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This paper explores stochastic differential equation models to characterize the growth dynamics of a stock modeled by generalized logistic growth under a general harvesting function. The latter function adds diminishing marginal productivity to effort increases, while the former incorporates several well‐known growth functions as particular instances. In order to obtain optimal policies, a Crank–Nicolson discretization scheme is derived for the general models. We employ the optimal control theory to determine the optimal harvesting policy for a specific harvested stock and compare the expected net present value of the harvester's total profit earned across harvesting policies. The main conclusion is that models with a higher growth rate result in greater profit values. [ABSTRACT FROM AUTHOR]

Highly important is a three-dimensional nonlinear partial differential equation because for many physical systems, one can, subject to suitable idealizations, formulate a differential equation that describes how the system changes in time. Thus, this article comprehensively reveals the investigation carried out on a (3+1)-dimensional generalized fifth-order Zakharov–Kuznetsov equation with power-law as well as dual power-law nonlinearities analytically, where the fifth-order term involved is regarded as a dispersion perturbation term. We utilize the well-celebrated Noether's theorem to comprehensively construct conserved currents of the underlying equation. A detailed Lie group analysis of the understudied model consisting of power-law nonlinearities is further performed. This involves performing reductions of the underlying models using their Lie point symmetries. In consequence, various invariants are found. In addition, the equation reduces to diverse ordinary differential equations using its point symmetries and consequently diverse solutions of interest were achieved. Moreover, we derive some solitary wave solutions by invoking the newly introduced logistic function technique for some particular cases of the equation under consideration. In consequence, we achieve some exponential function solutions. In addition, the physical meaning of the results is put on the front burner by revealing the wave dynamics of these solutions via graphical depictions. Finally, the significance of the robust and detailed findings in the work are further corroborated with various real-world applications. [ABSTRACT FROM AUTHOR]

With the evolution of the Internet of Things (IoT) several users take part in different applications via sensors. The foremost confront here remains in selecting the most confidential users or sensors in the edge computing system of the IoT. Here, both the end-users and the edge servers are likely to be malicious or compromised sensors. Several works have been contributed to identifying and isolating the malicious end-users or edge servers. Our work concentrates on the security aspects of edge servers of IoT. The Frank-Wolfe Optimal Service Requests (FWOSR) algorithm is utilized to evaluate the boundaries or limits of the logistic regression model, in which the convex problem under a linear approximation is solved for weight sparsity (i.e. several user requests competing for closest edge server) to avoid over-fitting in the supervised machine learning process. We design a Frank Wolfe Supervised Machine Learning (FWSL) technique to choose an optimal edge server and further minimize the computational and communication costs between the user requests and the edge server. Next, Dirichlet Gaussian Blocked Gibbs Vicinity-based Authentication model for location-based services in Cloud networks is proposed. Here, the vicinity-based authentication is implemented based on Received Signal Strength Indicators (RSSI), MAC address and packet arrival time. With this, the authentication accuracy is improved by introducing the Gaussian function in the vicinity test and provides flexible vicinity range control by taking into account multiple locations. Simulation and experiment are also conducted to validate the computational cost, communication cost, time complexity and detection error rate. [ABSTRACT FROM AUTHOR]

This paper focuses on an attraction-repulsion chemotaxis system coupled with a logistic source,$ \begin{equation*} \left\{ \begin{split} &u_t = \Delta u-\chi\nabla\cdot(u\nabla v)+\xi\nabla\cdot(u\nabla w)+f(u),&\quad&x\in\Omega, t>0,\\ &0 = \Delta v-\beta v+\alpha u,&\quad&x\in\Omega, t>0,\\ &0 = \Delta w-\delta w+\gamma u,&\quad&x\in\Omega, t>0,\\ &\frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = 0,&\quad&x\in\partial\Omega, t>0,\\ &u(x,0) = u_0(x),&\quad&x\in\Omega, \end{split} \right. \end{equation*} $in a smoothly bounded domain $ \Omega \subset \mathbb{R}^N $ ($ N\geq1 $), where $ f(u) \leq a - bu^r $ with $ a > 0 $, $ b > 0 $ and $ r \geq 1 $. The constants $ \chi $, $ \xi $, $ \alpha $, $ \beta $, $ \delta $ and $ \gamma $ are all positive. We establish the existence of a unique, globally bounded classical solution for this system under specific conditions: when $ N \leq 3 $ and $ r \geq 1 $, or when $ N \geq 4 $ and $ r = \frac{2N-2}{N} $, assuming that $ \xi\gamma = \chi\alpha $. By imposing additional assumptions on the coefficients, we also derive the asymptotic behavior of the solution. Notably, our proof technique relies heavily on the use of $ u \ln u $, a term typically employed in low-dimensional settings. We extend its applicability to arbitrary dimensions, thereby improving upon previous results reported in [17,33] which require $ r>\frac{2N-2}{N} $ for $ N\geq 1 $ and $ r = \frac{3}{2} $ specifically for $ N = 4 $. [ABSTRACT FROM AUTHOR]

The logistic map is a nonlinear difference equation well studied in the literature, used to model self-limiting growth in certain populations. It is known that, under certain regularity conditions, the stochastic logistic map, where the parameter is varied according to a specified distribution, has a unique invariant distribution. In these cases we can compare the long-term behaviour of the stochastic system with that of the deterministic system evaluated at the average parameter value. Here we examine the relationship between the mean of the stochastic logistic equation and the mean of orbits of the deterministic logistic equation at the expected value of the parameter. We formally prove that, in some cases, the addition of noise is beneficial to the populations, in the sense that it increases the mean, while for other ranges of parameters it is detrimental. A conjecture based on numerical evidence is presented at the end. [ABSTRACT FROM AUTHOR]

Logistic equations play a pivotal role in the study of any nonlinear evolution process exhibiting growth and saturation. The interest for the phenomenology they rule goes well beyond physical processes and covers many aspects of ecology, population growth, economy... According to such a broad range of applications, there are different forms of functions and distributions which are recognized as generalized logistics. Sometimes they are obtained by fitting procedures. Therefore, criteria might be needed to infer the associated nonlinear differential equations, useful to guess "hidden" evolution mechanisms. In this article we analyze different forms of logistic functions and use simple means to reconstruct the differential equation they satisfy. Our study includes also differential equations containing nonstandard forms of derivative operators, like those of the Laguerre type. [ABSTRACT FROM AUTHOR]

Several extensions of the basic scalar variable logistic density to the multivariate and matrix-variate cases, in the real and complex domains, are given where the extended forms end up in extended zeta functions. Several cases of multivariate and matrix-variate Bayesian procedures, in the real and complex domains, are also given. It is pointed out that there are a range of applications of Gaussian and Wishart-based matrix-variate distributions in the complex domain in multi-look data from radar and sonar. It is hoped that the distributions derived in this paper will be highly useful in such applications in physics, engineering, statistics and communication problems, because, in the real scalar case, a logistic model is seen to be more appropriate compared to a Gaussian model in many industrial applications. Hence, logistic-based multivariate and matrix-variate distributions, especially in the complex domain, are expected to perform better where Gaussian and Wishart-based distributions are currently used. [ABSTRACT FROM AUTHOR]

We consider a nonlinear Dirichlet problem driven by the (p (z) , q) -Laplacian and with a logistic reaction of the equidiffusive type. Under a nonlinearity condition on a quotient map, we show existence and uniqueness of positive solutions and the result is global in parameter λ. If the monotonicity condition on the quotient map is not true, we can no longer guarantee uniqueness, but we can show the existence of a minimal solution u λ * and establish the monotonicity of the map λ ⟼ u λ * and its asymptotic behaviour as the parameter λ decreases to the critical value λ ^ 1 (q) > 0 (the principal eigenvalue of (− Δ q , W 0 1 , q (Ω)) ). [ABSTRACT FROM AUTHOR]

In this paper, we focus on solving nonlinear fractional logistic differential equation within the Caputo–Fabrizio fractional derivative operator. First, we present a new numerical scheme called new decomposition transform method (NDTM) for solving general nonlinear fractional differential equations. The proposed scheme is obtained by combining the Jafari transform method and a new decomposition method. Furthermore, we establish the uniqueness theorem of the solution using Banach's fixed point theorem. Finally, after the convergence analysis of the NDTM, we provide two numerical examples of nonlinear fractional logistic differential equations involving the Caputo–Fabrizio fractional derivative operator to validate the results. The obtained results have shown that the new method for analytical solutions is simple to implement and very effective for analyzing the complex fractional problems that arise in related fields of science and engineering. [ABSTRACT FROM AUTHOR]

Modeling individual growth in marine species for aquaculture encounters many difficulties when the species pauses its growth but resumes its later after the disrupting phenomenon (environmental or physiological) has been overcome. Seasonal or oscillatory growth has been addressed by modifying existing models, such as von Bertalanffy and Gompertz, to include an oscillatory component in this study. The novelty of this study lies in the fractal approach used to analyze growth using multiple logistic functions. Three commercially farmed marine species were studied, including shellfish, crustacea, and finfish. The oscillatory version of the von Bertalanffy model as well as double and triple logistic functions were used for analysis. The best model was selected using the information theory, Specifically the Akaike criterion (AIC) and the Bayesian criterion (BIC). Normal and log-normal distributions of error were assumed. The triple logistic model with log-normal distribution in the error structure was found to be the best model to describe the growth pattern of the three commercially farmed species as it obtained the lowest AIC. Overall, this study concludes that the fractal approach is the most effective way to describe growth in farmed species, including shellfish, crustacean, and finfish. [ABSTRACT FROM AUTHOR]

We investigate the transient dynamics of the Fisher equation under nonlinear diffusion and fractional operators. Firstly, we investigate the effects of the nonlinear diffusivity parameter in the integer-order Fisher equation, by considering a Gaussian distribution as the initial condition. Measuring the spread of the Gaussian distribution by u (0 , t) − 2 , our results show that the solution reaches a steady state governed by the parameters present in the logistic function in Fisher's equation. The initial transient is an anomalous diffusion process, but a power law cannot describe the whole transient. In this sense, the main novelty of this work is to show that a q-exponential function gives a better description of the transient dynamics. In addition to this result, we extend the Fisher equation via non-integer operators. As a fractional definition, we employ the Caputo fractional derivative and use a discretized system for the numerical approach according to finite difference schemes. We consider the numerical solutions in three scenarios: fractional differential operators acting in time, space, and in both variables. Our results show that the time to reach the steady solution strongly depends on the fractional order of the differential operator, with more influence by the time operator. Our main finding shows that a generalized q-exponential, present in the Tsallis formalism, describes the transient dynamics. The adjustment parameters of the q-exponential depend on the fractional order, connecting the generalized thermostatistics with the anomalous relaxation promoted by the fractional operators in time and space. [ABSTRACT FROM AUTHOR]

To prevent sensitive images shared on social media from being stolen, researchers are seeking to find and innovate image encryption algorithms. Since attackers attempt to exploit it to access encrypted images, the problem of generating keys as the first stage of any encryption algorithm is regarded as a critical problem. This issue was dealt with in this paper through the use of the logistic map. This map must be calculated using specific conditions and special parameters to maintain its chaos. The Diffie-Hellman key exchange algorithm was modified to address this problem since the shared keys now consist of a 16-element vector. The first two elements will be used for a logistic map. The entire vector was used to create a self-invertible matrix. The encryption level entails two operations: the first is matrix multiplication with a vector, and the second is a bitwise XOR operation between two matrices. A proposed encryption algorithm is capable of producing encrypted images that are challenging to decrypt through the use of stringent security tests. The outcomes of each test show how well the proposed encryption algorithm performs compared with other algorithms and how to resist the attacks. [ABSTRACT FROM AUTHOR]

We systematically investigate the performance of numerical methods to solve Fisher's equation, which contains a linear diffusion term and a nonlinear logistic term. The usual explicit finite difference algorithms are only conditionally stable for this equation, and they can yield concentrations below zero or above one, even if they are stable. Here, we collect the stable and explicit algorithms, most of which we invented recently. All of them are unconditionally dynamically consistent for Fisher's equation; thus, the concentration remains in the unit interval for arbitrary parameters. We perform tests in the cases of 1D and 2D systems to explore how the errors depend on the coefficient of the nonlinear term, the stiffness ratio, and the anisotropy of the system. We also measure running times and recommend which algorithms should be used in specific circumstances. [ABSTRACT FROM AUTHOR]

The soft-max function, a well-known extension of the logistic function, has been extensively utilized in numerous stochastic classification methodologies, such as linear differential analysis, soft-max extrapolation, naive Bayes detectors, and neural networks. The focus of this study is the development of soft-max based fuzzy aggregation operators (AOs) for Pythagorean fuzzy sets (PyFS), capitalizing on the benefits provided by the soft-max function. In addition to introducing these novel AOs, we also present a comprehensive approach to multi-attribute decision-making (MADM) that employs the proposed operators. To demonstrate the efficacy and applicability of our MADM method, we applied it to a real-world problem involving Pythagorean fuzzy data. The analysis of supplier selection has been extensively examined in many academic works as a crucial component of supply chain management (SCM), recognised as a significant MADM challenge. The process of choosing healthcare suppliers is a pivotal element that has the potential to greatly influence the efficacy and calibre of healthcare provisions. In addition, we given a numerical example to rigorously evaluate the accuracy and dependability of the proposed procedures. This examination demonstrates the effectiveness and potential of our proposed soft-max based AOs and their applicability in Pythagorean fuzzy environments. [ABSTRACT FROM AUTHOR]

In this article, some mathematical properties of (i, x0)-generalized logistic-type function are presented. This four-parameter generalized function can be considered as a statistical phenomenon enhancing more vigorous survival analysis models. Moreover, the behaviors of the relevant parametric functions obtained are examined with graphics using computer programming language Python 3.9. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic profiles of positive solutions for diffusive logistic equations. The aim is to study the sharp effect of linear growth and nonlinear function. Both the classical reaction-diffusion equation and nonlocal dispersal equation are investigated. Our main results reveal that the linear and nonlinear parts of reaction term play quite different roles in the study of positive solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic stability and the hair-trigger effect for Cauchy problem of the following parabolic-parabolic Keller-Segel system with logistic term$ \begin{equation} \left\{ \begin{aligned} &u_{t} = \Delta u-\chi \nabla\cdot\left (u\nabla v \right)+u\left(a-b u\right)&x\in \mathbb{R}^{N},\, t>0,\\ &\tau v_{t} = \Delta v+\lambda u-\mu v&x\in \mathbb{R}^{N},\, t>0, \end{aligned} \right. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)\end{equation} $where $ \chi $, $ a $, $ b $, $ \lambda $, $ \mu $ and $ \tau $ are positive constants and $ N $ is a positive integer. To this end, for small $ \chi $, we firstly obtain the global boundedness of solution by loop-argument based on $ L^p-L^q $ estimates of heat semigroup, with which we can further obtain the asymptotic stability of the positive constant equilibria in $ L^\infty(\mathbb R^N) $ for any initial data with positive lower bound. Moreover, for the special case $ \tau = 1 $, if $ \int_{B(x,\delta)}\ln u_0(s)ds\in L^\infty(\mathbb R^N) $ for some $ \delta>0 $, by constructing localized Lyapunov type functional, the solutions are shown to converge to the positive constant equilibria uniformly on any compact subset of $ \mathbb R^N $, which is known as the hair-trigger effect. Our contribution lies in the generalization of the results on asymptotic stability from the special case $ \tau = 1 $([31]) to any $ \tau>0 $, and the generalization of classical results on hair-trigger effect for Fisher-KPP equation to Keller-Segel system. [ABSTRACT FROM AUTHOR]

A closed chain of oscillators can be considered a model for ring-shaped ecosystems, such as atolls or the coastal zones of inland reservoirs. We use the logistic map, which is often referred to as an archetypical example of how complex dynamics can arise from very simple nonlinear equations, as a model for a separate oscillator in the chain. We present an original algorithm that allows us to find solutions to the spatiotemporal logistic equation quite efficiently or to state with certainty that there are no such solutions. Based on the Shannon formula, we propose formulas for estimating the spatial and temporal entropy, which allow us to classify our solutions as regular or irregular. We show that regular solutions can occur within the Malthus parameter region that corresponds to the irregular dynamics of a solitary logistic map. [ABSTRACT FROM AUTHOR]

This work proposes a new mathematical model describing the dynamics of growing bacterial cultures. The model, described by a first order non-linear differential equation, as a generalization of the logistic equation, was compared with the most studied mathematical models. All models were numerically implemented and fitted to the experimental data, collected from the incubation of a bacterial strain of Pseudomonas fluorescens, to obtain the growth parameters. The experimental data showed the lowest fit error for both the Baranyi–Roberts and new models, which turned out to be equivalent. Simulations of the fitting algorithm were also implemented and repeated for a large number of initial guesses of the parameters, chosen in order to test the fitting and convergence performances. The innovative feature that makes the new model easier to use than Baranyi–Roberts model is definitely its simple and manageable analytical form and its good performance in terms of convergence time. [ABSTRACT FROM AUTHOR]

In this work, we will give a novel method to construct a continuous approximation of the Exponential, Logistic, and Gaussian functions that allow us to do a handmade drawing of their graphs for which there is no accuracy of drawing at elementary levels (even at advanced ones!). This method arises from solving the elementary ordinary differential equation x'(t) = ax(t) combined with a suitable piecewise constant argument. The proposed approximation will allow us to generate several numerical schemes in an elementary way, generalizing the classical ones as, Euler’s schemes. No sophisticated mathematical tools are needed. [ABSTRACT FROM AUTHOR]

Differential equation compartmental models are crucial tools for forecasting and analyzing disease trajectories. Among these models, those dealing with only susceptible and infectious individuals are particularly useful as they offer closed-form expressions for solutions, namely the logistic equation. However, the logistic equation has limited ability to describe disease trajectories since its solutions must converge monotonically to either the diseasefree or endemic equilibrium, depending on the parameters. Unfortunately, many diseases exhibit periodic cycles, and thus, do not converge to equilibria. To address this limitation, we developed a generalized susceptible-infectious-susceptible compartmental model capable of accurately incorporating the duration of infection distribution and describing both periodic and non-periodic disease trajectories. We characterized how our model's parameters influence its behavior and applied the model to predict gonorrhea incidence in the US, using Akaike Information Criteria to inform on its merit relative to the traditional SIS model. The significance of our work lies in providing a novel susceptible-infectedsusceptible model whose solutions can have closed-form expressions that may be periodic or non-periodic depending on the parameterization. Our work thus provides disease modelers with a straightforward way to investigate the potential periodic behavior of many diseases and thereby may aid ongoing efforts to prevent recurrent outbreaks. [ABSTRACT FROM AUTHOR]

In this paper, we address a distributed control problem for a system of partial differential equations describing the evolution of a tumor that takes the biological mechanism of chemotaxis into account. The system describing the evolution is obtained as a nontrivial combination of a Cahn–Hilliard type system accounting for the segregation between tumor cells and healthy cells, with a Keller–Segel type equation accounting for the evolution of a nutrient species and modeling the chemotaxis phenomenon. First, we develop a robust mathematical background that allows us to analyze an associated optimal control problem. This analysis forced us to select a source term of logistic type in the nutrient equation and to restrict the analysis to the case of two space dimensions. Then, the existence of an optimal control and first-order necessary conditions for optimality are established. [ABSTRACT FROM AUTHOR]

In this work, we study a multi-patch model, where the patches are coupled by asymmetrical migration terms, and each patch follows a logistic law under the assumption that some growth rates are much larger than the other. First, for Two-patch model where one growth rate is much larger than the second one, the total equilibrium population is greater or smaller than the sum of two carrying capacities for all migration rates. Second, we consider Three-patch model in the two cases: (i) where two growth rates are much larger than the third one, and (ii) where one growth rate is much larger than the other two. For both cases, we give a complete classification of all possible situations under which the fragmentation can lead to a total equilibrium population greater or smaller than the sum of the three carrying capacities. Finally, in the general case, we consider the model of n patches with the assumption that: (i) all growth rates but one are much larger than the n th growth rate, (ii) two blocks where the growth rates of the first block are much larger than that of the second one. For the first case, we give a complete classification of all possible situations under which the fragmentation can lead to a total equilibrium population greater or smaller than the sum of the n carrying capacities, and in the second case, we construct a reduced model and we prove its global stability. [ABSTRACT FROM AUTHOR]

This paper aims to highlight the differences between two dynamical processes which produce similar bifurcation diagrams. We establish some differences and state some open questions. In addition, we present a tool to analyze the complexity of fractional difference equations by using topological and permutation entropies. [ABSTRACT FROM AUTHOR]

The fact that very small changes cause unpredictable big changes in nature has led to the chaos theory. Many researchers focusing on chaos have improved this theory in several directions. One of the prominent areas in chaos theory is controlled chaos. Due to the usefulness of controlled chaos, it has been used in many working areas as the iteration methods of fixed point theory. In this study, we have developed a chaos controlling mechanism using the Ishikawa iteration method. Firstly, Ishikawa fixed point type controlling mechanism is defined and the theorems determining the control parameters are proved. Afterward, the results of Ishikawa fixed point type controlling mechanism were obtained by taking the logistic map as an example. In addition, by using the MATLAB software program, the convergence, periodical behavior and chaotic regions occurring outside the range of the parameters forming the controlling mechanism have been investigated and numerical simulations studied. Subsequently, the effectiveness of the stability intervals obtained for the fixed point and periodic fixed points of the Ishikawa fixed point controlling mechanism by applying the Lyapunov exponent is demonstrated. Finally, we have given an application as control chaos of Ishikawa traffic model. [ABSTRACT FROM AUTHOR]

Power series, as an important means to analyze functions in different complex settings, are employed in various applied areas to solve differential equations and nonlinear problems and provide the assessment of intervals of convergence. Accordingly, the fuzzy logistic differential equation using the Caputo operator has been studied in this paper. Accordingly, the fuzzy logistic differential equation using the Caputo operator has been studied in this paper. The generalized Hukuhara difference and the generalized Hukuhara derivative are also used, and a power series representation is proposed for the solution of the fuzzy fractional logistic equation. Afterward, power series coefficients are obtained using a recursive formula. Finally, numerical experiments and illustrated results of the computations are presented to allow for more realistic decisions reflecting high complexity and underlying uncertainty. Thus, the numerical computations in our study reveal the effectiveness and accuracy of the power series method. Therefore, it is found that the fuzzy solution converges to the deterministic solution when uncertainty decreases, and, based on the technical analyses, it has been demonstrated that the results obtained are more fundamental in preventing geometric growth in nonlinear phenomena where uncertainties emerge due to impreciseness and inexactness. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Keller-Segel chemotaxis model with self- and cross-diffusion terms and a logistic source. This system consists of a fully nonlinear reaction-diffusion equation with additional cross-diffusion. We establish some high-order finite difference schemes for solving one- and two-dimensional problems. The truncation error remainder correction method and fourth-order Padé compact schemes are employed to approximate the spatial and temporal derivatives, respectively. It is shown that the numerical schemes yield second-order accuracy in time and fourth-order accuracy in space. Some numerical experiments are demonstrated to verify the accuracy and reliability of the proposed schemes. Furthermore, the blow-up phenomenon and bacterial pattern formation are numerically simulated. [ABSTRACT FROM AUTHOR]

In survey sampling, we aspire to obtain sound and consistent responses, which are not achieved while dealing with sensitive issues. Frequently, respondents give elusive responses to sensitive questions, so we employ randomized response techniques that facilitate finding an appropriate proportion of socially sensitive characteristics. In the present study, we proposed a hidden logit estimation method using Huang, Warner, and Mangat's randomized response techniques. This study depicts that the estimates become closer to the standard logits as the values of p increase. We found that the hidden logit estimates obtained by the Huang randomized response technique were nearer to the parametric values, in contrast to the other existing techniques, and we demonstrate an increase in accuracy as well. The simulation-based AIC and SIC values are used to assess model performance. We found that the Huang model is the best model for the proposed hidden logit method. This paper contributes towards the application of logistic models in the case of sensitive or socially stigmatized issues. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a new and efficient computational method based on the hybrid of block‐pulse functions and shifted Legendre polynomials together with their exact operational vector of integration and stochastic operational matrix of integration with respect to the multifractional Brownian motion to approximate solutions of a class of nonlinear stochastic differential equations driven by multifractional Gaussian noise. The presented method transforms problems under consideration into systems of nonlinear equations which can be simply solved by the Newton method. Moreover, the convergence analysis of the presented method is investigated. Finally, the efficiency of the new method is illustrated by solving the stochastic logistic equation and three test problems. [ABSTRACT FROM AUTHOR]

Here we research the univariate quantitative approximation, ordinary and fractional, of Banach space valued continuous functions on a compact interval or all the real line by quasi-interpolation Banach space valued neural network operators. These approximations are derived by establishing Jackson type inequalities involving the modulus of continuity of the engaged function or its Banach space valued high order derivative of fractional derivatives. Our operators are defined by using a density function generated by a q-deformed and λ-parametrized A-generalized logistic function, which is a sigmoid function. The approximations are pointwise and of the uniform norm. The related Banach space valued feed-forward neural networks are with one hidden layer. [ABSTRACT FROM AUTHOR]

We examine a multi-patch model of a population connected by nonlinear asymmetrical migration, where the population grows logistically on each patch. Utilizing the theory of cooperative differential systems, we prove the global stability of the model. In cases of perfect mixing, where migration rates approach infinity, the total population follows a logistic law with a carrying capacity that is distinct from the sum of carrying capacities and is influenced by migration terms. Furthermore, we establish conditions under which fragmentation and nonlinear asymmetrical migration can lead to a total equilibrium population that is either greater or smaller than the sum of carrying capacities. Finally, for the two-patch model, we classify the model parameter space to determine if nonlinear dispersal is beneficial or detrimental to the sum of two carrying capacities. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the generalized fractional maps of the orders 0 < α < 1. Commonly used in publications, fractional and fractional difference maps of the orders 0 < α < 1 belong to this class of maps. As an example, we numerically solve the equations, which define asymptotically periodic points to draw the bifurcation diagrams for the fractional difference logistic map with α = 0.5. For periods more than four (T > 4), these bifurcation diagrams are significantly different from the bifurcation diagrams obtained after 10 5 iterations on individual trajectories. We present examples of transition to chaos on individual trajectories with positive and zero Lyapunov exponents. We derive the algebraic equations, which allow the calculation of bifurcation points of generalized fractional maps. We use these equations to calculate the bifurcation points for the fractional and fractional difference logistic maps with α = 0.5. The results of our numerical simulations allow us to make a conjecture that the cascade of bifurcations scenarios of transition to chaos in generalized fractional maps and regular maps are similar, and the value of the generalized fractional Feigenbaum constant δ f is the same as the value of the regular Feigenbaum constant δ = 4.669 .... [ABSTRACT FROM AUTHOR]

The problems of estimation and classification for two logistic populations with a common location and different scale parameters are considered. The MLEs of the associated parameters are derived by solving a system of non-linear equations numerically as they do not have closed-form expressions. The asymptotic confidence intervals and bootstrap confidence intervals are derived numerically. Bayes estimators for the associated parameters using Lindley's approximation method with respect to three types of priors, namely the vague prior, Jeffrey's prior and conjugate prior, are also derived numerically. Further, Bayes estimators using the Markov chain Monte Carlo (MCMC) method that uses the Metropolis-Hastings algorithm are also derived. Moreover, using these MCMC samples, the highest posterior density (HPD) credible confidence intervals are also derived for the associated parameters. The point estimators are compared through their bias and mean squared error, whereas the interval estimators are compared through coverage probabilities and expected lengths using the Monte-Carlo simulation method numerically. Based on these estimators, certain classification rules are derived to classify a new observation into one of the two logistic populations under the same model set-up. The expected probability of misclassification for each classification rule is computed numerically to evaluate their performances. Finally, two real-life examples are considered where the datasets have been satisfactorily modeled by using the logistic distribution with a common location, and the estimation and classification methodologies have been demonstrated. [ABSTRACT FROM AUTHOR]

We modeled biological growth using a random differential equation (RDE), where the initial condition is a random variable, and the growth rate is a suitable stochastic process. These assumptions let us obtain a model that represents well the random growth process observed in nature, where only a few individuals of the population reach the maximal size of the species, and the growth curve for every individual behaves randomly. Since we assumed that the initial condition is a random variable, we assigned a priori density, and we performed Bayesian inference to update the initial condition's density of the RDE. The Karhunen–Loeve expansion was then used to approximate the random coefficient of the RDE. Then, using the RDE's approximations, we estimated the density f(p, t). Finally, we fitted this model to the biological growth of the giant electric ray (or Cortez electric ray) Narcine entemedor. Simulations of the solution of the random logistic equation were performed to construct a curve that describes the solutions' mean for each time. As a result, we estimated confidence intervals for the mean growth that described reasonably well the observed data. We fit the proposed model with a training dataset, and the model is tested with a different dataset. The model selection is performed with the square of the errors. [ABSTRACT FROM AUTHOR]

The general framework for quantitative technical analysis of market prices is revisited and extended. The concept of a global time-translation invariance and its spontaneous violation and restoration is introduced and discussed. We find that different temporal patterns leading to some famous crashes (e.g., bubbles, hockey sticks, etc.) exhibit analogous probabilistic distributions found only in the time series for the stock market indices. A number of examples of crashes are presented. We stress that our goal here is to study the crash as a particular phenomenon created by spontaneous time-translation symmetry breaking/restoration. We ask only "how to calculate and interpret the probabilistic pattern which we encounter in the day preceding crash, and how to calculate the typical market reactions to shock?". [ABSTRACT FROM AUTHOR]

In this paper, we study the periodic solutions for some diffusive logistic equations. The main aim is to investigate the sharp different effects of local (Laplace) diffusion and nonlocal dispersal on the positive periodic solutions. Our result reveals that the spatial degeneracy plays different roles between reaction-diffusion equation and nonlocal dispersal equation. [ABSTRACT FROM AUTHOR]

To prove the existence of piecewise continuous solutions to a logistic quasi-periodic differential system with impulses (whose coefficients have rationally independent periods), this system is divided into a differential equation and a difference equation. The quasi-periodicity of a function is proved by showing that this function is the uniform limit of a series of trigonometric polynomials with a finite total number of frequencies. The asymptotically stable quasi-periodic positive and piecewise continuous solution is proved to exist and to be unique. Quasi-periodic variation of the environment leads to a quasi-periodic growth of the population size in the sense that the rationally independent frequencies of the system are also frequencies of the quasi-periodic solution. The positive solutions have a repeated behavior similar to that of the quasi-periodic solution for a sufficiently long time due to asymptotical stability. The separation of the continuous-discrete system into a differential equation and a difference equation is a method of proving the existence of a quasi-periodic solution with perturbed coefficients of the impulsive system. [ABSTRACT FROM AUTHOR]

Asymptotic methods are used to study solutions of a modified logistic delay equation containing a large parameter. A result on the existence and stability of a relaxation cycle is given. [ABSTRACT FROM AUTHOR]

We consider the stationary problem for a diffusive logistic equation with the homogeneous Dirichlet boundary condition. Concerning the corresponding Neumann problem, Wei-Ming Ni proposed a question as follows: Maximizing the ratio of the total masses of species to resources. For this question, Bai, He and Li [1] showed that the supremum of the ratio is 3 in the one dimensional case, and the author and Kuto [14] showed that the supremum is infinity in the multi-dimensional ball. In this paper, we prove that the same results still hold true for the Dirichlet problem, and our proof is based on the sub-super solution method. Moreover, we show the differences of the solutions corresponding to the maximizing sequence between the one- and two-dimensional domains. [ABSTRACT FROM AUTHOR]

We propose a new type of diffuse interface model describing the evolution of a tumor mass under the effects of a chemical substance (e.g., a nutrient or a drug). The process is described by utilizing the variables φ , an order parameter representing the local proportion of tumor cells, and σ , representing the concentration of the chemical. The order parameter φ is assumed to satisfy a suitable form of the Cahn–Hilliard equation with mass source and logarithmic potential of Flory–Huggins type (or generalizations of it). The chemical concentration σ satisfies a reaction-diffusion equation where the cross-diffusion term has the same expression as in the celebrated Keller–Segel model. In this respect, the model we propose represents a new coupling between the Cahn–Hilliard equation and a subsystem of the Keller–Segel model. We believe that, compared to other models, this choice is more effective in capturing the chemotactic effects that may occur in tumor growth dynamics (chemically induced tumor evolution and consumption of nutrient/drug by tumor cells). Note that, in order to prevent finite time blowup of σ , we assume a chemical source term of logistic type. Our main mathematical result is devoted to proving existence of weak solutions in a rather general setting that covers both the two- and three- dimensional cases. Under more restrictive assumptions on coefficient and data, and in some cases on the spatial dimension, we prove various regularity results. Finally, in a proper class of smooth solutions we show uniqueness and continuous dependence on the initial data in a number of significant cases. [ABSTRACT FROM AUTHOR]

In this paper, we consider a general single species model in a heterogeneous environment of n patches (n ≥ 2), where each patch follows a generalized logistic law. First, we prove the global stability of the model. Second, in the case of perfect mixing, i.e. when the migration rate tends to infinity, the total population follows a generalized logistic law with a carrying capacity which in general is different from the sum of the n carrying capacities. Next, we give some properties of the total equilibrium population and we compute its derivative at no dispersal. In some particular cases, we determine the conditions under which fragmentation and migration can lead to a total equilibrium population which might be greater or smaller than the sum of the n carrying capacities. Finally, we study an example of two-patch model where the first patch follows a logistic law and the second a Richard's law, we give a complete classification of the model parameter space as to whether dispersal is beneficial or detrimental to the sumof two carrying capacities. [ABSTRACT FROM AUTHOR]

We propose a projection-based test to check logistic regression models when the dimension of the covariate vector may be divergent. The proposed test achieves a reduction in dimension, and the proposed method behaves as if only a single covariate is present. The test is shown to be consistent and can detect root-n local alternatives. We derive the asymptotic distribution of the proposed test under the null hypothesis and establish the test's asymptotic behavior under the local and global alternatives. The numerical performance is remarkably attractive comparing to the existing methods. Real examples are presented for illustration. for this article are available online. [ABSTRACT FROM AUTHOR]

In this paper, we introduce logistic equations with Stieltjes derivatives and provide explicit solution formulas. As an application, we present a population model which involves intraspecific competition, periods of hibernation, as well as seasonal reproductive cycles. We also deal with various forms of Stieltjes integral equations, and find the corresponding logistic equations. We show that our work extends earlier results for dynamic equations on time scales, which served as an inspiration for this paper. [ABSTRACT FROM AUTHOR]

This work develops a regularized least absolute deviation-based sparse identification of dynamics (RLAD-SID) method to address outlier problems in the classical metric-based loss function and the sparsity constraint framework. Our method uses absolute derivation loss as a substitute of Euclidean loss. Moreover, a corresponding computationally efficient optimization algorithm is derived on the basis of the alternating direction method of multipliers due to the non-smoothness of both the new proposed loss function and the regularization term. Numerical experiments are performed to evaluate the effectiveness of RLAD-SID using several exemplary nonlinear dynamical systems, such as the van der Pol equation, the Lorenz system, and the 1D discrete logistic map. Furthermore, detailed numerical comparisons are provided with other existing methods in metric-based sparse regression. Numerical results demonstrate that (1) RLAD-SID shows significant robustness toward a large outlier and (2) RLAD-SID can be seen as a particular metric-based sparse regression strategy that exhibits the effectiveness of the metric-based sparse regression framework for solving outlier problems in a dynamical system identification. [ABSTRACT FROM AUTHOR]

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This work aims to propose a new model named Gompertz-Von Bertalanffy bicompartmental (GVB), a combination of the models Gompertz and Von Bertalanffy. The GVB models is applied to fit the kinetic curve of cumulative gas production (CGP) of four foods (SS - sunflower silage; CS - corn silage; and the mixtures 340SS - 660 gkg-1 of corn silage and 340 gkg-1 of sunflower silage; and 660SS - 340 gkg-1 of corn silage and 660 gkg-1 of sunflower silage). The GVB fit is compared to models Logistic-Von Bertalanffy bicompartmental (LVB) and bicompartmental logistic (BL). All the process studied employed the semiautomatic "in vitro" technique of producing gases used in ruminant nutrition. The gas production readout was performed at times 2, 4, 6, 8, 10, 12, 15, 19, 24, 30, 48, 72, and 96 h. The data generated were used to estimate the models' parameters by the least squared method with the iterative Gauss-Newton process. The data fit quality of the models was verified using the adjusted coefficient of determination criterion (Raj. ²), mean residual square (MRS), Akaike information criterion (AIC), and mean absolute deviation (MAD). Among the analyzed models, the LVB model presented the best quality of fit evaluators for CS. In contrast, the GVB model showed better quality of fit to describe CGP over time for 340SS, 660SS, and SS, presenting the highest values of (Raj.²) and the lowest values of MSR, AIC, and MAD. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamical behavior of a one-dimensional piecewise map based on the logistic map, where generalized multistability can be observed. The proposed system has the unique property that the function is symmetric with respect to the origin but not its behavior, furthermore this system can display three types of multistability, and chaos for both, monostable and bistable behaviors. The stability analysis of the proposed system is presented. We describe the structure of bistable regions in the bifurcation diagram. Particular attention is paid to the chaotic regions. Corresponding to coexisting attractors, three scenarios of coexisting attractors, namely fixed points, periodic orbits, and chaotic attractors, can be found, which are unreported behaviors in discrete chaotic systems. The mechanism that leads to multistability phenomenon including pitchfork bifurcation, period-halving bifurcations, and the coexisting invariant sets is demonstrated. Furthermore, the Lyapunov exponent is analyzed with the type of multistability distinguished for a given set of parameters. [ABSTRACT FROM AUTHOR]