Robert A. Desharnais

NONLINEAR DEMOGRAPHIC DYNAMICS: MATHEMATICAL MODELS, STATISTICAL METHODS, AND BIOLOGICAL EXPERIMENTS'

Our approach to testing nonlinear population theory is to connect rigorously mathematical models with data by means of statistical methods for nonlinear time series. We begin by deriving a biologically based demographic model. The mathematical analysis identifies boundaries in parameter space where stable equilibria bifurcate to periodic 2-cy- cles and aperiodic motion on invariant loops. The statistical analysis, based on a stochastic version of the demographic model, provides procedures for parameter estimation, hypothesis testing, and model evaluation. Experiments using the flour beetle Tribolium yield the time series data. A three-dimensional map of larval, pupal, and adult numbers forecasts four possible population behaviors: extinction, equilibria, periodicities, and aperiodic motion including chaos. This study documents the nonlinear prediction of periodic 2-cycles in laboratory cultures of Tribolium and represents a new interdisciplinary approach to un- derstanding nonlinear ecological dynamics.

Transitions in population dynamics: Equilibria to periodic cycles to aperiodic cycles

1. We experimentally set adult mortality rates, μ a , in laboratory cultures of the flour beetle Tribolium at values predicted by a biologically based, nonlinear mathematical model to place the cultures in regions of different asymptotic dynamics. 2. Analyses of time-series residuals indicated that the stochastic stage-structured model described the data quite well. Using the model and maximum-likelihood parameter estimates, stability boundaries and bifurcation diagrams were calculated for two genetic strains. 3. The predicted transitions in dynamics were observed in the experimental cultures. The parameter estimates placed the control and μ a = 0.04 treatments in the region of stable equilibria. As adult mortality was increased, there was a transition in the dynamics. At μ a = 0.27 and 0.50 the populations were located in the two-cycle region. With μ a = 0.73 one genetic strain was close to a two-cycle boundary while the other strain underwent another transition and was in a region of equilibrium. In the μ a = 0.96 treatment both strains were close to the boundary at which a bifurcation to aperiodicities occurs; one strain was just outside this boundary, the other just inside the boundary. 4. The rigorous statistical verification of the predicted shifts in dynamical behaviour provides convincing evidence for the relevance of nonlinear mathematics in population biology.

HISTORY AND CURRENT DEVELOPMENT OF A PARADIGM OF PREDATION IN ROCKY INTERTIDAL COMMUNITIES

A paradigm is a set of mutually supportive hypotheses that provides a frame of reference within a field. In 1962, Kuhn proposed that paradigms form within the dual contexts of empirical evidence and intellectual history. Facts potentially contradictory to a paradigm may not be recognized until they are observed repeatedly and incorporated as supportive evidence into a new theory. Support for this interpretation can be found in the history of a paradigm of predation in rocky intertidal communities. Hypotheses were developed in the contexts of innovative field experiments and historical arguments of competition theory. The resulting paradigm proposed that predators restrict populations of competitively dominant prey to refuges. Different types of refuge, or no refuge, prevail in different areas of the intertidal zone, accounting for patterns of prey distribution, the coexistence of natural enemies, and the local maintenance of diversity. An ensuing period of criticism made reference to potential contradictions. Rather than by predation alone, prey abundances are determined largely by an interplay of varying rates of predation and prey production. Furthermore, prey refuges are neither necessary nor sufficient to explain all observed instances of local coexistence of predators and prey. We present a model in which intertidal boundaries of prey are set by equilibria between predation and prey production. Predation and prey productivity vary with environmental gradients and with explicit spatial configurations of the prey. This synthesis relies on insights of the original paradigm, incorporates the contradictory observations, and depends on novel capabilities afforded by spatially explicit computer simulations. The resulting synthesis provides explanations for distinctive aspects of zonation, including abrupt prey boundaries in continuous gradients of predation, and converging upper and lower prey boundaries in gradients of decreasing wave exposure.

An interdisciplinary approach to understanding nonlinear ecological dynamics

Abstract We describe a research program which covers a spectrum of activities essential to testing nonlinear population theory: from the translation of the biology into the formal language of mathematics, to the analysis of mathematical models, to the development and application of statistical techniques for the analysis of data, to the design and implementation of biological experiments. The statistical analyses, mathematics, and biology are thoroughly integrated. We review several aspects of our current research effort that demonstrate this integration.

Power spectra reveal the influence of stochasticity on nonlinear population dynamics

Stochasticity alters the nonlinear dynamics of inherently cycling populations. The power spectrum can describe and explain the impacts of stochasticity. We fitted models to short observed time series of flour beetle populations in the frequency domain, then used a well fitting stochastic mechanistic model to generate detailed predictions of population spectra. Some predicted spectral peaks represent periodic phenomena induced or modified by stochasticity and were experimentally confirmed. For one experimental treatment, linearization theory explained that these peaks represent overcompensatory decay of deviations from deterministic oscillation. In another treatment, stochasticity caused frequent directional phase shifting around a cyclic attractor. This directional phase shifting was not explained by linearization theory and modified the periodicity of the system. If field systems exhibit directional phase shifting, then changing the intensity of demographic or environmental noise while holding constant the structure of the noise can change the main frequency of population fluctuations.

Quantitative in situ hybridization to measure single-cell changes in vasopressin and oxytocin mRNA levels after osmotic stimulation.

Summary1.The measurement of cellular mRNA content by quantitativein situ hybridization is a valuable approach to the study of gene expression in brain since this tissue exhibits a high degree of phenotypic heterogeneity.2.The cellular content of vasopressin and oxytocin mRNA in hypothalamo-neurohypophysial system neurons was altered by maintaining rats for 24 hr on 2% sodium chloride water.3.Statistical and graphical techniques were then used to analyze cell by cell how mRNA levels were altered as a result of osmotic stimulation. We propose that the negative binomial probability distribution is a suitable model to describe how mRNA content varies across a defined cell population. For both measures of oxytocin and vasopressin mRNA levels, maximum-likelihood estimation indicated that this model adequately described empirical findings obtained from rats drinking tap water or salt water.4.Both graphical and statistical analyses suggested how the defined neural system responds to osmotic stimulation: mRNA content was altered as a multiplicative function of “initial state.” The utility and limitations of the quantitative approach are discussed.

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.

Complex equilibria in the maintenance of boundaries: experiments with mussel beds

Stationary boundaries of sedentary species may belie dynamic processes that form them. Our aim was to test an implication of an evolving body of theory, that such boundaries are manifestations of complex regulatory dynamics. On rocky shores of British Columbia, large-scale field experiments altered the densities of predatory sea stars (Pisaster ochraceus), causing shifts in the location of the lower vertical boundaries of their prey, sea mussels (Mytilus californianus). While control mussel beds remained unchanged, experimental reductions of sea star densities caused the downward extension of the lower boundaries, and experimental increases in sea stars densities caused the upward recession of the lower boundary well into the zone presumed to be a spatial refuge from predation. Cleared plots prepared within the initial boundaries were recolonized to varying degrees, depending on predator densities. After 30 months, plots on sea star removal sites showed high densities of adult mussels, control plots showed intermediate densities, and sea star addition plots showed only a sparse cover of alternative prey. Observations by divers at high tide showed that as small prey were depleted progressively from removal, to control, to addition sites, correspondingly larger mussels were attacked, including very large individuals comprising the lower boundary of addition sites. The findings contradict classic theory of zonation based on static prey refuges and support an alternative theory in which boundaries are maintained by complex, spatially structured equilibria.

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.

Expansion of Paneth Cell Population in Response to Enteric Salmonella enterica Serovar Typhimurium Infection

ABSTRACT Paneth cells residing at the base of the small intestinal crypts contribute to the mucosal intestinal first line defense by secreting granules filled with antimicrobial polypeptides including lysozyme. These cells derive from the columnar intestinal stem cell located at position 0 and the transit amplifying cell located at position +4 in the crypts. We have previously shown that Salmonella enterica serovar Typhimurium (ST), a leading cause of gastrointestinal infections in humans, effects an overall reduction of lysozyme in the small intestine. To extend this work, we examined small-intestinal tissue sections at various time points after ST infection to quantify and localize expression of lysozyme and assess Paneth cell abundance, apoptosis, and the expression of Paneth cell differentiation markers. In response to infection with ST, the intestinal Paneth cell-specific lysozyme content, the number of lysozyme-positive Paneth cells, and the number of granules per Paneth cell decreased. However, this was accompanied by increases in the total number of Paneth cells and the frequency of mitotic events in crypts, by increased staining for the proliferation marker PCNA, primarily at the crypt side walls where the transit amplifying cell resides and not at the crypt base, and by apoptotic events in villi. Furthermore, we found a time-dependent upregulation of first β-catenin, followed by EphB3, and lastly Sox9 in response to ST, which was not observed after infection with a Salmonella pathogenicity island 1 mutant deficient in type III secretion. Our data strongly suggest that, in response to ST infection, a Paneth cell differentiation program is initiated that leads to an expansion of the Paneth cell population and that the transit amplifying cell is likely the main progenitor responder. Infection-induced expansion of the Paneth cell population may represent an acute intestinal inflammatory response similar to neutrophilia in systemic infection.