THE WELL-POSEDNESS OF SIZE STRUCTURED POPULATION MODELS WITH DENSITY DEPENDENT PARAMETERS

Thewell-posednessofnonlinearsizestructuredpopulationmodelsisstudied.Thenonlinearitiesareintroducedbyassumingthevitalparameters(thebirthrate,thedeathrate,andthegrowthrate)tobedensitydependent.TheidealadoptedhereisbasedonthemethodofGurtinandMacCamy[4]usedfornonlinearage-dependentpopulationmodels.Thenetreproductivenumberisintroducedandusedtodeterminethelocalandglobalstabilityoftrivialequilibrium.Thestabilityconditionsoftrivialequilibriumareobtained.     

Stable controls in age-dependent population dynamics

The classical age-dependent population model is considered, in which mortality depends on total population and fertility is age-dependent. It is shown that in general such systems are not completely controllable with respect to a control variable in the mortality function, but that in certain circumstances a suitable control can be found to hold the population at a specified level.     

Computerization of a population projection model based on generalized age-dependent branching processes: a connection with the Leslie matrix

The population projection model based on generalized age-dependent branching processes developed by Mode and Busby (1981) involves the solution of a large number of renewal type equations. It is shown that these equations may be solved recursively. Such a solution has two implications. One is that the projection model may be very efficiently computerized. Second, the recursive algorithm developed has striking similarities to two traditional methods of population projection used by demographers: the Leslie matrix and cohort component methods. The results presented here associate traditional projection techniques with the theory of age-dependent branching processes.     

Age-structured optimal control in population economics

This paper brings both intertemporal and age-dependent features to a theory of population policy at the macro-level. A Lotka-type renewal model of population dynamics is combined with a Solow/Ramsey economy. We consider a social planner who maximizes an aggregate intertemporal utility function which depends on per capita consumption. As control policies we consider migration and saving rate (both age-dependent). By using a new maximum principle for age-structured control systems we derive meaningful results for the optimal migration and saving rate in an aging population. The model used in the numerical calculations is calibrated for Austria.     

Viable controls in age-dependent population dynamics

The classical age-dependent population model is considered, where the mortality depends on total population and the fertility is age dependent. Conditions are determined under which such a system may be steered to a specific population level and held there.     

On the Gurtin-MacCamy conjecture concerning the stability of a simple nonlinear age-dependent population model

The conjecture that an age-dependent population model, involving a survival function dependent only on the population and a fertility function dependent on population and exponentially on age, could not entail Hopf bifurcation into stable orbits is shown to be incorrect.     

Single-locus gametophytic incompatibility in autotetraploids

It is known that a single-locus gametophytic self-incompatibility (GSI) system can persist with just two distinct alleles in an autotetraploid population, in contrast to diploid GSI systems, assuming "competitive interaction" in which heteroallelic pollen is universally compatible. The steady-state population structure of a GSI system in autotetraploids was investigated in an undivided population assuming "competitive interaction." A deterministic model was developed to predict the frequencies of genotypes with two, three, or four distinct S alleles, assuming no mutation or population subdivision. The model showed that unlike in diploid GSI systems, the limiting values of the frequencies of genotype classes do not minimize pollen wastage.     

Persistent distribution functions for the dispersion of structured populations

This paper primarily expounds upon the problem of persistent age-state distribution functions for the dispersion of structured populations. A general model is introduced, based on the following assumptions: 1) the state of an individual of age a is characterized by a set of random variables X₁, X₂,…, X_Q (weight, size, etc.) obeying a phenomenological master equation; 2) the birth function λ depends on the age a' of parents and on the state variables X₁,…, X_Q of the newborns; 3) the mortality function is composed of two additive terms—the first contribution depends only on age while the second contribution depends on the total population density; 4) the population diffuses to avoid crowding. These hypotheses define a nonlinear population model for which time- and space-persistent age-state distribution functions eventually may occur even if the total population density is time- and space-dependent. A biological interpretation of the main results is given in terms of the distribution function of the state vector at birth. In the last part of the paper a generalized model is presented, assuming that the behavior of an individual is described by a system of age-dependent master equations [29].     

Optimization of harvesting age in an integral age-dependent model of population dynamics

The paper deals with optimal control in a linear integral age-dependent model of population dynamics. A problem for maximizing the harvesting return on a finite time horizon is formulated and analyzed. The optimal controls are the harvesting age and the rate of population removal by harvesting. The gradient and necessary condition for an extremum are derived. A qualitative analysis of the problem is provided. The model shows the presence of a zero-investment period. A preliminary asymptotic analysis indicates possible turnpike properties of the optimal harvesting age. Biological interpretation of all results is provided.     

Separable models in age-dependent population dynamics

A class of population models is considered in which the parameters such as fecundity, mortality and interaction coefficients are assumed to be age-dependent. Conditions for the existence, stability and global attractivity of steady-state and periodic solutions are derived. The dependence of these solutions on the maturation periods is analyzed. These results are applied to specific single and multiple population models. It is shown that periodic solutions cannot occur in a general class of single population age-dependent models. Conditions are derived that determine whether increasing the maturation period has a stabilizing effect. In specific cases, it is shown that any number of switches in stability can occur as the maturation period is increased. An example is given of predator-prey model where each one of these stability switches corresponds to a stable steady state losing its stability via a Hopf bifurcation to a periodic solution and regaining its stability upon further increase of the maturation period.     

A model of HIV-1 transmission for urban areas of Africa

A model of the transmission dynamics of HIV-1, appropriate to urban areas of Africa, is presented and its behaviour explored through numerical studies. The model is a two-sex model with age-dependent demographic and behavioural parameters. Adults are classified by age, sex, risk group, and epidemiologic status. HIV-1 is transmitted to adults heterosexually, and to infants and children vertically and parenterally. Numerical studies show that, while AIDS will slow population growth, growth rates do not become negative for reasonable parameter values. The sex and age patterns of infection are explored, as is the potential economic impact of changes in the sex and age composition of the population.     

Estimation of population size and survival of sheep blowfly,Lucilia cuprina,in the field from serial recoveries of marked flies affected by weather dispersal and age-dependent trappability

A model is presented for analysis of mark-recapture data of mobile insects which, unlike the Lincoln Index, does not require marked individuals to remain within the sampling area or to mix uniformly with the wild population. The model assumes a single or multiple releases of marked insects from the centre of the sampling area and that captured individuals are not returned to the population. Dispersal rates of marked insects are estimable from serial recaptures and, for catches that are either unaffected by or have been corrected for weather effects, the model also provides estimates of mortality and age-dependent trappability. Application of the model is illustrated using mark-recapture data for adults of the Australian sheep blowfly Lucilia cuprina.     

Prey-taxis destabilizes homogeneous stationary state in spatial Gause–Kolmogorov-type model for predator–prey system

We consider a continuous taxis-diffusion-reaction system of partial-differential equations describing spatiotemporal dynamics of a predator–prey system. The local kinetics of the system is defined by general Gause–Kolmogorov-type model. The predator ability to pursue the prey is modelled by the Patlak–Keller–Segel taxis model, assuming that movement velocities of predators are proportional to the gradients of specific cues emitted by prey (e.g., odour, pheromones, exometabolites). The linear stability analysis of the model showed that the non-trivial homogeneous stationary regime of the model becomes unstable with respect to small heterogeneous perturbations with increase of prey-taxis activity; an Andronov–Hopf bifurcation occurs in the system when the taxis coefficient of predator exceeds its critical bifurcation value that exists for all admissible values of model parameters. These findings generalize earlier results obtained for particular cases of the Gause–Kolmogorov-type model assuming logistic reproduction of the prey population and the Holling types I and II functional responses of the predator population. Numerical simulations with theta-logistic growth of the prey population and the Ivlev functional response of predators illustrate and support results of the analytical study.     

An Inhomogeneous Markov Model to Fit the Migration Process

In this paper we study the migration process considering an inhomogeneous Markov model. This is a certain condition to investigate age-dependent population distributions, where the transition probabilities are not constant. We consider also a death process for a population alive in a region at age t and, as a result of this, combined transition probabilities between the states of the concerning Markov chain. The model has non-stationary distribution for t →∞, because the condition of ergodicity does not hold.     

Modeling of selection processes with age-dependent birth and death rates

Two simple models for the competition and selection in age-dependent populations are developed and analyzed mathematically. Following Eigen, competition is introduced by the condition of constant overall-number of the population. In the first model this condition is satisfied by regulation of a dilution flux and in the second case by regulation of a food density. The calculation of maximal fitness is given explicitly for both situations. It is shown that fitness depends in a complicated way on the age-dependence of the birth and death rates. Therefore species have to develop special aging strategies in order to survive in a population under selection pressure. In general, early reproduction is of advantage and increases fitness.     

Estimating Genetic Variability with Restriction Endonucleases

The estimation of the amount of sequence variation in samples of homologous DNA segments is considered. The data are assumed to have been obtained by restriction endonuclease digestion of the segments, from which the numbers and frequencies of the cleavage sites in the sample are determined. An estimator, p, of the proportion of sites that are polymorphic in the sample is derived without assuming any particular population genetic model for the evolution of the population. The estimator is very close to the EWENS, SPIELMAN and HARRIS (1981) estimator that was derived with the symmetric WRIGHT-FISHER neutral mode. ENGELS (1981) has also recently proposed an estimator of the same quantity, and he arrived at his estimator without assuming a particular population genetic model. The sampling variance of p and ENGELS' estimator are derived. It is found that the sampling variance of p is lower than the sampling variance of ENGELS' estimator. Also, the sampling variance of theta, an estimate of theta (=4Nu) is obtained for the symmetric WRIGHT-FISHER neutral model with free recombination and with no recombination.     

Stochastic dynamic population model for northern corn rootworm (Coleoptera: Chrysomelidae).

A stochastic dynamic population model for the complete life cycle of northern corn rootworm, Diabrotica barberi Smith & Lawrence, is described. Adult population dynamics from emergence to oviposition are based on a published single-season model for which temperature-dependent development and age-dependent advancement determine adult population dynamics and oviposition. Randomly generated daily temperatures make this model component stochastic. Stochastic hatch is 50+/-8%. A stochastic nonlinear density-dependent larval survival model is estimated using field data from artificial infestation experiments. A regional model of corn phenology is estimated to incorporate the effect of dispersal on adult mortality. Random daily weather is generated using parameters for Brookings, SD. Model performance is evaluated with deterministic simulations, which show that the population converges to zero unless adult mortality is reduced by the availability of corn pollen from the regional model of corn phenology. Stochastic model performance is evaluated with stochastic daily weather, egg hatch, and larval survival in various combinations. Sensitivity analysis is conducted to evaluate model responsiveness to each parameter. Model results are generally consistent with published data.     

Methanogenesis from acetate: enrichment studies.

An acetate enrichment culture was initiated by inoculating anaerobic sludge from a mesophilic methane digestor into a mineral salts medium with calcium acetate as the sole carbon and energy source. This enrichment was maintained indefinitely by weekly transfer into medium of the same composition. A study of this enrichment disclosed an unexpected age-dependent inhibition of methanogenesis by H2 and formate which apparently differed from the inhibition by chloroform and benzyl viologen. This age-dependent inhibition indicated that microbial interactions of the mixed enrichment population may play a regulatory role in methane formation. Futhermore, stimulation of methanogenesis in the acetate enrichment by addition of yeast extract showed a nutrient limitation which indicated that syntrophic interactions leading to formation of growth factors may also occur. A model is presented to illustrate the possible interrelationships between methanogenic and nonmethanogenic bacteria in their growth and formation of methane and carbon dioxide from acetate.     

Estimation by capture-recapture of recruitment and dispersal over several sites

Dispersal in animal populations is intimately linked with accession to reproduction, i.e. recruitment, and population regulation. Dispersal processes are thus a key component of population dynamics to the same extent as reproduction or mortality processes. Despite the growing interest in spatial aspects of population dynamics, the methodology for estimating dispersal, in particular in relation with recruitment, is limited. In many animal populations, in particular vertebrates, the impossibility of following individuals over space and time in an exhaustive way leads to the need to frame the estimation of dispersal in the context of capture-recapture methodology. We present here a class of age-dependent multistate capture-recapture models for the simultaneous estimation of natal dispersal, breeding dispersal, and age-dependent recruitment. These models are suitable for populations in which individuals are marked at birth and then recaptured over several sites. Under simple constraints, they can be used in populations where non-breeders are not observed, as is often the case with colonial waterbirds monitored on their breeding grounds. Biological questions can be addressed by comparing models differing in structure, according to the generalized linear model philosophy broadly used in capture-recapture methodology. We illustrate the potential of this approach by an analysis of recruitment and dispersal in the roseate tern Sterna dougallii .     

Cell population dynamics during the differentiative phase of tissue development

The dynamics of a cell population whose numbers are growing exponentially have been described well by a mathematical model based on the theory of age-dependent branching processes. Such a model, however, does not cover the period following exponential growth when cell differentiation curtails population size. This paper offers an extension to the branching process model to remedy this deficiency. The extended model is ideal for describing embryonic growth; its use is illustrated with data from embryonic retina. The model offers a better computational framework for the interpretation of a variety of data (growth curves of cell numbers, DNA histograms, thymidine labelling indices, FLM curves, BUdR-labelled mitoses curves) because age-distributions can be calculated at any stage of development, not just during exponential growth. Proportions of cells in the various phases of the cell cycle can be computed as growth slows. Such calculations show the gradual transition from a population dominated by cells which are young with respect to cell cycle age to one dominated by those which are old, and the effects such biases have on the proportions of cells in each phase.