Modeling the population dynamics of annual plants with seed bank and density dependent effects

A model is proposed for the population dynamics of an annual plant (Sesbania vesicaria) with a seed bank (i.e. in which a proportion of seeds remain dormant for at least one year). A simple linear matrix model is deduced from the life cycle graph. The dominant eigenvalue of the projection matrix is estimated from demographic parameters derived from field studies. The estimated values for population growth rate () indicates that the study population should be experiencing a rapid exponential increase, but this was not the case in our population.The addition of density dependent effects on seedling survivorship and adult fecundity, effects for which field studies provide evidence, considerably improves our model. Depending on the demographic parameters, the model leads to stable equilibrium, oscillations, or chaos. Study of the behaviour of this model in the parameter space shows that the existence of a seed bank allows higher among-year variation of adult fecundity, without leaving the region of demographic stability. Field data obtained over 3 years confirm this prediction.     

Demography when history matters: construction and analysis of second-order matrix population models

History matters when individual prior conditions contain important information about the fate of individuals. We present a general framework for demographic models which incorporates the effects of history on population dynamics. The framework incorporates prior condition into the i-state variable and includes an algorithm for constructing the population projection matrix from information on current state dynamics as a function of prior condition. Three biologically motivated classes of prior condition are included: prior stages, linear functions of current and prior stages, and equivalence classes of prior stages. Taking advantage of the matrix formulation of the model, we show how to calculate sensitivity and elasticity of any demographic outcome. Prior condition effects are a source of inter-individual variation in vital rates, i.e., individual heterogeneity. As an example, we construct and analyze a second-order model of Lathyrus vernus, a long-lived herb. We present population growth rate, the stable population distribution, the reproductive value vector, and the elasticity of λ to changes in the second-order transition rates. We quantify the contribution of prior conditions to the total heterogeneity in the stable population of Lathyrus using the entropy of the stable distribution.     

Le modele matriciel deterministe de Leslie et ses applications en dynamique des populations

The Leslie matrix model (Leslie, 1945) for discrete population growth has been modified and used several times in population dynamics. A review is given of the basic model (n _t + 1 = An _t) and of its principal modifications. The modifications relating to the influences of internal or external factors to the population are studied with greater detail. The same applies to models where the population is divided in stages rather than in age classes.In the same line, Hadjibiros (1975, 1976) studied a more general model, where the transition matrix corresponds to a population divided in stages of unequal duration, and where the time varying influences of the internal or external factors are included. This model offers the possibility of computer simulation of the population dynamics of a natural population represented by a demographic vector.Some kind of relationship is proposed between the general matrix model and the Lefkovitch matrix (Lefkovitch, 1964, 1965); the elements of which, obtained by regression techniques from census data, describe the biological dependance of a stage i at time t + 1 on a stage j at time t.

---

---

Travail réalisé dans le cadre du contrat DGRST n° 74 7 0452: Modélisation des systèmes litière-sol (Responsables: C. Geri et J. P. Cancela da Fonseca).     

The epidemic threshold of vector-borne diseases with seasonality

Cutaneous leishmaniasis is a vector-borne disease transmitted to humans by sandflies. In this paper, we develop a mathematical model which takes into account the seasonality of the vector population and the distribution of the latent period from infection to symptoms in humans. Parameters are fitted to real data from the province of Chichaoua, Morocco. We also introduce a generalization of the definition of the basic reproduction number R ₀ which is adapted to periodic environments. This R ₀ is estimated numerically for the epidemic in Chichaoua; 1.94. The model suggests that the epidemic could be stopped if the vector population were reduced by a factor 3.76.     

Dealing with Uncertainty in Spatially Explicit Population Models

It has been argued that spatially explicit population models (SEPMs) cannot provide reliable guidance for conservation biology because of the difficulty of obtaining direct estimates for their demographic and dispersal parameters and because of error propagation. We argue that appropriate model calibration procedures can access additional sources of information, compensating the lack of direct parameter estimates. Our objective is to show how model calibration using population-level data can facilitate the construction of SEPMs that produce reliable predictions for conservation even when direct parameter estimates are inadequate. We constructed a spatially explicit and individual-based population model for the dynamics of brown bears (Ursus arctos) after a reintroduction program in Austria. To calibrate the model we developed a procedure that compared the simulated population dynamics with distinct features of the known population dynamics (=patterns). This procedure detected model parameterizations that did not reproduce the known dynamics. Global sensitivity analysis of the uncalibrated model revealed high uncertainty in most model predictions due to large parameter uncertainties (coefficients of variation CV 0.8). However, the calibrated model yielded predictions with considerably reduced uncertainty (CV 0.2). A pattern or a combination of various patterns that embed information on the entire model dynamics can reduce the uncertainty in model predictions, and the application of different patterns with high information content yields the same model predictions. In contrast, a pattern that does not embed information on the entire population dynamics (e.g., bear observations taken from sub-areas of the study area) does not reduce uncertainty in model predictions. Because population-level data for defining (multiple) patterns are often available, our approach could be applied widely.     

Are tsetse fly populations close to equilibrium?

Glossina or tsetse flies, the vectors of sleeping sickness, form a unique group of insects with remarkable characteristics. They are viviparous with a slow rhythm of reproduction (one larva approximately every 10 days) determined by the regular ovulation of alternate ovaries. This unusual physiology enables the age of the females to be estimated by examining the ovaries.The resulting ovarian age structure of tsetse fly populations has been used to develop research into the demography of tsetse flies. Several authors have proposed methods of estimating population growth rates from ovarian age distribution data. However, such methods are applicable only when the growth rate () is equal to 1 (i.e. the intrinsic rate of increase r is equal to 0). In fact, in this type of estimation, the adult survival rate a (or equivalently the mortality rate) cannot be dissociated from the growth rate.Other independently determined demographic parameters must be used to remove this lack of identiflability. We have built a matrix model of the dynamics of tsetse fly populations which enables the growth rate to be calculated from the pupal survival rate, the pupal period and the adult survival rate. Assuming that the age-groups of the population studied have reached a stable distribution, it is possible to calculate the probabilities for the observed sample of belonging to each of the age-groups, to construct a likelihood function and thus to obtain an estimate of the apparent survival rate = a/ If the pupal survival rate and the pupal period are known, a and can then be calculated from .The application of this method to data collected for over two annual cycles in a savannah habitat (Burkina-Faso) showed a high overall stability in the populations of Glossina palpalis gambiensis. Seasonal fluctuations could be easily interpreted as being the result of climatic changes between the dry and rainy seasons.     

Global asymptotic stability of the size distribution in probabilistic models of the cell cycle

Probabilistic models of the cell cycle maintain that cell generation time is a random variable given by some distribution function, and that the probability of cell division per unit time is a function only of cell age (and not, for instance, of cell size). Given the probability density, f(t), for time spent in the random compartment of the cell cycle, we derive a recursion relation for _n(x), the probability density for cell size at birth in a sample of cells in generation n. For the case of exponential growth of cells, the recursion relation has no steady-state solution. For the case of linear cell growth, we show that there exists a unique, globally asymptotically stable, steady-state birth size distribution, _*(x). For the special case of the transition probability model, we display _*(x) explicitly.This work was supported by the National Science Foundation under grants MCS8301104 (to J.J.T.) and MCS8300559 (to K.B.H.), and by the National Institutes of Health under grant GM27629 (to J.J.T.).     

Density dependence and risk of extinction in a small population of sea otters

Sea otters (Enhydra lutris (L.)) were hunted to extinction off the coast of Washington State early in the 20th century. A new population was established by translocations from Alaska in 1969 and 1970. The population, currently numbering at least 550 animals, A major threat to the population is the ongoing risk of majour oil spills in sea otter habitat. We apply population models to census and demographic data in order to evaluate the status of the population. We fit several density dependent models to test for density dependence and determine plausible values for the carrying capacity (K) by comparing model goodness of fit to an exponential model. Model fits were compared using Akaike Information Criterion (AIC). A significant negative relationship was found between the population growth rate and population size (r ²=0.27, F=5.57, df=16, p<0.05), suggesting density dependence in Washington state sea otters. Information criterion statistics suggest that the model is the most parsimonious, followed closely by the logistic Beverton–Holt model. Values of K ranged from 612 to 759 with best-fit parameter estimates for the Beverton–Holt model including 0.26 for r and 612 for K. The latest (2001) population index count (555) puts the population at 87–92% of the estimated carrying capacity, above the suggested range for optimum sustainable population (OSP). Elasticity analysis was conducted to examine the effects of proportional changes in vital rates on the population growth rate (). The elasticity values indicate the population is most sensitive to changes in survival rates (particularly adult survival).     

A demographic study of deer browsing impacts on Trillium grandiflorum

When white-tailed deer populations reach high densities, they have negative and often dramatic effects on forest herb populations. However, it is not clear how deer affect the demographic processes of plant populations. We examined how the structure and dynamics of Trillium grandiflorum (Michx.) Salisb. populations are affected by deer browsing in the Upper Great Lakes region by sampling populations from nine study sites in a forested landscape in 1998 and 1999. We constructed a stage-based matrix population model for the regional population. Our model indicated that the long-term growth rate of the population to be –3.56% per year ( = 0.965). Mortality rates were highest for seeds (97.5%) followed by seedlings (29.1%), and lower for all remaining stage classes (4.9 to 8.5%). The observed stage distribution significantly differed from the stable stage distribution, and the damping ratio ( = 1.103) indicated the population would not reach its stable stage distribution anytime soon. In the absence of deer browsing, the long-term growth rate would improve to between –3.46% and –1.61% per year. A moderate drought during the study could account for the negative population growth rate, but deer browsing accelerates the rate of decline. Population growth is most sensitive to the proportion of plants remaining in the nonflowering stage, and deer browsing reduces this proportion. Browsing damage was relatively low in this study (5.4% of stems in 1998, 11.5% in 1999) compared to another study of browsing impacts on T. grandiflorum, indicating deer could have far more severe demographic consequences in populations subject to higher levels of browsing.     

Multivariate statistical analysis of net diatom species distributions in the Southwestern Atlantic and Indian Ocean

Summary Vertical net haul diatom assemblages from near South Georgia, and from between Africa and Antarctica, were examined and compared. Variation among South Georgia stations was examined by principal component, cluster and canonical discriminant analyses. Diatom distributions provide evidence for at least two distinct water masses. The region north of the island is characterized by neritic, temperate diatoms and by an assemblage with low species diversity. The region south of the island is characterized by oceanic, antarctic species and relatively high species diversity. The regions are most distinct to the west of the island, intergrading east of the island. Within the north-south division, five station groupings were detected on the basis of distribution of dominant net diatoms. By comparing classical species ecological categorizations to results of principal component analysis, a neritic-oceanic factor was identified from net diatom distributions. This factor was common to both areas in spite of the fact that Biscoe and Agulhas collections were from different seasons.     

Variable generation times and Darwinian fitness measures

Summary Reproductive value (RV) and net reproductive output (R _o) are frequently used fitness measures. We argue that they are only appropriate when intervals between reproductive events are fixed, as they are dimensionless generation-to-generation scalings with units offspring per parent. A fitness measure should account for two different effects of a decrease in generation time: (1) increased survival due to shorter exposure to mortality agents and (2) increased frequency of reproduction.R _o andRV deal with the first of these two effects, while a measure with a physical dimensionper time [T^–1] is needed to account for the second. The Malthusian growth parameter,r, meets this requirement and in situations where time to reproduction is variable, we propose, the instantaneous rate of spread of descendants (from an individual) be used instead ofR _o. As an alternative toRV, we suggest using the instantaneous difference = –r, wherer is the population rate of increase. WhileRV andR _o are dimensionless ratios, , and areper time rates which are appropriate in accounting for alterations in generation time.     

Population parameters and harvest risks for polar bears (<Emphasis Type="Italic">Ursus maritimus</Emphasis>) of Kane Basin,Canada and Greenland

We estimated demographic parameters and current harvest risks for a population of polar bears (Ursus maritimus Phipps) inhabiting northern Smith Sound and Kane Basin, Canada and Greenland. Our demographic analysis included a detailed assessment of age- and sex-specific survival and recruitment from 141 marked polar bears, using information contained within the standing age distribution of captures and mark-recapture analysis. Total survival rates for females were: 0.374 ± 0.180 (cubs), 0.686 ± 0.157 (ages 1–4), and 0.967 ± 0.043 (ages 5+). Mean litter size was 1.67 ± 0.08 cubs. Females did not reproduce until at least age 6, which is late compared to other populations of polar bears. The model-averaged, mark–recapture estimate of mean abundance (±1 SE) for years 1994–1997 was 164 ± 35 bears. We incorporated demographic parameters and their variances into a harvest risk analysis (i.e., a stochastic, harvested population viability analysis, PVA). Results suggest that polar bears in the region were severely over-harvested during the mark–recapture interval (1992–1997). The current status of the population is unknown.     

On the process of stabilization in the renewal model: approximations for the time to convergence

The transient behaviour of the renewal model leading to the stable age distribution is studied for weakly skewed net maternity functions (found in human as well as in some animal populations). The study, which is partly based on heuristic arguments, first provides approximate expressions for the damping constant and the circular frequency (in terms of the moments of the net maternity function) belonging to the principal oscillatory component of the birth trajectory. The time to stability (defined as the time interval after which the principal oscillatory component has become less than a certain fraction of the stable solution) is then determined in two cases: For the genesis model and for a stable population in which the net reproduction rate is reduced to one. The results are applied to a problem which arises in the mass rearing of pest insects.     

Extinction risk of a meta-population: aggregation approach

Aggregation of variables of a complex mathematical model with realistic structure gives a simplified model which is more suitable than the original one when the amount of data for parameter estimation is limited. Here we explore use of a formula derived for a single unstructured population (canonical model) in predicting the extinction time for a population living in multiple habitats. In particular we focus multiple populations each following logistic growth with demographic and environmental stochasticities, and examine how the mean extinction time depends on the migration and environmental correlation. When migration rate and/or environmental correlation are very large or very small, we may express the mean extinction time exactly using the formula with properly modified parameters. When parameters are of intermediate magnitude, we generate a Monte Carlo time series of the population size for the realistic structured model, estimate the "effective parameters" by fitting the time series to the canonical model, and then calculate the mean extinction time using the formula for a single population. The mean extinction time predicted by the formula was close to those obtained from direct computer simulation of structured models. We conclude that the formula for an unstructured single-population model has good approximation capability and can be applicable in estimating the extinction risk of the structured meta-population model for a limited data set.     

Intensity of microcarrier collisions in turbulent flow

A model is developed, allowing estimation of the share of inelastic interparticle collisions in total energy dissipation for stirred suspensions. The model is restricted to equal-sized, rigid, spherical particles of the same density as the surrounding Newtonian fluid. A number of simplifying assumptions had to be made in developing the model. According to the developed model, the share of collisions in energy dissipation is small.List of Symbols b parameter in velocity distribution function (Eq. (28)) - c _K factor in Kolmogoroff spectrum law (Eq. (20)) - D _t(r _p ) m²/s characteristic dispersivity at particle radius scale (Eq. (13)) - E(k, t) m³/s² energy spectrum as function of k and t (Eq. (16)) - E _K (k) m³/s² energy spectrum as function of k in Kolmogoroff-region (Eq. (20)) - E _p dimensionless mean kinetic energy of a colliding particle (Eq. (36)) - E _cp dimensionless kinetic energy exchange in a collision (Eq. (37)) - G(x, s) dimensionless energy spectrum as function of x and s (Eq. (16)) - G _B(x) dimensionless energy spectrum as function of x for boundary region (Eq. (29)) - G _K(x) dimensionless energy spectrum as function of x for Kolmogoroff-region (Eq. (21)) - g m/s² gravitational acceleration - I _cp dimensionless collision intensity per particle (Eq. (38)) - I _cv dimensionless volumetric collision intensity (Eq. (39)) - k l/m reciprocal of length scale of velocity fluctuations (Eq. (17)) - K dimensionless viscosity (Eq. (13)) - n(2) dimensionless particle collision rate (Eq. (12)) - n(r) l/s particle exchange rate as function of distance from observatory particle center (Eq. (7)) - r m vector describing position relative to observatory particle center (Eq. (2)) - r m scalar distance to observatory particle center (Eq. (3)) - r _pm particle radius (Eq. (1)) - s dimensionless time (Eq. (10)) - SC kg/ms³ Severity of collision (Eq. (1)) - t s time (Eq. (2)) - u(r, t) m/s velocity vector as function of position vector and time (Eq. (2)) - u(r, t) m/s magnitude of velocity vector as function of position vector and time (Eq. (3)) - u _r(r, t) m/s radial component of velocity vector as function of position vector and time (Eq. (3)) - u _r (r, t) m/s magnitude of radial component of velocity vector as function of position vector and time (Eq. (3)) - u (r, t) m/s latitudinal component of velocity vector as function of position vector and time (Eq. (3)) - u (r, t) m/s magnitude of latitudinal component of velocity vector as function of position vector and time (Eq. (3)) - u (r, t) m/s longitudinal component of velocity vector as function of position vector and time (Eq. (3)) - u (r, t) m/s magnitude of longitudinal component of velocity vector as function of position vector and time (Eq. (3)) - u _gsm/s superficial gas velocity - u(r) m/s root mean square velocity as function of distance from observatory particle center (Eq. (3)) - u_r(r) m/s root mean square radial velocity component as function of distance from observatory particle center (Eq. (4)) - u (r) m/s root mean square latitudinal velocity component as function of distance from observatory particle center (Eq. (4)) - u (r) m/s Root mean square longitudinal velocity component as function of distance from observatory particle center (Eq. (4)) - w(x) dimensionless root mean square velocity as function of dimensionless distance from observatory particle center (Eq. (11)) - V _pm³ particle volume (Eq. (36)) - w(2) dimensionless root mean square collision velocity (Eq. (34)) - w ^* parameter in boundary layer velocity equation (Eq. (24)) - x dimensionless distance to particle center (Eq. (9)) - x ^* value of x where G _Band G _K-curves touch (Eq. (32)) - x _K dimensionless micro-scale (Kolmogoroff-scale) of turbulence (Eq. (15)) - volumetric particle hold-up - m²/s³ energy dissipation per unit of mass - m²/s kinematic viscosity - kg/m³ density - (r) m³/s fluid-exchange rate as function of distance to observatory particle center - Latitudinal co-ordinate (Eq. (5)) - Longitudinal co-ordinate (Eq. (5))     

A learned neural network that simulates properties of the neuronal population vector

This paper considers steady-state and timedependent characteristics of the response of the hidden-layer neurons in a dynamic model for the neural network trained through supervised learning to perform transformation of input signals into output signals. This transformation is set up so as to correspond to variation in the directions of two-dimensional vectors and is treated as creation by the network of a movement direction in response to a stimulus direction. The input vector is encoded in the state of the input layer at the initial instant of time, and the output vector in the state of the output layer at great values of time. After the network has been trained on examples of the input-output relation, the hidden neurons turn out to be broadly tuned to direction. The corresponding dependence for their activity is approximated with a smooth function, whose maximum allows some preferred direction to be attributed to each neuron. If each hidden neuron is assigned a vector pointing in its preferred direction, then any arbitrarily chosen direction can be characterized by an imaginary neuronal population vector (Georgopoulos et al. 1986) defined as the sum of the vectors of preferred direction for the neurons, with the weights equal to their activities for the chosen direction. It is demonstrated that, although hidden neurons are broadly tuned to direction, the population vector points in a direction congruent with that of the input vector at the initial moment of time and accurately predicts the direction of the output vector at great values of time. In between, the population vector turns continuously from the one direction towards the other. The dynamic and stationary properties of the population vector of the hidden-layer neurons, as obtained within the framework of the model in question, show a close similarity to the experimentally observed (Georgopoulos et al. 1986; Georgopoulos et al. 1989) behaviour of the population vector constructed in the same manner on the ensemble of motor cortex neurons sensitive to a certain type of movement.     

Globally asymptotic properties of proliferating cell populations

This paper presents a general model for the cell division cycle in a population of cells. Three hypotheses are used: (1) There is a substance (mitogen) produced by cells which is necessary for mitosis; (2) The probability of mitosis is a function of mitogen levels; and (3) At mitosis each daughter cell receives exactly one-half of the mitogen present in the mother cell. With these hypotheses we derive expressions for the and curves, the distributions of mitogen and cell cycle times, and the correlation coefficients between mother-daughter (_md) and sister-sister (_ss) cell cycle times.The distribution of mitogen levels is shown to be given by the solution to an integral equation, and under very mild assumptions we prove that this distribution is globally asymptotically stable. We further show that the limiting logarithmic slopes of (t) and (t) are equal and constant, and that _md0 while _ss0. These results are in accord with the experimental results in many different cell lines. Further, the transition probability model of the cell cycle is shown to be a simple special case of the model presented here.