Shandelle M. Henson

Some Discrete Competition Models and the Competitive Exclusion Principle

A difference equation model, called that Leslie/Gower model, played a key historical role in laboratory experiments that helped establish the “competitive exclusion principle” in ecology. We show that this model has the same dynamic scenarios as the famous Lotka/Volterra (differential equation) competition model. It is less well known that some anomalous results from the experiments seem to contradict the exclusion principle and Lotka/Volterra dynamics. We give an example of a competition model that has non-Lotka/Volterra dynamics that are consistent with the anomalous case.

A Periodically Forced Beverton-Holt Equation

has a unique positive equilibrium K and all solutions with x0 . 0 approach K as t !1: This equation (known as the Beverton–Holt equation) arises in applications to population dynamics, and in that context K is the “carrying capacity” and r is the “inherent growth rate”. A modification of this equation that arises in the study of populations living in a periodically (seasonally) fluctuating environment replaces the constant carrying capacity K by a periodic sequence Kt of positive carrying capacities.

Global Dynamics of Some Periodically Forced, Monotone Difference Equations ∗

We study a class of periodically forced, monotone difference equations motivated by applications from population dynamics. We give conditions under which there exists a globally attracting cycle and conditions under which the attracting cycle is attenuant

Multiple mixed-type attractors in a competition model

We show that a discrete-time, two-species competition model with Ricker (exponential) nonlinearities can exhibit multiple mixed-type attractors. By this is meant dynamic scenarios in which there are simultaneously present both coexistence attractors (in which both species are present) and exclusion attractors (in which one species is absent). Recent studies have investigated the inclusion of life-cycle stages in competition models as a casual mechanism for the existence of these kinds of multiple attractors. In this paper we investigate the role of nonlinearities in competition models without life-cycle stages.

Multiple attractors and resonance in periodically forced population models

Oscillating discrete autonomous dynamical systems admit multiple oscillatory solutions in the advent of periodic forcing. The multiple cycles are out of phase, and some of their averages may resonate with the forcing amplitude while others attenuate. In application to population biology, populations with stable inherent oscillations in constant habitats are predicted to develop multiple attracting oscillatory final states in the presence of habitat periodicity. The average total population size may resonate or attenuate with the amplitude of the environmental fluctuation depending on the initial population size. The theory has been tested successfully in the laboratory by subjecting cultures of the flour beetle Tribolium to habitat periodicity of various amplitudes.

Hierarchical models of intra-specific competition: scramble versus contest

Hierarchical structured models for scramble and contest intraspecific competition are derived. The dynamical consequences of the two modes of competition are studied under the assumption that both populations divide up the same amount of a limiting resource at equal population levels. A comparison of equilibrium levels and their resiliences is made in order to determine which mode of competition is more advantageous. It is found that the concavity of the resource uptake rate is an important determining factor. Under certain circumstances contest competition is more advantageous for a population while under other circumstances scramble competition is more advantageous.

A chaotic attractor in ecology: theory and experimental data

Chaos has now been documented in a laboratory population. In controlled laboratory experiments, cultures of flour beetles (Tribolium castaneum) undergo bifurcations in their dynamics as demographic parameters are manipulated. These bifurcations, including a specific route to chaos, are predicted by a well-validated deterministic model called the “LPA model”. The LPA model is based on the nonlinear interactions among the life cycle stages of the beetle (larva, pupa and adult). A stochastic version of the model accounts for the deviations of data from the deterministic model and provides the means for parameterization and rigorous statistical validation. The chaotic attractor of the deterministic LPA model and the stationary distribution of the stochastic LPA model describe the experimental data in phase space with striking accuracy. In addition, model-predicted temporal patterns on the attractor are observed in the data. This paper gives a brief account of the interdisciplinary effort that obtained these results.

PHASE SWITCHING IN POPULATION CYCLES

Oscillatory populations may exhibit a phase change in which, for example, a high–low periodic pattern switches to a low–high pattern. We propose that phase shifts correspond to stochastic jumps between basins of attraction in an appropriate phase space which associates the different phases of a periodic cycle with distinct attractors. This mechanism accounts for two-cycle phase shifts and the occurrence of asynchronous replicates in experimental cultures of Tribolium.

Anatomy of a chaotic attractor: Subtle model-predicted patterns revealed in population data

Mathematically, chaotic dynamics are not devoid of order but display episodes of near-cyclic temporal patterns. This is illustrated, in interesting ways, in the case of chaotic biological populations. Despite the individual nature of organisms and the noisy nature of biological time series, subtle temporal patterns have been detected. By using data drawn from chaotic insect populations, we show quantitatively that chaos manifests itself as a tapestry of identifiable and predictable patterns woven together by stochasticity. We show too that the mixture of patterns an experimentalist can expect to see depends on the scale of the system under study.

Nonlinear Stochastic Population Dynamics: The Flour Beetle Tribolium as an Effective Tool of Discovery

When observation and theory collide, scientists turn to carefully designed experiments for resolution. Their motivation is especially high in the case of biological systems, which are typically far too complex to be grasped by observation and theory alone. The best procedure, as in the rest of science, is first to simplify the system, then to hold it more or less constant while varying the important parameters one or two at a time to see what happens. —Edward O. Wilson (2002)