Logistic population growth under random dispersal

A hyperbolastic type-I diffusion process: Parameter estimation by means of the firefly algorithm

A stochastic diffusion process, whose mean function is a hyperbolastic curve of type I, is presented. The main characteristics of the process are studied and the problem of maximum likelihood estimation for the parameters of the process is considered. To this end, the firefly metaheuristic optimization algorithm is applied after bounding the parametric space by a stagewise procedure. Some examples based on simulated sample paths and real data illustrate this development.

DYNAMICS OF A NONAUTONOMOUS SYSTEM WITH IMPULSIVE OUTPUT

In this paper, an impulsive exploitation of single species modelled by a general nonautonomous equation is considered. The persistence of the population can be obtained under sufficiently weak conditions and also be distinguished from conditions that cause extinction. These attributes resemble corresponding features of the related autonomous population growth model and improve the results of [6–8].

A comparison of persistence-time estimation for discrete and continuous stochastic population models that include demographic and environmental variability

A discrete-time Markov chain model, a continuous-time Markov chain model, and a stochastic differential equation model are compared for a population experiencing demographic and environmental variability. It is assumed that the environment produces random changes in the per capita birth and death rates, which are independent from the inherent random (demographic) variations in the number of births and deaths for any time interval. An existence and uniqueness result is proved for the stochastic differential equation system. Similarities between the models are demonstrated analytically and computational results are provided to show that estimated persistence times for the three stochastic models are generally in good agreement when the models satisfy certain consistency conditions.

A comparison of three different stochastic population models with regard to persistence time

Results are summarized from the literature on three commonly used stochastic population models with regard to persistence time. In addition, several new results are introduced to clearly illustrate similarities between the models. Specifically, the relations between the mean persistence time and higher-order moments for discrete-time Markov chain models, continuous-time Markov chain models, and stochastic differential equation models are compared for populations experiencing demographic variability. Similarities between the models are demonstrated analytically, and computational results are provided to show that estimated persistence times for the three stochastic models are generally in good agreement when the models are consistently formulated. As an example, the three stochastic models are applied to a population satisfying logistic growth. Logistic growth is interesting as different birth and death rates can yield the same logistic differential equation. However, the persistence behavior of the population is strongly dependent on the explicit forms for the birth and death rates. Computational results demonstrate how dramatically the mean persistence time can vary for different populations that experience the same logistic growth.

World and regional populations

Non-linear regression indicates that the logistic model provides excellent fits to world and continental human population data of the period 1950-1985. If the present logistic regime persists, the world population is likely to double in 47 years. Moreover, the eventual world population would be close to its final value of 23.8 billion by the year 2200, with about 11 billion in Asia and 10.5 billion in Africa. An approximately 95% confidence interval for the eventual world population is found to be between 11.9 and 35.7 billion. The logistic estimates of future populations are much higher than those considered most likely by the United Nations.

A nonautonomous model of population growth

With x = population size, the nonautonomous equation x = xf(t, x) provides a very general description of population growth in which any of the many factors that influence the growth rate may vary through time. If there is some fixed length of time (usually long) such that during any interval of this length the population experiences environmental variability representative of the variation that occurs in all time, then definite conclusions about the population's long-term behavior apply. Specifically, conditions that produce population persistence can be distinguished from conditions that cause extinction, and the difference between any pair of solutions eventually converges to zero. These attributes resemble corresponding features of the related autonomous population growth model x = xf(x).