Comment on "On the regulation of populations of mammals, birds, fish, and insects" III

Temporal resolution of birth rate analysis in zooplankton and its implications for identifying strong interactions in ecology

Expanding on Haeckel's classical definition, ecology can be defined as the study of strong and weak interactions between the organism and the environment, hence the need for identifying strong interactions as major drivers of population and community dynamics. The solution to this problem is facilitated by the fact that the frequency distribution of interaction strengths is highly skewed, resulting in few or, according to Liebig's law of the minimum, just one strong interaction. However, a single strong interaction often remains elusive. One of the reasons may be that, due to the ever-present dynamics of ecological systems, a single strong interaction is likely to exist only on relatively short time intervals, so methods with sufficient temporal resolution are required. In this paper, we study the temporal resolution of contribution analysis of birth rate in zooplankton, a method to assess the relative strength of bottom-up (food) versus top-down (predation) effects. Birth rate is estimated by the Edmondson-Paloheimo model. Our test system is a population of the cladoceran Bosmina longirostris inhabiting a small northern lake with few planktivorous predators, and thus likely controlled by food. We find that the method's temporal resolution in detecting bottom-up effects corresponds well to the species' generation time, and the latter seems comparable to the lifetime of a single strong interaction. This enables one to capture a single strong interaction "on the fly," right during its time of existence. We suggest that this feature, the temporal resolution of about the lifetime of a single strong interaction, may be a generally desirable property for any method, not only the one studied here, intended to identify and assess strong interactions. Success in disentangling strong interactions in ecological communities, and thus solving one of the key issues in ecology, may critically depend on the temporal resolution of the methods used.

Gene Drive Dynamics in Natural Populations: The Importance of Density Dependence, Space, and Sex

The spread of synthetic gene drives is often discussed in the context of panmictic populations connected by gene flow and described with simple deterministic models. Under such assumptions, an entire species could be altered by releasing a single individual carrying an invasive gene drive, such as a standard homing drive. While this remains a theoretical possibility, gene drive spread in natural populations is more complex and merits a more realistic assessment. The fate of any gene drive released in a population would be inextricably linked to the population's ecology. Given the uncertainty often involved in ecological assessment of natural populations, understanding the sensitivity of gene drive spread to important ecological factors is critical. Here we review how different forms of density dependence, spatial heterogeneity, and mating behaviors can impact the spread of self-sustaining gene drives. We highlight specific aspects of gene drive dynamics and the target populations that need further research.

A simple approximation of moments of the quasi-equilibrium distribution of an extended stochastic theta-logistic model with non-integer powers

The stochastic versions of the logistic and extended logistic growth models are applied successfully to explain many real-life population dynamics and share a central body of literature in stochastic modeling of ecological systems. To understand the randomness in the population dynamics of the underlying processes completely, it is important to have a clear idea about the quasi-equilibrium distribution and its moments. Bartlett et al. (1960) took a pioneering attempt for estimating the moments of the quasi-equilibrium distribution of the stochastic logistic model. Matis and Kiffe (1996) obtain a set of more accurate and elegant approximations for the mean, variance and skewness of the quasi-equilibrium distribution of the same model using cumulant truncation method. The method is extended for stochastic power law logistic family by the same and several other authors (Nasell, 2003; Singh and Hespanha, 2007). Cumulant truncation and some alternative methods e.g. saddle point approximation, derivative matching approach can be applied if the powers involved in the extended logistic set up are integers, although plenty of evidence is available for non-integer powers in many practical situations (Sibly et al., 2005). In this paper, we develop a set of new approximations for mean, variance and skewness of the quasi-equilibrium distribution under more general family of growth curves, which is applicable for both integer and non-integer powers. The deterministic counterpart of this family of models captures both monotonic and non-monotonic behavior of the per capita growth rate, of which theta-logistic is a special case. The approximations accurately estimate the first three order moments of the quasi-equilibrium distribution. The proposed method is illustrated with simulated data and real data from global population dynamics database.

Simultaneous identification of growth law and estimation of its rate parameter for biological growth data: a new approach

Scientific formalizations of the notion of growth and measurement of the rate of growth in living organisms are age-old problems. The most frequently used metric, "Average Relative Growth Rate" is invariant under the choice of the underlying growth model. Theoretically, the estimated rate parameter and relative growth rate remain constant for all mutually exclusive and exhaustive time intervals if the underlying law is exponential but not for other common growth laws (e.g., logistic, Gompertz, power, general logistic). We propose a new growth metric specific to a particular growth law and show that it is capable of identifying the underlying growth model. The metric remains constant over different time intervals if the underlying law is true, while the extent of its variation reflects the departure of the assumed model from the true one. We propose a new estimator of the relative growth rate, which is more sensitive to the true underlying model than the existing one. The advantage of using this is that it can detect crucial intervals where the growth process is erratic and unusual. It may help experimental scientists to study more closely the effect of the parameters responsible for the growth of the organism/population under study.

Strength of density feedback in census data increases from slow to fast life histories

Life-history theory predicts an increasing rate of population growth among species arranged along a continuum from slow to fast life histories. We examine the effects of this continuum on density-feedback strength estimated using long-term census data from >700 vertebrates, invertebrates, and plants. Four life-history traits (Age at first reproduction, Body size, Fertility, Longevity) were related statistically to Gompertz strength of density feedback using generalized linear mixed-effects models and multi-model inference. Life-history traits alone explained 10 to 30% of the variation in strength across species (after controlling for time-series length and phylogenetic nonindependence). Effect sizes were largest for body size in mammals and longevity in birds, and density feedback was consistently stronger for smaller-bodied and shorter-lived species. Overcompensatory density feedback (strength <-1) occurred in 20% of species, predominantly at the fast end of the life-history continuum, implying relatively high population variability. These results support the idea that life history leaves an evolutionary signal in long-term population trends as inferred from census data. Where there is a lack of detailed demographic data, broad life-history information can inform management and conservation decisions about rebound capacity from low numbers, and propensity to fluctuate, of arrays of species in areas planned for development, harvesting, protection, and population recovery.

Density dependence: an ecological Tower of Babel

The concept of density dependence represents the effect of changing population size on demographic rates and captures the demographic role of social and trophic mechanisms (e.g. competition, cooperation, parasitism or predation). Ecologists have coined more than 60 terms to denote different statistical and semantic properties of this concept, resulting in a formidable lexicon of synonymies and polysemies. We have examined the vocabulary of density dependence used in the modern ecological literature from the foundational lexicon developed by Smith, Allee, Haldane, Neave and Varley. A few simple rules suffice to abate terminological inconsistency and to enhance the biological meaning of this important concept. Correct citation of original references by ecologists and research journals could ameliorate terminological standards in our discipline and avoid linguistic confusion of mathematically and theoretically complex patterns.

Likelihood ridges and multimodality in population growth rate models

A central problem in population ecology is to use time series data to estimate the form of density dependence in the per capita growth rate (pgr). This is often accomplished with phenomenological models such as the theta-Ricker or generalized Beverton-Holt. Using the theta-Ricker model as a simple but flexible description of density dependence, we apply theory and simulations to show how multimodality and ridges in the likelihood surface can emerge even in the absence of model misspecification or observation error. The message for model fitting of real data is to consider the likelihood surface in detail, check whether the best-fit model is located on a likelihood ridge and, if so, evaluate predictive differences of biologically plausible models along the ridge. We present a detailed analysis of a focal data set showing how multimodality and ridges emerge in practice for fits of several parametric models, including a state-space model with explicit accommodation of observation error. Best-fit models for these data are biologically dubious beyond the range of the data, and likelihood ratio confidence regions include a wide range of more biologically plausible models. We demonstrate the broad relevance of these findings by presenting analyses of 25 additional data sets spanning a wide range of taxa. The results here are relevant to information-theoretic and Bayesian methods, which also rely on likelihoods. Beyond presentation of best-fit models and confidence regions around individual parameters, effort toward understanding features of the likelihood surface will help ensure the most robust translation from statistical analysis to biological interpretation.

Adaptive changes in prey vulnerability shape the response of predator populations to mortality

Simple models are used to explore how adaptive changes in prey vulnerability alter the population response of their predator to increased mortality. If the mortality is an imposed harvest, the change in prey vulnerability also influences the relationship between harvest effort and yield of the predator. The models assume that different prey phenotypes share a single resource, but have different vulnerabilities to the predator. Decreased vulnerability is assumed to decrease resource consumption rate. Adaptive change may occur by phenotypic changes in the traits of a single species or by shifts in the abundances of a pair of coexisting species or morphs. The response of the predator population is influenced by the shape of the predator's functional response, the shape of resource density dependence, and the shape of the tradeoff between vulnerability and food intake in the prey. Given a linear predator functional response, adaptive prey defense tends to produce a decelerating decline in predator population size with increased mortality. Prey defense may also greatly increase the range of mortality rates that allow predator persistence. If the predator has a type-2 response with a significant handling time, adaptive prey defense may have a greater variety of effects on the predator's response to mortality, sometimes producing alternative attractors, population cycles, or increased mean predator density. Situations in which there is disruptive selection on prey defense often imply a bimodal change in yield as a function of harvesting effort, with a minimum at intermediate effort. These results argue against using single-species models of density dependent growth to manage predatory species, and illustrate the importance of incorporating anti-predator behavior into models in applied population ecology.

Is spread of invasive species regulated? Using ecological theory to interpret statistical analysis

We investigate a recent proposal that invasive species display patterns of spatial "spread regulation" analogous to density-dependent regulation of population abundances. While invasive species do offer valuable tests of ecological theories about spatial spread, we argue that the statistical approach used in the study is not useful, and that the proposed definition of "spread regulation" is likely to be confusing. While concepts of negative feedbacks in spatial spread may be reasonable, the proposed definition of "spread regulation" encompasses accelerating, constant, or decelerating spread. There is no compelling biological or practical reason to adopt such a definition. Moreover, we show that the statistical patterns (from time series of ratios of newly to recently invaded sites) proposed as evidence of spread regulation are predictable from basic diffusion models or other common models of constant spread with some stochasticity in dynamics and/or observations. Because such a wide range of processes would generate the observed patterns, no clear biological conclusions emerge from the proposed approach to spread analysis. When regarded in the context of the impacts and management of invasive species, the proposed regulation concept has the potential to create costly misunderstandings.