arXiv subjects

Torsten Lindström

Publications and source records attributed to Torsten Lindström.

5 recordsLinked to original sources

arXiv2026

Ecological systems in a modeling perspective

May (1974,1976) opened the debate on whether biological populations might exhibit nonlinear dynamics and chaos. However, it has in general been difficult to verify nonlinear dynamics in biological populations. There are many reports concerning problems with this issue and some of them can be traced back to Hassell, Lawton, and May (1976) and Morris (1990) Our objective is not a discussion of the presence of nonlinear dynamics in biological populations. Instead, we analyze whether ecological census data can be used for validating nonlinearities at all. We choose our models and our situation so that as much as possible can be done rigorously with by hand computations. We consider a clearly nonlinear chemostat based model that is isolated. Some noise must be considered, and we choose a minimal approach: Only noise originating from the fact that ecological populations remain finite is considered, cf. Bailey (1964). Not only the interacting populations but also collected data sets tend to remain finite. Collection of long data sets might be associated with huge costs in ecology. Examples of exceptionally long and carefully studied ecological time series are those collected by Nicholson (1954) and Utida (1957). These data sets contain a few hundred data points, and we use this as a guideline for when an ecological time series should be considered exceptionally long in this chapter.

q-bio.OT↗

arXiv2022

On the stochastic engine of transmittable diseases in exponentially growing populations

The purpose of this paper is to analyze the mechanism for the interplay of deterministic and stochastic models for contagious diseases. Deterministic models for contagious diseases are prone to predict global stability. Small natural birth and death rates in comparison to disease parameters like the contact rate and the removal rate ensures that the globally stable endemic equilibrium corresponds to a tiny average proportion of infected individuals. Asymptotic equilibrium levels corresponding to low numbers of individuals invalidate the deterministic results. Diffusion effects force frequency functions of the stochastic model to possess similar stability properties as the deterministic model. Particular simulations of the stochastic model predict, however, oscillatory patterns. Small and isolated populations show longer periods, more violent oscillations, and larger probabilities of extinction. We prove that evolution maximizes the infectiousness of the disease as measured by the ability to increase the proportion of infected individuals. This holds provided the stochastic oscillations are moderate enough to keep the proportion of susceptible individuals near a deterministic equilibrium. We close our paper with a discussion of the herd-immunity concept and stress its close relation to vaccination-programs.

q-bio.PE↗

arXiv2019

Destabilization, stabilization, and multiple attractors in saturated mixotrophic environments

The ability of mixotrophs to combine phototrophy and phagotrophy is now well recognized and found to have important implications for ecosystem dynamics. In this paper we examine the dynamical consequences of the invasion of mixotrophs in a model that is a limiting case of the chemostat. The model is a hybrid of a competition model describing the competition between populations of autotroph and mixotroph for limiting resources, and a predator-prey type model describing the interaction between populations of autotroph and herbivore. Our results show that mixotrophs are able to invade in both autotrophic environments and environments described by interactions between autotrophs and herbivores. The interaction between autotrophs and herbivores might be in equilibrium or cycle. We find that invading mixotrophs have the ability to both stabilize and destabilize autotroph-herbivore dynamics depending on the competitive ability of mixotrophs. Moreover the invasion of mixotrophs can also result in multiple attractors. Therefore, our results reveal important consequences of mixotrophic invasions in ecosystems depending on environmental conditions.

math.DS↗

arXiv2015

Relationships between the delayed chemostat and the delayed logistic equation

The logistic equation is often considered as a simplification of the chemostat (Kooi, Boer, Kooijman (1998)). However, the global qualitative properties of the delayed single species chemostat are completely known, see e. g. Smith (2011). Such properties still remain open for the delayed logistic equation, see e. g. B{á}nheley, Czendes, Kristin, and Neymaier (2014) despite that they have been announced a long time ago (Wright (1955)). We may therefore ask whether the logistic equation really is a simplification and what information about the chemostat actually is contained. We discuss the links between these equations and conclude that they are less clear in the delayed case than they are in the non-delayed case. We keep the presentation as elementary as possible.

math.DS↗

arXiv2014

Logistic approximations and their consequences to bifurcations patterns and long-run dynamics

On infinitesimally short time interval various processes contributing to population change tend to operate independently so that we can simply add their contributions (Metz and Diekmann (1986)). This is one of the cornerstones for differential equations modeling in general. Complicated models for processes interacting in a complex manner may be built up, and not only in population dynamics. The principle holds as long as the various contributions are taken into account exactly. In this paper we discuss commonly used approximations that may lead to dependency terms affecting the long run qualitative behavior of the involved equations. We prove that these terms do not produce such effects in the simplest and most interesting biological case, but the general case is left open.

math.DS↗