The fitting of the logistic equation to the rate of increase of population density

Though there are many problems on the usefulness of the logistic curve, it may be necessary to examine before discussing these problems whether or not the actual data fit to the theoretical values. It has been clarified in this paper that the relation between the population density and its rate of increase per individual described by the differential equation (1) is represented by a straight line on a finite difference diagram on which N_i+1−N_i/N_i values are plotted against N_i+1. Utilizing this linear relation we may examine the fittness of the logistic curve to the actual data and when it is fitted we may estimate the parameters of the logistic equation by (5) and (6). The result of the application of this method to the experimental populations of azuki bean weevil indicates that the relation between parent and progeny densities fits well to the logistic type as has been proved byFujita andUtida (1953) who utilized the linear reltion between 1/R+2σ and parent density where R is the apparent rate of reproduction and σ is a constant dependent primarily upon the length of adult life (0≦σ≦1).     

The logistic equation with a diffusionally coupled delay

The asymptotic behaviour of a logistic equation with diffusion on a bounded region and a diffusionally coupled delay is investigated. An equivelent parabolic system is derived for certain types of delays. Using a Layapunov functional, sufficient conditions for the global asymptotic stability of the constant steady state are obtained. When the global stability is lost, using Hopf's bifurcation theory, existence of travelling waves is shown for ring-like and periodic one dimensional habitats.     

An application of the logistic equation to the population dynamics of salt-marsh gastropods

The logistic equation for population growth N = K/(1 + be^−rt) with modifications for environmentally influenced changes in habitat carrying capacity K, and species intrinsic growth rate r, can satisfactorily simulate density levels of real populations of herbivores. The model assumes that carrying capacity is a function of incident solar energy while intrinsic growth rate is a function of temperature and salinity.Test animals were Cerithidea and Assiminea from Mission Bay, San Diego, California.     

A stochastic logistic diffusion equation

A formal stochastic partial differential equation is introduced as a model for the diffusion of a biological population with a fluctuating birthrate in an environment with a finite local carrying capacity. A unique solution is constructed for a related integral equation, and a perturbation expansion is derived when the fluctuations in the birthrate are multiplied by a small parameter.     

The decline of the Serengeti Thomson's gazelle population

Summary The population of Thomson's gazelles in the Serengeti National Park, Tanzania has declined by almost two thirds over a 13 year period. In the early 1970s, numbers stood at 0.66 million animals but had decreased to less than 0.25 million animals in 1985 as estimated by 5 different censuses using two different counting techniques. Predation, interspecific competition and disease are all factors that could have contributed to this decline, and at least one of these factors, predation, could now prevent the Thomson's gazelle population from increasing.     

An alternative formulation for a delayed logistic equation

We derive an alternative expression for a delayed logistic equation, assuming that the rate of change of the population depends on three components: growth, death, and intraspecific competition, with the delay in the growth component. In our formulation, we incorporate the delay in the growth term in a manner consistent with the rate of instantaneous decline in the population given by the model. We provide a complete global analysis, showing that, unlike the dynamics of the classical logistic delay differential equation (DDE) model, no sustained oscillations are possible. Just as for the classical logistic ordinary differential equation (ODE) growth model, all solutions approach a globally asymptotically stable equilibrium. However, unlike both the logistic ODE and DDE growth models, the value of this equilibrium depends on all of the parameters, including the delay, and there is a threshold that determines whether the population survives or dies out. In particular, if the delay is too long, the population dies out. When the population survives, i.e., the attracting equilibrium has a positive value, we explore how this value depends on the parameters. When this value is positive, solutions of our DDE model seem to be well approximated by solutions of the logistic ODE growth model with this carrying capacity and an appropriate choice for the intrinsic growth rate that is independent of the initial conditions.     

Detecting population expansion and decline using microsatellites

This article considers a demographic model where a population varies in size either linearly or exponentially. The genealogical history of microsatellite data sampled from this population can be described using coalescent theory. A method is presented whereby the posterior probability distribution of the genealogical and demographic parameters can be estimated using Markov chain Monte Carlo simulations. The likelihood surface for the demographic parameters is complicated and its general features are described. The method is then applied to published microsatellite data from two populations. Data from the northern hairy-nosed wombat show strong evidence of decline. Data from European humans show weak evidence of expansion.     

Global behaviour of age-dependent logistic population models

We study the large time behaviour of a nonlinear population model with a general logistic term. It is proved that every solution must have a limit when time becomes infinite. We present conditions that guarantee the boundedness of the solution. Furthermore, we prove that in general no oscillation is possible for the total number of population. This is in sharp contrast to the linear case.This work was carried out with the aid of a grant from the International Development Centre, Ottawa, Canada     

A logistic branching process for population genetics

A logistic (regulated population size) branching process population genetic model is presented. It is a modification of both the Wright-Fisher and (unconstrained) branching process models, and shares several properties including the coalescent time and shape, and structure of the coalescent process with those models. An important feature of the model is that population size fluctuation and regulation are intrinsic to the model rather than externally imposed. A consequence of this model is that the fluctuation in population size enhances the prospects for fixation of a beneficial mutation with constant relative viability, which is contrary to a result for the Wright-Fisher model with fluctuating population size. Explanation of this result follows from distinguishing between expected and realized viabilities, in addition to the contrast between absolute and relative viabilities.     

Comparison of logistic equations for population growth.

Two different forms of the logistic equation for population growth appear in the ecological literature. In the form of the logistic equation that appears in recent ecology textbooks the parameters are the instantaneous rate of natural increase per individual and the carrying capacity of the environment. In the form of the logistic equation that appears in some older literature the parameters are the instantaneous birth rate per individual and the carrying capacity. The decision whether to use one form or the other depends on which form of the equation is biologically more realistic. In this study the form of the logistic equation in which the instantaneous birth rate per individual is a parameter is shown to be more realistic in terms of the birth and death processes of population growth. Application of the logistic equation to calculate yield from an exploited fish population also shows that the parameters must be the instantaneous birth rate per individual and the carrying capacity.     

逻辑斯谛增长导致的复杂种群动态及其实验验证

教科书通常用“S”型曲线表示逻辑斯谛增长模型导致的种群动态,事实上,这一模型可以产生包括稳定平衡、周期性振荡、混沌等多种多样的种群动态模式。介绍了如何用Kicker图示的方法分析逻辑斯谛增长可能导致的种群动态模式。同时介绍了相关的实验研究案例,这些实验工作大都是以生活史周期较短的昆虫为材料在相对简单的实验室环境中完成的。     

On a conjecture about the periodic solution of the logistic equation

Summary A conjecture is proved about the periodic solution of the logistic equation with periodic forcing term; when the tracking capacity increases so does the mean population size. A more general case is considered which contains this conjecture as a special case.     

Description of the delayed microbial growth by an extended logistic equation

Logistic equations are suitable for describing microbial growth. By means of VERHULST'S logistic equation, the adaptation to sigmoid-shaped curves of growth improves with a falling ratio C_xo/C_{x, max} < 0.2, if there is no lag-phase. The known logistic equations do not take into account any lag-phase behaviour, so that noticeable deviations in the model adaptation result in this range. Therefore, an extended logistic equation of rate is proposed by which any occuring lag-time is expressed by a 1st-order lag-term. The corresponding time law allows a very good adaptation of curves of delayed growth behaviour, and changes into VERHULST'S logistic equation for a lag-time t_L = 0. Application is facilitated by instructions for the numerical determination.