Thomas R. Kiffe

ON APPROXIMATING THE MOMENTS OF THE EQUILIBRIUM DISTRIBUTION OF A STOCHASTIC LOGISTIC MODEL

The classical Verhulst-Pearl logistic equation is a widely used deterministic population growth model. Bartlett, Gower, and Leslie (1960, Biometrika 47, 1-11) develop an analogous stochastic logistic model. They obtain the equilibrium distribution of the population size numerically, and derive elegant approximations for the mean, variance, and skewness of this equilibrium distribution for the model. This paper derives new approximations for these moments. The approximations are simple analytical expressions and in general are more accurate than the Bartlett et al. approximations. The approximations are illustrated with a growth model for Africanized honey bee populations and also with well-known examples in the literature.

A simple saddlepoint approximation for the equilibrium distribution of the stochastic logistic model of population growth

Abstract The deterministic logistic model of population growth and its notion of an equilibrium ‘carrying capacity’ are widely used in the ecological sciences. Leading texts also present a stochastic formulation of the model and discuss the concept and calculation of an equilibrium population size distribution. This paper describes a new method of finding accurate approximating distributions. Recently, cumulant approximations for the equilibrium distribution of this model were derived [Biometrics 52 (1996) 980], and separately a simple saddlepoint (SP) method of approximating distributions using exact cumulants was presented [J. Math. Appl. Med. Biol. 15 (1998) 41]. This paper proposed using the SP method with the new approximate cumulants, which are readily obtained from the assumed birth and death rates. The method is shown to be quite accurate with three test cases, namely on a classic model proposed by Pielou [Mathematical Ecology, New York, Wiley, p. 304] and on two African bee models proposed previously by the authors [Biometrics 52 (1996) 980; Theor. Popul. Biol. 53 (1998) 16]. Because the new method is also relatively simple to apply, it is expected that its use will lead to a more widespread utilization of the stochastic model in ecological modeling.

Describing the spread of biological populations using stochastic compartmental models with births

This paper derives new models for describing the spread of biological populations in space and time from classical birth-death-migration processes. The spatial aspect is incorporated using compartmental analysis and is developed for two spatial areas (or compartments). The exact bivariate distributions for such processes are intractable; hence approximating distributions are constructed by matching cumulants. A basic Markovian model with exponential waiting times between births is investigated first. The individual effects of swarming, multiple births, and Erlang distributed waiting times, all of which enhance the biological realism, are investigated. A full model which includes all of these effects is then studied. The models are illustrated with observed data on the spread of the Africanized honey bee in French Guiana. A full model with swarming, with an average of 2.64 colonies per swarming episode, and with waiting times following an Erlang distribution with shape parameter 5 is found to provide the best description of the observed data. The methodology is very general and should have broad application for other biological population models involving dispersal and growth.

On interacting bee/mite populations: a stochastic model with analysis using cumulant truncation

Parasitism by the Varroa mite has had recent drastic impact on both managed and feral bee colonies. This paper proposes a stochastic population dynamics model for interacting African bee colony and Varroa mite populations. Cumulant truncation procedures are used to obtain approximate transient cumulant functions, unconstrained by the usual assumption of bivariate Normality, for an assumed large-scale model. The apparent size of the variance and skewness functions suggest the importance of the proposed truncation procedure which retains some higher-order cumulants, but determining the accuracy of the approximations is problematical. A smaller-scale bee/Varroa mite model is hence proposed and investigated. The accuracy for the means is exceptional, for the second-order cumulants is moderate, and for some third-order cumulants is poor. Notwithstanding the poor accuracy of a skewness approximation, the saddlepoint approximations for the marginal transient population size distributions are excellent. The cumulant truncation methodology is very general, and research is continuing in its application to this new class of host-parasite models.

Application of population growth models based on cumulative size to pecan aphids

Models for aphid population growth based on cumulative (past) population size have been developed with both a deterministic formulation and a stochastic formulation. This article applies these mechanistic models to analyze a large dataset on pecan aphid. The models yield symmetric and right-skewed curves, which differ qualitatively from the observed data which tend to be left-skewed. Nevertheless this model-based analysis of data shows that the models are (1) useful representations of the data, (2) biologically reasonable descriptions of aphid population dynamics, and (3) a statistically powerful basis for the analysis of experimental data. The models also provide user-friendly predictive equations for aphid abundance. The deterministic model gives accurate point predictions of the peak and final cumulative population sizes, and the stochastic model gives, in addition, an accurate interval estimate of cumulative size. The models should be applicable to numerous other aphid abundance curves, as all aphid species have a similar reproductive mechanism.

Estimating parameters for birth-death-migration models from spatio-temporal abundance data: case of muskrat spread in The Netherlands

This article uses a stochastic birth-death-migration (BDM) process for describing the spread of muskrats in 11 provinces in the Netherlands during their invasion periods from 1968 to 1991. Mean value functions of the BDM model are derived for muskrat abundance and then transformed to describe catch. The mean value function of an immigration-birth model, without migration, is first fitted separately to the data from each province. The mean value functions of the BDM models are then fitted for groups of adjacent provinces. The article demonstrates that individual migration and net birth-rate parameters are estimable, with good precision for most provinces, from these spatio-temporal abundance data. The separate models without migration fit the data well, but the net birth rates are generally underestimated for donor provinces and overestimated for recipient provinces. The BDM models often fit the data better, and their estimated net birth rates from the provinces are closer together. The methods are general, and should have wide applicability to invasion data for other plant and animal species.

An abstract Volterra integral equation in a reflexive Banach space

This paper discusses the existence, uniqueness, and asymptotic behavior of solutions to the equation u(t) + ∝0t a(t − s) Au(s) ds = f(t), where A is a maximal monotone operator mapping the reflexive Banach space V into its dual V′.

Generalized aphid population growth models with immigration and cumulative-size dependent dynamics.

Mechanistic models in which the per-capita death rate of a population is proportional to cumulative past size have been shown to describe adequately the population size curves for a number of aphid species. Such previous cumulative-sized based models have not included immigration. The inclusion of immigration is suggested biologically as local aphid populations are initiated by migration of winged aphids and as reproduction is temperature-dependent. This paper investigates two models with constant immigration, one with continuous immigration and the other with restricted immigration. Cases of the latter are relatively simple to fit to data. The results from these two immigration models are compared for data sets on the mustard aphid in India.

Stochastic compartment models with Prendville growth rates

The Prendville growth mechanism, which assumes a linearly decreasing population growth rate, was generalized to a multicompartment system by Parthasarathy and Kumar. This article shows that their solution is only an approximate one. A diagnostic test is presented to indicate parameter vectors for which this first approximation is not very accurate. A second approximation, which overcomes some limitations of the first one, is developed and illustrated.

A Volterra Equation with a Nonconvolution Kernel

This paper is concerned with the asymptotic behavior of solutions of the Volterra integral equation \[x(t) + \int_0^t {a(t,\tau )g(x(\tau ))d\tau = f(t)} ,\quad 0 \leqq t < \infty \] If