A study of a Malthusian parameter in relation to some stochastic models of human reproduction

The paper is divided into six sections and is devoted to a study of a Malthusian parameter in relation to some stochastic models of human reproduction. In Section 1, some of the motivations underlying the study are discussed, and in Section 2 some literature on the stochastic model of population growth underlying the foundations of the paper is briefly reviewed. Section 3, which lays the foundations for the study of a more complicated model in Section 4, is devoted to the study of the Malthusian parameter in relation to a stochastic model of human reproduction formulated as a terminating renewal process. In Section 4 the Malthusian parameter is studied in relation to a terminating Markov renewal model of human reproduction, stemming from the work of Perrin and Sheps (1964). Among the mathematical results of independent interest in this section is a complete spectral decomposition of the Laplace-Stieltjes transform of the semi-Markov transition matrix in the model of Perrin and Sheps. Section 5 is devoted to the discussion of a mathematical method which allows accomodating in the model the time taken by an individual to reach reproductive age, and Section 6 ends the paper by supplying bounds for the Malthusian parameter which are valid under quite general conditions. Possible applications of the results in evaluating what influences a population policy may have on population growth are also discussed.     

The effect of skewed distributions of vital statistics on growth of age-structured populations

Given that the variance of vital statistics can influence population projections, it seems reasonable that positive skew as observed for the distribution of larval survivorship of spruce budworms might also have a significant effect on stochastic projections of population growth. Simulations of population growth, using variable survivorship for a single age class, demonstrate that shape of the distribution of survivorship influences the outcome of stochastic population growth, and therefore is important for evolutionary and ecological theory. Unfortunately, empirical distributions of survivorships or fecundities for single life history stages are rare in the current literature.     

Survival of small populations under demographic stochasticity.

We estimate the mean time to extinction of small populations in an environment with constant carrying capacity but under stochastic demography. In particular, we investigate the interaction of stochastic variation in fecundity and sex ratio under several different schemes of density dependent population growth regimes. The methods used include Markov chain theory, Monte Carlo simulations, and numerical simulations based on Markov chain theory. We find a strongly enhanced extinction risk if stochasticity in sex ratio and fluctuating population size act simultaneously as compared to the case where each mechanism acts alone. The distribution of extinction times deviates slightly from a geometric one, in particular for short extinction times. We also find that whether maximization of intrinsic growth rate decreases the risk of extinction or not depends strongly on the population regulation mechanism. If the population growth regime reduces populations above the carrying capacity to a size below the carrying capacity for large r (overshooting) then the extinction risk increases if the growth rate deviates from an optimal r-value.     

An agent-based modelling approach to estimate error in gyrodactylid population growth

Comparative studies of gyrodactylid monogeneans on different host species or strains rely upon the observation of growth on individual fish maintained within a common environment, summarised using maximum likelihood statistical approaches. Here we describe an agent-based model of gyrodactylid population growth, which we use to evaluate errors due to stochastic reproductive variation in such experimental studies. Parameters for the model use available fecundity and mortality data derived from previously published life tables of Gyrodactylus salaris, and use a new data set of fecundity and mortality statistics for this species on the Neva stock of Atlantic salmon, Salmo salar. Mortality data were analysed using a mark-recapture analysis software package, allowing maximum-likelihood estimation of daily survivorship and mortality. We consistently found that a constant age-specific mortality schedule was most appropriate for G. salaris in experimental datasets, with a daily survivorship of 0.84 at 13°C. This, however, gave unrealistically low population growth rates when used as parameters in the model, and a schedule of constantly increasing mortality was chosen as the best compromise for the model. The model also predicted a realistic age structure for the simulated populations, with 0.32 of the population not yet having given birth for the first time (pre-first birth). The model demonstrated that the population growth rate can be a useful parameter for comparing gyrodactylid populations when these are larger than 20-30 individuals, but that stochastic error rendered the parameter unusable in smaller populations. It also showed that the declining parasite population growth rate typically observed during the course of G. salaris infections cannot be explained through stochastic error and must therefore have a biological basis. Finally, the study showed that most gyrodactylid-host studies of this type are too small to detect subtle differences in local adaptation of gyrodactylid monogeneans between fish stocks.     

Dramatic shifts in Hawaiian monk seal distribution predicted from divergent regional trends

Total estimated abundance of Hawaiian monk seals was just 1,161 individuals in 2008 and this number is decreasing. Most monk seals reside in the remote Northwestern Hawaiian Islands (NWHI) where the decline is approximately 4%/yr, whereas relatively fewer seals currently occupy the main Hawaiian Islands (MHI). It is widely accepted that the MHI population is increasing, although there are no formal estimates of total abundance, population growth rate or vital rates. This lack of information has hampered efforts to anticipate future scenarios and plan conservation measures. We present the first estimates of MHI monk seal survival and age‐specific reproductive rates. Using these rates, a conservative estimate of current MHI abundance and a previously published stochastic simulation model, we estimate the MHI population growth rate and projected abundance trend. Analogous estimates for the NWHI are derived from a much richer data set. Estimated survival from weaning to age 1 yr is 77% in the MHI, much higher than recent NWHI estimates ranging from 42% to 57%. Moreover, MHI females begin reproducing at a younger age and attain higher birth rates than observed in the NWHI. The estimated MHI intrinsic rate of population growth is 1.07 compared to a 0.89–0.96 range in the NWHI. Assuming an initial abundance of 152 animals in the MHI, projections indicate that if current demographic trends continue, abundance in the NWHI and MHI will equalize in approximately 15 yr. These results underscore the imperative to mitigate the NWHI decline while devoting conservation efforts to foster population growth in the MHI, where documented threats including fishery interactions, direct killing, and disease could rapidly undo the current fragile positive trend.     

Contributions of high- and low-quality patches to a metapopulation with stochastic disturbance

Studies of time-invariant matrix metapopulation models indicate that metapopulation growth rate is usually more sensitive to the vital rates of individuals in high-quality (i.e., good) patches than in low-quality (i.e., bad) patches. This suggests that, given a choice, management efforts should focus on good rather than bad patches. Here, we examine the sensitivity of metapopulation growth rate for a two-patch matrix metapopulation model with and without stochastic disturbance and found cases where managers can more efficiently increase metapopulation growth rate by focusing efforts on the bad patch. In our model, net reproductive rate differs between the two patches so that in the absence of dispersal, one patch is high quality and the other low quality. Disturbance, when present, reduces net reproductive rate with equal frequency and intensity in both patches. The stochastic disturbance model gives qualitatively similar results to the deterministic model. In most cases, metapopulation growth rate was elastic to changes in net reproductive rate of individuals in the good patch than the bad patch. However, when the majority of individuals are located in the bad patch, metapopulation growth rate can be most elastic to net reproductive rate in the bad patch. We expand the model to include two stages and parameterize the patches using data for the softshell clam, Mya arenaria. With a two-stage demographic model, the elasticities of metapopulation growth rate to parameters in the bad patch increase, while elasticities to the same parameters in the good patch decrease. Metapopulation growth rate is most elastic to adult survival in the population of the good patch for all scenarios we examine. If the majority of the metapopulation is located in the bad patch, the elasticity to parameters of that population increase but do not surpass elasticity to parameters in the good patch. This model can be expanded to include additional patches, multiple stages, stochastic dispersal, and complex demography.     

Stochastic modeling of the transmission of respiratory syncytial virus (RSV) in the region of Valencia,Spain

In this paper, we study the dynamics of the transmission of respiratory syncytial virus (RSV) in the population using stochastic models. The stochastic models are developed introducing stochastic perturbations on the demographic parameter as well as on the transmission rate of the RSV. Numerical simulations of the deterministic and stochastic models are performed in order to understand the effect of fluctuating birth rate and transmission rate of the RSV on the population dynamics. The numerical solutions of stochastic models are calculated using Euler-Maruyama and Milstein schemes, and confidence intervals for stochastic solutions are given using Monte-Carlo method. Analysis of the numerical results reveals that perturbations on the transmission rate are more decisive in the dynamics of RSV than perturbations on demographic parameters. In addition, the stochastic models show the advantage of reproducing more effectively the noisy RSV hospitalization data. It is concluded that these stochastic models are a viable option to provide a realistic modeling of the RSV dynamics on the population.     

Cladoceran birth and death rates estimates: experimental comparisons of egg-ratio methods

I. Birth and death rates of natural cladoceran populations cannot be measured directly. Estimates of these population parameters must be calculated using methods that make assumptions about the form of population growth. These methods generally assume that the population has a stable age distribution.
2. To assess the effect of variable age distributions, we tested six egg ratio methods for estimating birth and death rates with data from thirty-seven laboratory populations of Daphnia pulicaria. The populations were grown under constant conditions, but the initial age distributions and egg ratios of the populations varied. Actual death rates were virtually zero, so the difference between the estimated and actual death rates measured the error in both birth and death rate estimates.
3. The results demonstrate that unstable population structures may produce large errors in the birth and death rates estimated by any of these methods. Among the methods tested, Taylor and Slatkin's formula and Paloheimo's formula were most reliable for the experimental data.
4. Further analyses of three of the methods were made using computer simulations of growth of age-structured populations with initially unstable age distributions. These analyses show that the time interval between sampling strongly influences the reliability of birth and death rate estimates. At a sampling interval of 2.5 days (equal to the duration of the egg stage), Paloheimo's formula was most accurate. At longer intervals (7.5–10 days), Taylor and Slatkin's formula which includes information on population structure was most accurate.     

Continuous-discrete models of the dynamics of an isolated population and of 2 competing species

The paper presents the analysis of various mathematical models for dynamics of isolated population and for competition between two species. It is assumed that mortality is continuous and birth of individuals of new generations takes place in certain fixed moments. Influence of winter upon the population dynamics and conditions of classic discrete model "deduction" of population dynamics (in particular, Moran-Ricker and Hassel's models) are investigated. Dynamic regimes of models under various assumptions about the birth and death rates upon the population states are also examined. Analysis of models of isolated population dynamics with nonoverlapping generations showed the density changes regularly if the birth rate is constant. Moreover, there exists a unique global stable level and population size stabilizes asymptotically at this equilibrium, i.e. cycle and chaotic regimes in various discrete models depend on correlation between individual productivity and population state in previous time. When the correlation is exponential upon mean population size the discrete Hassel model is realized. Modification of basis model, based on the assumption that during winter survival/death changes are constant, showed that population size at global level is stable. Generally, the dependence of population rate upon "winter parameters" has nonlinear character. Nonparametric models of competition between two species does not vary if the individual productivity is constant. In a phase space there are several stable stationary states and population stabilizes at one or other level asymptotically. So, in discrete models of competition between two species oscillation can be explained by dependence of population growth rate on the population size at previous times.     

Model life table fitting by maximum likelihood estimation: a procedure to reconstruct paleodemographic characteristics from skeletal age distributions

A procedure is presented that uses the regression coefficients for the Coale and Demeny west model life tables to model selected demographic characteristics from skeletal age-at-death distributions. Model death distributions were constructed and compared to a given skeletal distribution, using methods of maximum likelihood estimation to determine the best fit. Two chi-square tests are employed to evaluate the degree of fit. The resulting model includes estimates of demographic characteristics including gross reproductive rate, crude birth rate and life expectancy. The procedure is applied to three archaeological skeletal samples as test cases: two from eastern North America and one from Mexico. These display a range of correspondence (between the best fitting model and the data) from good to poor. The proposed procedure is a potentially powerful tool for both reconstructing paleodemographic rates and illuminating differences between typical human patterns and those found in archaeological populations.     

Climate Change and Human Disturbance Can Lead to Local Extinction of Alpine Rock Ptarmigan: New Insight from the Western Italian Alps

Alpine grouses are particularly vulnerable to climate change due to their adaptation to extreme conditions and to their relict distributions in the Alps where global warming has been particularly marked in the last half century. Grouses are also currently threatened by habitat modification and human disturbance, and an assessment of the impact of multiple stressors is needed to predict the fate of Alpine populations of these birds in the next decades. We estimated the effect of climate change and human disturbance on a rock ptarmigan population living in the western Italian Alps by combining an empirical population modelling approach and stochastic simulations of the population dynamics under the a1B climate scenario and two different disturbance scenarios, represented by the development of a ski resort, through 2050.The early appearance of snow-free ground in the previous spring had a favorable effect on the rock ptarmigan population, probably through a higher reproductive success. On the contrary, delayed snowfall in autumn had a negative effect possibly due to a mismatch in time to molt to white winter plumage which increases predation risk. The regional climate model PROTHEUS does not foresee any significant change in snowmelt date in the study area, while the start date of continuous snow cover is expected to be significantly delayed. The net effect in the stochastic projections is a more or less pronounced (depending on the model used) decline in the studied population. The addition of extra-mortality due to collision with ski-lift wires led the population to fatal consequences in most projections. Should these results be confirmed by larger studies the conservation of Alpine populations would deserve more attention. To counterbalance the effects of climate change, the reduction of all causes of death should be pursued, through a strict preservation of the habitats in the present area of occurrence.     

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.     

Reinstatement of traditional mowing regimes counteracts population senescence in the rare perennial Primula vulgaris

Question: Traditional management of grassland verges or ditch banks included mowing as a way to provide additional harvesting of hay. Nowadays, such sites are often left unmanaged, as mowing verges is no longer profitable in modern agricultural systems. Are vulnerable plant species able to withstand competition with the surrounding vegetation and maintain viable populations under these circumstances? How do they respond to reinstatement of traditional mowing regimes? Location: Oedelem, northwestern Belgium. Methods: To investigate the effect of reinstatement of the rare perennial Primula vulgaris, demography and adult plant performance were monitored in a grassland verge between 1999 and 2003 under different mowing regimes. Year transitions between life stages were analysed with matrix population models. To disentangle the contributions of the deviations in different life stage transitions to the variation in overall population growth rate, life table response experiments were used. Results: Both management and year had a strong impact on demographic traits of P. vulgaris. If plots were left unmanaged, lower plant performance and declining population growth rates were observed. While population growth rates differed significantly between mowing regimes, mowing of plots only in July did not differ from mowing in July and October in terms of vegetative and reproductive output of adults. Mowing twice a year appeared to be most efficient in increasing population growth rate both by raising recruitment and growth of individuals into large reproductive adults. Conclusions: Large P. vulgaris populations show a good ability to recover from recent abandonment of traditional management regimes. By mowing twice a year, managers are able to target vital rates that are most influential: growth and flowering of adult individuals.     

Stochastic gene expression in switching environments

Organisms are known to adapt to regularly varying environments. However, in most cases, the fluctuations of the environment are irregular and stochastic, alternating between favorable and unfavorable regimes, so that cells must cope with an uncertain future. A possible response is population diversification. We assume here that the cell population is divided into two groups, corresponding to two phenotypes, having distinct growth rates, and that cells can switch randomly their phenotypes. In static environments, the net growth rate is maximized when the population is homogeneously composed of cells having the largest growth rate. In random environments, growth rates fluctuate and observations reveal that sometimes heterogeneous populations have a larger net growth rate than homogeneous ones, a fact illustrated recently through Monte-Carlo simulations based on a birth and migration process in a random environment. We study this process mathematically by focusing on the proportion f(t) of cells having the largest growth rate at time t, and give explicitly the related steady state distribution π. We also prove the convergence of empirical averages along trajectories to the first moment , and provide efficient numerical methods for computing .      

Demographics of an Experimentally Released Population of Elk in Great Smoky Mountains National Park

ABSTRACT We assessed the potential for reestablishing elk (Cervus elaphus) in Great Smoky Mountains National Park (GSMNP), USA, by estimating vital rates of experimentally released animals from 2001 to 2006. Annual survival rates for calves ranged from 0.333 to 1.0 and averaged 0.592. Annual survival for subadult and adult elk (i.e., ≥ 1 yr of age) ranged from 0.690 to 0.933, depending on age and sex. We used those and other vital rates to model projected population growth and viability using a stochastic individual-based model. The annual growth rate (λ) of the modeled population over a 25-year period averaged 0.996 and declined from 1.059 the first year to 0.990 at year 25. The modeled population failed to attain a positive 25-year mean growth rate in 46.0% of the projections. Poor calf recruitment was an important determinant of low population growth. Predation by black bears (Ursus americanus) was the dominant calf mortality factor. Most of the variance of growth projections was due to demographic variation resulting from the small population size (n = 61). Management actions such as predator control may help increase calf recruitment, but our projections suggest that the GSMNP elk population may be at risk for some time because of high demographic variation.     

Dietary fibre for gestating sows: effects on parturition progress, behaviour, litter and sow performance

In pig production, parturition progress is a key event for sow's reproductive performance, evaluated by piglet survival and piglets' performance. The aim of this study was to investigate the impact of feeding a high-fibre (HF) diet during gestation on parturition progress and reproductive performance of sows. Forty-two primiparous sows (Large-White × Landrace crossbred) were fed during gestation either a control diet (C diet; 2.40 kg/day, 3.2% crude fibre, in % of dry matter (DM)), or a HF diet (2.80 kg/day, 12.4% crude fibre, in % of DM). All sows received 33 MJ digestible energy per day. Continuous video recordings were done on the parturition day to determine postural changes (standing, sitting, lying) and behavioural activities (nesting behaviour, uterine contractions, restlessness, social behaviour towards piglets) during parturition. Duration of parturition and individual birth intervals were also measured. Piglets' growth was evaluated by weekly weighing from birth until weaning, at 26.5 days of age. Sows were weighed and backfat thickness was measured at mating, on day 105 of gestation, on the 1st day post partum, and at weaning. Durations of parturition and of birth intervals were not affected by the gestation diet and averaged 211 ± 12 min and 16.5 ± 0.9 min (mean ± s.e.), respectively. During the parturition progress, the gestation diet did not affect the frequency and the time devoted to postural and behavioural activities. Dietary treatment during gestation did not influence duration of gestation and weaning-to-oestrus interval, as well as litter size, and number of stillborn and weaned piglets. Piglet weight at birth did not differ between gestation dietary treatments but piglets nursed by HF sows showed a 13.5% greater growth rate during the 1st week of life (P < 0.01) and tended to be heavier at weaning (P = 0.06) compared with C piglets. The HF sows were leaner at the end of gestation (P < 0.05), but variations of sows' weight during gestation and lactation were not affected by the gestation diet. All sows lost the same amount of backfat thickness during lactation. During lactation, the average daily feed intake was not significantly affected by the gestation diet. This study shows that substituting a control diet for a HF diet during gestation has limited effects on farrowing progress and reproductive performance, but improved piglets' growth rate during the 1st week of life and tended to increase their live weight at weaning.     

The context dependence of assortative mating: a demonstration with conspecific salmonid populations

Assortative mating is thought to play a key role in reproductive isolation. However, most experimental studies of assortative mating do not take place in multiple natural environments, and hence, they ignore its potential context dependence. We implemented an experiment in which two populations of brown trout (Salmo trutta) with different natural flow regimes were placed into semi‐natural stream channels under two different artificial flow regimes. Natural reproduction was allowed, and reproductive isolation was measured by means of parentage assignment to compare within‐population vs. between‐population male–female mating and relative offspring production. For both metrics, reproductive isolation was highly context dependent: no isolation was evident under one flow regime, but strong isolation was evident under the other flow regime. These patterns were fully driven by variance in the mating success of males from one of the two populations. Our results highlight how reproductive isolation through assortative mating can be strongly context dependent, which could have dramatic consequences for patterns of gene flow and speciation under environmental change.     

Novel bivariate moment-closure approximations

Nonlinear stochastic models are typically intractable to analytic solutions and hence, moment-closure schemes are used to provide approximations to these models. Existing closure approximations are often unable to describe transient aspects caused by extinction behaviour in a stochastic process. Recent work has tackled this problem in the univariate case. In this study, we address this problem by introducing novel bivariate moment-closure methods based on mixture distributions. Novel closure approximations are developed, based on the beta-binomial, zero-modified distributions and the log-Normal, designed to capture the behaviour of the stochastic SIS model with varying population size, around the threshold between persistence and extinction of disease. The idea of conditional dependence between variables of interest underlies these mixture approximations. In the first approximation, we assume that the distribution of infectives (I) conditional on population size (N) is governed by the beta-binomial and for the second form, we assume that I is governed by zero-modified beta-binomial distribution where in either case N follows a log-Normal distribution. We analyse the impact of coupling and inter-dependency between population variables on the behaviour of the approximations developed. Thus, the approximations are applied in two situations in the case of the SIS model where: (1) the death rate is independent of disease status; and (2) the death rate is disease-dependent. Comparison with simulation shows that these mixture approximations are able to predict disease extinction behaviour and describe transient aspects of the process.     

Estimating the growth of a newly established moose population using reproductive value

Estimating the population growth rate and environmental stochasticity of long-lived species is difficult because annual variation in population size is influenced by temporal autocorrelations caused by fluctuations in the age-structure. Here we use the dynamics of the reproductive value to estimate the long-term growth rate s and the environmental variance of a moose population that recently colonized the island of Vega in northern Norway. We show that the population growth rate was high (ŝ=0.26). The major stochastic influences on the population dynamics were due to demographic stochasticity, whereas the environmental variance was not significantly different from 0. This supports the suggestion that population growth rates of polytocous ungulates are high, and that demographic stochasticity must be assessed when estimating the growth of small ungulate populations.