Torsten Lindström

CONTINUOUS VERSUS DISCRETE SINGLE SPECIES POPULATION MODELS WITH ADJUSTABLE REPRODUCTIVE STRATEGIES

We investigate population models with both continuous and discrete elements. Birth is assumed to occur at discrete instants of time whereas death and competition for resources and space occur continuously during the season. We compare the dynamics of such discrete-continuous hybrid models with the dynamics of purely discrete models where within-season mortality and competition are modelled directly as discrete events. We show that non-monotone discrete single-species maps cannot be derived from unstructured competition processes. This result is well known in the case of fixed reproductive strategies and our results extend this to the case of adjustable reproductive strategies. It is also shown that the most commonly used non-monotone discrete maps can be derived from structured competition processes.

A predator-prey model with optimal suppression of reproduction in the prey

Several time-discrete ecological models are formulated and analyzed. Common to these models is that a trade-off between reproductive effort and competitive ability is assumed. We investigate how behavioral adjustment (suppression of reproduction) affects the population dynamics.

Qualitative analysis of a predator-prey system with limit cycles

Fairly regular multiannual microtine rodent cycles are observed in boreal Fennoscandia. In the southern parts of Fennoscandia these multiannual cycles are not observed. It has been proposed that these cycles may be stabilized by generalist predation in the south.We show that if the half-saturation of the generalist predators is high compared to the number of small rodents the cycles are likely to be stabilized by generalist predation as observed. We give examples showing that if the half-saturation of the generalist predators is low compared to the number of small rodents, then multiple equilibria and multiple limit cycles may occur as the generalist predator density increases.

Global stability of a model for competing predators: an extension of the Ardito & Ricciardi Lyapunov function

The author proves conditions for global stability of a model for competing predators by using Lyapunov functions, and gives several examples

An elementary approach to dynamics and bifurcations of skew tent maps

In this paper, the dynamics of skew tent maps are classified in terms of two bifurcation parameters. In time series analysis such maps are usually referred to as continuous threshold autoregressive models (TAR(1)-models) after Tong (Non-Linear Time Series, Clarendon Press, Oxford, UK, 1990). This study contains results simplifying the use of TAR(1)-models considerably, e.g. if a periodic attractor exists it is unique. On the other hand, we also claim that care must be exercised when TAR models are used. In fact, they possess a very special type of dynamical pattern with respect to the bifurcation parameters and their transition to chaos is far from standard.

On the appearance of multiple attractors in discrete food-chains

Abstract Klebanoff and Hastings [A. Klebanoff, A. Hastings, Chaos in three species food chains, Journal of Mathematical Biology 32 (1994) 427–451] detected cases of multiple attractors in continuous food-chains. In this paper we discuss similar phenomena in the discrete food-chains introduced by Lindstrom [T. Lindstrom, On the dynamics of discrete food-chains: low- and high-frequency behavior and chaos, Journal of Mathematical Biology 45 (2002) 396–418]. The results imply that the dynamical properties including species persistence may change due to disturbances that do not involve changes in the environmental parameters. Thus, there are possibilities that species may be eradicated or start to oscillate at different frequencies without any changes in the environment. Since this is now shown to hold in both seasonal and non-seasonal environments, we expect that this is a rather general property of ecosystems.

Conditional Reproductive Strategies Under Variable Environmental Conditions

Within the framework of adaptive dynamics we consider the evolution by natural selection of reproductive strategies in which individuals may adjust their reproductive behaviour in response to changing environmental conditions. As a specific example we considered a discrete-time model in which possible fluctuations in the environmental conditions are caused by predator—prey interaction. Our main findings include: (1) Coexistence between two fixed strategies (i.e., strategies that do not adjust to changing environmental conditions) is impossible; there exists a best fixed strategy, which invades and ousts all other fixed strategies. (2) A necessary condition for conditional (adjustable) strategies to evolve is that there are fluctuations in the environmental conditions. Predator—prey interactions may cause such fluctuations and under natural assumptions there exists an optimal conditional strategy which is uninvadable and invades and ousts all other strategies.

A Rosenzweig-MacArthur (1963) Criterion for the Chemostat

The Rosenzweig-MacArthur (1963) criterion is a graphical criterion that has been widely used for elucidating the local stability properties of the Gause (1934) type predator-prey systems. It has not been stated whether a similar criterion holds for models with explicit resource dynamics (Kooi et al. (1998)), like the chemostat model. In this paper we use the implicit function theorem and implicit derivatives for proving that a similar graphical criterion holds under chemostat conditions, too.

Stability switches in discrete food-chain problems

In an already classical paper, May [ 1976] pointed out that simple nonlinear models in ecology can possess complicated dynamics. A long debate concerning the possible existence of chaotic ecologies ...

Dynamical properties of maps fitted to data in the noise-free limit

We argue that any attempt to classify dynamical properties from nonlinear finite time-series data requires a mechanistic model fitting the data better than piecewise linear models according to standard model selection criteria. Such a procedure seems necessary but still not sufficient.