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# Spectral Analyses of Ecological Dynamics

Reuman, D.C., **Desharnais, R.A.**, Costantino, R.F., Ahmad, O.S., Cohen, J.E. 2006. Power spectra reveal the influence of stochasticity on nonlinear population dynamics. *Proceedings of the National Academy of Sciences USA* **103**: 18860-18865. [Abstract ] [PDF (468kB)]

**Abstract:**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.

Reuman, D.C., Costantino, R.F., **Desharnais, R.A.**, Cohen, J.E. 2008. Color of environmental noise affects the nonlinear dynamics of cycling, stage-structured populations. *Ecology Letters* **11**: 820-830. [Abstract ] [PDF (468kB)]

**Abstract:**Populations fluctuate because of their internal dynamics, which can be nonlinear and stochastic, and in response to environmental variation. Theory predicts how the colour of environmental stochasticity affects population means, variances and correlations with the environment over time. The theory has not been tested for cycling populations, commonly observed in field systems. We applied noise of different colours to cycling laboratory beetle populations, holding other statistical properties of the noise fixed. Theory was largely validated, but failed to predict observations in sufficient detail. The main period of population cycling was shifted up to 33% by the colour of environmental stochasticity. Noise colour affected population means, variances and dominant periodicities differently for populations that cycled in different ways without noise. Our results show that changes in the colour of climatic variability, partly caused by humans, may affect the main periodicity of cycling populations, possibly impacting industry, pest management and conservation.

**Desharnais, R.A.**, Reuman, D.C., Costantino, R.F., and Cohen, JE. 2018. Temporal scale of environmental correlations affects ecological synchrony. *Ecology Letters* **21**: 1800-1811. [Abstract ] [PDF (1.41MB)]

**Abstract:**Population densities of a species measured in different locations are often correlated over time, a phenomenon referred to as synchrony. Synchrony results from dispersal of individuals among locations and spatially correlated environmental variation, among other causes. Synchrony is often measured by a correlation coefficient. However, synchrony can vary with timescale. We demonstrate theoretically and experimentally that the timescale-specificity of environmental correlation affects the overall magnitude and timescale-specificity of synchrony, and that these effects are modified by population dispersal. Our laboratory experiments linked populations of flour beetles by changes in habitat size and dispersal. Linear filter theory, applied to a metapopulation model for the experimental system, predicted the observed timescale-specific effects. The timescales at which environmental covariation occurs can affect the population dynamics of species in fragmented habitats.

# Nonlinear Population Dynamics

**Desharnais, R.A.**, and Liu, L. 1987. Stable demographic limit cycles in laboratory populations of *Tribolium castaneum*. *Journal of Animal Ecology* **56**: 885-906. [Abstract ] [PDF (2.21MB)]

**Abstract:**(1) We present a general population matrix model in which the age-specific depend upon the age structure of the population. The fecundity and survivorship age-class are assumed to decrease exponentially at rates which depend on the each age-class. We specialize this model to describe the physiological and behavioural interactions among eggs, larvae, pupae and adults in laboratory populations of

*Tribolium*flour beetles. (2) A non-trivial equilibrium age structure exists provided the population can grow without density dependence. If such an equilibrium exists, it is unique. We linearize the model in the neighbourhood of its equilibrium and state the necessary and sufficient conditions for local asymptotic stability. (3) Using several simplifying assumptions, we estimate the parameters of the model using data from our own work and from the literature. With these estimates we predict the existence of an unstable equilibrium age structure. (4) Computer simulations are used to compare the behaviour of the model with census data from experimental populations of

*Tribolium castaneum*. After 70 days of culture, the experimental populations were subjected to demographic perturbations. Both the simulations and the experimental populations exhibit stable oscillations. In general, there is good agreement between the model and the data. (5) We simulated the model using a variety of parameter values. We show how each parameter affects the equilibrium and stability of the model. Increases in the rates of mortality or rates of egg and pupal cannibalism by adults are stabilizing, while high rates of fecundity or egg cannibalism by larvae lead to demographic oscillations. For each parameter, we obtain numerical estimates of the threshold between a stable and unstable point equilibrium. (6) Considering the variation in the rates of survivorship, reproduction, and cannibalism reported in the literature for different species and genetic strains of

*Tribolium*under different environmental conditions, we conclude that laboratory populations of

*Tribolium*can exhibit dynamic behaviours ranging from stable equilibria to demographic limit cycles.

Dennis, B., **Desharnais, R.A.**, Cushing, J.M., and Costantino, R.F. 1995. Nonlinear demographic dynamics: mathematical models, statistical methods, and biological experiments. *Ecological Monographs* **65**: 261-281. [Abstract ] [PDF (2.84MB)]

**Abstract:**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-cycles 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 understanding nonlinear ecological dynamics.

Costantino, R.F., Cushing, J.M., Dennis, B., and **Desharnais, R.A.** 1995. Experimentally induced transitions in the dynamic behavior of insect populations. *Nature* **375**: 227-230. [Abstract ] [PDF (7.22kB)]

**Abstract:**Simple nonlinear models can generate fixed points, periodic cycles and aperiodic oscillations in population abundance without any external environmental variation. Another familiar theoretical result is that shifts in demographic parameters (such as survival or fecundity) can move a population from one of these behaviours to another. Unfortunately, empirical evidence to support these theoretical possibilities is scarce. We report here a joint theoretical and experimental study to test the hypothesis that changes in demographic parameters cause predictable changes in the nature of population fluctuations. Specifically, we developed a simple model describing population growth in the flour beetle

*Tribolium*. We then predicted, using standard mathematical techniques to analyse the model, that changes in adult mortality would produce substantial shifts in population dynamic behaviour. Finally, by experimentally manipulating the adult mortality rate we observed changes in the dynamics from stable fixed points to periodic cycles to aperiodic oscillations that corresponded to the transitions forecast by the mathematical model.

Cushing, J.M., Dennis, B., **Desharnais, R.A.**, and Costantino, R.F. 1996. An interdisciplinary approach to understanding nonlinear ecological dynamics. *Ecological Modelling* **92**: 111-119. [Abstract ] [PDF (1.06MB)]

**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.

Costantino, R.F., **Desharnais, R.A.**, Cushing, J.M., and Dennis, B. 1997. Chaotic dynamics in an insect population. *Science* **275**: 389-391. [Abstract ] [PDF (825kB)]

**Abstract:**A nonlinear demographic model was used to predict the population dynamics of the flour beetle

*Tribolium*under laboratory conditions and to establish the experimental protocol that would reveal chaotic behavior. With the adult mortality rate experimentally set high, the dynamics of animal abundance changed from equilibrium to quasiperiodic cycles to chaos as adult-stage recruitment rates were experimentally manipulated. These transitions in dynamics corresponded to those predicted by the mathematical model. Phase-space graphs of the data together with the deterministic model attractors provide convincing evidence of transitions to chaos.

**Desharnais, R.A.**, Costantino, R.F., Cushing, J.M., and Dennis, B. 1997. Estimating chaos in an insect population. Reply to Perry et al. *Science* **276**: 1881-1882. [PDF (318kB)]

Dennis, B., **Desharnais, R.A.**, Cushing, J.M., and Costantino, R.F. 1997. Transitions in population dynamics: equilibria to periodic cycles to aperiodic cycles. *Journal of Animal Ecology* **66**: 704-729. [Abstract ] [PDF (825kB)]

**Abstract:**1. We experimentally set adult mortality rates,

*μ*, in laboratory cultures of the flour beetle

_{a}*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*= 0.04 treatments in the region of stable equilibria. As adult mortality was increased, there was a transition in the dynamics. At

_{a}*μ*= 0.27 and 0.50 the populations were located in the two-cycle region. With

_{a}*μ*= 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

_{a}*μ*= 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.

_{a}**Desharnais, R.A.** 1997. Population dynamics of *Tribolium*. Pages 303-328 in *Structured Population Models in Marine, Terrestrial, and Freshwater Systems*, S. Tuljapurkar and H. Caswell (editors), Chapman & Hall, New York. [PDF (2.07MB)]

Costantino, R.F., Cushing, J.M., Dennis, B., **Desharnais, R.A.**, and Henson, S.M. 1998. Resonant population cycles in alternating habitats. *Bulletin of Mathematical Biology* **60**: 247-273. [Abstract ] [PDF (252kB)]

**Abstract:**Experiments with the flour beetle

*Tribolium*have revealed that animal numbers were larger in cultures grown in a periodically fluctuating volume of medium than in cultures grown in a constant volume of the same average size. In this paper we derive and analyze a discrete stage-structured mathematical model that explains this phenomenon as a kind of resonance effect. Habitat volume is incorporated into the model by the assumption that all rates of cannibalism (larvae on eggs, adults on eggs and pupae) are inversely proportional to the volume of the culture medium. We tested this modeling assumption by conducting and statistically analyzing laboratory experiments. For parameter estimates derived from experimental data, our model indeed predicts, under certain circumstances, a larger (cycle-average) total population abundance when the habitat volume periodically fluctuates than when the habitat volume is held constant at the average volume. The model also correctly predicts certain phase relationships and transient dynamics observed in data. The analyses involve a thorough integration of mathematics, statistical methods, biological details and experimental data.

Cushing, J.M., Dennis, B., **Desharnais, R.A.**, and Costantino, R.F. 1998. Moving toward an unstable equilibrium: saddle nodes in population systems. *Journal of Animal Ecology* **67**: 298-306. [Abstract ] [PDF (282kB)]

**Abstract:**1. We identify an unstable equilibrium with a two-dimensional stable manifold and a one-dimensional unstable manifold in a three-state variable (larva, pupa, adult) insect population growth model. 2. The saddle node forecasts that the time series of some initial numbers of larvae, pupae and adults are drawn closely to the unstable equilibrium before approaching the asymptotic stable attractor (a two-cycle), while the time series of other initial points are not. 3. Using two quantitative indices, we examine time series from a

*Tribolium*experiment for evidence of the predicted saddle node. We conclude that a saddle node accounts for the transient dynamics in these data and for the differences between the transient behaviour of different replicates of the same experiment.

Cushing, J.M., Costantino, R.F., Dennis, B., **Desharnais, R.A.**, and Henson, S.M. 1998. Nonlinear population dynamics: models, experiments, and data. *Journal of Theoretical Biology* **194**: 1-9. [PDF (168kB)]

Henson, S.M., Cushing, J.M., Costantino, R.F., Dennis, B., and **Desharnais, R.A.** 1998. Phase switching in population cycles. *Proceedings of the Royal Society* **265**: 2229-2234. [Abstract ] [PDF (246kB)]

**Abstract:**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 di¡erent phases of a periodic cycle with distinct attractors. This mechanism accounts for two-cycle phase shifts and the occurrence of asyn-chronous replicates in experimental cultures of

*Tribolium*.

Henson, S. M., Costantino, R. F., Cushing, J. M., Dennis, B., and **Desharnais, R. A.** 1999. Multiple attractors and population dynamics in periodic habitats. *Bulletin of Mathematical Biology* **61**: 1121-1149. [Abstract ] [PDF (485kB)]

**Abstract:**Mathematical models predict that a population which oscillates in the absence of time-dependent factors can develop multiple attracting final states in the advent of periodic forcing. A periodically-forced, stage-structured mathematical model predicted the transient and asymptotic behaviors of

*Tribolium*(flour beetle) populations cultured in periodic habitats of fluctuating flour volume. Predictions included multiple (2-cycle) attractors, resonance and attenuation phenomena, and saddle influences. Stochasticity, combined with the deterministic effects of an unstable ’saddle cycle’ separating the two stable cycles, is used to explain the observed transients and final states of the experimental cultures. In experimental regimes containing multiple attractors, the presence of unstable invariant sets, as well as stochasticity and the nature, location, and size of basins of attraction, are all central to the interpretation of data.

Cushing, J.M., Henson, S.M., **Desharnais, R.A.**, Dennis, B., Costantino, R.F., and King, A. 2001. A chaotic attractor in ecology: theory and experimental data. *Chaos, Solitons, and Fractals* **12**: 219-234. [Abstract ] [PDF (4.53MB)]

**Abstract:**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.

Dennis, B., **Desharnais, R.A.**, Cushing, J.M., Henson, S.M. and Costantino, R.F. 2001. Estimating chaos and complex dynamics in an insect population. *Ecological Monographs* **71**: 277-303. [Abstract ] [PDF (499kB)]

**Abstract:**A defining hypothesis of theoretical ecology during the past century has been that population fluctuations might largely be explained by relatively low-dimensional, nonlinear ecological interactions, provided such interactions could be correctly identified and modeled. The realization in recent decades that such nonlinear interactions might result in chaos and other exotic dynamic behaviors has been exciting but tantalizing, in that attributing the fluctuations of a particular real population to the complex dynamics of a particular mathematical model has proved to be an elusive goal. We experimentally tested a model-predicted sequence of transitions (bifurcations) in the dynamic behavior of a population from stable equilibria to quasiperiodic and periodic cycles to chaos to three-cycles using cultures of the flour beetle

*Tribolium*. The predictions arose from a system of difference equations (the LPA model) describing the nonlinear life-stage interactions, predominantly cannibalism. We built a stochastic version of the model incorporating demographic variability and obtained conditional least-squares estimates for the model parameters. We generated 2000 "bootstrapped data Sets" with a time-series bootstrap technique, and for each set we reestimated the model parameters. The resulting 2000 bootstrapped parameter vectors were used to obtain confidence intervals for the model parameters and estimated distributions of the Liapunov exponents for the deterministic portion (the skeleton) of the model as well as for the full stochastic model. Frequency distributions of estimated dynamic behaviors of the skeleton at each experimental treatment were produced. For one treatment, over 83% of the bootstrapped parameter estimates corresponded to chaotic attractors, and the remainder of the estimates yielded high-period cycles. The low-dimensional skeleton accounted for at least 90% of the variability in the population abundances and accurately described the responses of populations to experimental demographic manipulations, including treatments for which the predicted dynamic behavior was chaos. Demographic stochasticity described the remaining noise quite well. We conclude that the fluctuations of experimental flour beetle populations are explained largely by known nonlinear forces involving cannibalistic-stage interactions. Claims of dynamic behavior such as periodic cycles or chaos must be accompanied by a consideration of the reliability of the estimated parameters and a realization that the population fluctuations are a blend of deterministic forces and stochastic events.

**Desharnais, R.A.**, Dennis, B., Cushing, J.M., Henson, S.M., and Costantino, R.F. 2001. Chaos and population control of insect outbreaks. *Ecology Letters* **4**: 229-235. [Abstract ] [PDF (1.19MB)]

**Abstract:**We used small perturbations in adult numbers to control large fluctuations in the chaotic demographic dynamics of laboratory populations of the flour beetle

*Tribolium castaneum*. A nonlinear mathematical model was used to identify a sensitive region of phase space where the addition of a few adult insects would result in a dampening of the life stage fluctuations. Three experimental treatments were applied: one in which perturbations were made whenever the populations were inside the sensitive region (“in-box treatment”), another where perturbations were made whenever the populations were outside the sensitive region (“out-box treatment”), and an unperturbed control. The in-box treatment caused a stabilization of insect densities at numbers well below the peak values exhibited by the out-box and control populations. This study demonstrates how small perturbations can be used to influence the chaotic dynamics of an ecological system.

Henson, S.M., Costantino, R.F., Cushing, J.M., **Desharnais, R.A.**, and Dennis, B. 2001. Lattice effects observed in chaotic dynamics of experimental populations. *Science* **294**: 602-605. [Abstract ] [PDF (574kB)]

**Abstract:**Animals and many plants are counted in discrete units. The collection of possible values (state space) of population numbers is thus a nonnegative integer lattice. Despite this fact, many mathematical population models assume a continuum of system states. The complex dynamics, such as chaos, often displayed by such continuous-state models have stimulated much ecological research; yet discrete-state models with bounded population size can display only cyclic behavior. Motivated by data from a population experiment, we compared the predictions of discrete-state and continuous-state population models. Neither the discrete- nor continuous-state models completely account for the data. Rather, the observed dynamics are explained by a stochastic blending of the chaotic dynamics predicted by the continuous-state model and the cyclic dynamics predicted by the discrete-state models. We suggest that such lattice effects could be an important component of natural population fluctuations.

Cushing, J.M., Costantino, R.F., Dennis, B., **Desharnais, R.A.**, and Henson, S.M. 2002. *Chaos in Ecology: Experimental Nonlinear Dynamics*. Academic Press, New York.

Henson, S.M., Costantino, R.F., **Desharnais, R.A.**, Cushing, J.M., and Dennis, B. 2002. Basins of attraction: population dynamics with two stable 4-cycles. *Oikos* **98**: 17-24. [Abstract ] [PDF (199kB)]

**Abstract:**We use the concepts of composite maps, basins of attraction, basin switching, and saddle fly-by's to make the ecological hypothesis of the existence of multiple attractors more accessible to experimental scrutiny. Specifically, in a periodically forced insect population growth model we identify multiple attractors, namely, two locally stable 4-cycles. Using the model-predicted basins of attraction, we examine data time series from a

*Tribolium*experiment for evidence of the multiple attractors. We conclude that the multiple attractor hypothesis together with demographic stochasticity accounts for the experimental observations.

King, A.A., **Desharnais, R.A.**, Henson, S.M., Costantino, R.F., Cushing, J.M., and Dennis, B. 2002. Random perturbations and lattice effects in chaotic population dynamics: Reply to Domokos and Scheuring. *Science* **297**: 2163a. [PDF (200kB)]

Dennis, B., **Desharnais, R.A.**, Cushing, J.M., Henson, S.H., and Costantino, R.F. 2003. Can Noise Induce Chaos? *Oikos* **102**: 329-340. [Abstract ] [PDF (136kB)]

**Abstract:**An important component of the mathematical definition of chaos is sensitivity to initial conditions. Sensitivity to initial conditions is usually measured in a deterministic model by the dominant Lyapunov exponent (LE), with chaos indicated by a positive LE. The sensitivity measure has been extended to stochastic models; however, it is possible for the stochastic Lyapunov exponent (SLE) to be positive when the LE of the underlying deterministic model is negative, and vice versa. This occurs because the LE is a long-term average over the deterministic attractor while the SLE is the long-term average over the stationary probability distribution. The property of sensitivity to initial conditions, uniquely associated with chaotic dynamics in deterministic systems, is widespread in stochastic systems because of time spent near repelling invariant sets (such as unstable equilibria and unstable cycles). Such sensitivity is due to a mechanism fundamentally different from deterministic chaos. Positive SLE's should therefore not be viewed as a hallmark of chaos. We develop examples of ecological population models in which contradictory LE and SLE values lead to confusion about whether or not the population fluctuations are primarily the result of chaotic dynamics. We suggest that “chaos” should retain its deterministic definition in light of the origins and spirit of the topic in ecology. While a stochastic system cannot then strictly be chaotic, chaotic dynamics can be revealed in stochastic systems through the strong influence of underlying deterministic chaotic invariant sets.

Edmunds, J., Cushing, J.M., Costantino, R.F., Henson, S.M., Dennis, B., and **Desharnais, R.A.** 2003. Park's *Tribolium* competition experiments: a non-equilibrium species coexistence hypothesis. *Journal of Animal Ecology* **72**: 703-712. [Abstract ] [PDF (278kB)]

**Abstract:**1. In this journal 35 years ago, P. H. Leslie, T. Park and D. B. Mertz reported competitive exclusion data for two

*Tribolium*species. It is less well-known that they also reported ‘difficult to interpret’ coexistence data. We suggest that the species exclusion and the species coexistence are consequences of a stable coexistence two-cycle in the presence of two stable competitive exclusion equilibria. 2. A stage-structured insect population model for two interacting species forecasts that as interspecific interaction is increased there occurs a sequence of dynamic changes (bifurcations) in which the classic Lotka–Volterra-type scenario with two stable competitive exclusion equilibria is altered abruptly to a novel scenario with three locally stable entities; namely, two competitive exclusion equilibria and a stable coexistence cycle. This scenario is novel in that it predicts the competitive coexistence of two nearly identical species on a single limiting resource and does so under circumstances of increased interspecific competition. This prediction is in contradiction to classical tenets of competition theory.

Henson, S.M., King, A.A., Costantino, R.F., Cushing, J.M., Dennis, B., and **Desharnais, R.A.** 2003. Explaining and predicting patterns in stochastic population systems. *Proceedings of the Royal Society of London B* **270**: 1549-1553. [Abstract ] [PDF (247kB)]

**Abstract:**Lattice effects in ecological time–series are patterns that arise because of the inherent discreteness of animal numbers. In this paper, we suggest a systematic approach for predicting lattice effects. We also show that an explanation of all the patterns in a population time–series may require more than one deterministic model, especially when the dynamics are complex.

King, A.A., Costantino, R.F., Cushing, J.M., Henson, S.H., **Desharnais, R.A.**, and Dennis, B. 2004. Anatomy of a chaotic attractor: Subtle model-predicted patterns revealed in population data. *Proceedings of the National Academy of Sciences USA* **101**: 408-413. [Abstract ] [PDF (698kB)]

**Abstract:**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.

**Desharnais, R.A.**, Edmunds, J., Costantino, R.F., and Henson, S.M. 2005. Species competition: uncertainty on a double invariant loop. *Journal of Difference Equations and Applications* **11**: 311-325. [Abstract ] [PDF (200kB)]

**Abstract:**The

*Tribolium*(flour beetle) competition experiments conducted by Park have been highly influential in ecology. We have previously shown that the dynamics of single-species

*Tribolium*populations can be well-described by the discrete-time, 3-dimensional larva–pupa–adult (LPA) model. Motivated by Park's experiments, we explore the dynamics of a 6-dimensional “competition LPA model” consisting of two LPA models coupled through cannibalism. The model predicts a double-loop coexistence attractor, as well as an unstable exclusion equilibrium with a 5-dimensional stable manifold that plays an important role in causing one of the species to go extinct in the presence of stochastic perturbations. We also present a stochastic version of the model, using binomial and Poisson distributions to describe the aggregation of demographic events within life stages. A novel “stochastic outcome diagram,” the stochastic counterpart to a bifurcation diagram, summarizes the model-predicted dynamics of uncertainty on the double-loop. We hypothesize that the model predictions provide an explanation for Park's data. This “stochastic double-loop hypothesis” is accessible to experimental verification.

Costantino, R.F., **Desharnais, R.A.**, Cushing, J.M., Dennis, B., Henson, S.M., and King, A.A. 2005. Nonlinear stochastic population dynamics: The flour beetle *Tribolium* as an effective tool of discovery. Chapter 4 in *Population Dynamics and Laboratory Ecology*, R.A. Desharnais (editor), Academic Press, New York. [Abstract ]

**Abstract:**This chapter discusses the message that in order to understand population fluctuations, deterministic and stochastic forces must be viewed as an integral part of the ecological system. The chapter explains the models, both animal and mathematical, and discusses how model parameters are estimated and the model validated. With the parameterized model in hand, It presents an overview of some of the nonlinear phenomena and related topics that have been documented in the experimental system: chaotic dynamics, population outbreaks, saddle nodes, phase switching, lattice effects, the anatomy of chaos and, finally, mechanistic models of stochasticity. Models of natural systems retain the status of hypotheses and are used only tentatively as building blocks in theories about population abundance patterns.

**Desharnais, R.A.** (editor). 2005. *Population Dynamics and Laboratory Ecology*. Advances in Ecological Research, Volume 37. Academic Press, New York.

**Desharnais, R.A.**, Costantino, R.F., Cushing, J.M., Henson, S.M., Dennis, B, and King, A.A. 2006. Experimental support for the scaling rule of demographic stochasticity. *Ecology Letters* **9**: 537-547. [Abstract ] [PDF (1.02MB)]

**Abstract:**A scaling rule of ecological theory, accepted but lacking experimental confirmation, is that the magnitude of fluctuations in population densities due to demographic stochasticity scales inversely with the square root of population numbers. This supposition is based on analyses of models exhibiting exponential growth or stable equilibria. Using two quantitative measures, we extend the scaling rule to situations in which population densities fluctuate due to nonlinear deterministic dynamics. These measures are applied to populations of the flour beetle

*Tribolium castaneum*that display chaotic dynamics in both 20-g and 60-g habitats. Populations cultured in the larger habitat exhibit a clarification of the deterministic dynamics, which follows the inverse square root rule. Lattice effects, a deterministic phenomenon caused by the discrete nature of individuals, can cause deviations from the scaling rule when population numbers are small. The scaling rule is robust to the probability distribution used to model demographic variation among individuals.

Costantino, R.F., and **Desharnais, R.A.** 2012. Chaos. Pages 126-131 in *Encyclopedia of Theoretical Ecology*, edited by A. Hastings and L.J. Gross, University of California Press.

Henson, S.M., **Desharnais, R.A.**, Funasaki, E.T., Galusha, J.G., Watson, J.W., Hayward, J.L. 2019. Predator-prey dynamics of bald eagles and glaucous-winged gulls at Protection Island, Washington, USA. *Ecology and Evolution* **9**: 3850-3867. [Abstract ] [PDF (1.37MB)]

**Abstract:**1. Bald eagle (

*Haliaeetus leucocephalus*) populations in North America rebounded in the latter part of the twentieth century, the result of tightened protection and outlawing of pesticides such as DDT. An unintended consequence of recovery may be a negative impact on seabirds. During the 1980s, few bald eagles disturbed a large glaucous-winged gull (

*Larus glaucescens*) colony on Protection Island, Washington, USA, in the Salish Sea. Breeding gull numbers in this colony rose nearly 50% during the 1980s and early 1990s. Beginning in the 1990s, a dramatic increase in bald eagle activity ensued within the colony, after which began a significant decline in gull numbers. 2. To examine whether trends in the gull colony could be explained by eagle activity, we fit a Lotka–Volterra-type predator–prey model to gull nest count data and Washington State eagle territory data collected in most years between 1980 and 2016. Both species were assumed to grow logistically in the absence of the other. 3. The model fits the data with generalized R2 = 0.82, supporting the hypothesis that gull dynamics were due largely to eagle population dynamics. 4. Point estimates of the model parameters indicated approach to stable coexistence. Within the 95% confidence intervals for the parameters, however, 11.0% of bootstrapped parameter vectors predicted gull colony extinction. 5. Our results suggest that the effects of bald eagle activity on the dynamics of a large gull colony were explained by a predator–prey relationship that included the possibility of coexistence but also the possibility of gull colony extinction. This study serves as a cautionary exploration of the future, not only for gulls on Protection Island, but for other seabirds in the Salish Sea. Managers should monitor numbers of nests in seabird colonies as well as eagle activity within colonies to document trends that may lead to colony extinction.

# Ecological Genetics

**Desharnais, R.A.**, and Costantino, R.F. 1980. Genetic analysis of a population of *Tribolium*. VII. Stability: Response to genetic and demographic perturbations. *Canadian Journal of Genetics and Cytology* 22: 577-589. [Abstract ]

**Abstract:**A biological counterpart to mathematical stability analysis was demonstrated using the

*Tribolium*model. The responses to deliberate demographic and genetic perturbations of

*T. castaneum*populations, initially homozygous for the corn oil sensitive (cos) allele and near demographic equilibrium, were examined experimentally. An equilibrium was said to be ecologically stable if it was locally stable to perturbations in population numbers alone and genetically stable if it was locally stable to changes in allele frequencies alone. A qualitative stability analysis of the within population age-class interactions indentified several conditions for stability, such as, the product of the rate of egg cannibalism by larvae times the rate of pupal cannibalism by adults must be greater than the product of the "net" fecundity of adults times the overall larval viability. The populations which were subjected to the addition or removal of genetically similar adults or immature life stages were ecologically stable. The cultures altered by the introduction of the + allele were genetically unstable to this perturbation and converged to an average polymorphic cos equilibrium allele frequency of 0.42. The incorporation of the + allele into populations initially homozygous for the cos allele was examined in terms of natural selection in a density-regulated population. The experimental data supported the theoretical prediction that selection is expected to move an ecologically stable system toward an evolutionarily stable strategy resulting in the maximization of adult numbers.

**Desharnais, R.A.**, and Costantino, R.F. 1982. Natural selection and fitness entropy in a density-regulated population. *Genetics* **101**: 317-329. [Abstract ] [PDF (656kB)]

**Abstract:**The entropy

*H*(

*p*

_{0},

*p**) of a population with the initial allele frequency

*p*

_{0}given the equilibrium polymorphic frequency

*p** has been proposed as a measure of natural selection. In the present paper, we have extended this concept to include a particular aspect of density-dependent selection. We compared size trajectory of a population initially at genetic equilibrium,

*N*'(

*t*), with the size trajectories of populations not initially at

*p**,

*N*(

*t*), but which do eventually converge to a common equilibrium allele frequency and equilibrium density,

*N**. The following experimentally-testable hypothesis was established: the total area defined by the difference between the trajectories of

*N*'(

*t*) and

*N*(

*t*) as they converge to

*N** is directly proportional to the fitness entropy when population size is transformed using the density-dependent fitness value. Two properties of this relationship were noted. First, it is independent of the magnitude of natural selection and, secondly, it does not depend upon the initial population density as long as the equilibrium and nonequilibrium populations have the same initial numbers. This hypothesis was evaluated with experimental data on the flour beetle

*Tribolium castaneum*.

**Desharnais, R.A.**, and Costantino, R.F. 1983. Natural selection and density-dependent population growth. *Genetics* **105**: 1029-1040. [Abstract ] [PDF (709kB)]

**Abstract:**Natural selection was studied in the context of density-dependent population growth using a single locus, continuous time model for the rates of change of population size and allele frequency. The maximization principle of density-dependent selection was applied to a class of fitness expressions with explicit recruitment and mortality terms. Three general results were obtained: First, at low population densities, the genetic basis of selection is the difference between the mean recruitment rate and the mean mortality rate. Second, at densities much higher than the equilibrium population size, selection is expected to act to minimize the mean mortality rate. Third, as the population approaches its equilibrium density, selection is predicted to maximize the ratio of the mean recruitment rate to the mean mortality rate.

**Desharnais, R.A.**, Foltz, D.W., and Zouros, E. 1985. Maintenance of genetic polymorphism under conditions of genotype-dependent growth and size-selective mortality. *Canadian Journal of Genetics and Cytology* **27**: 279-288. [Abstract ]

**Abstract:**Associations between heterozygosity at one or more electrophoretically detected enzyme loci and growth rate have been reported for several species of plants and animals, including several commercially important species of finfish and shellfish. The general pattern is for heterozygotes to grow faster than homozygotes, although there is some variation in growth response even within a species. Regardless of the physiological or biochemical basis of genotype-dependent growth, polymorphism at a locus affecting growth rate in an overdominant manner may be lost if larger individuals have a greater mortality rate than smaller ones. In an exploited population, mortality of this sort is likely to result from size-selective fishing pressure. Using a continuous-time single-locus model of natural selection, we have related the maintenance of polymorphism at a locus to two measures of fishing effort: ß, the legal minimum size below which there is no mortality, and f, an instantaneous mortality rate owing to fishing (above the legal minimum size). We considered two different models of fishing mortality. In model 1, fishing mortality above the legal minimum size is constant; in model 2, fishing mortality is a linear function of size (above ß). Numerical analysis of model 1 indicates that maintenance of polymorphism requires either a low rate of fishing mortality or a value of ß that is close to zero or close to the maximum attainable size. Analysis of model 2 gives similar results, suggesting that the conclusions are not dependent on the exact form of the mortality function.Key words: heterozygosity, growth, size, mortality.

**Desharnais, R.A.** 1986. Natural selection, fitness entropy, and the dynamics of coevolution. *Theoretical Population Biology* **30**: 309-340. [Abstract ]

**Abstract:**The coevolutionary dynamics of interacting populations were studied by combining continuous time Lotka-Volterra models of population growth with single-locus genetic models of weak selection. The effects of natural selection on population growth were evaluated using Ginzburg's fitness entropy function as a measure of the deviation of a population's initial allele frequencies from their polymorphic equilibrium values. This entropy measure was used to relate the dynamics of a community composed of evolving populations to the dynamics of a “reference community” whose populations are initially in genetic equilibrium. Specifically, a quantity called the “selective difference area” was defined as the total difference between the population size trajectories of a reference and evolving population. The selective difference area represents the amount of extra life a species would realize if the entire community were at genetic equilibrium. It was shown that this selective difference area is a simple linear function of the initial fitness entropies of each species. This prediction is independent of the strength of selection and holds for any arbitrary set of initial population densities. Numerical examples were presented to illustrate the results. Under the assumption of weak selection, a generalization for arbitrary population growth models was outlined.

**Desharnais, R.A.**, and Cohen, J.E. 1986. Life not lived due to disequilibrium in heterogeneous age-structured populations. *Theoretical Population Biology* **29**: 385-406. [Abstract ]

**Abstract:**Three models of age-structured populations with demographically heterogeneous subpopulations are analyzed. In the first model, each subpopulation has its own age-specific vital rates which are fixed in time. In the second model, the vital rates of each subpopulation are uniformly inhibited by increasing total numbers of individuals. In the third, the vital rates of groups of subpopulations are inhibited by the total numbers of individuals in other groups of subpopulations with an intensity that depends on the interacting pair of groups. Three functions are defined to measure disequilibrium in the subpopulation frequencies, subpopulation age structures, and total population size. For the first model, we show that disequilibrium will shift the trajectory of the total numbers of individuals forward or backward in time by an asymptotic constant that is proportional to the sum of the disequilibrium measures. For the second model, we establish sufficient conditions for the existence of a globally stable equilibrium and we show that disequilibrium will result in a finite loss or gain in life which is proportional to the sum of the disequilibrium measures. For the last model, we show that the loss or gain in life for each group of subpopulations is a linear combination over all groups of the sums of the three disequilibrium measures. We illustrate these results with numerical examples and give possible biological interpretations of the models. We relate these new results to previous work on the cost of natural selection and measures of demographic disequilibrium.

Costantino, R.F., and **Desharnais, R.A.** 1991. *Population Dynamics and the Tribolium Model: Genetics and Demography*. Springer-Verlag, New York.

de Vries, C., **Desharnais R.A.**, Caswell, H. 2020. A matrix model for density-dependent selection in stage-classified populations, with application to pesticide resistance in *Tribolium*. *Ecological Modelling* **416**: 1-12. [Abstract ] [PDF (2.68MB)]

**Abstract:**The study of eco-evolutionary dynamics is based on the idea that ecological and evolutionary processes may operate on the same, or very similar, time scales, and that interactions of ecological and evolutionary processes may have important consequences. Here we develop a model that combines Mendelian population genetics with nonlinear demography to create a truly eco-evolutionary model. We use the vec-permutation matrix approach, classifying individuals by stage and genotype. The demographic component is female dominant and density-dependent. The genetic component includes random mating by stage and genotype, and arbitrary effects of genotype on the demographic phenotype. Mutation is neglected. The result is a nonlinear matrix population model that projects stage × genotype dynamics. We show that the results can include bifurcations of population dynamics driven by the response to selection. We present analytical criteria that determine whether one allele excludes the other or if they persist in a protected polymorphism. The analysis is based on local stability analysis of the homozygous boundary equilibria. As an example, we use a density-dependent stage-classified model of the our beetle

*Tribolium castaneum*. Our model permits arbitrary life-cycle complexity and nonlinearity.

*Tribolium*has developed resistance to the pesticide malathion due to a dominant allele at a single autosomal locus. Using parameters reported from laboratory experiments, we show that the model successfully describes the dynamics of both resistant and susceptible homozygotes, and the outcome of a selection experiment containing both alleles. Stability analysis of the boundary equilibria conrms that the resistant allele excludes the susceptible allele, even in the absence of malathion, agreeing with previously reported results.

# Stochastic Population Growth

Costantino, R. F., and **Desharnais, R.A.** 1981. Gamma distributions of adult numbers for *Tribolium* populations in the region of their steady states. *Journal of Animal Ecology* **50**: 667-681. [Abstract ] PDF (1.47MB)]

**Abstract:**(1) A gamma steady-state probability distribution was established for adult numbers in continuously growing populations of the flour beetle

*Tribolium*. The derivation of the distribution was based on a general stochastic model of population growth with three biological entities: adult inhibition of immatures, pupal productivity, and death-rate among adults. (2) The hypothesis of a gamma steady-state distribution was tested using thirteen observed frequency distributions of adult numbers for

*Tribolium castaneum*and

*Tribolium confusum*. The data, in general, supported the gamma steady-state hypothesis. (3) Using the theoretical gamma probability distribution an attempt was made to explain the

*Tribolium*data and identify testable hypotheses. For instance, the observed differences in the mean and variance of the adult steady-state distributions for the genetic strains of

*T. castaneum*and

*T. confusum*were qualitatively consistent with the theoretical predictions based upon differences in cannibalistic rates, pupal productivity and adult death-rates. The expected effects of stochasticity on the mean and variance were also discussed. (4) The stochastic differential equation of population growth was viewed using the Ito and Stratonovich definitions of stochastic integrals. Both procedures yielded gamma steady-state distributions; however, the conditions for the existence of the stationary densities were different. With stochastic variation in the death-rate among adults, the existence conditions for the Ito calculus was that the magnitude of the stochastic variance must be less than twice the difference between the number of pupae produced per parent and the death-rate among adults. For the stratonovich calculus, the stationary density always existed.

**Desharnais, R.A.**, and Costantino, R.F. 1982. Approach to equilibrium and the steady-state probability distribution of adult numbers in *Tribolium brevicornis*. *American Naturalist* **119**: 102-111. [Abstract ] [PDF (701kB)]

**Abstract:**The rate of population growth in adult numbers, A, for the flour beetle

*Tribolium*was characterized by the mathematical model

*dA*/

*dt*=

*BA*exp(-

*CA*) -

*DA*with the biological entities pupal productivity,

*B*, adult inhibition of the immature life stages,

*C*, and the death rate among the adults,

*D*. A local stability analysis of the equilibrium

*A** = log(

*B*/(

*D*)/

*C*revealed that the eigenvalue

*λ*=

*D*log(

*D*/

*B*) and

*A** was stable if

*B*>

*D*. The time it takes for a perturbation to decay was evaluated using the time constant 1/|

*λ*|. The changes in adult numbers were then viewed as a stochastic birth-death process. The numbers of adults were found to asymptotically assume a constant mean value of

*A** = log(

*B*/

*D*)/

*C*and a constant variance of

*V** = 1/

*C*. Equations were established for the approach of the mean and variance to their respective equilibrium values together with the steady-state probability distribution of adult numbers. Formulas to estimate

*A**,

*B*,

*D*,

*C*, and

*λ*were obtained based on the adult population size data. Experimental observations on

*T. brevicornis*showed a good correspondence to the theoretical construct.

**Desharnais, R.A.**, and Costantino, R.F. 1985. Genetic analysis of a population of *Tribolium*. VIII. The stationary stochastic dynamics of adult numbers. *Canadian Journal of Genetics and Cytology* **27**: 341-350. [Abstract ] [PDF (670kB)]

**Abstract:**A stochastic differential equation model for the rate of change of adult numbers was used to analyze data from two genetically differentiated groups of laboratory populations of the flour beetle,

*Tribolium castaneum*. The first group of populations was homozygous for the corn oil sensitive allele and the other group was polymorphic at this genetic locus. Previous work has shown that mean larval viability is increased in the polymorphic populations. This suggests that the average number of potential recruits was higher in the polymorphic populations when compared to the homozygous ones. The stochastic model was used to derive predictions for the stationary distributions and the serial correlations at demographic equilibrium. These predictions were evaluated using 66 weeks of census data from both groups of populations while adult numbers were fluctuating in the region of their steady states. The data supported the following theoretical predictions: (i) the fluctuations in adult numbers can be described using a gamma probability density function; (ii) increased recruitment in the polymorphic populations results in a larger mean and variance for adult numbers; (iii) the autocorrelation of adult numbers decays exponentially with time; and (iv) increased recruitment in the polymorphic populations results in a faster rate of decay in the autocorrelations. These results suggest that genetically based fitness differences among populations can be reflected in the stationary stochastic dynamics of population size.

**Desharnais, R.A.**, Dennis, B., and Costantino, R.F. 1990. Genetic analysis of a population of *Tribolium*. IX. Maximization of population size and the concept of a stochastic equilibrium. *Genome* **33**: 571-580. [Abstract ] [PDF (795kB)]

**Abstract:**Motivated by the genetic hypothesis that natural selection results in the maximization of the equilibrium population size, the authors quantified this latter equilibrium for laboratory populations of the stored products pest

*Tribolium castaneum*using the gamma probability density function and different frequencies of the maize oil-sensitive (cos) allele. The gamma density function adequately described all observed distributions of adult numbers for different treatments, but, contrary to the theory, statistical comparisons of the fitted distribution indicated that polymorphic populations did not converge to the same identical distribution and that the polymorphic populations were intermediate in population size to the two homozygous groups.

# Spatially-Mediated Dynamics in Marine Benthic Communities

Wilson, W.G., Nisbet, R.M., Ross, A.H., Robles, C., and **Desharnais, R.A.** 1996. Abrupt population changes along smooth environmental gradients. *Bulletin of Mathematical Biology* **58**: 907-922. [Abstract ] [PDF (11.01MB)]

**Abstract:**Populations often exhibit abrupt changes in abundance associated with a smooth, continuous change in some component of their environment, with the abruptness usually attributed to inter-specific interactions or physical extremes. This paper presents a spatially explicit single-species population model in which intra-specific interactions alone are responsible for such an abrupt change. The essential mechanism involves cooperation in both colonization (through enhanced recruitment near other individuals) and mortality (protection through a “safety-in-numbers” interaction). Large fluctuations in population density would likely be observable near the transition region.

Robles, C.D., Alvarado, M.A., and **Desharnais, R.A.** 2001. The shifting balance of of littoral predator-prey interaction in regimes of hydrodynamic stress. *Oecologia* **128**: 142-152. [Abstract ] [PDF (338kB)]

**Abstract:**Above lowshore levels of wave-beaten rocky shores, desiccation from tidal exposure and hydrodynamics stresses from wave action are thought to create refuges from predation, allowing concentrations of sedentary prey such as mussel beds. Underwater time-lapse photography on rocky shores in Southern California revealed that dense aggregations of spiny lobsters prey on mussels during nocturnal high tides. In contradiction of the refuge hypothesis, the densest aggregations occurred on midshore levels of the most wave-exposed site, a semi-protected site showed intermediate densities, and a protected site showed only sparse numbers of lobsters. On wave-beaten shores, the lobsters' high mobility and rapid prey handling allowed them to exploit intertidal prey in the brief period at extreme high tide, when both desiccation and hydrodynamic stresses were at a minimum. The spatial differences in lobster densities were, however, positively related to the recruitment rates of juvenile mussels, the preferred prey. A field experiment demonstrated that predation by lobsters within a mussel bed affects the age/size structure of the bed without changing primary percent coverage. Therefore, concentrations of adult prey on some wave-swept sites appear to result from elevated rates of prey recruitment that surpass rates of predation, rather than absolute refuges from predation.

Robles, C. and **Desharnais, R.A.** 2002. History and current development of a paradigm of predation in rocky intertidal communities. Ecology 83: 1521-1536. [Abstract ] [PDF (407kB)]

**Abstract:**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.

Donalson, D.D., **Desharnais, R.A.**, Robles, C.D., and Nisbet, R. 2003. Spatial dynamics of a benthic community: Applying multiple models to a single system. Pages 429-444 in *Handbook of Scaling Methods in Aquatic Ecology: Measurement, Analysis and Simulation*, edited by L. Seuront and P. Strutton. CRC Press, Boca Raton, Florida. [PDF (1.18MB)]

Robles, C.D., **Desharnais, R.A.**, Garza, C., Donahue, M.J., Martinez, C.A. 2009. Complex equilibria in the maintenance of boundaries: experiments with mussel beds. *Ecology* **90**: 985-995. [Abstract ] [PDF (5.42MB)]

**Abstract:**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.

Robles, C.D., Garza, C., **Desharnais, R.A.**, Donahue, M.J. 2010. Landscape patterns in boundary intensity: a case study of mussel beds. *Landscape Ecology* **25**: 745-759. [Abstract ] [PDF (873kB)]

**Abstract:**This work examines the proposition that positive interactions among neighboring individuals within a population may produce landscape patterns in boundary intensity. The large scale patterns emerge because the interactions favor an aggregated distribution in the face of a potential limiting factor, and the strength of that factor varies over the landscape. The consequences of spatially varying neighborhood processes were explored using cellular automata simulating the structure of mussel beds in 2-dimensional intertidal landscapes, each characterized by a vertical gradient of tidal immersion and a horizontal gradient of wave energy. Running the model with and without the neighborhood processes demonstrated that the facilitating neighborhood processes elevate intensity above that caused by the gradients, and consequently abrupt (high intensity) boundaries emerged in the midst of gradual environmental variation. Trends generated on the 2-D landscape by the model were compared with those in photo-mosaics of intertidal mussel beds, Mytilus californianus on rocky shores of the British Columbia. The analysis involved interpolation of boundary locations using a spatially-constrained cluster algorithm, and then estimation of the corresponding boundary intensities using a landscape index aggregation (CLUMPY). The general similarity between predicted and real trends in intensity over the wave energy gradients suggests that spatially varying neighborhood processes determine much of the landscape scale variation in boundary intensity, while certain discrepancies (e.g. a more rapid rise of observed intensities with increasing wave exposure) suggest modifications of the theory and new empirical work.

Donahue, M.J., **Desharnais, R.A.**, Robles, C.D., Arriola, P. 2011. Mussel bed boundaries as dynamic equilibria: thresholds, phase shifts, and alternative states. *American Naturalist* **178**: 612-625. [Abstract ] [PDF (15.29MB)]

**Abstract:**Ecological thresholds are manifested as a sudden shift in state of community composition. Recent reviews emphasize the distinction between thresholds due to phase shifts-a shift in the location of an equilibrium-and those due to alternative states-a switch between two equilibria. Here, we consider the boundary of intertidal mussel beds as an ecological threshold and demonstrate that both types of thresholds may exist simultaneously and in close proximity on the landscape. The discrete lower boundary of intertidal mussel beds was long considered a fixed spatial refuge from sea star predators; that is, the upper limit of sea star predation, determined by desiccation tolerance, fixed the lower boundary of the mussel bed. However, recent field experiments have revealed the operation of equilibrium processes that maintain the vertical position of these boundaries. Here, we cast analytical and simulation models in a landscape framework to show how the discrete lower boundary of the mussel bed is a dynamic predator-prey equilibrium, how the character of that boundary depends on its location in the landscape, and how boundary formation is robust to the scale of local interactions.

# Other Academic

**Desharnais, R.A.** 1986. The advantages of APL for population modeling. Pages 14-27 in *APL as a Tool of Thought* (D. McCormick and J. Freeman, eds.). Association for Computing Machinery Special Interest Group on APL, New York.

McCabe, J.T., **Desharnais, R.A.**, and Pfaff, D.W. 1989. Graphical and statistical approaches to data analysis for *in situ* hybridization. Pages 822-848 in *Methods in Enzymology, Vol. 168, Hormone Action, Part K: Neuroendocrine Peptides.* P.M. Conn, editor. Academic Press, New York. [Reprinted as pages 107-133 in *Selected Methods in Enzymology, Neuroendocrine Peptides* P.M. Conn, editor. Academic Press, New York.]

McCabe, J.T., Kawata, M., Sano, Y., Pfaff, D.W., and **Desharnais, R.A.** 1990. Quantitative *in situ* hybridization to measure single-cell changes in vasopressin and oxytocin mRNA levels after osmotic stimulation. *Cellular and Molecular Neurobiology* **10**: 59-71. [Abstract ] [PDF (904kB)]

**Abstract:**1. The measurement of cellular mRNA content by quantitative

*in 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.

**Desharnais, R.A.** 1990. Review of *Population Harvesting: Demographic Models of Fish, Forest, and Animal Resources* by W.M. Getz and R.G. Haight. *Quarterly Review of Biology* **65**: 375.

Clark, S.J., and **Desharnais, R.A.** 1998. Honest answers to embarrassing questions: detecting cheating in the randomized response model. *Psychological Methods* **3**: 160-168. [Abstract ] [PDF (660kB)]

**Abstract:**Surveys and questionnaires are frequently used by psychologists, social scientists, and epidemiologists to collect data about behavior, attitudes, emotions, and so on. However, when asked about sensitive topics such as their sexual behavior or illegal activity, some respondents lie or refuse to answer. The randomized response method was developed to reduce these evasive answer biases by guaranteeing subject privacy. However, the method has been criticized as being susceptible to cheaters, that is, respondents who do not answer as directed by the randomizing device. Here the authors show that by splitting the sample into 2 groups and assigning each group a different randomization probability, it is possible to detect whether significant cheating is occurring and to estimate its extent while simultaneously protecting the identity of cheaters and those who may have engaged in sensitive behaviors.

Brown, A., **Desharnais, R.A.**, Roy, B.C., Malik, S., and Gomez, F.A. 2005. Optimization of conditions for flow-through partial-filling affinity capillary electrophoresis to estimate binding constants of ligands to receptors. *Analytica Chimica Acta* **540**: 403-410. [Abstract ] [PDF (418kB)]

**Abstract:**This work details the determination of the minimal injection time of ligand required in flow-through partial-filling affinity capillary electrophoresis (FTPFACE) to estimate binding constants of ligands to receptors. Two model systems are examined in this study: carbonic anhydrase B (CAB, EC 4.2.1.1) and arylsulfonamides, and vancomycin from Streptomyces orientalis and d-Ala-d-Ala peptides. Using CAB, a minimal injection time of 0.07 min at high pressure was determined that provided for the accurate and reproducible measurement of binding constants. In the FTPFACE technique, the capillary is first partially filled with a zone of ligand followed by a sample plug containing receptor and non-interacting standards. Upon application of a voltage the receptor and standards flow into the zone of ligand where a dynamic equilibrium is achieved between receptor and ligand. Continued electrophoresis results in the receptor and standards flowing through the domain of the ligand plug prior to detection. Analysis of the change in the relative migration time ratio (RMTR) of the receptor, relative to the non-interacting standards, as a function of the concentration of ligand, yields a value for the binding constant. In the present study, variable injection times of 4-carboxybenzenesulfonamide (CBSA) were examined to determine the minimal injection time needed to establish an equilibrium between CAB and ligand. A mathematical relationship was derived that correlated injection time and ligand concentration to the change in RMTR and comparisons made between the experimental and calculated values. Binding constants were obtained for a series of arylsulfonamide ligands and d-Ala-d-Ala terminus peptides to CAB and Van, respectively. The results support the use of FTPFACE to estimate affinity constants under variable experimental conditions.

Martinez, J.G., Waldon, M., Huang, Q., Alvarez, S., Oren, A., Sandoval, N., Du, M., Feimeng, Z., Zenz, A., Lohner, K., **Desharnais, R.A.**, Porter, E. 2009. Membrane-targeted synergistic activity of docosahexaenoic acid and lysozyme against *Pseudomonas aeruginosa*. *Biochemical Journal* **419**: 193-200. [Abstract ] [PDF (446kB)]

**Abstract:**Antimicrobial polypeptides, including lysozymes, have membrane perturbing activity and are well-documented effector molecules of innate immunity. In cystic fibrosis, a hereditary disease with frequent lung infection with

*Pseudomonas aeruginosa*, the non-esterified fatty acid DA (docosahexaenoic acid), but not OA (oleic acid), is decreased, and DA supplementation has been shown to improve the clinical condition in these patients. We hypothesized that DA may, either alone or in conjunction with lysozyme, exert antibacterial action against

*Ps. aeruginosa*. We found that DA and lysozyme synergistically inhibit the metabolic activity of Ps. aeruginosa, in contrast with OA. Electron microscopy and equilibrium dialysis suggest that DA accumulates in the bacterial membrane in the presence of lysozyme. Surface plasmon resonance with live bacteria and differential scanning calorimetry studies with bacterial model membranes reveal that, initially, DA facilitates lysozyme incorporation into the membrane, which in turn allows influx of more DA, leading to bacterial cell death. The present study elucidates a molecular basis for the synergistic action of non-esterified fatty acids and antimicrobial polypeptides, which may be dysfunctional in cystic fibrosis.

Lee, J.T., Jansen, M., Yilma, A.N., Nguyen, A., **Desharnais, R.A.**, and Porter, E. 2010. Antimicrobial lipids: novel innate defense molecules are elevated in sinus secretions of patients with chronic rhinosinusitis. *American Journal of Rhinology and Allergy* **24**: 99-104. [Abstract ] [PDF (374kB)]

**Abstract:**

*Background:*Airway secretions possess intrinsic antimicrobial properties that contribute to the innate host defense of the respiratory tract. These microbicidal capabilities have largely been attributed to the presence of antibacterial polypeptides. However, recent investigation has indicated that host-derived lipids including cholesteryl esters also exhibit antimicrobial properties. The purpose of this study was to determine whether sinus secretions contain such antimicrobial lipids and to compare the lipid composition in patients with and without chronic rhinosinusitis (CRS).

*Methods:*Maxillary sinus fluid was obtained via antral lavage from subjects with (seven patients) and without (nine patients) a history of CRS. After specimen collection, total lipid was extracted according to Bligh and Dyer (Bligh EG and Dyer WJ, A rapid method of total lipid extraction and purification, Can J Biochem Physiol 37:911–918, 1959) and lipid profiles were obtained by reverse phase high-performance liquid chromatography on an amide-embedded C18 column. In addition, the neutrophil-specific antimicrobial peptides human neutrophil peptides 1–3 (HNP1–3) were quantified by Western immunoblotting.

*Results:*Lipids, including cholesteryl esters, were identified in the maxillary sinus secretions of patients with and without CRS. However, levels of lipid composition differed between the two groups with CRS patients exhibiting greater amounts of all classes of lipids, reaching over 10-fold higher concentration when compared with non-CRS patients. This increase was independent of HNP1–3 content.

*Conclusion:*Sinus secretions of patients with CRS appear to show elevated levels of antimicrobial lipids compared with controls independent from neutrophil influx. This up-regulation suggests that host-derived lipids act as mediators of mucosal immunity in CRS. Further study is necessary to determine if such antimicrobial lipids function alone or synergistically with antibacterial peptides in conferring such inherent microbicidal properties.

Rodriguez, N.R., Eloi, M. D., Huynh, A., Dominguez, T., Cheung Lam, A. H., Molina-Carcamo, D., Naser, Z., **Desharnais, R.A.**, Salzman, N. H., and Porter, E. 2012. Expansion of paneth cell in response to enteric *Salmonella enterica* serovar Typhimurium infection. *Infection and Immunity* **80**: 266-275. [Abstract ] [PDF (1.26MB)]

**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 ß-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.

Kiwata, J., Anouseyan, R., **Desharnais, R.A.**, Cornwell, A., Khodiguian, N., and Porter, E. 2014. Effects of aerobic exercise on lipid-effector molecules of the innate immune response. *Medicine & Science in Sports & Exercise* **46**: 506-512. [Abstract ] [PDF (208kB)]

**Abstract:**

*INTRODUCTION:*Consistent, moderate-to-vigorous intensity exercise has been associated with a lower risk of upper respiratory tract infection (URI). However, the molecular basis for this apparent protection has not yet been fully resolved. Host-derived lipids such as cholesteryl esters (CEs) have emerged as important effector molecules of innate defense against infections. Here, we compared antimicrobial CEs in nasal fluid before and after moderate-to-vigorous exercise between active and inactive subjects.

*METHODS:*Nasal fluid was collected from fourteen healthy, recreationally-active subjects (32 ± 11 yr, 7 males, 7 females) and 14 healthy, inactive subjects (25 ± 3 yr, 7 males, 7 females) before and after treadmill exercise at 70% heart rate reserve. Nasal fluid was analyzed for lysozyme, cholesteryl linoleate (CL), cholesteryl arachidonate (CA), and albumin (Alb) concentrations.

*RESULTS:*Baseline concentrations (means ± SEM, inactive vs. active) of lysozyme (117.7±31.1 µg/mL vs. 122.9±15.5 µg/mL), CL+CA (15.3±1.8 µg/mL vs.26.2±10.05 µg/mL), and albumin (156.6± 54.5µg/mL vs.126.9±32.8 µg/mL) were similar to previously reported levels and did not differ significantly between study groups. However, post-exercise, CL+CA concentration was significantly lower in inactive compared to active subjects (7.8 ± 1.5 µg/mL vs. 20.1 ± 4.8 µg/mL, p = 0.036) dropping below the antimicrobial effective range. Once adjusted to albumin concentrations the changes were no longer significant, suggesting that plasma transudation accounted for the increased CA+CL concentration post-exercise in the active group relative to the inactive group.

*CONCLUSION:*Moderate-to-vigorous aerobic exercise acutely decreases the antimicrobial CE response in inactive subjects, but does not modify baseline levels of CEs between active and inactive subjects. This suggests that compared to active individuals, inactive individuals may be at greater risk for URI immediately post-exercise.

Ma, D.C. , Yoon, A.J., Faull, K.F., **Desharnais, R.A.** , Zemanick, E.T., and Porter, E. 2015. Cholesteryl esters are elevated in the lipid fraction of bronchoalveolar lavage fluid collected from pediatric cystic fibrosis patients. *PLOS ONE* (DOI: 10.1371/journal.pone.0125326). [Abstract ] [PDF (585kB)]

**Abstract:**

*Background:*Host-derived lipids including cholesteryl esters (CEs) such as cholesteryl linoleate have emerged as important antibacterial effectors of innate immunity in the airways and cholesteryl linoleate has been found elevated in the context of inflammation. Cystic fibrosis (CF) patients suffer from chronic infection and severe inflammation in the airways. Here, we identified and quantified CEs in bronchoalveolar lavage fluid (BALF) from CF patients and non-CF disease controls, and tested whether CE concentrations are linked to the disease.

*Materials and Methods:*CEs in BALF from 6 pediatric subjects with CF and 7 pediatric subjects with non-CF chronic lung disease were quantified by mass spectral analysis using liquid chromatography coupled with tandem mass spectrometry and multiple reaction monitoring. BALFs were also examined for total lipid, total protein, albumin, and, as a marker for inflammation, human neutrophil peptide (HNP) 1–3 concentrations. Statistical analysis was conducted after log 10 transformation of the data.

*Results:*Total lipid/protein ratio was reduced in CF BALF (p = 0.018) but the concentrations of CEs, including cholesteryl linoleate, were elevated in the total lipid fraction in CF BALF compared to non-CF disease controls (p < 0.050). In addition, the concentrations of CEs and HNP1-3 correlated with one another (p < 0.050).

*Conclusions:*The data suggests that the lipid composition of BALF is altered in CF with less total lipid relative to protein but with increased CE concentrations in the lipid fraction, likely contributed by inflammation. Future longitudinal studies may reveal the suitability of CEs as a novel biomarker for CF disease activity which may provide new information on the lipid mediated pathophysiology of the disease.

Apodaca, J.C., **Desharnais, R.A.**, Mitchell, L.J. 2021. The effect of the Safer at Home order on the frequency of DUI breath alcohol tests in Los Angeles County. *Journal of Forensic Sciences.* doi.org/10.1111/1556-4029.14687. [Abstract ] [PDF (835kB)]

**Abstract:**With Los Angeles County having a population size of just over 10 million and an additional 471,000 people who commute into Los Angeles County for employment, many drivers are at risk of being injured or killed in an alcohol-impaired driving collision. On March 19, 2020, the County of Los Angeles issued the Safer at Home order as a result of the COVID-19 pandemic. This curtailed driving and decreased the number of breath alcohol tests that were conducted in Los Angeles County. The number of breath tests conducted in January–February of 2019 and 2020 and March–April of 2019 and 2020 were evaluated using Fisher's exact test and analysis of variance. There was a statistically significant decrease in the overall number of breath tests conducted in Los Angeles County in March–April of 2020. There was also a significant decrease in the number of collisions where DUI was a factor. Accounting for changes in traffic volumes, the number of breath tests per vehicle miles driven also decreased significantly. Since the Safer at Home order closed all non-essential services such as bars and restaurants, there is indirect data on the relative contribution of liquor-serving establishments, and to some extent large social gatherings, to the incidence of drunk driving. Taking into account traffic volume, it was determined that the odds of encountering an intoxicated driver decreased by approximately 23% during the Safer at Home period. This information could help policy-makers determine the likely effectiveness of various countermeasures to prevent drunk driving.

# Teaching and Curriculum Development

**Desharnais, R.A.**, and Jefferson, M. 1994. *Lecture Notes for Principles of Biology III.* Burgess Publishing, Edina, Minnesota.

Novak, G.A., and **Desharnais, R.A.** 1994. Integrating the electronic desktop into the science curriculum. *Selected Papers from the National Conference on Teaching and Learning* **5**: 121-127.

Novak, G.A., and **Desharnais, R.A.** 1998. Virtual courseware for science education. *Selected Papers from the Ninth International Conference on College Teaching and Learning* **9**: 99-107.

**Desharnais, R.A.**, and Novak, G.A. 1998. Virtual courseware for science education. *Syllabus* **12**: 54-60.

**Desharnais, R.A.** 1999. Learning by doing with Biology Labs On-Line. *Strategies for Success* **31**: 4-5.

Palladino, M.A., **Desharnais, R.A.**, and Bell, J. 2001. *Student Lab Manual for Biology Labs On-Line*. Benjamin Cummings, San Francisco.

Palladino, M.A., **Desharnais, R.A.**, and Bell, J. 2001. *Instructor's Lab Manual for Biology Labs On-Line*. Benjamin Cummings, San Francisco.

**Desharnais, R.A.**, and Escorza-Treviño, S. 2003. *Lab Manual for Biometrics.* California State University, Los Angeles.

Limson, M., Witzlib, C., and **Desharnais, R.A.** 2007. Using web-based simulations to promote inquiry. *ScienceScope* **30**: 36-42. [PDF (303kB)]

**Desharnais, R.A.**¸and Limson, M. 2007. Designing and implementing Virtual Courseware to promote inquiry-based learning. *Journal of Online Learning and Teaching* **3**: 30-39. [Abstract ] [PDF (1.63MB)]

**Abstract:**Web-based learning objects continue to evolve as technological advances enhance our ability to create and share highquality learning resources. An important class of learning objects are simulations intended to supplement traditional science instruction. After several years of experience in this endeavor, the Virtual Courseware Project has arrived at a set of ten design principles that it uses to guide its development of new webbased learning activities. These guiding principles place an emphasis on educational standards, openended inquirybased learning, scientific methodology, critical thinking, and an intuitive and interactive user interface that includes linear tours, assessment tools, and documentation. These design principles are exemplified in Drosophila, an activity for learning the genetics of inheritance.

Narguizian, P. and **Desharnais, R.A.** 2012. Virtual Courseware: web-based simulations for promoting inquiry-based teaching and learning. *California Classroom Science*: Web link.

Son, J., Narguizian, P.¸ Beltz, D., and **Desharnais, R.A.** 2016. Comparing physical, virtual, and hybrid flipped Labs for general education biology. *Online Learning* **20**: 228-243. [Abstract ] [PDF (217kB)]

**Abstract:**The purpose of this study was to examine the impact on learning, attitudes, and costs in a redesigned general education undergraduate biology course that implemented web-based virtual labs (VLs) to replace traditional physical labs (PLs). Over an academic year, two new modes of VL instruction were compared to the traditional PL offering: (1) all VL with an in-person help center (VL-A) and (2) a hybrid flipped VL model where online labs alternated with in-person labs every week (VL-H). All three lab types included a face-to-face lecture with the same materials. Engaging inquiry-based exercises were developed for each VL activity in which students were provided background information, guided through a series of basic experiments, encouraged to design their own experiments, and required to produce a simple scientific report that was delivered for evaluation electronically. The VL-A group had the highest proportion of repeatable grades (below a C, 2.0 grade points). Students in the VL-H group achieved significantly better grades compared to the other lab instruction groups. The VL-H group also experienced statistically significant favorable shifts in their self-reported attitudes towards biology. The personnel costs for the VL-A and VL-H models were 29% and 63% of the PL model, respectively, allowing more sections to be offered. These results suggest that carefully designed online lab opportunities can result in higher student grades and more favorable attitudes towards science while reducing costs compared to traditional labs.

# Poetry, Essays, & Short Stories

Romanko, K.A., and **Desharnais, R.A.** 1990. Confessions of a technophile (humor). *ComputorEdge* **8**: 56-58.

**Desharnais, R.A.** 1996. Strange attractors: chaos theory as a catalyst for the collaboration of science and art (essay). *Collaborative Inquiry in a Postmodern Era: A Cat's Cradle* **2**: 27-35.

Romanko, K.A., and **Desharnais, R.A.** 2005. The Quantum Casino (poem). *Astropoetica*, Issue. 3.3, Fall.

Romanko, K.A., and **Desharnais, R.A.** 2006. Damnodynamics: The Science of Hell (poem). *Poe Little Thing*, Issue 4, Winter.

**Desharnais, R.A.** 2009. “The Shadow Dancer.” Short story in *Cinema Spec: Tales of Hollywood and Fantasy*, edited by Karen A. Romanko. Raven Electrick Ink, Los Angeles.