Gabriela Kirlinger

Permanence in Lotka-Volterra equations: linked prey-predator systems

Abstract A population-dynamical system is called permanent if all species survive, provided they are initially present. More precisely, a system is called permanent if there exists some level k > 0 such that if the number x i (0) of species i at time 0 ispositive for i = 1, 2, …, n , then x i ( t ) > k for all sufficiently large times t . In this paper conditions for permanence in prey-predator systems linked by interspecific competition of prey are deduced.

Permanence of some ecological systems with several predator and one prey species

The stability criterion used in the following is “permanence”. Permanence means that all trajectories starting in the interior are ultimately bounded away from the boundary and that this bound is independent of the initial values. Hence sufficiently small fluctuations cannot lead to extinction of any species. In the following we deal with one-prey, two-predator resp. one-prey, three-predator systems and a one-prey, two-predator, one-top-predator system with three trophic levels. It turns out that the characterization of permanence for such models described by Lotka-Volterra dynamics is rather simple and elegant.

Two predators feeding on two prey species: A result on permanence

For biological populations the precise asymptotic behavior of the corresponding dynamic system is probably less important than the question of extinction and survival of species. An ecological differential equation is called permanent if there exists some level k greater than 0 such that if the number xi(0) of species i at time 0 is positive for i = 1,2, ..., n then xi(t) greater than k for all sufficiently large times t Characterizations for permanence in a four-species prey-predator system modeled by the Lotka-Volterra equation are presented. The method used is based on a combination of two well-known approaches to dealing with permanence. An interesting feature is the occurrence of heteroclinic cycles.

High-Order Stiff ODE Solvers via Automatic Differentiation and Rational Prediction

A class of higher order methods is investigated which can be viewed as implicit Taylor series methods based on Hermite quadratures. Improved automatic differentiation techniques for the claculation of the Taylor-coefficients and their Jacobians are used. A new rational predictor is used which can allow for larger step sizes on stiff problems.

A note on convergence concepts for stiff problems

Most convergence concepts for discretizations of nonlinear stiff initial value problems are based on one-sided Lipschitz continuity. Therefore only those stiff problems that admit moderately sized one-sided Lipschitz constants are covered in a satisfactory way by the respective theory. In the present note we show that the assumption of moderately sized one-sided Lipschitz constants is violated for many stiff problems. We recall some convergence results that are not based on one-sided Lipschitz constants; the concept of singular perturbations is one of the key issues. Numerical experience with stiff problems that are not covered by available convergence results is reported.ZusammenfassungDie meisten Konvergenzkonzepte für Diskretisierungen nichtlinearer steifer Anfangswertprobleme basieren auf dem Begriff der einseitigen Lipschitz-Stetigkeit. Folglich sind durch diese theoretischen Konzepte nur steife Probleme mit moderater einseitiger Lipschitzkonstante abgedeckt. In der vorliegenden Arbeit zeigen wir, daß die Annahme moderater einseitiger Lipschitzkonstanten für viele steife Probleme verletzt ist. Wir weisen auf einige Konvergenzresultate hin, die nicht auf einseitigen Lipschitzkonstanten basieren; die Konzepte der singulären Störungstheorie sind hier von wesentlicher Relevanz. Wir berichten über einige numerische Erfahrungen mit steifen Problemen, die durch keine existierende Konvergenztheorie abgedeckt sind.

Modern convergence theory for stiff initial-value problems

Abstract In this paper we give a brief review of available theoretical results about convergence and error structures for discretizations of stiff initial-value problems. We point out limitations of the various approaches and discuss some recent developments.

An extension of B-convergence for Runge-Kutta methods

Abstract The well-known concepts of B-stability and B-convergence for the analysis of one-step methods applied to stiff initial value problems are based on the notion of one-sided Lipschitz continuity. In a recent paper (Auzinger et al. (1990)) the authors have pointed out that the one-sided Lipschitz constant m must often be expected to be very large (positive and of the order of magnitude of the stiff eigenvalues) despite a (globally) well-conditioned behavior of the underlying problem. As a consequence, the existing B-theory suffers from considerable restrictions; e.g., not even linear systems with time-dependent coefficients are satisfactorily covered. The purpose of the present paper is to fill this gap; for implicit Runge-Kutta methods we extend the B-convergence theory such as to be valid for a class of non-autonomous weakly nonlinear stiff systems; reference to the (potentially large) one-sided Lipschitz constant is avoided. Unique solvability of the system of algebraic equations is shown, and global error bounds are derived.

Asymptotic error expansions for stiff equations: applications

In a series of foregoing papers we have studied the structure of the global discretization error for the implicit Euler scheme and the implicit midpoint and trapezoidal rules applied to a general class of nonlinear stiff initial value problems. Full asymptotic error expansions (in the conventional sense) exist only in special situations; for the general case, asymptotic expansions in a weaker sense have been derived. In the present paper we demonstrate how these results can be used for an analysis of acceleration techniques applied to stiff problems. In particular, extrapolation and defect correction algorithms are considered. Various numerical results are presented and discussed.ZusammenfassungIn einer Reihe vorangegangener Arbeiten wurde die Struktur des globalen Diskretisierungsfehlers für das implizite Eulerverfahren sowie die implizite Mittelpunkts- und Trapezregel bei anwendung auf eine allgemeine Klasse nichtlinearer steifer Anfangswertprobleme untersucht. Volle asymptotische Entwicklungen (im konventionellen Sinn) existieren nur in speziellen Situationen; für den allgemeinen Fall wurden asymptotische Fehlerentwicklungen in einem schwächeren Sinn hergeleitet. In der vorliegenden Arbeit wird gezeigt, wie beschleunigte Algorithmen, angewendet auf steife Probleme, mit Hilfe der erwähnten Resultate analysiert werden können. Im besonderen werden Extrapolation und die Methode der Defektkorrektur betrachtet. Verschiedenste numerische Resultate werden präsentiert und ausführlich diskutiert.

Extending convergence theory for nonlinear stiff problems part I

Existing convergence concepts for the analysis of discretizations of nonlinear stiff problems suffer from considerable drawbacks. Our intention is to extend the convergence theory to a relevant class of nonlinear problems, where stiffness is axiomatically characterized in natural geometric terms.Our results will be presented in a series of papers. In the present paper (Part I) we motivate the need for such an extension of the existing theory, and our approach is illustrated by means of a convergence argument for the Implicit Euler scheme.

Linear multistep methods applied to stiff initial value problems-A survey

The numerical approximation of solutions of differential equations has been and continues to be one of the principal concerns of numerical analysis. Linear multistep methods and, in particular, backward differentiation formulae (BDFs) are frequently used for the numerical integration of stiff initial value problems. Such stiff problems appear in a variety of applications. While the intuitive meaning of stiffness is clear to all specialists, there has been much controversy about its correct mathematical definition. We present a historical development of the concept of stiffness. A survey of convergence results for special classes of stiff problems based on these different concepts of stiffness is given, e.g., for linear, stiff systems, problems in singular perturbation form, nonautonomous stiff systems, and rather general nonlinear stiff problems. Different approaches proving convergence of linear multistep methods applied to stiff initial value problems are introduced. It is further indicated that the corresponding proofs for singular perturbation problems are compatible with a nonlinear transformation and thus convergence of a quite general class of nonlinear problems seems to be covered.