Dimitri Breda

University of Udine | UNIUD · Department of Mathematics, Computer Science and Physics

We numerically address the stability analysis of linear age-structured population models with nonlocal diffusion, which arise naturally in describing dynamics of infectious diseases. Compared to Laplace diffusion, models with nonlocal diffusion are more challenging, since the associated semigroups have no regularizing properties in the spatial vari...

Physiologically structured population models are typically formulated as a partial differential equation of transport type for the density, with a boundary condition describing the birth of new individuals. Here we develop numerical bifurcation methods by combining pseudospectral approximate reduction to a finite dimensional system with the use of...

Delays appear always more frequently in applications, ranging, e.g., from population dynamics to automatic control, where the study of steady states is undoubtedly of major concern. As many other dynamical systems, those generated by nonlinear delay equations usually obey the celebrated principle of linearized stability. Therefore, hyperbolic equil...

A numerical method based on pseudospectral collocation is proposed to approximate the eigenvalues of evolution operators for linear renewal equations, which are retarded functional equations of Volterra type. Rigorous error and convergence analyses are provided, together with numerical tests. The outcome is an efficient and reliable tool which can...

Building from a continuous-time host-parasitoid model introduced by Murdoch et al. (Am Nat 129:263-282, 1987), we study the dynamics of a 2 host-parasitoid model assuming, for the sake of simplicity, that larval stages have a fixed duration. If each host is subjected to density-dependent mortality in its larval stage, we obtain explicit conditions...

We show, by way of an example, that numerical bifurcation tools for ODE yield reliable bifurcation diagrams when applied to the pseudospectral approximation of a one-parameter family of nonlinear renewal equations. The example resembles logistic- and Ricker-type population equations and exhibits transcritical, Hopf and period doubling bifurcations....

The growth of a population subject to maturation delay is mod-eled by using either a discrete delay or a delay continuously distributed over the population. The occurrence of stability switches (stable-unstable-stable) of the positive equilibrium as the delay increases is investigated in both cases. Necessary and suffcient conditions are provided b...

We apply the pseudospectral discretization approach to nonlinear delay models described by delay differential equations, renewal equations, or systems of coupled renewal equations and delay differential equations. The aim is to derive ordinary differential equations and to investigate the stability and bifurcation of equilibria of the original mode...

The aim of this chapter is to introduce basic notation and definitions, together with solvability theorems for Cauchy problems for DDEs and a remark on continuous dependence on the data.

All the following tests and applications refer to the notation and structure of model (7. 2), i.e., from the user’s point of view as explained in Sect. 7. 1.

The central subject of this chapter is the stability analysis of the zero solution of linear periodic DDEs.

We focus our attention on linear autonomous DDEs and on the analysis of the stability properties of the zero solution.

This work deals with physiologically structured populations of the Daphnia type. Their biological modeling poses several computational challenges. In such models, indeed, the evolution of a size structured consumer described by a Volterra functional equation (VFE) is coupled to the evolution of an unstructured resource described by a delay differen...

The IG approach consists in approximating the space \(X\) with a finite dimensional linear space \(X_{M}\), called the discretization of \(X\) of index \(M\), and the infinitesimal generator \(\fancyscript{A}\) with a finite dimensional linear operator \(\fancyscript{A}_{M}:X_{M}\rightarrow X_{M}\), called the discretization of \(\fancyscript{A}\)...

The book is provided with the following three MATLAB codes: myDDE.m; eigAM.m; eigTMN.m; freely available, [48].

In the SO approach, the eigenvalues of an evolution operator.

We consider Lyapunov exponents and Sacker–Sell spectrum for linear, nonautonomous retarded functional differential equations posed on an appropriate Hilbert space. A numerical method is proposed to approximate such quantities, based on the reduction to finite dimension of the evolution family associated to the system, to which a classic discrete QR...

We are interested in the asymptotic stability of equilibria of structured populations modelled in terms of systems of Volterra functional equations coupled with delay differential equations. The standard approach based on studying the characteristic equation of the linearized system is often involved or even unattainable. Therefore, we propose and...

In order to investigate the local stability of both equilibria and periodic orbits of delayed dynamical systems we employ the numerical method recently proposed by the authors for discretizing the associated evolution family. The objective is the efficient computation of stability charts for varying or uncertain system parameters. A benchmark set o...

The aim of this paper is to show that a large class of epidemic models, with both demography and non-permanent immunity incorporated in a rather general manner, can be mathematically formulated as a scalar renewal equation for the force of infection.

The subject of this paper is the analysis of the equibria of a SIR type epidemic model, which is taken as a case study among the wide family of dynamical systems of infinite dimension. For this class of systems both the determination of the stationary solutions and the analysis of their local asymptotic stability are often unattainable theoreticall...

SIR age-structured models are very often used as a basic model of epidemic spread. Yet, their behaviour, under generic assumptions on contact rates between different age classes, is not completely known, and, in the most detailed analysis so far, Inaba (1990) was able to prove uniqueness of the endemic equilibrium only under a rather restrictive co...

We address numerically the question of the asymptotic stability of equilibria for a Gurtin–MacCamy model with age-dependent spatial diffusion. The problem reduces to the study of a finite number of simpler models without diffusion, which are parametrized by the eigenvalues of the Laplacian operator. Here the approach in Breda et al. (2007, Stabilit...

This work is devoted to the analytic study of the characteristic roots oftextitscalar autonomous delay differential equations with either real or complex coefficients. The focus is placed on the robust analysis of the position of the roots in the complex plane with respect to the variation of the coefficients, with the final aim of obtaining suitab...

In this paper a numerical scheme to discretize the solution operators of linear time invariant - time delay systems is proposed and analyzed. Following previous work of the authors on the classic state space of continuous functions, here the focus is on working in product Hilbert state spaces. The method is based on a combination of collocation and...

This paper aims at comparing the pseudospectral method and discrete geometric approach for modeling quantization effects in nanoscale devices. To this purpose, we implemented a simulation tool, based on both methods, to solve a self-consistent Schrödinger–Poisson coupled problem for a 2-D electron carrier confinement according to the effective mass...

This paper deals with the approximation of the eigenvalues of evolution operators for linear retarded functional differential equations through the reduction to finite dimensional operators by a pseudospectral collocation. Fundamental applications such as determination of asymptotic stability of equilibria and periodic solutions of nonlinear autono...

We study an S-I type epidemic model in an age-structured population, with mortality due to the disease. A threshold quantity is found that controls the stability of the disease-free equilibrium and guarantees the existence of an endemic equilibrium. We obtain conditions on the age-dependence of the susceptibility to infection that imply the uniquen...

We present a two delays SEIR epidemic model with a saturation incidence rate. One delay is the time taken by the infected individuals to become infectious (i.e. capable to infect a susceptible individual), the second delay is the time taken by an infectious individual to be removed from the infection. By iterative schemes and the comparison princip...

This paper presents an in-detail investigation of the possible advantages related to the use of the pseudospectral (PS) method for the efficient description of the carrier quantization in nanoscale n- and p-MOS transistors. To this purpose, we have implemented, by using both the finite-difference (FD) and PS methods, self-consistent Schrödinger-Poi...

First order linear time invariant and time delayed dynamics of neutral type is taken into account with three rationally independent delays. There are two main contributions of this study: (a) It is the first complete treatment in the literature, on the stability analysis of systems with three delays. We use a recent procedure, the Cluster Treatment...

This paper presents a systematic comparison between the numerical efficiency of the pseudo-spectral (PS) and finite difference (FD) methods for the solution of eigenvalue problems related to both n and p-MOS transistors, with different geometries and carrier dimensionalities. Our results indicate remarkable advantages of the PS compared to the FD m...

A predator–prey model of Beddington–DeAngelis type with maturation and gestation delays is formulated and analyzed. This two-delay model is similar to the stage-structured model by Liu and Beretta [S. Liu, E. Beretta, Stage-structured predator–prey Model with the Beddington–DeAngelis functional response, SIAM J. Appl. Math. 66 (2006) 1101–1129] but...

A general class of linear and nonautonomous delay differential equations with initial data in a separable Hilbert space is treated. The classic questions of existence, uniqueness, and regularity of solutions are addressed. Moreover, the semigroup approach typically adopted in the autonomous case for continuous initial functions is extended, and thu...

In this paper the question of asymptotic stability for retarded functional reaction diffusion equations is faced. Due to the infinite dimension of the problem a numerical approach is necessary. Here we propose a technique based on a pseudospectral discretization in time and on a spectral discretization in space of the infinitesimal generator associ...

in this paper we give an account of the basic facts to be considered when one attempts to discretize the semigroup of solution operators for Linear Time Invariant - Time Delay Systems (LTI-TDS). Two main approaches are presented, namely pseudospectral and spectral, based respectively on classic interpolation when the state space is C = C(-τ,0;C) an...

This work is devoted to the analytic study of the characteristic roots of scalar autonomous Delay Differential Equations (DDEs) with complex coefficients. The focus is placed on the robust analysis of the position of the roots in C with respect to the variation of the coefficients, with the final aim of obtaining suitable representations for the re...

This paper deals with the approximation of the spectrum of linear and nonautonomous delay differential equations through the reduction of the relevant evolution semigroup from infinite to finite dimension. The focus is placed on classic collocation, even though the requirements that a numerical scheme has to fulfill in order to allow for a correct...

in this paper we give an account of the basic facts to be considered when one attempts to discretize the semigroup of solution operators for Linear Time Invariant - Time Delay Systems (LTI-TDS). Two main approaches are presented, namely pseudospectral and spectral, based respectively on classic interpolation when the state space is C = C(-Τ, 0; ℂ)...

A new efficient algorithm for the computation of z = constant level curves of surfaces z = f(x,y) is proposed and tested on several examples. The set of z-level curves in a given rectangle of the (x,y)-plane is obtained by evaluating f on a first coarse square grid which is then adaptively refined by triangulation to eventually match a desired tole...

In the recent years the authors developed numerical schemes to detect the stability properties of different classes of systems involving delayed terms. The base of all methods is the use of pseudospectral differentiation techniques in order to get numerical approximations of the relevant characteristic eigenvalues. This chapter is aimed to present...

The stability of an equilibrium point of a dynamical system is determined by the position in the complex plane of the so-called characteristic values of the linearization around the equilibrium. This paper presents an approach for the computation of characteristic values of partial differential equations of evolution involving time delay, which is...

We study an S--I type epidemic model in an age-structured population, with mortality due to the disease. A threshold quantity is found that controls the stability of the disease-free equilibrium and guarantees the existence of an endemic equilibrium. We obtain conditions on the age-dependence of the susceptibility to infection that imply the unique...

In this paper we propose a numerical scheme to investigate the stability of steady states of the nonlinear Gurtin–MacCamy system, which is a basic model in population dynamics. In fact the analysis of stability is usually performed by the study of transcendental characteristic equations that are too difficult to approach by analytical methods. The...

We study an S-I type epidemic model in an age-structured population, with mortality due to the disease. A threshold quantity R1 is found that controls the stability of the disease-free equilibrium, and guarantees the existence of an endemic equilibrium. Conditions on the age-dependence of the susceptibility to infection are obtained that imply the...

The stability of an equilibrium point of a dynamical system is determined by the position in the complex plane of the so-called characteristic values of the linearization around the equilibrium. This work presents an approach for the computation of characteristic values of partial differential equations of evolution involving time delay, based on a...

First order linear time invariant dynamics of neutral type is taken into account with two time delays. A unique feature is included in this study, from the stability perspective: terms that add neutral dynamics into a scalar retarded type characteristic equation studied by Hale, J., et al. in 1993,. The stability posture of this new dynamics can be...

In this paper a numerical scheme to investigate the stability of linear models of age-structured population dynamics is studied. The method is based on the discretization of the infinitesimal generator associated to the semigroup of the solution operator by using pseudospectral differencing techniques, hence following the approach recently proposed...

in this work we address the question of asymptotic stability of linear delay differential equations (DDEs) with time periodic coefficients, a class which is recognized to be fundamental in machining tool. Since the dynamics of such a class of delay systems is governed by the dominant eigenvalues (multipliers) of the monodromy operator associated to...

In this paper a new method for the numerical computation of characteristic roots for linear autonomous systems of Delay Differential Equations (DDEs) is proposed. The new approach enlarges the class of methods recently developed (see [SIAM J. Numer. Anal. 40 (2002) 629; D. Breda, Methods for numerical computation of characteristic roots for delay d...

By taking as a “prototype problem” a one-delay linear autonomous system of delay differential equations we present the problem of computing the characteristic roots of a retarded functional differential equation as an eigenvalue problem for a derivative operator with non-local boundary conditions given by the particular system considered. This theo...

In the last few years the authors developed numerical schemes to detect the stability of different classes of systems involving delayed terms. The base of all methodsis the use of pseudospectral differentiation techniques. This interactive paper aims to be a proper media either to sum up the results achieved in this field and to present a collectio...

In this work we address the question of asymptotic stability of linear delay differential equations (DDEs) with time periodic coefficients, a class which is recognized to be fundamental in machining tool. Since the dynamics of such a class of delay systems is governed by the dominant eigenvalues (multipliers) of the monodromy operator associated to...

In [D. Breda, S. Maset, and R. Vermiglio, IMA J. Numer. Anal., 24 (2004), pp. 1--19.] and [D. Breda, The Infinitesimal Generator Approach for the Computation of Characteristic Roots for Delay Differential Equations Using BDF Methods, Research report UDMI RR17/2002, Dipartimento di Matematica e Informatica, Università degli Studi di Udine, Udine, It...

A new efficient algorithm for the computation of the stability chart of linear time delay systems is proposed and tested on several examples. The stability chart is obtained by investigating the 2d-parameter space by a first coarse square grid which is then adaptively refined by triangulation to match the desired tolerance. This leads to a consider...

A new approach to computing the rightmost characteristic roots of linear Delay Differential Equations (DDEs) with multiple discrete and distributed delays is presented. It is based on the discretization of the infinitesimal generator of the solution operators semigroup and it avoids the use of the characteristic equation. The approximated roots are...

This paper is a collection of tests about the numerical computation of characteristic roots for linear delay differential equations (DDEs) with multiple discrete and distributed delays. Two different approaches are tested, based on the discretization of the infinitesimal generator of the solution operators semigroup associated to the DDE and of the...