The logistic equation and population decline

This short paper discusses the unsuitability of the logistic equation (as it is usually derived) as a model of population decline, suggests a simple, more general alternative, and considers the difference between Lotka's original development and the more usual derivation based on the concept of carrying capacity.     

Modelling Holling type II functional response in deterministic and stochastic food chain models with mass conservation

The Rosenzweig-MacArthur predator-prey model is the building block in modeling food chain, food webs and ecosystems. There are a number of hidden assumptions involved in the derivation. For instance the prey population growth is logistic without predation but also with predation. In order to reveal these we will start with modelling a resource-predator-prey system in a closed spatially homogeneous environment. This allows us to keep track of the nutrient flow. With an instantaneous remineralisation of the products excreted in the environment by the populations and dead body mass there is conservation of mass. This allows for a model dimension reduction and yields the mass balance predator-prey model. When furthermore the searching and handling processes are much faster that the population changing rates, the trophic interaction is described by a Holling type II functional response, also assumed in the Rosenzweig-MacArthur model. The derivation uses an extended deterministic model with number of searching and handling predators as model variables where the ratio of the predator/prey body masses is used as a mechanistic time-scale parameter. This extended model is also used as a starting point for the derivation of a stochastic model. We will investigate the stochastic effects of random switching between searching and handling of the predators and predator dying. Prey growth by consumption of ambient resources is still deterministic and therefore the stochastic model is hybrid. The transient dynamics is studied by numerical Monte Carlo simulations and also the quasi-equilibrium distribution for the population quantities is calculated. The body mass of the prey individual is the scaling parameter in the stochastic model formulation. This allows for a quantification of the mean-field approximation criterion for the justification of replacement of the stochastic by a deterministic model.     

Genome-wide analysis of single nucleotide variants allows for robust and accurate assessment of clonal derivation in cell lines used to produce biologics

A clonally derived (or “monoclonal”) cell line is a cell population derived from a single progenitor cell. Clonally derived cell lines are required for many biotechnological applications. For instance, recombinant mammalian cells used to produce therapeutic proteins are expected by regulatory authorities to be clonally derived. Assurance of clonal derivation (or “clonality”) is usually obtained from the characterization of the procedure used for cell cloning, for instance by assessing the success rate of single-cell sorting but not by assessing the cell line itself. We have developed a method to assess clonal derivation directly from the genetic makeup of cells. The genomic test of clonality is based on whole-genome sequencing and statistical analysis of single nucleotide variants. This approach quantifies the clonal fractions present in nonclonal samples and it provides a measure of the probability that a cell line is derived from a single cell. Upon experimental validation of the test, we show that it is highly accurate and that it can robustly detect minor clonal fractions of as little as 1% of the cell population. Moreover, we find that it is applicable to various cell line development protocols. This approach can simplify development protocols and shorten timelines while ensuring clonal derivation with high confidence.     

Survival in continuous structured populations models

The extended McKendrick-von Foerster structured population model is employed to derive a nonautonomous ordinary differential equation model of a population. The derivation assumes that the individual life history can be delineated into several physiological stages. We study the persistence of the population when the model is autonomous and base the nonautonomous survival analysis on the autonomous case and a comparison principle. A brief excursion into alternate life history strategies is presented.This work was supported in part by the U.S. Environmental Protection Agency under cooperative agreement CR 813353010     

The mechanistic basis of discrete-time population models: The role of resource partitioning and spatial aggregation

The purpose of this paper is to present a unified view to understand mechanistic basis of various discrete-time population models from the viewpoints of resource partitioning and spatial aggregation of individuals. A first-principles derivation is presented of a new population model which incorporates both scramble and contest competition using a site-based framework in which individuals are distributed over discrete resource sites. The derived model has parameters relating to the way of resource partitioning and the degree of spatial aggregation of individuals, respectively. The model becomes various population models in various limits in these parameters. This model thus provides a unified view to understand how various population models are interrelated. The dependence of the stability of the model on the parameters is also examined.     

Development and Validation of a Risk-Score Model for Type 2 Diabetes: A Cohort Study of a Rural Adult Chinese Population

Some global models to predict the risk of diabetes may not be applicable to local populations. We aimed to develop and validate a score to predict type 2 diabetes mellitus (T2DM) in a rural adult Chinese population. Data for a cohort of 12,849 participants were randomly divided into derivation (n = 11,564) and validation (n = 1285) datasets. A questionnaire interview and physical and blood biochemical examinations were performed at baseline (July to August 2007 and July to August 2008) and follow-up (July to August 2013 and July to October 2014). A Cox regression model was used to weigh each variable in the derivation dataset. For each significant variable, a score was calculated by multiplying β by 100 and rounding to the nearest integer. Age, body mass index, triglycerides and fasting plasma glucose (scores 3, 12, 24 and 76, respectively) were predictors of incident T2DM. The model accuracy was assessed by the area under the receiver operating characteristic curve (AUC), with optimal cut-off value 936. With the derivation dataset, sensitivity, specificity and AUC of the model were 66.7%, 74.0% and 0.768 (95% CI 0.760–0.776), respectively. With the validation dataset, the performance of the model was superior to the Chinese (simple), FINDRISC, Oman and IDRS models of T2DM risk but equivalent to the Framingham model, which is widely applicable in a variety of populations. Our model for predicting 6-year risk of T2DM could be used in a rural adult Chinese population.     

Cancer risk models and preselection for screening

Objective: The invitation to population screening is based on age criteria in many countries. Screening is not offered to younger or older participants, because the benefits in these age groups do not outweigh the harms. One could argue that it is not so much age that determines the benefits but the risk of developing preclinical and treatable cancer. Cancer risk varies with age but is also affected by other factors. Methods: We performed a systematic review for risk models for the three types of cancer for which population screening programs exist: breast, cervical and colon cancer. We used an evaluation scheme that distinguishes three phases of model development: model derivation, validation and impact analysis. Data were collected in August 2010. Results: We identified two colorectal, four breast and three cervix cancer risk models. One colorectal, four breast and none of the cervix cancer models have been externally validated. We could not identify evaluations of the impact on population screening effectiveness. Conclusion: We conclude that risk models for the pre-selection of screening have been developed. These models could improve the pre-selection for screening, help in making personal decisions about participation, and reduce adverse effects of population screening. The validity of this hypothesis, as well as practicalities and issues of equity and reliability, have to be tested in further studies.     

On the Mechanistic Derivation of Various Discrete-Time Population Models

We present a derivation of various discrete-time population models within a single unifying mechanistic context. By systematically varying the within-year patterns of reproduction and aggression between individuals we can derive various discrete-time population models. These models include classical examples such as the Ricker model, the Beverton-Holt model, the Skellam model, the Hassell model, and others. Some of these models until now lacked a good mechanistic interpretation or have been derived in a different context. By using this mechanistic approach, the model parameters can be interpreted in terms of individual behavior.     

Multinomial-Sampling Models for Random Genetic Drift

Three different derivations of models with multinomial sampling of genotypes in a finite population are presented. The three derivations correspond to the operation of random drift through population regulation, conditioning on the total number of progeny, and culling, respectively. Generations are discrete and nonoverlapping; the diploid population mates at random. Each derivation applies to a single multiallelic locus in a monoecious or dioecious population; in the latter case, the locus may be autosomal or X-linked. Mutation and viability selection are arbitrary; there are no fertility differences. In a monoecious population, the model yields the Wright-Fisher model (i.e., multinomial sampling of genes) if and only if the viabilities are multiplicative. In a dioecious population, the analogous reduction does not occur even for pure random drift. Thus, multinomial sampling of genotypes generally does not lead to multinomial sampling of genes. Although the Wright-Fisher model probably lacks a sound biological basis and may be inaccurate for small populations, it is usually (perhaps always) a good approximation for genotypic multinomial sampling in large populations.     

Randomization Modelling of the Crossover Experiment for Clinical Trials

The two-period cross-over experiment for clinical trials has been examined by several writers following a Gaussian linear model approach. Some authors have expressed interest in the “derivation of the finite permutation model” and have pointed out that the randomization approach to modeling the two-period cross-over design “would highlight the importance of randomizing the subjects to the two groups as a basis for inference”. However, in the literature, there is no development of the randomization approach to this important design. In this paper, after a statement of the experimental design and formulation of the observation random variables of the finite population, two additive randomization models—one with residual effects, the other without—which are the analogues of Grizzle's Gaussian models, are derived. Statistical inference is developed for these randomization models and the results are compared with those of the corresponding Gaussian models. Also, exact inference based upon Fischer's approach is presented.     

Numerical solution of structured population models

Numerical methods are presented for a general mass-structured population model with demographic rates that depend on individual mass, time, and total population mass. Several types of numerical methods are described, Eulerian methods, implicit methods, and the method of characteristics. These methods are compared for a sample problem with an exact solution. The preferred numerical technique combines the method of characteristics with an adaptive grid. Numerical solution of model equations developed for mosquitofish illustrates this method and demonstrates how seasons can play a dominant role in shaping population development.     

Modeling erythropoiesis subject to malaria infection

A mathematical model of erythropoiesis subject to malaria infection is developed by combining ideas from previous models that addressed only one of the two phenomena. The nature of the model allows one to account for suppression of erythropoiesis by the toxin hemozoin, which is a by-product of digested hemoglobin. Following the derivation of the model, numerical simulations are performed and show that the number of parasites produced per bursting erythrocyte has the most significant effect of erythropoiesis. It is also shown that removing hemozoin may be an effective method for aiding the recovery of the erythrocyte population, but is not effective in maintaining a healthy population in the early stages of infection. The second half of the paper introduces an implicit finite difference scheme that was used to perform the simulations previously mentioned. An existence-uniqueness result is then provided via the numerical method.     

On the spread of epidemics in a closed heterogeneous population

Heterogeneity is an important property of any population experiencing a disease. Here we apply general methods of the theory of heterogeneous populations to the simplest mathematical models in epidemiology. In particular, an SIR (susceptible-infective-removed) model is formulated and analyzed when susceptibility to or infectivity of a particular disease is distributed. It is shown that a heterogeneous model can be reduced to a homogeneous model with a nonlinear transmission function, which is given in explicit form. The widely used power transmission function is deduced from the model with distributed susceptibility and infectivity with the initial gamma-distribution of the disease parameters. Therefore, a mechanistic derivation of the phenomenological model, which is believed to mimic reality with high accuracy, is provided. The equation for the final size of an epidemic for an arbitrary initial distribution of susceptibility is found. The implications of population heterogeneity are discussed, in particular, it is pointed out that usual moment-closure methods can lead to erroneous conclusions if applied for the study of the long-term behavior of the models.     

The Dynamics of Sex Ratio Evolution: From the Gene Perspective to Multilevel Selection

The new dynamical game theoretic model of sex ratio evolution emphasizes the role of males as passive carriers of sex ratio genes. This shows inconsistency between population genetic models of sex ratio evolution and classical strategic models. In this work a novel technique of change of coordinates will be applied to the new model. This will reveal new aspects of the modelled phenomenon which cannot be shown or proven in the original formulation. The underlying goal is to describe the dynamics of selection of particular genes in the entire population, instead of in the same sex subpopulation, as in the previous paper and earlier population genetics approaches. This allows for analytical derivation of the unbiased strategic model from the model with rigorous non-simplified genetics. In effect, an alternative system of replicator equations is derived. It contains two subsystems: the first describes changes in gene frequencies (this is an alternative unbiased formalization of the Fisher-Dusing argument), whereas the second describes changes in the sex ratios in subpopulations of carriers of genes for each strategy. An intriguing analytical result of this work is that the fitness of a gene depends on the current sex ratio in the subpopulation of its carriers, not on the encoded individual strategy. Thus, the argument of the gene fitness function is not constant but is determined by the trajectory of the sex ratio among carriers of that gene. This aspect of the modelled phenomenon cannot be revealed by the static analysis. Dynamics of the sex ratio among gene carriers is driven by a dynamic “tug of war” between female carriers expressing the encoded strategic trait value and random partners of male carriers expressing the average population strategy (a primary sex ratio). This mechanism can be called “double-level selection”. Therefore, gene interest perspective leads to multi-level selection.     

Cell volume distributions reveal cell growth rates and division times

A population of cells in culture displays a range of phenotypic responses even when those cells are derived from a single cell and are exposed to a homogeneous environment. Phenotypic variability can have a number of sources including the variable rates at which individual cells within the population grow and divide. We have examined how such variations contribute to population responses by measuring cell volumes within genetically identical populations of cells where individual members of the population are continuously growing and dividing, and we have derived a function describing the stationary distribution of cell volumes that arises from these dynamics. The model includes stochastic parameters for the variability in cell cycle times and growth rates for individual cells in a proliferating cell line. We used the model to analyze the volume distributions obtained for two different cell lines and one cell line in the absence and presence of aphidicolin, a DNA polymerase inhibitor. The derivation and application of the model allows one to relate the stationary population distribution of cell volumes to extrinsic biological noise present in growing and dividing cell cultures.     

A quantitative genetic model for mixed diploid and triploid hybrid progenies in tree breeding and evolution

Interspecific hybridization has played a critical role in tree evolution and breeding. The findings of triploidy in forest trees stimulate the development of a quantitative genetic model to estimate the nature of gene action. The model is based on clonally replicated triploid progenies derived from a two-level population and individual-within-population mating design in which offspring have a double dose of alleles from the parent and a single dose of alleles from the other parent. With the same genetic assumptions of a diploid model, except non-Mendelian behavior at meiosis, and the experimental variances estimated from a linear statistical model, total genetic variances in the triploid progenies are separated into additive, dominance, and epistatic components. In addition, by combining the new model with the already existing model based on disomic expression, the partitioning of additive, dominant, and epistatic variances can be obtained for a mixed diploid/triploid F₁ progeny population. This paper provides an alternative technique to study the modes of quantitative inheritance for outcrossing, long-lived forest trees in which inbred lines cannot be easily generated. The accuracy for estimating gene action using this technique is discussed.     

On a Reproductive Stage-Structured Model for the Population Dynamics of the Malaria Vector

A reproductive stage-structured deterministic differential equation model for the population dynamics of the human malaria vector is derived and analysed. The model captures the gonotrophic and behavioural life characteristics of the female Anopheles sp. mosquito and takes into consideration the fact that for the purposes of reproduction, the female Anopheles sp. mosquito must visit and bite humans (or animals) to harvest necessary proteins from blood that it needs for the development of its eggs. Focusing on mosquitoes that feed exclusively on humans, our results indicate the existence of a threshold parameter, the vectorial reproduction number, whose size increases with increasing number of gonotrophic cycles, and is also affected by the female mosquito’s birth rate, its attraction and visitation rate to human residences, and its contact rate with humans. A stability analysis of the model indicates that the mosquito can establish itself in the environment if and only if the value of the vectorial reproduction number exceeds unity and that mosquito eradication is possible if the vectorial reproduction number is less than unity, since, then, the trivial steady state which always exist is unique and is globally and asymptotically stable. When a persistent vector population steady state exists, it is locally and asymptotically stable for a range of reproduction numbers, but can also be driven to instability via a Hopf bifurcation as the reproduction number increases further away from unity. The model derivation identifies and characterizes control parameters relating to activities such as human-mosquito contact and the mosquito’s survival chances between blood meals and egg laying. Our results show that the total mosquito population size increases with increasing number of gonotrophic cycles. Therefore understanding the fundamental aspects of the mosquito’s behaviour provides a pathway for the study of human-mosquito contact and mosquito population control. Control of the mosquito population densities would ultimately lead to malaria control.     

Bottom-up derivation of discrete-time population models with the Allee effect

This paper presents a framework in which various single-species discrete-time population models exhibiting the Allee effect are derived from first principles. Here, the Allee effect means a reduction in individual fitness at low population sizes. The derivation is based on the distribution of female and male individuals among discrete resource sites, in addition to competitive and cooperative interaction among individuals. These derivations show how the derived population models depend on the type and the intensity of competition, and the degree of clustering of individuals. Along with these models exhibiting the Allee effect, this paper also presents first-principles derivation of population models without the Allee effect which include a parameter relating to the intensity of competition.     

A note on effective population size with overlapping generations

A simple derivation is given for a formula obtained previously for the effective size of random-mating populations with overlapping generations. The effective population size is the same as that for a population with discrete generations having the same variance of lifetime family size and the same number of individuals entering the population per generation.     

The ecological genetics of conditional strategies

We develop a quantitative genetic model for conditional strategies that incorporates the ecological realism of previous strategic models. Similar to strategic models, the results show that environmental heterogeneity, cue reliability, and environment-dependent fitness trade-offs for the alternative tactics of the conditional strategy interact to determine when conditional strategies will be favored and that conditional strategies should be a common form of adaptive variation in nature. The results also show that conditional and unconditional development can be maintained in one of two ways: by frequency-dependent selection or by the maintenance of genetic variation that exceeds the threshold for induction. We then modified the model to take into account variance in exposures to the environmental cue as well as variance in response to the cue, which allows a derivation of a dose-response curve. Here the results showed that increasing the genetic variance for response both flattens and shifts the dose-response curve. Finally, we modify the model to derive the dose-response curve for a population polymorphic for a gene that blocks expression of the conditional strategy. We illustrate the utility of the model by application to predator-induced defense in an intertidal barnacle and compare the results with phenotypic models of selection.