Introduction to functions and models- LOGISTIC GROWTH MODELS

Physica A: Statistical Mechanics and its Applications, 2015

h i g h l i g h t s

Revista Brasileira de Ensino de Física, 2017

In this work, some phenomenological growth models based only on the population information (macroscopic level) are deduced in an intuitive way. These models, for instance Verhulst, Gompertz and Bertalanffy-Richards models, are introduced in such a way that all the parameters involved have a physical interpretation. A model based on the interaction (distance dependent) between the individuals (microscopic level) is also presented. This microscopic model have some phenomenological models as particular cases. In this approach, the Verhulst model represents the situation in which all the individuals interact in the same way, regardless of the distance between them (mean field approach). Other phenomenological models are retrieved from the microscopic model according to two quantities: i) the way that the interaction decays as a function the distance between two individuals and ii) the dimension of the spatial structure formed by the individuals of the population. This microscopic model ...

Oikos, 2008

Biomathematics, 2006

We introduce the most common quantitative approaches to population dynamics and ecology, emphasizing the different theoretical foundations and assumptions. These populations can be aggregates of cells, simple unicellular organisms, plants or animals. The basic types of biological interactions are analysed: consumer-resource, prey-predation, competition and mutualism. Some of the modern developments associated with the concepts of chaos, quasi-periodicity, and structural stability are discussed. To describe short-and long-range population dispersal, the integral equation approach is derived, and some of its consequences are analysed. We derive the standard McKendrick age-structured density dependent model, and a particular solution of the McKendrick equation is obtained by elementary methods. The existence of demography growth cycles is discussed, and the differences between mitotic and sexual reproduction types are analysed.

Journal of Mathematical Biology, 1983

Various biological phenomena lead to single species models where the relative rate of increase is a non-monotone function of the density, i.e. depensation models. A brief survey of the literature and some new models are given. A nonlinear nonautonomous O.D.E. is proposed as a general depensation model in a periodically fluctuating environment. Results on existence, multiplicity and global stability of periodic solutions are given.

2011

This article examines the popular logistic model of growth from three perspectives: its sensitivity to initial conditions; its relationship to analogous difference equation models; and the formulation of stochastic models of population growth where the mean population size satisfies the logistic relationship. The results indicate that the appealing sigmoid logistic curve is sensitive to initial conditions and care must be exercised in developing difference equation models which display the same appealing long term behavior as the logistic growth curve. It is shown that although the logistic model is appealing in terms of its simplicity its realism is questionable in terms of the degree to which it reflects demographically accepted assumptions about the probabilities of individual births and deaths in the growth of a population. In particular this lack of realism has serious implications for the computer simulation of stochastic birth and death processes where the mean population siz...

2019

Dynamics among central sources (hubs) providing a resource and large number of components enjoying and contributing to this resource describes many real life situations. Modeling, controlling, and balancing this dynamics is a general problem that arises in many scientific disciplines. We analyze a stochastic dynamical system exhibiting this dynamics with a multiplicative noise. We show that this model can be solved exactly by passing to variables that describe the mass ratio between the components and the hub. We derive a deterministic equation for the average mass ratio. This equation describes logistic growth. We derive the full phase diagram of the model and identify three regimes by calculating the sample and moment Lyapunov exponent of the system. The first regime describes full balance between the non-hub components and the hub, in the second regime the entire resource is concentrated mainly in the hub, and in the third regime the resource is localized on a few non-hub compone...

Biophysics, 2019

Application of mathematical modeling for analysis of natural phenomena relies on model reduction techniques, which inevitably raises the question of whether the results of simulation reflect real processes. This work analyzes problems that arise when the results obtained by mathematical modeling of population processes are compared to data collected by monitoring of natural ecosystems. The source of these problems is that the type of dependencies between variables that describe the population dynamics, as well as the choice of numerical values assigned to the parameters of the mathematical model, are often impossible to justify, even based on the monitoring data from a particular ecosystem. This paper proposes an approach to mathematical modeling that would take the impact of the entire complex of biotic and abiotic factors on the population dynamics into account. Its central feature is consideration of ecosystem monitoring data and incorporating them directly into mathematical models of population dynamics. This approach would make it possible, in particular, to evaluate the extent to which individual environmental factors influence both the variations in population abundance recorded during monitoring and those characteristics of population processes that are not directly measured during monitoring, but are obtained by mathematical modeling.

AppliedMath

A general population model with variable carrying capacity consisting of a coupled system of nonlinear ordinary differential equations is proposed, and a procedure for obtaining analytical solutions for three broad classes of models is provided. A particular case is when the population and carrying capacity per capita growth rates are proportional. As an example, a generalised Thornley–France model is given. Further examples are given when the growth rates are not proportional. A criterion when inflexion may occur is also provided, and results of numerical simulations are presented.