Actuarial senescence in laboratory and field populations of Lepidoptera

1. Recent observations of actuarial senescence – an increase in mortality rate with age – have challenged the assertion that the brevity of adult insect life spans precludes ageing. 2. Here the rate of senescence in 22 species of Lepidoptera was quantified by fitting demographic models to adult survivorship data drawn from a range of field and laboratory studies. 3. Senescence was evident in all 22 species investigated, with a model of age‐related mortality consistently fitting the survivorship curves significantly better than an alternative model which assumes constant mortality. 4. The rates of senescence varied significantly among species. The rates of senescence also differed significantly between sexes for all species tested, but not in a consistent way.     

Effect of density on age-specific mortality in Drosophila: a density supplementation experiment

Age-specific mortality rates were studied at two adult density levels in four inbred lines of Drosophila melanogaster. In experimental populations, adult densities were maintained at constant levels throughout the experiment by replacing dead flies with live, marked mutants. In control populations, densities declined naturally as the cohorts aged. For all experimental populations the best mortality model is the two-stage Gompertz model, with slower mortality acceleration at older ages. Flies in the experimental populations generally lived longer than flies in control populations, regardless of sex, genotype, or initial density level. The data demonstrate that deceleration of age-specific mortality rates at older ages is not caused by declining cohort densities. Mortality deceleration is a real phenomenon that raises serious questions about the evolution of senescence.     

Evaluation of Fourier-transform infrared spectroscopy for data capture in predictive microbiology

Chemical changes in the medium, induced by the fermentative species Lactobacillus plantarum and Lactobacillus brevis and by the enzymatic action of a proteolytic, spoilage species, Yarrowia lipolytica, were analysed using Fourier-transform i.r. spectroscopy (FTIR). Changes in the absorbance data over time could be modelled using one of the more current predictive, mathematical models of microbial growth, such as the Gompertz equation. Moreover, a linear correlation between FTIR data (expressed as absorbance of some selected peaks) and viability data (expressed as log₁₀ c.f.u./g or ml) was observed during the fermentation process, both for L. plantarum and L. brevis.     

Nonlinearity in interspecific interactions in response to climate change: Cod and haddock as an example

Climate change has profound ecological effects, yet our understanding of how trophic interactions among species are affected by climate change is still patchy. The sympatric Atlantic haddock and cod are co‐occurring across the North Atlantic. They compete for food at younger stages and thereafter the former is preyed by the latter. Climate change might affect the interaction and coexistence of these two species. Particularly, the increase in sea temperature (ST) has been shown to affect distribution, population growth and trophic interactions in marine systems. We used 33‐year long time series of haddock and cod abundances estimates from two data sources (acoustic and trawl survey) to analyse the dynamic effect of climate on the coexistence of these two sympatric species in the Arcto‐Boreal Barents Sea. Using a Bayesian state‐space threshold model, we demonstrated that long‐term climate variation, as expressed by changes of ST, affected species demography through different influences on density‐independent processes. The interaction between cod and haddock has shifted in the last two decades due to an increase in ST, altering the equilibrium abundances and the dynamics of the system. During warm years (ST over ca. 4°C), the increase in the cod abundance negatively affected haddock abundance while it did not during cold years. This change in interactions therefore changed the equilibrium population size with a higher population size during warm years. Our analyses show that long‐term climate change in the Arcto‐Boreal system can generate differences in the equilibrium conditions of species assemblages.     

Combining Gompertzian growth and cell population dynamics

A two-compartment model of cancer cells population dynamics proposed by Gyllenberg and Webb includes transition rates between proliferating and quiescent cells as non-specified functions of the total population, N. We define the net inter-compartmental transition rate function: Phi(N). We assume that the total cell population follows the Gompertz growth model, as it is most often empirically found and derive Phi(N). The Gyllenberg-Webb transition functions are shown to be characteristically related through Phi(N). Effectively, this leads to a hybrid model for which we find the explicit analytical solutions for proliferating and quiescent cell populations, and the relations among model parameters. Several classes of solutions are examined. Our model predicts that the number of proliferating cells may increase along with the total number of cells, but the proliferating fraction appears to be a continuously decreasing function. The net transition rate of cells is shown to retain direction from the proliferating into the quiescent compartment. The death rate parameter for quiescent cell population is shown to be a factor in determining the proliferation level for a particular Gompertz growth curve.     

A Microenvironment Based Model of Antimitotic Therapy of Gompertzian Tumor Growth

A model of tumor growth, based on two-compartment cell population dynamics, and an overall Gompertzian growth has been previously developed. The main feature of the model is an inter-compartmental transfer function that describes the net exchange between proliferating (P) and quiescent (Q) cells and yields Gompertzian growth for tumor cell population N = P + Q. Model parameters provide for cell reproduction and cell death. This model is further developed here and modified to simulate antimitotic therapy. Therapy decreases the reproduction-rate constant and increases the death-rate constant of proliferating cells with no direct effect on quiescent cells. The model results in a system of two ODE equations (in N and P/N) that has an analytical solution. Net tumor growth depends on support from the microenvironment. Indirectly, this is manifested in the transfer function, which depends on the proliferation ratio, P/N. Antimitotic therapy will change P/N, and the tumor responds by slowing the transfer rate from P to Q. While the cellular effects of therapy are modeled as dependent only on antimitotic activity of the drug, the tumor response also depends on the tumor age and any previous therapies—after therapy, it is not the same tumor. The strength of therapy is simulated by the parameter λ, which is the ratio of therapy induced net proliferation rate constant versus the original. A pharmacodynamic factor inversely proportional to tumor size is implemented. Various chemotherapy regimens are simulated and the outcomes of therapy administered at different time points in the life history of the tumor are explored. Our analysis shows: (1) for a constant total dose administered, a decreasing dose schedule is marginally superior to an increasing or constant scheme, with more pronounced benefit for faster growing tumors, (2) the minimum dose to stop tumor growth is age dependent, and (3) a dose-dense schedule is favored. Faster growing tumors respond better to dose density.     

Stochastic Gompertz model of tumour cell growth

In this communication, based upon the deterministic Gompertz law of cell growth, a stochastic model in tumour growth is proposed. This model takes account of both cell fission and mortality too. The corresponding density function of the size of the tumour cells obeys a functional Fokker--Planck equation which can be solved analytically. It is found that the density function exhibits an interesting "multi-peak" structure generated by cell fission as time evolves. Within this framework the action of therapy is also examined by simply incorporating a therapy term into the deterministic cell growth term.     

The clam sipho as indicator for growth indices in the soft-shell clam <Emphasis Type="Italic">Mya arenaria</Emphasis>

Due to the species’ deep burrowing behaviour, growth parameters of the soft-shell clam Mya arenaria L. 1758 are difficult to estimate, especially in deeper habitats which are not directly accessible. In this study, we analysed 192 specimens of M. arenaria and found, contrary to results of most other studies that the sigmoid Gompertz growth model (GGM) is appropriate to describe the growth. Predictions from this model confirm the finding of a life span up to 8 years and as a consequence of this, a maximum shell-length of 60 mm. Individual growth rates calculated from the first deviation indicate a nearly exponential growth in young individuals. The relationship between the clam sipho-width and age, shell-length and biomass conform to the GGM. This is a new approach to assess growth of M. arenaria and to solve practical problems that arise in field studies.     

Modelling and predicting the spatial distribution of tree root density in heterogeneous forest ecosystems

Background and Aims In mountain ecosystems, predicting root density in three dimensions (3-D) is highly challenging due to the spatial heterogeneity of forest communities. This study presents a simple and semi-mechanistic model, named ChaMRoots, that predicts root interception density (RID, number of roots m^–2). ChaMRoots hypothesizes that RID at a given point is affected by the presence of roots from surrounding trees forming a polygon shape.Methods The model comprises three sub-models for predicting: (1) the spatial heterogeneity – RID of the finest roots in the top soil layer as a function of tree basal area at breast height, and the distance between the tree and a given point; (2) the diameter spectrum – the distribution of RID as a function of root diameter up to 50 mm thick; and (3) the vertical profile – the distribution of RID as a function of soil depth. The RID data used for fitting in the model were measured in two uneven-aged mountain forest ecosystems in the French Alps. These sites differ in tree density and species composition.Key Results In general, the validation of each sub-model indicated that all sub-models of ChaMRoots had good fits. The model achieved a highly satisfactory compromise between the number of aerial input parameters and the fit to the observed data.Conclusions The semi-mechanistic ChaMRoots model focuses on the spatial distribution of root density at the tree cluster scale, in contrast to the majority of published root models, which function at the level of the individual. Based on easy-to-measure characteristics, simple forest inventory protocols and three sub-models, it achieves a good compromise between the complexity of the case study area and that of the global model structure. ChaMRoots can be easily coupled with spatially explicit individual-based forest dynamics models and thus provides a highly transferable approach for modelling 3-D root spatial distribution in complex forest ecosystems.     

Deciphering death: a commentary on Gompertz (1825) ‘On the nature of the function expressive of the law of human mortality,and on a new mode of determining the value of life contingencies’

In 1825, the actuary Benjamin Gompertz read a paper, ‘On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies’, to the Royal Society in which he showed that over much of the adult human lifespan, age-specific mortality rates increased in an exponential manner. Gompertz''s work played an important role in shaping the emerging statistical science that underpins the pricing of life insurance and annuities. Latterly, as the subject of ageing itself became the focus of scientific study, the Gompertz model provided a powerful stimulus to examine the patterns of death across the life course not only in humans but also in a wide range of other organisms. The idea that the Gompertz model might constitute a fundamental ‘law of mortality’ has given way to the recognition that other patterns exist, not only across the species range but also in advanced old age. Nevertheless, Gompertz''s way of representing the function expressive of the pattern of much of adult mortality retains considerable relevance for studying the factors that influence the intrinsic biology of ageing. This commentary was written to celebrate the 350th anniversary of the journal Philosophical Transactions of the Royal Society.