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Exponential model for population growth (exponential model) is a foundation to evaluate population dynamics in Population Ecology field. Here, we used a didactic experiment to teach exponential model for an undergraduate course of Population Ecology. We built nine populations of weevils with three different initial population sizes: eight, 16, and 32 individuals with three replicates each. We provided equal food resource availability, and counted their population sizes weekly for 12 weeks. We estimated the intrinsic growth rate (i.e., r parameter), by trials and errors with an exponential model build in an Excel spreadsheet. The population growth rate (i.e., dN/dt parameter) was estimated using r values. Replicates with eight and 16 individuals reached the highest values of r and dN/dt, while replicates with 32 individuals reached the lowest values. Beyond of exponential model, two density dependency issues acting in populations were observed. First, in the lowest initial population sizes we observed the effect of demographic stochasticity acting in both r and dN/dt in one of the three populations. Second, we observed the intraspecific competition reducing r values in largest initial populations. Therefore, we highlight the importance of didactic experiment into learning exponential model in Population Ecology course, both for teaching and learning practices.    

Stochasticity is a core component of ecology, as it underlies key processes that structure and create variability in nature. Despite its fundamental importance in ecological systems, the concept is often treated as synonymous with unpredictability in community ecology, and studies tend to focus on single forms of stochasticity rather than taking a more holistic view. This has led to multiple narratives for how stochasticity mediates community dynamics. Here, we present a framework that describes how different forms of stochasticity (notably demographic and environmental stochasticity) combine to provide underlying and predictable structure in diverse communities. This framework builds on the deep ecological understanding of stochastic processes acting at individual and population levels and in modules of a few interacting species. We support our framework with a mathematical model that we use to synthesize key literature, demonstrating that stochasticity is more than simple uncertainty. Rather, stochasticity has profound and predictable effects on community dynamics that are critical for understanding how diversity is maintained. We propose next steps that ecologists might use to explore the role of stochasticity for structuring communities in theoretical and empirical systems, and thereby enhance our understanding of community dynamics

Dispersal as well as population growth is a key demographic process that determines population dynamics. However, determining the effects of environmental covariates on dispersal from spatial‐temporal abundance proxy data is challenging owing to the complexity of model specification for directional dispersal permeability and the extremely high computational loads for numerical integration. In this paper, we present a case study estimating how environmental covariates affect the dispersal of Japanese sika deer by developing a spatially explicit state‐space matrix model coupled with an improved numerical integration technique (Markov chain Monte Carlo with particle filters). In particular, we explored the environmental drivers of inhomogeneous range expansion, characteristic of animals with short dispersal. Our model framework successfully reproduced the complex population dynamics of sika deer, including rapid changes in densely populated areas and distribution fronts within a decade. Furthermore, our results revealed that the inhomogeneous range expansion of sika deer seemed to be primarily caused by the dispersal process (i.e., movement barriers in fragmented forests) rather than population growth. Our state‐space matrix model enables the inference of population dynamics for a broad range of organisms, even those with low dispersal ability, in heterogeneous landscapes, and could address many pressing issues in conservation biology and ecosystem management. We present a case study estimating how environmental elements affect the dispersal of Japanese sika deer by developing a spatially explicit state‐space matrix model coupled with an improved numerical integration technique. Our results revealed that the inhomogeneous range expansion of sika deer was primarily caused by the dispersal process (i.e., movement barriers in fragmented forests) rather than population growth.

Ecological drift causes species abundances to fluctuate randomly, lowering diversity within communities and increasing differences among otherwise equivalent communities. Despite broad interest in ecological drift, ecologists have little experimental evidence of its consequences in nature, where competitive forces modulate species abundances. We manipulated drift by imposing 40-fold variation in the size of experimentally assembled annual plant communities and holding their edge-to-interior ratios comparable. Drift over three generations was greater than predicted by neutral models, causing high extinction rates and fast divergence in composition among smaller communities. Competitive asymmetries drove populations of most species to small enough sizes that demographic stochasticity could markedly influence dynamics, increasing the importance of drift in communities. The strong effects of drift occurred despite stabilizing niche differences, which cause species to have greater population growth rates when at low local abundance. Overall, the importance of ecological drift appears greater in non-neutral communities than previously recognized, and varies with community size and the type and strength of density dependence.

Savannas are highly variable systems, and predicting variation, especially in the tree layer, represents a major unresolved challenge for forecasting biosphere responses to global change. Prediction to date has focused on disentangling interactions between resource limitation and chronic disturbances to identify what determines local savanna vegetation heterogeneity. By focusing at too fine a scale, this approach overlooks: sample size limitation arising fromsparse tree distributions; stochasticity in demographic and environmental processes that is preserved as heterogeneity among tree populations with slow dynamics; and spatial self-organization. Renewedfocus on large (1–50 ha) permanent plots and on spatial patterns of tree-layer variability at even larger landscape spatial scales (≥1000s of ha) promises to resolve these limitations, consistent with the goal of predicting large-scale biosphere responses to global change.

Ecological drift can override the effects of deterministic niche selection on small populations and drive the assembly of some ecological communities. We tested this hypothesis with a unique data set sampled identically in 200 streams in two regions (tropical Brazil and boreal Finland) that differ in macroinvertebrate community size by fivefold. Null models allowed us to estimate the magnitude to which β-diversity deviates from the expectation under a random assembly process while taking differences in richness and relative abundance into account, i.e., β-deviation. We found that both abundance- and incidence-based β-diversity was negatively related to community size only in Brazil. Also, β-diversity of small tropical communities was closer to stochastic expectations compared with b-diversity of large communities. We suggest that ecological drift may drive variation in some small communities by changing the expected outcome of niche selection, increasing the chances of species with low abundance and narrow distribution to occur in some communities. Habitat destruction, overexploitation, pollution, and reductions in connectivity have been reducing the size of biological communities. These environmental pressures might make smaller communities more vulnerable to novel conditions and render community dynamics more unpredictable. Incorporation of community size into ecological models should provide conceptual and applied insights into a better understanding of the processes driving biodiversity.

Temporal asynchrony among species helps diversity to stabilize ecosystem functioning, but identifying the mechanisms that determine synchrony remains a challenge. Here, we refine and test theory showing that synchrony depends on three factors: species responses to environmental variation, interspecific interactions, and demographic stochasticity. We then conduct simulation experiments with empirical population models to quantify the relative influence of these factors on the synchrony of dominant species in five semiarid grasslands. We found that the average synchrony of per capita growth rates, which can range from 0 (perfect asynchrony) to 1 (perfect synchrony), was higher when environmental variation was present (0.62) rather than absent (0.43). Removing interspecific interactions and demographic stochasticity had small effects on synchrony. For the dominant species in these plant communities, where species interactions and demographic stochasticity have little influence, synchrony reflects the covariance in species' responses to the environment.

Contemporary studies of species coexistence are underpinned by deterministic models that assume that competing species have continuous (i.e., noninteger) densities, live in infinitely large landscapes, and coexist over infinite time horizons. By contrast, in nature, species are composed of discrete individuals subject to demographic stochasticity and occur in habitats of finite size where extinctions occur in finite time. One consequence of these discrepancies is that metrics of species' coexistence derived from deterministic theory may be unreliable predictors of the duration of species coexistence in nature. These coexistence metrics include invasion growth rates and niche and fitness differences, which are now commonly applied in theoretical and empirical studies of species coexistence. In this study, we tested the efficacy of deterministic coexistence metrics on the duration of species coexistence in a finite world. We introduce new theoretical and computational methods to estimate coexistence times in stochastic counterparts of classic deterministic models of competition. Importantly, we parameterized this model using experimental field data for 90 pairwise combinations of 18 species of annual plants, allowing us to derive biologically informed estimates of coexistence times for a natural system. Strikingly, we found that for species expected to deterministically coexist, community sizes containing only 10 individuals had predicted coexistence times of more than 1000 years. We also found that invasion growth rates explained 60% of the variation in intrinsic coexistence times, reinforcing their general usefulness in studies of coexistence. However, only by integrating information on both invasion growth rates and species' equilibrium population sizes could most (>99%) of the variation in species coexistence times be explained. This integration was achieved with demographically uncoupled single-species models solely determined by the invasion growth rates and equilibrium population sizes. Moreover, because of a complex relationship between niche overlap/fitness differences and equilibrium population sizes, increasing niche overlap and increasing fitness differences did not always result in decreasing coexistence times, as deterministic theory would predict. Nevertheless, our results tend to support the informed use of deterministic theory for understanding the duration of species' coexistence while highlighting the need to incorporate information on species' equilibrium population sizes in addition to invasion growth rates.

Although most studies of factors contributing to successful establishment and spread of non-native species have focused on species traits and characteristics (both biotic and abiotic), increasing empirical and statistical evidence implicates propagule pressure—propagule sizes, propagule numbers, and temporal and spatial patterns of propagule arrival—as important in both facets of invasion. Increasing propagule size enhances establishment probability primarily by lessening effects of demographic stochasticity, whereas propagule number acts primarily by diminishing impacts of environmental stochasticity. A continuing rain of propagules, particularly from a variety of sources, may erase or vitiate the expected genetic bottleneck for invasions initiated by few individuals (as most are), thereby enhancing likelihood of survival. For a few species, recent molecular evidence suggests ongoing propagule pressure aids an invasion to spread by introducing genetic variation adaptive for new areas and habitats. This phenomenon may also explain some time lags between establishment of a non-native species and its spread to become an invasive pest.

A major question in ecology is how age-specific variation in demographic parameters influences population dynamics. Based on long-term studies of growing populations of birds and mammals, we analyze population dynamics by using fluctuations in the total reproductive value of the population. This enables us to account for random fluctuations in age distribution. The influence of demographic and environmental stochasticity on the population dynamics of a species decreased with generation time. Variation in age-specific contributions to total reproductive value and to stochastic components of population dynamics was correlated with the position of the species along the slow-fast continuum of life-history variation. Younger age classes relative to the generation time accounted for larger contributions to the total reproductive value and to demographic stochasticity in \"slow\" than in \"fast\" species, in which many age classes contributed more equally. In contrast, fluctuations in population growth rate attributable to stochastic environmental variation involved a larger proportion of all age classes independent of life history. Thus, changes in population growth rates can be surprisingly well explained by basic species-specific life-history characteristics.