Volterra Difference Equations Research Articles

FRACTAL-FRACTIONAL SIRS MODEL FOR THE DISEASE DYNAMICS IN BOTH PREY AND PREDATOR WITH SINGULAR AND NONSINGULAR KERNELS

In this study, we consider a mathematical model for the disease dynamics in both prey and predator by considering the Susceptible–Infected–Recovered–Susceptible (SIRS) model with the prey–predator Lotka–Volterra differential equations. Carrying capacity, predation, migration, and immunity loss are also taken into account for both species. Using the law of mass action, the physical model is transformed into a nonlinear coupled system of ODEs. The classical/integer order ODE system is then generalized through the fractal-fractional differential operators of power law and Mittag-Leffler kernels. For the model under consideration, we additionally check for positivity, boundedness, the basic reproduction number, and equilibrium points. Graphical results are obtained by use of a numerical approach, and the existence and uniqueness of the solution are also established theoretically. The graphical solutions has been displayed via 2D and 3D phase plots. It has been shown that when the fractional order reduces, the amplitude of the chaotic attractor dynamics shrinks, along with the range of limit cycles and periodic trajectories while a drop in the fractal dimension parameter causes an increase in the time period of the chaotic attractor dynamics. This study not only improves the understanding of epidemic breakouts in predator–prey systems, but also highlights the efficiency of fractal-fractional calculus in ecological modeling.

Periodic Points, Stability, Bifurcations, and Transition to Chaos in Generalized Fractional Maps

This paper is a review of the recent advances in the investigation of the properties of generalized fractional maps. These maps are Volterra difference equations of the convolution type with power-law-like kernels. They include fractional and fractional difference maps of a constant or variable order. The main objective of this article is to describe the criteria of stability, equations defining periodic points and bifurcations, and transition to chaos that are common for all generalized fractional maps.

Multi-dimensional almost automorphic type sequences and applications

Abstract In this paper, we investigate several new classes of multi-dimensional almost automorphic type sequences and focus on their applications to various difference equations involving Volterra difference equations. We provide many structural results, illustrative examples and open problems about the notion under consideration.

Zeros transfer for recursively defined polynomials

The zeros of D’Arcais polynomials, also known as Nekrasov–Okounkov polynomials, dictate the vanishing of the Fourier coefficients of powers of the Dedekind eta functions. These polynomials satisfy difference equations of hereditary type with non-constant coefficients. We relate the D’Arcais polynomials to polynomials satisfying a Volterra difference equation of convolution type. We obtain results on the transfer of the location of the zeros. As an application, we obtain an identity between Chebyshev polynomials of the second kind and 1-associated Laguerre polynomials. We obtain a new version of the Lehmer conjecture and bounds for the zeros of the Hermite polynomials.

Generalized almost periodic solutions of Volterra difference equations

In this paper, we investigate several new classes of generalized $\rho$-almost periodic sequences in the multi-dimensional setting. We specifically analyze the class of Levitan $\rho$-almost periodic sequences and the class of remotely $\rho$-almost periodic sequences. We provide many important applications of the established theoretical results to the abstract Volterra difference equations.

Bohl Theorem for Volterra Equations

This work deals with the preservation of exponential stability under small perturbations for Volterra differential equations. The so-called Bohl-Perron type stability theorems for these systems are also studied.

Proportional stochastic generalized Lotka–Volterra model with an application to learning microbial community structures

Inferring microbial community structure based on temporal metagenomics data is an important goal in microbiome studies. The deterministic generalized Lotka–Volterra (GLV) differential equations have been commonly used to model the dynamics of microbial taxa. However, these approaches fail to take random environmental fluctuations into account and usually ignore the compositional nature of relative abundance data, which may deteriorate the estimates. In this article, we consider the microbial dynamics in terms of relative abundances by introducing a reference taxon, and propose a new proportional stochastic GLV (pSGLV) differential equation model, where the random perturbations of Brownian motion in this model can naturally account for the external environmental effects on the microbial community. We establish conditions and show some mathematical properties of the solutions including general existence and uniqueness, stochastic ultimate boundedness, stochastic permanence, the existence of stationary distribution, and ergodicity property. We further develop approximate maximum likelihood estimators (AMLEs) based on discrete observations and systematically investigate the consistency and asymptotic normality of the proposed estimators. At last, numerical simulations support our theoretical findings and our method is demonstrated through an application to the well-known “moving picture” temporal microbial dataset.

Approximate Analytic Solution to the Three Species Lotka – Volterra Differential Equation Model

This paper provides an approximate analytic solution to the three species Lotka – Volterra differential equations by symbolic regression. The approximate analytic solution through symbolic regression is made as close as desired to the actual analytic solution by using the Jacobian system. This is proposed as the equilibrium will be stabilized if and only if the real parts of each of the eigenvalues are negative. As a result, the symbolic regression approach is found to provide an approximation to the faster convergence that can be expected with a more refined Euler numerical approach.

Numerical analysis of the Volterra differential equations via a combination of evolutionary algorithms with moving least squares and finite difference methods

In this paper, the Volterra differential equations governing on the predator-prey dynamics is numerically solved through the combination of evolutionary algorithms with Moving Least Squares as well as the Finite Difference Method. Among the numerous evolutionary algorithms, the Genetic Algorithms and Particle Swarm Optimization, as two powerful and effective approaches, are selected. The penalty method is chosen to impose the initial conditions on the objective function, while the Moving Least Squares is implemented to interpolate the derivatives in the differential equations. In order to illustrate the effectiveness of the Moving Least Squares, the results are compared with those obtained via the Finite Difference Method based evolutionary algorithms. Moreover, the results are associated with those of a numerical method and an analytical scheme to evaluate the obtained populations of the prey and the predator. These comparisons clearly illustrate the reliability and efficiency of the proposed methodology to solve nonlinear differential equations.

Optimal Speed of Response in the Lotka–Volterra Controlled System

We consider a controlled system of Lotka–Volterra differential equations that describes the evolution of two interrelated populations of predators and prey. The system contains two control variables, which are chosen so as to minimize the time of transition to a stationary point. The control functions and the corresponding trajectories of motion in the state space are constructed and their optimality is substantiated.

Inequalities and sufficient conditions for exponential stability and instability for nonlinear Volterra difference equations with variable delay

Inequalities and sufficient conditions that lead to exponential stability of the zero solution of the variable delay nonlinear Volterra difference equation\(\begin{eqnarray*}x(n+1)=a(n)h(x(n))+\sum^{n-1}_{s=n-g(n)}b(n,s)h(x(s))\end{eqnarray*}\) are obtained. Lyapunov functionals are constructed and employed in obtaining the main results. A criterion for the instability of the zero solution is also provided. The results generalizes some results in the literature.

Asymptotically ordinary linear Volterra difference equations with infinite delay

• A linear Volterra difference equation with infinite delay and sufficiently small coefficients is shown to be asymptotically equivalent to a linear ordinary difference equation. • The coefficient matrix of the corresponding ordinary difference equation can be written as a limit of successive approximations. • The eigenvalues of the approximating matrices converge to the characteristic roots of the Volterra difference equation at an exponential rate. A class of linear Volterra difference equations with infinite delay is considered. It is shown that if the coefficient matrices are sufficiently small, then the Volterra difference equation is asymptotically equivalent to a linear ordinary difference equation at infinity. The coefficient matrix of the ordinary difference equation is a solution of an associated matrix equation which can be obtained by successive approximations. The eigenvalues of the approximating matrices converge exponentially to the characteristic roots of the Volterra difference equation. As a corollary, an efficient new method is obtained for the computation of the characteristic roots with an explicit error estimate.

Long-time behavior of numerical solutions to nonlinear fractional ODEs

In this work, we study the long time behavior, including asymptotic contractivity and dissipativity, of the solutions to several numerical methods for fractional ordinary differential equations (F-ODEs). The existing algebraic contractivity and dissipativity rates of the solutions to the scalar F-ODEs are first improved. In order to study the long time behavior of numerical solutions to fractional backward differential formulas (F-BDFs), two crucial analytical techniques are developed, with the first one for the discrete version of the fractional generalization of the traditional Leibniz rule, and the other for the algebraic decay rate of the solution to a linear Volterra difference equation. By means of these auxiliary tools and some natural conditions, the solutions to F-BDFs are shown to be contractive and dissipative, and also preserve the exact contractivity rate of the continuous solutions. Two typical F-BDFs, based on the Grünwald–Letnikov formula and L1 method respectively, are studied. For high order F-BDFs, including convolution quadrature schemes based on classical second order BDF and product integration schemes based on quadratic interpolation approximation, their numerical contractivity and dissipativity are also developed under some slightly stronger conditions. Numerical experiments are presented to validate the long time qualitative characteristics of the solutions to F-BDFs, revealing very different decay rates of the numerical solutions in terms of the the initial values between F-ODEs and integer ODEs and demonstrating the superiority of the structure-preserving numerical methods.

Asymptotic behavior of solutions of second-order difference equations of Volterra type

In this paper we investigate the Volterra difference equation of the form $ \D(r_n\D x_n)=b_n+\sum_{k=1}^{n}K(n,k)f(x_k). $ We establish sufficient conditions for the existence of a solution $x$ of the above equation with the property $ x_n=y_n+\o(n^s), $ where $y$ is a given solution of the equation $\D(r_n\D y_n)=b_n$ and $s$ is nonpositive real number. We also obtain sufficient conditions for the existence of asymptotically periodic solutions.

The large contraction principle and existence of periodic solutions for infinite delay Volterra difference equations

In this article, we establish sufficient conditions for the existence of periodic solutions of a nonlinear infinite delay Volterra difference equation: $$\Delta x(n) = p(n) + b(n)h(x(n)) + \sum^{n}_{k = -\infty}B(n, k)g(x(k)).$$ We employ a Krasnosel'ski\u{i} type fixed point theorem, originally proved by Burton. The primary sufficient condition is not verifiable in terms of the parameters of the difference equation, and so we provide three applications in which the primary sufficient condition is verified.

Pseudo Asymptotically Periodic Solutions for Volterra Difference Equations of Convolution Type

In this paper, the author studies the existence and uniqueness of discrete pseudo asymptotically periodic solutions for nonlinear Volterra difference equations of convolution type, where the nonlinear perturbation is considered as Lipschitz condition or non-Lipschitz case, respectively. The results are a consequence of application of different fixed point theorems, namely, the contraction mapping principle, the Leray-Schauder alternative theorem and Matkowski’s fixed point technique.

On the existence and uniqueness of (N,lambda )-periodic solutions to a class of Volterra difference equations

In this paper we introduce the class of (N,lambda )-periodic vector-valued sequences and show several notable properties of this new class. This class includes periodic, anti-periodic, Bloch and unbounded sequences. Furthermore, we show the existence and uniqueness of (N, lambda )-periodic solutions to the following class of Volterra difference equations with infinite delay: \t\t\tu(n+1)=α∑j=−∞na(n−j)u(j)+f(n,u(n)),n∈Z,α∈C,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ u(n+1)=\\alpha \\sum_{j=-\\infty }^{n}a(n-j)u(j)+f \\bigl(n,u(n) \\bigr), \\quad n \\in \\mathbb{Z}, \\alpha \\in \\mathbb{C}, $$\\end{document} where the kernel a and the nonlinear term f satisfy suitable conditions.

Qualitative theory for Volterra difference equations

We investigate lp boundedness, the topological structure of solutions set and the asymptotic periodicity of Volterra functional difference equations. The theoretical results are complemented with a set of applications.

A note on exact finite difference schemes for modified Lotka–Volterra differential equations

We construct a class of modified Lotka–Volterra ordinary differential equations (ODE’s) and show that a nonlinear change of the dependent variables transform them into a set of coupled, linear ODE’s. Using the latter equations, we calculate the corresponding exact finite difference schemes using a technique given by Mickens. Next, we show how to reconfigure these relations to obtain the exact finite difference representation of the original modified, nonlinear Lotka–Volterra ODE’s.

Chaotic predation scheme for age-clustered one predator–one prey Lotka–Volterra

Lotka–Volterra differential equations deal with modeling of predator and prey populations and interrelation of population sizes in a continuous domain. Since all populations, be they predators or prays, are only the estimates in consequence of sampling process, this paper initially concerns with discretization schemes of one predator–prey Lotka–Volterra systems. Second-order Runge–Kutta approximation is used to discretize the original differential equations for flexibility to manipulate the system parameters. Subsequent to discretization, a novel predation scheme is introduced to enhance the rationalization of original model with age cluster-based and detailed rules which are based on reproductivity, predation ability, predation essentiality, food provision and consumption. Aging is also simulated with transitive structure of the algorithms that the alive individuals are getting older and changing clusters after the predation scheme is operated. Experiments revealed that our model exhibits not only fluctuations like the original model but also stable trajectories and fractal structure depending on the model parameters. Therefore, the main novelty of this paper briefly is the discovery of chaotic one predator–one prey system exhibiting chaotic behavior for a narrow interval and revealing a strange attractor which is very unique.