Population Growth Equation Research Articles

The Equations of War

Previous articleNext article No AccessReportsThe Equations of WarStephen BeckermanStephen Beckerman Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by Current Anthropology Volume 32, Number 5Dec., 1991 Sponsored by the Wenner-Gren Foundation for Anthropological Research Article DOIhttps://doi.org/10.1086/204013 Views: 4Total views on this site Citations: 5Citations are reported from Crossref Copyright 1991 The Wenner-Gren Foundation for Anthropological ResearchPDF download Crossref reports the following articles citing this article:A. Ghasemabadi, M. H. Rahmani Doust Hopf Bifurcation and Stability Analysis of Delayed Lotka–Volterra Predator–Prey Model Having Disease for Both Existing Species, (Jun 2021): 155–166.https://doi.org/10.1007/978-981-16-1402-6_13Charles H. Anderton, John R. Carter Principles of Conflict Economics, 1 (Apr 2019).https://doi.org/10.1017/9781316875506Atena Ghasemabadi Stability and bifurcation in a generalized delay prey–predator model, Nonlinear Dynamics 90, no.44 (Nov 2017): 2239–2251.https://doi.org/10.1007/s11071-017-3431-3S.A. Belbas, W.H. Schmidt Optimal control of impulsive Volterra equations with variable impulse times, Applied Mathematics and Computation 214, no.22 (Aug 2009): 353–369.https://doi.org/10.1016/j.amc.2009.04.066George R. Milner An Osteological Perspective on Prehistoric Warfare, (Jan 1995): 221–244.https://doi.org/10.1007/978-1-4899-1310-4_10

Modelling the population dynamics of different aphid genotypes in plant variety mixtures

Mathematical models demonstrate that intermixing plants of a susceptible variety among resistant plants within a field may reduce the rate at which resistance-adapted (virulent) aphid genotypes develop. This process should occur if the initial proportion of the virulent aphids is small, if the aphids randomly immigrate to fields, and if their population is subsequently depressed by insecticide application or natural enemies. In these conditions the avirulent aphids will usually reach the field first providing conditions for an increase in mortality factors which then suppress later arriving virulent aphids. The effects of various factors on the magnitude of this result were found using two different modelling approaches. Each approach involved a within season ecological model in which aphids reproduced parthenogenetically and a between season one locus genetic model in which the allele for virulence was assumed to be either dominant or recessive. The first approach, using deterministic Lotka-Volterra equations, demonstrated that if the initial proportion of virulence was below a critical value, virulence would quickly be eliminated from the population even if the avirulent aphids had no selective advantage. The critical value was determined by the proportion of susceptible plants, the level of resistance, the probability of aphids entering the field, the aphid population growth rate, the time when mortality factors appeared, and the cost of virulence. The second approach expanded on the previous model by introducing stochastic effects, using more realistic population growth equations, and following spatial and movement patterns of the organisms. The general conclusion from the stochastic model was similar to that of the deterministic model, with the additional result that most parameters did not greatly effect the outcome. The most important parameters were the percent susceptible plants in the field, the level of resistance, and the aphid population growth rate. In the genetic section of both models the type of inheritance of virulence made only small differences in these results if the initial proportion of the virulent phenotype was the same.

Intra- and Interspecific Interactions in Low Density Populations in Resource-Rich Habitats

Parasites on the gills of marine fish are used to study intra- and interspecific interactions.(...) Evidence is given that most animals species are likely to occur at low densities in resource-rich habitats, and that conclusions based on the study of gill parasites can be applied to them. In particular, a discussion of a diversification equation modelled on the logistic equation for population growth shows that there is a vast number of vacant niches. The general conclusion is that the major problem for most animals is not to avoid competition but to find suitable sites for feeding and mating

Construction of population growth equations in the presence of viability constraints

A mathematical method based on the G-projection of differential inclusions is used to construct dynamical models of population biology. We suppose that the system under study, not being limited by resources, may be described by a control system $$\dot x\left( t \right) = f\left( {x\left( t \right),u\left( t \right)} \right)$$ \] where u is a control describing the choice of resources. Then considering the constraints that the system must satisfy we define a viability set K. Since there may not exist a control u(·) such that the corresponding solution satisfies x(t) e K, we have to change the dynamics of the control system to get a viable solution. Using the G-projection we introduce so-called “projected” control system $$\dot x\left( t \right) = \Pi _{T_K }^G f\left( {x\left( t \right),u\left( t \right)} \right)$$ that has a viable solution. The projected system has usually simpler dynamics than traditional models used in population biology.

Ziebur's matrix equation for population growth

AbstractThe paper examines a matrix equation given by Ziebur [6] for the growth of a population in which the birth-rate and death-rate are age-dependent. For convenience the population was sub-divided into four age groups, with the same birth-rate and death-rate for individuals in a particular group, and the matrix equation relates the numbers in each sub-division in consecutive years. This avoids delay terms and makes it easier to modify the growth equation but it is shown that the form suggested by Ziebur for the transition matrix leads to some difficulties.

Periodic McKendrick equations for age-structered population growth

With the averaged net reproductive rate used as a bifurcation parameter, the existence of a local parameterized branch of time-periodic solutions of the McKendrick equations is proved under the assumption that the death and fertility rates suffers small-amplitude time periodicities. The required linear theory is developed and the results are illustrated by means of a simple example in which fertility varies cosinusoidally in time.

Small Parameters in Structured Population Models and the Trotter–Kato Theorem

The justification of some (often implicit) limit arguments used in the development of structured population models is discussed via two examples. The first example shows how a pair of sink-source terms may transform into a side condition relating the appearance of individuals in the interior of the individual state space to the outflow of individuals at its boundary. The second example considers the usual equation for size-dependent population growth in which it is implicitly assumed that discrete finitely-sized young are produced from infinitesimal contributions by all potential parents. The main mathematical tool for dealing with these examples is the Trotter–Kato theorem for one-parameter semigroups of bounded linear operators.

Travelling waves for forced Fisher’s equation

Publisher Summary This chapter discusses travelling waves for forced Fisher's equation and describes Fisher's partial differential equation for model population growth. The method of Liapunov–Schmidt and the bifurcation from a simple eigenvalue theorem of Crandall and Rabinowitz imply that there are solutions of the form where the function satisfies the asymptotic. The saddle point theory of periodic differential equations can be used to obtain the existence of a transverse heteroclinic point and infinitely many waves.

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.

Global branches of equilibrium solutions of the McKendrick equations for age-structured population growth

The existence of positive equilibrium solutions of the McKendrick equations for the dynamics of an age-structured population is studied as a bifurcation phenomenon using the inherent net reproductive rate n as a bifurcation parameter. Under only continuity assumptions on the density dependent death and tertility rates, it is shown that a global continuum of positive equilibria exists in a certain Banach space. This continuum connects from the trivial solution at n = 1 to the boundary of the domain on which the problem is posed. Results concerning the spectrum are given. In particular, some circumstances are described under which positive equilibria exist for all n values greater than the critical value n = 1.

Qualitative behaviour of stochastic delay equations with a bounded memory

A dichotomy is proved concerning recurrence properties of the solution of certain stochastic delay equations. If the solution process is recurrent, there exists an invariant measure π on the state space C which is unique (up to a multiplicative constant) and the tail-field is trivial. If π happens to be a probability measure, then for every initial condition, the distribution of the process converges to it as t→∞. We will formulate a sufficient condition for the existence of an invariant probability measure (ipm) in icrnia of Lyapunov junctionals and give two examples, one Heing the stochastic-delay version of the famous logistic equation of population growth. Finally we study approximations of delay equations by Markov chains.

Analytical solutions for distributions of chemotactic bacteria

This paper reports general and specialized results on analytical solutions to the governing phenomenological equations for chemotactic redistribution and population growth of motile bacteria. It is shown that the number of bacteria cells per unit volume, b, is proportional to a certain prescribed function of s, the concentration of the critical substrate chemotactic agent, for steady-state solutions through an arbitrary spatial region with a boundary that is impermeable to bacteria cell transport. Moreover, it is demonstrated that the steady-state solution for b and s is unique for a prescribed total number of bacteria cells in the spatial region and a generic Robin boundary condition on s. The latter solution can be approximated to desired accuracy in terms of the Poisson-Green's function associated with the spatial region. Also, as shown by example, closed-form exact steady-state solutions are obtainable for certain consumption rate functions and geometrically symmetric spatial regions. A solutional procedure is formulated for the initialvalue problem in cases for which significant population growth is present and bacteria cell redistribution due to motility and chemotactic flow proceeds slowly relative to the diffusion of the chemoattractant substrate. Finally, a remarkably simple exact analytical solution is reported for a steadily propagating plane-wave which features motility, chemotactic motion and bacteria population growth regulated by substrate diffusion.

Distribution of Thalli in a Population of the Epiphytic Lichen Lobaria oregana and a Model of Population Dynamics and Production

Data on weight-class distribution, growth rate and asexual propagule (lobule) release are used to analyze populations of the nitrogen-fixing lichen Lobaria oregana in old-growth Douglas fir canopies of western Oregon. A life-table model prepared from observed data is used to produce an estimate of net annual production of biomass, and to analyze the response of a population of this lichen to possible environmental perturbations. The weight-class distribution is similar in each offour trees studied and at different levels within the trees, but differed among the different components of the trees. Survivorship curves (log survivorship versus estimated thallus age) are similar for populations in all trees studied. Estimated net annual production is 31.1% of the standing crop of Lobaria oregana biomass. Mathematical manipulation of matrix formulations of the life-table model indicates that populations would be stable if they recruited 10% of the lobules released each year. In such a configuration, the model population returns to its original size and distribution within 30 years following a simulated storm that removes 90% of the largest thalli and momentarily increases lobule production. Study of growth and annual production of lichens has emphasized individual thalli (reviewed by Hale 1973 and Armstrong 1976) or total standing crops of several species in one habitat (reviewed in Pike 1978). There are virtually no studies of the structure of whole lichen populations or the dynamics within them. Pike (197 1) looked at relative numbers and total weight of thalli of Usnea subfloridana Stirt. on oak and ash twigs of different ages in Oregon in order to estimate annual production, but did not look at the dynamics of growth of individual thalli. Yarranton (1975) examined the change in distribution of biomass and numbers of podetia of Cladina stellaris (Opiz.) Brodo in successional populations on burnt sites 25 to 55 years of age in Ontario, Canada. Data on the distribution of size classes of podetia were not analyzed because of difficulty in estimating the effect of clonal fragmentation. Armstrong (1978) has examined the pattern of dispersion of Parmelia glabratula ssp. fuliginosa (Fr. ex Duby) Laund. on slate rock surfaces. He found larger thalli at the top than bottom of the substrate, but identical growth rates among thalli, concluding that colonization had taken place from top to bottom. Further 007-2745/83/309-331 $2.35/0 This content downloaded from 157.55.39.45 on Tue, 19 Jul 2016 05:14:23 UTC All use subject to http://about.jstor.org/terms 310 THE BRYOLOGIST [Volume 86 studies by Armstrong (1981) have shown that substrate topography plays a major role in the entrapment of propagules and their establishment as thalli, and that only a small percentage of propagules become established as thalli. The dynamics of populations of organisms (growth, mortality, reproduction, potential stability and potential response to environmental perturbations) can be examined by analysis of survivorship curves and life tables (Harper 1977). These types of analyses traditionally have been used to study organisms (insects, birds, seed plants) whose populations show a natural division into ageable groups, and lichens have not been studied in this manner. Lichens are long-lived, reproduce by fragmentation and do not have life histories that fall into specific age classes (Harper & White 1974). Thus, it has been difficult to establish age-specific functions for mortality and reproduction in such populations. Watson's (1979) study of the variability of populations of mosses is unique, using survivorship curves to analyze the age-class distribution of cryptogams. She was able to show differences among species and among populations of the same species in different environments. Lefkovitch (1965) showed that a modified formulation of life-table analysis developed by Leslie (1945) can be used for a population of organisms grouped in unequal-aged stages. Hartshorn (1975) has used this method in describing the population dynamics of a tropical tree species. Leslie (1945) showed that the dominant root (XI) of a matrix of (age-specific) coefficients is equal to er, where r is the intrinsic rate of natural increase in the exponential equation of population growth,

Optimal control of a multi-cohort fishery

Abstract Models of optimal renewable resource exploitation typically assume an equation for total population growth over time irrespective of the age distribution of the stock. By means of the theory of parametric optimal control, this paper outlines a model of a fishery with population varying in both age and time. Instead of an optimal steady-state harvest we derive an optimal harvest age distribution with lower utilization of younger members of the population.

Gamma Distributions of Adult Numbers for Tribolium Populations in the Region of Their Steady States

SUMMARY (1) A gamma steady-state probability distribution was established for adult numbers in continuously growing populations of the flour beetle Tribolium. The derivation of the distribution was based on a general stochastic model of population growth with three biological entities: adult inhibition of immatures, pupal productivity, and death-rate among adults. (2) The hypothesis of a gamma steady-state distribution was tested using thirteen observed frequency distributions of adult numbers for Tribolium castaneum and Tribolium confusum. The data, in general, supported the gamma steady-state hypothesis. (3) Using the theoretical gamma probability distribution an attempt was made to explain the Tribolium data and identify testable hypotheses. For instance, the observed differences in the mean and variance of the adult steady-state distributions for the genetic strains of T. castaneum and T. confusum were qualitatively consistent with the theoretical predictions based upon differences in cannibalistic rates, pupal productivity and adult death-rates. The expected effects of stochasticity on the mean and variance were also discussed. (4) The stochastic differential equation of population growth was viewed using the Ito and Stratonovich definitions of stochastic integrals. Both procedures yielded gamma steady-state distributions; however, the conditions for the existence of the stationary densities were different. With stochastic variation in the death-rate among adults, the existence conditions for the Ito calculus was that the magnitude of the stochastic variance must be less than twice the difference between the number of pupae produced per parent and the death-rate among adults. For the stratonovich calculus, the stationary density always existed.

R- AND K-SELECTION IN DROSOPHILA PSEUDOOBSCURA.

Of the ecological forces which shape evolution of life histories, competition seems especially important. When competition is absent, those genotypes which reproduce more rapidly and prolifically are favored; when resources are limiting, the advantage goes instead to those best able to compete. MacArthur and Wilson (1967) termed the extreme cases rand K-selection, respectively, after terms in the widely-used logistic equation of population growth. They understood that environments imposing pure ror K-selection were only ideals, ends of a continuum, and that most actual situations would be somewhere intermediate. A characteristic set of traits is thought to result from selection for one extreme or the other: among other things, r-selected populations should develop more rapidly, have a higher intrinsic rate of increase (r), reproduce earlier, develop to a smaller size, invest less in each offspring, and tend toward semelparity (i.e., to reproduce only one time); K-selected species should have a higher carrying capacity (K), invest more energy into each offspring, develop more slowly but eventually attain a larger size, and tend toward neoteny and iteroparity (i.e., to reproduce more than once). Comprehensive discussions of the rationale and history of these expectations may be found in the reviews by Pianka (1970), Stearns (1976), and Gould (1977). In spite of this assumed correspondence between ecological circumstances and type of life history, we know of no demonstration that r-selection results in one life history, while K-selection results in

The effect of time lags on the stability of the equilibrium state of a population growth equation

A scalar integrodifferential equation is considered which describes a single self-regulating species. Three results are presented towards showing that the ‘carrying capacity’ equilibrium state becomes unstable as the self-regulating mechanism acts after a longer time lag.

The qualitative analysis of a difference equation of population growth.

The difference equation f b :[0,1]−[0,1] defined by f b (x)=b x(1−x) is studied. In particular complete qualitative information is obtained for the parameter value b=3.83. For example the number of fixed points of (f b )i is given by $$N_i = 1 + \left( {\frac{{1 + \sqrt 5 }}{2}} \right)^i + \left( {\frac{{1 - \sqrt 5 }}{2}} \right)^i$$

On the stability of the stationary state of a population growth equation with time-lag.

If in the Verhulst equation for population growth the reproduction factor depends on the history then the equilibrium may become unstable and oscillations and even non-constant periodic solutions may occur. It is shown that the equilibrium is unstable if the reproduction factor at time t is, up to a sufficiently large factor, an arbitrary average of the population densities in the interval (t-2, t-1).

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.