Derivation of a Logistic Equation for Organizations,and its Expansion into a Competitive Organizations Simulation

One of the earliest and most famous of the models that produce chaotic behaviour is the Logistic equation. It has a long history of use in economics and organization science studies. In those studies, the applicability of the equation is generally assumed rather than derived from first principles, with only conjecture offered as to the identity of the parameters.This paper shows a deductive derivation of a Logistic equation for organizations in a competitive economy. The construct is based on a system that consists of one or more organizations, each with its own cost, productivity, and reinvestment parameters, and each having its individual population of employees.The model is chaotic, and demonstrates some fascinating characteristics when organizations with parameters that individually would generate chaotic behaviour are mixed with organizations with parameters that individually would generate complex or stable behaviour. Some such mixed systems tend to initially behave like a complex or chaotic system, but transition over time to match the behaviour of the organization with the most “stable” parameters.If a system is at equilibrium, changing one organization’s parameters can result in a burst of oscillations or chaotic activity while the system transitions to a new equilibrium. However, this activity can be delayed or might not occur at all.Adding a number of real world complications to the Organization Logistic equation created a deterministic time-step simulation of an economic system. This simulation was also found to exhibit chaotic behaviour, nonmonotonicity, and the Butterfly Effect, as well as spontaneous bankruptcies.The possibility that a competitive economic system might be inherently chaotic deserves further investigation. A broader insight is that the Scientific Method is not an appropriate scientific paradigm under which to grow social science knowledge if those social science systems are governed by chaotic mechanisms.Alan Zimm graduated from UCLA with a B.S. in physics. He served in the United States Navy for 20 years as a nuclear power qualified surface warfare officer, reaching the rank of Commander. He received a M.S. degree in Operations Research from the Naval Postgraduate School, and a D.P.A. in Public Administration with emphasis on policy analysis and strategic planning from the University of Southern California. Since 1993 he has worked at The Johns Hopkins University Applied Physics Laboratory involved with warfare analysis and the modeling of complex human systems. In 1999 he was awarded the Arleigh Burke Award from the United States Naval Institute and in 2001 a Distinguished Citation Award from the University of Southern California School of Policy, Planning, and Development.     

Kolmogorov流动模型的七模截断及其混沌特性

为了给出Kolmogorov流动模型中混沌行为的数学描述,选取常数k=3,重新对描述该模型的Navier-Stokes方程进行截断,得到了一个新的七维混沌系统．数值模拟了控制参数在一定范围内变化时方程组的基本动力学行为和混沌轨线,分析了其混沌特性．一方面证实了具有湍流特性的数学对象归因于低维混沌吸引子,另一方面有利于更好地了解湍流流动产生的机理.     

Effect of toxic substance on delayed competitive allelopathic phytoplankton system with varying parameters through stability and bifurcation analysis

We have studied the combined effect of toxicant and fluctuation of the biological parameters on the dynamical behaviors of a delayed two-species competitive system with imprecise biological parameters. Due to the global increase of harmful phytoplankton blooms, the study of dynamic interactions between two competing phytoplankton species in the presence of toxic substances is an active field of research now days. The ordinary mathematical formulation of models for two competing phytoplankton species, when one or both the species liberate toxic substances, is unable to capture the oscillatory and highly variable growth of phytoplankton populations. The deterministic model never predicts the sudden localized behavior of certain species. These obstacles of mathematical modeling can be overcomed if we include interval variability of biological parameters in our modeling approach. In this investigation, we construct imprecise models of allelopathic interactions between two competing phytoplankton species as a parametric differential equation model. We incorporate the effect of toxicant on the species in two different cases known as toxic inhibition and toxic stimulatory system. We have discussed the existence of various equilibrium points and stability of the system at these equilibrium points. In case of toxic stimulatory system, the delay model exhibits a stable limit cycle oscillation. Analytical findings are supported through exhaustive numerical simulations.     

Fast game dynamics coupled to slow population dynamics: A single population with hawk-dove strategies

We study a model of a population subdivided into two subpopulations corresponding to hawk and dove tactics. It is assumed that the hawk and dove individuals compete for a resource every Day, I.e., at a fast time scale. This fast part of the model is coupled to a slow part which describes the growth of the subpopulations and the long term effects of the encounters between the individuals which must fight to have an access to the resource. We aggregate the model into a single equation for the total population. It is shown that in the case of a constant game matrix, the total population grows according to a logistic curve whose τ and K parameters are related to the coefficients of the hawk-dove game matrix. Our result shows that high equilibrium density populations are mainly doves, whereas low equilibrium density populations are mainly hawks. We also study the case of a density dependent game matrix for which the gain is linearly decreasing with the total density.     

Dynamics with riddled basins of attraction in models of interacting populations

Recently it has been shown that when there are chaotic attractors whose basins are such that every point in the basin has pieces of another attractors's basin arbitrarily nearby, the basins are said to be riddled. A key requirement for the occurrence of a riddled basin is the loss of transverse stability of an invariant subspace, of dimension less than the full space, containing a chaotic attractor. This type of complex dynamics has been found in simple models of interacting populations for which the invariant subspace is defined by the extinction of one species. The characterizations and implications of these behaviors for population ecology are discussed.     

A stationary population model with an interior interface-type boundary

We propose a stationary system that might be regarded as a migration model of some population abandoning their original place of abode and becoming part of another population, once they reach the interface boundary. To do so, we show a model where each population follows a logistic equation in their own environment while assuming spatial heterogeneities. Moreover, both populations are coupled through the common boundary, which acts as a permeable membrane on which their flow moves in and out. The main goal we face in this work will be to describe the precise interplay between the stationary solutions with respect to the parameters involved in the problem, in particular the growth rate of the populations and the coupling parameter involved on the boundary where the interchange of flux is taking place.     

Dynamical behaviors of a discrete two-dimensional competitive system exactly driven by the large centre

In this paper, a new discrete large-sub-center system is obtained by using the Euler and nonstandard discretization methods for the corresponding continuous system. It is surprised that all dynamic behaviors of the discrete system are exactly driven by the large-center equation, for example, the stabilities, the bifurcations, the period-doubling orbits, and the chaotic dynamics, etc. Additionally, the global asymptotical stability, the existence of exact 2-periodic solutions, the flip bifurcation theorem, and the invariant set of the sub-center equation is also given. These results reveal far richer dynamics of the discrete model compared with the continuous model. Through numerical simulation, we can observe some complex dynamic behaviors, such as period-doubling cascade, periodic windows, chaotic dynamics, etc. Especially, our theoretical results are also showed by those numerical simulations.     

Complex dynamics in physical pendulum equation with suspension axis vibrations

The physical pendulum equation with suspension axis vibrations is investigated. By using Melnikov＇s method, we prove the conditions for the existence of chaos under periodic perturbations. By using second-order averaging method and Melinikov＇s method, we give the conditions for the existence of chaos in an averaged system under quasi-periodic perturbations for Ω = nω ＋ εv, n = 1 - 4, where ν is not rational to ω. We are not able to prove the existence of chaos for n = 5 - 15, but show the chaotic behavior for n = 5 by numerical simulation. By numerical simulation we check on our theoretical analysis and further exhibit the complex dynamical behavior, including the bifurcation and reverse bifurcation from period-one to period-two orbits; the onset of chaos, the entire chaotic region without periodic windows, chaotic regions with complex periodic windows or with complex quasi-periodic windows; chaotic behaviors suddenly disappearing, or converting to period-one orbit which means that the system can be stabilized to periodic motion by adjusting bifurcation parameters α, δ, f0 and Ω; and the onset of invariant torus or quasi-periodic behaviors, the entire invariant torus region or quasi-periodic region without periodic window, quasi-periodic behaviors or invariant torus behaviors suddenly disappearing or converting to periodic orbit; and the jumping behaviors which including from period- one orbit to anther period-one orbit, from quasi-periodic set to another quasi-periodic set; and the interleaving occurrence of chaotic behaviors and invariant torus behaviors or quasi-periodic behaviors; and the interior crisis; and the symmetry breaking of period-one orbit; and the different nice chaotic attractors. However, we haven＇t find the cascades of period-doubling bifurcations under the quasi-periodic perturbations and show the differences of dynamical behaviors and technics of research between the periodic perturbations and quasi-periodic perturbations.     

A HABITAT BASED MODEL FOR THE DISTRIBUTION OF FOREST INTERIOR NESTING BIRDS IN A FRAGMENTED LANDSCAPE

ABSTRACT. Increased awareness of the plight of many forest dwelling species has made necessary the development of methods for projecting the spatial distribution of these populations. This is particularly important for populations that currently occupy forest fragments and that are likely to be exposed to further disruption of their natural habitat. In this paper we develop a model for predicting the distribution of a bird population that evolved as forest interior dwellers. This model uses as its basis knowledge of the relationship between demographic characteristics of the population and the qualities of the habitat where individuals reside. We make the assumption that individuals will be naturally drawn to areas where they might expect greater reproductive success and repelled from areas where there is a high degree of intraspecific competition (high density). We apply the model to the ovenbird population in a large region of the Midwest. We use the model to examine the relative extent to which the surplus production from two major source areas supports extensive sink populations. The basic diffusion model parameterized by county forest cover data projects a population distribution which compares favorably with the results from the breeding bird count.     

Quorum Sensing in Populations of Spatially Extended Chaotic Oscillators Coupled Indirectly via a Heterogeneous Environment

Many biological and chemical systems could be modeled by a population of oscillators coupled indirectly via a dynamical environment. Essentially, the environment by which the individual element communicates with each other is heterogeneous. Nevertheless, most of previous works considered the homogeneous case only. Here we investigated the dynamical behaviors in a population of spatially distributed chaotic oscillators immersed in a heterogeneous environment. Various dynamical synchronization states (such as oscillation death, phase synchronization, and complete synchronized oscillation) as well as their transitions were explored. In particular, we uncovered a non-traditional quorum sensing transition: increasing the population density leaded to a transition from oscillation death to synchronized oscillation at first, but further increasing the density resulted in degeneration from complete synchronization to phase synchronization or even from phase synchronization to desynchronization. The underlying mechanism of this finding was attributed to the dual roles played by the population density. What’s more, by treating the environment as another component of the oscillator, the full system was then effectively equivalent to a locally coupled system. This fact allowed us to utilize the master stability functions approach to predict the occurrence of complete synchronization oscillation, which agreed with that from the direct numerical integration of the system. The potential candidates for the experimental realization of our model were also discussed.     

Dynamical properties of discrete Lotka–Volterra equations

A discrete version of the Lotka–Volterra differential equations for competing population species is analyzed in detail in much the same way as the discrete form of the logistic equation has been investigated as a source of bifurcation phenomena and chaotic dynamics. It is found that in addition to the logistic dynamics – ranging from very simple to manifestly chaotic regimes in terms of governing parameters – the discrete Lotka–Volterra equations exhibit their own brands of bifurcation and chaos that are essentially two-dimensional in nature. In particular, it is shown that the system exhibits “twisted horseshoe” dynamics associated with a strange invariant set for certain parameter ranges.     

Two critical times for the SIR model

We consider the SIR model and study the first time the number of infected individuals begins to decrease and the first time this population is below a given threshold. We interpret these times as functions of the initial susceptible and infected populations and characterize them as solutions of a certain partial differential equation. This allows us to obtain integral representations of these times and in turn to estimate them precisely for large populations.     

Bifurcations of periodic solutions and chaos in Duffing-van der Pol equation with one external forcing

The Duffing-Van der Pol equation withfifth nonlinear-restoring force and one external forcing term isinvestigated in detail: the existence and bifurcations of harmonicand second-order subharmonic, and third-order subharmonic,third-order superharmonic and $m$-order subharmonic under smallperturbations are obtained by using second-order averaging methodand subharmonic Melnikov function; the threshold values of existenceof chaotic motion are obtained by using Melnikov method. Thenumerical simulation results including the influences of periodicand quasi-periodic and all parameters exhibit more new complexdynamical behaviors. We show that the reverse period-doublingbifurcation to chaos, period-doubling bifurcation to chaos,quasi-periodic orbits route to chaos, onset of chaos, and chaossuddenly disappearing, and chaos suddenly converting to periodorbits, different chaotic regions with a great abundance of periodicwindows (periods:1,2,3,4,5,7,9,10,13,15,17,19,21,25,29,31,37,41, andso on), and more wide period-one window, and varied chaoticattractors including small size and maximum Lyapunov exponentapproximate to zero but positive, and the symmetry-breaking ofperiodic orbits. In particular, the system can leave chaotic regionto periodic motion by adjusting the parameters $p, \beta, \gamma, f$and $\omega$, which can be considered as a control strategy.     

A METAPHYSIOLOGICAL POPULATION MODEL OF STORAGE IN VARIABLE ENVIRONMENTS

ABSTRACT. We use mechanistic arguments to generalize a hierarchical metaphysiological approach developed by one of us to modeling biological populations (Getz, [1991, 1993]) and extend the approach to include a storage component in the population. We model the growth of single species and consumer-resource interactions, both with and without storage. Our approach unifies modeling storage across trophic levels and is much simpler and more efficient to implement numerically than individual based approaches or population approaches that include integral, delay, or partial differential equation components in the model. Using intake functions (i.e., functional responses) that include the effects of interference competition, we apply the model to a hypothetical herbivore feeding on a resource that fluctuates seasonally and demonstrate the importance of a flow from storage that buffers the population against periods when resources are scarce or absent. We also apply the model to a hypothetical plant population that is driven by fluctuating resources and demonstrate the importance of a translocation flow from storage at the end of a dormant season, corresponding to periods when resources are most scarce. Finally, we couple these two populations for the case where the herbivore feeds exclusively on non-storage biomass, and demonstrate how the population dynamics can be affected by the rates at which buffering and translocation flows transfer from storage to active tissue in the herbivore and plant populations. In particular, for certain buffering and translocation flow rates, 1-year unimodal, 2-year bimodal, and 2-year unimodal cycles can emerge in the same herbivore population.     

Structured populations with distributed recruitment: from PDE to delay formulation

In this work, first, we consider a physiologically structured population model with a distributed recruitment process. That is, our model allows newly recruited individuals to enter the population at all possible individual states, in principle. The model can be naturally formulated as a first‐order partial integro‐differential equation, and it has been studied extensively. In particular, it is well posed on the biologically relevant state space of Lebesgue integrable functions. We also formulate a delayed integral equation (renewal equation) for the distributed birth rate of the population. We aim to illustrate the connection between the partial integro‐differential and the delayed integral equation formulation of the model utilising a recent spectral theoretic result. In particular, we consider the equivalence of the steady state problems in the two different formulations, which then lead us to characterise irreducibility of the semigroup governing the linear partial integro‐differential equation. Furthermore, using the method of characteristics, we investigate the connection between the time‐dependent problems. In particular, we prove that any (non‐negative) solution of the delayed integral equation determines a (non‐negative) solution of the partial differential equation and vice versa. The results obtained for the particular distributed states at birth model then lead us to present some very general results, which establish the equivalence between a general class of partial differential and delay equation, modelling physiologically structured populations. Copyright © 2016 John Wiley & Sons, Ltd.     

Stability results for a size-structured population model with resources-dependence and inflow

It is undoubted that the survival of individuals of populations is dependent on resources (e.g., foods). We formulate a system of integro-differential equations to model the dynamics of a size-structured and resources-dependent population, a kind of inflow of newborn individuals from external environment is considered. The resource-dependence is incorporated through the size growth, mortality, fertility and feeding rates of the target population. The existence of the stationary size distributions are discussed, and the linear stability is investigated by means of the semigroup theory and the characteristic equation technique, some sufficient conditions for stability/instability of stationary states are obtained, and two examples and the corresponding simulations are presented.     

Bifurcations and Chaos in Duffing Equation

The Duffing equation with even-odd asymmetrical nonlinear-restoring force and one external forcingis investigated.The conditions of existence of primary resonance,second-order,third-order subharmonics,m-order subharmonics and chaos are given by using the second-averaging method,the Melnikov method andbifurcation theory.Numerical simulations including bifurcation diagram,bifurcation surfaces and phase portraitsshow the consistence with the theoretical analysis.The numerical results also exhibit new dynamical behaviorsincluding onset of chaos,chaos suddenly disappearing to periodic orbit,cascades of inverse period-doublingbifurcations,period-doubling bifurcation,symmetry period-doubling bifurcations of period-3 orbit,symmetry-breaking of periodic orbits,interleaving occurrence of chaotic behaviors and period-one orbit,a great abundanceof periodic windows in transient chaotic regions with interior crises and boundary crisis and varied chaoticattractors.Our results show that many dynamical behaviors are strictly departure from the behaviors of theDuffing equation with odd-nonlinear restoring force.     

Evolutionary dynamics of phenotype-structured populations: from individual-level mechanisms to population-level consequences

Epigenetic mechanisms are increasingly recognised as integral to the adaptation of species that face environmental changes. In particular, empirical work has provided important insights into the contribution of epigenetic mechanisms to the persistence of clonal species, from which a number of verbal explanations have emerged that are suited to logical testing by proof-of-concept mathematical models. Here, we present a stochastic agent-based model and a related deterministic integrodifferential equation model for the evolution of a phenotype-structured population composed of asexually-reproducing and competing organisms which are exposed to novel environmental conditions. This setting has relevance to the study of biological systems where colonising asexual populations must survive and rapidly adapt to hostile environments, like pathogenesis, invasion and tumour metastasis. We explore how evolution might proceed when epigenetic variation in gene expression can change the reproductive capacity of individuals within the population in the new environment. Simulations and analyses of our models clarify the conditions under which certain evolutionary paths are possible and illustrate that while epigenetic mechanisms may facilitate adaptation in asexual species faced with environmental change, they can also lead to a type of “epigenetic load” and contribute to extinction. Moreover, our results offer a formal basis for the claim that constant environments favour individuals with low rates of stochastic phenotypic variation. Finally, our model provides a “proof of concept” of the verbal hypothesis that phenotypic stability is a key driver in rescuing the adaptive potential of an asexual lineage and supports the notion that intense selection pressure can, to an extent, offset the deleterious effects of high phenotypic instability and biased epimutations, and steer an asexual population back from the brink of an evolutionary dead end.     

低维混沌时序非线性动力系统的预测方法及其应用研究

主要研究由低维混沌时序所确定的非线性动力系统的预测方法及其应用。在国外学者研究工作的基础上,应用一种非线性混沌模型在相空间内对时序进行重构工作,先通过改进的最小二乘方法来估计模型的参数,满足一定精度后,再采用最优化方法来估计模型的参数,并用所求得的混沌时序模型在其相空间内对时序的未来值进行预测。给出了非常有代表性的实例对文中模型和算法进行验证。结果发现采用该算法能较准确地求得模型的参数,在相空间中对混沌时序进行预测,将传统方法中的外推变成了相空间中的内插,及选取最佳的模型阶数等工作都能增加预测的准确程度,且混沌时序不可能进行长期的预测。     

Special dynamic behaviors of a temporal chaotic system

When dynamic behaviors of temporal chaotic system are analyzed,we find that a temporal chaotic system has not only genetic dynamic behaviors of chaotic reflection,but also has phenomena influencing two chaotic attractors by original values.Along with the system parameters changing to certain value,the system will appear a break in chaotic region,and jump to another orbit of attractors.When it is opposite that the system parameters change direction,the temporal chaotic system appears complicated chaotic behaviors.