In contrast to the single species models that were extensively studied in the 1970s and 1980s, predator–prey models give rise to long-period oscillations, and even systems with stable equilibria can display oscillatory transients with a regular frequency. Many fluctuating populations appear to be governed by such interactions. However, predator–prey models have been poorly studied with respect to the interaction of nonlinear dynamics, noise, and system identification. I use simulated data from a simple host–parasitoid model to investigate these issues. The addition of even a modest amount of noise to a stable equilibrium produces enough structured variation to allow reasonably accurate parameter estimation. Despite the fact that more-or-less regular cycles are generated by adding noise to any of the classes of deterministic attractor (stable equilibrium, periodic and quasiperiodic orbits, and chaos), the underlying dynamics can usually be distinguished, especially with the aid of the mechanistic model. However, many of the time series can also be fit quite well by a wrong model, and the fitted wrong model usually misidentifies the underlying attractor. Only the chaotic time series convincingly rejected the wrong model in favor of the true one. Thus chaotic population dynamics offer the best chance for successfully identifying underlying regulatory mechanisms and attractors.

Small populations are often at risk of extinction through processes that are effectively stochastic. Prediction of this extinction risk requires that the observed temporal variation in demographic rates be accurately partitioned between demographic stochasticity (variation among individuals) and environmental stochasticity (variation among years, correlated across individuals). However, studies of population viability analysis that include both forms of stochasticity parameterize the magnitude of environmental stochasticity incorrectly (they overestimate it). I describe and evaluate tests (1) to determine whether all the year-to-year variation in observed survivorship can be explained by demographic stochasticity alone, and (2) if not, to estimate the magnitude of environmental stochasticity in survival. The first issue can be resolved with a G test. I used simulated data to show that this test has an appropriate type I error rate, unless the individual survival probability is either very low or very high. Small amounts of environmental stochasticity often are not detected by the G test (type II error), but the hypothesis of demographic stochasticity alone is consistently rejected when environmental stochasticity is large. In contrast, estimating the magnitude of environmental stochasticity requires explicit hypotheses about the nature of the underlying variation, but I provide a flexible framework in which many such hypotheses can be examined. In particular, I show, using simulated data, that if the temporal variability in individual survival probabilities is distributed according to a beta distribution, then the maximum likelihood estimate of the variance of the survival probability is biased, but in a consistent and correctable way. The estimate obtained with my method is usually superior to an estimate that assumes that all of the variation in the observed survivorship is due to environmental stochasticity. I show how to include deterministic sources of variability, such as density dependence, and how to apply different assumptions about the underlying environmental stochasticity. I illustrate these tests with data from a population of Acorn Woodpeckers (Melanerpes formicivorus). With these data, I can determine that there is strong environmental stochasticity in juvenile survival, whereas variation in adult survival can be explained either by density dependence or by weak environmental stochasticity.

Spatial extent can have two important consequences for population dynamics: It can generatespatial structure, in which individuals interact more intensely with neighbors than with more distant conspecifics, and it allows forenvironmental heterogeneity, in which habitat quality varies spatially. Studies of these features are difficult to interpret because the models are complex and sometimes idiosyncratic. Here we analyze one of the simplest possible spatial population models, to understand the mathematical basis for the observed patterns: two patches coupled by dispersal, with dynamics in each patch governed by the logistic map. With suitable choices of parameters, this model can represent spatial structure, environmental heterogeneity, or both in combination. We synthesize previous work and new analyses on this model, with two goals: to provide a comprehensive baseline to aid our understanding of more complex spatial models, and to generate predictions about the effects of spatial structure and environmental heterogeneity on population dynamics. Spatial structure alone can generate positive, negative, or zero spatial correlations between patches when dispersal rates are high, medium, or low relative to the complexity of the local dynamics. It can also lead to quasiperiodicity and hyperchaos, which are not present in the nonspatial model. With density-independent dispersal, spatial structure cannot destabilize equilibria or periodic orbits that would be stable in the absence of space. When densities in the two patches are uncorrelated, the probability that the population in a patch reaches extreme low densities is reduced relative to the same patch in isolation; this “rescue effect” would reduce the probability of metapopulation extinction beyond the simple effect of spreading of risk. Pure environmental heterogeneity always produces positive spatial correlations. The dynamics of the entire population is approximated by a nonspatial model with mean patch characteristics. This approximation worsens as the difference between the patches increases and the dispersal rate decreases: Under extreme conditions, destabilization of equilibria and periodic orbits occurs at mean parameter values lower than those predicted by the mean parameters. Apparent within-patch dynamics are distorted: The local population appears to have the wrong growth parameter and a constant number of immigrants (or emigrants) per generation. Adding environmental heterogeneity to spatial structure increases the occurrence of spatially correlated population dynamics, but the resulting temporal dynamics are more complex than would be predicted by the mean parameter values. The three classes of spatial pattern (positive, negative, and zero correlation), while still mathematically distinct, become increasingly similar phenomenologically.

Humans play a critical role in the dispersal of exotic invasive species. Estimating pathways for non‐native species by human vectors is a major challenge to invasion biologists, as well as federal, state and regional resource managers. Focusing on dispersal pathways that are available to not just one, but a number of species, allows for the efficient inspection and possible reduction of many exotic species introductions. Transient recreational boating has been used as an estimate of invasion pressure to inland freshwater bodies, and used to predict prior and future species invasions. Specifically, recreational boating traffic is used to predict human-mediated aquatic invasion in the Midwestern United States through the use of spatial interaction models called gravity models. California and Nevada contain some of the largest and most recreationally utilized lakes, rivers and reservoirs in the Western United States. These waterways attract millions of visitor days by boaters not only from within the region, but all over the United States and are currently experiencing increasing numbers of non‐native species introductions from the Midwestern U.S.

This report describes aspects of dispersal of an aquatic invasive plant, Eurasian watermilfoil, both within and between fresh water bodies by recreational boating. This study addresses the question of habitat and/or dispersal limitation for watermilfoil by assessing the movement of recreational boaters within Lake Tahoe, and between Lake Tahoe and other locations, as well as characterizing nearshore habitat locations in highly visited boating destinations. Additionally, this report examines the nature of recreational boater movement data, and the impacts of boater preference as well as the impact of the spatial aspect of data gathering from one versus many locations. Specifically, this report presents the following: 1) an examination of the use of transportation models known as gravity models to describe recreational boater traffic to inland waterways in California and Nevada, 2) an analysis of waterway access point habitat quality as it relates to Eurasian watermilfoil, and 3) the invasion of Eurasian watermilfoil within Lake Tahoe, and how that relates to within-lake boater movement and habitat variables associated with invaded and uninvaded sites within Lake Tahoe.

Lively debate continues over whether marine reserves can lead to increased fishery yields when compared to conventional, quota-based management, apparently driven by differences in the complexity and biological richness of the models being used. In an influential article, Hastings and Botsford used an analytically tractable, spatially implicit, non-age-structured model to assert that reserves are typically incapable of increasing yields relative to conventional management, regardless of the type (pre- or post-dispersal, involving adults and/or larvae) or functional form (Ricker or Beverton-Holt) of density dependence present. A recent numerical (simulation) model by Gaylord et al. concludes that reserves can enhance yield compared to conventional management, a result the authors attribute to their spatially-explicit evaluation of stage-structured adult growth, survivability and fecundity; and intercohort (adult-on-larvae) post-dispersal density dependent population dynamics. Here we demonstrate that the increased model complexity is not responsible for the different conclusions. We analyze a spatially-implicit model without stage structure that incorporates intercohort post-dispersal density dependence. In this simple model we still find annual extirpation of adult populations outside reserves due to fishing to enhance larval recruitment there, allowing for increased yields compared to those achieved when harvest is evenly spread across the entire domain under conventional management. Consideration of neither spatially-explicit dispersal dynamics nor stage-structure in adult demographics is required for reserves to substantially improve yield beyond that attainable under conventional management. In contrast, consideration of within cohort post-dispersal density dependence among larva during settlement in an otherwise identical model generates equivalence in yield between the two management strategies. These results recast a common message characterizing the relative benefit of reserve versus non-reserve management from "equivalence at best" to "potentially improved".

Demographic stochasticity is almost universally modeled as sampling variance in a homogeneous population, although it is defined as arising from random variation among individuals. This can lead to serious misestimation of the extinction risk in small populations. Here, we derive analytical expressions showing that the misestimation for each demographic parameter is exactly (in the case of survival) or approximately (in the case of fecundity) proportional to the among-individual variance in that parameter. We also show why this misestimation depends on systematic variation among individuals, rather than random variation. These results indicate that correctly assessing the importance of demographic stochasticity requires (1) an estimate of the variance in each demographic parameter; (2) information on the qualitative shape (convex or concave) of the mean–variance relationship; and (3) information on the mechanisms generating among-individual variation. An important consequence is that almost all population viability analyses (PVAs) overestimate the importance of demographic stochasticity and, therefore, the risk of extinction.