Logistic Differential Equation Research Articles

Growth and Motility of Gut Commensal Escherichia Coli in Health and Disease

Despite the growth of E. coli is comparatively well investigated, there is still a lack of knowledge on growth of gut commensal E. coli in health and disease.Our previous investigations have shown that there is an obvious deviation from the norm in the gut microbiota of Armenian Crohn's disease (CD) patients. The proportions of members of the CFB lineages were similar in the healthy and diseased individuals, while the increased numbers of Enterobacteriaceae members in fecal samples were found. It has been shown that growth of commensal E. coli isolates from the inflamed gut differed from those of E. coli in healthy individuals.In this study we inspected the effect of metronidazole (the antibiotic used in CD therapy) on growth and motility of commensal E. coli from the healthy and CD voluntaries.27 patients with CD and 35 healthy individuals were enrolled in this study. The logistic differential equation of Verhulst has been used to characterize the growth of commensal E. coli isolates.Results have demonstrated the increased numbers of E. coli isolates in feces of subjects with active CD. In vitro investigations of growth and motility of gut commensal E. coli isolates have shown that addition of metronidazole to the culture medium does not affect the growth parameters of isolates of healthy people. While statistically significant differences in growth parameters were obtained for patients' isolates in the culture medium with metronidazole, in contrast to the medium without it.The presented data show that the chronic inflammation and drugs used during the inflammation effect on physiological characteristics of gut commensal bacteria.

On a discretization process of fractional-order Logistic differential equation

Abstract In this work we introduce a discretization process to discretize fractional-order differential equations. First of all, we consider the fractional-order Logistic differential equation then, we consider the corresponding fractional-order Logistic differential equation with piecewise constant arguments and we apply the proposed discretization on it. The stability of the fixed points of the resultant dynamical system and the Lyapunov exponent are investigated. Finally, we study some dynamic behavior of the resultant systems such as bifurcation and chaos.

Exploring dynamical systems and chaos using the logistic map model of population change

The logistic map difference equation is encountered in the theoretical ecology literature as a mathematical model of population change for organisms with non-overlapping generations and density-dependent dynamics influenced solely by intraspecific interactions. This article presents the logistic map as a simple model suitable for introducing students to the properties of dynamical systems including periodic orbits, bifurcations, and deterministic chaos. After a brief historical and mathematical introduction to models of population change and the logistic map, the article summarizes the logistic map activities I teach in my introductory physics laboratories for non-physics majors. The logistic map laboratory introduces the many bioscience students in my courses to a foundational model in population ecology that has inspired ecologists to recognize the importance of nonlinear dynamics in real populations. Although I use this activity in courses for non-majors, the logistic map model of population change could also be taught to physics majors to introduce properties of dynamical systems while demonstrating an application of mathematical modeling outside of traditional physics.

Evolution Model Analysis for Comprehensive Transportation Structure

According to the issue of comprehensive transportation structure the authors establish the logistic differential equation model to describe the competition process of railway transport and highway transport. By the stability analysis of the model the evolution trends between railway transport and highway transport are obtained. Using MATLAB software the model simulates the evolution trends under the different parameters conditions.

Evolution Model Analysis for Comprehensive Transportation Structure

According to the issue of comprehensive transportation structure the authors establish the logistic differential equation model to describe the competition process of railway transport and highway transport. By the stability analysis of the model the evolution trends between railway transport and highway transport are obtained. Using MATLAB software the model simulates the evolution trends under the different parameters conditions.

The role of multiple modeling perspectives in students' learning of exponential growth

The exponential is among the most important family functions in mathematics; the foundation for the solution of linear differential equations, linear difference equations, and stochastic processes. However there is little research and superficial agreement on how the concepts of exponential growth are learned and/or should be taught initially. In order to investigate these issues, I preformed a teaching experiment with two high school students, which focused on building understandings of exponential growth leading up to the (nonlinear) logistic differential equation model. In this paper, I highlight some of the ways of thinking used by participants in this teaching experiment. From these results I discuss how mathematicians using exponential growth routinely make use of multiple--sometimes contradictory--ways of thinking, as well as the danger that these multiple ways of thinking are not being made distinct to students.

Growth peculiarities of commensal Escherichia coli isolates from the gut microflora in Crohn’s disease patients

Verhulst's logistic differential equation, popular in mathematical ecology, is used in modeling of population growth, neural networks, statistics, reaction models, Fermi distribution, modeling of tumor growth, etc. We used this function to characterize growth of commensal Escherichia coli isolates from gut microflora in Crohn's disease patients. The results of our investigations show differences in growth parameters of commensal E. coli, isolated from the gut microflora in Crohn's disease patients and healthy volunteers; it is most likely explained by the influence of chronic inflammatory processes on growth and reproduction of these bacteria. It has been established that the used mathematical model well characterizes growth of patients' gut E. coli isolates, and it can be important for the expedient probiotics' application during the disease.

On the Logistic Equation in the Complex Plane

The famous logistic differential equation is studied in the complex plane. The method used is based on a functional analytic technique which provides a unique solution of the ordinary differential equation (ODE) under consideration in H 2(𝔻) or H 1(𝔻) and gives rise to an equivalent difference equation for which a unique solution is established in ℓ2 or ℓ1. For the derivation of the solution of the logistic differential equation this discrete equivalent equation is used. The obtained solution is analytic in {z ∈ ℂ: |z| <T}, T > 0. Numerical experiments were also performed using the classical 4th order Runge–Kutta method. The obtained results were compared for real solutions as well as for solutions of the form y(t) = u(t) + iv(t), t ∈ ℝ. For t ∈ ℂ the solution derived by the present method, seems to have singularities, that is, points where it ceases to be analytic, in certain sectors of the complex plane. These sectors, depending on the values of the involved parameters, can move at different directions, join forming common sectors, or pass through each other and continue moving independently. Moreover, the real and imaginary part of the solution seem to exhibit oscillatory behavior near these sectors.

Dynamics of a Single Species in a Fluctuating Environment under Periodic Yield Harvesting

We discuss the effect of a periodic yield harvesting on a single species population whose dynamics in a fluctuating environment is described by the logistic differential equation with periodic coefficients. This problem was studied by Brauer and Sánchez (2003) who attempted the proof of the existence of two positive periodic solutions; the flaw in their argument is corrected. We obtain estimates for positive attracting and repelling periodic solutions and describe behavior of other solutions. Extinction and blow-up times are evaluated for solutions with small and large initial data; dependence of the number of periodic solutions on the parameterσassociated with the intensity of harvesting is explored. Asσgrows, the number of periodic solutions drops from two to zero. We provide bounds for the bifurcation parameter whose value in practice can be efficiently approximated numerically.

On Approximate Solutions for Fractional Logistic Differential Equation

A new approximate formula of the fractional derivatives is derived. The proposed formula is based on the generalized Laguerre polynomials. Global approximations to functions defined on a semi-infinite interval are constructed. The fractional derivatives are presented in terms of Caputo sense. Special attention is given to study the convergence analysis and estimate an error upper bound of the presented formula. The new spectral Laguerre collocation method is presented for solving fractional Logistic differential equation (FLDE). The properties of Laguerre polynomials approximation are used to reduce FLDE to solve a system of algebraic equations which is solved using a suitable numerical method. Numerical results are provided to confirm the theoretical results and the efficiency of the proposed method.

Hopf Bifurcation and Stability Analysis for A Delayed Logistic Equation

This paper deals with the stability analysis and theHopf bifurcation at the equilibrium points of a logistic delay differential equation.By applying the Halanay inequality, the local stability of the logistic differential equation is discussed. The stability of bifurcation periodic solutions and the direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Numerical examples show interesting nonlinear behavior of the logistic differential equation at the end of the paper.

FTT:Power : A global model of the power sector with induced technological change and natural resource depletion

This work introduces a model of Future Technology Transformations for the power sector (FTT:Power), a representation of global power systems based on market competition, induced technological change (ITC) and natural resource use and depletion. It is the first component of a family of sectoral bottom-up models of technology, designed for integration into the global macroeconometric model E3MG. ITC occurs as a result of technological learning produced by cumulative investment and leads to highly nonlinear, irreversible and path dependent technological transitions. The model uses a dynamic coupled set of logistic differential equations. As opposed to traditional bottom-up energy models based on systems optimisation, such differential equations offer an appropriate treatment of the times and structure of change involved in sectoral technology transformations, as well as a much reduced computational load. Resource use and depletion are represented by local cost–supply curves, which give rise to different regional energy landscapes. The model is explored for a single global region using two simple scenarios, a baseline and a mitigation case where the price of carbon is gradually increased. While a constant price of carbon leads to a stagnant system, mitigation produces successive technology transitions leading towards the gradual decarbonisation of the global power sector.

Alternative approach to modeling bacterial lag time, using logistic regression as a function of time, temperature, pH, and sodium chloride concentration.

The objective of this study was to develop a probabilistic model to predict the end of lag time (λ) during the growth of Bacillus cereus vegetative cells as a function of temperature, pH, and salt concentration using logistic regression. The developed λ model was subsequently combined with a logistic differential equation to simulate bacterial numbers over time. To develop a novel model for λ, we determined whether bacterial growth had begun, i.e., whether λ had ended, at each time point during the growth kinetics. The growth of B. cereus was evaluated by optical density (OD) measurements in culture media for various pHs (5.5 ∼ 7.0) and salt concentrations (0.5 ∼ 2.0%) at static temperatures (10 ∼ 20°C). The probability of the end of λ was modeled using dichotomous judgments obtained at each OD measurement point concerning whether a significant increase had been observed. The probability of the end of λ was described as a function of time, temperature, pH, and salt concentration and showed a high goodness of fit. The λ model was validated with independent data sets of B. cereus growth in culture media and foods, indicating acceptable performance. Furthermore, the λ model, in combination with a logistic differential equation, enabled a simulation of the population of B. cereus in various foods over time at static and/or fluctuating temperatures with high accuracy. Thus, this newly developed modeling procedure enables the description of λ using observable environmental parameters without any conceptual assumptions and the simulation of bacterial numbers over time with the use of a logistic differential equation.

A new algorithm for solving differential equations

Different analytic methods have been proposed to solve differential equations, so far. In this paper, a novel analytic method that efficiently solves ODEs is presented. This method requires only the calculation of the first Adomian polynomial, namely A0, and does not need to solve the functional equation in each iteration, as well as provides less computational work than other existing methods. Some important ordinary differential equations including the Lane–Emden equation of index m, the logistic nonlinear differential equation, and the Riccati equation are considered to illustrate the efficiency of the proposed algorithm. Copyright © 2012 John Wiley & Sons, Ltd.

Which Pure Chaos Model Will Describe IDX Composite (Jakarta Composite Index (JCI) of Indonesia Stock Exchange (IDX)) Better?

Is it possible to approach a time series by modeling it through a predicted model of a fully deterministic and pure chaotic model? IDX Composite (Jakarta Composite Index) time series will be examined whether it can be explained by one of renowned fully deterministic and pure chaotic models or their variants.Daily return data of Jakarta Composite Index (JCI) of Indonesia Stock Exchange from January 05, 1988 to June 30, 2009 are observed. “Compass rose” pattern of scatter diagram will be observed. Chosen renowned fully deterministic and pure chaotic models are Logistic map equation, Henon map and Mackey Glass delay differential equation which have low dimensionality. The initial value return of JCI is applied to three chaotic models to create the chaotic series that have the same initial conditions with JCI return. The length of data points of chaotic series is set to be the same length of JCI series.In addition to the scatter diagram comparisons other three tests are applied to all chosen chaos models and JCI series. BDS Statistic, R/S Analysis and Lyapunov Exponent of Wolf et al (1985) algorithm are the used test tools to investigating some characteristics of chaotic system: nonlinear, persistence (fractal) and sensitive dependence on initial condition. Performance results of chaos characteristic tests on chosen models will be explored to approach predicted chaotic model of JCI. The objective of this research result is to find - if any - the best approached and fitted chaos model to describe JCI better as an approximation model. Understanding of chaotic attributes - nonlinear, fractal and sensitive dependence on initial condition - will provide better analysis of non periodicity of crisis on Indonesia Stock Exchange as well as of phenomenon of crisis, order and disorder on Indonesia Stock Exchange.

Sourcing for parameter estimation and study of logistic differential equation

We present two simulation activities for students to generate real data and several data sources for the purpose of estimating parameters in the logistic differential equation model.

Modelling Students’ Affinity for Lecture Attendance Using Differential Equations: The Case of Students of Kwame Nkrumah University of Science and Technology

Research Context: The aim of the study was to investigate students’ affinity for lecture attendance using differential equations and to explore reasons why students absent themselves from lectures. Research Methods: The study employed a quantitative methodology which was adopted in order to allow the researchers to gather more precise and quantifiable information on students’ perception about lecture attendance. Questionnaires were administered continuously to a cross-section of students numbering five hundred to ascertain their affinity for lecture attendance and reasons why they attend or do not attend lectures. Conclusion: A logistic differential equation had been used to model students’ affinity for lecture attendance. The calculated or predicted values of students who regularly attended lectures were obtained by using logistic differential equation. The predicted values were then compared with the observed values and analyzed. From Tables 1,2,3 and the graphs Case Study British Journal of Education, Society & Behavioural Science, 2(4): 389-401, 2012 390 in Figs. 1 and 2 which were obtained by the usage of MATLAB, it was seen that the observed values and the predicted values almost converged to the same values, signifying that the model is good for the prediction of students’ preference for lecture attendance. It was interesting to note that the reasons that drive the majority of students to lectures are to find out what they are supposed to learn, not to miss important information and to find out about assessment task. The study also revealed that some students use on-line notes as a substitute for lecture attendance.

Numerical Studies for Fractional‐Order Logistic Differential Equation with Two Different Delays

A numerical method for solving the fractional‐order logistic differential equation with two different delays (FOLE) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce FOLE to a system of algebraic equations. Special attention is given to study the convergence and the error estimate of the presented method. Numerical illustrations are presented to demonstrate utility of the proposed method. Chaotic behavior is observed and the smallest fractional order for the chaotic behavior is obtained. Also, FOLE is studied using variational iteration method (VIM) and the fractional complex transform is introduced to convert fractional Logistic equation to its differential partner, so that its variational iteration algorithm can be simply constructed. Numerical experiment is presented to illustrate the validity and the great potential of both proposed techniques.

Maps serving the combined coupling for use in environmental models and their behaviour in the presence of dynamical noise

Many physical, biological as well as the environmental problems, can be described by the dynamics of driven coupled oscillators. In order to study their behaviour as a function of coupling strength and nonlinearity, we considered dynamics of two maps serving the combined coupling (diffusive and linear) in the above fields. Firstly, we have considered a logistic difference equation on extended domain that is a part of the maps, that is discussed using its bifurcation diagram, Lyapunov exponent, sample as well as the permutation entropy. Secondly we have performed the dynamical analysis of the coupled maps using Lyapunov exponent and cross sample entropy in dependence on two coupling parameters. Further, we investigated how dynamical noise can affects the structure of their bifurcation diagrams. It was done (i) by the noise entering in two specific ways, that disturbs either the logistic parameter on extended domain or (ii) by an additive “shock” to the state variables. Finally, we demonstrated the effect of forcing by parametric noise, introduced in all maps’ parameter, on Lyapunov exponent of coupled maps.

Extinction in nonautonomous competitive Lotka–Volterra systems with infinite delay

The principle of competitive exclusion is extended to n -species nonautonomous Lotka–Volterra competition systems of differential equations with infinite delay. It is shown that if the coefficients are bounded, continuous and satisfy certain inequalities, then any solution with initial function in an appropriate space will have n − 1 of its components tend to zero, while the remaining one will stabilize at a certain solution of a logistic differential equation.