Meta‐analysis of intrinsic rates of increase and carrying capacity of populations affected by toxic and other stressors

Abstract

Most of the thousands of substances and species that are of concern for environmental management will not be investigated empirically at ecologically relevant levels because of financial, practical, and ethical constraints. To allow risk assessment for these less well‐known categories, we have developed a mechanistic model with classical equations from toxicology and ecology. The parameters are linked to well‐known properties, such as the octanol‐water partition ratio Kow, acute lethal (body) concentrations, and organism size. This allows estimation of intrinsic rates of increase r and carrying capacity K over a wide range of substances and species. The model was calibrated with parameter values (μ ± 95% confidence interval) obtained in reviews and validated by a meta‐analysis with largely independent data from 200 laboratory experiments. For single substances, the 5 to 95% interval of the observations on intrinsic rates of increase overlapped with the range predicted by the model. Model and experiments independently indicated that population growth ceased below 1% of the acute median lethal concentration in about 5% of the cases. Exceptional values and possible explanations were identified. The reduction of the carrying capacity K was nearly proportional to the inhibition of the population growth r. Population‐level effects of mixtures as estimated by concentration addition were confirmed by observations in the experiments. The impact of a toxicant and another stressor could generally be described by response multiplication, with the exception of cases with extreme stress. Data sets on population laboratory experiments are biased to metals and crustaceans. This field will benefit from empirical studies on chemicals, conditions, and species, identified as risky by the model. Other implications of the model for environmental management and research are discussed.

Carrying capacity in a heterogeneous environment with habitat connectivity

“…However, deeper physiological mechanisms are also clearly at work in causing this correlation, particularly when manipulated by stressors as in the studies analysed by Hendriks et al . (). For example, our inability to predict the effect of a growth inhibitory drug, Cyh, on carrying capacity highlights the need for future theoretical work on the mechanistic/physiological basis of an r‐K correlation.…”

Dietary traces of neonicotinoid pesticides as a cause of population declines in honey bees: an evaluation by Hill's epidemiological criteria

“…The coherence criterion asks whether invoking a factor as the cause of a particular phenomenon conflicts with established knowledge. There is no conflict inherent in the proposition that dietary intake of an insecticidal chemical, such as a neonicotinoid, could harm honey bees sufficiently to cause a population decline because xenobiotics have this effect on other species 46. However, the parameters of inequalities (3) and (6) cannot be populated with well‐justified, quantitative values that specify a threshold for the generation of sufficient harm.…”

Modelling the impact of toxic and disturbance stress on white-tailed eagle (Haliaeetus albicilla) populations

“…The basic equations are briefly summarised, as details can be found elsewhere (Hendriks and Enserink 1996; Hendriks et al 2005). The number of breeding individuals in a given year  N ( t ) is calculated from the population in the preceding year  N ( t − Δt ) by\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ N\left( t \right) = N\left( {t - \Updelta t} \right) + \frac{{r\left( {C,t} \right)}}{{r\left( {0,t} \right)}} \cdot r\left( {0,t} \right) \cdot N\left( {t - \Updelta t} \right) \cdot \Updelta t \cdot \left( {1 - \frac{{N\left( {t - \Updelta t} \right)}}{N(\infty )}} \right) $$\end{document}where r ( C,t ) and r ( 0,t ) represent the rates of increase at time t under contaminated conditions and under reference conditions respectively, N (∞) represents the carrying capacity, and the time step Δt was set at 1 year.…”

“…The ratio between the rate of increase r ( C ) at concentration C and the rate of increase under reference conditions r ( 0 ) is calculated according to (Hendriks and Enserink 1996; Hendriks et al 2005)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\frac{r\left( C \right)}{r\left( 0 \right)} = \frac{{ - \ln \left( {1 + \left( {\tfrac{C}{LC50}} \right)^{1/\beta } } \right) - \ln \left( {1 + \left( {\tfrac{C}{EC50}} \right)^{1/\beta } } \right)}}{R\left( 0 \right)} + 1} $$\end{document}with R ( 0 ) as the lifetime fecundity, i.e. the average number of offspring per individual per generation time, calculated as (Birch 1948; Hendriks and Enserink 1996)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {R\left( 0 \right)} = \sum\limits_{0}^{{a_{\max } }} {l\left( a \right)} \cdot m\left( a \right) \cdot da $$\end{document} Non-toxic environmental stressors, like disturbance, may affect the population size N ( t ) by further reducing the rate of increase r ( C ) (Hendriks et al 2005). Under the assumption that effects of toxic stress and disturbance are purely additive, the population fractions unaffected by either stressor can be multiplied to determine the population fraction unaffected by both stressors combined (Traas et al 2002).…”