How does temporal variability affect predictions of weed population numbers?

1. Population models that are used to predict weed population dynamics or the impact of control measures on weed abundance typically ignore temporal variability in life-history parameters and control measures, and utilize mean arithmetic population growth rates to predict population abundance.
2. We demonstrate that the persistence of weeds in a stochastically varying environment depends on the geometric mean population growth rate being greater than zero, rather than the arithmetic mean population growth rate being greater than zero.
3. In a stochastically varying environment we show that temporal variability in fecundity, germination and survivorship will tend to decrease population size, relative to predictions based on arithmetic means. Conversely, variability in competitive effects and weed control will tend to increase population size, relative to predictions based on arithmetic mean values. The distinction between these two sets of parameters is that increases in the former will increase population growth rate, whereas increases in the latter will decrease it.
4. We argue that population models based on arithmetic mean population growth rates will tend to over-estimate population size. Numerical simulations indicate that this bias may be considerable.
5. Since short-term studies cannot, in general, estimate the geometric mean growth rate of a population we suggest several approaches for estimating the degree of bias in the predictions of models owing to the effects of variability. Accounting for such variability is necessary since current models for the dynamics of weed populations are based on arithmetic mean measures of population growth and hence likely to be biased.     

The impact of mortality on predator population size and stability in systems with stage-structured prey

The relationships between a predator population's mortality rate and its population size and stability are investigated for several simple predator-prey models with stage-structured prey populations. Several alternative models are considered; these differ in their assumptions about the nature of density dependence in the prey's population growth; the nature of stage-transitions; and the stage-selectivity of the predator. Instability occurs at high, rather than low predator mortality rates in most models with highly stage-selective predation; this is the opposite of the effect of mortality on stability in models with homogeneous prey populations. Stage-selective predation also increases the range of parameters that lead to a stable equilibrium. The results suggest that it may be common for a stable predator population to increase in abundance as its own mortality rate increases in stable systems, provided that the predator has a saturating functional response. Sufficiently strong density dependence in the prey generally reverses this outcome, and results in a decrease in predator population size with increasing predator mortality rate. Stability is decreased when the juvenile stage has a fixed duration, but population increases with increasing mortality are still observed in large areas of stable parameter space. This raises two coupled questions which are as yet unanswered; (1) do such increases in population size with higher mortality actually occur in nature; and (2) if not, what prevents them from occurring? Stage-structured prey and stage-related predation can also reverse the 'paradox of enrichment', leading to stability rather than instability when prey growth is increased.     

Disease transmission models with density-dependent demographics

The models considered for the spread of an infectious disease in a population are of SIRS or SIS type with a standard incidence expression. The varying population size is described by a modification of the logistic differential equation which includes a term for disease-related deaths. The models have density-dependent restricted growth due to a decreasing birth rate and an increasing death rate as the population size increases towards its carrying capacity. Thresholds, equilibria and stability are determined for the systems of ordinary differential equations for each model. The persistence of the infectious disease and disease-related deaths can lead to a new equilibrium population size below the carrying capacity and can even cause the population to become extinct.Research supported in part by Centers for Disease Control contract 200-87-0515     

Environmental noise and population dynamics of the ciliated protozoa Tetrahymena thermophila in aquatic microcosms

Population theory predicts that the reddened environmental noise, especially in combination with high population growth rate, reddens population dynamics, increases population variability and strengthens environment–population correlation. We tested these predictions with axenic populations of ciliated protozoa Tetrahymena thermophila. Populations with low and high growth rate were cultured in a stable environment, and in environments with sublethal temperature fluctuations that had blue, white and red spectra (i.e. negatively autocorrelated, uncorrelated, or positively autocorrelated, respectively). Population size and biomass of individuals were determined at 3-h intervals for 18 days.
Dynamics of all populations were reddened, suggesting that internal mechanisms can redden the population spectra. However, population dynamics were reddest, variability highest, and environment–population correlation strongest in the red environment as predicted. Contrary to theoretical predictions and previous empirical findings, population growth rate (r_max being equal to 0.05 and 0.3 h⁻¹) had no effect on population dynamics.
Mean cell size and variability of cell size were affected by the presence and type of environmental noise suggesting that the physiological consequences of variability depend on colour. Environmental variability decreased mean population size and biomass and the decrease was strongest in rapidly fluctuating blue and white environments. The latter finding implies that rapid fluctuations are physiologically stressful, an effect that is not accounted for in the basic population models.     

The Probability of Fixation in Populations of Changing Size

The rate of adaptive evolution of a population ultimately depends on the rate of incorporation of beneficial mutations. Even beneficial mutations may, however, be lost from a population since mutant individuals may, by chance, fail to reproduce. In this paper, we calculate the probability of fixation of beneficial mutations that occur in populations of changing size. We examine a number of demographic models, including a population whose size changes once, a population experiencing exponential growth or decline, one that is experiencing logistic growth or decline, and a population that fluctuates in size. The results are based on a branching process model but are shown to be approximate solutions to the diffusion equation describing changes in the probability of fixation over time. Using the diffusion equation, the probability of fixation of deleterious alleles can also be determined for populations that are changing in size. The results developed in this paper can be used to estimate the fixation flux, defined as the rate at which beneficial alleles fix within a population. The fixation flux measures the rate of adaptive evolution of a population and, as we shall see, depends strongly on changes that occur in population size.     

Impact of natural enemies on obligately cooperative breeders

Obligately cooperative breeders (cooperators) display a negative growth rate once they fall below a minimum density. Constraints imposed by natural enemies, such as predators or competitors, may push cooperator groups closer to this threshold, thus increasing the risk that stochastic fluctuations will drive them below it. This may indirectly drive these groups to extinction, thereby increasing the risk of population extinction. In this paper, we construct mathematical models of the dynamics of groups of cooperators and non-cooperators in the presence of two types of enemies: enemies whose dynamics do not depend on the dynamics of their victim (e.g., amensal competitor, generalist predator) and those whose dynamics do. In the latter case, we distinguish positive (e.g., specialist predator) and negative (e.g., bilateral competitor) reciprocal effects. These models correspond to the classical amensal, predation and competition models, in the presence of an Allee effect. We then develop the models to study consequences at the population level. By comparing models with or without an Allee effect, we show that enemies decrease the group size of cooperators more than that of non-cooperators, and this increases their group extinction risk. We also demonstrate how an Allee effect at a lower dynamical level can have consequences at a higher level: inverse density dependence at the group level generated lower population sizes and higher risks of population extinction. Our results also suggest that demographic compensation can be achieved by cooperators through an increased intrinsic growth rate, or by decreasing the enemy constraint. Both of these types of compensation have been observed in empirical studies of cooperators.     

Elasticity Analysis of Green Sturgeon Life History

I provide an analysis of a simplified life history model for green sturgeon, Acipenser medirostris, based on published and recent estimates of reproduction and growth rates and survival rates from life history theory. The deterministic life cycle models serve as a tool for qualitative analysis of the impacts of perturbations on green sturgeon, including harvest regulations based on minimum and maximum size limits (“slot limits”). Elasticity analysis of models with two alternative age–length relationships give similar results, with a high sensitivity of population growth rate to changes in the survival rate of subadult and adult fish. A dramatic increase in the survival of young of the year sturgeon or annual egg production is required to compensate for relatively low levels of fishing mortality. Peak reproductive values occur from ages 25 to 40. An increase or decrease in the maximum and minimum size limits can have a profound effect on the elasticity of population growth to changes in the annual survival rate of age classes specified by the slot, due to changes in the number of age classes of subadults and adults that are available for harvest. This analysis provides managers with a simple tool to assess the relative impacts of alternative harvest regulations. In general, green sturgeon follow life history patterns similar to other sturgeon, but species-specific demographic information is needed to produce more complex assessment and viability analysis models.     

Can Ingestion of Lead Shot and Poisons Change Population Trends of Three European Birds: Grey Partridge,Common Buzzard,and Red Kite?

Little is known about the magnitude of the effects of lead shot ingestion alone or combined with poisons (e.g., in bait or seeds/granules containing pesticides) on population size, growth, and extinction of non-waterbird avian species that ingest these substances. We used population models to create example scenarios demonstrating how changes in these parameters might affect three susceptible species: grey partridge (Perdix perdix), common buzzard (Buteo buteo), and red kite (Milvus milvus). We added or subtracted estimates of mortality due to lead shot ingestion (4–16% of mortality, depending on species) and poisons (4–46% of mortality) reported in the UK or France to observed mortality of studied populations after models were calibrated to observed population trends. Observed trends were decreasing for partridge (in continental Europe), stable for buzzard (in Germany), and increasing for red kite (in Wales). Although lead shot ingestion and poison at modeled levels did not change the trend direction for the three species, they reduced population size and slowed population growth. Lead shot ingestion at modeled rates reduced population size of partridges by 10%, and when combined with bait and pesticide poisons, by 18%. For buzzards, decrease in mean population size by lead shot and poisons combined was much smaller (≤ 1%). The red kite population has been recovering; however, modeled lead shot ingestion reduced its annual growth rate from 6.5% to 4%, slowing recovery. If mortality from poisoned baits could be removed, the kite population could potentially increase at a rapid annual rate of 12%. The effects are somewhat higher if ingestion of these substances additionally causes sublethal reproductive impairment. These results have uncertainty but suggest that declining or recovering populations are most sensitive to lead shot or poison ingestion, and removal of poisoned baits can have a positive impact on recovering raptor populations that frequently feed on carrion.     

Impact of climate change factors on the clonal sedge Carex bigelown: implications for population growth and vegetative spread

Hypothesized life-cycle responses to climate change for the arctic, clonal perennial Carex bigelown are constructed using a range of earlier observations and experiments together with new information from monitoring and an environmental perturbation study These data suggest, that under current climate change scenarios, increases in CO₂, temperature and nutrient availability would promote growth in a qualitatively similar way The evidence suggests that both tiller size and daughter tiller production will increase, and be shifted towards production of phalanx tillers which have a greater propensity for flowering Furthermore, age at tillering as well as tiller life span may decrease, whereas survival of younger age classes might be higher Mathematical models using experimental data incorporating these hypotheses were used to a) integrate the various responses and to calculate the order of magnitude of changes in population growth rate (γ). and b) to explore the implications of responses in individual demographic parameters for population growth rate The models suggest that population growth rate following climate change might increase significantly, but not un-realistically so. with the younger, larger, guerilla ullers being the most important tiller categones in contributing to X The rate of vegetative spread is calculated to more than double, while cyclical trends in flowering and populauon growth are predicted to decrease substantially     

Long-term trends in the population size of king penguins at Crozet archipelago: environmental variability and density dependence?

We examined the growth rate of the breeding population of king penguins of Crozet archipelago over 41 years. Most colonies showed positive growth rates from the 1960s. However, there was evidence for a decrease in the larger colonies since the early 1990s, and for lower growth rates in the smaller colonies during the 1990s. The overall population size was estimated using log linear models, and the average annual growth rate was +6.3% for the 41-year period. Four change points were detected in the annual growth rate: +21.1% during 1978–1985, +4.3% during 1985–1995, –19.2% during 1995–1999, and +10.9% during 1999–2003. Time-series analyses of the population-size estimates and the relationship between growth rate and population size both indicated density-dependence in population growth rate. Variations in population sizes are also discussed in relation to environmental fluctuations. Our results suggest that important changes occurred over the past 10 years.     

MUTATIONAL MELTDOWN IN LABORATORY YEAST POPULATIONS

Abstract.— In small or repeatedly bottlenecked populations, mutations are expected to accumulate by genetic drift, causing fitness declines. In mutational meltdown models, such fitness declines further reduce population size, thus accelerating additional mutation accumulation and leading to extinction. Because the rate of mutation accumulation is determined partly by the mutation rate, the risk and rate of meltdown are predicted to increase with increasing mutation rate. We established 12 replicate populations of Saccharomyces cerevisiae from each of two isogenic strains whose genomewide mutation rates differ by approximately two orders of magnitude. Each population was transferred daily by a fixed dilution that resulted in an effective population size near 250. Fitness declines that reduce growth rates were expected to reduce the numbers of cells transferred after dilution, thus reducing population size and leading to mutational meltdown. Through 175 daily transfers and approximately 2900 generations, two extinctions occurred, both in populations with elevated mutation rates. For one of these populations there is direct evidence that extinction resulted from mutational meltdown: Extinction immediately followed a major fitness decline, and it recurred consistently in replicate populations reestablished from a sample frozen after this fitness decline, but not in populations founded from a predecline sample. Wild‐type populations showed no trend to decrease in size and, on average, they increased in fitness.     

Using otolith weight to age fish

The problem of determining and verifying ages of fish, from populations having a considerable variation in size at age, has been investigated using the relationship between otolith size and fish size, which has been shown by several authors to be influenced by growth rate. In such a population of Sardinella aurita Val. an index of age can be obtained for individual fish by calculating the equivalent otolith weight at a particular fish length, using the otolith weight–fish length relationship determined for each age group. This statistic not only permits a much greater proportion of fish to be assigned ages than is possible with otolith reading alone, but also enables the age groups to be verified as year classes. However, it is concluded that, although appropriate models based on otolith-fish size relationships can predict age for groups of fish in which growth rates are known or can be assumed to be consistent, such techniques have a limited application in ageing fish from wild populations with highly variable growth rates.     

Continuous-discrete models of the dynamics of an isolated population and of 2 competing species

The paper presents the analysis of various mathematical models for dynamics of isolated population and for competition between two species. It is assumed that mortality is continuous and birth of individuals of new generations takes place in certain fixed moments. Influence of winter upon the population dynamics and conditions of classic discrete model "deduction" of population dynamics (in particular, Moran-Ricker and Hassel's models) are investigated. Dynamic regimes of models under various assumptions about the birth and death rates upon the population states are also examined. Analysis of models of isolated population dynamics with nonoverlapping generations showed the density changes regularly if the birth rate is constant. Moreover, there exists a unique global stable level and population size stabilizes asymptotically at this equilibrium, i.e. cycle and chaotic regimes in various discrete models depend on correlation between individual productivity and population state in previous time. When the correlation is exponential upon mean population size the discrete Hassel model is realized. Modification of basis model, based on the assumption that during winter survival/death changes are constant, showed that population size at global level is stable. Generally, the dependence of population rate upon "winter parameters" has nonlinear character. Nonparametric models of competition between two species does not vary if the individual productivity is constant. In a phase space there are several stable stationary states and population stabilizes at one or other level asymptotically. So, in discrete models of competition between two species oscillation can be explained by dependence of population growth rate on the population size at previous times.     

Predicted steady-state cell size distributions for various growth models

The question of how an individual bacterial cell grows during its life cycle remains controversial. In 1962 Collins and Richmond derived a very general expression relating the size distributions of newborn, dividing and extant cells in steady-state growth and their growth rate; it represents the most powerful framework currently available for the analysis of bacterial growth kinetics. The Collins-Richmond equation is in effect a statement of the conservation of cell numbers for populations in steady-state exponential growth. It has usually been used to calculate the growth rate from a measured cell size distribution under various assumptions regarding the dividing and newborn cell distributions, but can also be applied in reverse--to compute the theoretical cell size distribution from a specified growth law. This has the advantage that it is not limited to models in which growth rate is a deterministic function of cell size, such as in simple exponential or linear growth, but permits evaluation of far more sophisticated hypotheses. Here we employed this reverse approach to obtain theoretical cell size distributions for two exponential and six linear growth models. The former differ as to whether there exists in each cell a minimal size that does not contribute to growth, the latter as to when the presumptive doubling of the growth rate takes place: in the linear age models, it is taken to occur at a particular cell age, at a fixed time prior to division, or at division itself; in the linear size models, the growth rate is considered to double with a constant probability from cell birth, with a constant probability but only after the cell has reached a minimal size, or after the minimal size has been attained but with a probability that increases linearly with cell size. Each model contains a small number of adjustable parameters but no assumptions other than that all cells obey the same growth law. In the present article, the various growth laws are described and rigorous mathematical expressions developed to predict the size distribution of extant cells in steady-state exponential growth; in the following paper, these predictions are tested against high-quality experimental data.     

Phanerozoic marine biodiversity follows a hyperbolic trend

Changes in marine biodiversity through the Phanerozoic correlate much better with hyperbolic model (widely used in demography and macrosociology) than with exponential and logistic models (traditionally used in population biology and extensively applied to fossil biodiversity as well). The latter models imply that changes in diversity are guided by a first-order positive feedback (more ancestors, more descendants) and/or a negative feedback arising from resource limitation. Hyperbolic model implies a second-order positive feedback. The hyperbolic pattern of the world population growth arises from a second-order positive feedback between the population size and the rate of technological growth. The hyperbolic character of biodiversity growth can be similarly accounted for by a feedback between the diversity and community structure complexity. The similarity between the curves of biodiversity and human population probably comes from the fact that both are derived from the interference of the hyperbolic trend with cyclical and stochastic dynamics.     

Perturbation approaches for integral projection models

Perturbation analysis of population models is fundamental to elucidating mechanisms of population dynamics and examining scenarios of change. The use of integral projection models (IPMs) has increased in the last decade, and while many of the tools and approaches developed for matrix models remain relevant, the nature of IPMs expands the framework of perturbation analysis, with different approaches often requiring important considerations. This article provides a review of – and practical guide to – different perturbation approaches for IPMs, formalizes methodologies for perturbing IPM size transition probabilities, and highlights areas where researchers should be particularly careful and critical when conducting and interpreting perturbation analysis. I use a simulated dataset to compare five hierarchical perturbation approaches for IPMs found within 63 published studies, and apply a combination of approaches to the example of an invasive perennial plant. Other perturbation approaches for IPMs are also highlighted. Most perturbation analyses for IPMs to date have focused on the response of the asymptotic population growth rate (λ) to changes in elements of the discretized projection kernel and/or the growth– survival and reproduction– recruitment sub‐kernels. Perturbations to vital rate functions and regression predictions underlying these kernels provide mechanistic insight, but are less common and can require important considerations regarding the perturbation of size transitions separate from survival and the nature of the state variable (used to represent size). The second most common approach is more specific to IPMs and examines the influence of vital rate regression parameters, each of which can have broad influence on the population growth rate. Researchers using IPMs have many perturbation options available and should carefully consider which approach or combination of approaches is most applicable and interpretable for their specific questions.     

Effect of population size on the prospect of species survival

Many recent studies have demonstrated a negative effect of small population size on single plant traits. However, not much is known about the actual consequences of reduced plant performance on the long-term prospect of species survival. I studied the effect of population size on population growth rate and survival probability in the rare perennial herbScorzonera hispanica occurring in fragmented grasslands. Its performance was measured using several traits related to reproduction in 21 populations ranging in size from 3 to 2475 plants. These data were then connected with data on full demography of the species from three of the studied populations. Two different matrix models differing in the number of transitions based on measurements in the populations differing in size were used to explore the relationship between population size and population growth rate. Both matrix models showed that despite the decline in seed production in small populations, population growth rate is never significantly different from one, and the populations could thus be expected to survive in the long run. Calculations of extinction probabilities that take into account demographic and environmental stochasticity, however, showed that populations below 100 flowering individuals have a high probability to become extinct. This demonstrates that demographic and environmental stochasticity is an important driver of the fate of small populations in this system. This study demonstrates that estimation of population growth rate can provide new insights into the effect of population size on population growth and survival. It also shows how matrix models enable the combination various pieces of information about the single populations into one overall measure, and may provide a useful tool for the standardization of studies on the effects of population size on population performance.     

Relative nonlinearity and permanence

We modify the commonly used invasibility concept for coexistence of species to the stronger concept of uniform invasibility. For two-species discrete-time competition and predator-prey models, we use this concept to find broad easily checked sufficient conditions for the rigorous concept of permanent coexistence. With these results, permanent coexistence becomes a tractable concept for many discrete-time population models. To understand how these conditions apply to nonpoint attractors, we generalize the concept of relative nonlinearity and use it to show how population fluctuations affect the long-term low-density growth rate (“the invasion rate”) of a species when it is invading the system consisting of the other species (“the resident”) at a single-species attractor. The concept of relative nonlinearity defines circumstances when this invasion rate is increased or decreased by resident population fluctuations arising from a nonpoint attractor. The presence and sign of relative nonlinearity is easily checked in models of interacting species. When relative nonlinearity is zero or positive, fluctuations cannot decrease the invasion rate. It follows that permanence is then determined by invasibility of the resident’s fixed points. However, when relative nonlinearity is negative, invasibility, and hence permanent coexistence, can be undermined by resident population fluctuations. These results are illustrated with specific two-species competition and predator-prey models of generic forms.     

Persistence of structured populations in random environments

Environmental fluctuations often have different impacts on individuals that differ in size, age, or spatial location. To understand how population structure, environmental fluctuations, and density-dependent interactions influence population dynamics, we provide a general theory for persistence for density-dependent matrix models in random environments. For populations with compensating density dependence, exhibiting “bounded” dynamics, and living in a stationary environment, we show that persistence is determined by the stochastic growth rate (alternatively, dominant Lyapunov exponent) when the population is rare. If this stochastic growth rate is negative, then the total population abundance goes to zero with probability one. If this stochastic growth rate is positive, there is a unique positive stationary distribution. Provided there are initially some individuals in the population, the population converges in distribution to this stationary distribution and the empirical measures almost surely converge to the distribution of the stationary distribution. For models with overcompensating density-dependence, weaker results are proven. Methods to estimate stochastic growth rates are presented. To illustrate the utility of these results, applications to unstructured, spatially structured, and stage-structured population models are given. For instance, we show that diffusively coupled sink populations can persist provided that within patch fitness is sufficiently variable in time but not strongly correlated across space.     

Demography_Lab,an educational application to evaluate population growth: Unstructured and matrix models

Training in Population Ecology asks for scalable applications capable of embarking students on a trip from basic concepts to the projection of populations under the various effects of density dependence and stochasticity. Demography_Lab is an educational tool for teaching Population Ecology aspiring to cover such a wide range of objectives. The application uses stochastic models to evaluate the future of populations. Demography_Lab may accommodate a wide range of life cycles and can construct models for populations with and without an age or stage structure. Difference equations are used for unstructured populations and matrix models for structured populations. Both types of models operate in discrete time. Models can be very simple, constructed with very limited demographic information or parameter‐rich, with a complex density‐dependence structure and detailed effects of the different sources of stochasticity. Demography_Lab allows for deterministic projections, asymptotic analysis, the extraction of confidence intervals for demographic parameters, and stochastic projections. Stochastic population growth is evaluated using up to three sources of stochasticity: environmental and demographic stochasticity and sampling error in obtaining the projection matrix. The user has full control on the effect of stochasticity on vital rates. The effect of the three sources of stochasticity may be evaluated independently for each vital rate. The user has also full control on density dependence. It may be included as a ceiling population size controlling the number of individuals in the population or it may be evaluated independently for each vital rate. Sensitivity analysis can be done for the asymptotic population growth rate or for the probability of extinction. Elasticity of the probability of extinction may be evaluated in response to changes in vital rates, and in response to changes in the intensity of density dependence and environmental stochasticity.