Linearized oscillations in population dynamics

A linearized oscillation theorem due to Kulenović, Ladas and Meimaridou (1987,Quart. appl. Math. XLV, 155–164) and an extension of it are applied to obtain the oscillation of solutions of several equations which have appeared in population dynamics. They include the logistic equation with several delays, Nicholson's blowflies model as described by Gurney, Blythe and Nisbet (1980,Nature, Lond. 287, 17–21) and the Lasota-Wazewska model of the red blood cell supply in an animal. We also developed a linearized oscillation result for difference equations and applied it to several equations taken from the biological literature.     

Temperature‐Dependent Growth and Life Cycle of Nemoura sichuanensis (Plecoptera: Nemouridae) in a Chinese Mountain Stream

Plecoptera constitute a numerically and ecologically significant component in mountain streams all over the world, but little is known of their life cycles in Asia. The life cycle of Nemoura sichuanensis and its relationship to water temperature was investigated during a 4‐year study in a headwater stream (known as the Jiuchong torrent) of the Xiangxi River in Central China. Size structure histograms suggest that the life cycle was univoltine, and the relationships between the growth of Nemoura sichuanensis, physiological time, and effective accumulated water temperature were described using logistic regressions. The growth pattern was generally similar within year classes but growth rates did vary between year‐classes. Our field data suggest a critical thermal threshold for emergence in Nemoura sichuanensis, that was close to 9 °C. The total number of physiological days required for completing larval development was 250 days. The effective accumulated water temperature was 2500 degree‐days in the field. Development during the life cycle increased somewhat linearly with the physiological time and the effective accumulated water temperature, but some non‐linear relationships were best developed by logistic equations. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Transient kinetics of yeast growth in batch and continous culture with an inhibitory carbon and energy source

Growth and acetate metabolism by Candida utilis (ATCC 9226) is reported for both acetate- and zinc-limited cultures in defined media. Acetate concentrations were varied from suboptimal to inhibitory levels in both types of media in differential shake flask culture and in batch and continuous cultures in stirred fermentors. Transient responses of steady-state cultures to small or large additions of concentrated sodium acetate, or to shifts in dilution rate or inlet acetate concentration are compared with one another and with simple mathematical models of growth and acetate metabolism. Exponential growth was observed during unrestricted growth (differential shake flask and batch cultures) with both types of media. Addition of acetate during unrestricted growth always caused lags and for larger pulses, lower specific growth rates were observed after exponential growth resumed. Inhibition by high acetate concentrations was much greater in acetate–limited than in zinc–limited cultures. During restricted growth (steady-state, continuous cultures), high acetate concentrations again consistently caused growth lags but stimulated, inhibited, or temporarily stopped acetage uptake. Qualitative agreement between the predictions of a simple mathematical model of acetate inhibition fitted to differential shake flask data and the observed transient data was surprisingly good.     

A mathematical model of the food-consumer system I. A case without food replenishment

A simple mathematical model was proposed to describe the dynamics of a food-consumer system. The model was based on the Logistic Theory and consisted of Eqs. (4), (5) and (6). The model was divided into the following three cases for further analyss; i) without food supply except at the initial time, ii) with continuous food supply at a constant rate, and iii) with food supply at varying rates. Only the first model was dealth with in this paper. The assumptions of the model 1 are that a definite amount of food is given only once at the initial time and only the feeding by animals is responsible for the decrease of food, and that the rate of decrease is proportional to the amount of animals. It is also assumed that the growth of animal population is represented by the logistic curve, and that the upper limit of the population is proportional to the amount of food at that time. For simplicity the parameters of basic differential equations are assumed to be constant throughout the time course. Analytical solutions of this non-linear model were given by Eqs. (8), (9), (10) and (11). The properties of time course of the food amount and consumer population were discussed from the mathematical and biological points of view. The method of the estimation of the three constants λ,b, and c from the experimental data was also suggested. Since we had no available data for animal populations, we applied the model, regarding reserve substance as x and new plant body as y, to the data of the initial growth of Azuki bean plant in the dark. This model is very simple, but it may be useful for analyzing the behavior of food-consumer system. And it may give some clue to the analysis of the more complex systems.     

On relationships between various distribution functions in balanced unicellular growth

The relationships between various size distributions in balanced exponential growth of a batch culture of microorganisms are presented. Starting from the partial differential integral equations (Eakmanet al., 1966; Fredricksonet al., 1967) derived for the growth of a microbial culture expressions are obtained for the growth rate of organisms of specific size and size range. These expressions were first obtained by Collins and Richmond (1962) by an entirely different method. Also derived are equations which link probability functions, which are basic to the growth of a microbial culture, with other size distributions that can be estimated experimentally.     

Kinematics of plant growth.

Many of the concepts and equations which have been used in the study of compressible fluids can be applied to problems of plant development. Growth field variables, i.e. functions of position in the plant and of time, can be specified in either Eulerian (spatial) or Lagrangian (material) terms. The two specifications coincide only when the spatial distribution of the variable is steady, and steady patterns are most likely to emerge when an apex is chosen as origin of the co-ordinate system. The growth field itself can be described locally by the magnitude and orientation of the principal axes of the rate of strain tensor and by the vorticity tensor. Material derivatives can be calculated if the temporal and spatial variation in both growth velocity, u (rate of displacement from a material origin), and the variable of interest are known. The equation of continuity shows the importance of including both growth velocity, u, and growth rate, ▽ ·u in estimates of local biosynthesis and transport rates in expanding tissue, although the classical continuity equation must be modified to accommodate the compartmentalized distributions characteristic of plant tissue. Relatively little information on spatial variation in plant organs can be found in the botanical literature, but the current availability of interactive computer graphics equipment suggests that analysis of the spatial distribution of growth rates at least is no longer difficult.     

A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model

Continuous-time birth-death Markov processes serve as useful models in population biology. When the birth-death rates are nonlinear, the time evolution of the first n order moments of the population is not closed, in the sense that it depends on moments of order higher than n. For analysis purposes, the time evolution of the first n order moments is often made to be closed by approximating these higher order moments as a nonlinear function of moments up to order n, which we refer to as the moment closure function. In this paper, a systematic procedure for constructing moment closure functions of arbitrary order is presented for the stochastic logistic model. We obtain the moment closure function by first assuming a certain separable form for it, and then matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) equations for some initial time and set of initial conditions. The separable structure ensures that the steady-state solutions for the approximate equations are unique, real and positive, while the derivative matching guarantees a good approximation, at least locally in time. Explicit formulas to construct these moment closure functions for arbitrary order of truncation n are provided with higher values of n leading to better approximations of the actual moment dynamics. A host of other moment closure functions previously proposed in the literature are also investigated. Among these we show that only the ones that achieve derivative matching provide a close approximation to the exact solution. Moreover, we improve the accuracy of several previously proposed moment closure functions by forcing derivative matching.     

A stochastic computer model for simulating population growth

A model is described for investigating the interactions of age-specific birth and death rates, age distribution and density-governing factors determining the growth form of single-species populations. It employs Monte Carlo techniques to simulate the births and deaths of individuals while density-governing factors are represented by simple algebraic equations relating survival and fecundity to population density. In all respects the model's behavior agrees with the results of more conventional mathematical approaches, including the logistic model andLotka's Law, which predicts a relationship betwen age-specific rates, rate of increase and age distribution. Situations involving exponential growth, three different age-independent density functions affecting survival, three affecting fecundity and their nine combinations were tested. The one function meeting the assumptions of the logistic model produced a logistic growth curve embodying the correct values or r_m and K. The others generated sigmoid curves to which arbitrary logistic curves could be fitted with varying success. Because of populational time lags, two of the functions affecting fecundity produced overshoots and damped oscillations during the initial approach to the steady state. The general behavior of age-dependent density functions is briefly explored and a complex example is described that produces population fluctuations by an egg cannibalism mechanism similar to that found in the flour beetle Tribolium. The model is free of inherent time lags found in other discrete time models yet these may be easily introduced. Because it manipulates separate individuals, the model may be combined readily with the Monte Carlo simulation models of population genetics to study eco-genetic phenomena.     

Description of the delayed microbial growth by an extended logistic equation

Logistic equations are suitable for describing microbial growth. By means of VERHULST'S logistic equation, the adaptation to sigmoid-shaped curves of growth improves with a falling ratio C_xo/C_{x, max} < 0.2, if there is no lag-phase. The known logistic equations do not take into account any lag-phase behaviour, so that noticeable deviations in the model adaptation result in this range. Therefore, an extended logistic equation of rate is proposed by which any occuring lag-time is expressed by a 1st-order lag-term. The corresponding time law allows a very good adaptation of curves of delayed growth behaviour, and changes into VERHULST'S logistic equation for a lag-time t_L = 0. Application is facilitated by instructions for the numerical determination.     

Estimate of population extinction risk and its application to ecological risk management

Environmental threats, such as habitat size reduction or environmental pollution, may not cause immediate extinction of a population but may shorten the expected time to extinction. We developed a method to estimate the mean time to extinction for a density-dependent population with environmental fluctuation and to compare the impacts of different risk factors. We first derived a formula of the mean extinction time for a population with logistic growth and environmental and demographic stochasticities expressed as a stochastic differential equation model (canonical model). The relative importance of different risk factors is evaluated by the decrease in the mean extinction time. We studied an approximated formula for the reduction in habitat size that enhances extinction risk by the same magnitude as a given decrease in survivorship caused by toxic chemical exposure. In a large population (large K) or in a slowly growing population (small r), a small decrease in survivorship can cause the extinction risk to increase, corresponding to a significant reduction in the habitat size. Finally, we studied an approximate maximum likelihood estimate of three parameters (intrinsic growth rate r, carrying capacity K, and environmental stochasticity σ ² _e ) from time series data. By Monte Carlo sampling, we can remove the bias very effectively and determine the confidence interval. We discuss here how the reliability of the estimate changes with the length of time series. If we know the intrinsic rate of population growth r, the mean extinction time is estimated quite accurately even when only a short time series is available for parameter estimation. Received: March 31, 1999 / Accepted: November 9, 1999     

Effects of different leaf traits on growth rates of insect herbivores on willows

We examined relative effects of traits of leaf quality of ten willow species (Salix: Salicaceae) on growth rates of five species of insect herbivores found in interior Alaska (a willow sawfly, Nematus calais; the tiger swallowtail butterfly, Papilio canadensis; and three species of chrysomelid beetles, Gonioctena occidentalis, Calligrapha verrucosa, and Chrysomela falsa). Leaf traits examined were water content, toughness, total nitrogen contnet, pubescence, and presence or absence of phenolic glycosides. Of ten Salix species, four species contain phenolic glycosides in their leaves. We examined relative effects of water content, toughness, and nitrogen content of the Salix leaves on larval growth rates at three different levels, i.e., on a single host species, between different host species, and between herbivore species. The within-host analyses showed that effects of water content, toughness and/or nitrogen content on herbivore growth rates were generally significant in early-season herbivores but not in late-season herbivores. For each herbivore species, differences in growth rates between hosts were not explained by differences in water content, toughness, or nitrogen content. The between-herbivore analysis showed that the interspecific difference in larval growth rates were related to difference in water and nitrogen content of the hosts. Pubescence of Salix leaves had little effects on herbivore growth rates. Presence of phenolic glycosides had a positive effects on growth rates of a specialist, N. calais, but no effect on the other specialist, Ch. falsa. Presence of phenolic glycosides had, in general, negative effects on growth rates of nonspecialists, G. occidentalis, C. verrucosa, and P. canadensis.     

Synthesis, cytotoxicity and human telomerase inhibition activities of a series of 1,2-heteroannelated anthraquinones and anthra[1,2-d]imidazole-6,11-dione homologues

A series of 1,2-heteroannelated anthraquinones and anthra[1,2-d]imidazole-6,11-dione tetracyclic analogues with different side chain were prepared using an various synthetic route via acylation, cyclization, condensation, and intramolecular heterocyclization. Tetracyclic system containing alkyl and aryl, aromatic and heterocyclic, linear and cyclic, polar and apolar, and basic and acids residues were incorporated. They were evaluated for their effects on telomerase activity, hTERT expression, cell proliferations, and in vitro cytotoxicity against NCI’s 60 cell line human tumor screen. Compounds 4, 11, 12, 14, 15, 16, 17, 19, 20, 23, 25, and 26 were selected by the NCI for one dose screening program and further studies on 4, 23 and 25 where the curves cross these lines represent the interpolated values to cause 50% growth inhibition (GI₅₀), total growth inhibition (TGI) and 50% cell killing (LC₅₀), respectively. Further studies did not reveal any compound that showed potent and significant on telomerase inhibitory activity and hTERT repressing ability. Comparative testing of these compounds in the NCI’s screen revealed varying levels of potency and differential cytotoxicity, apparently related to the unsaturation levels in and substitution patterns on the core ring system. It appeared that addition of a fourth planar aromatic system to a tricyclic chromophore might enhances potent cytotoxic agents, at a level equivalent to a second side chain in one of the tricyclic series. Although the exact mechanism of how this pharmacophore contributes to its activity is still unclear, however, the group in the extended arm of the tetracyclic system might contribute to proper binding to the residues within the grove of G-quadruplex structure.     

Estimation of growth regulation in natural populations by extended family of growth curve models with fractional order derivative: Case studies from the global population dynamics database

Estimating the trend in population time series data using growth curve models is a central idea in population ecology. Several models, mainly governed by differential or difference equations, have been applied to real data sets to identify general growth pattern and make predictions. In this article, we analyze ecological time series data by fitting mathematical models governed by fractional differential equations (FDE). The order of the FDE (α) is used to quantify the evidence of memory in the population processes. The application of FDE is exemplified by analyzing time series data on two bird species Phalacrocorax carbo (Great cormorant) and Parus bicolor (Tufted titmouse) and two mammal species Castor canadensis (Beaver) and Ursus americanus (American black bear) extracted from the global population dynamics database. Five different population growth models were fitted to these data; density-independent exponential, negative density-dependent logistic and θ-logistic model, positive density-dependent exponential Allee and strong Allee model. Both ordinary and fractional derivative representations of these models were fitted to the time series data. Markov chain Monte Carlo (MCMC) method was used to estimate the model parameters and Akaike information criterion was used to select the best model. By estimating the return rate for each of the time series, we have shown that populations governed by FDE with a small value of α (high level of memory) return to the stable equilibrium faster. This demonstrates a synergistic interplay between memory and stability in natural populations.     

Change in individual growth rate and its link to gill-net fishing in two sympatric whitefish species

Size-selective fishing is expected to affect traits such as individual growth rate, but the relationship between the fishery-linked selection differentials and the corresponding phenotypic changes is not well understood. We analysed a 25-year monitoring survey of sympatric populations of the two Alpine whitefish Coregonus albellus and C. fatioi. We determined the fishing-induced selection differentials on growth rates, the actual change of growth rates over time, and potential indicators of reproductive strategies that may change over time. We found marked declines in adult growth rate and significant selection differentials that may partly explain the observed declines. However, when comparing the two sympatric species, the selection differentials on adult growth were stronger in C. albellus while the decline in adult growth rate seemed more pronounced in C. fatioi. Moreover, the selection differential on juvenile growth was significant in C. albellus but not in C. fatioi, while a significant reduction in juvenile growth over the last 25 years was only found in C. fatioi. Our results suggest that size-selective fishing affects the genetics for individual growth in these whitefish, and that the link between selection differentials and phenotypic changes is influenced by species-specific factors.     

Spiking neural network simulation: numerical integration with the Parker-Sochacki method

Mathematical neuronal models are normally expressed using differential equations. The Parker-Sochacki method is a new technique for the numerical integration of differential equations applicable to many neuronal models. Using this method, the solution order can be adapted according to the local conditions at each time step, enabling adaptive error control without changing the integration timestep. The method has been limited to polynomial equations, but we present division and power operations that expand its scope. We apply the Parker-Sochacki method to the Izhikevich ‘simple’ model and a Hodgkin-Huxley type neuron, comparing the results with those obtained using the Runge-Kutta and Bulirsch-Stoer methods. Benchmark simulations demonstrate an improved speed/accuracy trade-off for the method relative to these established techniques.

Homeostatic control of thyroxin concentration expressed by a set of linear differential equations

The purpose of this work is to express current concepts on the relationship between the rates of secretion of thyroxin and of thyroid stimulating hormone (TSH) by a set of linear differential equations (two attempts have been made previously in this direction; cf. Roston,Bull. Math. Biophysics,21, 271–282, 1959; Danziger and Elmergreen,Bull. Math. Biophysics,16, 15–21, 1954), and to show that the solutions to these equations fulfill two criteria: that they correctly express the previously observed behavior of thyroxin and TSH, and that they allow certain predictions to be made which are amenable to experimental verification or disproval by currently existing techniques. This mathematical model is necessarily only an approximation of reality.     

Calculation of the cumulative distribution function of the time to a small observable tumor

Multistage mathematical models of carcinogenesis (when applied to tumor incidence data) have historically assumed that the growth kinetics of cells in the malignant state are disregarded and the formation of a single malignant cell is equated with the emergence of a detectable tumor. The justification of this simplification is, from a mathematical point of view, to make the estimation of tumor incidence rates tractable. However, analytical forms are not mandatory in the estimation of tumor incidence rates. Portier et al. (1996b, Math. Biosci. 135, 129–146) have demonstrated the utility of the Kolmogorov backward equations in numerically calculating tumor incidence. By extending their results, the cumulative distribution function of the time to a small observable tumor may be numerically obtained.     

1-(2-((Z)-6-(2-(Trifluoromethyl)phenyl)hexa-3-en-1,5-diynyl)phenyl)piperidin-2-one as a new potent apoptosis agent

Compounds 4a–f, 5a–f and 6–9, showed significant growth inhibition activity against human tumor cell lines. Of these compounds, 1-(2-((Z)-6-(2-(trifluoromethyl)phenyl)hexa-3-en-1,5-diynyl)phenyl)piperidin-2-one (8) displayed the most potent growth inhibition activity. Compound 8 also arrested cancer cells in G2/M phase and induced apoptosis via activation of caspase-3 and -9. According to western-blotting analysis, compound 8 can up-regulate Bax, down-regulate Bcl-2 and XIAP, as well as promote cytochrome c release.     

Spatiotemporal dynamics of two generic predator–prey models

We present the analysis of two reaction–diffusion systems modelling predator–prey interactions, where the predator displays the Holling type II functional response, and in the absence of predators, the prey growth is logistic. The local analysis is based on the application of qualitative theory for ordinary differential equations and dynamical systems, while the global well-posedness depends on invariant sets and differential inequalities. The key result is an L ^∞-stability estimate, which depends on a polynomial growth condition for the kinetics. The existence of an a priori L ^p -estimate, uniform in time, for all p≥1, implies L ^∞-uniform bounds, given any nonnegative L ^∞-initial data. The applicability of the L ^∞-estimate to general reaction–diffusion systems is discussed, and how the continuous results can be mimicked in the discrete case, leading to stability estimates for a Galerkin finite-element method with piecewise linear continuous basis functions. In order to verify the biological wave phenomena of solutions, numerical results are presented in two-space dimensions, which have interesting ecological implications as they demonstrate that solutions can be ‘trapped’ in an invariant region of phase space.