer while the numbers ofp-SRB were one or two orders lower than those ofl-SRB anda-SRB. A sediment suspension was fractionated into four fractions (<1, 1–10, 10–94 and >94 μm). The highest proportions ofl-SRB anda-SRB were found in the 10–94 μm fraction: 66–97% forl-SRB and 53–98% fora-SRB. The highest proportion ofp-SRB was found in the >94 μm fraction (70–74%). These results indicate that most SRB were associated with sediment particles. One isolate from an acetate-utilizing enrichment culture was similar toDesulfotomaculum acetoxidans, a spore-forming sulfate-reducing bacterium. When lactate and sulfate were added to sediment samples,l-SRB anda-SRB in the <10 μm-fraction grew more rapidly than those in whole sediment for the first 2 days. This result suggests that nutrients uptake by free-living and small particle-associated (<10 μm) SRB is higher than that by SRB associated with larger particles.     

Facultative hypothermia as a thermoregulatory strategy in the phyllostomid bats, Carollia perspicillata and Sturnira lilium

The present study questions whether hypothermia is an artifact due to captivity-induced stress or a thermoregulatory strategy for bats of the neotropical family Phyllostomidae. In Guanacaste, Costa Rica, Carollia perspicillata and Sturnira lilium exhibited a bimodal distribution of body temperatures when submitted to an ambient temperature of 21 °C. Body temperature was highly correlated with body mass in both species. C. perspicillata of mass ≥20 g and S. lilium of mass ≥17 g remained normothermic (body temperature >37 °C), whereas at masses below 18 g and 13 g, respectively, >80% of individuals were hypothermic (body temperature ≤32 °C). In two treatment groups for each species, we restricted food intake to ca. 20% of body mass on either night 1 or night 4 following capture. Hypothermia was significantly related to food-restriction, but not time in captivity. Metabolic rate (ml O₂ ·  g⁻¹ h⁻¹) at ambient temperature = 21 °C was MR = e ^{(–2.11 + 0.101 Tb)} (r ² = 0.7, P < 0.001) for C. perspicillata and MR = e ^{(−2.62 + 0.115 Tb)} (r ² = 0.89) for S. lilium. Free-ranging, radio tagged C. perspicillata exhibited daily depression of body temperature to 33–34 °C. We conclude that hypothermia is an thermoregulatory strategy that allows phyllostomid bats to adjust metabolic rate to feeding success and the level of fat stores. Accepted: 20 August 1996     

A fitness function for optimal life history in relation to density effects,competition, predation and stability of the environment

A fitness function (function maximized under natural selection) is studied in a population model in which the growth of a population is suppressed by crowding, density-independent continuous mortality (by euryphagous predators) and periodic disturbances. The dynamics of the population density between occurrence of disturbance can be expressed as,dN/dt=(F(N/K)−D)N, whereN is the population density,K is the carrying capacity,D is the density-independent continuous mortality, andF is the growth regulation factor described as a function of crowding (N/K). The period of disturbance isS. The survival rate under disturbance isu. It is concluded that the fitness function is (approximately) a product of competitive ability (C), carrying capacity, and degree of saturation, and is given byCKF ⁻¹(D−(lnu)/S). The degree of saturation is the inverse function of regulation factor (F) at the death rate due to predators and disturbance. I assume a population in which density is regulated only through survival. In this case, a low survival rate at the critical age-group means a high value ofCKF ⁻¹(D−(lnu)/S). Therefore, the reciprocal of the density-dependent survival rate at critical age-group is a measure of the fitness function. Using this measure, I predict the optimal age (body size) at first reproduction of a species of salamander. I also found that fitness calculated from observed values ofl(x) andm(x) includes a tautology. When the concept of fitness function is compared with the ESS method, the latter is more flexible. However, there is a possibility that an ESS is at the minimum of fitness function.     

Diffusion with attrition

This article treats the problem of the sharp front observed when a diffusing substance interacts irreversibly with binding sites within the medium. The model consists of two simultaneous partial differential equations that are nonlinear and cannot be solved in closed form. The parameters are the diffusion coefficient D in the direction under consideration (x), the interaction constant k, the binding-site concentration μ and the boundary concentration of the diffusing ion c ₀. Our aim is to develop methods to enable the estimation of these parameters from the experimental data. An analytical solution for the case k → ∞, as found by others, is given first and then a finite element analysis package is used to obtain numerical solutions for the general case. Graphs are presented to illustrate the effects of the various parameters. Simple graphical procedures are described to compute μ and c ₀. The position of the advancing front ξ then provides, together with μ, a way to estimate D. A mathematical identity relating D and x and a second one involving D, k and t help to reduce the complexity of the problem. A new, measurable quantity S(t) is defined as where f is the total concentration (free + bound) of the diffusing ion at time t, and detailed plots are furnished that permit the computation of k directly from S(t), μ and D. The accuracy with which such methods can be expected to determine the various parameters of the model is considered at some length. Finally, in a concluding section, we simulate typical experimental data, examine the validity of our methods, and see how their accuracy is affected by controlled amounts of various kinds of noise.     

Flight of the honey bee VI: energetics of wind tunnel exhaustion flights at defined fuel content,speed adaptation and aerodynamics

To gain information on extended flight energetics, quasi-natural flight conditions imitating steady horizontal flight were set by combining the tetheredflight wind-tunnel method with the exhaustion-flight method. The bees were suspended from a two-component aerodynamic balance at different, near optimum body angle of attack and were allowed to choose their own speed: their body mass and body weight was determined before and after a flight; their speed, lift, wingbeat frequency and total flight time were measured throughout a flight. These values were used to determine thrust, resultant aerodynamic force (magnitude and tilting angle), Reynolds number, total flight distance and total flight impulse. Flights in which lift was body weight were mostly obtained. Bees, flown to complete exhausion, were refed with 5, 10, 15 or 20 l of a 1.28-mol·l^-1 glucose solution (energy content w=18.5, 37.0, 55.5 or 74.0 J) and again flown to complete exhaustion at an ambient temperature of 25±1.5°C by a flight of known duration such that the calculation of absolute and relative metabolic power was possible. Mean body mass after exhaustion was 76.49±3.52 mg. During long term flights of 7.47–31.30 min similar changes in flight velocity, lift, thrust, aerodynamic force, wingbeat frequency and tilting angle took place, independent of the volume of feeding solution. After increasing rapidly within 15 s a more or less steady phase of 60–80% of total flight time, showing only a slight decrease, was followed by a steeper, more irregular decrease, finally reaching 0 within 20–30 s. In steady phases lift was nearly equal to resultant aerodynamic force; tilting angle was 79.8±4.0°, thrust to lift radio did not vary, thrust was 18.0±7.4% of lift, lift was somewhat higher/equal/lower than body mass in 61.3%, 16.1%, 22.6% of all totally analysable flights (n=31). The following parameters were varied as functions of volume of feeding solution (5–20 l in steps of 5 l) and energy content. (18.5–74.0 J in steps of 18.5 J): total flight time, velocity, total flight distance, mean lift, thrust, mean resultant aerodynamic force, tilting angle, total flight impulse, wingbeat frequency, metabolic power and metabolic power related to body mass, the latter related to empty, full and mean (=100 mg) body mass. The following positive correlations were found: L=1.069·10^-9 f ^2.538; R=1.629·10^-9 f ^2.464; P _m=7.079·10^-8 f ^2.456; P _m=0.008v+0.008; P _m=18.996L+0.022; P _m=19.782R+0.021; P _m=82.143T+0.028; P _m=1.245·bm _f ^1.424 ; P _{mrel e}=6.471·bm _f ^1.040 ; =83.248+0.385. The following negative correlations were found: V=3.939–0.032; T=1.324·10^-4–0.038·10^-4. Statistically significant correlations were not found in T(f), L(), R(), f(), P _m(bm _e), P _{m rel e}(bm _e), P _{m rel f}(bm _e), P _{m rel f}(bm _f).Abbreviations A(m²) frontal area - bl(m) body length - bm(mg) body mass - c(mol·1^-1) glucose concentration of feeding solution - c _D (dimensionless) drag coefficient, related to A - D(N) drag - F _w(N) body weight - F _wp weight of paper fragment lost at flight start - f wingbeat frequency (s^-1) - g(=9.81 m·s^-2) gravitational acceleration - I(Ns)=R(t) dt total impulse of a flight - L(N) lift vertical sustaining force component - P _m(J·s^-1=W) metabolic power - P_{m ret} (W·g^-1) metabolic power, related to body mass - R(N) resultant aerodynamic force - Re v·bl·v ^-1 (dimensionless) Reynolds number, related to body length - s(m) v(t) dt virtual flight distance of a flight - s(km) total virtual flight distance - T (N) thrust horizontal force component of horizontal flight - T _a (°C) ambient temperature - t(s) time - t _tot (s or min) total flight time - v(m·s^-1) flight velocity - v(l) volume of feeding solution - W (J) energy and energy content of V - ( °) body angle of attack between body longitudinal axis and flow direction - ( °) tilting angle ( 90°) between R and the horizont in horizontal flight v(=1.53·10^-5m²·s^-1 for air at 25°) kinematic viscosity - (=1.2 kg·m^-3 at 25°C) air density     

Interaction of maturation delay and nonlinear birth in population and epidemic models

 A population with birth rate function B(N) N and linear death rate for the adult stage is assumed to have a maturation delay T>0. Thus the growth equation N′(t)=B(N(t−T)) N(t−T) e⁻ d ₁ T−dN(t) governs the adult population, with the death rate in previous life stages d ₁≧0. Standard assumptions are made on B(N) so that a unique equilibrium N _e exists. When B(N) N is not monotone, the delay T can qualitatively change the dynamics. For some fixed values of the parameters with d ₁>0, as T increases the equilibrium N _e can switch from being stable to unstable (with numerically observed periodic solutions) and then back to stable. When disease that does not cause death is introduced into the population, a threshold parameter R ₀ is identified. When R ₀<1, the disease dies out; when R ₀>1, the disease remains endemic, either tending to an equilibrium value or oscillating about this value. Numerical simulations indicate that oscillations can also be induced by disease related death in a model with maturation delay. Received: 2 November 1998 / Revised version: 26 February 1999     

Log-relative growth: A new dendrochronological approach to study diameter growth in Cedrela odorata and Juglans neotropica,Central Forest,Peru

Many studies regarding growth in diameter at breast height (D) in trees suffer from several problems, including heteroscedasticity, temporal autocorrelation and very low statistical adjustments. In growth ring studies, growth models are sometimes omitted, presenting only a mean curve or smoothings, while studies that use models often do not address the above mentioned problems. For these reasons, this paper proposes a new approach to the classical modeling of D = f(t), where t is age (years), using the logarithmic transformation of the relative growth rate ln(1/D)(dD/dt) = ln f(D, A),where A is the asymptote of D based on the differential growth rate model of von Bertalanffy. High statistically significant adjustments for Cedrela odorata (ME = 65%, model efficiency, ME, an analogous to R² but for non-linear regressions), and Juglans neotropica (ME = 78%) were obtained and met all regression assumptions. These equations were integrated to obtain D = f(t) for both species, followed by self and independent validation. Based on these equations, different life history and silviculture traits were calculated for both species. This procedure does not appear to have been previously used in the study of tree growth.     

Theoretical study of isotope effects on the stereodynamics of H<Superscript>+</Superscript>+HD and its isotopic variant D<Superscript>+</Superscript>+HD

The stereodynamics for H⁺+HD and its isotopic variant D⁺+HD were studied with a quasi-classical trajectory (QCT) method at a collision energy of 0.7 eV on the ground 1¹A′ potential energy surface (PES). The polarization-dependent differential cross-sections (PDDCSs) in the center-of-mass frame are presented here. Furthermore, the distribution of the angle between k and j′, p(θ _r ) and the distribution of the dihedral angle p(ϕ _r ) were calculated and are discussed. The results indicate that isotopic substitution exerts substantial effects on the differential cross-section and the product’s rotational polarization.     

Temporal patterns of water flux in trees and lianas in a Panamanian moist forest

Using constant heat sap flow sensors, xylem water fluxes in ten tree species and two liana species were monitored for 5–10 days during the beginning of the wet season in May, 1993. For a subset of the trees, a branch was also monitored at the top of the crown for 5 days. Xylem flux (J _S) was related diurnally in all plants to vapor pressure deficit (D) measured within the upper-third of the canopy, and to incoming shortwave radiation R _S above the canopy. Cross-correlation analysis was used to estimate time lags between diurnal patterns of J _S and D or R _S, and between J _S in stems and branches. The maximum correlation coefficient from cross-correlation of J _S with R _S (range=0.57–0.92) was often higher than the maximum of J _S with D (range=0.43–0.89), indicating that diurnal J _S was more dependent on R _S than D. Time lags (lag corresponding to maximum correlation) of J _S at stem-base with D was shorter (0–45 min) than with radiation (5–115 min), highly variable within a species, and uncorrelated to the height or exposure of tree crowns or liana in the canopy. On a stand level, not accounting for the diel lag between stem sap flux and canopy flux resulted in errors in estimated canopy transpiration of up to 30%. Received: 19 October 1998 / Accepted: 8 June 1999     

Comparative study of immunobiological activities ofPseudomonas aeruginosa andBrucella melitensis lipopolysaccharides

The toxic activity ofBrucella melitensis andPseudomonas aeruginosa lipopolysaccharides as well as their behavior as immunogens, mitogens, and interferon inducers have been studied. Although their toxicities were very similar, the former molecule was incapable of eliciting a primary immune response in mice. Rabbit hyperimmunization gave titers half of those obtained withP. aeruginosa lipopolysaccharide. Optimal mitogenic responses of spleen cell cultures were obtained using 10–50 μg/ml and 50–100 μg/ml ofPseudomonas andBrucella lipopolysaccharide, respectively, giving the latter a lower stimulation of³H-thymidine uptake. Interferon titers induced in chickens byBrucella lipopolysaccharide were three times lower than those obtained withPseudomonas lipopolysaccharide.