A stochastic two-phase growth model

This article proposes a stochastic growth model that starts as a Yule process and is subsequently joined with a Prendiville process when the population attains certain prescribed critical size. In other words, the model assumes exponential growth in an early stage and logistic growth later on to reflect growth retardation caused by overcrowding. In the case that the population starts with a single unit, closed form expressions are given for the distribution of the population size and for the mean and variance functions of the process. Numerical solutions are briefly discussed for the process that starts with more than one unit.     

Cell Growth and Size Homeostasis in Silico

Cell growth in size is a complex process coordinated by intrinsic and environmental signals. In a research work performed by a different group, size distributions of an exponentially growing population of mammalian cells were used to infer cell-growth rate in size. The results suggested that cell growth was neither linear nor exponential, but subject to size-dependent regulation. To explain the observed growth pattern, we built a mathematical model in which growth rate was regulated by the relative amount of mRNA and ribosomes in a cell. Under the growth model and a stochastic division rule, we simulated the evolution of a population of cells. Both the sampled growth rate and size distribution from this in silico population agreed well with experimental data. To explore the model space, alternative growth models and division rules were studied. This work may serve as a starting point to understand the mechanisms behind cell growth and size regulation using predictive models.     

Cell Growth and Size Homeostasis in Silico

Cell growth in size is a complex process coordinated by intrinsic and environmental signals. In a research work performed by a different group, size distributions of an exponentially growing population of mammalian cells were used to infer cell-growth rate in size. The results suggested that cell growth was neither linear nor exponential, but subject to size-dependent regulation. To explain the observed growth pattern, we built a mathematical model in which growth rate was regulated by the relative amount of mRNA and ribosomes in a cell. Under the growth model and a stochastic division rule, we simulated the evolution of a population of cells. Both the sampled growth rate and size distribution from this in silico population agreed well with experimental data. To explore the model space, alternative growth models and division rules were studied. This work may serve as a starting point to understand the mechanisms behind cell growth and size regulation using predictive models.     

A study of some diffusion models of population growth

The stochastic differential equations of many diffusion processes which arise in studies of population growth in random environments can be transformed, if the Stratonovich stochastic calculus is employed, to the equation of the Wiener process. If the transformation function has certain properties then the transition probability density function and quantities relating to the time to first attain a given population size can be obtained from the known results for the Wiener process. Some other random growth processes can be derived from the Ornstein-Uhlenbeck process. These transformation methods are applied to the random processes of Malthusian growth, Pearl-Verhulst logistic growth and a recent model of density independent growth due to Levins.     

Extinction of populations due to inbreeding depression with demographic disturbances

The process of population extinction due to inbreeding depression with constant demographic disturbances every generation is analysed using a population genetic and demographic model. The demographic disturbances introduced into the model represent loss of population size that is induced by any kind of human activities, e.g. through hunting and destruction of habitats. The genetic heterozygosity among recessive deleterious genes and the population size are assumed to be in equilibrium before the demographic disturbances start. The effects of deleterious mutations are represented by decreases in the growth rate and carrying capacity of a population. Numerical simulations indicate rapid extinction due to synergistic interaction between inbreeding depression and declining population size for realistic ranges of per-locus mutation rate, equilibrium population size, intrinsic rate of population growth, and strength of demographic disturbances. Large populations at equilibrium are more liable to extinction when disturbed due to inbreeding depression than small populations. This is a consequence of the fact that large populations maintain more recessive deleterious mutations than small populations. The rapid extinction predicted in the present study indicates the importance of the demographic history of a population in relation to extinction due to inbreeding depression.     

The Probability of Fixation in Populations of Changing Size

The rate of adaptive evolution of a population ultimately depends on the rate of incorporation of beneficial mutations. Even beneficial mutations may, however, be lost from a population since mutant individuals may, by chance, fail to reproduce. In this paper, we calculate the probability of fixation of beneficial mutations that occur in populations of changing size. We examine a number of demographic models, including a population whose size changes once, a population experiencing exponential growth or decline, one that is experiencing logistic growth or decline, and a population that fluctuates in size. The results are based on a branching process model but are shown to be approximate solutions to the diffusion equation describing changes in the probability of fixation over time. Using the diffusion equation, the probability of fixation of deleterious alleles can also be determined for populations that are changing in size. The results developed in this paper can be used to estimate the fixation flux, defined as the rate at which beneficial alleles fix within a population. The fixation flux measures the rate of adaptive evolution of a population and, as we shall see, depends strongly on changes that occur in population size.     

Process and measurement errors of population size: their mutual effects on precision and bias of estimates for demographic parameters

Knowing the parameters of population growth and regulation is fundamental for answering many ecological questions and the successful implementation of conservation strategies. Moreover, detecting a population trend is often a legal obligation. Yet, inherent process and measurement errors aggravate the ability to estimate these parameters from population time-series. We use numerical simulations to explore how the lengths of the time-series, process and measurement error influence estimates of demographic parameters. We first generate time-series of population sizes with given demographic parameters for density-dependent stochastic population growth, but assume that these population sizes are estimated with measurement errors. We then fit parameters for population growth, habitat capacity, total error and long-term trends to the ‘measured’ time-series data using non-linear regression. The length of the time-series and measurement error introduce a substantial bias in the estimates for population growth rate and to a lesser degree on estimates for habitat capacity, while process error has little effect on parameter bias. The total error term of the statistical model is dominated by process error as long as the latter is larger than the measurement error. A decline in population size is difficult to document as soon as either error becomes moderate, trends are not very pronounced, and time-series are short (<10–15 seasons). Detecting an annual decline of 1% within 6-year reporting periods, as required for the European Union for the species of Community Interest, appears unachievable.     

SIZE STRUCTURE AND DYNAMICS IN A POPULATION OF SARGASSUM MUTICUM (PHAEOPHYCEAE)

Sargassum muticum (Yendo) Fensholt is an introduced brown seaweed with a very distinctive seasonal growth cycle on European shores. The present study links the dynamics of a population of S. muticum with the seasonal growth cycle of the species and the density-dependent processes operating throughout this cycle. Results indicate that both growth cycle and intraspecific competition influenced the structure and population dynamics. Size inequality increased during the slow growth phase (autumn–winter) of the 2-year study. Mechanisms generating inequality of size could be the existence of asymmetric competition and the inherent differences in growth rates between old (regenerated) and new thalli (recruits). Inequality of size distributions decreased progressively during the last months of the growth phase (spring–summer) and could be related to a process of self-thinning. There was a negative biomass–density relationship (as a measure of biomass accumulation-driven mortality) that confirms the importance of self-thinning as a major demographic factor in the S. muticum population.     

Nestboxes and immigration drive the growth of an urban Peregrine Falcon Falco peregrinus population

Drivers of wildlife population dynamics are generally numerous and interacting. Some of these drivers may impact demographic processes that are difficult to estimate, such as immigration into the focal population. Populations may furthermore be small and subject to demographic stochasticity. All of these factors contribute to blur the causal relationship between past management action and current population trends. The urban Peregrine Falcon Falco peregrinus population in Cape Town, South Africa, increased from three pairs in 1997 to 18 pairs in 2010. Nestboxes were installed over this period to manage the interface between new urban pairs of Falcons and the human users of colonized buildings, and incidentally to improve breeding success. We used integrated population models (IPMs) formally to combine information from a capture–mark–recapture study, monitoring of reproductive success and counts of population size. As all local demographic processes were directly observed, the IPM approach also allowed us to estimate immigration by difference. The provision of nestboxes, as a possible stimulant of population growth, improved breeding success and accounted for an estimated 3–26% of the population increase. The most important driver of growth, however, was immigration. Despite low sample sizes, the IPM approach allowed us to obtain relatively precise estimates of the population‐level impact of nestbox deployment. The goal of conservation interventions is often to increase population size, so the effectiveness of such interventions should ideally be assessed at the population level. IPMs are powerful tools in this context for combining demographic information that may be limited due to small population size or practical constraints on monitoring. Our study quantitatively documented both the immigration process that led to growth of a small population and the effect of a management action that helped the process.     

French wasps in the New World: experimental biological control introductions reveal a demographic Allee effect

Many populations introduced into a novel environment fail to establish. One underlying process is the Allee effect, i.e., the difficulty of individuals to survive and reproduce when rare, and the consequently low or negative population growth. Although observations showing a positive relation between initial population size and establishment probability suggest that the Allee effect could be widespread in biological invasions, experimental tests are scarce. Here, we used a biological control program against Diuraphis noxia (Mordvilko) (Hemiptera: Aphididae) in the United States to manipulate initial population size of the introduced parasitoid Aphelinus asychis Walker (Hymenoptera: Aphelinidae) originating from France. For eight populations and three generations after introduction, we studied spatial distribution and spread, density, mate-finding, and population growth. Dispersal was lower in small populations during the first generation. Smaller initial population size nonetheless resulted in lower density during the three generations studied. The proportion of mated females and the population sex ratio were not affected by initial population size or population density. Net reproductive rate decreased with density within each generation, suggesting negative density-dependence. But for a given density, net reproductive rate was smaller in populations initiated with few individuals than in populations initiated with many individuals. Hence, our results demonstrate a demographic Allee effect. Mate-finding is excluded as an underlying mechanism, and other component Allee effects may have been overwhelmed by negative density-dependence in reproduction. Impact of generalist predators could provide one potential explanation for the relationship between initial population size and net reproductive rate. However, the continuing effect of initial population size on population growth suggests genetic processes may have been involved in the observed demographic Allee effect.     

Directionality theory: an empirical study of an entropic principle in life-history evolution

Understanding the relationship between ecological constraints and life-history properties constitutes a central problem in evolutionary ecology. Directionality theory, a model of the evolutionary process based on demographic entropy, a measure of the uncertainty in the age of the mother of a randomly chosen newborn, provides an analytical framework for addressing this problem. The theory predicts that in populations that spend the greater part of their evolutionary history in the stationary growth phase (equilibrium species), entropy will increase. Equilibrium species will be characterized by high iteroparity and strong demographic stability. In populations that spend the greater part of their evolutionary history in the exponential growth phase (opportunistic species), entropy will decrease when population size is large, and will undergo random variation when population size is small. Opportunistic species will be characterized by weak iteroparity and weak demographic stability when population size is large, and random variations in these attributes when population size is small. This paper assesses the validity of these predictions by employing a demographic dataset of 66 species of perennial plants. This empirical analysis is consistent with directionality theory and provides support for its significance as an explanatory and predictive model of life-history evolution.     

Distribution of nucleotide differences between two randomly chosen cistrons in a population of variable size

The distribution of the number of nucleotide differences between two randomly chosen cistrons in a finite population is studied here when the population size changes from generation to generation. When genetic variability is measured by heterozygosity (i.e., the probability that two cistrons are different), by the probability that two cistrons differ at two or more nucleotide sites, or by mean number of site differences between cistrons, it is seen that in a population going through a small bottleneck all of these measures decline rapidly but, as soon as population size becomes large, they start to increase owing to new mutations. The amount of reduction in these measures depends not only on the size of bottleneck but also on the rate of population growth. The implications of this study explaining the observed variations in the rates of amino acid substitutions during the evolutionary process are also discussed.     

Spatial and temporal scale of density-dependent body growth and its implications for recruitment, population dynamics and management of stream-dwelling salmonid populations

Density-dependent variations in body growth and size have important consequences for the population dynamics of stream-dwelling salmonid populations, since body size is related to a variety of ecologically relevant characteristics. These include survival and fecundity, competitive and predatory abilities, and foraging behavior. However, little work has been done to understand how density-dependent body growth varies across temporal and spatial scales and when this compensatory process is relevant for recruitment and population dynamics of stream-dwelling salmonids. Increased intra- or inter-cohort competition reduces growth rates of juveniles. Both within- and among-cohort differences at the juvenile stage are likely to be maintained through the lifetime. Limited movement or dispersal can lead to subdivision of a population into several local populations with independent dynamics. The spatial and temporal variation in movement and the patchy distribution of resources make fish likely to experience density-dependence across location, life-stage, and season. The relaxation of density-dependent suppression of body growth at low densities constitutes a potential mechanism for salmonids to persist in the face of environmental perturbation and may contribute to explaining the peculiar resilience to population collapses often showed by salmonids. The inclusion of density-dependent growth in population models may increase the usefulness of model predictions in management contexts. Models not accounting for density-dependent growth may underestimate the recovery potential of resident salmonid populations when they collapse to low densities.     

Population growth with randomly distributed jumps

 The growth of populations with continuous deterministic and random jump components is treated. Three special models in which random jumps occur at the time of events of a Poisson process and admit formal explicit solutions are considered: A) Logistic growth with random disasters having exponentially distributed amplitudes; B) Logistic growth with random disasters causing the removal of a uniformly distributed fraction of the population size; and C) Exponential decay with sudden increases (bonanzas) in the population and with each increase being an exponentially distributed fraction of the current population. Asymptotic and numerical methods are employed to determine the mean extinction time for the population, qualitatively and quantitatively. For Model A, this time becomes exponentially large as the carrying capacity becomes much larger than the mean disaster size. Implications for colonizing species for Model A are discussed. For Models B and C, the practical notion of a small, but positive, effective extinction level is chosen, and in these cases the expected extinction time rises rapidly with population size, yet at less than an e xponentially large order. Received 21 June 1996; received in revised form 17 February 1997     

The growth and decline of a population of the spruce aphid Elatobium abietinum during a three year study,and the changing pattern of fecundity,recruitment and alary polymorphism in a Northern Ireland Forest

The population size and structure of the green spruce aphid was followed throughout the spring — summer cycle on the same group of trees in a low-elevation coastal Sitka spruce forest for three consecutive years. The relationship between the pattern of change and the phenology of bud burst, which heralds a marked change in needle sap quality, suggests that yearly differences in the winter temperature regime may affect the duration of the population growth phase and hence the peak numbers attained in late spring.An index of population growth rate was sufficiently sensitive to aphid fecundity during the population cycles of two years to suggest that the changing rate of fertility was the decisive process governing changes in population size. The commitment of aphids to alate development was greater than that recorded elsewhere in Britain but did little to effect population decline since the contribution of alatae to larviposition was substantial while seasonally pulsed.     

Steady size distributions for cells in one-dimensional plant tissues

We consider a population of cells growing and dividing steadily without mortality, so that the total cell population is increasing, but the proportion of cells in any size class remains constant. The cell division process is non-deterministic in the sense that both the size at which a cell divides, and the proportions into which it divides, are described by probability density functions. We derive expressions for the steady size/birth-size distribution (and the corresponding size/age distribution) in terms of the cell birth-size distribution, in the particular case of one-dimensional growth in plant organs, where the relative growth rate is the same for all cells but may vary with time. This birth-size distribution is shown to be the principal eigenfunction of a Fredholm integral operator. Some special cases of the cell birth-size distribution are then solved using analytical techniques, and in more realistic examples, the eigen-function is found using a simple, generally applicable numerical iteration.     

Changes in reproductive strategy in the ruffe during a period of establishment in a new habitat

The pattern of maturation, body size and fecundity was examined in a population of ruffe ( Gymnocephalus cernuus L.) three times during a period of rapid growth, and eventual stabilization, following its introduction to a new habitat. When the ruffe were less common, maturing ruffe were relatively large and immature ruffe relatively small, compared with when the ruffe were abundant. Intermediate ruffe population size showed a maturation pattern intermediate between these two extremes. It is suggested that this pattern of maturation is a response of the ruffe population to changing growth opportunity induced by changing intraspecific competition. This fluctuating maturation pattern is interpreted in terms of a threshold-dependent maturation trigger, operating on the rate of accumulation of energy and a trade-off between somatic growth and gonad development. When the ruffe population was large, high intraspecific competition resulted in low opportunity for growth; only fish with the highest rate of food acquisition were able to mature in a given year–the investment in gonadal tissue reducing somatic growth. When the ruffe population was low, the high rate of energy acquisition in the population resulted in the triggering of maturation, even at small size, only the very smallest fish remaining immature. High growth opportunity allowed maturing fish to develop gonad and maintain somatic growth. The pattern of size related fecundity also changed over the three periods. When growth opportunity was low, size related fecundity was greater than when opportunity for growth was high. This suggests that maturing females faced with poor growth conditions compensated by increasing egg number for a given body size either by decreasing egg size or by increasing total investment in ovarian tissue.     

On the therapy effect for a stochastic growth Gompertz-type model

We consider a diffusion model based on a generalized Gompertz deterministic growth in which carrying capacity depends on the initial size of the population. The drift of the resulting process is then modified by introducing a time-dependent function, called "therapy", in order to model the effect of an exogenous factor. The transition probability density function and the related moments for the proposed process are obtained. A study of the influence of the therapy on several characteristics of the model is performed. The first-passage-time problem through time-dependent boundaries is also analyzed. Finally, an application to real data concerning a rabbit population subject to particular therapies is presented.     

Modelling the lineage of a growing population as an age-dependent branching process

A new model for population growth as a branching process is described. It requires knowledge of the population's age-specific survivorship and fecundity curves. The total population size at any time is calculated by summing across all branches of the tree representing the population's lineage and thus the model also allows the lineage of the developing population to be traced. Use of the model is demonstrated by a hypothetical example. The model could be of particular value in tracing the phylogenetic development of populations and/or species.     

Functional mapping of quantitative trait loci that interact with the hg mutation to regulate growth trajectories in mice

The high growth (hg) mutation increases body size in mice by 30-50%. Given the complexity of the genetic regulation of animal growth, it is likely that the effect of this major locus is mediated by other quantitative trait loci (QTL) with smaller effects within a web of gene interactions. In this article, we extend our functional mapping model to characterize modifier QTL that interact with the hg locus during ontogenetic growth. Our model is derived within the maximum-likelihood context, incorporated by mathematical aspects of growth laws and implemented with the EM algorithm. In an F2 population founded by a congenic high growth (HG) line and non-HG line, a highly additive effect due to the hg gene was detected on growth trajectories. Three QTL located on chromosomes 2 and X were identified to trigger significant additive and/or dominant effects on the process of growth. The most significant finding made from our model is that these QTL interact with the hg locus to affect the shapes of the growth process. Our model provides a powerful means for understanding the genetic architecture and regulation of growth rate and body size in mammals.