G. Kirlinger

Bio: G. Kirlinger is an academic researcher from Vienna University of Technology. The author has contributed to research in topics: Stiff equation & Initial value problem. The author has an hindex of 8, co-authored 17 publications receiving 263 citations. Previous affiliations of G. Kirlinger include University of Vienna.

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.

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

Abstract: 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.

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

Abstract: 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.

Abstract: 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 >0 such that if the number x i (0) of species i at time 0 is positive for i =1,2, … , n then x i ( t )> 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.

Solving Ordinary Differential Equations

Abstract: Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Permanence and the assembly of ecological communities

Abstract: A numerical technique for assembly of ecological communities of Lotka- Volterra form is described. The technique is based upon a global criterion for coexistence of species known as permanence. This provides a relatively fast and accurate method to determine the sequence of communities that develops when species are drawn sequentially and in an arbitrary order from a regional pool of species. Steps in the assembly sequence that cannot be resolved by this method are determined by numerical integration. The results are as follows. (1) At each step in an assembly sequence, a species that succeeds in invading when rare persists in the resulting community even if one or more of the resident species becomes extinct. (2) Assembly sequences are terminated with a community that is unin- vadable by any of the remaining species from the pool. The number of these endpoints is small, even when the species pool is large. (3) In some cases, the final community cannot be reassembled from the species left in it; other species, which are absent at the end, are needed for the endpoint to be reached. (4) Invasion resistance builds up in three stages during an assembly sequence. Over much of the sequence, invasion resistance shows little if any increase; during this period, species composition continues to change until the se- quence happens to land on an endpoint. (5) Communities assembled from large species pools are more resistant to invasion than those assembled from small species pools.

Permanence and the dynamics of biological systems.

Abstract: A basic question in mathematical biology concerns the long-term survival of each component, which might typically be a population in an ecological context, of a system of interacting components. Many criteria have been used to define the notion of long-term survival. We consider here the subject of permanence, i.e., the study of the long-term survival of each species in a set of populations. These situations may often be modeled successfully by dynamical systems and have led to the development of some interesting mathematical techniques and results. Our intention here is to describe these and to consider their application to several of the most frequently used models occurring in mathematical biology. We particularly wish to include and cover those models leading to problems that are essentially infinite dimensional, for example reaction-diffusion equations, and to make the discussion accessible to a wide audience, we include a chapter outlining the fundamental theory of these.