In this paper we propose and study a hybrid discrete-continuous mathematical model of collective motion under alignment and chemotaxis effect. Starting from paper [23], in which the Cucker-Smale model [22] was coupled with other cell mechanisms, to describe the cell migration and self-organization in the zebrafish lateral line primordium, we introduce a simplified model in which the coupling between an alignment and chemotaxis mechanism acts on a system of interacting particles. In particular we rely on a hybrid description in which the agents are discrete entities, while the chemoattractant is considered as a continuous signal. The proposed model is then studied both from an analytical and a numerical point of view. From the analytic point of view we prove, globally in time, existence and uniqueness of the solution. Then, the asymptotic behaviour of a linearised version of the system is investigated. Through a suitable Lyapunov functional we show that for t -> +infinity, the migrating aggregate exponentially converges to a state in which all the particles have a same position with zero velocity. Finally, we present a comparison between the analytical findings and some numerical results, concerning the behaviour of the full nonlinear system.

The dynamic behaviour of rigid blocks subjected to support excitation is represented by discontinuous differential equations with state jumps. In the numerical simulation of these systems, the jump times corresponding to the numerical trajectory do not coincide with the ones of the given problem. When multiple state jumps occur, this approximation may affect the accuracy of the solution and even cause an order reduction in the method. Focus here is on the error behaviour in the numerical dynamic. The basic idea is to investigate how the error propagates in successive impacts by decomposing the numerical integration process of the overall system into a sequence of discretized perturbed problems.

The integral representation of some biological phenomena consists in Volterra equations whose kernels involve a convolution term plus a non convolution one. Some significative applications arise in linearised models of cell migration and collective motion, as described in Di Costanzo et al. (Discrete Contin. Dyn. Syst. Ser. B 25 (2020) 443-472), Etchegaray et al. (Integral Methods in Science and Engineering (2015)), Grec et al. (J. Theor. Biol. 452 (2018) 35-46) where the asymptotic behaviour of the analytical solution has been extensively investigated. Here we consider this type of problems from a numerical point of view and we study the asymptotic dynamics of numerical approximations by linear multistep methods. Through a suitable reformulation of the equation, we collect all the non convolution parts of the kernel into a generalized forcing function, and we transform the problem into a convolution one. This allows us to exploit the theory developed in Lubich (IMA J. Numer. Anal. 3 (1983) 439-465) and based on discrete variants of Paley-Wiener theorem. The main effort consists in the numerical treatment of the generalized forcing term, which will be analysed under suitable assumptions. Furthermore, in cases of interest, we connect the results to the behaviour of the analytical solution.

In some important biological phenomena Volterra integral and integrodifferential equations represent an appropriate mathematical model for the description of the dynamics involved (see e.g. [1], and the bibliography therein). In most cases, the kernels of these equations are of convolution type, however, some recent applications, as cell migration and collective motion [4-5], are characterized by kernels with a quasi-convolution form, namely involving a convolution contribution plus a non-convolution term. We focus on problems of this type and exploit some known results about convolution equations [2, 3], in order to describe the asymptotic dynamics of numerical approximations and connect the results to the behaviour of the analytical solution

We consider homogeneous linear Volterra Discrete Equations and we study the asymptotic behaviour of their solutions under hypothesis on the sign of the coefficients and of the first- and second-order differences. The results are then used to analyse the numerical stability of some classes of Volterra integrodifferential equations.

We analyze the stability of convolution quadrature methods for weakly singular Volterra integral equations with respect to a linear test equation. We prove that the asymptotic behavior of the numerical solution replicates the one of the continuous problem under some restriction on the stepsize. Numerical examples illustrate the theoretical results.

The equations describing engineering and real-life models are usually derived in an approximated way. Thus, in most cases it is necessary to deal with equations containing some kind of perturbation. In this paper we consider fractional dfferential equations and study the eects on the continuous and numerical solution, of perturbations on the given function, over long-time intervals. Some bounds on the global error are also determined.

The dynamic behaviour of rigid blocks subjected to support excitation is represented by discontinuous differential equations with state jumps, which are not known in advance. In the numerical simulation of these systems, the jump times corresponding to the numerical trajectory do not coincide with the ones of the given problem. When multiple state jumps occur, this approximation may affect the accuracy of the solution and even cause an order reduction in the method. Focus here is on the stability and convergence properties of the numerical dynamic. The basic idea is to investigate how the error propagates in successive impacts by decomposing the numerical integration process of the overall system into a sequence of discretized perturbed problems.

Within the theoretical framework of the numerical stability analysis for the Volterra integral equations, we consider a new class of test problems and we study the long-time behavior of the numerical solution obtained by direct quadrature methods as a function of the stepsize. Furthermore, we analyze how the numerical solution responds to certain perturbations in the kernel.

Volterra Integral Equations (VIEs) arise in many problems of real life, as, for example, feedback control theory, population dynamics and fluid dynamics. A reliable numerical simulation of these phenomena requires a careful analysis of the long time behavior of the numerical solution. Here we develop a numerical stability theory for Direct Quadrature (DQ) methods which applies to a quite general and representative class of problems. We obtain stability results under some conditions on the stepsize and, in particular cases, unconditional stability for DQ methods of whatever order. (C) 2017 IMACS. Published by Elsevier B.V. All rights reserved.

We analyze the stability of the zero solution to Volterra equations on time scales with respect to two classes of bounded perturbations. We obtain sufficient conditions on the kernel which include some known results for continuous and for discrete equations. In order to check the applicability of these conditions, we apply the theory to a test example.

In this paper, to complete the global dynamics of a multi-strains SIS epidemic model, we establish a precise result on coexistence for the cases of the partial and complete duplicated multiple largest reproduction ratios for this model.

We study a nonlinear boundary value problem involving a nonlocal (integral) operator in the coefficients of the unknown function. Provided sufficient conditions for the existence and uniqueness of the solution, for its approximation, we propose a numerical method consisting of a classical discretization of the problem and an algorithm to solve the resulting nonlocal and nonlinear algebraic system by means of some iterative procedures. The second order of convergence is assured by different sufficient conditions, which can be alternatively used in dependence on the given data. The theoretical results are confirmed by several numerical tests. (C) 2015 Elsevier B.V. All rights reserved.

An important topic in the numerical analysis of Volterra integral equations is the stability theory. The main results known in theliterature have been obtained on linear test equations or, at least, on nonlinear equations with convolution kernel. Here, we considerVolterra integral equations with Hammerstein nonlinearity, not necessarily of convolution type, and we study the error equation forDirect Quadrature methods with respect to bounded perturbations. For a class of Direct Quadrature methods, we obtain conditionson the stepsize h for the numerical solution to behave stably and we report numerical examples which show the robustness of thisnonlinear stability theory.

The modeling of various physical questions often leads to nonlinear boundary value problems involving a nonlocal operator, which depends on the unknown function in the entire domain, rather than at a single point. In order to answer an open question posed by J.R. Cannon and D.J. Galiffa, we study the numerical solution of a special class of nonlocal nonlinear boundary value problems, which involve the integral of the unknown solution over the integration domain. Starting from Cannon and Galiffa's results, we provide other sufficient conditions for the unique solvability and a more general convergence theorem. Moreover, we suggest different iterative procedures to handle the nonlocal nonlinearity of the discrete problem and show their performances by some numerical tests.

In a recent paper we proposed a numerical method to solve a non-standard non-linear second order integro-differential boundary value problem. Here, we answer two questions remained open: we state the order of convergence of this method and provide some sufficient conditions for the uniqueness of the solution both of the discrete and the continuous problem. Finally, we compare the performances of the method for different choices of the iteration procedure to solve the non-standard nonlinearity.

We consider Volterra integral equations on time scales and present our study about the long time behavior of their solutions. We provide sufficient conditions for the stability and investigate the convergence properties when the kernel of the equations vanishes at infinity.

In this paper we consider Volterra integral equations on time scales and describe our study about the long time behavior of their solutions. We provide sufficient conditions for the stability under constant perturbations by using the direct Lyapunov method and we present some examples of application.

The paper deals with the numerical solution of a nonstandard Sturm-Liouville boundary value problem on the half line where the coefficients of the differential terms depend on the unknown function by means of a scalar integral operator. By using a finite difference discretization, a truncated quadrature rule and an iterative procedure, we construct a numerical method, whose convergence is proved. The order of convergence and the truncation at infinity are also discussed. Finally, some numerical tests are given to show the performance of the method. © 2013 Elsevier B.V. All rights reserved.