Decrease, Population growth rate

A population can be defined as a group of interbreeding organisms, all of the same species, occupying a specific space during a specific time (Vandermeer and Goldberg 2003, Gotelli 2008).  As the population is the biological level of organization that is often the focus of ecological risk assessments, population growth rate (and hence population size over time) is important to consider within the context of applied conservation practices.

If N is the size of the population and t is time, then the population growth rate (dN/dt) is proportional to the instantaneous rate of increase, r, which measures the per capita rate of population increase over a short time interval. Therefore, r, is a difference between the instantaneous birth rate (number of births per individual per unit of time; b) and the instantaneous death rate (number of deaths per individual per unit of time; d) [Equation 1]. Because  r is an instantaneous rate, its units can be changed via division.  For example, as there are 24 hours in a day, an r of 24 individuals/(individual x day) is equal to an r of 1 individual/(individual/hour) (Caswell 2001, Vandermeer and Goldberg 2003, Gotelli 2008, Murray and Sandercock 2020). 

Equation 1:  r = b - d

This key event refers to scenarios where r < 0 (instantaneous death rate exceeds instantaneous birth rate).

Examining r in the context of population growth rate:

● A population will decrease to extinction when the instantaneous death rate exceeds the instantaneous birth rate (r < 0).  

           ● The smaller the value of r below 1, the faster the population will decrease to zero.  

● A population will increase when resources are available and the instantaneous birth rate exceeds the instantaneous death rate (r > 0)

           ● The larger the value that r exceeds 1, the faster the population can increase over time      

● A population will neither increase or decrease when the population growth rate equals 0 (either due to N = 0, or if the per capita birth and death rates are exactly balanced).  For example, the per capita birth and death rates could become exactly balanced due to density dependence and/or to the effect of a stressor that reduces survival and/or reproduction (Caswell 2001, Vandermeer and Goldberg 2003, Gotelli 2008, Murray and Sandercock 2020).     

Effects incurred on a population from a chemical or non-chemical stressor could have an impact directly upon birth rate (reproduction) and/or death rate (survival), thereby causing a decline in population growth rate.  

● Example of direct effect on r:  Exposure to 17b-trenbolone reduced reproduction (i.e., reduced b) in the fathead minnow over 21 days at water concentrations ranging from 0.0015 to about 41 mg/L (Ankley et al. 2001; Miller and Ankley 2004).             

Alternatively, a stressor could indirectly impact survival and/or reproduction.  

● Example of indirect effect on r:  Exposure of non-sexually differentiated early life stage fathead minnow to the fungicide prochloraz has been shown to produce male-biased sex ratios based on gonad differentiation, and resulted in projected change in population growth rate (decrease in reproduction due to a decrease in females and thus recruitment) using a population model. (Holbech et al., 2012; Miller et al. 2022)

Density dependence can be an important consideration:

● The effect of density dependence depends upon the quantity of resources present within a landscape.  A change in available resources could increase or decrease the effect of density dependence and therefore cause a change in population growth rate via indirectly impacting survival and/or reproduction.  

● This concept could be thought of in terms of community level interactions whereby one species is not impacted but a competitor species is impacted by a chemical stressor resulting in a greater availability of resources for the unimpacted species.  In this scenario, the impacted species would experience a decline in population growth rate. The unimpacted species would experience an increase in population growth rate (due to a smaller density dependent effect upon population growth rate for that species).       

Closed versus open systems:

● The above discussion relates to closed systems (there is no movement of individuals between population sites) and thus a declining population growth rate cannot be augmented by immigration.  

● When individuals depart (emigrate out of a population) the loss will diminish population growth rate.  

Population growth rate applies to all organisms, both sexes, and all life stages.

Population growth rate (instantaneous growth rate) can be measured by sampling a population over an interval of time (i.e. from time t = 0 to time t = 1).  The interval of time should be selected to correspond to the life history of the species of interest (i.e. will be different for rapidly growing versus slow growing populations). The population growth rate, r, can be determined by taking the difference (subtracting) between the initial population size, N_t=0 (population size at time t=0), and the population size at the end of the interval, N_t=1 (population size at time t = 1), and then subsequently dividing by the initial population size. 

Equation 2:  r = (N_t=1 - N_t=0) / N_t=0

The diversity of forms, sizes, and life histories among species has led to the development of a vast number of field techniques for estimation of population size and thus population growth over time (Bookhout 1994, McComb et al. 2021).  

● For stationary species an observational strategy may involve dividing a habitat into units. After setting up the units, samples are performed throughout the habitat at a select number of units (determined using a statistical sampling design) over a time interval (at time t = 0 and again at time t = 1), and the total number of organisms within each unit are counted. The numbers recorded are assumed to be representative for the habitat overall, and can be used to estimate the population growth rate within the entire habitat over the time interval.  

● For species that are mobile throughout a large range, a strategy such as using a mark-recapture method may be employed (i.e. tags, bands, transmitters) to determine a count over a time interval (at time = 0 and again at time =1).   

Population growth rate can also be estimated using mathematical model constructs (for example, ranging from simple differential equations to complex age or stage structured matrix projection models and individual based modeling approaches), and may assume a linear or nonlinear population increase over time (Caswell 2001, Vandermeer and Goldberg 2003, Gotelli 2008, Murray and Sandercock 2020). The AOP framework can be used to support the translation of pathway-specific mechanistic data into responses relevant to population models and output from the population models, such as changing (declining) population growth rate, can be used to assess and manage risks of chemicals (Kramer et al. 2011). As such, this translational capability can increase the capacity and efficiency of safety assessments both for single chemicals and chemical mixtures (Kramer et al. 2011).  

Some examples of modeling constructs used to investigate population growth rate:

● A modeling construct could be based upon laboratory toxicity tests to determine effect(s) that are then linked to the population model and used to estimate decline in population growth rate.  Miller et al. (2007) used concentration–response data from short term reproductive assays with fathead minnow (Pimephales promelas) exposed to endocrine disrupting chemicals in combination with a population model to examine projected alterations in population growth rate.  

● A model construct could be based upon a combination of effects-based monitoring at field sites (informed by an AOP) and a population model.  Miller et al. (2015) applied a population model informed by an AOP to project declines in population growth rate for white suckers (Catostomus commersoni) using observed changes in sex steroid synthesis in fish exposed to a complex pulp and paper mill effluent in Jackfish Bay, Ontario, Canada. Furthermore, a model construct could be comprised of a series of quantitative models using KERs that culminates in the estimation of change (decline) in population growth rate.  

● A quantitative adverse outcome pathway (qAOP) has been defined as a mathematical construct that models the dose–response or response–response relationships of all KERs described in an AOP (Conolly et al. 2017, Perkins et al. 2019). Conolly et al. (2017) developed a qAOP using data generated with the aromatase inhibitor fadrozole as a stressor and then used it to predict potential population‐level impacts (including decline in population growth rate). The qAOP modeled aromatase inhibition (the molecular initiating event) leading to reproductive dysfunction in fathead minnow (Pimephales promelas) using 3 computational models: a hypothalamus–pituitary–gonadal axis model (based on ordinary differential equations) of aromatase inhibition leading to decreased vitellogenin production (Cheng et al. 2016), a stochastic model of oocyte growth dynamics relating vitellogenin levels to clutch size and spawning intervals (Watanabe et al. 2016), and a population model (Miller et al. 2007).

● Dynamic energy budget (DEB) models offer a methodology that reverse engineers stressor effects on growth, reproduction, and/or survival into modular characterizations related to the acquisition and processing of energy resources (Nisbet et al. 2000, Nisbet et al. 2011).  Murphy et al. (2018) developed a conceptual model to link DEB and AOP models by interpreting AOP key events as measures of damage-inducing processes affecting DEB variables and rates.

● Endogenous Lifecycle Models (ELMs), capture the endogenous lifecycle processes of growth, development, survival, and reproduction and integrate these to estimate and predict expected fitness (Etterson and Ankley, 2021).  AOPs can be used to inform ELMs of effects of chemical stressors on the vital rates that determine fitness, and to decide what hierarchical models of endogenous systems should be included within an ELM (Etterson and Ankley, 2021).

Consideration of population size and changes in population size over time is potentially relevant to all living organisms.

Maintenance of sustainable fish and wildlife populations (i.e., adequate to ensure long-term delivery of valued ecosystem services) is a widely accepted regulatory goal upon which risk assessments and risk management decisions are based.

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