Advances in Difference Equations and Discrete Dynamical Systems

It is known that the behavior of a nonlinear semelparous Leslie matrix model with the basic reproduction number close to one can be approximated by a solution of a Lotka-Volterra differential equation. Furthermore, even in multi-species cases, a similar approximation works as long as every species is semelparous. This paper gives a mathematical basis to this approximation and shows that Lotka-Volterra equations are helpful to study a certain bifurcation problem of multi-species semelparous population models. With the help of this approximation method, we find an example of coexistence of two biennial populations with temporal segregation. This example provides a new mechanism of producing population cycles.

We give examples of dichotomy spectra for nonautonomous linear difference equations in infinite-dimensional spaces. Particular focus is on the spectrum of integrodifference equations having compact coefficients. Concrete systems with explicitly known spectra are discussed for several purposes: (1) They yield reference examples for numerical approximation schemes. (2) The asymptotic behavior of spectral intervals is tackled illustrating their merging.

Positive density-dependence occurs when individuals experience increased survivorship, growth, or reproduction with increased population densities. Mechanisms leading to these positive relationships include mate limitation, saturating predation risk, and cooperative breeding and foraging. Individuals within these populations may differ in age, size, or geographic location and thereby structure these populations. Here, I study structured population models accounting for positive density-dependence and environmental stochasticity i.e. random fluctuations in the demographic rates of the population. Under an accessibility assumption (roughly, stochastic fluctuations can lead to populations getting small and large), these models are shown to exhibit a dynamical trichotomy: (i) for all initial conditions, the population goes asymptotically extinct with probability one, (ii) for all positive initial conditions, the population persists and asymptotically exhibits unbounded growth, and (iii) for all positive initial conditions, there is a positive probability of asymptotic extinction and a complementary positive probability of unbounded growth. The main results are illustrated with applications to spatially structured populations with an Allee effect and age-structured populations experiencing mate limitation.

In this paper we discuss connections between the evolutionary games on graphs and replicator equations. On the traditional examples of social dilemma games we introduce the basic ideas of replicator dynamics and the mathematical concepts behind evolutionary games on graphs. We show that the stability regions of evolutionary games on complete graphs with the sequential and synchronous updating with deterministic imitation dynamics converge to the stability regions of replicator equations. Finally, we show that by a finer choice of a time scale and a stochastic imitation dynamic update rule not only the stability regions but also the trajectories of evolutionary games on graphs converge to those of replicator equations.

The main results of the paper are complimenting, extending and improving several earlier results obtained for Halanay type discrete difference inequalities. The novel idea is that the discrete time results are derived from our recent related continuous time results by using suitable delay differential inequalities with piecewise constant arguments. The sharpness of the results are illustrated by examples.

We consider the Cauchy problems of nonlinear partial differential equations of the normal form in the class of the analytic functions. We apply semi-discrete finite difference approximation which discretizes the problems only with respect to the time variable, and we give a result about convergence. The main result shows convergence of consistent finite difference schemes even without stability, and therefore shows independence between stability and convergence for finite difference schemes. Our theoretical result can be realized numerically on multiple-precision arithmetic environments.

We show the relation between the Jacobs-de Leeuw–Glicksberg decomposition of semigroups and the spectral decomposition of the Markov operators which are induced by stochastic difference equations by using our new results.

The purpose of paper is to analyze the behavior of the error in the iterative method. Especially, we are interested in the classical iterative method such as SOR method and its preconditioning techniques to solve the linear system \(A{u}={q}\). In order to accelerate convergence, many researchers proposed several preconditioners [4‐8]. There is also preconditioner available for both classical iterative and Krylov subspace methods. We focus on the behavior of error to find a good preconditioner. We treat difference equation derived from partial differential equation(PDE), because the coefficient matrix given by using difference approximation is easy to investigate. By examining the behavior of the error, we choose an effective preconditioner, and show the numerical results.

We deal with a four-dimensional nonlinear difference system with deviating arguments in the paper. The first equation of the system is of a neutral type. We study oscillatory and nonoscillatory solutions of neutral difference systems and their asymptotic properties. We establish sufficient conditions for the system to have strongly monotone solutions or Kneser solutions and then sufficient conditions for the system to have property B.

This chapter is dedicated to the study of the positive invariance of polyhedral sets with respect to dynamical systems described by discrete-time delay difference equations (DDEs). Set invariance in the original state-space, also referred to as \(\mathscr {D}\)-invariance, leads to conservative definitions due to its delay independent property. This limitation makes the \(\mathscr {D}\)-invariant sets only applicable to a limited class of systems. However, there exists a degree of freedom in the state-space transformations which can enable the positive invariant set-characterizations. In this work we revisit the set factorizations and extend their use in order to establish flexible set-theoretic analysis tools. With linear algebra structural results, it is shown that similarity transformations are a key element in the characterization of low complexity invariant sets within the class of convex polyhedral candidates. In short, it is shown that we can construct, in a low dimensional state-space, an invariant set for a dynamical system governed by a delay difference equation. The basic idea which enables the construction is a simple change of coordinates for the DDE. The obtained \(\mathscr {D}\)-invariant set exists in the new coordinates even if its existence necessary conditions are not fulfilled in the original state-space. This proves that the \(\mathscr {D}\)-invariance notion is dependent on the state-space representation of the dynamics. It is worth to recall as a term of comparison that the positive invariance for delay-free dynamics is independent of the state-space realization.

The present paper is considered a two-dimensional difference system: where all coefficients are real-valued sequences; p and \(p^*\) are positive numbers satisfying \(1/p + 1/p^* = 1\); and \(\phi _p(x) = |x|^{p-2}x\) for \(x \ne 0\), and \(\phi _p(0) = 0\). The aim of this paper is to clarify that uniform asymptotic stability and exponential stability are equivalent for the above system. To illustrate the obtained results, an example is given. In addition, a figure of a solution orbit which is drawn by a computer is also attached for a deeper understanding.

It is known that the exponential growth rate of every positive solution of a Poincaré difference equation is a nonnegative eigenvalue of the limiting equation with a positive eigenvector. In this note we show how this discrete result implies its continuous counterpart.

In this note we review some of the latest research on the qualitative analysis of solutions of difference equations using fixed point theory and Lyapunov functionals. It turns out that the use of fixed point theory alleviates some of the difficulties that arise from the use of Lyapunov functionals. Using fixed point theory requires us to find a mapping from suitable spaces that is a solution of the given difference equation. Once the suitable mapping is constructed there will be many fixed pint theorems to use, depending on the given equation, that yield a fixed point of that mapping and satisfies our initial value problem. In some cases a regular contraction argument will not be suitable and hence we replace it with what we call Large Contraction.

In this paper, we consider a nonlinear protocol for a structured time-varying synchronous multi-agent system in which an opinion sharing dynamics is presented by non-autonomous polynomial stochastic operators associated with high-order stochastic hyper-matrices. We show that the proposed nonlinear protocol generates the Krause mean process. We provide a criterion to establish a consensus in the multi-agent system under the proposed nonlinear protocol.

A mathematical epidemic discrete equation, which appears as a model for the spread of disease-causing, is treated. In this paper, we consider the asymptotic stability of a discrete SIR epidemic model by using the classical linearization method and some Liapunov functions.

Our purpose here is to establish nonoscillation criteria for the second-order linear difference equations of the form Here, \(\{r_n\}\) and \(\{p_n\}\) are sequences of real numbers and \(\Delta _a\) is the weighted difference operator defined by \(\Delta _ax_n = x_{n+1} - ax_n\) with any positive constant a. A certain sequence determined from the constant a and two sequences \(\{r_n\}\) and \(\{p_n\}\) plays an important role in the results obtained. To be a little more precise, what should be paid attention to is a weighted sum of two adjacent terms of the sequence. The main tools for the proof of our results are Sturm’s separation theorem and the Riccati transformation method. Our results are compared with several previous works by using some specific examples.

We consider a two-stage voluntary provision model where individuals in a family contribute to a pure public good and/or a household public good, and an altruistic parent makes a non-negative income transfer to his or her child. The subgame perfect equilibrium derived in the model is analyzed using two evolutionary dynamics games (i.e., replicator dynamics and best response dynamics). As a result, the equilibria with ex-post transfers and pre-committed transfers coexist, and are unstable in the settings of replicator dynamics as well as best response dynamics, whereas the monomorphic states (i.e., all families undertake either ex-post or pre-committed transfers) are stable. An income redistribution policy does not alter the real allocations in the settings of both evolutionary dynamics games, because the resulting real allocations depend on only the total income of society and not on the distribution of individual income.

In this article we consider the following system of piecewise linear difference equations: \(x_{n+1}=|x_{n}|-y_{n}-1\) and \(y_{n+1} = x_{n}+|y_n|-1\) where the initial condition \((x_{0}, y_{0})\) is an element of \( \{(x, 0): x > \frac{3}{2}\}\) and \(x_0\) is not in a sequence of intervals \(B_n = \{x : \frac{2^{2n+1}-1}{2^{2n}} < x \le \frac{2^{2n+2}-1}{2^{2n+1}} \} \) for any integer n. We show that the solution to the system is eventually one of two particular prime period 4 solutions.