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DEVELOPING INTEGRAL PROJECTION MODELS FOR ECOTOXICOLOGY 1

0 – Front matter 2

0.1 - Authors and affiliations 3

N.L. Pollesch1, K.M. Flynn1, S.M. Kadlec1, J.A. Swintek2 and M. A. Etterson1 4

1 USEPA Office of Research and Development, Mid-Continent Ecology Division, 6201 Congdon Blvd, 5

Duluth, MN, USA 55804 6

2 Badger Technical Services, Duluth, MN, USA 55804 7

0.2 - Abstract 8

This paper presents a first application of integral projection models (IPMs) to ecotoxicology. In many 9

ecosystems, especially aquatic ecosystems, size plays a critical role in the factors that determine an 10

individual’s ability to survive and reproduce. In aquatic ecotoxicology, size measures are informative of 11

both realized and potential acute and chronic effects of chemical exposure. This paper demonstrates 12

how chemical and non-chemical effects on growth, survival, and reproduction can be linked to 13

population-level impacts using size-structured IPMs. The modeling approach was developed with the 14

goals and constraints of ecological risk assessors in mind, who are tasked with estimating the effects of 15

chemical exposures to wildlife populations in a data-limited environment. The included case study is a 16

collection of daily IPMs parameterized for the annual cycle of fathead minnow (Pimephales promelas) 17

which motivated the development of modelling techniques for seasonal, iteroparous reproduction and 18

size-dependent over-winter survival. Effects of a time-variable chemical exposure were incorporated 19

using a simplified threshold-exceedance model and a more detailed toxicokinetic-toxicodynamic model. 20

Results demonstrate that size-structured IPMs provide a promising framework for synthesizing 21


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ecotoxicologically relevant data and theory to explore assumptions of chemical effects and the resulting 22

population-level impact. 23

0.3 – Keywords 24

Integral projection model, aquatic ecotoxicology, ecological risk assessment, fathead minnow 25

(Pimephales promelas), size structured population model, toxicity translation 26

0.4 – Highlights 27

• Integral projection models (IPMs) are shown to be a promising approach for studying the 28

population level effects of natural and anthropogenic stressors considered in ecotoxicological 29

applications 30

• Fathead minnow (P. promelas) life history is used, with seasonal batch-spawning and over-31

winter survival built into daily IPM transition kernels 32

• Realistic time-variable chemical exposure and effects are linked to IPMs and results of threshold 33

exposure and toxicokinetic-toxicodynamic effects models are explored 34

• IPMs are developed with consideration given to the applications and data availability constraints 35

of ecological risk assessors 36

1 - Introduction 37

Wildlife populations are challenged by a diverse suite of natural and anthropogenic stressors. 38

Contaminant exposure is a part of the anthropogenic impact felt in every ecosystem on the planet (Kang 39

et al., 2012; Daly and Wania, 2004; Wania and Mackay, 1993). Ecotoxicologists and ecological risk 40

assessors work to understand these impacts and to estimate effects of exposure. A major challenge in 41

ecological risk assessment is one of extrapolation, including lab-to-field, individual-to-population, cross-42

species, and cross-chemical. Empirical observations to support ecological risk assessment can be limited 43


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by both practical and ethical constraints. Therefore, ecological risk assessors may supplement 44

laboratory-derived observation with models to estimate the effects of chemical exposures on 45

populations (Etterson et al., 2017). These include chemical exposure models based on the fate and 46

transport of chemicals, chemical effect models that range from simple exposure-response curves to 47

detailed multi-compartment toxicokinetic-toxicodynamic (TK-TD) models. These models also include 48

population and community models, sometimes referred to as toxicity translators, that estimate the 49

impacts of chemical exposure at the levels of biological organization most often aligned with regulatory 50

goals for wildlife protection (Kramer et al., 2011; Bennett & Etterson, 2007). 51

There is a long history of developing mathematical models to study aquatic population dynamics. From 52

Ricker’s (1940) foundational methods using ordinary differential equations, to current efforts utilizing 53

increasingly sophisticated mathematical and statistical approaches (e.g. Engen et al., 2018), we have 54

seen the better part of a century of research in this area. Models derived for aquatic populations, and 55

specifically fish, range from individual-based models to partial differential equations to matrix projection 56

models (MPMs) (Benoît & Rochet, 2004; Gleason & Nacci, 2001; Law, et al., 2009). Applications of these 57

models span from determining the effects of harvest in managed fisheries to the effects of 58

environmental contaminants on population viability (Miller et al., 2007; White et al., 2016). 59

Structured population models link population dynamics to one or more traits of individuals, such as size, 60

age, sex, or spatial location. The structuring variable depends on modeling objectives and available data 61

and each choice of structuring variable(s) has an associated set of advantages and drawbacks (Collie et 62

al., 2014). Common approaches to structured population models are MPMs, meta-population ordinary 63

differential equation models, and more recently, integral projection models (Ellner et al., 2016; 64

Tuljapurkar & Caswell, 2012; Akҫakaya, 2000). Size-structured IPMs have been applied in a wide variety 65

of areas; however, relatively few applications of IPMS for modeling fish populations are currently 66

available in the literature (e.g. Erickson et al., 2017; White et al., 2016). 67


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A goal of this research was to explore the use of size-structured IPMs for ecotoxicological applications, 68

and specifically for fish populations exposed to chemical stressors. For fish, gape-limited predation and 69

size-dependent over-winter survival are two mechanisms linking size to individual survival; reproductive 70

maturity and fecundity can also be a function of size (Urban, 2007; Danylchuk & Tonn, 2001; Divino & 71

Tonn, 2007; Sogard 1997), and growth is a measure of change in size, which can be modified by external 72

stressors. Endpoints from toxicological studies are also often organized as acute and chronic effects on 73

growth, reproduction, and survival. Therefore, in addition to straightforward modification of growth, 74

reproduction, and survival functions to fit basic life-history considerations, modification due to size-75

dependent effects of chemical exposure made size-structured IPM an intriguing formalism to study. 76

Consideration of the constraints on data availability and adequacy is a necessary step in model 77

development and implementation (Getz et al., 2018). Empirical size measurements from laboratory and 78

field studies are often more readily observed and measured than other structuring variables such as age, 79

or developmental stage. For example, age-structured fish population models are common with otolith 80

ring counting as an established way to measure fish age. Yet this method to measure age requires 81

extensive training and destructive sampling. In contrast, fish weight and length measurements are 82

collected more easily and do not entail destructive sampling. The relative availability of empirical size 83

observations to support model parameterization is a contributing factor motivating the exploration of 84

size-structured IPMs for ecotoxicological applications. 85

2 - Background and motivation 86

2.1 - Ecological risk assessment context and modeling goals 87

The utility of population models for assessing the risks to fish and wildlife from contaminant exposure 88

has been recognized for at least a half century (Young, 1968). However, the adoption of population 89

models for ecological risk assessment by regulatory agencies has been slow, in part because available 90


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models may not be commensurate with available toxicological data (Forbes et al., 2016; Raimondo et al. 91

2018). Recently, the United States Environmental Protection Agency (USEPA) Office of Pesticide 92

Programs has begun to adopt models that evaluate risk from a population perspective as part of their 93

tiered ecological risk assessment (ERA) process (Etterson et al., 2017), which includes three general 94

steps: problem formulation, analysis, and risk characterization. These steps may be repeated in iterative 95

tiers along a continuum of decreasing generality and increasing realism, halting if a lower-tier 96

assessment suggests little or no ecological risk. At each tier, USEPA scientists assess the fate of the 97

pesticide and the risks the pesticide poses to the environment. Models are used in each tier and, like the 98

risk assessment, move from general (tier 1) to realistic (tier 3) as greater precision and understanding of 99

risk is required. These risk assessments consider major transport pathways (e.g., surface runoff, 100

groundwater transport) from sites where the pesticide is applied and degradation using data submitted 101

by registrants for pesticide active ingredients and their formulated products. 102

For fish, the first tier of ERA at USEPA often starts with the Pesticide in Water Calculator (PWC, 2016) 103

which provides a 30-year hourly time series of estimated chemical concentrations in a hypothetical 104

agricultural pond. Tier I assessments for fish summarize PWC output by calculating annual summary 105

statistics on a daily (acute) and 60d (chronic) basis as the 90th percentile of these annual values 106

(Thursby et al., 2018). These conservative measures of exposure are compared to toxicity values 107

resulting from standardized toxicity tests submitted as part of the pesticide registration process as risk 108

quotients (RQs), which are ratios of estimated exposure concentrations to estimated effect 109

concentrations. In general, effect determinations are derived from two toxicity tests, a 96-hour Fish 110

Acute Toxicity Test (USEPA, 1996) that provides an estimate of the aqueous concentration lethal to 50% 111

of tested fish (i.e., the LC50) and endpoints from a 30-day Fish Early Life Stage Test (USEPA, 2016) with 112

developing embryo/larval fish that provides estimates of maximum aqueous concentrations that do not 113

impair growth and development of young fish (i.e., the No Observed Adverse Effect Concentration, or 114


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NOAEC). RQs are compared to a priori levels of concern (LOC) to determine whether there is risk of 115

adverse effects. For acute RQs, the LOC = 0.5, whereas for chronic RQs the LOC = 1 for non-listed 116

species. When RQ > LOC, the risk assessment may proceed to a higher tier at which point more realistic 117

and detailed modeling approaches, including the use of population models, such as the IPMs presented 118

here, would be considered. 119

2.1.1 - Effects of chemical exposure on survival 120

During standard toxicity tests, cohorts of laboratory-reared fish are exposed under controlled conditions 121

to constant concentrations of a single chemical, and observations of survival, growth, and reproduction 122

are used to produce statistical estimates of concentrations likely to result in, for example, the death of 123

50% of fish during constant exposure for 96 hours (96hr LC50), or a 20% reduction in growth or 124

reproductive output over constant exposure for 30 days (30-d EC20). However, fish populations in 125

natural waters are exposed to highly time-variable concentrations of pesticides, depending on field 126

application rates and timing relative to precipitation events. Currently, risk assessors make conservative 127

assumptions about concentrations to which fish populations are exposed. These include peak predicted 128

concentrations or time-weighted average concentrations. These exposure profiles are then paired with 129

laboratory-measured effects, often at constant concentrations, to estimate the impact of the exposure 130

on populations. Effects of pulsed exposures may not be accurately predicted by effects of constant 131

exposures, given natural mechanisms that cause delayed consequences and either increased or 132

decreased sensitivity to subsequent pulses (Thursby et al., 2018; Kadlec et al., in prep). Existing 133

environmental exposure models as well as pulsed toxic effects models can be used to estimate 134

ecological risk on their own. However, when these models are linked with mechanistic population 135

models, such as the IPMs presented, refined population-level inferences can be made. 136


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The IPMs developed in this paper use output from established models of chemical exposure and effect. 137

USEPA’s Pesticide in Water Calculator (PWC version 1.52; PWC, 2016), which can produce hourly 138

predicted environmental concentrations for specific pesticides in a variety of agricultural and 139

hydrological scenarios (PWC, 2016), was chosen as the exposure model. The General Unified Threshold 140

model of Survival (GUTS; Albert et al., 2016; Jager et al., 2011), a simplified TK-TD model, was chosen as 141

an approach for predicting daily acute effects on fish survival under variable duration and concentration 142

exposure scenarios. GUTS has recently been endorsed for use within risk assessment by the European 143

Food Safety Authority (EFSA Panel, 2018). Concurrent research at USEPA demonstrates the 144

experimental work required to efficiently parameterize and evaluate pulsed exposure models for 145

fathead minnow and specific chemicals (Kadlec et al., in prep). Linking IPMs to existing models for 146

exposure and effect, such as GUTS and the PWC, that risk assessors have familiarity with was pursued 147

with the goal of incorporating the use of IPMs into established ERA workflows. 148

2.2 - The fathead minnow (Pimephales promelas) as a study species 149

The fathead minnow has been well studied in the field and is an important test species in toxicology. 150

Extensive exposure studies for a variety of chemicals have been undertaken using fathead minnows; 151

Ankley & Villeneuve (2006) provide a review of the fathead minnow’s use and significance in toxicology. 152

The fathead minnow is a small, relatively short-lived species native to a large part of North America. It is 153

an iteroparous batch (fractional) spawner, meaning that once females reach reproductive maturity, they 154

spawn multiple times throughout the spawning season with varying numbers of spawning attempts and 155

various inter-spawn intervals (Jensen et al., 2001). Divino & Tonn (2007) explored the impact of hatch 156

time on survival and recruitment by collecting size measurements for both pre- and post-winter 157

individuals. The availability of data on fathead minnow growth, reproduction, and survival, the existing 158

information about responses of fathead minnow to a variety of chemical stressors, and the capacity to 159


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conduct experiments directly designed to support this modeling effort made the fathead minnow an 160

ideal species to explore the use of IPMs for modeling fish populations in ecotoxicology. 161

2.2.1 - Observations to support growth models for fathead minnow 162

The growth model for fathead minnow used for the IPM was developed from a large dataset of growth 163

measurements, including wet weight, dry weight, and length, collected from fish reared in clean water 164

for 288 days (Swintek et al., 2019). For the first 36 days post-hatch (dph), growth measurements were 165

collected approximately every two days. From 36 dph to 78 dph, growth measurements were collected 166

weekly, and from 78 dph to 155 dph, growth measurements were collected every two weeks. From 155 167

dph to 246 dph, a reproductive assessment was conducted, and no growth measurements were 168

collected. From 246 dph to 288 dph, growth measurements were collected every 10 days. This data was 169

used to statistically estimate the optimal parameters of growth transitions. Specifically, von Bertalanffy 170

(vB) and Weibull growth curves were explored for fit within a size-dependent growth transition function. 171

Fathead minnow is sexually dimorphic, with males growing larger in size than females, so growth 172

parameters were determined for both sexes (see Appendix A). 173

2.2.2 - Winter effect as a non-chemical stressor 174

The stressors associated with the winter season have a size-dependent effect on fathead minnow 175

recruitment in its northern range. Winter conditions induce a variety of changes associated with lower 176

temperatures, decreased oxygen, and shorter photoperiod and studies have pointed to winter 177

conditions as being an important factor to consider in aquatic toxicological applications (Driedger et al., 178

2010; Danylchuk and Tonn, 2006; Lemly 1995). In a study of early- versus late-hatching fathead 179

minnows, it was found that fish that hatched earlier in the spawning season were more likely to survive 180

winter compared with those produced later in the season (Divino and Tonn, 2007). It was noted in 181

Divino and Tonn (2007) that 16 mm was the threshold size below which no individuals survived winter. 182


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They also found that reproductive timing and size-dependent winter survival resulted in a shift of the 183

population size-distribution over winter, with the early-hatching cohort contributing greater biomass to 184

the following year’s population than their later-hatching counterparts (Divino and Tonn, 2007). The 185

observations of Divino and Tonn (2007) on over-winter survival together with the observations of 186

Markus (1934), on winter as creating a no-growth scenario, underly the assumptions used to derive the 187

winter transition kernel for fathead minnow. 188

3 - Model and methods 189

3.1 - IPM kernel structure 190

IPMs evolve size distributions of populations, 𝑛(𝑧, 𝑡), in discrete times steps, 𝑛(𝑧, 𝑡) → 𝑛(𝑧′, 𝑡 + 1) →191

⋯, by integrating the product of the population distribution against a transition kernel, 𝐾(𝑧′, 𝑧). For 192

readers that are unfamiliar with IPMs, we give a brief introduction, however for those interested in 193

more details, we suggest Ellner et al., (2016) and Merow et al., (2014). 194

A standard form for an IPM is as follows, 195

(EQ 1) 𝑛(𝑧′, 𝑡 + 1) = ∫ 𝐾(𝑧′, 𝑧)𝑛(𝑧, 𝑡)𝑈

𝐿𝑑𝑧. 196

The size distributions evolve through time and integrating the size distribution across the range of sizes 197

gives the total population, 𝑁(𝑡), at time 𝑡, ∫ 𝑛(𝑧, 𝑡)𝑑𝑧 = 𝑁(𝑡)𝑈

𝐿. 198

The transition kernel is usually written as the sum of two components, the growth/survival component, 199

𝑃(𝑧′, 𝑧), and the reproduction component, 𝑅(𝑧′, 𝑧). Equations 2 and 3 show how the kernel is broken 200

down into its components, 201

𝑛(𝑧′, 𝑡 + 1) = ∫ 𝐾(𝑧′, 𝑧)𝑛(𝑧, 𝑡)𝑈

𝐿𝑑𝑧, 202

(EQ 2) = ∫ (𝑃(𝑧′, 𝑧) + 𝑅(𝑧′, 𝑧))𝑛(𝑧, 𝑡)𝑈

𝐿𝑑𝑧, 203

(EQ 3) = ∫ (𝐺(𝑧′, 𝑧)𝑆(𝑧) +1

2𝑝𝐵(𝑧)𝐵(𝑧)𝐺0(𝑧

′)𝑆(𝑧0)) 𝑛(𝑧, 𝑡)𝑈

𝐿𝑑𝑧. 204


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Brief descriptions of transition kernel functions from Equation 3 are as follows: 𝐺(𝑧′, 𝑧) is the adult 205

growth transition function, 𝑆(𝑧), is the survival probability, 𝑝𝐵(𝑧), is the probability of spawning, 𝐵(𝑧) is 206

the number of hatchlings per spawn, 𝐺0(𝑧′), is hatchling growth transition function, and 𝑆(𝑧0) is the 207

survival probability for size at hatch, 𝑧0. The 1

2 in the reproduction kernel owes to the assumption of a 208

1:1 sex 209

210


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ratio between males and females and that in this model, only females are tracked. In our model, we 211

have chosen to use a daily time-step and a post-reproductive census structure (Figure 1) 212

213

214

The daily time-step was chosen to match the temporal resolution of much of the toxicity and exposure 215

data available for the model. Most standard toxicity tests for fish, like the 96hr acute test (USEPA, 1996), 216

the fathead minnow short-term reproductive assay (USEPA, 2011), and the early life stage toxicity test 217

(USEPA, 2016) are conducted over short time frames and may include daily observations of survival and 218

reproduction. Further, a daily time-step allows the direct linking with high-resolution chemical exposure 219

and TK-TD models that predict impacted survival during chemical exposures that fluctuate daily. 220

3.2 - IPM kernel functions 221

To construct a model that incorporates the effect of chemical and overwintering stressors, we begin by 222

defining a set of demographic functions. These functions define a preliminary transition kernel and 223

modifications of these functions are made to explore different scenarios and capture specific life-history 224

traits. 225

Figure 1: The general form of a life-cycle diagram for a post-reproductive census IPM structure

(modified from Ellner et al., 2016). Order of events in integral projection models determines the form

of the projection kernel for the specified time-step. Due to seasonal reproduction and the daily time-

step used, this life-cycle representation applies to reproductive days only, and outside of the

reproductive season would be simplified to just include survival and growth components.


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The initial transition kernel functions have the following definitions: 226

Adult growth transition, 𝐺(z′, 𝑧), 227

(EQ 4) 𝐺(𝑧′, 𝑧) ∼ 𝑁𝑜𝑟𝑚𝑎𝑙(𝜇𝐺(𝑧), 𝜎𝐺2(𝜇𝐺)), 228

The mean growth, 𝜇𝐺(𝑧), is the size-dependent vB growth function, common to modeling fish growth 229

(Quinn and Deriso, 1999), and which has been used in IPM models previously (Erickson et al. 2017; 230

White et al. 2016), 231

(EQ 5) 𝜇𝐺(𝑧) = 𝑧𝑖𝑛𝑓 − (𝑧𝑖𝑛𝑓 − 𝑧)𝑒(−𝜅𝐺). 232

The variance term, 𝜎𝐺2, has the following form, 233

(EQ 6) 𝜎𝐺2(𝜇𝐺(𝑧)) = 𝑣𝑎𝑟𝜅𝐺

(𝑧𝑖𝑛𝑓 − 𝜇𝐺(𝑧))2. 234

Survival probability, 𝑆(𝑧), is assumed to be of logistic-form (Erickson et al., 2017), 235

(EQ 7) 𝑆(𝑧) = 𝑆𝑚𝑖𝑛 +(𝑆𝑚𝑎𝑥−𝑆𝑚𝑖𝑛)

1+𝑒(𝑆𝑏(𝑆𝑎−𝑧)). 236

Probability of reproducing, 𝑝𝐵(𝑧), is a size threshold function, 237

(EQ 8) 𝑝𝐵(𝑧, 𝑡) = {0 𝑧 < 𝑧𝑟𝑒𝑝𝑟𝑜

𝑝𝑠𝑝𝑎𝑤𝑛(𝑡) 𝑧 ≥ 𝑧𝑟𝑒𝑝𝑟𝑜. 238

Hatchlings per spawn, 𝐵(𝑧) is, also a size threshold function, 239

(EQ 9) 𝐵(𝑧) = {0 𝑧 < 𝑧𝑟𝑒𝑝𝑟𝑜

𝐵𝑠𝑝𝑎𝑤𝑛 𝑧 ≥ 𝑧𝑟𝑒𝑝𝑟𝑜, 240

Spawning probability, 𝑝𝑠𝑝𝑎𝑤𝑛 , is determined daily based on species life-history. For fathead minnow, a 241

batch spawning algorithm (Supplemental Table 1) is used to estimate daily spawning probabilities within 242

the spawning season. The estimated number of surviving hatchlings per spawn, 𝐵𝑠𝑝𝑎𝑤𝑛, was derived 243

from the average annual reproductive output, number of spawning attempts, and estimated 244

recruitment rates (Markus, 1934). It is assumed that hatchlings per spawn is uniform above the 245

reproductively mature size, as shown in Equation 9. 246


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Hatchling growth transition, 𝐺0(z′), 247

(EQ 10) 𝐺0(𝑧′) ∼ 𝑁𝑜𝑟𝑚𝑎𝑙(𝜇𝐺0, 𝜎𝐺0

2 (𝜇𝐺0)), 248

(EQ 11) 𝜇𝐺0= 𝑧𝑖𝑛𝑓 − (𝑧𝑖𝑛𝑓 − 𝑧ℎ𝑎𝑡𝑐ℎ)𝑒

(−𝜅𝐺), 249

(EQ 12) 𝜎𝐺0

2 (𝜇𝐺0) = 𝜇𝐺0

𝑣𝑎𝑟𝜖𝐺, 250

is similar to adult growth, 𝐺(z′, 𝑧), except that all growth starts from the same size, 𝑧ℎ𝑎𝑡𝑐ℎ. Due to 251

starting size being fixed, there is a slightly simpler variance term, 𝜎𝐺02 . Appendix A contains details on 252

growth transition function derivation and parameterization. Table 1 provides symbols, descriptions, 253

values, and references for the parameters used in the transition kernels. 254

255


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Symbol Description Value Reference

𝒛𝒉𝒂𝒕𝒄𝒉 Size at hatch 5.6 𝑚𝑚

Experimentally Determined (Appendix A Table 4)

𝒛𝒊𝒏𝒇 Maximum size 74 𝑚𝑚


𝜿𝑮 vonBertalanffy curvature 0.009


𝒗𝒂𝒓𝜿𝑮 𝜅𝐺 variance

5.94 ∗ 10−7 Experimentally Determined (Appendix A Table 4)

𝑺𝒎𝒊𝒏 Minimum daily survival probability 0.9949

Dervied, Vandenbos et al., 2006

𝑺𝒎𝒂𝒙 Maximum daily survival probability 0.9992

Derived, Divino & Tonn, 2007

𝑺𝒂 Logistic survival parameter associated with transition rate between 𝑆𝑚𝑖𝑛 and 𝑆𝑚𝑎𝑥

. 5 Calibrated1

𝑺𝒃 Logistic survival parameter associated with inflection

35 Calibrated1

𝒛𝒓𝒆𝒑𝒓𝒐 Reproductively mature size 44 𝑚𝑚

Estimated from experimental observation

𝒑𝒔𝒑𝒂𝒘𝒏 Spawning probability Varies by simulation

Modeled – (Supplemental Table 1)

𝑩𝒔𝒑𝒂𝒘𝒏 Hatchlings per spawn 187.5

Derived, Markus, 1934; Vandenbos et al., 2006

𝒗𝒂𝒓𝝐𝑮 𝜖𝐺 variance

0.96 Experimentally Determined (Appendix A Table 4)

𝒛𝒘𝒊𝒏𝒕𝒆𝒓 Winter size cutoff 16 𝑚𝑚

Divino & Tonn, 2007

- Inter-spawn interval 3 𝑑𝑎𝑦𝑠

Jensen et al., 2001

- Max Spawns per season 12

Markus, 1934; Jensen et al., 2001

𝑺𝒅 Survival decrement Varies by simulation

Modeled TCEM and GUTS (See Section 3.6)

- Threshold Concentration for 50% reduction in survival

2.5 𝑚𝑔/𝐿 Modeled GUTS (See Section 43.6 and Kadlec et al., in prep)

256

257

Table 1: Parameter symbols, descriptions, values, and references. This table includes the parameter

values for kernel transition functions as well as relevant values for other model components. 1Survival parameters 𝑆𝑎 and 𝑆𝑏 were not literature available and were calibrated to reflect juvenile to

adult size transition.


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3.3 - Deriving the growth transition functions: 258

Standard non-linear regression techniques were used to estimate parameters for both Weibull and vB 259

growth functions. For IPM application, equations of size as a function of hatch time were modified to 260

equations of growth as a function of size. The delta method was used to calculate the variance around 261

the estimates of size for the new parametrizations. Analytical results were verified by simulation to 262

determine fit and any conditions that needed to be imposed to ensure accuracy. Weibull and vB for the 263

mean daily growth increment, 𝜇𝐺 in Equation 5, were compared. It was found that the vB growth model 264

as the mean for the normally distributed size transitions not only performed better in the growth 265

simulations than Weibull, but also had a simpler expression for the variance terms for both adult and 266

newly hatched individuals. Therefore, vB was used for mean growth transitions in our model. The 267

associated set of best-fit parameters are available in Appendix A. Supplemental Figure 1 shows the 268

mean and standard deviation for the daily growth increment by size for the Baseline model. 269

3.4 - Incorporating iteroparity 270

Batch spawning species spawn multiple times throughout the reproductive season with some 271

proportion of the reproductively capable females depositing eggs each day of the season. This behavior 272

has been observed and measured for fathead minnow in the field and the laboratory (Markus, 1934; 273

Jensen et al., 2001). We incorporate batch spawning into the model by reparameterization of the daily 274

transition kernels. Assumptions underlying our batch spawning module include (1) a defined spawning 275

season, outside of which no reproduction occurs, (2) each day within the spawning season has a 276

different probability of reproduction based on the inter-spawn interval and the number of spawns per 277

season (3) the average number of hatchlings per spawn is used to introduce offspring into the 278

population, and (4) the introduction of hatchlings happens immediately, ignoring the time for embryo 279

development (approximately 4 days, Jensen et al., 2001). In the absence of information on population-280


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wide distributions on timing of spawning initiation and resolved data on distributions of numbers of 281

spawns per individual, uniform distributions were assumed. The set of assumptions that informed the 282

batch spawning algorithm is based on observations on the inter-spawn interval, number of spawns per 283

season, and number of eggs per spawn as reported in Gale & Buynak, 1982, Jensen et al., 2001, and 284

Markus, 1934. The algorithm pseudo-code and parameters used in deriving the daily spawning 285

probabilities are given in Supplemental Table 1. An example of the output of the batch spawning 286

algorithm that is used in the simulations is provided in Supplemental Figure 2. 287

3.5 - Winter transition kernels 288

To incorporate the effect of winter, we derived a winter-day transition kernel based on the following 289

assumptions: there is no growth or reproduction during the winter season (𝑅(𝑧′, 𝑧) ≡ 0) and survival is 290

size-dependent such that there is a minimum size below which survival probability is 0. For individuals 291

above the threshold, survival is assumed to be the same as non-winter. For our simulations, winter 292

began on December 21st and ended on March 20th. These assumptions lead to the following transition 293

kernel, 𝐾𝑊(𝑧′, 𝑧) = 𝐺𝑊(𝑧′, 𝑧)𝑆𝑊(𝑧) where, 294

(EQ 13) 𝐺𝑊(𝑧′, 𝑧) = {1 𝑧′ = 𝑧0 𝑒𝑙𝑠𝑒

, 295

and the preliminary survival function was modified to include a size threshold, such that probability of 296

survival is 0 below the size 𝑧𝑤𝑖𝑛𝑡𝑒𝑟, 297

(EQ 14) 𝑆𝑊(𝑧) = {𝑆(𝑧) 𝑧 > 𝑧𝑤𝑖𝑛𝑡𝑒𝑟

0 𝑒𝑙𝑠𝑒. 298

3.6 - Incorporating effects of chemical exposure 299

A realistic time-variable annual exposure generated by the PWC for a simulated agricultural application 300

of diazinon was used for simulation (PWC, 2016). This exposure scenario was interpreted for acute 301

effects on survival using two different chemical effects models. GUTS is an ordinary differential 302


17

equations (ODE)-based TK-TD model that takes as input experimental data on the time-course of 303

survival due to exposure and provides predictions of daily changes in survival probability. This was used 304

along with a threshold concentration exceedance model (TCEM) for comparison. The TCEM assumes 305

that if the concentration of the chemical exceeds a specified threshold on a given day, the exceedance 306

results in background survival probability decremented correspondingly. For example, if an LC50 is used 307

as the threshold, the expected change in background survival probability is assumed to be decreased by 308

50% for that day. Supplemental Figure 3 provides a diagram for how chemical exposure is incorporated 309

in the IPM. The TCEM and GUTS models represent a range in effect model complexity used in ERA 310

(Schmolke et al.,2017), with the TCEM as a simplistic effect model and GUTS as a more resolved TK-TD 311

model. 312

The R package “morse” (Baudrot et al., 2019) was used to fit GUTS models with empirical time-course of 313

survival data on the acute effects of diazinon exposure on juvenile fathead minnow. The fitted GUTS 314

models were used to predict changes in survival due to exposure. Full details on the concurrent 315

exposure experiments supporting modeling are being made available in Kadlec et al., (in prep). The R 316

package “morse” was also used to determine a 96hr LC50 value from the same experimental 317

observations used to parameterize the GUTS models. The 96hr LC50 determined this way was used as 318

the threshold concentration in the TCEM. Changes in survival probability predicted by GUTS and TCEM 319

from the PWC exposure scenario were incorporated into the IPMs as daily survival decrements from the 320

background survival probabilities (Figure 2). 321

322


18

323

324

325

EXPOSURE SCENARIO AND ACUTE EFFECTS

Figure 2: Simulated exposure scenario and the modeled acute effect responses. Based on a modeled

exposure profile, daily survival decrements were calculated using a TK-TD GUTS model, parameterized

by laboratory data for P. promelas. The TCEM model used a calculated 96hr LC50 value so that when

chemical concentrations exceeded the threshold, a survival decrement of 50% was assumed for the

day. Note: Many days exposure concentrations are negligible as are the effects, however May through

June has the largest concentrations and corresponding effects predicted.


19

Whether interpreted by GUTS or TCEM, the acute effects from chemical exposure on survival are 326

incorporated into the transition kernel with a modification to the survival function, 𝑆𝐸(𝑧). Assuming a 327

size-independent effect of chemical exposure with associated survival decrement, 𝑆𝑑(𝑡), yields the 328

following, 329

(EQ 15) 𝑆𝐸(𝑧, 𝑡) = 𝑆𝑚𝑖𝑛 +(𝑆𝑚𝑖𝑛−𝑆𝑚𝑎𝑥)

1+𝑒(𝑆𝑏(𝑆𝑎−𝑧))

− 𝑆𝑑(𝑡). 330

For some chemicals, fathead minnow sensitivity to exposure is related to developmental stage (Mayes 331

et al., 1983; McKim, 1977); therefore, using size as a proxy for development, it is useful to explore how 332

size-dependent sensitivity can affect population-level outcomes. To accomplish this, a size-dependent 333

toxicity modifier function was introduced, denoted 𝜓𝑥(𝑧), and used as follows, 334

(EQ 16) 𝑆𝐸𝜓(𝑧, 𝑡) = 𝑆𝑚𝑖𝑛 +

(𝑆𝑚𝑖𝑛−𝑆𝑚𝑎𝑥)

1+𝑒(𝑆𝑏(𝑆𝑎−𝑧))

− 𝑆𝑑(𝑡)𝜓𝑥(𝑧). 335

This concept of a size-dependent toxicity modifier is flexible and can be adjusted to context. For 336

demonstration purposes, a step-function toxicity modifier was used, denoted 𝜓𝑆𝑇𝑃(𝑧). This functional 337

form, although simple, corresponds well with data available (e.g. Mayes et al., 1983) for size/stage 338

dependent toxic response that are frequently reported at discrete age/stage. The step-function toxicity 339

modifier, 𝜓𝑆𝑇𝑃(𝑧), uses the standard form, 340

(EQ 17) 𝜓𝑆𝑇𝑃(𝑧) = ∑ 𝑚𝑖𝟙{Ω𝑖}(𝑧)𝑛

𝑖=0 . 341

Where Ω𝑖 is the size range for toxicity multiplier 𝑚𝑖 and Ω𝑖 constitute a partition of the size range. 342

3.7 - Numerical kernel discretization 343

As is common practice, the IPM transition kernels were solved numerically to provide a discretized daily 344

transition kernel. Frequently, the mid-point rule is sufficient for numerical integration of the transition 345

kernel (Merow et al., 2014). However, in this application changes in size due to growth are very small 346


20

over a single day, and thus nearly deterministic. Therefore, a non-standard numerical integration 347

approach was needed. The numerical integration technique provided in Ellner et al. (2016) was 348

implemented which uses a gaussian quadrature rule (default = order 3) and difference in the cumulative 349

distribution functions (CDFs) of the probability distributions, as opposed to the probability distribution 350

functions (PDFs) that would be used in a standard mid-point rule implementation for IPMs (see Ellner et 351

al., 2016 chapter 6.8) The effects of using 𝑛 = 100, 200, 300, 400, and 500 size classes were explored 352

and it was determined that discretizing the size range into 𝑛 = 300 size classes struck a balance 353

between numerical accuracy and computation time, and so was used for all simulations. 354

Another challenge to the small growth increment manifested in the transitions of larger-sized 355

individuals. According to vB growth, as individuals get larger, their size-specific growth rate decreases 356

(Supplemental Figure 1). Coupling this with a variance that is tied to the growth increment, therefore 357

also very small (Equation 6), meant that for very large individuals, the probability of transition to the 358

largest size class was approximately 0. This led to an artificial maximum achievable size that differed 359

from 𝑧𝑖𝑛𝑓. To address this, a floor of 1.0E-5 used in the variance term. However, with the floor 360

implemented for variance, a small amount of eviction presented itself in the model. Eviction is 361

a phenomenon in IPMs where growth transitions take individuals outside of the size range 362

limits of integration, 𝑈 and 𝐿 (Equation 1). Williams et al. (2012) suggest methods to address 363

eviction in IPM. One such method is to use a PDF truncated to the size range (𝑧ℎ𝑎𝑡𝑐ℎ, 𝑧𝑖𝑛𝑓) for 364

growth transitions, we implemented this method which corrected the eviction. 365

3.8 - Simulations 366

Simulated scenarios using the size-structured IPM for fathead minnow were chosen to demonstrate the 367

flexibility of the modeling approach and to explore different methods to incorporate acute effects of 368

chemical exposure. Two different chemical effects models, GUTS and TCEM, using an exposure time-369


21

series output from the PWC were investigated (Figure 2). The assumption of size-independent toxicity 370

was compared to threshold size-dependent effects through use of the toxicity modifier. The effect of 371

winter was simulated, and the Winter scenario was additionally used to explore the effect of simulation 372

start date on modeled outcomes. Each scenario is composed of 365 potentially different daily transition 373

kernels with components modified for the daily context. Table 2 provides a summary of simulated 374

scenarios and their features. Table 3 provides examples of the transition kernels used for different days 375

in the scenarios modeled. 376

Simulation Scenario ATTRIBUTES

SCENARIOS

BASELINE

WINTER CHEMICAL EXPOSURE

Calendar Year Start Jan1-Dec31

Reproductive Season Start May22-May21

GUTS-Uniform

GUTS- Step

TCEM- Uniform

TCEM- Step

P. Promelas life history

Daily time step

1 year simulation

Batch spawning May 22 – Aug 22

PWC EXPOSURE MODEL

GUTS EFFECT MODEL TCEM Effect model TOXICITY MODIFIER: 𝝍𝒙(𝒛) = 𝟏

TOXICITY MODIFIER: 𝝍𝒙(𝒛) = 𝝍𝑺𝑻𝑷(𝒛)

WINTER SEASON DEC 21 – MAR 1

Jan 1st Start Date

May 22nd Start Date

377

Table 2: Visualization of simulated scenarios organized by the scenario attributes included. Detailed

descriptions of simulated scenarios are provided in the text. Results of simulated scenarios are given in

Table 4


22

378

SCENARIO DAILY TRANSITION KERNELS, 𝑲(𝒛′, 𝒛) EQUATION REFERENCE

BASELINE

Non-reproductive season 𝐺(𝑧′, 𝑧)𝑆(𝑧) 4,7

Reproductive season 𝐺(𝑧′, 𝑧)𝑆(𝑧) + 𝑝𝐵(𝑧)𝐵(𝑧)𝐺0(𝑧′)𝑆(𝑧ℎ𝑎𝑡𝑐ℎ) 4,7,8,9,10

WINTER 𝐺𝑊(𝑧′, 𝑧)𝑆𝑊(𝑧) 13,14

EXPOSURE

Non-reproductive season 𝐺(𝑧′, 𝑧)𝑆𝐸𝜓(𝑧) 4,16

Reproductive season 𝐺(𝑧′, 𝑧)𝑆𝐸𝜓(𝑧) + 𝑝𝐵(𝑧)𝐵(𝑧)𝐺0(𝑧′)𝑆𝐸𝜓(𝑧ℎ𝑎𝑡𝑐ℎ) 4,8,9,10,16

Table 3: Each day in each scenario simulated is created from different daily transition kernels.

Examples of transition kernels used for the different scenarios simulated are provided and references

to their definitions are provided.


23

3.9 - Model endpoints and analyses 379

Multiple population-level endpoints were calculated from the discretized daily transition kernels and an 380

annual discretized transition kernel was computed for each scenario as the product of the daily 381

discretized transition kernels. Comparing modeled numbers of individuals in a specific natural 382

population was not an objective for this initial effort; instead modeled scenarios were compared to a 383

baseline to explore impacts of different modeling assumptions and choices. The Baseline scenario, 384

without any additional stressors, served as the point of comparison for all other simulations; whenever 385

possible, results of scenarios were given as a percent change from Baseline. Results presented include 386

summary information from the discretized transition kernels as well as results of simulated population 387

trajectories. To simulate population trajectories, unless otherwise noted, each simulated year began on 388

January 1st, with 100 individuals uniformly distributed in size between the lower (𝑧ℎ𝑎𝑡𝑐ℎ) and upper 389

(𝑧𝑖𝑛𝑓) size ranges. Dominant eigenvalues, 𝜆, and average stable sizes, 𝜁,̅ were calculated. Where 𝜁̅ =390

∑ 𝑣𝑖𝑚𝑖𝑖 , with 𝑣 as the normalized right eigenvector associated with the dominant eigenvalue and 𝑚 as 391

the vector of mid-points of the size classes from the numerical discretization routine. In addition to 392

asymptotic analyses, maximum and minimum column sums, denoted as α and 𝜔, and referred to as 393

minimum and maximum growth potentials, respectively, are given as summaries of discretized kernels 394

as well. These values bound growth potential 𝑁(𝑡 + 1)/𝑁(𝑡), where 𝑁 is population size, and are 395

informative along with 𝜆 to infer baseline and stressor effects on growth potential (Stott et al., 2011). A 396

benefit of reporting minimum and maximum column sums is that their information on growth potential 397

does not rely on the population being near the stable size distribution. 398

3.10 - Sensitivity and elasticity analyses 399

The asymptotic growth rate of the population, 𝜆, is used as a summary endpoint to determine long-term 400

growth potential of the population exhibited in the model. Therefore, we wanted to understand which, 401


24

if any, model components had disproportionate roles in determining values of 𝜆. We carried out 402

sensitivity and elasticity analyses for each of the kernel component functions. Specifically, for each of 403

the scenarios and each daily transition kernel, we investigated the sensitivity and elasticity of 𝜆 with 404

respect to changes in the survival, growth, and reproductive kernel functions, 𝑆, 𝐺, 𝑃𝐵 , 𝐵, and 𝐺0. Our 405

analyses for sensitivity and elasticity accounted for the full spectrum of sizes 𝑧. Specifically, sensitivity 406

was calculated as, 407

(EQ 18) 𝜕𝜆

𝜕𝑓= ∬(

𝜕𝜆

𝜕𝐾(𝑧′,𝑧)) (

𝜕𝐾(𝑧′ ,𝑧)

𝜕𝑓) 𝑑𝑧′𝑑𝑧, 408

where 𝑓 was 𝑆(𝑧), 𝐺(𝑧′, 𝑧), 𝑃𝐵(𝑧), 𝐵(𝑧), 𝐺0(𝑧′). The standard formula for elasticities, (

𝜕 log𝜆

𝜕 log𝑓), was used. 409

Sensitivities and elasticities of 𝜆 with respect to a function 𝑓 are denoted as 𝑆𝑒𝑛𝑠(𝑓) and 𝐸𝑙𝑎𝑠(𝑓), 410

respectively. To compute sensitivities and elasticities we used the discretized forms of the daily 411

transition kernels, 𝐾(𝑧′, 𝑧). Sensitivity and elasticity measures were computed for each of the 5 kernel 412

functions, within each of the 365 daily transition kernels, for each scenario explored. 413

4 - Results 414

A summary of simulation results from the discretized annual transition kernels is provided in Table 4. 415

Figure 3 provides a visual of the growth potentials, 𝛼 and 𝜔, for the annual transitions for each 416

simulation. A summary of the analyses of discretized daily transition kernels is given in the Supplement 417

(Supplemental Table 2). Plots of the time series of average sizes and numbers of individuals by day for 418

each scenario are also provided in the Supplement (Supplemental Figures 4 through 10). 419


25

ANNUAL DISCRETIZED TRANSITION KERNEL SUMMARY

COLOR KEY: % BASELINE

BASELINE


(+)1

00

%-

80

%

80

%-

60

%

60

% -

40

%

40

% -

20

%

20

% -

0%

Calendar Year Start Jan1-Dec31

Reproductive Season Start May22-May21

GUTS- Uniform

GUTS- Step TCEM-

Uniform TCEM- Step

Simulated Population Size – 𝑵

Units: # of individuals 23130 18131 16341 12479 17347 2889 9582

𝑵/𝑵𝑩𝒂𝒔𝒆𝒍𝒊𝒏𝒆 - 78.39% 70.65% 53.95% 75.00% 12.49% 41.43%

Dominant Eigenvalue – 𝝀

Unitless 248.41 248.40 248.40 134.03 186.30 31.03 102.91

𝝀/𝝀𝑩𝒂𝒔𝒆𝒍𝒊𝒏𝒆 - 100.00% 100.00% 53.95% 75.00% 12.49% 41.43%

Mean Stable Size - 𝜻 ̅

Units: mm 60.14 58.83 65.30 60.14 60.14 60.14 60.15

�̅�/�̅�𝑩𝒂𝒔𝒆𝒍𝒊𝒏𝒆 - 97.82% 108.59% 100.00% 100.00% 100.00% 100.02%

Maximum Annual Growth Potential – 𝜶

Unitless 248.41 248.41 248.40 134.03 186.30 31.03 102.91

𝜶 𝜶𝑩𝒂𝒔𝒆𝒍𝒊𝒏𝒆⁄ - 100.00% 100.00% 53.95% 75.00% 12.49% 41.43%

Minimum Annual Growth Potential – 𝝎

Unitless 181.35 0 1.20 97.85 136.01 22.66 75.13

𝝎/𝝎𝑩𝒂𝒔𝒆𝒍𝒊𝒏𝒆 - 0.00% 0.66% 53.95% 75.00% 12.49% 41.43%

420

421

422

423

Table 4: Summaries of annual discretized transition kernels. annual discretized transition matrices were

generated as the product of the daily transition matrices for each scenario simulated. Dominant

eigenvalues, average size at the stable size distribution, resulting population after simulation

(simulation began with 100 uniformly distributed individuals), maximum column sum, and minimum

column sum provided. Results also given as a percentage of the baseline scenario. Summaries are

given to two decimal places in order to show the sometimes slight differences between kernels.


26

424

425

426

Figure 3: Growth potentials for annual transitions by simulation scenario. Baseline and the Winter

scenarios are identical in their maximum growth potential but differ in their minimum due to over-

winter mortality leading to 0 column sums. The exposure scenarios have lower growth potentials and a

smaller difference between α and ω, leading to less variability in population growth outcomes based on

size distributions.


27

4.1 - Baseline simulation 427

The Baseline scenario demonstrated expected progression to larger size classes during the non-428

reproduction season and large shifts towards smaller size classes during reproductive season, ranging 429

from May 22nd to Aug 22nd (ordinal day 143-235) as shown in Figure 4. The discretized annual transition 430

kernel had a dominant eigenvalue of 248.41. The associated stable size distribution was determined, 431

and average stable size 𝑤𝑎𝑠 𝜁̅ = 60.14 mm. The annual maximum column sum, 𝛼, was 248.41 and the 432

minimum was value of 𝜔 was 181.35. At the end of the Baseline simulation, the initial population of 100 433

individuals had grown to 23130 individuals. 434

435


28

436

437

438

BASELINE DAILY SIZE DISTRIBUTIONS

Figure 4: Size distributions by day for the Baseline scenario. Starting at a uniform distribution of

individuals, visualization shows growth transition to larger size classes, until reproduction, when

density shifts to offspring. During and post reproduction, new cohort transitions to larger sizes classes.


29

4.2 - Over-winter simulation 439

The Winter transition kernel was used for a winter season lasting from Dec 21st to March 20th. Winter 440

was simulated in the absence of chemical exposure (Figure 5) and resulted in a dominant eigenvalue of 441

the discretized annual transition kernel of 248.40, which was just below the Baseline (< 1% decrease). 442

The stable average size 𝜁̅ = 58.83 mm, was a 2% decrease from the Baseline. α for the annual transition 443

was 248.41, (100% of Baseline), and 𝜔 was 0, due to the Winter survival probability of 0 for individuals 444

below the size cut off, 𝑧𝑤𝑖𝑛𝑡𝑒𝑟. Despite the similar summary values for the discretized annual transition 445

matrices between the Winter and Baseline scenarios, the simulated population growth was different. At 446

the end of the Winter scenario with Jan 1st simulation start date, the population was 18131 individuals, 447

which was a 21% decrease from the Baseline. 448

We also explored the effect of changing the Winter simulation start date from Jan 1st to May 22nd, to 449

correspond with the first day of reproduction. Holding all other model parameters constant, in the 450

simulation that started on May 22nd, the dominant eigenvalues for the discretized annual transition 451

kernels were identical. However, the stable size distribution was different (Figure 6), with 𝜁̅ = 65.30 452

mm, a 9% increase from the Baseline. The simulated population was 16341 at the end of the 453

simulations, which is a 29% decrease from Baseline and smaller than the Jan 1st Winter simulation. The 454

maximum and minimum column sums for the discretized annual transition kernel were 𝛼 =248.40 and 455

𝜔 =1.20, respectively, and both differed from the Jan 1st Winter simulation. 𝛼 was slightly smaller than 456

Baseline, but still almost identical, and 𝜔 showed a 99% decrease from Baseline (Table 4). 457


30

458

459

460

461

WINTER DAILY SIZE DISTRIBUTIONS

Figure 5: Size distributions by day for the Winter scenario. Starting at a uniform distribution of

individuals, there is an immediate loss of density in size classes smaller than the winter size threshold.

Visualization also shows the effect of the no growth assumption during winter, where above the winter

size-cutoff, no transition to larger classes occurs until winter ends, and the distribution is essentially

frozen. Reproductive season shifts density to smaller sizes similarly to baseline. The onset of winter

later in the year (OD=355) once again freezes the size distribution.


31

462

463

Figure 6: Comparison of stable size distributions for the two different Winter scenarios. The

distribution of the scenario that assumes a May 22nd start shows a shift to larger size classes compared

to the Jan 1st simulation start.


32

4.3 - Chemical exposure simulations 464

Chemical stressor scenarios were simulated with two different kinds of chemical effects models, GUTS 465

as the TK-TD model and TCEM as the threshold response model. Two different assumptions of size-466

dependent toxicity: a uniformly toxic response and size-dependent response used by specifying 𝜓𝑆𝑇𝑃(𝑧) 467

such that juveniles (𝑧 < 44 mm) are twice as sensitive as adults. The information used to determine 468

toxic effects were the results of an exposure to juvenile fish, thus 𝜓𝑆𝑇𝑃(𝑧) = 1 for 𝑧 < 44 mm and 1

2 for 469

𝑧 ≥ 44 mm. The same toxicity modifier was used for all days in the simulation. 470

4.3.1 - GUTS effect model 471

Under the assumption of uniform toxicity, 𝜓𝑥(𝑧) = 1, the discretized annual transition kernel had a 472

dominant eigenvalue of 134.03, which was 46% decrease from the Baseline scenario. The maximum and 473

minimum column sums from the discretized annual transition kernel were 𝛼 = 134.03 and 𝜔 = 97.85, 474

respectively, and both were 46% decrease from of the Baseline. The stable size distribution had a mean 475

of 𝜁̅ = 60.14 mm, showing no difference from Baseline. 476

Under the assumption that adults were half as sensitive as juveniles, the dominant eigenvalue of the 477

discretized annual transition kernel was 186.30, a 25% decrease from Baseline. The associated stable 478

size distribution was the same as the GUTS-Uniform scenario, with a mean of 𝜁̅ = 60.14 mm. Column 479

sums from the annual discretized kernel were 𝛼 = 186.30 and 𝜔 = 136.01, showing higher growth 480

potential than the GUTS-Uniform scenario, but still a 25% decrease from Baseline. 481

4.3.2 - TCEM effect model 482

The TCEM approach models toxic effects only when chemical concentrations exceed a specified 483

threshold (2.5 mg/L). There were 3 days within the exposure scenario where concentrations exceeded 484

the threshold concentration. For the TCEM scenario assuming size-uniform toxicity, the discretized 485

annual transition kernel had a dominant eigenvalue of 31.03, which was an 87% decrease from the 486

Baseline scenario. The mean of the stable size distribution for the annual transition was the same as the 487


33

Baseline scenario, 60.14𝑚𝑚. Growth potential in the discretized annual transition kernel was between 488

𝛼 = 31.03 and 𝜔 = 22.66, each showing an 88% decrease from Baseline. 489

For TCEM with size-dependent toxicity, the discretized annual transition kernel had a dominant 490

eigenvalue of 102.91, a 60% decrease from Baseline. The average stable size was once again the same 491

as Baseline (60.15 mm). Minimum and maximum column sums for the discretized annual transition 492

kernel were 𝜔 =75.13 and 𝛼 = 102.91, respectively, and each were a 59% decrease from Baseline. 493

4.4 - Sensitivity and elasticity analyses 494

Results presented and discussed are limited to elasticities for their relative comparability. However, 495

results and a brief discussion of the sensitivity analysis are included in Appendix B. 496

Within the Baseline scenario, trends in elasticities of 𝜆 tracked the reproductive season. During the 497

reproductive season, 𝐸𝑙𝑎𝑠(𝐺) > 𝐸𝑙𝑎𝑠(𝑆), with (𝐸𝑙𝑎𝑠(𝑆)

𝐸𝑙𝑎𝑠(𝐺)) reaching its minimum of .95 on ordinal day 191 498

(July 10th), corresponding to peak reproduction. 𝐸𝑙𝑎𝑠(𝑝𝐵) and 𝐸𝑙𝑎𝑠(𝐵) were identical for all days, and 499

elasticities for 𝑝𝐵, 𝐵, and 𝐺0 were 0 outside of the reproductive season. Within the reproductive season 500

however, 𝐸𝑙𝑎𝑠(𝐺0) was greater than 𝐸𝑙𝑎𝑠(𝑝𝐵) and 𝐸𝑙𝑎𝑠(𝐵), but only slightly, and all were much less 501

than 𝐸𝑙𝑎𝑠(𝑆) and 𝐸𝑙𝑎𝑠(𝐺). To give an idea of magnitude, (𝐸𝑙𝑎𝑠(𝐺0)

𝐸𝑙𝑎𝑠(𝑆)) was at most .094 and (

𝐸𝑙𝑎𝑠(𝐺0)

𝐸𝑙𝑎𝑠(𝐺)) was 502

at most .09 (Figure 7). 503

Elasticities within the Winter scenario followed the same trends as the Baseline, however the magnitude 504

of 𝐸𝑙𝑎𝑠(𝑆) and 𝐸𝑙𝑎𝑠(𝐺) were smaller in the Winter scenario during the winter season (Dec 21st- Mar 505

20th). The ratio of 𝐸𝑙𝑎𝑠(𝑆) to 𝐸𝑙𝑎𝑠(𝐺) was, however, identical during winter and the non-winter, non-506

reproductive season, with 𝐸𝑙𝑎𝑠(𝑆) just slightly larger than 𝐸𝑙𝑎𝑠(𝐺) during the non-reproductive season. 507


34

508

509

510

ELASTICITIES OF LAMBDA TO KERNEL FUNCTIONS

Figure 7: Elasticities of daily λ values with respect to the various kernel components for the Baseline

and Winter scenarios. All elasticities of the reproduction kernel components are 0 outside of the

reproductive season. 𝑝𝐵 , 𝐵, and 𝐺0 are the kernel functions to which λ has the least elasticity during

the reproductive season. In general, elasticities of λ were higher for growth than survival within the

reproductive season and outside reproductive season survival elasticities were higher than but just

slightly growth.


35

Elasticity trends held for the GUTS and TCEM exposure scenarios. However, the introduction of survival 511

decrements had the potential to reverse the ordering of elasticities with respect to growth and survival 512

during the reproductive season, so that 𝐸𝑙𝑎𝑠(𝑆) > 𝐸𝑙𝑎𝑠(𝐺) during the reproductive season. 513

Specifically, in the GUTS scenarios during the largest exposure decrement on May 29th (OD 149) 514

𝐸𝑙𝑎𝑠(𝑆) > 𝐸𝑙𝑎𝑠(𝐺), but otherwise the trend of 𝐸𝑙𝑎𝑠(𝐺) > 𝐸𝑙𝑎𝑠(𝑆) during reproduction and 𝐸𝑙𝑎𝑠(𝑆) >515

𝐸𝑙𝑎𝑠(𝐺) outside of reproduction held. The TCEM scenarios had a similar trend. However, in this case 516

the switching of sensitivities during the reproductive season occurred on both May 25th and 29th, 517

corresponding to both threshold exceedances that happened during the reproductive season. 518

5 - Discussion 519

The IPM approach was found to be suitable to the incorporation of natural and anthropogenic stressors 520

relevant to ecotoxicological applications. Daily IPMs were created to match the life history of fathead 521

minnow, including seasonal reproduction and batch spawning. Size-independent effects of chemical 522

exposure predicted by TK-TD and threshold models were mapped to daily survival rates of individuals 523

under fluctuating realistic exposure profiles. A method to incorporate size-dependent effects of 524

exposure was introduced and simulations were undertaken to explore the role of multiple modelling 525

assumptions and scenarios compared to a baseline scenario. Simulations results showed the significant 526

influence chemical exposure can have as well on population levels, as well as quantifying the differences 527

at the population level from different exposure effect models, like GUTS and TCEM. The IPMs presented 528

show the ability of the modeling approach to synthesize multiple ecotoxicologically relevant factors and 529

illustrate the type of population level inference that can be undertaken using IPMs. 530

Daily transition kernels had dominant eigenvalues that were slightly less than one during the non-531

reproductive season, and greater than one within the reproductive season. As expected, the high 532

fecundity of fathead minnow shifted densities to the smaller size classes during the reproductive season 533

(Figure 4). The cumulative effect of the 365 daily transition kernels was explored by creating an annual 534


36

discretized transition kernel as the product of the daily transition kernels. Based on literature values, a 535

single female can reproduce up to 12 times each spawning season with nearly 200 female hatchlings per 536

spawn, so the growth potential demonstrated in the model was biologically plausible for a single season. 537

The large values of 𝜆 and 𝛼 for the annual transition matrices demonstrated the ability of the species to 538

grow rapidly. However, the inclusion of density dependent transition kernel functions will be critical for 539

projecting into longer time periods than the annual projections shown here. Elasticity analysis showed 540

that growth, 𝐺(𝑧′, 𝑧), had a larger influence on 𝜆 than the other kernel functions. The daily time-step 541

led to small daily growth transitions, necessitating non-standard IPM numerical solution techniques 542

(Ellner et al., 2016). However, the challenges associated with this small growth increment and eviction 543

in the largest size-classes were accommodated with minimal modifications. 544

5.1 - Inclusion of winter 545

The key assumptions for the Winter scenario were no growth and a threshold cutoff size for survival. 546

Although values for 𝜆 and 𝛼 were nearly identical between Winter and Baseline scenarios, simulations 547

showed that including a winter season led to a 22% decrease on the total population. Furthermore, the 548

Winter scenarios had annual 𝜔 values that were much smaller than Baseline, indicating that realizable 549

population growth is more dependent on the size-distribution of individuals in the Winter scenario than 550

in the Baseline scenario. From an ecotoxicological standpoint, this indicates that chemicals causing 551

decreased growth may have greater impacts on fish populations experiencing winter, especially if 552

significant proportions of the recruitment class do not exceed the size necessary for over-winter 553

survival. 554

Consistent with this finding, field observations of winter effects on recruitment of fathead minnow 555

showed shifts towards more density in larger size classes when fall and spring populations were 556

measured, due to the size-selective survival pressure that winter can exert (Divino & Tonn, 2007). Our 557


37

model predicted the same shift, although the pre- and post-winter shift to a larger sized population was 558

small. However, if the entire year is considered, the inclusion of winter, compared to Baseline, showed 559

a shift to, on average, smaller-sized individuals in the population. Due to size-selective mortality skewing 560

the population towards larger individuals in the absence of growth during winter, the overall shift to 561

smaller individuals is likely due to the no-growth condition that winter imposes. This result 562

demonstrates that short-term shifts in size distributions due to seasonally effects may not be 563

representative of overall trends when seasonality is included. 564

The inclusion of winter and its associated non-chemical stressor provides motivation to discuss the 565

impact of the date a simulated year begins. Specifically, if the simulation begins on Jan 1st, as it does in 566

the examples presented, the stable size distribution for the year will be different than if the annual cycle 567

begins on the first day of spring, or the first day of reproduction. However, the dominant eigenvalue for 568

different simulation start dates are identical. To explore this a little more, let 𝐾𝑖 be the discretized 569

transition kernel for the 𝑖𝑡ℎ ordinal day of the year, and let 𝜆 and 𝑣 be the dominant eigenvalue-570

eigenvector pair for the annual transition kernel starting on Jan 1st, then, 571

𝐾1𝐾2 ⋯𝐾365𝑣 = 𝜆𝑣. 572

If the starting date is reordered, such that the simulation begins on Dec 31st, then, 573

𝐾365𝐾1𝐾2 ⋯𝐾364(𝐾365𝑣) = 𝜆(𝐾365𝑣), 580

and now the (𝐾365𝑣) is the new eigenvector for the same eigenvalue 𝜆 with the year starting on Dec 574

31st. This process can be generalized by left multiplying by the product ∏ 𝐾𝑗365𝑗=𝑖 , for any 𝑖 ∈ [1,365] to 575

begin the year on the 𝑖𝑡ℎ ordinal day, and the eigenvector associated with the eigenvalue 𝜆 is then 576

(∏ 𝐾𝑗365𝑗=𝑖 )𝑣. This demonstrates that a cyclic reordering of daily transition matrices, such as what takes 577

place when alternate starting dates are chosen, does not change the dominant eigenvalue. However, as 578

shown, it does change the eigenvector, and thus the stable size distribution associated with it. This 579


38

change in asymptotic size structure is environmentally relevant, as different size-selective pressures are 581

encountered by populations. Thus, an understanding of the effect of simulation timing on results is 582

important for appropriate interpretation of modeled outputs. 583

5.2 - Population impacts of chemical exposure 584

A standard workflow for the incorporation of acute effects on survival in an IPM was established in this 585

research. This workflow utilized a publicly available chemical exposure simulation model, the PWC 586

(PWC, 2016). The effects of time-variable exposure scenarios were simulated with a threshold chemical 587

exceedance model and a publicly available TK-TD model, GUTS (Albert, 2016). Size-dependence was 588

simulated within each effects model through the introduction of a threshold size-dependent toxicity 589

modifier 𝜓𝑆𝑇𝐸𝑃(𝑧). The threshold chemical exceedance model scenarios showed the largest impacts at 590

the population level due to exposure. Chemical concentrations exceeded the 96hr LC50 on three 591

occasions, yet this resulted in an 88% population decrease from Baseline. The range of growth 592

potentials was smallest in this scenario as well (Figure 3). The survival effects predicted by the GUTS 593

effect model were smaller than those of the threshold model, but still influential, with a simulated 594

population decrease of 46% from Baseline. Both effects models were parameterized using data for 595

juvenile fish; therefore, the inclusion of a size-dependent toxicity modifier assuming that adults were 596

less sensitive than juveniles decreased the effects of the chemical exposure for the population. 597

Including effects from chemical exposures also demonstrated the potential to reorder the relative 598

elasticities of 𝜆 with respect to the kernel functions. The general trends in the Baseline model for 599

elasticities was that during reproduction 𝜆 was more responsive to perturbations to growth than survival 600

during the reproductive season. However, with a large enough survival decrement from chemical 601

exposure, the survival function showed the larger elasticity. 602


39

5.3 - Data availability for Ecological Risk Assessment 603

It is often the case that ecological risk assessment takes place in a data-limited environment. As such, 604

the detailed time course of survival data for P. promelas that was used to parameterize GUTS will often 605

not be available. The TCEM model uses only the exposure profile and a 96hr LC50 value to estimate 606

survival decrements. Although the differences in these effects model assumptions will be intimately 607

tied to the ecological and exposure context, these scenarios provide an example quantification of the 608

differences in effect modeling assumptions and demonstrate that this modeling approach can be used 609

to explore the effects of these assumptions. The vB growth function used for mean size transitions is a 610

commonly used for fish growth. As such, values are available for multiple fish species to parameterize 611

vB growth functions in the literature and through online databases such as fishbase.org (Froese & Pauly, 612

2019). Data availability for model parameterization can be a challenge for model application, the IPM 613

approach has the flexibility to be parameterized with limited data and refined should more resolved 614

data become available. 615

6 - Conclusions 616

The integral projection modeling approach developed here allowed for the incorporation of changes to 617

growth, reproduction, and survival based on natural and anthropogenic stressors. Given the use of a 618

daily time-step, this approach is especially suited to exploring the role that the timing of life-history 619

events (e.g. reproduction) and stressors have in influencing population growth trajectories. The 620

modular approach to kernel construction also makes this model adaptable to altering any of the 621

demographic components, such as the family of probabilistic growth transitions, the functional form of 622

the size-dependent survival probability, and any of the functions determining reproduction. This model 623

was constructed with ecotoxicological applications in mind, and specifically, ecological risk assessment. 624

Therefore, many assumptions made in the model can be modified based on the data available and 625


40

application context, including modification to life-history patterns of other aquatic organisms or other 626

fish species. We found the IPM approach amenable to ecotoxicological applications. 627

This model development proceeded in parallel with experimental support for parameterization 628

(Appendix A and Kadlec et al., in prep). Due to this synergistic effort, many model parameters were able 629

to be directly measured and validated. One result of this was high confidence in the growth transition 630

kernel components. Fathead minnow is a standard test species for toxicity testing, so toxicological and 631

reproduction data were available in the literature. Size-dependent survival of fathead minnow 632

represented by the logistic function (Equation 7) was parameterized based on average lifespan, size at 633

juvenile to adult transition, and the field-estimated probability to survive from hatch to adult size. 634

However, individual-level data to validate the parameterization of our size-dependent survival functions 635

was not available. Given the role that the survival function plays in dynamics, experimentation or field 636

observations of survival across a spectrum of sizes would be desirable to further refine this aspect of the 637

model. 638

Annual population growth rates of the magnitude observed in this model will very quickly lead to 639

biologically unrealistic population estimates. Given the large population growth potential of many 640

aquatic species, including density dependence in the transition kernel will be useful if the application 641

calls for estimation of the population over longer time periods. Techniques to include density 642

dependence in the transition kernel have been established for IPMs (Eager et al., 2014) and inclusion of 643

density dependence is anticipated as development of the model continues. These high growth rates are 644

also a function of the data sources for growth and survival, which include laboratory observations where 645

survival and growth rates will be higher than in wild populations. 646

Chronic effects of chemical exposure, such as those that affect growth or reproduction, can be especially 647

challenging for chemical risk assessment. A strength of the IPM approach presented is that vital rates 648


41

can be explicitly linked to size. Therefore, if data were available to modify/re-parameterize growth 649

transitions during exposure, this approach would be well-suited to incorporate chronic effects of 650

exposure on growth. The ability to incorporate chronic effects is one of the outstanding aspects of using 651

an IPM approach. Thus, future development of this IPM will likely include modifications to reproductive 652

and growth functions based on sublethal exposures. 653

The timing of chemical exposure can also play an important role in determining the magnitude of 654

impacts of chemical exposures on fish populations. For example, if newly hatched offspring are more 655

sensitive to chemical exposure than adult fish, one would expect chemical exposure during and directly 656

following the reproductive season to have a larger impact than exposure just prior to reproduction, 657

when the population is comprised of mostly larger-sized individuals. The ability of the model to 658

differentiate between differently timed events is tied to the model’s inclusion of size-dependent survival 659

effects. Efforts to refine size-dependent responses to chemical exposure, whether through re-660

parameterization of the kernel functions during exposure or through the expansion of the size-661

dependent toxicity modifier (𝜓𝑥(𝑧)) concept, should not only improve mortality predictions in general, 662

but also the ability of the model to discriminate between scenarios with differently timed exposures. 663

The introduction of the size dependent toxicity modifier should enable straightforward inclusion of size-664

dependent effects towards this goal. 665

Wildlife populations experience a wide variety of natural and anthropogenic stressors. These stressors 666

can have acute or chronic effects on population growth and viability. The development of modeling 667

tools that can be used in the data-limited environment, such as ecological risk assessment, is necessary 668

to understand the potential of stressors to impact wildlife populations. The IPMs introduced here serve 669

as an example of their potential within ecotoxicology. It is our goal that approaches introduced in this 670

paper will provide a new set of structured-modeling tools to support ecological risk assessment of 671

chemicals for environmental protection of wildlife populations. 672


42

7 - Acknowledgements 673

This work represents a parallel effort of laboratory experimentation as well as theoretical and 674

computational model development. We would like to gratefully acknowledge the contributions of 675

Victoria Kurker and Frank Whiteman for their laboratory support. We would also like to express our 676

gratitude to Richie Erickson for reviewing the model’s R implementation and Steve Ellner for reviewing 677

the model’s theoretical construction, including the analysis and guidance on the number of size classes 678

to use. We are grateful to Jeffery Divino for supplying data on field measurements related to growth and 679

survival of P. promelas and to Kris Garber for providing insight into the current challenges associated 680

with the ecological risk assessment process. Mention of trade names or commercial products does not 681

constitute endorsement or recommendation for use by USEPA. Likewise, the contents of the article 682

neither constitute nor necessarily reflect official policy of the USEPA. 683

8 – Data and code availability 684

External data sources used to support model development are all cited within the text. Additional 685

toxicological data can be found in with our corresponding paper, Kadlec et al., (in prep), at USEPA’s 686

Science Hub (https://catalog.data.gov/harvest/epa-sciencehub) or upon request. Supporting R code is 687

undergoing development into an R package and can be made available upon request of the authors. 688

9 - Declaration of conflicting interests 689

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or 690

publication of this article. 691

10 - Funding 692

This work was supported by the U.S. EPA Office of Research and Development 693

694

https://catalog.data.gov/harvest/epa-sciencehub

https://catalog.data.gov/harvest/epa-sciencehub


43

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US Environmental Protection Agency (EPA). 2016. Ecological Effects Test Guidelines. OCSPP 850.1400 824 Fish Early Life Stage Toxicity Test. EPA 712-C-008-October 2016. Washington DC. 825

Vandenbos, R. E., Tonn, W. M., & Boss, S. M. (2006). Cascading life-history interactions: alternative 826 density-dependent pathways drive recruitment dynamics in a freshwater fish. Oecologia, 148(4), 827 573-582. 828

Wania, F., & Mackay, D. (1993). Global fractionation and cold condensation of low volatility 829 organochlorine compounds in polar regions. Ambio, 10-18. 830

White, J. W., Nickols, K. J., Malone, D., Carr, M. H., Starr, R. M., Cordoleani, F., … Botsford, L. W. (2016). 831 Fitting state-space integral projection models to size-structured time series data to estimate 832 unknown parameters. Ecological Applications, 26(8), 2675–2692. http://doi.org/10.1002/eap.1398 833

Williams, J. L., Miller, T.EX., & Ellner, S.P. (2012) Avoiding unintentional eviction from integral projection 834 models. Ecology 93.9 (2012): 2008-2014. 835

Young, H. (1968). A consideration of insecticide effects on hypothetical avian populations. Ecology, 836 49(5), 991-994. 837

838


47

12 - Supplementary material 839

840

Supplemental Figure 1: Mean daily growth transitions by size (mm). The mean growth increment

decreases as individuals grow larger, following the vonBertalanffy growth model. The standard

deviation around the growth increment also decreases as the growth increment shrinks for larger

individuals.

BASELINE MEAN DAILY GROWTH INCREMENTS


48

841

Batch-spawning Algorithm Compute 𝑑𝑎𝑖𝑙𝑦 𝑠𝑝𝑎𝑤𝑛𝑖𝑛𝑔 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦

Inputs: 𝑆𝑝𝑎𝑤𝑛𝑖𝑛𝑔 𝑠𝑡𝑎𝑟𝑡 𝑑𝑎𝑡𝑒, 𝑠𝑝𝑎𝑤𝑛𝑖𝑛𝑔 𝑒𝑛𝑑 𝑑𝑎𝑡𝑒,𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑠𝑝𝑎𝑤𝑛𝑖𝑛𝑔 𝑎𝑡𝑡𝑒𝑚𝑝𝑡𝑠, 𝑖𝑛𝑡𝑒𝑟 −𝑠𝑝𝑎𝑤𝑛 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙, 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙𝑠

Outputs: 𝑑𝑎𝑖𝑙𝑦 𝑠𝑝𝑎𝑤𝑛𝑖𝑛𝑔 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑣𝑒𝑐𝑡𝑜𝑟

Begin procedure

1: for 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒊𝒏𝒅𝒊𝒗𝒊𝒅𝒖𝒂𝒍𝒔

2: sample 𝒔𝒑𝒂𝒘𝒏𝒊𝒏𝒈 𝒂𝒕𝒕𝒆𝒎𝒑𝒕𝒔 uniformly between 0 and 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑠𝑝𝑎𝑤𝑛𝑖𝑛𝑔 𝑎𝑡𝑡𝑒𝑚𝑝𝑡𝑠

3: determine 𝒊𝒏𝒅𝒊𝒗𝒊𝒅𝒖𝒂𝒍𝒔’ 𝒔𝒑𝒂𝒘𝒏𝒊𝒏𝒈 𝒑𝒆𝒓𝒊𝒐𝒅 = (1 + 𝑖𝑛𝑡𝑒𝑟𝑠𝑝𝑎𝑤𝑛 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙) ∗𝑠𝑝𝑎𝑤𝑛𝑖𝑛𝑔 𝑎𝑡𝑡𝑒𝑚𝑝𝑡𝑠

4: determine 𝒊𝒏𝒅𝒊𝒗𝒊𝒅𝒖𝒂𝒍𝒔′𝒔𝒑𝒂𝒘𝒏𝒊𝒏𝒈 𝒔𝒄𝒉𝒆𝒅𝒖𝒍𝒆: sample individuals’ spawning start date uniformly from possible spawning dates based on 𝑠𝑝𝑎𝑤𝑛𝑖𝑛𝑔 𝑠𝑡𝑎𝑟𝑡 𝑑𝑎𝑡𝑒,𝑠𝑝𝑎𝑤𝑛𝑖𝑛𝑔 𝑒𝑛𝑑 𝑑𝑎𝑡𝑒, and 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙𝑠’ 𝑠𝑝𝑎𝑤𝑛𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑

5: determine 𝒕𝒐𝒕𝒂𝒍 𝒅𝒂𝒊𝒍𝒚 𝒔𝒑𝒂𝒘𝒏𝒔 : map 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙𝑠′ 𝑠𝑝𝑎𝑤𝑛𝑖𝑛𝑔 𝑠𝑐ℎ𝑒𝑑𝑢𝑙𝑒 onto ordinal day and add all attempts per day

6: return 𝒅𝒂𝒊𝒍𝒚 𝒔𝒑𝒂𝒘𝒏𝒊𝒏𝒈 𝒑𝒓𝒐𝒃𝒂𝒃𝒊𝒍𝒊𝒕𝒚 𝒗𝒆𝒄𝒕𝒐𝒓 = 𝑡𝑜𝑡𝑎𝑙 𝑑𝑎𝑖𝑙𝑦 𝑠𝑝𝑎𝑤𝑛𝑠

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙𝑠

End procedure

Parameters used for simulations:

𝑺𝒑𝒂𝒘𝒏𝒊𝒏𝒈 𝒔𝒕𝒂𝒓𝒕 𝒅𝒂𝒕𝒆: May 22nd

𝑺𝒑𝒂𝒘𝒏𝒊𝒏𝒈 𝒆𝒏𝒅 𝒅𝒂𝒕𝒆: August 22nd

𝑴𝒂𝒙𝒊𝒎𝒖𝒎 𝒔𝒑𝒂𝒘𝒏𝒊𝒏𝒈 𝒂𝒕𝒕𝒆𝒎𝒑𝒕𝒔: 12 (Markus, 1934; Jensen et al., 2001)

𝑰𝒏𝒕𝒆𝒓 − 𝒔𝒑𝒂𝒘𝒏 𝒊𝒏𝒕𝒆𝒓𝒗𝒂𝒍: 3 Days (Jensen et al., 2001)

𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒊𝒏𝒅𝒊𝒗𝒊𝒅𝒖𝒂𝒍𝒔: 100

842

843

844

Supplemental Table 1: Description of the batch spawning algorithm used to compute the daily

spawning probabilities for an iteroparous spawner. Parameter values used are representative of P.

promelas life history. Assumptions of uniform distribution of number of spawns and spawning start

date can be refined if information becomes available.


49

845

846

BASELINE DAILY SPAWNING PROBABILITY

Supplemental Figure 2: Daily spawning probability time-series as an example output from the batch

spawning algorithm. A lower probability of spawning emerges toward the beginning and ending of the

spawning season, and probability of spawning is highest during the middle of the spawning season.

This general trend for daily spawning probabilities is a result of an assumption of uniform probability of

spawning attempts and uniform distribution of spawning start dates. Details of batch spawning

algorithm are given in Supplemental Table 1.


50

847

848

849

Supplemental Figure 3 A conceptual diagram to show linkages between the PWC exposure model,

GUTS or TCEM effects model, and the IPM model.


51

850

851

DAILY DISCRETIZED TRANSITION

KERNEL SUMMARY

SIMULATION SCENARIOS

BASELINE


Calendar Year Start

Jan1-Dec31

Reproductive Season Start

May22-May21 GUTS-

Uniform GUTS- Step

TCEM- Uniform

TCEM- Step

DAILY DOMINANT EIGENVALUES SUMMARY

𝑴𝑰𝑵(𝝀𝑫⃗⃗ ⃗⃗ )

𝑴𝑨𝑿(𝝀𝑫⃗⃗ ⃗⃗ )

𝑴𝑬𝑨𝑵(𝝀𝑫⃗⃗ ⃗⃗ )

0.9999166

1.04999

1.011535

0.9999166

1.04999

1.011535

0.9999166

1.04999

1.011535

0.6545586

1.04999

1.010057

0.8272376

1.04999

1.010675

0.499916

1.04999

1.007317

0.7499166

1.04999

1.009279

DAILY MAXIMUM COLUMN SUMS SUMMARY

𝑴𝑰𝑵(𝜶𝑫⃗⃗ ⃗⃗ ⃗)

𝑴𝑨𝑿(𝜶𝑫⃗⃗ ⃗⃗ ⃗)

𝑴𝑬𝑨𝑵(𝜶𝑫⃗⃗ ⃗⃗ ⃗)

0.9999166

5.9592

1.917933

0.9999166

5.9592

1.917933

0.9999166

5.9592

1.917933

0.6545586

5.9592

1.915897

0.827237

5.9592

1.916629

0.499916

5.9592

1.910186

0.7499166

5.9592

1.912241

DAILY MINIMUM COLUMN SUMS SUMMARY

𝑴𝑰𝑵(𝝎𝑫⃗⃗⃗⃗⃗⃗ )

𝑴𝑨𝑿(𝝎𝑫⃗⃗⃗⃗⃗⃗ )

𝑴𝑬𝑨𝑵(𝝎𝑫⃗⃗⃗⃗⃗⃗ )

0.9949533

0.9949533

0.9949533

0

0.9949533

0.7496224

0

0.9949533

0.7496224

0.6495954

0.9949533

0.9934896

0.6495954

0.9949533

0.9934896

0.4949533

0.9949533

0.9908437

0.4949533

0.9949533

0.9908437

DAILY DOMINANT EIGENVALUES: 𝝀𝑫⃗⃗ ⃗⃗ = {𝝀𝟏, 𝝀𝟐, … , 𝝀𝟑𝟔𝟓}

DAILY MAXIMUM COLUMN SUM: 𝜶𝑫⃗⃗ ⃗⃗ ⃗ = {𝜶𝟏, 𝜶𝟐, … , 𝜶𝟑𝟔𝟓} DAILY MINIMUM COLUMN SUM: 𝝎𝑫⃗⃗⃗⃗⃗⃗ = {𝝎𝟏, 𝝎𝟐, … ,𝝎𝟑𝟔𝟓}

Supplemental Table 2: Daily discretized transition kernel summary. Given the daily-time step,

minimum, maximum, and mean values for the daily discretized transition kernels are provided for the

dominant eigenvalues, the maximum column sum, and the minimum column sum.


52

852

853

Supplemental Figure 4: Time series of number of individuals and average individual size from Baseline

scenario simulation.


53

854

Supplemental Figure 5: Time series of number of individuals and average individual size from Winter

scenario simulation.


54

855

Supplemental Figure 6: Time series of number of individuals and average individual size from Winter

scenario simulation with a Reproductive season starting date to simulation (May 22nd)


55

856

Supplemental Figure 7: Time series of number of individuals and average individual size from GUTS

scenario simulation with a size-independent (Uniform) survival decrement.


56

857

Supplemental Figure 8: Time series of number of individuals and average individual size from GUTS

scenario simulation with a step function toxicity modifier on the survival decrement.


57

858

Supplemental Figure 9: Time series of number of individuals and average individual size from TCEM

scenario simulation with a size-independent (Uniform) survival decrement.


58

859

Supplemental Figure 10: Time series of number of individuals and average individual size from TCEM

scenario simulation with a step function toxicity modifier on the survival decrement.


59

Appendix A: Deriving probability distributions of growth from size using 860

Weibull and von Bertalanffy growth functions 861

Appendix B: Additional sensitivity and elasticity analysis results to 862

accompany Developing integral projection models for ecotoxicology 863


1

Appendix A - Deriving probability distributions of growth from size using

Weibull and von Bertalanffy growth functions From Developing Integral Projection Models for Aquatic Ecotoxicology, Pollesch et al.

This appendix is focused on developing probability density functions that can be used to account for

probabilistic growth transitions for use in an IPM with Fathead minnow as the case study (Pimephales

promelas). Standard non-linear regression techniques were used to attain parameter estimates for

growth modeled using both the Weibull and von Bertalanffy growth curves. Further, growth curves were

developed from equations of size, calculated from time of hatch to equations of size calculated from a

(non-hatch) time based on the current size of the organism. The delta method was used to calculate the

variance around the estimates of size for the new parametrizations for size to size transitions. The

analytical results presented were tested numerically (by simulation) to determine both the accuracy of

the results and any conditions that need to be imposed to ensure that the results remain accurate. The

simulations reveal the analytical results attain an acceptable accuracy as long as the current size of the

organism used in the calculations is between the estimates used for both asymptotic size or hatch size,

with the results becoming less accurate as the current size approaches either extreme.

Two growth curves (the von Bertalanffy and the Weibull) are used to estimate mid-lifetime growth,

which is growth from time starting at current size (𝑧) instead of growth from a time starting at hatch.

The report does this by defining a probability density function, 𝐺(𝑧′, 𝑧), where 𝑧′ is the size after growth

at time step, 𝜏. Based on the assumption of the distribution of the parameters, 𝐺(𝑧′, 𝑧) would follow the

normal distribution with a mean calculated from one of the mid-lifetime growth curves and a variance

calculated using the delta method. This appendix also includes the probability density function for the

size of offspring, 𝑔(𝑧, 𝜏𝑔), which is growth over a time step, 𝜏𝑔, starting from hatch. Since 𝑔(𝑧, 𝜏𝑔) is

measured from hatch it also follows a normal distribution, however it’s mean follows the standard

parametrization of the growth curve being used and its variance is dictated by the residual variance of


2

the model fit. The equations for both mid-lifetime and hatchling growth are derived for both von

Bertalanffy and the Weibull in their full form and for the version of each growth curve where hatch size

(𝐿0) and maximum asymptotic size (𝐿∞) are treated as constants. Estimates for every parameter and

their covariance’s for every model for both male and female fathead minnows are also all supplied.

The (full) von Bertalanffy to model growth yields a probability density function, 𝐺(𝑧′, 𝑧), for the size (𝑧′)

after a growth time step (𝜏) given a starting size of (𝑧) of:

Equation 1:

𝐺(𝑧′, 𝑧)~𝑁(𝜇 = 𝑉𝐵(𝑧, 𝜏), 𝜎2 = 𝜎𝑉𝐵 | 𝑧2 ) where

𝑉𝐵(𝑧, 𝜏) = 𝜇 = 𝐿∞ − (𝐿∞ − 𝑧) ∗ 𝑒−(𝑘∗𝜏) and

𝜎𝑉𝐵 | 𝑧2 = (

𝜇−𝑧

𝐿∞−𝑧)2𝜎𝐿∞

2 + 2𝜏(𝐿∞ − 𝜇) (𝜇−𝑧

𝐿∞−𝑧) 𝜎𝑘,𝐿∞

+ 𝜏2(𝐿∞ − 𝜇)2𝜎𝑘2

Where 𝐿∞ is the mean population asymptotic size, 𝑘 controls the growth rate with respect to size, 𝜎𝑥2 is

the variance of 𝑥, and 𝜎𝑥,𝑦 is the covariance between 𝑥 and 𝑦. The probability density function for the

size of the offspring, 𝑔(𝑧, 𝜏𝑔) using the (full von Bertalanffy) is shown in equation 2.

Equation 2:

𝑔(𝑧′, 𝜏𝑔)~𝑁(𝜇 = 𝑉𝐵(𝜏𝑔), 𝜎2 = 𝜎𝜀 2 ∗ 𝜇) where

𝑉𝐵(𝜏𝑔) = 𝐿∞ − (𝐿∞ − 𝐿0) ∗ 𝑒−(𝑘∗𝜏𝑔)

The additional symbol (𝐿0) in equation 2 represents the mean population size at hatch.

The parameter estimates for the von Bertalanffy growth curve for length of male/female fathead

minnows are listed in Tables 1 and 2.


3

Table 1: Full von Bertalanffy Estimates for Male Fathead Minnows

Parameter Estimates

𝑳∞ 𝑳𝟎 𝒌 𝝈𝜺

𝟐 75.4 3.13 0.011 0.42

Covariance

𝑳∞ 𝑳𝟎 𝒌 𝝈𝜺𝟐

L∞ 6.73E+00 8.69E-01 2.16E-03 1.16E-03

L0 8.69E-01 1.11E+00 7.50E-04 4.97E-05

𝒌 -2.16E-03 -7.50E-04 1.16E-06 5.34E-07

σε2 -1.16E-03 4.97E-05 5.34E-07 1.25E-02

Table 2: Full von Bertalanffy Estimates for Female Fathead Minnows

Parameter Estimates


𝟐 58.9 2.61 0.016 0.35

Covariance


𝑳∞ 3.32E+00 4.50E-01 -1.71E-03 1.75E-02

𝑳𝟎 4.50E-01 1.09E+00 -1.03E-03 -1.26E-02

𝒌 -1.71E-03 -1.03E-03 2.01E-06 -8.44E-06

𝝈𝜺𝟐 1.75E-02 -1.26E-02 -8.44E-06 4.82E-03

In addition to the (full) von Bertalanffy, 𝐿∞ and 𝐿0 have the potential to be treated as fixed values

instead of random variables. This changes the meaning of 𝐿∞ and 𝐿0 from mean asymptotic population

sizes to absolute asymptotic population size. As in no organism can be larger then 𝐿∞ or smaller then 𝐿0.

When 𝐿∞ and 𝐿0 are treated as fixed values the variance term associated with 𝐺(𝑧′, 𝑧), 𝜎𝑉𝐵 | 𝑧2 ,

simplifies drastically and changes equation 1 into equation 3. Equation 2 remains unchanged, however

treating 𝐿∞ and 𝐿0 as a constant instead of a random variable changes the values for both the


4

parameter and covariance estimates (Tables 3 and 4).

Equation 3:

𝐺(𝑧′, 𝑧)~𝑁(𝜇 = 𝑉𝐵(𝑧, 𝜏), 𝜎2 = 𝜎𝑉𝐵 | 𝑧2 ) where

𝑉𝐵(𝑧, 𝜏) = 𝜇 = 𝐿∞ − (𝐿∞ − 𝑧) ∗ 𝑒−(𝑘∗𝜏) and

𝜎𝑉𝐵 | 𝑧2 = 𝜏2(𝐿∞ − 𝜇)2𝜎𝑘

2

Table 3: von Bertalanffy (Fixed L∞ and L0) Estimates for Male Fathead Minnows

Parameter Estimates


𝟐 86 5.6 0.008 0.58

Covariance


L∞ 0 0 0 0

L0 0 0 0 0

𝒌 0 0 1.93E-07 -1.27E-08

σε2 0 0 -1.27E-08 2.43E-02

Table 4: von Bertalanffy (Fixed 𝐿∞ and 𝐿0) for Female Fathead Minnows

Parameter Estimates


𝟐 74 5.6 0.009 0.96

Covariance


𝑳∞ 0 0 0 0

𝑳𝟎 0 0 0 0

𝒌 0 0 5.94E-07 -3.22E-05

𝝈𝜺𝟐 0 0 -3.22E-05 2.21E-03

The (full) Weibull which, compared to the von Bertalanffy, ends up with a much more complex variance

structure. This is because the Weibull, unlike the von Bertalanffy, cannot be easily reparametrized in


5

terms of current size (𝑧) and instead the variable representing time since hatch (𝑡) needs to be

recalculated based on 𝑧. This leads to a re-parametrization of the Weibull (𝑊𝑒𝑖) using the time it would

take to reach the current size (𝑇𝑏𝑖𝑜); as shown in equation 4. This is all due to the additional parameter

the Weibull has over the von Bertalanffy (𝑣) which controls the effect of 𝑘 over time.

Equation 4:

𝑊𝑒𝑖(𝑡 + 𝜏) = 𝐿∞ − (𝐿∞ − 𝐿0) ∗ 𝑒−((𝑘(𝑡+𝜏))^(𝑣))

𝑊𝑒𝑖(𝜏 ; 𝑧) = 𝐿∞ − (𝐿∞ − 𝐿0) ∗ 𝑒−((𝑘(𝑇𝑏𝑖𝑜+𝜏))

𝑣)

𝑇𝑏𝑖𝑜(𝑧) =1

𝑘(−ln (

(𝐿∞ − 𝑧)

(𝐿∞ − 𝐿0)))

1𝑣⁄

𝐺(𝑧′, 𝑧) and 𝑔(𝑧′, 𝜏𝑔) using the (full) Weibull are shown in equations 5 and 6 below.

Equation 5:

𝐺(𝑧′, 𝑧)~𝑁(𝜇 = 𝑊𝑒𝑖(𝑧, 𝜏), 𝜎2 = 𝜎𝑊𝑒𝑖 | 𝑧2 ) where

𝑊𝑒𝑖(𝑧, 𝜏) = 𝜇 = 𝐿∞ − (𝐿∞ − 𝐿0) ∗ 𝑒−((𝑘(𝑇𝑏𝑖𝑜+𝜏))^𝑣) and

𝜎𝑊𝑒𝑖 | 𝑧2 =

(𝜕𝑊𝑒𝑖

𝜕𝐿∞)2∗ 𝜎𝐿∞

2 + (𝜕𝑊𝑒𝑖

𝜕𝐿0)2∗ 𝜎𝐿0

2 + (𝜕𝑊𝑒𝑖

𝜕𝑘)2∗ 𝜎𝑘

2 + (𝜕𝑊𝑒𝑖

𝜕𝑣)2∗ 𝜎𝑣

2 +

2 ∗ (𝜕𝑊𝑒𝑖

𝜕𝐿∞) ∗ (

𝜕𝑊𝑒𝑖

𝜕𝐿0) ∗ 𝜎𝐿∞,𝐿0 + 2 ∗ (

𝜕𝑊𝑒𝑖

𝜕𝐿∞) ∗ (

𝜕𝑊𝑒𝑖

𝜕𝑘) ∗ 𝜎𝐿∞,𝑘 + 2 ∗ (

𝜕𝑊𝑒𝑖

𝜕𝐿∞) ∗ (

𝜕𝑊𝑒𝑖

𝜕𝑘) ∗ 𝜎𝐿∞,𝑘 +

2 ∗ (𝜕𝑊𝑒𝑖

𝜕𝐿∞) ∗ (

𝜕𝑊𝑒𝑖

𝜕𝑣) ∗ 𝜎𝐿∞,𝑣 + 2 ∗ (

𝜕𝑊𝑒𝑖

𝜕𝐿0) ∗ (

𝜕𝑊𝑒𝑖

𝜕𝑘) ∗ 𝜎𝐿0,𝑘 + 2(

𝜕𝑊𝑒𝑖

𝜕𝐿0) ∗ (

𝜕𝑊𝑒𝑖

𝜕𝑣) ∗ 𝜎𝐿0,𝑣 +

2 ∗ (𝜕𝑊𝑒𝑖

𝜕𝑘) ∗ (

𝜕𝑊𝑒𝑖

𝜕𝑣) ∗ 𝜎𝑘,𝑣

where:


6

𝜕𝑊𝑒𝑖

𝜕𝐿∞= 1 − (1 +

(𝑧 − 𝐿0) ∗ 𝑇𝑏𝑖𝑜1−𝑣 ∗ (𝑇𝑏𝑖𝑜 + 𝜏)𝑣−1

𝐿∞ − 𝑧) ∗ (

𝐿∞ − 𝜇

𝐿∞ − 𝐿0)

𝜕𝑊𝑒𝑖

𝜕𝐿0= (1 − 𝑇𝑏𝑖𝑜

1−𝑣 ∗ (𝑇𝑏𝑖𝑜 + 𝜏)𝑣−1) ∗ (𝐿∞ − 𝜇

𝐿∞ − 𝐿0)

𝜕𝑊𝑒𝑖

𝜕𝑘= 𝑣 ∗ 𝜏 ∗ 𝑘𝑣−1 ∗ (𝑇𝑏𝑖𝑜 + 𝜏)𝑣−1 ∗ (𝐿∞ − 𝜇)

𝜕𝑊𝑒𝑖

𝜕𝑣= −𝑘𝑣 ∗ (𝑇𝑏𝑖𝑜 + 𝜏)𝑣 ∗ ln(

(𝑘 ∗ 𝑇𝑏𝑖𝑜)𝑇𝑏𝑖𝑜

𝑇𝑏𝑖𝑜+𝜏

𝑘 ∗ (𝑇𝑏𝑖𝑜 + 𝜏)) ∗ (𝐿∞ − 𝜇)

Equation 6:

𝑔(𝑧′, 𝜏𝑔)~𝑁(𝜇 = 𝑊𝑒𝑖(𝜏𝑔), 𝜎2 = 𝜎𝜀 2 ∗ 𝜇) where

𝑊𝑒𝑖(𝜏𝑔) = 𝜇 = 𝐿∞ − (𝐿∞ − 𝐿0) ∗ 𝑒−((𝑘𝜏𝑔)^𝑣)

Tables 5 and 6 show the parameter estimates for the full Weibull.

Table 5: Full Weibull Estimates for Male Fathead Minnows

Parameter Estimates

𝑳∞ 𝑳𝟎 𝒌 𝒗 𝝈𝜺

𝟐 72.8 4.68 0.013 1.15 0.40

Covariance

𝑳∞ 𝑳𝟎 𝒌 𝒗 𝝈𝜺𝟐

𝑳∞ 5.22E+00 -1.02E+00 -1.68E-03 -8.00E-02 3.93E-02

𝑳𝟎 -1.02E+00 2.06E+00 2.56E-04 1.04E-01 -1.25E-02

𝒌 -1.68E-03 2.56E-04 1.02E-06 2.88E-05 -1.78E-05

𝒗 -8.00E-02 1.04E-01 2.88E-05 6.27E-03 -2.99E-03

𝝈𝜺𝟐 3.93E-02 -1.25E-02 -1.78E-05 -2.99E-03 9.48E-03

Table 6: Full Weibull Estimates for Female Fathead Minnows

Parameter Estimates


𝟐 57.2 5.13 0.018 1.29 0.30


7

Covariance


𝑳∞ 2.92E+00 -3.60E-01 -1.48E-03 -7.89E-02 -2.38E-03

𝑳𝟎 -3.60E-01 1.55E+00 -3.26E-04 1.19E-01 1.18E-02

𝒌 -1.48E-03 -3.26E-04 1.79E-06 5.53E-05 -9.39E-07

𝒗 -7.89E-02 1.19E-01 5.53E-05 1.77E-02 1.03E-03

𝝈𝜺𝟐 -2.38E-03 1.18E-02 -9.39E-07 1.03E-03 7.04E-03

Treating 𝐿∞ and 𝐿0 as fixed values instead of random variables vastly simplifies the variance term

associated with 𝐺(𝑧′, 𝑧), 𝜎𝑊𝑒𝑖 | 𝑧2 , and changes equation 5 into equation 7. Equation 6 (governing the

size of offspring) remains unchanged. The new parameter estimates associated with treating 𝐿∞ and 𝐿0

as constants instead of a random variable are listed in Tables 7 and 8.

Equation 6:

𝐺(𝑧′, 𝑧)~𝑁(𝜇 = 𝑊𝑒𝑖(𝑧, 𝜏), 𝜎2 = 𝜎𝑊𝑒𝑖 | 𝑧2 ) where

𝑊𝑒𝑖(𝑧, 𝜏) = 𝜇 = 𝐿∞ − (𝐿∞ − 𝐿0) ∗ 𝑒−((𝑘(𝑇𝑏𝑖𝑜+𝜏))^𝑣) and

𝜎𝑊𝑒𝑖 | 𝑧2 = (

𝜕𝑊𝑒𝑖

𝜕𝑘)2

∗ 𝜎𝑘2 + (

𝜕𝑊𝑒𝑖

𝜕𝑣)2

∗ 𝜎𝑣2 + 2 ∗ (

𝜕𝑊𝑒𝑖

𝜕𝑘) ∗ (

𝜕𝑊𝑒𝑖

𝜕𝑣) ∗ 𝜎𝑘,𝑣

𝜎𝑊𝑒𝑖 | 𝑧2 = 𝑘2𝑣 ∗ (𝑇𝑏𝑖𝑜 + 𝜏)2𝑣 ∗ (𝐿∞ − 𝜇)2 ∗

[

𝑣2 ∗ 𝜏2

𝑘2 ∗ (𝑇𝑏𝑖𝑜 + 𝜏)2∗ 𝜎𝑘

2 +

ln𝟐 ((𝑘 ∗ 𝑇𝑏𝑖𝑜)

𝑇𝑏𝑖𝑜𝑇𝑏𝑖𝑜+𝜏

𝑘 ∗ (𝑇𝑏𝑖𝑜 + 𝜏)) ∗ 𝜎𝑣

2 +

−2 ∗ 𝑣 ∗ 𝜏

𝑘 ∗ (𝑇𝑏𝑖𝑜 + 𝜏)∗ (

(𝑘 ∗ 𝑇𝑏𝑖𝑜)𝑇𝑏𝑖𝑜

𝑇𝑏𝑖𝑜+𝜏

𝑘 ∗ (𝑇𝑏𝑖𝑜 + 𝜏)) ∗ 𝜎𝑘,𝑣

]

Table 7: Weibull (Fixed L∞ and L0) Estimates for Male Fathead Minnows

Parameter Estimates


𝟐


8

86 5.6 0.008 0.96 0.59

Covariance


𝑳∞ 0 0 0 0 0

𝑳𝟎 0 0 0 0 0

𝒌 0 0 2.79E-07 1.50E-05 -5.24E-06

𝒗 0 0 1.50E-05 2.17E-03 -9.80E-04

𝝈𝜺𝟐 0 0 -5.24E-06 -9.80E-04 2.20E-02

Table 8: Weibull (Fixed L∞ and L0) Estimates for Male Fathead Minnows

Parameter Estimates


𝟐 74 5.6 0.008 0.85 0.80

Covariance


𝑳∞ 0 0 0 0 0

𝑳𝟎 0 0 0 0 0

𝒌 0 0 2.79E-07 1.50E-05 -5.24E-06

𝒗 0 0 1.50E-05 2.17E-03 -9.80E-04

𝝈𝜺𝟐 0 0 -5.24E-06 -9.80E-04 2.20E-02

The delta method is used to calculate the variance components of 𝐺(𝑧′, 𝑧). The delta method loosely

states; if there is a set of variables (𝜃) in which the difference between their estimate (𝜃) and their true

value is multivariate normal, then the difference between a function of 𝜃, 𝑓(𝜃), and the same function

of the estimates 𝑓(𝜃), converges in distribution to normality with mean 0 and variance ∇𝑓(𝜃)𝑇 ∗Σ

𝑛∗

∇𝑓(𝜃). Where ∇ is the gradient operator, 𝑇 is the transpose operator, Σ is the covariance matrix

associated with 𝜃, and 𝑛 is the number of observations used to calculate 𝜃. This is often written as:

Equation 7:

𝑓(𝜃) − 𝑓(𝜃)𝑑→𝑁(0, ∇𝑓(𝜃)𝑇 ∗

Σ

𝑛∗ ∇𝑓(𝜃))


9

This method is useful for non-linear regression as it is used to calculate the confidence intervals around

functions being estimated. An often arising challenge with the delta method is the fact that it operates

under “asymptotic convergence”, so if there is not enough data then the delta method inaccurately, and

can often (compared to bootstrapping) underestimate the confidence intervals around 𝑔(𝜃).

The parameter estimates for each growth curve by (Tables 1-8) were attained within MATLAB using the

fminunc function by numeric optimization of the likelihood function associated with the distributional

assumption listed in equation 7.

Equation 7:

𝑌𝑖 − 𝑓(𝜃)~𝑁(0, 𝝈𝜺𝟐 ∗ 𝑌𝑖)

Where 𝑌𝑖 is an observation and 𝑓(𝜃) is either the Weibull or the von Bertalanffy. Equation 7 uses what

mixed effect models within R call a “fixed variance structure”. A fixed variance structure is used in place

of a log transformation when compensating for the specific type of heteroscedasticity of variance that

occurs when the variance is a proportion of the mean of the predicted value instead of a constant that is

independent of the mean of the predicted value.

The covariance matrix (Σ) was attained by inverting the Hessian matrix (ℋ) calculated numerically as

part of the output from fminunc. Since the covariance attained from inverting ℋ is the estimate of the

covariance matrix for the population mean (Σ 𝑛⁄ ), it is further multiplied by the average number of

observations at each time (�̅�, Table 9) to scale up the variance to the individual level (Σ). Equation 8

shows estimate for the individual level estimator of the covariance matrix of 𝑓(𝜃) which values are

shown in tables 1-8 under “Covariance”.


10

Table 9: Average number of observations at each time (�̅�).

�̅� Fathead Minnows Medaka

Male 13.93 24.31

Female 12.82 24.23

Equation 8:

Σ̂ = �̅� ∗ ℋ−1

Parameter estimates for models where 𝐿∞ and 𝐿0 are treated as constants instead of random variables

(tables {3, 4, 7, 8, 11, 12, 15, 16}) are calculated using the same methodology as the parameter

estimates for the full models. The only difference was that 𝐿∞ and 𝐿0 were calculated using the

maximum or minimum values (respectively) of each data set. The exception was fork length for male

fathead minnows where the maximum value of the data set was 106mm with a second highest value of

86mm. In this case 86mm was chosen for the value of 𝐿∞ since using 106mm for 𝐿∞ produced models

that provided a poorer fit of the data (Figure 1).

Of note, since the maximum (or minimum) in a data set is a biased estimator for the maximum (or

minimum) of a population it is not always not used. Often, a correction factor is applied to the estimate

to correct for the bias. For instance, if the population comes from a uniform distribution with a

Figure 1: Simulations of fathead minnow fork length using 𝐿∞ = 86 (left) and 𝐿∞

= 106 (right) using the Weibull growth curve.


11

minimum value of 0, then the estimator for the maximum value of the population based on N (the

number of samples) is corrected by a factor of (𝑁 + 1) 𝑁⁄ . However, the correction factor is dependent

on the underlying distribution. Since the distribution of 𝐿∞ is unknown or assumed to be normal which

means the maximum value for 𝐿∞ for the population may not exist; for simplicity the maximum (or

minimum) of a data set is used of the estimator for the maximum (or minimum) of the population.

Finally, the derivatives for use in ∇𝑔(𝜃) were calculated using Wolfram Alpha

(http://www.wolframalpha.com) which was checked against a numerical differentiation of each growth

function. As long as 𝑧 remained between 𝐿0̂ and 𝐿∞̂ the analytical derivative matched the numeric one.

However, when 𝑧 was outside that range the analytical derivative (from Wolfram Alpha) could fail and

even produce complex numbers. The derivatives were then further simplified by hand and checked

numerically again against the derivatives in their original form.

The analytical results presented in equations 1-6 are tested numerically by simulation. The full

simulation procedure is as follows:

1) Use every combination of 𝑧 ∈ { 5, 6, 10, 25, 50, 70, 80} and 𝜏 ∈ {1, 10, 25, 50, 100}.

2) Simulate 1000 sets of parameters assuming the parameter are multivariate normal with

means and covariance attained from parameter estimates for male fathead minnows.

3) Use the sets of parameters to generate growth curve going from time t = 0 units to t =400

units in steps of 0.01 units.

4) For each growth curve:

a. Find the time corresponding to the supremum of values of the growth curve less

than or equal to 𝑧.

b. Output the value of the growth curve 𝜏 units in time later as 𝑧′

5) Calculate the mean and variance of 𝑧′.

http://www.wolframalpha.com/

http://www.wolframalpha.com/


12

6) Treat steps 2-6 as one simulation and repeat them a 1000 times.

7) Average of the means and the variances of 𝑧′ over the 1000 simulations.

The results can be split up into 2 groups based on 𝑧. The first group contained intermediate values of 𝑧

where the results from the variances calculated by the delta method agreed with the variances attained

from the simulation. The second group, containing the extreme values of 𝑧 where the variances

calculated by the delta method did not always agree with the simulations. Tables 10-13 display the

results of the simulations for each growth curve considered.

Table 10: Full von Bertalanffy; percent difference between simulation estimates and analytical estimates.

Mean Variance

𝑧 𝜏⁄ 1 10 25 50 100 𝑧 𝜏⁄ 1 10 25 50 100

5 0.51 -0.02 -0.26 -0.40 -0.49 5 584.19 -6.39 -3.39 0.59 9.24

6 0.05 -0.20 -0.34 -0.44 -0.50 6 46.69 -0.26 1.22 3.79 11.88

10 -0.01 -0.29 -0.52 -0.73 -0.90 10 1.96 1.53 2.48 5.48 13.83

25 -0.01 -0.16 -0.34 -0.52 -0.66 25 3.39 3.21 4.96 8.37 13.12

50 -0.01 -0.09 -0.20 -0.32 -0.41 50 9.66 8.70 8.28 6.54 1.77

70 0.00 0.04 0.10 0.20 0.37 70 -36.27 -36.66 -36.63 -36.24 -35.27

80 0.09 0.87 2.02 3.60 5.81 80 -94.29 -94.06 -93.74 -92.97 -91.48

Note; green indicates less than 1% in absolute value, yellow between 1% and 10%, and red between

10% and 25%. For this model 𝐿∞ = 75.4 and 𝐿0 = 3.13.

Table 11: von Bertalanffy (fixed 𝐿∞ and 𝐿0); percent difference between simulation estimates and analytical estimates.

Mean Variance

𝑧 𝜏⁄ 1 10 25 50 100 𝑧 𝜏⁄ 1 10 25 50 100

5 10.53 4.89 2.43 1.17 0.39 5 -1.49 -1.49 -1.45 -1.35 -0.87

6 0.05 0.01 -0.03 -0.08 -0.14 6 1.14 0.10 0.07 0.18 0.64

10 0.03 0.01 -0.02 -0.06 -0.12 10 1.08 0.09 0.06 0.15 0.58

25 0.01 0.00 -0.01 -0.04 -0.09 25 1.14 0.10 0.07 0.17 0.58

50 0.00 0.00 0.00 -0.02 -0.04 50 1.13 0.10 0.07 0.16 0.60

70 0.00 0.00 0.00 -0.01 -0.02 70 1.14 0.10 0.07 0.15 0.60


13

80 0.00 0.00 0.00 0.00 -0.01 80 1.11 0.08 0.05 0.14 0.56


10% and 25%. For this model 𝐿∞ = 86 and 𝐿0 = 5.6.

Table 12: Full Weibull; percent difference between simulation estimates and analytical estimates.

Mean Variance

𝑧 𝜏⁄ 1 10 25 50 100 𝑧 𝜏⁄ 1 10 25 50 100

5* -3.58 -2.88 -1.67 -0.91 -0.17 5* -93.56 -50.98 -28.17 6.15 11.54

6 -0.60 -1.13 -0.84 -0.53 -0.01 6 -77.66 -36.28 -20.52 -1.46 6.76

10 -0.05 -0.05 -0.03 0.03 0.41 10 -15.86 -14.30 -13.27 -11.85 4.38

25 -0.01 -0.01 0.00 0.00 0.49 25 0.44 1.33 1.48 1.56 2.36

50 0.00 0.01 0.02 0.04 0.50 50 0.51 0.88 1.23 1.24 -7.39

70 -0.01 -0.02 -0.03 -0.05 -0.44 70 57.30 74.55 74.71 80.51 83.58

80 NA NA NA NA NA 80 NA NA NA NA NA

The simulation could not reach values of 𝑧 or 𝑧′ of 80mm, so entries for 𝑧 = 80 are listed as NA. Note;

green indicates less than 1% in absolute value, yellow between 1% and 10%, and red between 10% and

25%. The (*) indicates that the real component of a complex number was used. For this model 𝐿∞ = 72.8

and 𝐿0 = 4.68.

Table 13: Weibull (fixed 𝐿∞ and 𝐿0); percent difference between simulation estimates and analytical estimates.

Mean Variance

𝑧 𝜏⁄ 1 10 25 50 100 𝑧 𝜏⁄ 1 10 25 50 100

5* -9.17 -7.04 -5.81 -4.03 -0.43 5* -81.87 21.88 22.13 19.41 -1.94

6 -0.26 -0.34 -0.33 -0.27 -0.02 6 -9.26 -5.28 -3.55 -2.00 -0.26

10 -0.07 -0.12 -0.14 -0.15 -0.03 10 -6.94 -5.10 -4.07 -2.77 0.03

25 -0.01 -0.02 -0.02 -0.03 0.00 25 -1.38 -0.75 -0.41 -0.22 0.16

50 0.00 -0.01 -0.01 -0.01 0.02 50 -1.11 -0.36 -0.30 -0.05 0.49

70 0.00 0.00 0.00 -0.01 0.01 70 -1.19 -0.76 -0.17 -0.38 1.02

80 0.00 -0.01 -0.01 -0.02 -0.09 80 34.43 35.54 36.38 36.92 49.36


10% and 25%. The (*) indicates that the real component of a complex number was used. For this model

𝐿∞ = 86 and 𝐿0 = 5.6.

Tables 10-13 show that as 𝑧 pushes against 𝐿∞ or 𝐿0 the delta method breaks down. This is because for

the intermediate values of 𝑧, the parameter estimates are, or are very close to, being independent of 𝑧.

This leads to the parameter estimate being approximately multivariate normal, which is a required


14

assumption of the delta method. However, as 𝑧 approaches 𝐿∞ or 𝐿0, the probability distributions of the

parameters start to depend on 𝑧. This effect is something similar to the Berkson's paradox. One example

of this is that 𝑧 aproaching 𝐿0̂ would cause 𝐿0 to change from a normal distribution to a normal

distribution truncated at 𝑧. von Bertalanffy with fixed values for 𝐿∞ and 𝐿0 both perform quite well,

providing support for their use in the IPM model parameterized for P. promelas.

In summary, four models for growth were presented here, along with both the parameter and

covariance estimates for each of the models. Equations for growth based on time measured from the

current size of the organism instead of the time from hatch along with using the delta method to derive

the variance around the estimate of size. Both the simulations and the equations for growth agree with

each other as long as 𝐿0 < 𝑧 < 𝐿∞, with the results from the equations diverging from the results of

the simulations as 𝑧 approaches either 𝐿0 or 𝐿∞. In general, the results from the equations aligned more

with the results from the simulations for simpler models (Von Bertalanffy; fixed 𝐿∞ and 𝐿0) then the

more complex models like the Weibull. This leads to a tradeoff between using a model that fits data

better verses a using a model that can better estimate mid-lifetime growth. The analytical equation for

growth presented, von Bertalanffy models with fixed 𝐿0 and 𝐿∞ was shown to accurately estimate mid-

lifetime growth when 𝐿0 < 𝑧 < 𝐿∞.


1

Appendix B – Additional sensitivity and elasticity analysis results to

accompany Developing Integral Projection Models for Ecotoxicology From Developing Integral Projection Models for Aquatic Ecotoxicology, Pollesch et al.

Elasticity analysis results were discussed in the main text of Developing Integral Projection Models for

Ecotoxicology and a brief discussion of trends in sensitivities is provided here. When sensitivities for the

different kernel component functions were compared, overall, the trends remained the same as the

elasticities. There were a couple of notable exceptions to this. Within the Baseline scenario, there was

a strict hierarchy of sensitivities. Specifically, for every day of the year, 𝑆𝑒𝑛𝑠(𝑃𝐵) > 𝑆𝑒𝑛𝑠(𝐺) >

𝑆𝑒𝑛𝑠(𝑆) > 𝑆𝑒𝑛𝑠(𝐺0) > 𝑆𝑒𝑛𝑠(𝐵). As is shown below (Figures 1 through 6), 𝑆𝑒𝑛𝑠(𝑃𝐵) emerges as the

kernel component function that 𝜆 is most sensitive too. Except for the Winter scenario, 𝜆 remained the

most sensitive to 𝑃𝐵 every day of the year. Within the Winter scenario, during the winter season,

sensitivities of 𝑃𝐵, 𝐵, and 𝐺0 were all identically 0, and we also found that 𝑆𝑒𝑛𝑠(𝑆) > 𝑆𝑒𝑛𝑠(𝐺) during

the winter season as well, although there were in the same ratio as the elasticities, and their difference

in magnitude was quite small. A few other exceptions to the sensitivity hierarchy emerged during the

exposure scenarios. These included all the GUTS scenarios, (𝑆𝑒𝑛𝑠(𝑆)

𝑆𝑒𝑛𝑠(𝐺)) > 1 on May 29th and 30th. In the

TCEM-Uniform scenario, there was also a switch to (𝑆𝑒𝑛𝑠(𝑆)

𝑆𝑒𝑛𝑠(𝐺)) > 1 on May 24th, 29th and in TCEM-Step on

May 29th. When it came to the exposure scenarios, the behavior of the sensitivities and elasticities of the

kernel components relative to each other were nearly identical, besides the diminished role of 𝑃𝐵(𝑧).


2

Figure 1: Sensitivities for each kernel function for Baseline and Winter simulation scenarios. In each

case, 𝑝𝐵 (blue) tends to be the function that λ is most sensitive to. Another general behavior is that λ is

more sensitive to growth than to survival during reproduction, but slightly more to survival than growth

during the non-reproductive season. Large differences in sensitivities exist between the reproductive

and non-reproductive season. Sensitivity to 𝑝𝐵 also shows a seasonal difference in the Winter scenario,

where it and all other reproductive kernel components are identically zero during the winter season.


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Figure 2: Sensitivities for each kernel function for GUTS simulation scenarios. In each case, 𝑝𝐵 (blue)

tends to be the function that 𝜆 is most sensitive to. Another general behavior is that 𝜆 is more sensitive

to growth than to survival during reproduction, but slightly more to survival than growth during the

non-reproductive season. Large differences in sensitivities exist between the reproductive and non-

reproductive season. Exposure in both GUTS scenarios has the potential to reverse the ordering of

sensitivities between 𝑆 and 𝐺 during the reproductive season, making 𝜆 more sensitive to survival than

growth, and does so on May 29th and 30th.


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Figure 3: Sensitivities for each kernel function for TCEM simulation scenarios. In each case, 𝑝𝐵 (blue)

tends to be the function that 𝜆 is most sensitive to. Another general behavior is that 𝜆 is more sensitive

to growth than to survival during reproduction, but slightly more to survival than growth during the

non-reproductive season. Large differences in sensitivities exist between the reproductive and non-

reproductive season. Exposure in both TCEM scenarios has the potential to reverse the ordering of

sensitivities between 𝑆 and 𝐺 during the reproductive season, making 𝜆 more sensitive to survival than

growth, and does so on May 25th and 29th in the Uniform scenario and May 29th in the Step scenario.


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Figure 4: Elasticity measures for kernel functions for the Baseline and Winter scenarios. Within the

reproductive season, 𝜆 is always more responsive to growth, 𝐺, with respect to elasticities, than to

survival, 𝑆. Outside of the reproductive season, these two are switched. With λ being slightly more

elastic to 𝑆 than to 𝐺. Unlike sensitivities, elasticities of 𝜆 with respect to 𝑝𝐵 are nearly identical to the

other reproductive kernel component functions.


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Figure 5: Elasticity measures for kernel functions for the GUTS simulation scenarios. During reproduction, 𝜆 is generally most responsive to growth, 𝐺, with respect to elasticities, which is slightly more than to survival, 𝑆. We see elasticities increase when there is a survival decrement. 𝜆 becomes more responsive to survival than to growth outside of the reproductive season and has the potential to become so during reproduction when there is the large survival decrement. Unlike sensitivities, elasticities of 𝜆 with respect to 𝑝𝐵 are nearly identical to the other reproductive kernel component functions.


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Figure 6: Elasticity measures for kernel functions for the TCEM simulation scenarios. During the reproductive season 𝜆 is generally most responsive to growth, 𝐺, which is slightly more than to survival, 𝑆. However, outside of the reproductive season, this switches, and elasticity for 𝑆 becomes larger than for 𝐺. 𝜆 becomes more responsive to survival than to growth outside of the reproductive season and has the potential to become so during reproduction when there is the large survival decrement. Unlike sensitivities, elasticities of 𝜆 with respect to 𝑝𝐵 are nearly identical to the other reproductive kernel component functions.

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