Beyond average: an experimental test of temperature variability on the population dynamics of <Emphasis Type="Italic">Tribolium confusum</Emphasis>

The relationship between ectotherm ecology and climatic conditions has been mainly evaluated in terms of average conditions. Average temperature is the more common climatic variable used in physiological and population studies, and its effect on individual and population-level processes is well understood. However, the intrinsic variability of thermal conditions calls attention to the potential effects that this variability could have in ecological systems. Regarding this point, two hypotheses are proposed. From the allocation principle, it may be inferred that if temperature variability is high enough to induce stress in the organisms, then this extra-cost should reduce the energetic budget for reproduction, which will be reflected in population parameters. Moreover, a mathematical property of non-linear functions, Jensen’s inequality, indicates that, in concave functions, like the temperature–reproduction performance function, variability reduces the expected value of the output variable, and again modifies population parameters. To test these hypotheses, experimental cultures of Tribolium confusum under two different thermal variability regimens were carried out. With these data, we fitted a simple population dynamics model to evaluate the predictions of our hypothesis. The results show that thermal variability reduces the maximum reproductive rate of the population but no other parameters such as carrying capacity or the nonlinear factor in a nonlinear version of the Ricker model, which confirms our hypotheses. This result has important consequences, such as the paradoxical increase in population variability under a decrease in thermal variability and the necessary incorporation of climatic variability to evaluate the net effect of climate change on the dynamics of natural populations.     

Macrophage response to <Emphasis Type="Italic">Mycobacteriumtuberculosis</Emphasis> infection

The immune response to Mycobacterium tuberculosis (Mtb) infection is the formation of multicellular lesions, or granolomas, in the lung of the individual. However, the structure of the granulomas and the spatial distribution of the immune cells within is not well understood. In this paper we develop a mathematical model investigating the early and initial immune response to Mtb. The model consists of coupled reaction-diffusion-advection partial differential equations governing the dynamics of the relevant macrophage and bacteria populations and a bacteria-produced chemokine. Our novel application of mathematical concepts of internal states and internal velocity allows us to begin to study this unique immunological structure. Volume changes resulting from proliferation and death terms generate a velocity field by which all cells are transported within the forming granuloma. We present numerical results for two distinct infection outcomes: controlled and uncontrolled granuloma growth. Using a simplified model we are able to analytically determine conditions under which the bacteria population decreases, representing early clearance of infection, or grows, representing the initial stages of granuloma formation.     

Modeling toxoplasmosis spread in cat populations under vaccination

In this paper we present an epidemiological model to study the transmission dynamics of toxoplasmosis in a cat population under a continuous vaccination schedule. We explore the dynamics of toxoplasmosis at the population level using a mathematical model that includes the effect of oocyst, since the probability of acquisition of Toxoplasma Gondii infection depends on the environmental load of the parasite. This model considers indirectly the infection of prey through the oocyst shedding by cats. We prove that the basic reproduction number R₀ is a threshold value that completely determines the global dynamics and the outcome of the disease. Numerical computer simulations are presented to investigate different scenarios. These simulations show the effectiveness of a constant vaccination program.     

Mathematical Modelling of the <Emphasis Type="Italic">Aux/IAA</Emphasis> Negative Feedback Loop

The hormone auxin is implicated in regulating a diverse range of developmental processes in plants. Auxin acts in part by inducing the Aux/IAA genes. The associated pathway comprises multiple negative feedback loops (whereby Aux/IAA proteins can repress Aux/IAA genes) that are disrupted by auxin mediating the turnover of Aux/IAA protein. In this paper, we develop a mathematical model of a single Aux/IAA negative feedback loop in a population of identical cells. The model has a single steady-state. We explore parameter space to uncover a number of dynamical regimes. In particular, we identify the ratio between the Aux/IAA protein and mRNA turnover rates as a key parameter in the model. When this ratio is sufficiently small, the system can evolve to a stable limit cycle, corresponding to an oscillation in Aux/IAA expression levels. Otherwise, the steady-state is either a stable-node or a stable-spiral. These observations may shed light on recent experimental results.     

Coordinated Development of Yeast Colonies: Quantitative Modeling of Diffusion‐Limited Growth – Part 2

A mathematical model was developed which described the growth of yeast colonies based on the assumptions that (i) these populations were built up of single cells whose proliferation was (ii) exclusively controlled by nutrient availability in the environment. The model was of a hybrid cellular automaton type and described discrete cells residing on a one‐dimensional lattice as well as on continuously distributed nutrients. Experimental results and numerical calculations were compared to elucidate under which cultivation conditions the diffusion‐limited growth (DLG) was the major construction principle in yeast colonies. Simulations were scaled to the growth of Yarrowia lipolytica and Candida boidinii colonies under carbon and nitrogen limitation. They showed that nutrient‐controlled growth of the individual cells resulted in DLG of the population. Quantitative predictions for the spatio‐temporal development of the cell‐density profile inside a growing yeast mycelium were compared to the growth characteristics of the model yeast mycelia. Only for the carbon‐limited growth of C. boidinii colonies on glucose as the limiting nutrient resource did the DLG model reproduce the cell‐density profile estimated at the end of the cultivation. Under all other cultivation conditions, strong discrepancies between calculations and experimental results were evident precluding DLG as the ruling regulatory mechanism. Thus, whether or not the development of a yeast population could be described by a DLG scenario, was strongly dependent on the particular cultivation conditions and the applied yeast species. In those cases for which the DLG hypothesis failed to explain the observed growth patterns, the underlying assumptions, i.e., the complete absence of nutrient translocation between the individual cells inside the yeast mycelia as well as the exclusively nutrient‐controlled proliferation of the cells, have to be reevaluated. The presented study demonstrated how the mathematical analysis of growth processes in yeast populations could assist the experimental identification of potential regulatory mechanisms.     

Computer simulation study on the cyclicities ofTribolium population dynamics

Under a set of assumptions, a mathematical model was constructed to investigate the effect of cannibalistic behavior and medium renewal schedule on Tribolium population dynamics, and the results of simulation analyses were presented to show how modelling can contribute to a better understanding of experimental study. The analyses of the present model, the main concern of which is the cyclicity in Tribolium population. showed that there are two distinct factors which create cyclicity, the cannibalistic behavior in Tribolium itself, and the medium renewal schedule. Cannibalism per se does not necessarily cause cyclicity, but the combinations of cannibalistic behavior among various life stages and their relative intensities among them can cause cyclicity and can also determine the period of cycle. External factors also can generate cyclicity, but their interaction with cannibalistic behavior has a more significant effect in modifying the period of cycle. Some discrepancies between the model and experimental results were discussed.     

Near‐infrared spectroscopy for metabolite quantification and species identification

Near‐infrared (NIR) spectroscopy is a high‐throughput method to analyze the near‐infrared region of the electromagnetic spectrum. It detects the absorption of light by molecular bonds and can be used with live insects. In this study, we investigate the accuracy of NIR spectroscopy in determining triglyceride level and species of wild‐caught Drosophila. We employ the chemometric approach to produce a multivariate calibration model. The multivariate calibration model is the mathematical relationship between the changes in NIR spectra and the property of interest as determined by the reference analytical method. Once the calibration model was developed, we used an independent set to validate the accuracy of the calibration model. The optimized calibration model for triglyceride quantification yielded coefficients of determination of 0.73 for the calibration test set and 0.70 for the independent test set. Simultaneously, we used NIR spectroscopy to discriminate two species of Drosophila. Flies from independent sets were correctly classified into Drosophila melanogaster and Drosophila simulans with accuracy higher than 80%. These results suggest that NIRS has the potential to be used as a high‐throughput screening method to assess a live individual insect's triglyceride level and taxonomic status.     

Genetic divergence among sympatric colour morphs of the Dalmatian wall lizard (<Emphasis Type="Italic">Podarcis melisellensis</Emphasis>)

If alternative phenotypes in polymorphic populations do not mate randomly, they can be used as model systems to study adaptive diversification and possibly the early stages of sympatric speciation. In this case, non random mating is expected to support genetic divergence among the different phenotypes. In the present study, we use population genetic analyses to test putatively neutral genetic divergence (of microsatellite loci) among three colour morphs of the lizard Podarcis melisellensis, which is associated with differences in male morphology, performance and behaviour. We found weak evidence of genetic divergence, indicating that gene flow is somewhat restricted among morphs and suggesting possible adaptive diversification.     

Multiregional Periodic Matrix for Modeling the Population Dynamics of Sardine (<Emphasis Type="Italic">Sardina pilchardus</Emphasis>) Along the Moroccan Atlantic Coast: Management Elements for Fisheries

In this paper, we present a deterministic time discrete mathematical model based on multiregional periodic matrices to describe the dynamics of Sardina pilchardus in the Central Atlantic area of the Moroccan coast. This model deals with two stages (immature and mature) and three spatial zones where sardines are supposed to migrate from one zone to another. The population dynamics is described by an autonomous recurrence equation N(t + 1) = A.N(t), where A is a positive matrix whose entries are estimated using data collected during biannual acoustic surveys carried out from 2001 to 2003 onboard the Norwegian research vessel “Dr Fridtjof Nansen”. The dominant eigenvalue λ of A that gives the long-term growth rate of fish population is smaller than one. This agrees with the stock decrease observed in the data collected. We show that λ is highly sensitive to the recruitment rate and much less sensitive to the reproduction rate. These results can clearly be used to define an efficient scenario in order to fight for instance against a stock decrease.     

Modelling the population dynamics of <Emphasis Type="Italic">Daphnia obtusa</Emphasis> (Kurz) in Lake Orta (N. Italy) under pre- and post-liming conditions

A mathematical matrix model was formulated to investigate the response of Daphnia obtusa population dynamics to the changes in the water chemistry of Lake Orta before and after the liming operation. Model parameters were estimated from experimental laboratory data. Model analysis showed that water chemistry changes induced by liming affected mainly egg survival and predicted the highest population growth at pH␣6. Whereas increased egg mortality heavily inhibits population growth rate, the model still predicts a long term tendency of the population to increase in number. However, both before and after the liming operation due to high food availability in the laboratory, egg production was higher under all experimental conditions than in the field. When food limitation is accounted for and more realistic, field based estimates of egg production are used, the model predicts the extinction of D. obtusa population in the lake. This suggests that the effects of water chemistry changes on egg mortality had a critical role in the disappearance of D. obtusa from Lake Orta and may even adequately explain the extinction of this population.     

Mathematical models for the <Emphasis Type="Italic">Aedes aegypti</Emphasis> dispersal dynamics: Travelling waves by wing and wind

Biological invasion is an important area of research in mathematical biology and more so if it concerns species which are vectors for diseases threatening the public health of large populations. That is certainly the case for Aedes aegypti and the dengue epidemics in South America. Without the prospect of an effective and cheap vaccine in the near future, any feasible public policy for controlling the dengue epidemics in tropical climates must necessarily include appropriate strategies for minimizing the mosquito population factor. The present paper discusses some mathematical models designed to describe A. aegypti’s vital and dispersal dynamics, aiming to highlight practical procedures for the minimization of its impact as a dengue vector. A continuous model including diffusion and advection shows the existence of a stable travelling wave in many situations and a numerical study relates the wavefront speed to a few crucial parameters. Strategies for invasion containment and its prediction based on measurable parameters are analysed.     

A mathematical model of population dynamics for Batesian mimicry system

We analyse a mathematical model of the population dynamics among a mimic, a corresponding model, and their common predator populations. Predator changes its search-and-attack probability by forming and losing its search image. It cannot distinguish the mimic from the model. Once a predator eats a model individual, it comes to omit both the model and the mimic species from its diet menu. If a predator eats a mimic individual, it comes to increase the search-and-attack probability for both model and mimic. The predator may lose the repulsive/attractive search image with a probability per day. By analysing our model, we can derive the mathematical condition for the persistence of model and mimic populations, and then get the result that the condition for the persistence of model population does not depend on the mimic population size, while the condition for the persistence of mimic population does depend the predator's memory of search image.     

The Africanized honey bee dispersal: A mathematical zoom

A general mathematical model for population dispersal featuring long range taxis is presented and exemplified by the dispersal episode of the Africanized honey bees (Apis mellifera adansonii) throughout the American Continent. The mathematical model is a discrete-time and nonlocal model represented by an integrodifference recursion. A newtaxis concept is defined and introduced into the mathematical model by an appropriate modification of the redistribution kernel. The model is capable of predicting the natural barrier for the expansion of the Africanized honey bees in the southern part of the Continent due to low winter temperatures. It also describes a sensitive expansion velocity with respect to the quality of resources, which can explain the AHB’s astounding spread rate, by using two different kinds of population dynamics strategies, one for a resourceful environment and the other for poor regions. An erratum to this article is available at .     

Estimating the selective advantage of mutant p53 tumour cells to repeated rounds of hypoxia

The tumour suppressor gene, p53, plays an important role in tumour development. Under low levels of oxygen (hypoxia), cells expressing wild-type p53 undergo programmed cell death (apoptosis), whereas cells expressing mutations in the p53 gene may survive and express angiogenic growth factors that stimulate tumour vascularization. Given that cells expressing mutations in the p53 gene have been observed in many forms of human tumour, it is important to understand how both wild-type and mutant cells react to hypoxic conditions. In this paper a mathematical model is presented to investigate the effects of alternating periods of hypoxia and normoxia (normal oxygen levels) on a population of wild-type and mutant p53 tumour cells. The model consists of three coupled ordinary differential equations that describe the densities of the two cell types and the oxygen concentration and, as such, may describe the growth of avascular tumours in vitro and/or in vivo. Numerical and analytical techniques are used to determine how changes in the system parameters influence the time at which mutant cells become dominant within the population. A feedback mechanism, which switches off the oxygen supply when the total cell density exceeds a threshold value, is introduced into the model to investigate the impact that vessel collapse (and the associated hypoxia) has on the time at which the mutant cells become dominant within vascular tumours growing in vivo. Using the model we can predict the time it takes for a subpopulation of mutant p53 tumour cells to become the dominant population within either an avascular tumour or a localized region of a vascular tumour. Based on independent experimental results, our model suggests that the mutant population becomes dominant more quickly in vivo than in vitro (12 days vs 17 days).     

Temperature-dependent Allee effects in a stage-structured model for <Emphasis Type="Italic">Bythotrephes</Emphasis> establishment

Whether the invasive freshwater cladoceran Bythotrephes longimanus can establish after introduction into a water body depends on several biotic and abiotic factors. Among these, water temperature is important because both development rates and mode of reproduction (parthenogenetic or sexual) in Bythotrephes are influenced by temperature. We built a stage-structured model for the population dynamics of Bythotrephes based on the temperature-dependency of events in its life cycle and used the density of resting eggs at the end of each year to track changes in population density. The model was parameterized using data from published laboratory experiments and data on the Bythotrephes population in Harp Lake, Canada, from 1994 to 2005. The parameterized model was then used to simulate the outcome of invasions with different initial resting egg densities under different temperature regimes. A strong Allee effect emerged from the model, i.e. there is a critical threshold density above which the population can establish and below which it goes extinct. We showed analytically that the existence of an Allee effect arises from the model structure and is therefore robust to the parameter values. An increase in temperature reduces the establishment threshold for introductions in the same year as well as for introductions in the previous years. We therefore hypothesize that climate warming might facilitate Bythotrephes invasions. Finally, we study how the establishment threshold is influenced by the timing of the introduction event and thus identify time periods during the year when lakes may be particularly susceptible to Bythotrephes invasions.     

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.     

Segregation of a population in an environment

A mathematical model for spatial patterns and the segregation of a population is presented. Individuals in a population are assumed to move at random under the influence of a given environment potential V(x). The notion of kinetic excitation K(x) and intensity excitation Q(x) of a population is introduced. Then equilibrium states of a population are defined through a macroscopic relation K(x) + Q(x) + V(x) = constant. The problem of finding out equilibrium distributions is reduced to an eigenvalue problem. It is shown that a population is segregated by the nodal surfaces of the eigenfunctions, if it is excited. Some applications of the model to biological and ecological problems are indicated.     

Estimation of the Power of the Kruskal-Wallis Test

Power calculations of a statistical test require that the underlying population distribution(s) be completely specified. Statisticians, in practice, may not have complete knowledge of the entire nature of the underlying distribution(s) and are at a loss for calculating the exact power of the test. In such cases, an estimate of the power would provide a suitable substitute. In this paper, we are interested in estimating the power of the Kruskal-Wallis one-way analysis of variance by ranks test for a location shift. We investigated an extension of a data-based power estimation method presented by Collings and Hamilton (1988), which requires no prior knowledge of the underlying population distributions other than necessary to perform the Kruskal-Wallis test for a location shift. This method utilizes bootstrapping techniques to produce a power estimate based on the empirical cumulative distribution functions of the sample data. We performed a simulation study of the extended power estimator under the conditions of k = 3 and k = 5 samples of equal sizes m = 10 and m = 20, with four underlying continuous distributions that possessed various location configurations. Our simulation study demonstates that the Extended Average × & Y power estimation method is a reliable estimator of the power of the Kruskal-Wallis test for k = 3 samples, and a more conservative to a mild overestimator of the true power for k = 5 samples.