We analyze the stability of the equilibria and bifurcations of a planar system of ordinary differential equations describing the product-resource interaction in a sustainable supply chain. While periodic behavior in supply chain models has often been documented either in systems of higher dimensions or in delayed systems, here a Hopf bifurcation arises in a planar and delay-free system. We show that the interior equilibrium loses stability through a supercritical Hopf bifurcation as the environmental capacity exceeds a critical threshold that depends on the maximum production rate, the resource level at which production reaches half its maximum rate, and the demand and remanufacturing rates. We derive this threshold explicitly and provide estimates for the amplitude and period of the periodic solution close to the Hopf bifurcation, by means of the resulting Hopf normal form. We then further substantiate our analytical results through numerical simulations.

We present a Runge-Kutta method of weak order 2 for the numerical time integration of stochastic delay differential equations. This scheme extends the class of second order Runge-Kutta methods introduced by A. Rößler in [SIAM J. Numer. Anal., 47(3):1713-1738, 2009] for stochastic ordinary differential equations. The proposed integrator is applicable to equations with discrete commensurable delays and is particularly efficient for problems involving multiple noise terms. Experimental confirmation of the weak order 2 is provided and MATLAB codes are freely available.

We consider the problem of discretizing evolution operators of linear delay equations with the aim of approximating their spectra, which is useful in investigating the stability properties of (nonlinear) equations via the principle of linearized stability. We develop a general convergence analysis based on a reformulation of the operators by means of a fixed-point equation, providing a list of hypotheses related to the regularization properties of the equation and the convergence of the chosen approximation techniques on suitable subspaces. This framework unifies the proofs for some methods based on pseudospectral discretization, which we present here in this new form. To exemplify the generality of the framework, we also apply it to a method of weighted residuals found in the literature, which was previously lacking a formal convergence analysis.

We considered a model for an infectious disease outbreak, when the depletion of susceptible individuals is negligible, and assumed that individuals adapt their behavior according to the information they receive about new cases. In line with the information index approach, we supposed that individuals react to past information according to a memory kernel that is continuously distributed in the past. We analyzed equilibria and their stability, with analytical results for selected cases. Thanks to the recently developed pseudospectral approximation of delay equations, we studied numerically the long-term dynamics of the model for memory kernels defined by gamma distributions with a general non-integer shape parameter, extending the analysis beyond what is allowed by the linear chain trick. In agreement with previous studies, we showed that behavior adaptation alone can cause sustained waves of infections even in an outbreak scenario, and notably in the absence of other processes like demographic turnover, seasonality, or waning immunity. Our analysis gives a more general insight into how the period and peak of epidemic waves depend on the shape of the memory kernel and how the level of minimal contact impacts the stability of the behavior-induced positive equilibrium.

We prove convergence of the spectral element method for piecewise polynomial collocation applied to periodic boundary value problems (BVP) for functional differential equations with possibly state-dependent delays. If the exact solution of the BVP has an analytic extension then the collocation solution converges geometrically. This means that the accuracy of the approximation is of order e^-ηm for some η>0 depending on the size of the mesh, when using polynomials of degree m. If the exact solution has a finite order of continuous differentiability then the collocation solution converges with this order. For functional differential equations with state-dependent delays the right-hand side cannot be expected to be differentiable with respect to its arguments in the classical sense, and analyticity of the solution does not necessarily follow from analyticity of the coefficients in the right-hand side. Thus, our geometric convergence statement assumes analyticity of the solution, rather than of the right-hand side.

In this survey, we provide an in-depth investigation of exponential Runge-Kutta methods for the numerical integration of initial-value problems. These methods offer a valuable synthesis between classical Runge-Kutta methods, introduced more than a century ago, and exponential integrators, which date back to the 1960s. This manuscript presents both a historical analysis of the development of these methods up to the present day and several examples aimed at making the topic accessible to a broad audience.

We prove convergence of piecewise polynomial collocation methods applied to periodic boundary value problems for functional differential equations with state-dependent delays. The state dependence of the delays leads to nonlinearities that are not locally Lipschitz continuous preventing the direct application of general abstract discretization theoretic frameworks. We employ a weaker form of differentiability, which we call mild differentiability, to prove that a locally unique solution of the functional differential equation is approximated by the solution of the discretized problem with the expected order.

We propose the use of exponential Runge-Kutta methods for the time integration of delay differential equations. The approach is based on their reformulation as abstract differential equations and the reduction of the latter to finite-dimensional systems of ordinary differential equations via pseudospectral discretization. We substantiate our results by means of some illustrative numerical simulations with EXPINT, a MATLAB package for exponential integrators. Copyright (c) 2024 The Authors.

Exponential Runge-Kutta methods for semi-linear ordinary differential equations can be extended to abstract differential equations, defined on Banach spaces. Thanks to the sun-star theory, both delay differential equations and renewal equations can be recast as abstract differential equations, which motivated the present work. The result was a general approach that allowed us to define the methods explicitly and analyze their convergence properties in a unifying way.

In a recent paper by Guglielmi and Hairer (SIADS 2015), an analysis in the $\varepsilon\to 0$ limit was proposed of regularized discontinuous ODEs in codimension-2 switching domains; this was obtained by studying a certain 2-dimensional system describing the so-called hidden dynamics. In particular, the existence of a unique limit solution was not proved in all cases, a few of which were labeled as ambiguous, and it was not clear whether or not the ambiguity could be resolved. In this paper, we show that it cannot be resolved in general. A first contribution of this paper is an illustration of the dependence of the limit solution on the form of the switching function. Considering the parameter dependence in the ambiguous class of discontinuous systems, a second contribution is a bifurcation analysis, revealing a range of possible behaviors. Finally, we investigate the sensitivity of solutions in the transition from codimension-2 domains to codimension-3 when there is a limit cycle in the hidden dynamics.

We extend the use of piecewise orthogonal collocation to computing periodic solutions of renewal equations, which are particularly important in modeling population dynamics. We prove convergence through a rigorous error analysis. Finally, we show some numerical experiments confirming the theoretical results, and a couple of applications in view of bifurcation analysis.

Stability and bifurcation analyses of delay equations represent fundamental challenges in many applications, emerging in important fields like, e.g., control theory and population dynamics. Exact approaches are usually unattainable due to the infinite dimension of the associated state space, so that resorting to numerical methods is unavoidable. This work is a survey on the use of collocation techniques to address the problems above from the numerical standpoint. As such it summarizes the main contributions in this context of the research of the authors in the last 15 years or so. Methods for the linearized stability analysis of equilibria and periodic solutions, as well as for the bifurcation analysis of general nonlinear systems are illustrated, together with an essential overview of relevant problems and potential developments.

The asymptotic stability of the null equilibrium of a linear population model with two physiological structures formulated as a first-order hyperbolic PDE is determined by the spectrum of its infinitesimal generator. In this paper, we propose a general numerical method to approximate this spectrum. In particular, we first reformulate the problem in the space of absolutely continuous functions in the sense of Carathéodory, so that the domain of the corresponding infinitesimal generator is defined by trivial boundary conditions. Via bivariate collocation, we discretize the reformulated operator as a finite-dimensional matrix, which can be used to approximate the spectrum of the original infinitesimal generator. Finally, we provide test examples illustrating the converging behavior of the approximated eigenvalues and eigenfunctions, and its dependence on the regularity of the model coefficients.

The problem of computing periodic solutions can be expressed as a boundary value problem and solved numerically via piecewise collocation. Here, we extend to renewal equations the corresponding method for retarded functional differential equations in (K. Engelborghs et al., SIAM J Sci Comput., 22 (2001), pp. 1593-1609). The theoretical proof of the convergence of the method has been recently provided in (A. Ando and D. Breda, SIAM J Numer Anal., 58 (2020), pp. 3010-3039) for retarded functional differential equations and in (A. Ando and D. Breda, submitted in 2021) for renewal equations and consists in both cases in applying the abstract framework in (S. Maset, Numer Math., 133 (2016), pp. 525-555) to a reformulation of the boundary value problem featuring an infinite-dimensional boundary condition. We show that, in the renewal case, the proof can also be carried out and even simplified when considering the standard formulation, defined by boundary conditions of finite dimension.

We describe a piecewise collocation method for computing periodic solutions of renewal equations, obtained as an extension of the corresponding method in [K. Engelborghs et al., SIAM J. Sci. Comput., 22 (2001), pp. 1593--1609] for retarded functional differential equations. Then, we rigorously prove its convergence under the abstract framework proposed in [S. Maset, Numer. Math., 133 (2016), pp. 525--555], as previously done in [A.A. and D.B., SIAM J. Numer. Anal., 58 (2020), pp. 3010--3039] for general retarded functional differential equations. Finally, we show some numerical experiments on models from populations dynamics which confirm the order of convergence obtained theoretically, as well as a few applications in view of bifurcation analysis.

We analyze the convergence of piecewise collocation methods for computing periodic solutions of general retarded functional differential equations under the abstract framework recently developed in [S. Maset, Numer. Math., 133 (2016), pp. 525--555], [S. Maset, SIAM J. Numer. Anal., 53 (2015), pp. 2771--2793], and [S. Maset, SIAM J. Numer. Anal., 53 (2015), pp. 2794--2821]. We rigorously show that a reformulation as a boundary value problem requires a proper infinite-dimensional boundary periodic condition in order to be amenable to such analysis. In this regard, we also highlight the role of the period acting as an unknown parameter, which is critical since it is directly linked to the course of time. Finally, we prove that the finite element method is convergent, while we limit ourselves to commenting on the infeasibility of this approach as far as the spectral element method is concerned.

Recently, many realistic models of structured populations are described through delay equations which involve quantities defined by the solutions of external problems. For instance, the size or survival probability of individuals may be described by ordinary differential equations, and their maturation age may be determined by a nonlinear condition. When treating these complex models with existing continuation approaches in view of analyzing stability and bifurcations, the external quantities are computed from scratch at every continuation step. As a result, the requirements from the computational point of view are often demanding. In this work we propose to improve the overall performance by investigating a suitable numerical treatment of the external problems in order to include the relevant variables into the continuation framework, thus exploiting their values computed at each previous step. We explore and test this internal continuation with prototype problems first. Then we apply it to a representative class of realistic models, demonstrating the superiority of the new approach in terms of computational time for a given accuracy threshold.

Collocation methods can be applied in different ways to delay models, e.g., to detect stability of equilibria, Hopf bifurcations and compute periodic solutions to name a few. On the one hand, piecewise polynomials can be used to approximate a periodic solution for some fixed values of the model parameters, possibly using an adaptive mesh. On the other hand, polynomial collocation can be used to reduce delay systems to systems of ordinary differential equations and established continuation tools are then applied to analyze stability and detect bifurcations. These techniques are particularly useful to treat realistic models describing structured populations, where delay differential equations are coupled with renewal equations and vital rates are given implicitly as solutions of external equations, which in turn change with model parameters. In this work we show how collocation can be used to improve the performance of continuation for such complex models and to compute periodic solutions of coupled problems.

A prototype SIR model with vaccination at birth is analyzed in terms of the stability of its endemic equilibrium. The information available on the disease influences the parents' decision on whether vaccinate or not. This information is modeled with a delay according to the Erlang distribution. The latter includes the degenerate case of fading memory as well as the limiting case of concentrated memory. The linear chain trick is the essential tool used to investigate the general case. Besides its novel analysis and that of the concentrated case, it is showed that through the linear chain trick a distributed delay approaches a discrete delay at a linear rate. A rigorous proof is given in terms of the eigenvalues of the associated linearized problems and extension to general models is also provided. The work is completed with several computations and relevant experimental results.