Design of a multidimensional density dependent matrix population model for assessing ecological risk to fish populations

Abstract

Modeling exposure and recovery of fish and wildlife populations after stressor mitigation serves as a basis for evaluating population status and remediation success. Herein, we develop a novel multidimensional density dependent matrix population model that analyzes both size-structure and age class-structure of the population simultaneously over time. This population modeling approach emphasizes application in conjunction with field monitoring efforts (e.g., through effects-based monitoring programs) and/or laboratory analysis to link effects due to chemical and/or nonchemical stressors to adverse outcomes in whole organisms and populations. For demonstration purposes, we applied the model to investigate population trajectories for Atlantic killifish (Fundulus heteroclitus) exposed to 112, 296, and 875 pg/g of 2,3,7,8-tetrachlorodibenzo-p-dioxin with effects on fertility and survival rates. The Atlantic killifish is an important and well-studied model organism for understanding the effects of pollutants and other stressors in estuarine and marine ecosystems. For each exposure concentration, the corresponding plots of total population size, population size structure, and age structure over time were generated. For example, exposure to 875 pg/g of 2,3,7,8-tetrachlorodibenzo-p-dioxin resulted in a 13.1% decline in population size after 2 years, a 22.1% decline in population size after 5 years, and a 27.9% decline in population size over 10 years with plots of all size classes and age classes exhibiting declines. The present study serves as an example of how multidimensional matrix population models are useful tools for ecological risk assessment because they integrate effects across the life cycle, provide a linkage between endpoints observed in the individual and ecological risk to the population as a whole, and project outcomes for multiple generations.

Introduction

Modeling exposure and recovery of fish and wildlife populations after stressor mitigation serves as a basis for evaluating population status and remediation success. Matrix population models are useful tools for ecological risk assessment because they integrate effects across the life cycle, provide a linkage between endpoints observed in the individual and ecological risk to the population as a whole, and project outcomes for many generations in the future. Matrix population models originally were designed to classify individuals by a single characteristic (Lewis 1942, Leslie 1945). Such a modeling approach describes population dynamics in terms of a single “i-state” or dimension (i.e. age) of individuals making up the population (Metz and Dickmann 1986, Roth and Caswell 2016). The model projects the distribution of the population as a set of possible outcomes assigned to the “i-state” or dimension over time (i.e. age classes), where individuals move among these possible outcomes (age classes) according to the vital rates presented in the Leslie matrix. However, as data collection methods have advanced and more sophisticated data sets are produced in the laboratory and field settings, multiple dimensions within demographic models will become increasingly important. A matrix modeling approach with an arbitrary number of dimensions has been described as a “hyperstate” model because it contains multiple i-states (Roth and Caswell, 2016). The ability to incorporate additional dimensions into matrix models (e.g., size structure if appropriate) will enable extension of demographic state spaces, as well as directly impact data usage to guide conservation and population management practices.

Herein, we used Atlantic killifish (Fundulus heteroclitus) toxicity and demographic data to demonstrate the value of a multidimensional density dependent logistic matrix population model. Atlantic killifish is widely occurring species, with habitat ranging along the Atlantic coast from US Florida to the Maritime Provinces of Canada, and is an important and well-studied model organism for understanding the effects of pollutants and other stressors in estuarine and marine ecosystems. We formulated a model that analyzes both size-structure and age class-structure of the population simultaneously, and is capable of producing analyses integrating multiple i-states of the population over time. The model is an expanded version of the single i-state density dependent models previously developed for fathead minnow (Pimephales promelas) and white sucker (Catostomus commersoni) exposed to endocrine disrupting chemicals (Ankley et al. 2008, Miller and Ankley 2004, Miller et al. 2007, Miller et al. 2013, Miller et al. 2015) to now allow for evaluation of multiple i-states simultaneously.

The resulting model has the ability to predict population-level outcomes, such as total population size, population recruitment, or population loss due to density dependence at each time step interval using direct measurements of vital rates of the impacted Atlantic killifish population. In addition, this modeling approach has the ability to project population age structure and size structure over any given length of time series of exposure. Further, this model allows for analysis that integrates multiple i-states, such as evaluation of size class structure within a given age class. We applied the model using data from a previously published study evaluating changes in vital rates for Atlantic killifish exposed to 2,3,7,8-tetrachlorodibenzo-p-dioxin, whereby exposure was found to negatively impact both fertility and survivorship (Munns et al . 1997).

Credible species-specific model constructs are needed to make linkages between responses occurring to individual organisms and their respective populations. The present study addresses the challenge of translating individual-level data to population-level responses. Our approach specifically allows estimation of changes in population trends for a selected fish species, demonstrated in this study for Atlantic killifish, under given exposure conditions that yield measurable changes in fecundity and survival. The population model derived and applied herein can be utilized in conjunction with field monitoring efforts (e.g., through effects-based monitoring programs) and/or laboratory analysis to link effects resulting from exposure to stressors to adverse outcomes in whole organisms and populations. Future deployment of the model could include other species and complex scenarios consisting of multiple stressors, whereby the model can be used to examine multiple dimensions of a population exposed to both chemical and/or non-chemical stressors. Accounting for density dependent population growth is a difficulty facing many ecotoxicological risk assessments. The density-dependent modeling construct described herein is a suitable methodology for translation of impacts across biological levels of organization from the individual to the population, which could be employed in a broader context to develop research and management strategies for assessing ecological risk and implementing restoration to impacted populations of fish and wildlife.

Methods

Model development

We build upon the population modeling approach originally described by Miller and Ankley (2004), which is a density-dependent logistic matrix model constructed using a Leslie projection matrix (Leslie 1945) in combination with the logistic equation. This approach has been used for analysis across multiple species and in combination with both laboratory based experimental design as well as in field applications, including effects based monitoring studies (Miller and Ankley 2004, Miller et al. 2007, Ankley et al. 2008, Miller et al. 2013, Miller et al. 2015):

$${\mathbf{\text{n}}}_{\mathit{t}+\mathbf{1}}=\text{exp}\left(-{\text{rPP}}_t/\text{K}\right){\mathbf{\text{M}}}_{\mathbf{1}}{\mathbf{\text{n}}}_{\mathit{t}}$$ (1)

$${\mathbf{\text{M}}}_{\mathbf{1}}=\mathbf{\text{M}}-{\mathbf{\text{C}}}_{\mathit{t}}$$ (2)

whereby ${\mathbf{\text{n}}}_{\mathit{t}+\mathbf{1}}$ is the vector of population age and size structure at time $t+1$, ${\mathbf{\text{n}}}_{\mathit{t}}$ is the vector of population age and size structure at time $t$, $r$ is the intrinsic rate of increase, ${\text{P}}_t$ is the population size at time $t$, $\text{K}$ is carrying capacity, and ${\mathbf{\text{M}}}_{\mathbf{1}}$ is the Leslie matrix containing vital rates (survivorship and fertility) that have been adjusted to include an age-specific and/or size-specific percentage reductions in fecundity and/or survival rates over the time step $t$ resulting from an exposure to chemical and/or nonchemical stressors. Thus, within Equation (2), ${\mathbf{\text{C}}}_{\mathit{t}}$ is a matrix representing the discount to vital rates resulting from exposure to the stressor(s) at time $\text{t}$. The matrix ${\mathbf{\text{C}}}_{\mathbf{\text{t}}}$ is represented by elements ${\text{c}}_{i,j},i=1,2\dots ,\text{n}$ and $j=1,2,\dots ,\text{n}$. Within the matrix ${\mathit{C}}_{\mathit{t}}$, ${\text{c}}_{1,j}={\alpha }_j\times F_j$, for $j=1,2,\dots ,\text{n}$, whereby ${\alpha }_j$ is the percentage discount to the fertility rate of age class $j\left({\text{F}}_j\right)$. Furthermore, ${\text{c}}_{i+1,i}={\beta }_i\times {\text{S}}_i$, for $i=1,2,\dots ,\text{n}-1$, whereby ${\beta }_i$ is the discount to the survival rate corresponding to age class $i\left({\text{S}}_i\right)$. All other elements of ${\mathbf{\text{C}}}_{\mathit{t}}$ are equal to 0. In addition, compensation is achieved by density dependence that acts cumulatively over life stages and generations to stabilize the population. Population regulation via compensatory responses must ultimately occur through density-dependent changes in rates of mortality or in reproductive success (Rose et al. 2001). The default assumption of no compensation would be so overly conservative that it would become impractical and economically inefficient as in such cases the fish populations would grow to indefinite size (Rose et al. 2001). The matrix population projection model originally proposed by Lewis (1942) and Leslie (1945) are limited in application because they describe exponential population growth. Both the total population and each age class within the population grow exponentially (Lewis 1942, Leslie 1945, Gotelli 1998, Caswell 2001). The logistic equation has been used for analysis of population projection within multiple studies of both fish and wildlife populations and is widely accepted as a model for density-dependent population growth (Maynard-Smith 1968, Roughgarden 1975, May 1976, Whittaker and Goodman 1979, Krebs 1985, Strebel 1985, Bayliss 1989, Burgman et al. 1993, Gotelli 1998). Within a population, density-dependent competition results from increased population size. Under the assumption of logistic population growth, fecundity and survival rates of all individuals across all age classes will be affected, exhibiting an increase or decrease based on the amount of food and resources available. The formulation of the model of Equation (1) and Equation (2) provides a link between life-history parameters that can be directly measured at the field site, the logistic equation, and Leslie’s original projection matrix.

Using birth pulse survival probabilities, birth pulse fertilities, and a prebreeding census (Gotelli 1998, Caswell 2001), formulation of the Leslie matrix as used in the present study can be written for eight non-overlapping size classes within 4 age classes (with 2 unique non-overlapping size classes per age class) as:

$$\mathbf{\text{M}}=\left|\begin{array}{cccccc}{\text{F}}_{1,3,1}& {\text{F}}_{1,4,1}& {\text{F}}_{2,5,1}& {\text{F}}_{2,6,1}& {\text{F}}_{3,7,1}& {\text{F}}_{3,8,1}\\ {\text{F}}_{1,3,2}& {\text{F}}_{1,4,2}& {\text{F}}_{2,5,2}& {\text{F}}_{2,6,2}& {\text{F}}_{3,7,2}& {\text{F}}_{3,8,2}\\ {\text{S}}_{1,3,5}& {\text{S}}_{1,4,5}& 0& 0& 0& 0\\ {\text{S}}_{1,3,6}& {\text{S}}_{1,4,6}& 0& 0& 0& 0\\ 0& 0& {\text{S}}_{2,5,7}& {\text{S}}_{2,6,7}& 0& 0\\ 0& 0& {\text{S}}_{2,5,8}& {\text{S}}_{2,6,8}& 0& 0\end{array}\right|$$ (3).

In examining the vital rates within the matrix $\mathbf{\text{M}}$, fertility rates are indexed first by the age class producing offspring, secondly by the size class of the fish producing offspring and, thirdly by the size class of the offspring produced. For example, ${\text{F}}_{3,7,1}$ corresponds to recruitment of offspring of size class 1 by age 3 fish that are in size class 7. Survival rates are indexed first by current age class, secondly by the current size class, and thirdly by the size class that the fish will move into over the time step t. Thus, ${\text{S}}_{2,6,8}$ would represent the survival rate of age 2 moving from size class 6 into size class 8 over the time step t. Within the matrix $\mathbf{\text{M}}$, fertility rates were calculated by adjusting fecundity for sex ratio, percentage of maturity, and age class 0 survival. The dominant eigenvalue of the Leslie matrix represents the finite rate of increase. Taking the natural log of the dominant eigenvalue of the Leslie matrix results in the intrinsic rate of increase, an estimate of the per capita rate of population increase associated with a population represented by the rates in the matrix.

In simple qualitative terms, the application of the density-dependent logistic matrix model of Equations (1) and (2) requires no additional parameters beyond what is found in a combined life and fecundity table that contains size class estimates by age, which is converted to the Leslie projection matrix. The model also requires an estimate of carrying capacity which is commonly obtained through GIS analysis (Mladenoff et al. 1995, Mladenoff et al. 1997) and an estimate of the effect of the chemical stressor on the vital rates of fertility and survivorship as found in the Leslie projection matrix. To provide an indication of relative impact, output of the model is expressed invariant of carrying capacity by plotting population size proportional to carrying capacity at each time step in the model, as opposed to evaluation on the basis of absolute numbers. In demonstrating model output in this manner, a value of 1.0 represents a population at carrying capacity and values between 0 and 1.0 represent a population below the carrying capacity threshold.

Model demonstration

We demonstrated this population model to investigate population trajectories for Atlantic killifish with dietary exposures to 112, 296, and 875 pg/g of 2,3,7,8-tetrachlorodibenzo-p-dioxin using the effects on fertility and survival rates as determined by Munns et al. (1997). In that study, changes in adult survival rate relative to control fish were observed at all levels of exposure, and changes in survival of age class 0 fish were documented at the highest exposure concentration (Table 1). Further, changes in fertility rate relative to control fish were not observed at an exposure concentration of 112 pg/g of 2,3,7,8-tetrachlorodibenzo-p-dioxin, but occurred across all breeding fish for exposures concentrations of 296 pg/g and 875 pg/g 2,3,7,8-tetrachlorodibenzo-p-dioxin relative to control fish (Table 1).

Table 1.

Effects on annual fertility and survival rates for Fundulus heteroclitus with exposures to 112, 296, and 875 pg/g of dioxin (Munns et al. 1997).

			Exposure Concentration
	112 pg/g Dioxin		296 pg/g Dioxin		875 pg/g Dioxin
	Change In Fertility Rate	Change In Survival Rate	Change In Fertility Rate	Change In Survival Rate	Change In Fertility Rate	Change In Survival Rate
	(Relative To Control)	(Relative To Control)	(Relative To Control)	(Relative To Control)	(Relative To Control)	(Relative To Control)
Age Class 0	Not Applicable	0	Not Applicable	0	Not Applicable	−0.0001724
Age Class 1	0	−0.01387	−0.05627	−0.004268	−0.1013	−0.04162
Age Class 2	0	−0.01387	−0.05624	−0.004268	−0.1001	−0.04162
Age Class 3	0	−0.01387	−0.0563	−0.004268	−0.09496	−0.04162

Notes:

^1.All exposures were dietary exposures.

^2.Effects of Age Class 0 comrise effects for embryos, larvae, and 28 day larvae.

^3.Age Class 0 fish do not breed and thus effects on fertility rate for Age Class 0 fish are not applicable.

A combined life and fecundity table that contains size class estimates by age for Atlantic killifish was constructed using observed data collected from ongoing field studies, and also using a series of previous studies with Atlantic killifish populations unexposed to chemical stress (See Table 2.) (Horton 1965, Fritz and Garside 1975, Valiela et al. 1977, Kneib and Stiven 1978, Taylor 1986). Female Atlantic killifish mature at age 1 year and reproduce annually with reproductive output that varies with size class. Egg production for a given size class was estimated using an allometric regression function derived from previous study by Kneib and Stiven (1978) relating size to egg production for Atlantic killifish.

Table 2:

Summary of observed annual survival and fecundity (eggs/female) of Fundulus heteroclitus by age and size (Horton 1965, Fritz and Garside 1975, Valiela et al. 1977, Kneib and Stiven 1978, Taylor 1986)

Summary of Observed Survival and Fecundity By Age and Size
Age	Size	Survival	Fecundity
0	<30mm	0.004	0
	>30mm To 40mm	0.006	0
1	>40mm to 50mm	0.4	283
	>50mm to 60mm	0.6	465
2	>60mm to 70mm	0.16	647
	>70mm to 80mm	0.24	829
3	>80mm To 90mm	0	1011
	> 90mm	0	1193

A Leslie projection matrix was developed for Atlantic killifish using birth-pulse fertilities and a prebreeding census (See Figure 1.) (Gotelli 1998, Caswell 2001). Following the methods of Miller and Ankley (2004), fertility rates within the matrix were calculated by adjusting fecundity for sex ratio, percentage of maturity, and age class 0 survival. The dominant eigenvalue of the Leslie matrix represents the finite rate of increase. Taking the natural log of the dominant eigenvalue of the Leslie matrix results in the intrinsic rate of increase, an estimate of the per capita rate of population increase associated with a population represented by the rates in the matrix (Gotelli 1998, Caswell 2001). The proportion of breeding female fish was estimated to be 50% in age class 1, 80% in age class 2, and 100% for age class 3. Further, survival beyond age class 3 was not allowed as a maximum age of 4 yr has been recorded for Atlantic killifish in the field (Horton 1965, Fritz and Garside 1975, Valiela et al. 1977, Kneib and Stiven 1978, Taylor 1986). Two non-overlapping size classes can be found within each age class. Over a given time step, there is a probability of 0.75 that a smaller fish will remain in the smaller size class for a given age class and a probability of 0.75 that a larger fish remain in the larger size class for a given age class (there is a probability of 0.25 of changing size class).

Figure 1.

Leslie projection matrix for Atlantic killifish using birth-pulse fertilities and a prebreeding census.

The Leslie matrix formulated for Atlantic killifish was incorporated within the population model of Equations (1) and (2) to project population status resulting from exposure to varying concentrations of 2,3,7,8-tetrachlorodibenzo-p-dioxin. For simplicity, the population was initiated at carrying capacity and at the stable age distribution as determined by finding an eigenvector (right eigenvector) associated with the dominant eigenvalue of the matrix (Leslie 1945, Caswell 2001). The population model was executed over a 10-yr simulation period, and results were recorded using an annual time step. Equilibrium population size was also determined for each exposure concentration. All model projections utilized the intrinsic rate of increase as calculated from the Leslie matrix.

Results

The summary of life and fecundity measures for Atlantic killifish (Table 2) was used to construct the corresponding Leslie projection matrix (Figure 1) that yielded an intrinsic rate of increase equal to 0.251. Exposure to 2,3,7,8-tetrachlorodibenzo-p-dioxin compromised the fecundity and survival of Atlantic killifish (see Table 1), and thus the annual vital rates of fertility and survival for the affected age and size classes within the Leslie matrix were adjusted accordingly to account for the exposure within model projections. Projected trends in total population size over time for an Atlantic killifish population existing at carrying capacity and subsequently exposed to varying levels of 2,3,7,8-tetrachlorodibenzo-p-dioxin including 112, 296, and 875 pg/g were calculated using the Leslie projection matrix (Figure 1) with Equation (2), and plotted (Figure 2). An Atlantic killifish population initially at carrying capacity and subsequently exposed to a concentration of 112 pg/g of 2,3,7,8tetrachlorodibenzo-p-dioxin resulted in only a minimal effect on the population with an approximate 1% decrease in population size after 2 years, a 1.8% reduction in population size after 5 years, a 2.3% reduction in population size after 10 years, and an equilibrium population size of 97.6%. In comparison, an Atlantic killifish population initially at carrying capacity and subsequently exposed to 296 pg/g of 2,3,7,8-tetrachlorodibenzo-p-dioxin experienced an approximate 6.1% decrease in population size after 2 years, a 10.3% reduction in population size after 5 years, a 12.8% reduction in population size after 10 years and an equilibrium population size of approximately 86.3%. Furthermore, an Atlantic killifish population initially at carrying capacity and subsequently exposed to 875 pg/g of 2,3,7,8-tetrachlorodibenzo-p-dioxin was calculated to have a 13.1% reduction in population size after 2 years, a 22.1% reduction in population size after 5 years, a 27.9% reduction in population size after 10 years, and an equilibrium population size of 69.3%.

Figure 2.

Total population size over time for an Atlantic killifish population existing at carrying capacity and subsequently exposed to varying levels of 2,3,7,8-tetrachlorodibenzo-p-dioxin including 112, 296, and 875 pg/g.

Model output consisting of time series trends in population status over time was also generated specific to a given dimension of the matrix to examine both patterns in age class structure and size class structure of the population associated with each exposure concentration. For example, trajectories for age-class structure over time for an Atlantic killifish population initially at carrying capacity and subsequently exposed to 296 pg/g of dioxin demonstrated a consistent decline across all age classes over ten years (Figure 3, Panel A). Similarly, trajectories for size structure over time for an Atlantic killifish population initially at carrying capacity and subsequently exposed to 296 pg/g of dioxin showed a corresponding decline across all age classes over ten years (Figure 3, Panel B).

Figure 3.

Panel A: Age class structure over time for an Atlantic killifish population initially at carrying capacity and subsequently exposed to 296 pg/g 2,3,7,8-tetrachlorodibenzo-p-dioxin. Panel B: Size class structure over time for an Atlantic killifish population initially at carrying capacity and subsequently exposed to 296 pg/g of 2,3,7,8-tetrachlorodibenzo-pdioxin.

In addition, model output was examined for a given combination of dimensions of the matrix in order to demonstrate patterns specific to a particular age and size of fish within the population when comparing the response to all exposure concentrations. As an example, the trends in abundance over time were compared for age class 1, size class 40 to 50mm fish within a population at carrying capacity and subsequently exposed to either 112, 296, or 875 pg/g of of 2,3,7,8-tetrachlorodibenzo-p-dioxin (Figure 4). As demonstrated, exposure to 112 pg/g of 2,3,7,8-tetrachlorodibenzo-p-dioxin resulted in only an approximate 1.9% decline in age class 1, size class 40 to 50mm fish from baseline conditions. However, exposures to 296 pg/g and 875 pg/g of 2,3,7,8-tetrachlorodibenzo-p-dioxin resulted in more substantial declines in age class 1, size class 40 to 50 mm fish equivalent to approximately 13.8% and 28.8% from baseline conditions respectively.

Figure 4.

Trends in abundance over time were compared for age class 1, size class 40–50cm fish within a population at carrying capacity and subsequently exposed to either 112, 296, or 875 pg/g of of 2,3,7,8-tetrachlorodibenzo-p-dioxin.

Discussion

Examining and comprehending how a population responds to stressors (both chemical and non-chemical) is of primary importance in assessing ecological risk. In order to be applicable for decision-making purposes, observations on individuals will need to be extrapolated to population level responses. Therefore, the development of modeling approaches that can be used together with field-based monitoring and/or laboratory analysis to translate responses measured at the level of the individual to impacts on populations is substantially important (Forbes et al. 2008, Barnthouse et al. 2009, Miller et al. 2013, Miller et al. 2015). Population models allow for individual-level observations to be related to effects on populations, and as current field monitoring efforts become more sophisticated with the capability of collecting individual level data comprising multiple attributes, the need for modeling approaches that can be used within multidimensional space will increase.

The study herein includes the formulation and demonstration of a novel multidimensional density dependent logistic matrix population model for Atlantic killifish exposed to a single chemical stressor, 2,3,7,8-tetrachlorodibenzo-p-dioxin. For each level of exposure including 112, 296, and 875 pg/g of of 2,3,7,8-tetrachlorodibenzo-p-dioxin, the total population as well as all age classes and size classes exhibited declines over time. This can be directly attributed to the effects on survivorship and fecundity documented for individual fish (Munns et al., 1997). Orchestration of the model showcased trends in age structure over time, size structure over time, and specific combination of age and size structure over time. The modeling construct demonstrated here provides value in the ability to examine multiple dimensions simultaneously (attributes of both size and age structure over time), as well as to examine effects unique to a given age and size of fish within the population. Moreover, it provides the ability to compare across particular subgroups within the population that are responding under a given stress load.

In moving beyond the model as demonstrated in the present study, further development should include a focus on the potential of killifish to adapt to exposure to 2,3,7,8-tetrachlorodibenzo-p-dioxin. Previous studies with some toxic chemicals (i.e. the PCB congener 3,3’4,4’,5 hexachlorobiphenyl or PCB 126) involving killifish populations residing in highly contaminated sites have documented the ability of killifish to tolerate and/or respond adaptively to chemical exposures (Van Veld and Nacci 2008, Nacci et al. 2010). In considering extension of the model of Equations (1) through (3) to include more than two individual characteristics (two individual dimensions), a logical next step will be to consider exposure history when applied within a time series that includes multiple generations of fish and (through adaptation) a difference in susceptibility to toxicity that varies by generation. Such a hyperstate model could provide a more accurate estimate of the impacts of chemically induced stress to the population, and would be needed to investigate the long term ecological impacts of chemical pollution. Further, examining population response to chemical stress that is specific to subsequent adapting generations of fish within a time series would enable a quantification associated with tolerance exhibited by wild fish.

The model developed here (Equation (1)) is designed to be utilized in combination with other models as a linked mathematical framework. For example, the model can be used as a component in the formulation of a quantitative Adverse Outcome Pathway (qAOP) consisting of a series of biologically based, computational models describing key event relationships linking a molecular initiating event (MIE) to an adverse outcome. For example, Conolly et al. (2017) presented a qAOP describing the linkage between inhibition of cytochrome P450 19A aromatase (the MIE) and population-level decreases in the fathead minnow (Pimephales promelas). The qAOP consisted of three linked computational models for the following: (a) the hypothalamic-pitutitary-gonadal axis in female FHMs, where aromatase inhibition decreases the conversion of testosterone to 17β-estradiol (E2), thereby reducing E2-dependent vitellogenin (VTG; egg yolk protein precursor) synthesis, (b) VTG-dependent egg development and spawning (fecundity), and (c) fecundity-dependent population projection using a density dependent matrix model (Miller and Ankley 2004, Miller et al. 2007, Conolly et al. 2017). As an additional example, the model can be applied to address resource limitation in combination with chemical stress as appropriate for linking together with a dynamic energy budget (DEB) model within a coupled modeling framework (Kooijman 2000, Muller and Nisbet 2000, Nisbet et al. 2000, Klanjscek et al. 2006). In this application, food availability and the resulting energy intake and budgeting are major factors affecting growth, survival and reproduction of individuals which in turn can be used for calculation of the vital rates that comprise the Leslie projection matrix within the density dependent logistic population model for projection of population dynamics for a wide range of parameter values and environmental conditions (Figure 5). Ecologically relevant key events within an adverse outcome pathway can be interpreted as measures of damage inducing processes within the DEB model with subsequent effects on population status (Murphy et al. 2018).

Figure 5.

Application of a density dependent logistic population model to address resource limitation in combination with chemical stress as appropriate for linking together with a dynamic energy budget model (DEB) within a coupled modeling framework (klanjscek et al. 2006).

Application of the modeling approach presented here could be expanded upon to include investigation of multiple stressors. For illustration purposes, we applied the population model under hypothetical scenarios that included exposure to 296 pg/g of dioxin in combination with an additional hypothetical stressor that reduced both fecundity and growth. Three different scenarios were modeled and trajectories for total population size over time were determined (Figure 6). Scenario 1 included exposure to 296 pg/g 2,3,7,8-tetrachlorodibenzo-p-dioxin in combination with an additional hypothetical stressor that results in a 5% further reduction to fecundity and a 5% increase in probability that a fish is in the smaller size class within each given age class. Scenario 2 included exposure to 296 pg/g 2,3,7,8-tetrachlorodibenzo-p-dioxin in combination with an additional hypothetical stressor that results in a 10% further reduction to fecundity and a 10% increase in probability that a fish is in the smaller size class within each given age class. Scenario 3 included exposure to 296 pg/g 2,3,7,8-tetrachlorodibenzo-p-dioxin in combination with a hypothetical stressor that results in a 15% further reduction to fecundity and a 15% increase in probability that a fish is in the smaller size class within each given age class. In addition to total population size, population age structure and population size structure were investigated for each scenario. For example, population trajectories for size-class structure and age class structure over time for Atlantic killifish with exposure to 296 pg/g 2,3,7,8-tetrachlorodibenzo-p-dioxin and exposure to additional stress that results in 15% further reduction to fecundity and a 15% increase in probability that a fish is in the smaller size class within each given age class (Scenario 3) are shown in Figure 7. Any number of additional stressors can be included for analysis using the modeling approach demonstrated here given that the cumulative effects on vital rates can be measured.

Figure 6.

Application the population model of Equation (1) under hypothetical multiple stressor scenarios. Scenario1 : Exposure to 296 pg/g 2,3,7,8-tetrachlorodibenzo-p-dioxin in combination with an additional hypothetical stressor that results in a 5% further reduction to fecundity and a 5% increase in probability that a fish is in the smaller size class within each given age class. Scenario 2: Exposure to 296 pg/g 2,3,7,8-tetrachlorodibenzo-p-dioxin in combination with an additional hypothetical stressor that results in a 10% further reduction to fecundity and a 10% increase in probability that a fish is in the smaller size class within each given age class. Scenario 3: Exposure to 296 pg/g 2,3,7,8-tetrachlorodibenzo-p-dioxin in combination with a hypothetical stressor that results in a 15% further reduction to fecundity and a 15% increase in probability that a fish is in the smaller size class within each given age class.

Figure 7.

Panel A: Age class structure over time for an Atlantic killifish population initially at carrying capacity and subsequently exposed to 296 pg/g 2,3,7,8-tetrachlorodibenzo-p-dioxin in combination with a hypothetical stressor that results in a 15% further reduction to fecundity and a 15% increase in probability that a fish is in the smaller size class within each given age class. Panel B: Size class structure over time for an Atlantic killifish population initially at carrying capacity and subsequently exposed to 296 pg/g 2,3,7,8-tetrachlorodibenzo-p-dioxin in combination with a hypothetical stressor that results in a 15% further reduction to fecundity and a 15% increase in probability that a fish is in the smaller size class within each given age class.

The model of Equations (1) through (3) utilized assumptions including a closed system and nonoverlapping size classes distributed among age classes. However, a metapopulation modeling approach (Galic et al. 2010) that integrates multi-patch spatial dynamics within Equation (1) can accommodate both immigration and emigration scenarios using the addition of population vectors to represent individuals of each age class entering or exiting the population at each time step. For example, expanding Equation (1) yields:

$${\mathbf{\text{n}}}_{t+\mathbf{1}}=\text{exp}\left(-{\text{rP}}_t/\text{K}\right){\mathbf{\text{M}}}_{\mathbf{1}}{\mathbf{\text{n}}}_{\mathit{t}}+{\mathbf{\text{i}}}_{\mathbf{\text{t}}}-{\mathbf{\text{e}}}_{\mathbf{\text{t}}}$$ (4)

where ${\mathbf{\text{i}}}_{\mathbf{\text{t}}}$ is a vector of age structure at time t entering the population through immigration, and ${\mathbf{\text{e}}}_{\mathbf{\text{t}}}$ is a vector of age structure at time t exiting the population through emigration. Application of a model construct whereby immigration and emigration fluxes are included will affect the resulting population trajectories because of the influence of the age structure, size, and exposure history of fish moving into and out of the population at each time step. In considering the use of overlapping size classes within the model formulation presented herein, the Leslie projection matrix of Equation (3) could be constructed to allow for size classes to overlap among the various age classes. As an example, a Leslie projection matrix, ${\mathbf{\text{M}}}_{\mathbf{2}}$, was formulated for a population comprised of 4 overlapping size classes and 4 age classes, whereby fish can only grow larger or stay the same size over one time step (Figure 8). In the same manner as with $\mathbf{\text{M}}$, within ${\mathbf{\text{M}}}_{\mathbf{2}}$ fertility rates are indexed first by the age class producing offspring, secondly by the size class of the fish producing offspring and, thirdly by the size class of the offspring produced. Also, survival rates are indexed first by current age class, secondly by the current size class, and thirdly by the size class that the fish will move into over the time step t. The matrix ${\mathbf{\text{M}}}_{\mathbf{2}}$ could then be utilized as a replacement for $\mathbf{\text{M}}$ within Equations (1) and (2) and to project simultaneously both size structure and age structure of the population over time.

Figure 8.

A Leslie projection matrix formulated for a population comprised of 4 overlapping size classes and 4 age classes.

Density dependence is the phenomenon by which the values of vital rates including survivorship and fecundity depend upon the density of the population (Burgman et al. 1993). Within a population, density-dependent competition results from increased population size, and fecundity and survival rates will decrease or increase based upon the amount of food and resources available to organisms. Within the present study, we represent density dependence using the logistic equation. When modeling populations whereby detailed information has been collected for the effect of density on individual age classes, Equation (1) offers the versatility of replacing exp $\left(-r{P}_t/K\right)$ with age-specific density-dependent functions. This could be accomplished in Equation (1) by replacing exp $\left(-r{P}_t/K\right)$ with a diagonal matrix of age specific density dependent functions. Alternatively, when data exists such that density dependent functions could be precisely estimated for each vital rate within the Leslie matrix, the Leslie matrix could then be separated into a matrix for births and a matrix for deaths (Goodman 1969, Usher 1969, Jensen 1974, 1995), and separate density dependent functions could be applied to each process. Although this approach seems to be most realistic theoretically, difficulty in parameter estimation related to each separate vital rate specific density dependent function will drastically increase the uncertainty of model predictions.

In summary, for the model development and application presented herein we chose to focus on Atlantic killifish, as it is a commonly occurring species with a wide ranging habitat that is routinely used as a model species for investigating the effects of stressors in aquatic systems. However, the model is easily adaptable to any species for which a life table containing vital rates of survivorship and fecundity, size and age measurements of the population, and the effect of stressors on vital rates has been collected. The model can be deployed in combination with both field and laboratory scenarios, and is capable of being applied with spatially explicit data collected within habitat suitability and restoration studies. By developing models, such as the one presented in the current study, that can accommodate variation in extended demographic state spaces and account for ecological factors such as density dependence, we can more easily move towards deployment of species-specific matrix models capable of characterizing population status based upon multiple attributes and make predictions of the ecological impact of stressors based upon a broader encompassment of available data.