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Mark Lewis

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Mark Lewis

Department of Mathematical and Statistical Sciences/Biological Sciences - Lewis Research Group

Department of Mathematical and Statistical Sciences
University of Alberta
632 Central Academic Building
Edmonton, AB T6G 2G1
Phone: 780-492-0197

  • Senior Canada Research Chair in Mathematical Biology

http://hdl.handle.net/10402/era.29483

Subject areas and related deposits

  • Aggregation

    • A nonlocal continuum model for biological aggregation.

      We construct a continuum model for biological aggregations in which individuals experience long-range social attraction and short range dispersal. For the case of one spatial dimension, we study the steady states analytically and numerically. There exist strongly nonlinear states with compact support and steep edges that correspond to localized biological aggregations, or clumps. These steady state clumps are approached through a dynamic coarsening process. In the limit of large population size, the clumps approach a constant density swarm with abrupt edges. We use energy arguments to understand the nonlinear selection of clump solutions, and to predict the internal density in the large population limit. The energy result holds in higher dimensions as well, and is demonstrated via numerical simulations in two dimensions.

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  • Allee effect

    • Dispersal data and the spread of invading organisms

      Models that describe the spread of invading organisms often assume that the dispersal distances of propagules are normally distributed. In contrast, measured dispersal curves are typically leptokurtic, not normal. In this paper, we consider a class of models, integrodifference equations, that directly incorporate detailed dispersal data as well as population growth dynamics. We provide explicit formulas for the speed of invasion for compensatory growth and for different choices of the propagule redistribution kernel and apply these formulas to the spread of D. pseudoobscura. We observe that: (1) the speed of invasion of a spreading population is extremely sensitive to the precise shape of the redistribution kernel and, in particular, to the tail of the distribution; (2) fat-tailed kernels can generate accelerating invasions rather than constant-speed travelling waves; (3) normal redistribution kernels (and by inference, many reaction-diffusion models) may grossly underestimate rates of spread of invading populations in comparison with models that incorporate more realistic leptokurtic distributions; and (4) the relative superiority of different redistribution kernels depends, in general, on the precise magnitude of the net reproductive rate. The addition of an Allee effect to an integrodifference equation may decrease the overall rate of spread. An Allee effect may also introduce a critical range; the population must surpass this spatial threshold in order to invade successfully. Fat-tailed kernels and Allee effects provide alternative explanations for the accelerating rates of spread observed for many invasions.

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  • Allee Effects

    • Allee effects, invasion pinning, and species' borders

      All species’ ranges are the result of successful past invasions. Thus, models of species’ invasions and their failure can provide insight into the formation of a species’ geographic range. Here, we study the properties of invasion models when a species cannot persist below a critical population density known as an “Allee threshold.” In both spatially continuous reaction-diffusion models and spatially discrete coupled ordinary-differential equation models, the Allee effect can cause an invasion to fail. In patchy landscapes (with dynamics described by the spatially discrete model), range limits caused by propagation failure (pinning) are stable over a wide range of parameters, whereas, in an uninterrupted habitat (with dynamics described by a spatially continuous model), the zero velocity solution is structurally unstable and thus unlikely to persist in nature. We derive conditions under which invasion waves are pinned in the discrete space model and discuss their implications for spatially complex dynamics, including critical phenomena, in ecological landscapes. Our results suggest caution when interpreting abrupt range limits as stemming either from competition between species or a hard environmental limit that cannot be crossed: under a wide range of plausible ecological conditions, species’ ranges may be limited by an Allee effect. Several example systems appear to fit our general model.

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  • Animal movement

    • Home range analysis using a mechanistic home range model

      The traditional models used to characterize animal home ranges have no mechanistic basis underlying their descriptions of space use, and as a result, the analysis of animal home ranges has primarily been a descriptive endeavor. In this paper, we characterize coyote (Canis latrans) home range patterns using partial differential equations for expected space use that are formally derived from underlying descriptions of individual movement behavior. To our knowledge, this is the first time that mechanistic models have been used to characterize animal home ranges. The results provide empirical support for a model formulation of movement response to scent marks, and suggest that having relocation data for individuals in adjacent groups is necessary to capture the spatial arrangement of home range boundaries. We then show how the model fits can be used to obtain predictions for individual movement and scent marking behavior and to predict changes in home range patterns. More generally, our findings illustrate how mechanistic models permit the development of a predictive theory for the relationship between movement behavior and animal spatial distribution.

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  • Aquaculture

    • Allee effects may slow the spread of parasites in a coastal marine ecosystem

      Allee effects are thought to mediate the dynamics of population colonization, particularly for invasive species. However, Allee effects acting on parasites have rarely been considered in the analogous process of infectious disease establishment and spread. We studied the colonization of uninfected wild juvenile Pacific salmon populations by ectoparasitic salmon lice (Lepeophtheirus salmonis) over a 4-year period. In a data set of 68, 376 fish, we observed 85 occurrences of precopular pair formation among 1, 259 preadult female and 613 adult male lice. The probability of pair formation was dependent on the local abundance of lice, but this mate limitation is likely offset somewhat by mate-searching dispersal of males among host fish. A mathematical model of macroparasite population dynamics that incorporates the empirical results suggest a high likelihood of a demographic Allee effect, which can cause the colonizing parasite populations to die out. These results may provide the first empirical evidence for Allee effects in a macroparasite. Furthermore, the data give a rare detailed view of Allee effects in colonization dynamics and suggest that Allee effects may dampen the spread of parasites in a coastal marine ecosystem.

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    • Fish farms, parasites, and predators: implications for salmon population dynamics

      For some salmon populations, the individual and population effects of sea lice (Lepeophtheirus salmonis) transmission from sea cage salmon farms is probably mediated by predation, which is a primary natural source of mortality of juvenile salmon. We examined how sea lice infestation affects predation risk and mortality of juvenile pink (Oncorhynchus gorbuscha) and chum (O. keta) salmon, and developed a mathematical model to assess the implications for population dynamics and conservation. A risk-taking experiment indicated that infected juvenile pink salmon accept a higher predation risk in order to obtain foraging opportunities. In a schooling experiment with juvenile chum salmon, infected individuals had increased nearest-neighbor distances and occupied peripheral positions in the school. Prey selection experiments with cutthroat trout (O. clarkii ) predators indicated that infection reduces the ability of juvenile pink salmon to evade a predatory strike. Group predation experiments with coho salmon (O. kisutch) feeding on juvenile pink or chum salmon indicated that predators selectively consume infected prey. The experimental results indicate that lice may increase the rate of prey capture but not the handling time of a predator. Based on this result, we developed a mathematical model of sea lice and salmon population dynamics in which parasitism affects the attack rate in a type II functional response. Analysis of the model indicates that: (1) the estimated mortality of wild juvenile salmon due to sea lice infestation is probably higher than previously thought; (2) predation can cause a simultaneous decline in sea louse abundance on wild fish and salmon productivity that could mislead managers and regulators; and (3) compensatory mortality occurs in the saturation region of the type II functional response where prey are abundant because predators increase mortality of parasites but not overall predation rates. These findings indicate that predation is an important component of salmon–louse dynamics and has implications for estimating mortality, reducing infection, and developing conservation policy.

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  • Aquatic invasions

    • Models of lake invasibility by Bythotrephes longimanus, a non-indigenous zooplankton.

      We built a family of hierarchical risk models for the spread of invasions by the spiny waterflea (Bythotrephes longimanus) in lakes in Ontario, Canada. Knowledge of covariates determining lake invasibility and ability to predict risk of future invasions may help to develop management policy and slow the invasions in the future. The models are based on two component submodels. The first component was a stochastic gravity submodel for the propagule pressure between lakes via recreational boaters. The second component was a submodel for establishment risk, given that the invader has already been introduced to a lake. This component was a logistic regression model, incorporating up to 17 measured covariates that describe the physical and chemical condition of the lake. Variants of the risk model, each incorporating different subsets of the covariates, were calibrated using presence/absence data from a 300-lake survey conducted in 2005–2006 by the Canadian Aquatic Invasive Species Network (CAISN). The predictive capacity of the best model was high, giving AUC values close to 0.94. Of the model covariates considered, the most important predictors of existing invasions were propagule pressure and lake pH, and, to lesser extents, phosphorus (P) and lake elevation. Our fitting of the propagule pressure submodel demonstrated a significant Allee effect for Bythotrephes. Our development of the establishment risk predictor showed that it is essential to account for temporal variability in lake physico-chemistry. We demonstrated that invasions of lake networks by the spiny waterflea follow highly predictable patterns which can be understood with a properly calibrated, hierarchical risk model.

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  • Aquatic organisms

    • Effects of Heterogeneity on Spread and Persistence in Rivers

      The question how aquatic populations persist in rivers when individuals are constantly lost due to downstream drift has been termed the “drift paradox.” Recent modeling approaches have revealed diffusion-mediated persistence as a solution. We study logistically growing populations with and without a benthic stage and consider spatially varying growth rates. We use idealized hydrodynamic equations to link river cross-sectional area to flow speed and assume heterogeneity in the form of alternating patches, i.e., piecewise constant conditions. We derive implicit formulae for the persistence boundary and for the dispersion relation of the wave speed. We explicitly discuss the influence of flow speed, cross-sectional area and benthic stage on both persistence and upstream invasion speed.

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  • Ballast water

    • A mechanistic model for understanding invasions: using the environment as a predictor of population success

      Aim We set out to develop a temperature-and salinity-dependent mechanistic population model for copepods that can be used to understand the role of environmental parameters in population growth or decline. Models are an important tool for understanding the dynamics of invasive species; our model can be used to determine an organism's niche and explore the potential for invasion of a new habitat. Location Strait of Georgia, British Columbia, Canada. Methods We developed a birth rate model to determine the environmental niche for an estuarine copepod. We conducted laboratory experiments to estimate demographic parameters over a range of temperatures and salinities for Eurytemora affinis collected from the Nanaimo Estuary, British Columbia (BC). The parameterized model was then used to explore what environmental conditions resulted in population growth vs. decline. We then re-parameterized our model using previously published data for E. affinis collected in the Seine Estuary, France (SE), and compared the dynamics of the two populations. Results We established regions in temperature-salinity space where E. affinis populations from BC would likely grow vs. decline. In general, the population from BC exhibited positive and higher intrinsic growth rates at higher temperatures and salinities. The population from SE exhibited positive and higher growth rates with increasing temperature and decreasing salinity. These different relationships with environmental parameters resulted in predictions of complex interactions among temperature, salinity and growth rates if the two subspecies inhabited the same estuary. Main conclusions We developed a new mechanistic model that describes population dynamics in terms of temperature and salinity. This model may prove especially useful in predicting the potential for invasion by copepods transported to Pacific north-west estuaries via ballast water, or in any system where an ecosystem is subject to invasion by a species that shares demographic characteristics with an established (sub) species.

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  • Bayes

    • Estimating Population Spread: What Can We Forecast and How Well?

      Recent literature on plant population spread advocates quantification of long-distance dispersal (LDD). These estimates could provide insights into rates of migration in response to climate change and rates of alien invasions. LDD information is not available for parameterization of current models because it is hard to obtain. We combine a new stochastic model with a flexible framework that permits assimilation of evidence that might be derived from a range of sources. Results are consistent with the prediction of traditional diffusion that population spread has a finite asymptotic velocity. Unlike traditional diffusion, spread is not well described by the mean; it is erratic. In contrast with deterministic models, our results show that inherent uncertainty, rather than parameter sensitivity, thwarts informative forecasts of spread velocity. Analysis shows that, because LDD is inherently unpredictable, even full knowledge of LDD parameters might not provide informative estimates of velocity for populations characterized by LDD. Although predictive distributions are too broad to provide precise estimates of spread rate, they are valuable for comparing spread potential among species and for identifying potential for invasion. Using combinations of dispersal data and the estimates provided by dispersal biologists that derive from multiple sources, the model predicts spread rates that are much slower than those from traditional (deterministic) fat-tailed models and from simulation models of spread, but for different reasons. Deterministic fat-tailed models overestimate spread rate, because they assume that fractions of individuals can rapidly occupy distant sites. Stochastic models recognize that distant colonization is limited to discrete individuals. Stochastic simulations of plant migration overestimate migration of trees, because they typically assume values of R-0 that are too large.

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  • Biocontrol

    • When can herbivores slow or reverse the spread of an invading plant? A test case from Mount St. Helens

      Here we study the spatial dynamics of a coinvading consumer-resource pair. We present a theoretical treatment with extensive empirical data from a long-studied field system in which native herbivorous insects attack a population of lupine plants recolonizing a primary successional landscape created by the 1980 volcanic eruption of Mount St. Helens. Using detailed data on the life history and interaction strengths of the lupine and one of its herbivores, we develop a system of integrodifference equations to study plant-herbivore invasion dynamics. Our analyses yield several new insights into the spatial dynamics of coinvasions. In particular, we demonstrate that aspects of plant population growth and the intensity of herbivory under low-density conditions can determine whether the plant population spreads across a landscape or is prevented from doing so by the herbivore. In addition, we characterize the existence of threshold levels of spatial extent and/or temporal advantage for the plant that together define critical values of “invasion momentum,” beyond which herbivores are unable to reverse a plant invasion. We conclude by discussing the implications of our findings for successional dynamics and the use of biological control agents to limit the spread of pest species.

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  • Biological invasions

    • Allee effect and control of lake system invasion.

      We consider the model of invasion prevention in a system of lakes that are connected via traffic of recreational boats. It is shown that, in presence of an Allee effect, the general optimal control problem can be reduced to a significantly simpler stationary optimization problem of optimal invasion stopping. We consider possible values of model parameters for zebra mussels. The general N-lake control problem has to be solved numerically, and we show a number of typical features of solutions: distribution of control efforts in space and optimal stopping configurations related with the clusters in lake connection structure.

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    • Minimizing invasion risk by reducing propagule pressure: a model for ballast-water exchange

      Biological invasions are a major and increasing agent of global biodiversity change. Theory and practice indicate that invasion risk can be diminished by reducing propagule pressure, or the quantity, quality, and frequency of introduced individuals. For aquatic invasions, the primary global invasion pathway is ballast-water transport, and the primary risk reduction strategy is currently open-ocean exchange. Exchange was developed with shipping between freshwater ports in mind, but the majority of shipping connects brackish and marine ports. A worldwide convention, adopted in 2004 by the International Maritime Organization, now mandates ballast-water exchange (or equivalent management) for its 164 member states. Will exchange be as effective in reducing invasion risk for euryhaline species (those capable of tolerating a wide range of salinity levels) in saltwater ports? Here we develop a simple mathematical framework for optimizing ballast-water exchange in terms of exchange level, timing, and species salinity tolerance. Our model shows that when species survival is worse in the post-exchange than in the pre-exchange water, exchange is always effective. However, when survival is equal or better following exchange, a critical level and timing are required for effective exchange. We illustrate the model’s applications with a variety of introduced marine and estuarine organisms.

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    • Prediction and error in multi-stage models for spread of aquatic non-indigenous species.

      Aim  Predictions of spread of non-indigenous species allow for greater efficiency in managing invasions by targeting areas for preventative measures. The invasion sequence is a useful concept in predictions of spread, as it allows us to test hypotheses about the transport and establishment of propagules in novel habitats. Our aims are twofold: (1) to develop and validate multi-stage invasion models for the introduced fishhook waterflea, Cercopagis pengoi, and (2) to assess how variability in the transport patterns of the propagules influences the accuracy and spatial extent for predictions of spread. Location  New York State, USA. Methods  We developed a two-stage model for the spread of C. pengoi. First, we developed a stochastic gravity model for dispersal based on surveys of recreational boat traffic in New York State as a proxy for propagule pressure. We then modelled the probability of establishment based on predicted levels of propagule pressure and measures of lakes’ physicochemistry. In addition, we used Monte Carlo simulations based on the gravity model to propagate variability in boater traffic through the establishment model to assess how uncertainty in dispersal influenced predictions of spread. Results  The amount recreationalists were willing to spend, lake area and population size of the city nearest to the destination lake were significant factors affecting boater traffic. In turn, boater traffic, lake area, specific conductance and turbidity were significant predictors of establishment. The inclusion of stochastic dispersal reduced the rate of false positives (i.e. incorrect prediction of an invasion) in detecting invasions at the upper 95% prediction interval for the probability of establishment. Main conclusions  Combinations of measures of propagule pressure, habitat suitability and stochastic dispersal allow for the most accurate predictions of spread. Further, multi-stage spread models may overestimate the extent of spread if stochasticity in early stages of the models is not considered.

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  • Biomaterials

    • The Mechanics of Lung Tissue under High-Frequency Ventilation

      High-frequency ventilation isa radical departure from conventional lung ventilation, with frequenciesgreater than 2Hz, and volumesp er breath much smaller than the anatomical deadspace. Its use has been shown to benefit premature infants and patients with severe respiratory distress, but a vital question concerns ventilator-induced damage to the lung tissue, and a clear protocol for the most effective treatment has not been identified. Mathematical modeling can help in understanding the mechanical effects of lung ventilation, and hence in establishing such a protocol. In this paper we describe the use of homogenization theory to predict the macroscopic behavior of lung tissue based upon the three dimensional microstructure of respiratory regions, making the simplifying assumption that the microstructure is periodic. This approach yields equations for macroscopic air flow, pressure, and tissue deformation, with parameters which can be determined from a specification of the tissue microstructure and its material properties. We are able to include an alternative hypothesis as to the dependence of lung tissue shear viscosity on the frequency of forcing, known as the structural damping hypothesis. We then show how, if we consider isotropic tissue, the parameters determining the macroscopic response of the tissue can be estimated from bulk measurements. Finally, we consider the solutions of the macroscopic system when we consider variations in just one spatial dimension. In particular, we demonstrate that the structural damping hypothesis leads to markedly different solution behavior.

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  • Bythotrephes

    • The spread, establishment and impacts of the spiny water flea, Bythotrephes longimanus, in temperate North America: a synopsis of the special issue.

      More than most sub-disciplines of ecology, the study of biological invasions is characterized by breadth rather than by depth. Studies of expanding ranges of invaders are common, as are post-invasion case studies, but we rarely have a deep understanding of the dynamics and regulators of the processes of invasion and resultant ecological transformations. This is unfortunate because such depth may well be needed to develop targeted, knowledge-based, management plans. In this collection we provide this needed depth of study of the key aspects of the invasion process for the spiny water flea, Bythotrephes longimanus. We do so by presenting the results of the work conducted by researchers in the Canadian Aquatic Invasive Species Network (CAISN), and several of their American and European collaborators over the past half decade. Given its rapid spread in the Great Lakes basin in North America, and the decreases in pelagic biodiversity that have ensued, the last decade has witnessed a surge of research on Bythotrephes. In this collection we learn much about mechanisms and dynamics of its spread, about the key role of humans in that spread, about the importance of Allee effects to establishment and persistence, about choices and parameterization of risk assessment models, about the value of comparing “effects” in native and invaded regions, about complex probable interactions of the invasion with impending changes in the climate, and about the regulators of the invader’s abundance and impacts. There should be much of interest in the collection for aquatic ecologists and invading species biologists alike.

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    • Temperature-dependent Allee effects in a stage-structured model for Bythotrephes establishment

      Whether the invasive freshwater cladoceran Bythotrephes longimanus can establish after introduction into a water body depends on several biotic and abiotic factors. Among these, water temperature is important because both development rates and mode of reproduction (parthenogenetic or sexual) in Bythotrephes are influenced by temperature. We built a stage-structured model for the population dynamics of Bythotrephes based on the temperature-dependency of events in its life cycle and used the density of resting eggs at the end of each year to track changes in population density. The model was parameterized using data from published laboratory experiments and data on the Bythotrephes population in Harp Lake, Canada, from 1994 to 2005. The parameterized model was then used to simulate the outcome of invasions with different initial resting egg densities under different temperature regimes. A strong Allee effect emerged from the model, i.e. there is a critical threshold density above which the population can establish and below which it goes extinct. We showed analytically that the existence of an Allee effect arises from the model structure and is therefore robust to the parameter values. An increase in temperature reduces the establishment threshold for introductions in the same year as well as for introductions in the previous years. We therefore hypothesize that climate warming might facilitate Bythotrephes invasions. Finally, we study how the establishment threshold is influenced by the timing of the introduction event and thus identify time periods during the year when lakes may be particularly susceptible to Bythotrephes invasions.

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  • Chronic wasting disease

    • Chronic wasting disease: on possible transmission mechanisms in deer.

      We develop a model for the spread of chronic wasting disease (CWD) in a mule deer (Odocoileus hemionus) population to assess possible mechanisms of disease transmission and parameterize it for the mule deer population in Alberta, Canada. We consider seven mechanisms of disease transmission corresponding to direct and indirect contacts that change with seasonal distribution and groupings of deer. We determine the minimum set of mechanisms from all possible combinations of mechanisms with different weights for duration of seasonal segregation of sexes that are able to reproduce the observed ratio of CWD prevalence in adult males and females of ∼2 and greater. Multiple mechanisms are likely to produce the ratio of male:female prevalence levels and include: (1) environmentally mediated transmission associated with higher food intake by males, (2) female to male transmission during mating of this polygamous species, (3) increased male susceptibility to CWD and (4) increased intensity of direct contacts within male social groups. All of these mechanisms belong to the class of frequency-dependent transmission. Also important is seasonality in deer social structure with an increasing ratio of prevalence in males:females under all mechanisms as the duration of sexual segregation increases throughout a year.

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  • Climate change

    • Predicting survival, reproduction and abundance of polar bears under climate change.

      Polar bear (Ursus maritimus) populations are predicted to be negatively affected by climate warming, but the timeframe and manner in which change to polar bear populations will occur remains unclear. Predictions incorporating climate change effects are necessary for proactive population management, the setting of optimal harvest quotas, and conservation status decisions. Such predictions are difficult to obtain from historic data directly because past and predicted environmental conditions differ substantially. Here, we explore how models can be used to predict polar bear population responses under climate change. We suggest the development of mechanistic models aimed at predicting reproduction and survival as a function of the environment. Such models can often be developed, parameterized, and tested under current environmental conditions. Model predictions for reproduction and survival under future conditions could then be input into demographic projection models to improve abundance predictions under climate change. We illustrate the approach using two examples. First, using an individual-based dynamic energy budget model, we estimate that 3–6% of adult males in Western Hudson Bay would die of starvation before the end of a 120 day summer fasting period but 28–48% would die if climate warming increases the fasting period to 180 days. Expected changes in survival are non-linear (sigmoid) as a function of fasting period length. Second, we use an encounter rate model to predict changes in female mating probability under sea ice area declines and declines in mate-searching efficiency due to habitat fragmentation. The model predicts that mating success will decline non-linearly if searching efficiency declines faster than habitat area, and increase non-linearly otherwise. Specifically for the Lancaster Sound population, we predict that female mating success would decline from 99% to 91% if searching efficiency declined twice as fast as sea ice area, and to 72% if searching efficiency declined four times as fast as area. Sea ice is a complex and dynamic habitat that is rapidly changing. Failure to incorporate climate change effects into population projections can result in flawed conservation assessments and management decisions.

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  • Coexistence

    • Partial differential equations in ecology: spatial interactions and population dynamics

      Most of the fundamental elements of ecology, ranging from individual behavior to species abundance, diversity, and population dynamics, exhibit spatial variation. Partial differential equation models provide a means of melding organism movement with population processes and have been used extensively to elucidate the effects of spatial variation on populations. While there has been an explosion of theoretical advances in partial differential equation models in the past two decades, this work has been generally neglected in mathematical ecology textbooks. Our goal in this paper is to make this literature accessible to experimental ecologists. Partial differential equations are used to model a variety of ecological phenomena; here we discuss dispersal, ecological invasions, critical patch size, dispersal-mediated coexistence, and diffusion-driven spatial patterning. These models emphasize that simple organism movement can produce striking large-scale patterns in homogeneous environments, and that in heterogeneous environments, movement of multiple species can change the outcome of competition or predation.

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  • Disease modelling

    • Wildlife disease elimination and density dependence.

      Disease control by managers is a crucial response to emerging wildlife epidemics, yet the means of control may be limited by the method of disease transmission. In particular, it is widely held that population reduction, while effective for controlling diseases that are subject to density-dependent (DD) transmission, is ineffective for controlling diseases that are subject to frequency-dependent (FD) transmission. We investigate control for horizontally transmitted diseases with FD transmission where the control is via culling or harvest that is non-selective with respect to infection and the population can compensate through DD recruitment or survival. Using a mathematical model, we show that culling or harvesting can eradicate the disease, even when transmission dynamics are FD. Eradication can be achieved under FD transmission when DD birth or recruitment induces compensatory growth of new, healthy individuals, which has the net effect of reducing disease prevalence by dilution. We also show that if harvest is used simultaneously with vaccination, and there is high enough transmission coefficient, application of both controls may be less efficient than vaccination alone. We illustrate the effects of these control approaches on disease prevalence for chronic wasting disease in deer where the disease is transmitted directly among deer and through the environment.

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  • Dynamic programming

    • Sequential decision-making in a variable environment: modeling elk movement in Yellowstone National Park as a dynamic game.

      We develop a suite of models with varying complexity to predict elk movement behavior during the winter on the Northern Range of Yellowstone National Park (YNP). The models range from a simple representation of optimal patch choice to a dynamic game, and we show how the underlying theory in each is related by the presence or absence of state- and frequency-dependence. We compare predictions from each of the models for three variables that are of basic and applied interest: elk survival, aggregation, and use of habitat outside YNP. Our results suggest that despite low overall forage depletion in the winter, frequency-dependence is crucial to the predictions for elk movement and distribution. Furthermore, frequency-dependence interacts with mass-dependence in the predicted outcome of elk decision-making. We use these results to show how models that treat single movement decisions in isolation from the seasonal sequence of decisions are insufficient to capture landscape scale behavior.

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  • Eastern North-America

  • Expansion

    • Dispersal, Population Growth, and the Allee Effect: Dynamics of the House Finch Invasion of Eastern North America

      Since about 1940, when they were first released in the new York City area, house finches (Carpodacus mexicanus) have multiplied explosively and colonized much of eastern North America. We take advantage of the richly detailed documentation of this biological invasion to construct a mathematical model that predicts the rate of population spread on the basis of readily measurable demographic parameters. We seek to improve on previous models by predicting a rate of spread that accelerates following an initial period of slower growth, a pattern of spread followed by house finches as well as a variety of other invading species. We postulate that an Allee effect - disproportionately lowered fecundity below a critical threshold density of abundance-is the mechanism leading to a slower rate of spread in the early stages of the invasion. Our integrodifference equation model also emphasizes the link between long-distance dispersal and the rate of population spread.

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  • Extreme events

  • Gravity models

    • Stochastic gravity models for modeling lake invasions.

      Freshwater aquatic systems in North America are being invaded by many different species, ranging from fish, mollusks, cladocerans to various bacteria and viruses. These invasions have serious ecological and economic impacts. Human activities such as recreational boating are an important pathway for dispersal. Gravity models are used to quantify the dispersal effect of human activity. Gravity models currently used in ecology are deterministic. This paper proposes the use of stochastic gravity models in ecology, which provides new capabilities both in model building and in potential model applications. These models allow us to use standard statistical inference tools such as maximum likelihood estimation and model selection based on information criteria. To facilitate prediction, we use only those covariates that are easily available from common data sources and can be forecasted in future. This is important for forecasting the spread of invasive species in geographical and temporal domain. The proposed model is portable, that is it can be used for estimating relative boater traffic and hence relative propagule pressure for the lakes not covered by current boater surveys. This makes our results broadly applicable to various invasion prediction and management models.

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  • Host–parasite population dynamics

    • Aquaculture-induced changes to dynamics of a migratory host and specialist parasite: a case study of pink salmon and sea lice

      Exchange of diseases between domesticated and wild animals is a rising concern for conservation. In the ocean, many species display life histories that separate juveniles from adults. For pink salmon (Oncorhynchus gorbuscha) and parasitic sea lice (Lepeophtheirus salmonis), infection of juvenile salmon in early marine life occurs near salmon sea-cage aquaculture sites and is associated with declining abundance of wild salmon. Here, we develop a theoretical model for the pink salmon/sea lice host–parasite system and use it to explore the effects of aquaculture hosts, acting as reservoirs, on dynamics. Because pink salmon have a 2-year lifespan, even- and odd-year lineages breed in alternate years in a given river. These lineages can have consistently different relative abundances, a phenomenon termed “line dominance”. These dominance relationships between host lineages serve as a useful probe for the dynamical effects of introducing aquaculture hosts into this host–parasite system. We demonstrate how parasite spillover (farm-to-wild transfer) and spillback (wild-to-farm transfer) with aquaculture hosts can either increase or decrease the line dominance in an affected wild population. The direction of the effect depends on the response of farms to wild-origin infection. If aquaculture parasites are managed to a constant abundance, independent of the intensity of infections from wild to farm, then line dominance increases. On the other hand, if wild-origin parasites on aquaculture hosts are proportionally controlled to their abundance then line dominance decreases.

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  • Integrodifference equations

    • Spatially-explicit matrix models: A mathematical analysis of stage-structured integrodifference equations.

      This paper is concerned with mathematical analysis of the ‘critical domain-size’ problem for structured populations. Space is introduced explicitly into matrix models for stage-structured populations. Movement of individuals is described by means of a dispersal kernel. The mathematical analysis investigates conditions for existence, stability and uniqueness of equilibrium solutions as well as some bifurcation behaviors. These mathematical results are linked to species persistence or extinction in connected habitats of different sizes or fragmented habitats; hence the framework is given for application of such models to ecology. Several approximations which reduce the complexity of integrodifference equations are given. A simple example is worked out to illustrate the analytical results and to compare the behavior of the integrodifference model to that of the approximations.

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  • Invasion

    • Climate and competition: The effect of moving range boundaries on habitat invasibility.

      Predictions for climate change include movement of temperature isoclines up to 1000 meters per year, and this is supported by recent empirical studies. This paper considers effects of a rapidly changing environment on competitive outcomes between species. The model is formulated as a system of nonlinear partial differential equations in a moving domain. Terms in the equations decribe competition interactions and random movement by individuals. Here the critical patch size and travelling wave speed for each species, calculated in the absence of competition and in a stationary habitat, play a role in determining the outcome of the process with competition and in a moving habitat. We demonstrate how habitat movement, coupled with edge effects, can open up a new niche for invaders that would be otherwise excluded.

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  • Lévy movement paths

  • Linear conjecture

    • Analysis of the linear determinacy for spread in cooperative models.

      The discrete-time recursion system \un+1=Q[\un] with \un(x) a vector of population distributions of species and Q an operator which models the growth, interaction, and migration of the species is considered. Previously known results are extended so that one can treat the local invasion of an equilibrium of cooperating species by a new species or mutant. It is found that, in general, the resulting change in the equilibrium density of each species spreads at its own asymptotic speed, with the speed of the invader the slowest of the speeds. Conditions on Q are given which insure that all species spread at the same asymptotic speed, and that this speed agrees with the more easily calculated speed of a linearized problem for the invader alone. If this is true we say that the recursion has a single speed and is linearly determinate. The conditions are such that they can be verified for a class of reaction-diffusion models.

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  • Lotka-Volterra

    • Spreading speed and linear determinacy for two-species competition models.

       One crucial measure of a species' invasiveness is the rate at which it spreads into a competitor's environment. A heuristic spread rate formula for a spatially explicit, two-species competition model relies on `linear determinacy' which equates spread rate in the full nonlinear model with spread rate in the system linearized about the leading edge of the invasion. However, linear determinacy is not always valid for two-species competition; it has been shown numerically that the formula only works for certain values of model parameters when the model is diffusive Lotka-Volterra competition [2]. This paper derives a set of sufficient conditions for linear determinacy in spatially explicit two-species competition models. These conditions can be interpreted as requiring sufficiently large dispersal of the invader relative to dispersal of the out-competed resident and sufficiently weak interactions between the resident and the invader. When these conditions are not satisfied, spread rate may exceed linearly determined predictions. The mathematical methods rely on the application of results established in a companion paper

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  • Matrix model

    • On net reproductive rate and the timing of reproductive output

      Understanding the relationship between life-history patterns and population growth is central to demographic studies. Here we derive a new method for calculating the timing of reproductive output, from which the generation time and its variance can also be calculated. The method is based on the explicit computation of the net reproductive rate (R ) using a new graphical approach. Using 0 nodding thistle, desert tortoise, creeping aven, and cat’s ear as examples, we show how R and the timing of reproduction is calculated 0 and interpreted, even in cases with complex life cycles. We show that the explicit R formula allows us to explore the effect of all repro- 0 ductive pathways in the life cycle, something that cannot be done with traditional analysis of the population growth rate (l). Additionally, we compare a recently published method for determining population persistence conditions with the condition R 1 1 and 0 show how the latter is simpler and more easily interpreted biologically. Using our calculation of the timing of reproductive output, we illustrate how this demographic measure can be used to understand the effects of life-history traits on population growth and control.

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  • Net reproductive rate

    • Identifying non-invasible habitats for marine copepods using temperature-dependent R0

      If a non-indigenous species is to thrive and become invasive it must first persist under its new set of environmental conditions. Net reproductive rate (R 0) represents the average number of female offspring produced by a female over its lifetime, and has been used as a metric of population persistence. We modeled R 0 as a function of ambient water temperature (T) for the invasive marine calanoid copepod Pseudodiaptomus marinus, which is introduced to west coast of North America from East Asia by ship ballast water. The model was based on temperature-dependent stage-structured population dynamics given by a system of ordinary differential equations. We proposed a methodology to identify habitats that are non-invasible for P. marinus using the threshold of R 0(T) < 1 in order to identify potentially invasible habitats. We parameterized the model using published data on P. marinus and applied R 0(T) to identify the range of non-invasible habitats in a global scale based on sea surface temperature data. The model predictions matched the field evidence of species occurrences well.

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    • R0 Analysis of a Spatiotemporal Model for a Stream Population

      Water resources worldwide require management to meet industrial, agricultural, and urban consumption needs. Management actions change the natural flow regime, which impacts the river ecosystem. Water managers are tasked with meeting water needs while mitigating ecosystem impacts. We develop process-oriented advection-diffusion-reaction equations that couple hydraulic flow to population growth, and we analyze them to assess the effect of water flow on population persistence. We present a new mathematical framework, based on the net reproductive rate R0 for advection-diffusion-reaction equations and on related measures. We apply the measures to population persistence in rivers under various flow regimes. This work lays the groundwork for connecting R0 to more complex models of spatially structured and interacting populations, as well as more detailed habitat and hydrological data.

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  • Nonindigenous species

    • Chance Establishment for Sexual, Semelparous Species: Overcoming the Allee Effect

      We formalize the establishment process for a sexual, semelparous organism through the use of hierarchical probability modeling from parameters of survival, probability of being female, probability of being fertilized, and expected fecundity.We show how to calculate the expected per capita growth rate and probability of extinction. An Allee effect is observed if the expected population growth rate decreases as the initial population size decreases. The model can be further extended as a stochastic process to evaluate the probability of extinction in subsequent generations. One of the novel results is the formulation of an analytical probability distribution for the next generation population size. As case studies, we use the Chinese mitten crab (Eriocheir sinensis) and the apple snail (Pomacea canaliculata), both of which appear on the World Conservation Union’s list of 100 worst invaders.We evaluate the strength of the Allee effect and conclude that apple snails experience a weak Allee effect and Chinese mitten crabs experience a strong Allee effect. We emphasize one scenario where the stochastic process reveals that invasion risk can be estimated by the probability of the survival of one fertilized female, because the expected fecundity for one surviving female overwhelms the system such that population persistence is almost certain.

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    • Waiting for invasions: a framework for the arrival of nonindigenous species

      The process of nonindigenous species (NIS) arrival has received limited theoretical consideration despite importance in predicting and preventing the establishment of NIS. We formulate a mechanistically based hierarchical model of NIS arrival and demonstrate simplifications leading to a marginal distribution of the number of surviving introduced individuals from parameters of survival probability and propagule pressure. The marginal distribution is extended as a stochastic process from which establishment emerges with a waiting time distribution. This provides a probability of NIS establishment within a specified period and may be useful for identifying patterns of successful invaders. However, estimates of both the propagule pressure and the individual survival probability are rarely available for NIS, making estimates of the probability of establishment difficult. Alternatively, researchers are able to measure proportional estimates of propagule pressure through models of NIS transport, such as gravity models, or of survival probability through habitat-matching indexes measuring the similarity between potentially occupied and native NIS ranges. Therefore, we formulate the relative waiting time between two locations and the probability of one location being invaded before the other.

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  • Nonlocal dispersal

    • The effect of dispersal patterns on stream populations

      Individuals in streams are constantly subject to predominantly unidirectional flow. The question of how these populations can persist in upper stream reaches is known as the “drift paradox.” We employ a general mechanistic movement-model framework and derive dispersal kernels for this situation. We derive thin- as well as fat-tailed kernels. We then introduce population dynamics and analyze the resultingin tegrodifferential equation. In particular, we study how the critical domain size and the invasion speed depend on the velocity of the stream flow. We give exact conditions under which a population can persist in a finite domain in the presence of stream flow, as well as conditions under which a population can spread against the direction of the flow. We find a critical stream velocity above which a population cannot persist in an arbitrarily large domain. At exactly the same stream velocity, the invasion speed against the flow becomes zero; for larger velocities, the population retreats with the flow.

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  • Nonlocal hyperbolic system

    • Complex spatial group patterns result from different animal communication mechanisms.

      We present previously undescribed spatial group patterns that emerge in a one-dimensional hyperbolic model for animal group formation and movement. The patterns result from the assumption that the interactions governing movement depend not only on distance between conspecifics, but also on how individuals receive information about their neighbors and the amount of information received. Some of these patterns are classical, such as stationary pulses, traveling waves, ripples, or traveling trains. However, most of the patterns have not been reported previously. We call these patterns zigzag pulses, semizigzag pulses, breathers, traveling breathers, and feathers.

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    • Modeling group formation and activity patters in self-organizing collectives of individuals.

      We construct and analyze a nonlocal continuum model for group formation with application to self-organizing collectives of animals in homogeneous environments. The model consists of a hyperbolic system of conservation laws, describing individual movement as a correlated random walk. The turning rates depend on three types of social forces: attraction toward other organisms, repulsion from them, and a tendency to align with neighbors. Linear analysis is used to study the role of the social interaction forces and their ranges in group formation. We demonstrate that the model can generate a wide range of patterns, including stationary pulses, traveling pulses, traveling trains, and a new type of solution that we call zigzag pulses. Moreover, numerical simulations suggest that all three social forces are required to account for the complex patterns observed in biological systems. We then use the model to study the transitions between daily animal activities that can be described by these different patterns.

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    • Weakly nonlinear analysis of a hyperbolic model for animal group formation.

      We consider an one-dimensional nonlocal hyperbolic model for group formation with application to self-organizing collectives of animals in homogeneous environments. Previous studies have shown that this model displays at least four complex spatial and spatiotemporal group patterns. Here, we use weakly nonlinear analysis to better understand the mechanisms involved in the formation of two of these patterns, namely stationary pulses and traveling trains. We show that both patterns arise through subcritical bifurcations from spatially homogeneous steady states. We then use these results to investigate the effect of two social interactions (attraction and alignment) on the structure of stationary and moving animal groups. While attraction makes the groups more compact, alignment has a dual effect, depending on whether the groups are stationary or moving. More precisely, increasing alignment makes the stationary groups compact, and the moving groups more elongated. Also, the results show the existence of a threshold for the total group density, above which, coordinated behaviors described by stationary and moving groups persist for a long time.

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  • Optimal control

    • Optimal control of biological invasions in lake networks.

      A metapopulation model for alien species invasion of a lake network is coupled with an economic model of prevention. The model restates a stochastic problem in deterministic terms. It provides a macroscopic description of the lake network with prevention methods controlling both the outflow of invaders at infected lakes and the inflow of invaders at uninfected lakes. Results indicate that optimal control implements no more than one of these methods at any moment in time. Typical optimal control measures change over time as the lake ecosystem becomes successively more invaded. Early control of outflow from infected lakes is replaced by later control of inflow to remaining uninfected lakes. Closed-loop control trajectories are analytically characterized in the phase-plane for a limiting case, while in general a simple and stable numerical algorithm is developed for solving the optimal control problem.

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  • Population density

    • Ecological chaos in the wake of invasion.

      Irregularities in observed population densities have traditionally been attributed to discretization of the underlying dynamics. We propose an alternative explanation by demonstrating the evolution of spatiotemporal chaos in reaction-diffusion models for predator-prey interactions. The chaos is generated naturally in the wake of invasive waves of predators. We discuss in detail the mechanism by which the chaos is generated. By considering a mathematical caricature of the predator-prey models, we go on to explain the dynamical origin of the irregular behavior and to justify our assertion that the behavior we present is a genuine example of spatiotemporal chaos.

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  • Population dynamics

    • Response of equilibrium states to spatial environmental heterogeneity in advective systems

      Much ecological research involves identifying connections between abiotic forcing and population densities or distributions. We present theory that describes this relationship for populations in media with strong unidirec- tional °ow (e.g., aquatic organisms in streams and rivers). Typically, equi- librium populations change in very di®erent ways in response to changes in demographic versus dispersal rates and to changes over local versus larger spatial scales. For populations in a mildly heterogeneous environment, there is a population \response length" that characterizes the distance downstream over which the impact of a point source perturbation is felt. The response length is also an important parameter for characterizing the response to non- point source disturbances at di®erent spatial scales. In the absence of density dependence, the response length is close to the mean distance traveled by an organism in its lifetime. Density-dependent demographic rates are likely to increase the response length from this default value, and density-dependent dispersal will reduce it. Indirect density dependence, mediated by predation, may also change the response length, the direction of change depending on the strength of the prey's tendency to °ee the predator.

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  • Predator-prey models

    • Pattern formation in prey-taxis systems

      In this paper, we consider spatial predator–prey models with diffusion and prey-taxis. We investigate necessary conditions for pattern formation using a variety of non-linear functional responses, linear and non-linear predator death terms, linear and non-linear prey-taxis sensitivities, and logistic growth or growth with an Allee effect for the prey.We identify combinations of the above non-linearities that lead to spatial pattern formation and we give numerical examples. It turns out that prey-taxis stabilizes the system and for large prey-taxis sensitivity we do not observe pattern formation. We also study and find necessary conditions for global stability for a type I functional response, logistic growth for the prey, non-linear predator death terms, and non-linear prey-taxis sensitivity.

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  • Predator–prey

    • How predation can slow, stop or reverse a prey invasion.

      Observations on Mount St Helens indicate that the spread of recolonizing lupin plants has been slowed due to the presence of insect herbivores and it is possible that the spread of lupins could be reversed in the future by intense insect herbivory [Fagan, W. F. and J. Bishop (2000). Trophic interactions during primary sucession: herbivores slow a plant reinvasion at Mount St. Helens. Amer. Nat. 155, 238–251]. In this paper we investigate mechanisms by which herbivory can contain the spatial spread of recolonizing plants. Our approach is to analyse a series of predator-prey reaction-diffusion models and spatially coupled ordinary differential equation models to derive conditions under which predation pressure can slow, stall or reverse a spatial invasion of prey. We focus on models where prey disperse more slowly than predators. We comment on the types of functional response which give such solutions, and the circumstances under which the models are appropriate.

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  • Predator–prey invasions

    • Invasion theory and biological control.

      Recent advances in the mathematical theory of invasion dynamics have much to offer to biological control. Here we synthesize several results concerning the spatiotemporal dynamics that occur when a biocontrol agent spreads into a population of an invading pest species. We outline conditions under which specialist and generalist predators can influence the density and rate of spatial spread of the pest, including the rather stringent conditions under which a specialist predator can successfully reverse a pest invasion. We next discuss the connections between long distance dispersal and invasive spread, emphasizing the different consequences of fast spreading pests and predators. Recent theory has considered the effects of population stage-structure on invasion dynamics, and we discuss how population demography affects the biological control of invading pests. Because low population densities generally characterize early stages of an invasion, we discuss the lessons invasion theory teaches concerning the detectability of invasions. Stochasticity and density-dependent dynamics are common features of many real invasions, influencing both the spatial character (e.g. patchiness) of pest invasions and the success of biocontrol agents. We conclude by outlining theoretical results delineating how stochastic effects and complex dynamics generated by density dependence can facilitate or impede biological pest control.

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  • Primary Succession

    • A stoichiometric model of early plant primary succession

      The relative importance of plant facilitation and competition during primary succession depends on the development of ecosystem nutrient pools, yet the interaction of these processes remains poorly understood. To explore how these mechanisms interact to drive successional dynamics, we devised a stoichiometric ecosystem-level model that considers the role of nitrogen and phosphorus limitation in plant primary succession. We applied this model to the primary plant community on Mount St. Helens, Washington State, to check the validity of the proposed mechanisms. Our results show that the plant community is colimited by nitrogen and phosphorus, and they confirm previous suggestions that the presence of a nitrogen-fixing legume, Lupinus lepidus, can enhance community biomass. In addition, the observed nutrient supply rates may promote alternative successional trajectories that depend on the initial plant abundances, which may explain the observed heterogeneity in community development. The model further indicates the importance of mineralization rates and other ecosystem parameters to successional rates. We conclude that a model framework based on ecological stoichiometry allows integration of key biotic processes that interact nonlinearly with biogeochemical aspects of succession. Extension of this approach will improve the understanding of the process of primary succession and its application to ecosystem rehabilitation.

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  • Seasonal environment

    • Seasonal influences on population spread and persistence in streams: spreading speeds.

      The drift paradox asks how stream-dwelling organisms can persist, without being washed out, when they are continuously subject to the unidirectional stream flow. To date, mathematical analyses of the stream paradox have investigated the interplay of growth, drift and flow needed for species persistence under the assumption that the stream environment is temporally constant. However, in reality, streams are subject to major seasonal variations in environmental factors that govern population growth and dispersal. We consider the influence of such seasonal variations on the drift paradox, using a time-periodic integrodifferential equation model. We establish upstream and downstream spreading speeds under the assumption of periodically fluctuating environments, and also show the existence of periodic traveling waves. The sign of the upstream spreading speed then determines persistence. Fluctuating environments are characterized by seasonal correlations between the flow, transfer rates, diffusion and settling rates, and we investigate the effect of such correlations on the population spread and persistence. We also show how results in this paper can formally connect to those for autonomous integrodifferential equations, through the appropriate weighted averaging methods. Finally, for a specific dispersal function, we show that the upstream spreading speed is nonnegative if and only if the critical domain size exists in this temporally fluctuating environment.

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  • Seasonality

    • Seasonal influences on population spread and persistence in streams: Critical domain size

      The critical domain size problem determines the size of the region of habitat needed to ensure population persistence. In this paper we address the critical domain size problem for seasonally fluctuating stream environments and determine how large a reach of suitable stream habitat is needed to ensure population persistence of a stream-dwelling species. Two key factors, not typically found in critical domain size problems, are fundamental in determining whether population can persist. These are the unidirectional nature of stream flow and seasonal fluctuations in the stream environment. We characterize the fluctuating environments in terms of seasonal correlations among the flow, transfer rates, diffusion, and settling rates, and we investigate the effect of such correlations on the critical domain size problem. We show how results for the seasonally fluctuating stream can formally be connected to those for autonomous integro-differential equations, through the appropriate weighted averaging methods.

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  • Spatial competition

    • Spatial patterns and coexistence mechanisms in systems with unidirectional flow.

      River ecosystems are the prime example of environments where unidirectional flow influences the dispersal of individuals. Spatial patterns of community composition and species replacement emerge from complex interplays of hydrological, geochemical, biological, and ecological factors. Local processes affecting algal dynamics are well understood, but a mechanistic basis for large scale emerging patterns is lacking. To understand how these patterns could emerge in rivers, we analyze a reaction–advection–diffusion model for two competitors in heterogeneous environments. The model supports waves that invade upstream up to a well-defined “upstream invasion limit”. We discuss how these waves are produced and present their key properties. We suggest that patterns of species replacement and coexistence along spatial axes reflect stalled waves, produced from diffusion, advection, and species interactions. Emergent spatial scales are plausible given parameter estimates for periphyton. Our results apply to other systems with unidirectional flow such as prevailing winds or climate-change scenarios.

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  • Species invasions

    • Modeling ships' ballast water as invasion threats to the Great Lakes.

      The spread of nonindigenous species in aquatic ecosystems provides an opportunity to develop new perspectives on the invasion process. In this paper we review existing invasion models, most of which were developed to describe invasions of terrestrial habitats, and propose an alternative that explores long-distance invasions mediated by discharge of contaminated ballast water by ships in-bound to the Great Lakes. Based on current knowledge of shipping traffic to the Great Lakes, our model predicts that mid-ocean exchange of ballast water lowers propagule delivery by approximately three to four orders of magnitude relative to unexchanged ballast water. Propagule pressure of individual ships that enter the Great Lakes loaded with cargo and which declare 'no ballast on board' (NOBOB) is typically one to two orders of magnitude higher than that of vessels that exchange ballast. Because NOBOB vessels dominate (~90%) inbound traffic into the Great Lakes, these vessels collectively appear to pose the greatest risk of new introductions mediated via ballast water.

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  • Structure

    • A body composition model to estimate mammalian energy stores and metabolic rates from body mass and body length, with application to polar bears

      Many species experience large fluctuations in food availability and depend on energy from fat and protein stores for survival, reproduction and growth. Body condition and, more specifically, energy stores thus constitute key variables in the life history of many species. Several indices exist to quantify body condition but none can provide the amount of stored energy. To estimate energy stores in mammals, we propose a body composition model that differentiates between structure and storage of an animal. We develop and parameterize the model specifically for polar bears (Ursus maritimus Phipps) but all concepts are general and the model could be easily adapted to other mammals. The model provides predictive equations to estimate structural mass, storage mass and storage energy from an appropriately chosen measure of body length and total body mass. The model also provides a means to estimate basal metabolic rates from body length and consecutive measurements of total body mass. Model estimates of body composition, structural mass, storage mass and energy density of 970 polar bears from Hudson Bay were consistent with the life history and physiology of polar bears. Metabolic rate estimates of fasting adult males derived from the body composition model corresponded closely to theoretically expected and experimentally measured metabolic rates. Our method is simple, noninvasive and provides considerably more information on the energetic status of individuals than currently available methods.

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