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The problem of testing in education is relevant for many countries. Testing solves three problems. The first task is to assess the quality of current training. The second task is to conduct a comparative analysis of learning outcomes. The third task is the management of the educational process in a particular educational institution and in the education sector. This determines the relevance of testing and the relevance of developing new methods for assessing test results. The article proposes a new method for assessing test results for different situations: “teacher– student”, computer test, virtual testing model, test on a mixed reality model and others. To solve the problem, a special quasi-sigmoidal function is introduced. It is analogous to the logistic function, but takes into account the peculiarities of real testing of students. The logistic function ranges from minus infinity to plus infinity. There are no negative assessments in education. The introduced function lies only in the positive range of the argument. It describes actual positive scores when testing students. The authors called this function the complexity function. With its help, the complexity of the subject is assessed according to the test results. To substantiate the method, the function of the logarithms of the odds, logistic regression and the resulting Rush method are investigated. The article notes two shortcomings of the Rush method. The testing principle has been defined for the new function, which is used to estimate complexity. The article introduces two new concepts: the test difficulty function and the integral test score. Integral assessment of testing is a smooth function and makes it possible to go from a stepwise dependence to a continuous one. The cumulative test score translates the point test results into a continuous function and creates a correlation between the scores. The results of an experiment with the participation of RTU MIREA students are presented. The experimental results are analyzed. The possibility of using the method in educational processes is shown. The method is an alternative to the Rush method.

Food industry effluents represent one of the major concerns when it comes to environmental impact; hence, their valorization through different chemical and biological routes has been suggested as a possible solution. The vast amount of organic and inorganic nutrients present in food industry effluents makes them suitable substrates for microbial growth. This study suggests two valorization routes for whey as dairy industry effluent and flotation wastewater from the wine industry through microbial conversion to biocontrol agents as value-added products. Cultivations of the biocontrol strain Bacillus sp. BioSol021 were performed in a 16 L bioreactor to monitor the bioprocess course and investigate bioprocess kinetics in terms of microbial growth, sugar substrate consumption and surfactin synthesis, as an antimicrobial lipopeptide. The produced biocontrol agents showed high levels of biocontrol activity against mycotoxigenic strains of Aspergillus flavus, followed by a significant reduction of sugar load of the investigated effluents by the producing microorganisms. With proven high potential of whey and winery flotation wastewater to be used as substrates for microbial growth, this study provides grounds for further optimization of the suggested valorization routes, mostly in terms of bioprocess conditions to achieve maximal techno-economical feasibility, energy saving and maximal reduction of effluents’ organic and inorganic burden.

Modelers have choices in how they approach a problem, with different approaches potentially leading to different outcomes. Sometimes one approach gives a consistently lower (or higher) result than another. The theorem and corollaries in this study show that if the logistic equation or, equivalently, the SI model, are perturbed at time zero by a range of values with mean zero, the resulting trajectories must average to a value below (for logistic and I) or above (for S) the solution with average initial condition. The proof of the theorem shows that this phenomenon is the result of algebraic properties of the nonlinear quadratic term, although we note it can be extended to a larger class of systems. More importantly it shows that the only necessary criterion is that the perturbations average to zero. The source of them and the properties of their distribution does not matter to the result of the theorem but does affect the magnitude of the proven difference.

Here, we study the logistic delay differential equation with two different delays. First of all, we disscuse the local stability and Hopf bifurcation conditons. The method of steps is used to get a discretized analogue of the original system. Local stability and bifurcation analysis of the discretized system is investigated. Finally, we carry out some numerical simulations such as bifurcation diagram, Lyapunov exponent and phase portraits to verify the theoretical results and to illustrate complex dynamics of the considered system.

A kinetic model has been proposed for the spread of epidemics, describing the dynamics of the variation in the number of disease-free, infected, and recovered (SIR) cases, based on a lag logistic equation. It has been established that this model predicts the possibility of existence of a quasi-steady-state mode of an epidemic in which the number of infected cases is constant due to the balance of the daily increment of infections and recoveries. Conditions have been identified under which such a mode can be a source of the advance of the second epidemic wave. The COVID-19 pandemic data were used to show the possibility of reliable forecasts based on this model of the spread of an epidemic for a period of up to two months.

Modern software development is based on a systems approach, in which a program or software complex is considered as a system of interacting software components. Models of software components are analogs of complex system subsystems. Therefore, a complex program is considered as a system of software components. The organization of the structure of software components affects the quality and result of the program. The organization of interaction between software components affects the efficiency of the program. An important factor in the system of software components is the life cycle, which determines the effectiveness and feasibility of using this program. Software differs from many complex systems and information systems in that it has the ability to increase its life cycle. Moreover, the need to increase the life cycle is characterized by two factors: external and internal. The internal factor arises due to the obsolescence of the program. In this case, it does not meet the new conditions, for example, a new operating system. The external factor arises from external influences in the form of interference or purposeful actions, such as computer viruses. The problem of creating the structure of software components of computing systems and information systems that ensure the duration of the life cycle in the presence of external influences is topical. The study of this problem contributes to the improvement of the technological base of computing systems and information systems that solve applied problems. The article presents a new life cycle model based on two models of growth and degradation. The article recommends a resource-based approach for life cycle assessment. As an analytical solution, it is proposed to use a logistic equation, which describes the mechanisms of the life cycle formation process quite well. The article discusses three types of resource in calculations: physical, technological and communicative. A general redundancy solution is proposed to create a network with the inclusion of a multigraph model.

In ecology and population biology, logistic equation is widely applied for simulating the population of organisms. By combining the logistic model with the low-density effect called Allee effect, several variations of mathematical expressions have been proposed. The upper half of the work was dedicated to establish a novel equation for highly flexible density effect model with Allee threshold. Allee effect has been rarely observed in microorganisms with asexual reproduction despite of theoretical studies. According to the exploitation ecosystem hypotheses, plants are believed to be insensitive to Allee effect. Taken together, knowledge on the existence of low-density effect in photosynthetic microorganisms is required for redefining the ecological theories emphasizing the photosynthetic organisms as the basis for food chains. Therefore, in the lower half of the present article, we report on the possible Allee effect in photo-autotrophic organisms, namely, green paramecia, and cyanobacteria. Optically monitored growth of green paramecia was shown to be regulated by Allee-like weak low-density effect under photo-autotrophic and photo-heterotrophic conditions. Insensitiveness of wild type cyanobacteria (Synechocystis sp. Strain PCC6803) to low-density effect was confirmed, as consistent with our empirical knowledge. In contrast, a mutant line of PCC6803 impaired with a photosynthesis-related pxcA gene was shown to be sensitive to typical Allee’s low-density effect (i.e. this line of cells failed to propagate at low cellular density while cells start logarithmic growth at relatively higher inoculating density). This is the first observation that single-gene mutation in an autotrophic organism alters the sensitivity to Allee effect.

For secure transmission of digital images, existing cryptographic algorithms transform coherent visual information into a noise-like appearance prompting an adversary of the presence of a possible cipher. This paper proposes an algorithm that produces a visually coherent and meaningful cipher image. The proposed algorithm consists of a permutation-substitution subroutine to obtain a partial cipher. The Arnold-3D map does the permutation, and a delayed logistic map performs the substitution in this subroutine. The hiding of this partial cipher is done in the reference image using an integer wavelet transform. The pixels of the partial cipher are embedded in the four sub-bands of the decomposed reference image as 4 to 1-pixel encoding using Cantor-like pairing function. In addition to the lossless encryption scheme, the integer nature of all the sub-bands in the wavelet decomposition and the invertible pairing function facilitates the perfect reconstruction of the reference image. One of the significant novelty of this work lies in the subtle use of simple pairing functions, which prohibits the unnecessary increase in the size of the cipher, thereby reducing the storage and transmission costs., Highlights • A novel compression and embedding scheme for generating VMCI. • Subtle use of pairing functions and Integer Wavelet transform in the scheme. • Facilitates perfect decryption/reconstruction of both the original/reference image. • The VMCI transmission cost is reduced by four times as compared to earlier schemes. • Robust and secure against traditional cryptanalysis and steganalysis.

Uvažujeme nelokální okrajové úlohy pro frakcionální laplacián se sublineárními nelinearitami a také nelinearitami logistického typu. Pro tyto úlohy dokazujeme existenci kladného řešení pomocí metody dolních a horních řešení. V článku se též zabýváme numerickými experimenty pomocí metody konečných prvků. We consider a class of nonlinear fractional Laplacian problems satisfying the homogeneous Dirichlet condition on the exterior of a bounded domain. We prove the existence of positive weak solution for classes of sublinear nonlinearities including logistic type. A method of sub- and supersolution, without monotone iteration, is established to prove our existence results. We also provide numerical bifurcation diagrams and the profile of positive solutions, corresponding to the theoretical results using the finite element method in one dimension.

We study a two-point boundary-value problem describing steady states of a population dynamics model with diffusion, logistic growth, and nonlinear density-dependent dispersal on the boundary. In particular, we focus on a model in which the population exhibits hump-shaped density-dependent dispersal on the boundary, and explore its effects on existence, uniqueness and multiplicity of steady states.

Few data are available regarding comprehensive or quantitative assessment of fish feed considering both the environmental and feeding impacts. Aiming to fill the gap, an experimental study to investigate the effects of three fish feeds on concentrations of nutrients and crucian carp (Carassius carassius) growth was conducted in laboratory aquariums in the presence and absence of prometryn. Results showed that weight gain rates of crucian carp treated with Tong Wei (TW) feed were 106.3% and 2.0% higher than that of Zhong Shan (ZS) and Zhong Liang (ZL) feeds, a possible explanation was that the quality of protein in TW feed was highest as evidenced by the protein efficiency ratios. Meanwhile, TW feed posed relatively lighter effects on water qualities (between ZL and ZS). Prometryn significantly inhibited the growth of crucian carp and thus affected concentrations of nutrients in water indirectly. The relationships between weight gain rates of fish and concentrations of nutrients in water (R2 = 0.929–0.990) were developed. In sum, this study suggested that it is realizable to obtain better fish growth performance with lesser degrading effects on water qualities by producing and selecting appropriate feed regardless of prometryn existence, and the developed equations could be used as a basis for future studies.

In this work, we discuss some pitfalls when solving differential equations with neural networks. Due to the highly nonlinear cost functional, local minima might be approximated by which functions may be obtained, that do not solve the problem. The main reason for these failures is a sensitivity on initial guesses for the nonlinear iteration. We apply known algorithms and corresponding implementations, including code snippets, and present an example and counter example for the logistic differential equations. These findings are further substantiated with variations in collocation points and learning rates.

In the present paper, we model the cumulative number of persons, reported to be infected with COVID-19 virus, by a sum of several logistic functions (the so-called multilogistic function). We introduce logistic wavelets and describe their properties in terms of Eulerian numbers. Moreover, we implement the logistic wavelets into Matlab’s Wavelet Toolbox and then we use the continuous wavelet transform (CWT) to estimate the parameters of the approximating multilogistic function. Using the examples of several countries, we show that this method is effective as a method of fitting a curve to existing data. However, it also has a predictive value, and, in particular, allows for an early assessment of the size of the emerging new wave of the epidemic, thus it can be used as an early warning method.

This paper proposes phenomenological equations that describe various aspects of aftershock evolution: elementary master equation, logistic equation, stochastic equation, and nonlinear diffusion equation. The elementary master equation is a first-order differential equation with a quadratic term. It is completely equivalent to Omori’s law. The equation allows us to introduce the idea of proper time of earthquake source “cooling down” after the main shock. Using the elementary master equation, one can pose and solve an inverse problem, the purpose of which is to measure the deactivation coefficient of an earthquake source. It has been found for the first time that the deactivation coefficient decreases with increasing magnitude of the main shock. The logistic equation is used to construct a phase portrait of a dynamical system simulating the evolution of aftershocks. The stochastic equation can be used to model fluctuation phenomena, and the nonlinear diffusion equation provides a framework for understanding the spatiotemporal distribution of aftershocks. Earthquake triads, which are a natural trinity of foreshocks, main shock, and aftershocks, are considered. Examples of the classical triad, the mirror triad, the symmetrical triad, as well as the Grande Terremoto Solitario, which can be considered as an anomalous symmetrical triad, are given. Prospects for further development of the phenomenology of earthquakes are outlined.

The behavior of all the solutions of the logistic equation with delay and&nbsp, diffusion in a sufficiently small positive neighborhood of the equilibrium state is studied. It is assumed that the Andronov–Hopf bifurcation conditions are met for the coefficients of the problem. Small perturbations of all coefficients are considered, including the delay coefficient and the coefficients of theboundary conditions. The conditions are studied when these perturbations depend on the spatial variable and when they are time-periodic functions. Equations on the central manifold are constructed as the main results. Their nonlocal dynamics determines the behavior of all the solutions of the original boundary value problem in a sufficiently small neighborhood of the&nbsp, equilibrium state. The ability to control the dynamics of the original problem using the phase change in the perturbing force is set. The numerical and analytical results regarding the dynamics of the system with parametric perturbation are obtained. The asymptotic formulas for the solutions of the original boundary value problem are given.

Mención Internacional en el título de doctor This thesis develops the discrete sub- and supersolutions method and applies it to prove the convergence of the nonlinear finite element method applied to the generalized di usive logistic equation...... Este trabajo ha sido desarrollado en el Departamento de Matemáticas de la Universidad Carlos III de Madrid (UC3M) bajo la dirección de la profesora Marcela Molina Meyer. Se contó con una beca de la UC3M de ayuda al estudio de máster de dos años de duración y posteriormente con un contrato de investigador en formación de la UC3M durante tres años más. Adicionalmente se recibió una ayuda de la UC3M para realizar una estancia de investigación, financiada por el programa "Ayudas para la movilidad de investigadores de la Universidad Carlos III de Madrid en centros de investigación nacionales o extranjeros, Convocatoria 2015. Modalidad B: investigadores en formación predoctoral". La estancia de investigación se realizó en el Departamento de Matematicas de Heriot-Watt University (Edimburgo). Además, esta tesis ha sido respaldada parcialmente por el Ministerio de Economía y Competitividad de España con el proyecto MTM2015-65899-P y por el Ministerio de Ciencia, Innovación y Universidades de España con el proyecto PGC2018-097104-B-100. Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid Presidente: John Werner Peter Volker.- Secretario: Julia María Novo Martín.- Vocal: Sara Pollock

A new SPICE compatible memristor model is presented which is based on the solution to the well-known Logistic equation. Previously published first order function models are limited in their ability to elucidate the memristor's oscillatory and chaotic behavior. Fractional order models are not SPICE friendly although they are known to demonstrate chaos. The relationship between the logistic equation and its iterated map is used to show how the proposed model explains the memristor's oscillatory and chaotic characteristic. It emulates empirically observed ohmic and semiconducting response to temperature. The utility of the model is tested by implementing it in a novel reliability and power aware relaxation oscillator design. A methodical approach to estimating the oscillation frequency of such highly nonlinear circuitry is presented. Computed results are validated against the relaxation oscillator implemented in an LTSpice testbench.

The ultimate goal of scientific research is the attempt to discover and predict future flows of social phenomena. However, predicting the future is not at all easy. Many phenomena in nature and society are described by researchers with linear models, but these phenomena are in fact non-linear. Many are dynamic systems, in which processes are susceptible to changes in time, are non-linear and can behave improperly. These irregularities can so increase that the behaviour of the dynamic system becomes completely unpredictable and so spontaneously goes into chaos. The scientific discipline, which will be discussed in this paper, is called non-linear dynamics. In this context, it is necessary to point out in particular a specific group of techniques that deal with the study of linear and non-linear time series, since they are of utmost importance for forecasting the future developmental flows of a social system. Modern technological technology has a rich arsenal of various analytical procedures that serve this purpose. For each case of forecasting the behaviour of a system, it is important that the empirical values of the variable are predicted to be chronologically built over a period of time. The data obtained in this way constitute a specific time series or series, and a group of statistical techniques for analysing that time series is called a time series analysis that can include deterministic but also non-deterministic data. For the analysis of linear data in the statistical methodology, special techniques have been developed that allow analysis of the time components. The classic model provides a convincing explanation of the four basic components of the time series, as well as their interrelations. These are: 1. Trend, 2. Seasonal variations, 3. cyclic component, and 4. Residual component. However, in a further discussion, we will not deal with this.

In this paper, chains of coupled logistic equations with delay are considered, and the local dynamics of these chains is investigated. A basic assumption is that the number of elements in the chain is large enough. This implies that the study of the original systems can be reduced to the study of a distributed integro–differential boundary value problem that is continuous with respect to the spatial variable. Three types of couplings of greatest interest are considered: diffusion, unidirectional, and fully connected. It is shown that the critical cases in the stability of the equilibrium state have an infinite dimension: infinitely many roots of the characteristic equation tend to the imaginary axis as the small parameter tends to zero, which characterizes the inverse of the number of elements of the chain. In the study of local dynamics in cases close to critical, analogues of normal forms are constructed, namely quasinormal forms, which are boundary value problems of Ginzburg–Landau type or, as in the case of fully connected systems, special nonlinear integro–differential equations. It is shown that the nonlocal solutions of the obtained quasinormal forms determine the principal terms of the asymptotics of solutions to the original problem from a small neighborhood of the equilibrium state.

Film mulching in combination with high plant density (PD) is a common agronomic technique in rainfed maize (Zea mays L.) production. However, the effects of combining colored plastic film mulching and PD on dry matter accumulation (DMA) dynamics and yield of spring maize have not been thoroughly elucidated to date. Thus, a 2-year field experiment was conducted with three mulching treatments (no mulching (M0), transparent plastic film mulching (M1), and black plastic film mulching (M2)) and five plant densities (60,000 (D1), 67,500 (D2), 75,000 (D3), 82,500 (D4), and 90,000 plants ha−1 (D5)). A logistic equation was used to simulate the DMA process of spring maize by taking the effective accumulated air temperature compensated by effective accumulated soil temperature as the independent variable. The results showed that compared with M0 treatment, the growth period of M1 and M2 treatments was preceded by 10 and 4 days in 2016, and 10 and 7 days in 2017, respectively. The corrected logistic equation performed well in the characterization of maize DMA process with its characteristic parameter (final DMA, a; maximum growth rate of DMA, GRmax; effective accumulated temperature under maximum growth rate of DMA, xinf; effective accumulated temperature when maize stops growing, xmax; effective accumulated temperature when maize enters the fast-growing period, x1). Plastic film color mainly affected DMA by influencing xinf. PD mainly affected DMA by affecting GRmax and x1. During the first slow growing period, the DMA of M1 treatment was the largest among the three mulching treatments, however, during the fast growing period, the DMA of M2 treatment accelerated and exceeded that of M1 treatment, resulting in the largest final DMA(a) and yield. When the PD was increased from D1 to D4, the maximum growth rate (GRmax) continued to increase, and the effective accumulated temperature when maize enters the fast growing period (x1) continued to decrease, which substantially increased the final DMA(a) and yield. The application of M2D4 treatment can harmonize the relevant factors to improve the DMA and yield of spring maize in rainfed regions of Northeast China.

As the extent of hypoxia in coastal waters increases, the survivorship of jellyfish polyps relative to that of competing sessile organisms often increases, enabling them to reproduce more prolifically, leading to a medusa bloom in the following year. Quantifying the population of polyps can be used to predict when these blooms will occur. We used a time-delayed logistic equation to quantify the response to variable dissolved oxygen (DO) concentrations and temperatures in a population of moon jellyfish (Aurelia coerulea) polyps on substrates that carried competing sessile organisms. The availability of substrate depends on the DO threshold for each competitor, and substrates only become available to the polyps during hypoxic periods. We used the median sublethal concentration (SLC50) thresholds of hypoxia for different groups of benthic organisms to calculate the DO-dependent survivorship of A. coerulea polyps competing on the substrate. Since the median lethal time (LT50) for cnidarians is close to 240 h, we chose a 10-day delay in the time-delayed logistic equation. The carrying capacity is determined every 10 days depending on DO concentrations and temperature. The polyps reproduce by budding at a temperature-dependent rate after settling on the substrate during the hypoxic period, and thus, the annual polyp reproduction rate is determined by multiplying the temperature-dependent budding rate by the DO-dependent survivorship. The duration of hypoxia is a key factor determining the polyp population, which can increase more as the duration of hypoxia increases. Modeling simulations were compared to observed data. In this model, the DO and temperature distribution data make it possible to quantify variations in the population of the A. coerulea polyps, which can be used to predict the abundance and appearance of medusa the following year.

We develop several formal analogies between the logistic equation and the spatially homogeneous and isotropic relativistic cosmology described by the Einstein–Friedmann equations. These analogies produce an effective Lagrangian and Hamiltonian and new symmetries for the logistic equation., Symmetry, 13 (4), ISSN:2073-8994

This paper considers a multi-patch model, where each patch follows a logistic law, and patches are coupled by asymmetrical migration terms. First, in the case of perfect mixing, i.e when the migration rate tends to infinity, the total population follows a logistic equation with a carrying capacity which in general is different from the sum of the n carrying capacities, and depends on the migration terms. Second, we determine, in some particular cases, the conditions under which fragmentation and asymmetrical migration can lead to a total equilibrium population greater or smaller than the sum of the carrying capacities. Finally, for the three-patch model, we show numerically the existence of at least three critical values of the migration rate for which the total equilibrium population equals the sum of the carrying capacities., Comment: 29 pages, 7 figures, 1 table. Section 5 added in version 2

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Центром внимания данной работы является понятие «устойчивое развитие», которое анализируется авторами с позиций прошлого, настоящего и будущего времени. Для количественной оценки устойчивого развития любой динамической системы необходимы математические модели. Первой математической моделью (в далеком прошлом — 1798 году), где были использованы арифметическая и геометрическая прогрессии для оценки будущего состояния планеты, была модель Мальтуса. Она была критически встречена и в целом не поддержана научным мировым сообществом. В 70-е годы прошлого столетия (настоящее время) был создан Римский клуб, одной из задач которого было привлечение мировой общественности к существующим в мире экологическим, экономическим, демографическим и другим проблемам путем создания математических динамических моделей и анализа полученных с их помощью результатов. В настоящее время исследователями делается упор на разработку критериев «устойчивого развития», на анализ и оценку «устойчивого развития» с точки зрения моделирования и с учетом выдвигаемых критериев. Авторы выдвигают гипотезу, что в будущем в методологическом плане при исследовании проблемы «устойчивое развитие» особое внимание будет уделено синергетике — науке о самоорганизации динамических систем. Логистическое уравнение было использовано для количественной оценки вклада туристического сектора в социально-экономическое развитие северных районов России и, соответственно, в устойчивое развитие региона., The focus of this work is the concept of “sustainable development”, which is analyzed by the authors from the perspective of the past, present and future. Mathematical models are needed to quantify the sustainable development of any dynamic system. The first mathematical model (in the distant past — 1798), where arithmetic and geometric progressions were used to estimate the future state of the planet, was the Malthus model. It was critically received and, in general, not supported by world academic society. In the 1970s (present), the Club of Rome was created, one of the tasks of which was to attract the attention of the world community to the existing environmental, economic, demographic and other problems in the world by creating mathematical dynamic models and analyzing the results obtained with their help. Currently, the researchers focus on the development of the criteria for “sustainable development”, on the analysis and evaluation of “sustainable development” from the point of view of modelling and taking into account the criteria put forward. The authors hypothesize that in the future, in terms of methodology, when studying the issue of “sustainable development”, special attention will be paid to synergetics, i.e. the science of self-organization of dynamic systems. The Ferhulst equation was used to quantify the contribution of the tourism sector on the socio-economic development of the Northern regions of Russia and, accordingly, on the sustainable development of the region.

This work introduces a deterministic approach to the martensite transformation curve. Martensite is a nucleation-controlled transformation that has two characteristics: autocatalysis and auto-accommodation. Only a small number of martensite units initially form owing to primary nucleation. These new units may cause the transformation of other units by autocatalysis. We call this kind of transformation chained autocatalysis. Moreover, as the transformation progresses, the auto-accommodation influences the arrangement of new units. This work assumes that the transformation-saturation relates to the exhaustion of the chained autocatalysis, which underlines the microstructure. To compare, we considered the KJMA’s extended-transformation concept that implies assuming exhaustion by impingement. Data from isothermal martensite transformations and anisothermal martensite transformations are used to validate the model. Those data comprised different grain sizes and carbon contents. The model is based upon Verhulst’s logistic concept. We propose that the model's high fitting-capability stems from its deterministic aspect combined with martensite’s self-similarity. Additionally, we suggest that chained autocatalysis controls the rate of martensite transformation. Therefore, the relaxation of transformation strains by plasticity assisted by mutual accommodation determines the transformation's martensite volume in the absence of post-propagation coarsening/coalescence.

© 2020 by the authors., The industrial production of marine bacteria with probiotic properties is limited by the excessive cost of the culture media adequate for their growth. The present work aimed to study the suitability of 30 marine media formulated with nitrogen sources (fish peptones) from different fish discards and seawater, for the growth of two marine probiotic bacteria (MPB), namely Phaeobacter sp. and Pseudomonas fluorescens. These fish peptones were produced from several discarded fish and by-products (heads, skins, and whole individuals of megrim, mackerel, gurnard, hake, etc.). In all cultivations, no significant differences were found on cell viability when the microorganism grew on commercial or alternative media. In relation to the biomass production, the growth of Phaeobacter sp. on waste media was commonly similar or a 20% lower than observed in the control cultures. For P. fluorescens, various peptones (skin peptones of pouting and blue whiting) showed even higher productive ability than commercial peptones. An initial economical evaluation revealed that low-cost media reduced until 120 times the cost of production of MPB., This research was funded by the projects CVMar+i (0302_CVMAR_I_1_P, POCTEP 2015) and Xunta de Galicia (Grupos de Potencial Crecimiento, IN607B 2018/19). This work was also supported by national funds through FCT/MEC (PIDDAC), project reference IF/00588/2015.

The objective of this work was to compare the performance of the nonlinear Gompertz, logistic, and von Bertalanffy equations, to describe the growth curve of immunocastrated male pigs from birth until reaching 140 kg body weight. Standard error, number of iterations, mean square of the error, and coefficient of determination were compared between the models. The logistic and von Bertalanffy equations do not accurately estimate the initial and slaughter weights. The Gompertz equation shows the highest reliability and is, therefore, the most suitable one to describe the growth curve of immunocastrated pigs from birth until 140 kg body weight is reached. Resumo: O objetivo deste trabalho foi comparar o desempenho das equações não lineares de Gompertz, logística e a de von Bertalanffy, para descrever a curva de crescimento de suínos machos imunocastrados, desde o nascimento até atingirem 140 kg de massa corporal. O erro-padrão, o número de iterações, o quadrado médio do erro e o coeficiente de determinação foram comparados entre os modelos. A equação logística e a de von Bertalanffy não estimam com precisão as massas inicial e de abate. A equação de Gompertz apresenta maior confiabilidade e é, portanto, a mais adequada para descrever a curva de crescimento de suínos imunocastrados, desde o nascimento até a obtenção de140 kg de massa corporal.

A mathematical model based on nonlinear ordinary differential equations is proposed for quantitative description of the outbreak of the novel coronavirus pandemic. The model possesses remarkable properties, such as as full integrability. The comparison with the public data shows that exact solutions of the model (with the correctly specified parameters) lead to the results, which are in good agreement with the measured data in China and Austria. Prediction of the total number of the COVID-19 cases is discussed and examples are presented using the measured data in Austria, France, and Poland. Some generalizations of the model are suggested as well.

We apply the recently developed theory of symmetry of stochastic differential equations to stochastic versions of the logistic equation; these may have environmental or demographical noise, or both—in which case we speak of the complete model. We study all these cases, both with constant and with non-constant noise amplitude, and show that the only one in which there are nontrivial symmetries is that of the stochastic logistic equation with (constant amplitude) environmental noise. In this case, the general theory of symmetry of stochastic differential equations is used to obtain an explicit integration, i.e., an explicit formula for the process in terms of any single realization of the driving Wiener process.

We present results on the mortality statistics of the COVID-19 epidemic in a number of countries. Our data analysis suggests classifying countries in five groups, (1) Western countries, (2) East Block, (3) developed Southeast Asian countries, (4) Northern Hemisphere developing countries and (5) Southern Hemisphere countries. Comparing the number of deaths per million inhabitants, a pattern emerges in which the Western countries exhibit the largest mortality rate. Furthermore, comparing the running cumulative death tolls as the same level of outbreak progress in different countries reveals several subgroups within the Western countries and further emphasises the difference between the five groups. Analysing the relationship between deaths per million and life expectancy in different countries, taken as a proxy of the preponderance of elderly people in the population, a main reason behind the relatively more severe COVID-19 epidemic in the Western countries is found to be their larger population of elderly people, with exceptions such as Norway and Japan, for which other factors seem to dominate. Our comparison between countries at the same level of outbreak progress allows us to identify and quantify a measure of efficiency of the level of stringency of confinement measures. We find that increasing the stringency from 20 to 60 decreases the death count by about 50 lives per million in a time window of 20 days. Finally, we perform logistic equation analyses of deaths as a means of tracking the dynamics of outbreaks in the “first wave” and estimating the associated ultimate mortality, using four different models to identify model error and robustness of results. This quantitative analysis allows us to assess the outbreak progress in different countries, differentiating between those that are at a quite advanced stage and close to the end of the epidemic from those that are still in the middle of it. This raises many questions in terms of organisation, preparedness, governance structure and so on., Nonlinear Dynamics, 101 (3), ISSN:0924-090X, ISSN:1573-269X

Выполнена математическая формализация уравнений модели, для построения которой использовалась концепция системы ресурспотребитель. В рассматриваемом случае потребителем выступает биомасса растений, а ресурсомраспределенные в узкой прикорневой зоне растений основные элементы его минерального питания (азот, фосфор и калий). Динамические уравнения модели следуют основному положению химической кинетики, согласно которому результат взаимодействия динамических переменных в системах рассматриваемого профиля определяется их произведением. Выполнена оценка параметров и установлена адекватность модели выборочным распределениям. В качестве выборочных распределений используется массив экспериментальных данных роста яровой пшеницы (Красноуфимская-100) на торфяной низинной почве, предварительная почвенная обработка которой проведена с помощью азотных, фосфорных и калийных удобрений., Modern mathematical models for the simulation of dynamics in the fertilizer-soil-plant system, the components of which are agricultural plants, soil microorganisms and elements of their mineral nutrition, are considered. Based on the analysis of the adopted provisions, a model that takes into account the relationships and the specific nature of the joint changes in its components has been developed. The mathematical formalization of the model equations is carried out, for the construction of which the concept of the resourceconsumer system was used. In this case, the consumer is the biomass of plants, and the content of the main elements of its mineral nutrition distributed in the narrow basal zone of plants is a resource. The dynamic equations of the model follow the basic principle of chemical kinetics, according to which the result of the interaction between dynamic variables in the systems of the profile in question is determined by their product. The equations also contain the self-limitation factor, which sets the growth rate of the curve for the logistic equation and the specific rate of the model variables saturation effect. Thus, the specific rate of change in biomass is determined by its natural growth rate, the weighted sum of the nutrient contents in plants, and intraspecific competition. The rate of change in the content of these elements per unit of biomass is proportional to their current content in the rhizosphere and to the factor of the mutual influence of the elements on each other. The parameters are estimated and the adequacy of the model to sample distributions is established. An array of experimental data on the growth of spring wheat (Krasnoufimskaya-100) on peat lowland soil, the preliminary soil treatment of which was carried out using nitrogen, phosphorus, and potassium fertilizers, is used as sample distributions. The coefficients obtained as a result of parameter estimation and the calculated distributions of model dynamic variables with a sufficiently high degree of adequacy correspond to their experimental distributions and reflect the real situation of the system evolution., №25 (2020)

In this work, the semi-analytical solution is studied for the diffusive logistic equation with both mixed instantaneous and delayed density. The domain of reaction–diffusion in one dimension is shown. Delay partial differential equation is approximated with a delay ordinary differential equation system by using the Galerkin technique method. Steady-state solutions and stability analysis as well as bifurcation diagrams are derived. The effect of diffusion parameter and delay values is comprehensively studied; as a result, both parameters can destabilize or stabilize the model. We obtained that the decrease in values of the Hopf bifurcations for growth rate is associated with an increase in delay values, whereas the diffusion parameter is increased. Furthermore, comparisons between the numerical simulations and semi-analytical results present a good agreement for all examples and figures of the Hopf bifurcations. Examples of limit cycle and phase-plane map are plotted to confirm the benefits and accuracy of semi-analytical solutions result. For periodic solutions, an asymptotic method is studied after the Hopf bifurcation point for both one- and two-term semi-analytical systems.

The superior provenance is a prerequisite for ecological restoration, and a better mastery on the growth rhythms of Ormosia species is fundamental to reforest effectively. For the selection of better provenance and the formulation of artificial cultivation methods, the height and the ground diameter of Ormosia hosiei Hemsl. et Wils., O. xylocarpa Chun ex L. Chen and O. henryi Prain seedlings of different provenances were monitored in the first year. The results showed that their dynamic growths presented a slow-fast-slow trend that fit a “S” growth curve by the logistic mathematical model, and the growth of Ormosia species seedlings of different provenances significantly differed. The accumulated growth increment of O. xylocarpa was the largest (averagely 45.50 cm) and the accumulated growth increment of O. henryi was the least (averagely 20.33 cm). Thus, O. hosiei of Jiujiang provenance, O. xylocarpa of Liping provenance and O. henryi of Longquan provenance have a stronger adaptability for future artificial cultivation in Jiangxi China. RESUMO A origem melhor é um pré-requisito para a restauração ecológica, e um melhor domínio dos ritmos de crescimento das espécies Ormosia é fundamental para reflorescimento com eficácia. Para a seleção de melhor origem e a formulação de métodos artificiais de cultivo, a altura e o diâmetro do solo de mudas Ormosia hosiei Hemsl. et Wils., O. xylocarpa Chun ex L. Chen e O. henryi Prain de diferentes origens foram monitoradas no primeiro ano. Os resultados mostraram que seus crescimentos dinâmicos tiveram uma tendência lenta-rápida-lenta que se correspondia a uma curva de crescimento “S” no modelo matemático logístico, e o crescimento de espécies de mudas Ormosia de diferentes origens diferiram significativamente. O incremento de crescimento acumulado de O. xylocarpa foi o maior (em média de 45,50 cm) e o incremento de crescimento acumulado de O. henryi foi o menor (em média de 20,33 cm). Assim, O. hosiei da origem de Jiujiang, O. xylocarpa da origem de Liping e O. henryi da origem de Longquan têm uma adaptabilidade mais forte para o futuro cultivo artificial em Jiangxi, China.

Highlights • The generalized logistic equation is used to interpret the COVID-19 epidemic data. • The model describes properly the growth in the number of confirmed cases with time. • The actual data and logistic curves are shown. • The logistic model can describe the number of sick people per day., The generalized logistic equation is used to interpret the COVID-19 epidemic data in several countries: Austria, Switzerland, the Netherlands, Italy, Turkey and South Korea. The model coefficients are calculated: the growth rate and the expected number of infected people, as well as the exponent indexes in the generalized logistic equation. It is shown that the dependence of the number of the infected people on time is well described on average by the logistic curve (within the framework of a simple or generalized logistic equation) with a determination coefficient exceeding 0.8. At the same time, the dependence of the number of the infected people per day on time has a very uneven character and can be described very roughly by the logistic curve. To describe it, it is necessary to take into account the dependence of the model coefficients on time or on the total number of cases. Variations, for example, of the growth rate can reach 60%. The variability spectra of the coefficients have characteristic peaks at periods of several days, which corresponds to the observed serial intervals. The use of the stochastic logistic equation is proposed to estimate the number of probable peaks in the coronavirus incidence.

Highlights • Criticism on the “Mass Hypothesis” in models of epidemic spreading. • Review of the Generalized Richards Model of epidemic outbreaks. • Review of the form of segregation named “herd behaviour” in the Trophic Web Theory. • Adaptation of “herd behaviour” to epidemic outbreak description, as a motivation to the Generalized Richards Model. • Interpretation of the exponents p and α of the Generalized Richards Model with particular reference to the COVID-19 case., Some ideas are presented about a geometric motivation of the apparent capacity of generalized logistic equations to describe the outbreak of quite many epidemics, possibly including that of the COVID-19 infection. This interpretation pivots on the complex, possibly fractal, structure of the locus describing the “contagion event set”, and on what can be learnt from the models of trophic webs with “herd behaviour”. Under the hypothesis that the total number of cases, as a function of time, is fitted by a solution of the Generalized Richards Model, it is argued that the exponents appearing in that differential equation, usually determined empirically, represent the geometric signature of the non-space filling, network-like locus on which contagious contacts take place.

The paper reports on application of the Gompertz model to describe the growth dynamics of COVID-19 cases during the first wave of the pandemic in different countries. Modeling has been performed for 23 countries: Australia, Austria, Belgium, Brazil, Great Britain, Germany, Denmark, Ireland, Spain, Italy, Canada, China, the Netherlands, Norway, Serbia, Turkey, France, Czech Republic, Switzerland, South Korea, USA, Mexico, and Japan. The model parameters are determined by regression analysis based on official World Health Organization data available for these countries. The comparison of the predictions given by the Gompertz model and the simple logistic model (i.e., Verhulst model) is performed allowing to conclude on the higher accuracy of the Gompertz model.

In this paper, we consider the generalized logistic equation with nonlocal reaction term $$ -\Delta u=u \biggl(\lambda +b \int_{\Omega }u^{r}\,dx-f(u) \biggr)\quad \text{in } \Omega,\qquad u>0 \quad \text{ in } \Omega,\qquad u=0 \quad \text{ on } \partial \Omega. $$ Using the bifurcation and sub-supersolution method, we obtain the non-existence, existence, and uniqueness of positive solutions for different parameters on the nonlocal terms. Our works about the nonlocal elliptic problem improve the results in the previous literature.

The years 2013 and 2014 have been marked by the most prolonged mass protest in the history of Bulgaria. In the present article the dynamics of the protest is investigated with the instruments of socio-physics, applied by an original own combination of known mathematical models, in order to describe its outbreak, dying, re-escalation and fluctuations in the number of protesters.

The work objective is the development of the mathematical model for analyzing the air pollution process. State reports published in the “Ecological Vestnik of Don” journal are taken as basic data. The target of research is the data on the automotive pollutant emission into the atmospheric air registered in Rostov-on-Don and Rostov Region as of from 2001 to 2010. The mathematical simulation of the air contamination process is performed through two approaches: 1) description by mathematical models of Gauze type; 2) improvement and forecasting of the air contamination process by Prony’s method; MathCad, MatLab program packages are used. The mathematical simulation results allow to analyze the contamination process and to identify its regularities.

In this paper the logistic growth model of spontaneous utilization of school library in Imo State, is presented. First‐order variables separable logistic equation is solved. The parameters that gave the best logistic curve for the data were determined. The number of male and female science students who utilized school library spontaneously also is presented. The logistic equation allows rigorous estimation of 2.34% growth rate of male science students’ spontaneous utilization of school library in Imo State. While that of female students is decreasing by 2.71%. The results revealed that male students are experiencing upward trend in the spontaneous utilization of school library while their female counterparts are experiencing downward trend. Therefore, it is recommended that science students should utilize the school library resources spontaneously without being induced to do so. They should also develop passion for reading because reading books pushes readers to use their imaginations and encourages creativity for scientific endeavours.

We consider the equation -\Delta u=au-b(x)u^{2}-ch(x)\quad\text{in }\Omega,\qquad u=0\quad\text{on }% \partial\Omega, where Ω is a smooth bounded domain in {\mathbb{R}^{N}} , {b(x)} and {h(x)} are nonnegative functions, and there exists {\Omega_{0}\subset\subset\Omega} such that {\{x:b(x)=0\}=\overline{\Omega}_{0}} . We investigate the existence of positive solutions of this equation for c large under the strong growth rate assumption {a\geq\lambda_{1}(\Omega_{0})} , where {\lambda_{1}(\Omega_{0})} is the first eigenvalue of the {-\Delta} in {\Omega_{0}} with Dirichlet boundary condition. We show that if {h\equiv 0} in {\Omega\setminus\overline{\Omega}_{0}} , then our equation has a unique positive solution for all c large, provided that a is in a right neighborhood of {\lambda_{1}(\Omega_{0})} . For this purpose, we prove and utilize some new results on the positive solution set of this equation in the weak growth rate case.

A controlled experiment was run to quantify emergence of Kochia indica under different temperature, salinity and seeding depth levels at Yazd National Salinity Research Center in 2013. Treatments were five day/night temperature regimes: 20/10, 25/15, 30/20, 35/25 and 40/30 °C, five salinity levels: 2, 6, 10, 14 and 18 dS m-1, and seeding depth on the surface (0 mm), 5, 10 and 15 mm. Final emergence percentage, emergence rate index and elapsed time (days) to reach 50% of the maximum emergence were measured. The results showed that the highest and lowest final emergence percentages were observed at 25/15°C and 40/30°C day/night, respectively. Final emergence percentages at salinity levels of 6, 10, 14 and 18 dS m-1were, respectively, 9, 22, 36 and 57% lower than 2 dS m-1. Final emergence percentages for 5, 10 and 15 mm seeding depths were, respectively, 30, 44 and 72% lower, as compared with the placement of seeds on the soil surface. Regression analysis showed that final emergence percentage linearly decreased with increase in salinity and seeding depth levels. However, elapsed time (days) to reach 50% of the maximum emergence (T50), increased as salinity and seeding depth increased, so that the highest T50 was obtained for 18 dS m-1 and seeding depth of 15 mm. Increase in salinity and seeding depth was associated with a significant decrease in emergence rate index. In addition, using a logistic regression equation, emergence rate of K. indica was quantified on each day after sowing for each temperature-salinity combination to predict the distribution range of the plant in these situations. &nbsp

Relevance of the publication is defined by the increasing value of the foreign trade operations in the economy of the Republic of Belarus. The problem of alignment of trade balance still remains a subject of intensive scientific search and is considered as the priority direction for development of national economy as for many years the tendency of excess of total payments of import over receipts of financial means from export of goods is observed. For the solution of the specified problem it is offered to introduce in economic practice and to adapt to domestic realities the definitions «fluctuation», «coherence», «divergence» and on the basis of their analysis to offer model of change of system of a foreign trade turnover which can be considered as one of mechanisms of an exit to zero balance.

[EN] We study time series generated by the parametric family of fractional discrete maps introduced by Wu and Baleanu, presenting an alternative way of introducing these maps. For the values of the parameters that yield chaotic time series, we have studied the Shannon entropy of the degree distribution of the natural and horizontal visibility graphs associated to these series. In these cases, the degree distribution can be fitted with a power law. We have also compared the Shannon entropy and the exponent of the power law fitting for the different values of the fractionary exponent and the scaling factor of the model. Our results illustrate a connection between the fractionary exponent and the scaling factor of the maps, with the respect to the onset of the chaos., J.A. Conejero is supported Ministerio de Economia y Competitividad Grant Project MTM2016-75963-P. Carlos Lizama is supported by CONICYT, under Fondecyt Grant number 1180041. Cristobal Rodero-Gomez is funded by European Commission H2020 research and Innovation programme under the Marie Sklodowska-Curie grant agreement No. 764738.

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) This work is dedicated to the study of dynamical properties of some partial differential equations (PDE, for short) of parabolic type, local or nonlocal ones. We prove existence, uniqueness and establish the asymptotic behavior, for large times, for two classes of PDE. Namely, a (nonlocal) parabolic equation with a Kirchhoff term and flux boundary condition and two parabolic PDE driven by the p-Laplacian with logistic terms involving potentials and weights which may be indefi nite and unbounded. First we prove existence and continuous dependence on data for the PDE with Kirchhoff term using Faedo-Galerkin method and a suitable change of variables. Some sufficient conditions are given to ensure uniqueness of solution. Concerning asymptotic behavior, we show that the omega limit set of each (semi) orbit contains, at least, one stationary solution. We then study stability of local minima of the associated energy functional showing first a result on asymptotic stability of a global minimum for the energy. We also prove a sufficient condition for the existence of isolated local minimum of the energy functional, which is proved to be an asymptotically stable stationary solution in a suitable neighborhood. Finally we determine the dynamics of positive solutions with bounded initial datum for two classes of parabolic PDE driven by the p-Laplacian with indefi nite and unbounded potentials and logistic sources having weights which are also indefi nite and unbounded. The boundary conditions appearing are also of flux type (linear or nonlinear). The asymptotic stability properties of the stationary solutions are described using principal eigenvalues of some elliptic eigenvalue problems involving a parameter of the original PDE. Those eigenvalue problems are also studied herein in order to be used as a tool for obtaining our results due to the low regularity assumptions on their coeficients from the viewpoint of the literature. Este trabalho é dedicado ao estudo da dinâmica de algumas equações diferenciais parciais (EDP, abreviadamente) parabólicas, locais ou não locais. Mais precisamente, provamos existência, unicidade e comportamento assintótico de solução, quando o tempo é grande, para uma EDP parabólica com termo de Kirchhoff e condição de fronteira tipo fluxo e duas classes de EDP parabólicas governadas pelo p-Laplaciano envolvendo termos logísticos e potenciais que podem ser indefi nidos e ilimitados. Inicialmente, provamos existência de solução e dependência contínua com respeito aos dados da EDP parabólica com termo de Kirchhoff utilizando o método de Faedo-Galerkin e uma mudança de variáveis adequada. Algumas condições sobre o termo não local são fornecidas de modo que unicidade de solução é assegurada. No que concerne comportamento assintótico de solução, mostramos que o conjunto Omega limite de cada (semi) órbita contém, pelo menos, uma solução estacionária. Estudamos estabilidade de mínimos do funcional energia associado e mostramos, sob certas condições, um resultado de estabilidade assintótica sobre mínimo global da energia. Também fornecemos uma condição su ficiente para existência de mínimo local isolado do funcional energia, o qual é inferido ser solução estacionária assintoticamente estável ao construirmos uma vizinhança em que toda solução que nela começa converge à aquele mínimo local isolado. Finalmente, estabelecemos a dinâmica das soluções positivas com dados iniciais limitados de duas classes de EDP parabólicas governadas pelo p-Laplaciano com potenciais indefi nidos e ilimitados e com termos logísticos possuindo pesos que também são inde finidos e ilimitados, além de condições de fronteira tipo fluxo (linear ou não linear). As propriedades de estabilidade assintótica das soluções estacionárias são descritas através da relação entre o parâmetro que aparece nas EDP e os autovalores principais de certos problemas de autovalores elípticos com potenciais e pesos indefi nidos e ilimitados. Tratamos esses problemas de autovalores em detalhes, devido a pouca regularidade exigida em seus coe cientes do ponto de vista da literatura, para que sejam utilizados no estudo das EDP de evolução. CNPq: 141488/2015-8

To overcome the limitations of the normal logistic equation, we aimed to improve the logistic model under hydrodynamic conditions for the examination of the responses of cyanobacterium, coupled turbulence mixing, and growth of cyanobacterium in population dynamics models. Selecting Microcystis aeruginosa and experimenting with the ideal conditions in a laboratory beaker, the chlorophyll-a concentration reached the corresponding maximum under each turbulent condition compared with the control. According to the experiment results, the theory of mass transfer, turbulence mixing, and the logistic equation are organically combined. The improved logistic growth model of Microcystis aeruginosa and competition growth model in the symbiont Scenedesmus quadricauda under turbulent conditions were established. Using the MATLAB multi-parameter surface fitting device, both models produced good fitting effects, with R &gt, 0.95, proving that the results fit the models, and demonstrating the relationship of the unity of nutrient transfer and algae growth affected by turbulence mixing. With continuous increases in turbulent mixing, the fitted curve became smoother and steadier. Algae stimulated by turbulence accelerate reproduction and fission to achieve population dominance. The improved logistic model quantitatively explains the Microcystis aeruginosa response to turbulence and provides a basis to represent ecological and biogeochemical processes in enclosed eutrophic water bodies.

Chickenpox is a common acute and highly contagious disease in childhood; moreover, there is currently no targeted treatment. Carrying out an early warning on chickenpox plays an important role in taking targeted measures in advance as well as preventing the outbreak of the disease. In recent years, the infectious disease dynamic model has been widely used in the research of various infectious diseases. The logistic differential equation model can well demonstrate the epidemic characteristics of epidemic outbreaks, gives the point at which the early epidemic rate changes from slow to fast. Therefore, our study aims to use the logistic differential equation model to explore the epidemic characteristics and early-warning time of varicella. Meanwhile, the data of varicella cases were collected from first week of 2008 to 52nd week of 2017 in Changsha. Finally, our study found that the logistic model can be well fitted with varicella data, besides the model illustrated that there are two peaks of varicella at each year in Changsha City. One is the peak in summer–autumn corresponding to the 8th–38th week; the other is in winter–spring corresponding to the time from the 38th to the seventh week next year. The ‘epidemic acceleration week’ average value of summer–autumn and winter–spring are about the 16th week (ranging from the 15th to 17th week) and 45th week (ranging from the 44th to 47th week), respectively. What is more, taking warning measures during the acceleration week, the preventive effect will be delayed; thus, we recommend intervene during recommended warning weeks which are the 15th and 44th weeks instead.

Trabajo Fin de Grado. Grado en matemáticas. Curso académico 2018-2019, [ES]Los sistemas dinámicos discretos han sido estudiados desde su introducción en 1892 por Henri Poincaré, en relación con el famoso problema de los tres cuerpos y continúan siendo investigados en nuestros días. A menudo se estudian mediante simulaciones por ordenador en todas las ramas de la ciencia. El famoso "efecto mariposa" pertenece al estudio del comportamiento caótico en sistemas dinámicos. Este trabajo presenta una introducción teórica a los sistemas dinámicos discretos, incluidas las propiedades caóticas, y finaliza con uno de los ejemplos más conocidos, la ecuación logística discreta, que modela el crecimiento de una población como una función f(x)=r x(1-x) respecto el parámetro de crecimiento r., [EN]Discrete dynamical systems have been studied since their introduction in 1892 by Henri Poincaré, relating to the famous three body problem, and continue to be the subject of current research. They are often studied via computer simulations in all branches of science. The well-known "butterfly effect" belongs to the study of chaotic behavior in dynamical systems. This work presents a theoretical introduction to discrete dynamical systems, including chaotic properties, and ends with one of the best known examples, the discrete logistic equation, modeling the growth of a population as a function f(x) = r x (1-x) of the growth parameter r.

A great deal of educational literature has focused on problem-based learning (PBL) in mathematics at the primary and secondary level, but arguably there is an even greater need for PBL in college math courses. We present a project centered around the Humans versus Zombies moderated tag game played on the Utah State University campus. We discuss the project in the context of an undergraduate differential equations course and discuss how the project was launched. We highlight examples of students mathematical models along with their verbal and written responses, as well as discussing assessment and student learning. Results are discussed in the context of higher and lower cognition levels as well as mathematical appreciation.