A new nonlinear model of age-dependent population growth

A new nonlinear age-structured population model is presented. Within its framework the occurence of time-persistent age distributions is possible, even if the population sizes are nonstationary. The age distribution as well as the moments of the generation index can be determined analytically. The proposed model is a nonlinear generalization of Lotka's theory of stable populations.     

Non-linear age-dependent population growth

Summary A non-linear problem arising from age-dependent population dynamics is studied. Existence and uniqueness and a priori bounds for the growth of population are proved. Moreover the existence and the stability of equilibrium age distributions is investigated.     

A stochastic computer model for simulating population growth

A model is described for investigating the interactions of age-specific birth and death rates, age distribution and density-governing factors determining the growth form of single-species populations. It employs Monte Carlo techniques to simulate the births and deaths of individuals while density-governing factors are represented by simple algebraic equations relating survival and fecundity to population density. In all respects the model's behavior agrees with the results of more conventional mathematical approaches, including the logistic model andLotka's Law, which predicts a relationship betwen age-specific rates, rate of increase and age distribution. Situations involving exponential growth, three different age-independent density functions affecting survival, three affecting fecundity and their nine combinations were tested. The one function meeting the assumptions of the logistic model produced a logistic growth curve embodying the correct values or r_m and K. The others generated sigmoid curves to which arbitrary logistic curves could be fitted with varying success. Because of populational time lags, two of the functions affecting fecundity produced overshoots and damped oscillations during the initial approach to the steady state. The general behavior of age-dependent density functions is briefly explored and a complex example is described that produces population fluctuations by an egg cannibalism mechanism similar to that found in the flour beetle Tribolium. The model is free of inherent time lags found in other discrete time models yet these may be easily introduced. Because it manipulates separate individuals, the model may be combined readily with the Monte Carlo simulation models of population genetics to study eco-genetic phenomena.     

Linear age-dependent population growth with harvesting

This paper studies the effect of harvesting a fraction of a population where the population growth is modelled by a linear age-dependent model, the Von Foerster equation. Two harvesting strategies are considered: the first is where a fraction of the population greater than agec is removed, and the second is where a fraction of the population of age greater thanc but less thanc+n is removed. In the case where the death rate and fertility rate are time independent, the effect of harvesting on the stable age distribution is examined. Research done at the University of New Mexico and partially supported there by NIH Grant No. RR-08139.     

Linear age-dependent population growth with seasonal harvesting

A population growth is modelled by the Von Foerster PDE with accompanying Lotka-Volterra integral equation describing the birth rate; the age specific death and fertility rates are assumed to depend only on age and not time. A harvesting policy where a fraction of the population of age greater than a given age is harvested for a fraction of a given season. This introduces a time dependence, but this difficulty is circumvented by devising approximate timeindependent models whose birthrates bracket the true birthrate-the standard renewal equation theory applies to the approximate models so quantitative results can be obtained.The author wishes to thank Professors J. J. Levin and R. K. Miller for some useful remarks. This research was partially supported by NIH Grant GMO 7661-02.     

Nonlinear age-dependent models for prediction of population growth

In this paper, some new algorithms are proposed to estimate parameter functions in nonlinear age-dependent population models by practical data. These algorithms together with a numerical method are applied to compute the human population using the data provided by the United Nations Demographic Yearbook.     

A model of age-dependent population dynamics providing simple criteria for growth or extinction.

The Gurtin-MacCamy model for age-dependent population dynamics is reduced to a single ordinary differential equation. (This is done by assuming a specific form of the mortality function, and by assuming fertility to be age-dependent only.) The model is then applied to a population of greater horseshoe bats.     

On the global stability of the logistic age-dependent population growth

We study an age-dependent population equation with a nonlinear death rate of logistic type. The global asymptotic stability of the null solution is investigated when R(0)<1. If R(0)>1 we get the existence of a nontrivial steady state that becomes asymptotically stable itself, while the null solution is unstable. The rate of decay is estimated.Supported in part by CNR-GNAFA     

Temporal autocorrelation and stochastic population growth

How much does environmental autocorrelation matter to the growth of structured populations in real life contexts? Interannual variances in vital rates certainly do, but it has been suggested that between‐year correlations may not. We present an analytical approximation to stochastic growth rate for multistate Markovian environments and show that it is accurate by testing it in two empirically based examples. We find that temporal autocorrelation has sizeable effect on growth rates of structured populations, larger in many cases than the effect of interannual variability. Our approximation defines a sensitivity to autocorrelated variability, showing how demographic damping and environmental pattern interact to determine a population's stochastic growth rate.     

Nonlinear stochastic modeling of aphid population growth

This paper develops a stochastic population size model for the black-margined pecan aphid. Prajneshu [Prajneshu, A nonlinear statistical model for aphid population growth. J. Indian Soc. Agric. Statist. 51 (1998), p. 73] proposes a novel nonlinear deterministic model for aphid abundance. The per capita death rate in his model is proportional to the cumulative population size, and the solution is a symmetric analytical function. This paper fits Prajneshu's deterministic model to data. An analogous stochastic model, in which both the current and the cumulative aphid counts are state variables, is then proposed. The bivariate solution of the model, with parameter values suggested by the data, is obtained by solving a large system of Kolmogorov equations. Differential equations are derived for the first and second order cumulants, and moment closure approximations are obtained for the means and variances by solving the set of only five equations. These approximations, which are simple for ecologists to calculate, are shown to give accurate predictions of the two endpoints of applied interest, namely (1) the peak aphid count and (2) the final cumulative aphid count.     

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.     

A stochastic model of a temperature-dependent population

The theoretical basis is developed for a population model which allows the use of constant temperature experimental data in predicting the size of an insect population for any variable temperature environment. The model is based on a stochastic analysis of an insect's mortality, development, and reproduction response to temperature. The key concept in the model is the utilization of a physiological time scale. Different temperatures affect the population by increasing an individual's physiological age by differing rates. Conditions for the temperature response properties are given which establish the validity of the model for variable temperature regimes. These conditions refer to the relationship between chronological and physiological age. Reasonable agreement between the model and field populations demonstrates the practicality of this approach.     

Non-linear age-dependent population diffusion

Summary A non-linear problem arising in the study of an age-dependent population diffusion is considered. Existence and uniqueness results together with a priori bounds for the growth of the population are obtained. Moreover the solutions are shown to depend continuously on the initial data.This work was done under the auspices of the G.N.A.F.A. of the National Research Council.     

An age-dependent epidemic model with application to measles

A combined epidemic-demographic model is developed which models the spread of an infectious disease throughout a population of constant size. The model allows for births, deaths, temporary or permanent immunity, and immunization. The relationship of this model to previously studied epidemic and demographic models is illustrated. An advantage of this model is that all epidemic and demographic parameters may be estimated. The stability of the equilibrium point corresponding to the elimination of the disease is studied and a threshold value is found which indicates whether the disease will die out or remain endemic in the population. The application of the model to measles indicates that immunization levels needed to reduce the incidence to near zero may not be as high as previously predicted.     

A stochastic model of animal growth

The plasticity of growth of animals in time, due to the resilience of their response to the ways they can be fed, suggests the difficulty of describing growth by a stochastic model in the time domain. A model is presented which avoids this difficulty by describing growth as a Markov process in the food-consumed domain, assuming that, at conception, (1) the maximum mature weight as a number α of biomass units of mass μ, and (2) the probability B of production of a biomass unit per unit of food consumed, are specified. Constancy of α, μ and B, as the animal feeds, is the basis of the proposed Markov process. The mean growth from infancy to maturity in the food-consumed domain is then the old law of diminishing returns empirically formulated first by Spillman (1924) for cattle and swine, and confirmed by Titus, Jull &; Hendricks (1934) for fowl, and by Parks (1972) across species from mice to steers. The solution also leads to the possibility that the distribution of weights in a population of growing animals of the same species, is related to the distribution of mature weights among the individuals. An experiment by Lister &; McCance (1967) with well-fed and severely undernourished pigs, shows the stability of growth in the foodconsumed domain compared to the plasticity in the time domain. Other implications of the model are discussed.     

Iteration and nonlinear equations of age-dependent population growth with a birth window

Several iteration schemes are developed for finding the age distribution satisfying a nonlinear McKendrick-Von Foerster equation of population growth. The scheme is based on the concept of matriarchal and temporal generation expansions developed by S.I. Rubinow, and it depends on the assumption that the birth and death rates depend on only a portion of the present population, not the entire population.     

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.     

A model for the age-dependent dynamics of growth of terrestrial mammals

Two hypothetical biological mechanisms were proposed that largely determine the dynamics of growth of overland mammals. The first mechanism is the maintenance of a spatial similarity of the anatomy of the organism during its growth, and the second is the maintenance of the stability of the internal environment of the organism. On the basis of the advanced hypothesis, a model of age-dependent dynamics of growth was constructed, and a differential equation describing the changes in the body mass with time was derived. According to this model, the dynamics of growth and the mass of an adult individual are determined by two energy constants that characterize the mechanism of food assimilation and the energy expenditures for the movement of the individual in space. The deviation between the solution obtained and the experimental data on age-dependent changes in the mass of the human body was on the average 6%. These insignificant deviations were explained in the framework of the proposed hypothesis, which indicates its validity.