Nomenclature.

To ease the understanding of mathematical notation, we applied the following rules of nomenclature. Larval traits were denoted by x if treated as species-specific and by y if treated as generic, with subscripts abbreviating the role of the trait. Variables with the subscript t were re-calculated every time-step. Variables with the subscript i were calculated separately for each size bin of potential prey items.

Growth physiology (equations 1–9).

To calculate growth, the two state variables standard length L and dry mass M were initialized, then a series of equations was iteratively solved for 24 hourly time steps, one night followed by one day (Tables 2–3). Initial L was set to x_len. Initial M and positive L-growth were determined according to the body shape trait x_body, which functioned as an upper limit to M/L³ and a reference point for larval condition (M/L³ is often termed Fulton’s condition factor). A loss of M never resulted in a loss of L. All L-changes were thus modeled as isometric growth, an approximation simplifying direct comparisons of x_body among different larval types. Changes in M were calculated from the balance of metabolic gains from ingested prey (with metabolic efficiency y_eff) and losses to active respiration. Respiration was modeled as a percentage of M lost per hour at 10°C (routine respiration x_res), corrected for ambient temperature by a Q₁₀ factor (x_rQ10). We converted respiration from units of oxygen uptake to units of dry mass using a conversion factor of 0.85 µg µl⁻¹, which is typical for young marine fish larvae [28]–[33]. Respiration was increased between sunrise and sunset to account for the cost of foraging activity (y_act). The ingestion of prey was limited by the lesser of either digestive capacity or foraging capacity during the daylight fraction of the time step. Digestive capacity was represented as a percentage of M evacuated per hour at 10°C (y_dig) corrected for ambient temperature by a Q10 factor (y_dQ10). Larvae encountering temperatures in excess of their upper thermal tolerance (x_tol) were excluded.

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Table 3. Quirks environmental factors and internal variables.

https://doi.org/10.1371/journal.pone.0098205.t003

Prey fields (equations 10–12).

Potential prey were binned into groups of characteristic length l_i, individual dry mass m_i, biomass concentration b_i, and numeric concentration c_i. The relationship among these variables was idealized as a normalized size spectrum sensu Platt and Denman [34], by assuming a linear slope s between the logarithm of normalized b_i (i.e., b_i divided by the width of the m_i bin) and the logarithm of m_i. This simplification was consistent with empirical data from diverse aquatic systems [35]–[37] and a growing body of literature on size spectrum theory [34], [36], [38]. Prey fields were specified in terms of l_i bins, b_total, and s. For each bin, c_i was equal to b_i divided by m_i. The distribution of b_total among prey size bins (according to s) was solved analytically. The present study used prey from 0.04 to 2 mm length in 196 discrete 0.01-mm bins. A power function fit to this size range of copepods [39]–[41] and protist plankton [42] was used to calculate m_i (n = 109, r² = 0.95, two outliers removed, 45% carbon content assumed when necessary).

Foraging behavior (equations 13–21).

Quirks modeled foraging capacity based on a predation sequence of proximity, encounter, pursuit, attack, and ingestion, and assumed optimal diet selection (accounting for handling time y_hand) [9], [43]. Proximity: The number of prey passing in proximity of larvae was calculated from prey concentration multiplied by the volume of a visually scanned cylinder. The radius of the cylinder represented larval visual ability (y_vis). The length of the cylinder represented the combined velocities of larval swimming (y_swim), prey swimming, and turbulence. Relative turbulent velocity between predator and prey (separated by encounter distance y_dist) was calculated according to “Kolmogorov 1941 theory” [44], [45], assuming a universal Kolmogorov constant of 0.53 [46], [47]. While this approach is theoretically only valid for larger separation distances, it has been empirically validated at the scale of larval fish predation events [48], [49]. Encounter: In addition to proximity, encounters required prey detection (observation success), which was varied as a type 2 functional response with half-detected prey length y_det, based on herring larvae reacting to prey of different sizes [15]. Pursuit: As observed in data of cod larvae feeding under turbulent conditions [49], relative pursuit success was linearly reduced from 100% at zero turbulence to 0% at a maximum corrigible turbulent velocity y_turb. Capture: Based on herring larvae attacking prey of various sizes [15], relative capture success was linearly reduced from 100% for infinitesimally small prey length to 0% for the maximum ingestible prey length (x_ing).

Larval traits.

Estimates for all traits were open to interpretation, due to the uncertainty, variability, and availability of empirical data. Some of the variability among published measurements reflected different environmental conditions. To realistically model maximum growth potential, measurements representing favorable conditions were used for traits related to morphology, such as larval shape (relating growth and condition) or ingestible prey size. To realistically model prey-limited growth and starvation, measurements characterizing low prey concentrations were used for foraging-related traits, such as swimming speed or metabolic efficiency (high efficiency for low rates of ingestion). In some cases, contradictory results were found in the literature, and averaged parameter estimates were used. Only six traits were deemed adequately studied and sufficiently variable among species to merit species-specific parameterization. Of these six, the traits x_res and x_rQ10 were estimated for E. encrasicolus based on measurements in two closely related species: Northern anchovy (Engraulis mordax) and Bay anchovy (Anchoa mitchilli). For young sprat larvae x_res and x_rQ10 were estimated by the herring parameters.

Generic digestion.

Quirks represented the complex process of digestion in a greatly simplified, mechanistic form. Conceptually, M-gain from digestion depended on how much prey fit inside the larval gut at one time, how quickly the gut contents were digested and evacuated (temperature dependent), and an efficiency term accounting for losses to excretion as well as the costs of digestion and growth. A literature review of these factors (Table 1) yielded too little information for species-specific parameterization, so generic traits were used for digestion. The trait y_eff = 67.5% combined an assimilation efficiency of 75% and specific dynamic action of 10% (75% · 90% = 67.5%). The trait y_dig = 2.5% incorporated a gut capacity of 5% M divided by an evacuation time of 2 h. For every 10°C increase in temperature, the gut evacuation rate was multiplied by a Q10 factor of y_dQ10 = 2.5.

Volumetric foraging rate.

Parameterizing the rate at which larvae effectively scanned their environment for prey was of particular importance. Two main types of estimates were available from the literature: the multiplication of visual fields by swimming speeds and the division of predator reaction rates by prey concentrations. In studies using the multiplication approach, video recordings of larval reactions to prey were used to measure the size and shape of larval visual fields. Then, the continuous (cruise predator) or saltatory (pause-travel predator) movements between successive visual fields were used to calculate volumetric foraging rate. This method generally involves the unrealistic assumption that larvae actually detect 100% of the potential prey organisms passing through their visual fields [50], and is likely to overestimate foraging rate. In studies using the division approach, the rate at which larvae pursued, attacked, or ingested prey at very low concentrations was recorded. Then, volumetric foraging rate was estimated by dividing the mean reaction rate (e.g., the number of strike postures per minute) by the prey concentration. This approach is likely to underestimate foraging rate, if any prey are detected but not pursued, pursued but not attacked, or attacked but not ingested. Various other sources of bias likely affect both approaches (prey type, size, and concentration; experimental water volume, turbidity, and turbulence; larval size, concentration, condition, and satiation). In synthesis of available experimental data for species of anchovy (A. mitchilli and E. mordax), cod, and herring, we chose a generic effectively scanned cylinder radius of y_vis = 0.6 L s⁻¹ and a generic effective swimming speed of y_swim = 0.75 L s⁻¹. This resulted in foraging rates of 0.5 l h⁻¹ for 5.5-mm anchovy, 1.0 l h⁻¹ for both 7-mm cod and sprat, and 6.7 l h⁻¹ for 13-mm herring larvae (not accounting for prey swimming, turbulence, or imperfect prey detection). For an illustration of the four larval types and their foraging rates, see Figure 1.

Validation.

We compiled a list of empirical studies estimating growth rates of young anchovy, cod, herring, and sprat larvae (Table 4), and used it to validate Quirks results. Data used for validation were independent from data used for parameterization except for one study yielding anchovy body shape [51] and one of five studies yielding generic metabolic efficiency [52]. Studies comparing different environmental conditions (including starvation experiments), reporting particularly high growth rates, or applying alternative methodologies were preferentially selected, and all those allowing for temperature, photoperiod, and prey concentration estimates were used. One sprat study by Dulčić [53] was included under the explicit assumption that growth had not been prey limited. The assumption was warranted, since Dulčić [53] replicated essentially identical and high growth measurements in two different years, despite a relatively cold temperature and a short photoperiod. The validation list included field and laboratory growth estimates based on larval length, mass, or protein content, otolith microstructure, and biochemical condition (RNA-DNA ratio). Quirks was set up to mimic the environmental conditions of each study and estimate growth. The slope s of the normalized size spectrum could not be calculated for most studies and was otherwise set to a default value of −1.2. This number has frequently been used as a reference point [36] ever since it was predicted from theoretical considerations of oligotrophic aquatic systems in equilibrium [34], [36]. For field studies, turbulent kinetic energy dissipation rate ε was set to 10⁻⁷ W kg⁻¹, approximating wind-driven turbulence at 30 m depth in storm conditions or near the surface in calm conditions [54], and sufficient to significantly influence larval fish foraging success [49], [55]. The default for laboratory studies was zero turbulence. Temperature was set to match experimental treatments (lab) or the surface layer (field). For several studies, photoperiod was estimated from date and latitude using the NOAA solar calculator, as implemented in the R package “maptools” [25]. For field studies, prey concentrations were matched to larval growth rates on a station-by-station basis when possible; otherwise, median prey concentrations were used.

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Table 4. List of studies used to validate Quirks growth rate estimates.

https://doi.org/10.1371/journal.pone.0098205.t004

Sensitivity analysis.

We measured the influence of endogenous larval traits and exogenous environmental factors on Quirks output in two series of individual parameter perturbations [5]. The first series quantified the sensitivity of modeled growth potential assuming ad libitum feeding. The second series examined parameter influence on the prey concentration required for M-growth of exactly 5% d⁻¹. For each parameter and larval type, we determined the parameter range for which the output changed by less than ±10%. For example, model sensitivity to a hypothetical trait x_hyp could be such that any value between 80% and 200% of x_hyp results in similar growth potential (G±10%). Perturbations were performed with respect to species-specific reference points, representing the environmental conditions that young exogenously feeding larvae would typically experience in the central North Sea. Temperature and photoperiod were set to average conditions two weeks after the approximate midpoint of the adult spawning season (Table 5). Turbulence and prey fields were approximated by ε = 10⁻⁷ W kg⁻¹ and s = −1.2, as in the validation runs. Since prey concentration was fixed in the first sensitivity analysis (to ensure ad libitum feeding), and functioned as the output variable of the second analysis, the importance of prey concentration was separately quantified. This was done by determining the minimum level required for positive growth (starvation point), 5% growth, and maximum growth (satiation point).

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Table 5. Comparison of anchovy (Engraulis encrasicolus), cod (Gadus morhua), autumn-spawned herring (Clupea harengus), and sprat (Sprattus sprattus) larvae in the North Sea.

https://doi.org/10.1371/journal.pone.0098205.t005

Optimal foraging conditions.

Here, we quantified the suitability of different prey sizes, prey-size distributions, and turbulence levels (increasing turbulence enhances encounters with prey, but decreases pursuit success) for young larvae. First, we examined encounters with each of the 196 modeled prey bins, in terms of the average ingested dry mass per encounter, standardized by handling time. This metric is used in foraging theory (and in Quirks) to determine the ranking of different prey types for inclusion in an optimal diet [9], [43]; from a predator’s perspective, it determines the optimal prey length. Second, we calculated a numerical solution for the value of s (normalized size spectrum slope) resulting in the lowest possible starvation (and satiation) point for each larval type. In the idealized model framework, s fully defines the size distribution of available plankton biomass, and thus also the optimal prey field. To further examine the effects of turbulence, we also calculated starvation and satiation points for ε-values of 10⁻¹² to 10⁻⁴ W kg⁻¹.

Model skill.

In the validation runs, there was a good match between growth predicted by Quirks and growth observed empirically (R² = 52%, Table 6, Figure 2). Further, the distribution of slopes between any two data points was centered near one (median = 0.72, interquartile range = 0.98), and the corresponding distribution of intercepts was centered at approximately zero (median = 0.04, interquartile range = 0.19). In other words, Quirks had surprisingly low bias (a slope of one and intercept of zero would indicate no systematic bias whatsoever). We did observe a statistically significant photoperiod effect, thus predicting growth at extreme photoperiods may require extra care. For example, Quirks predicted much too low growth potential (∼6% vs. ∼29% d⁻¹) for sprat larvae in the northern Adriatic Sea in January (∼11°C, ∼9 h daylight) [53]. At the other extreme, Quirks predicted a growth rate of 24% vs. the observed 7% d⁻¹ for cod larvae reared in continuous daylight at 8°C [56]. Here, modeled growth potential may not have been grossly inaccurate, because another study of cod larvae at the same 8°C temperature (and ∼15 h daylight) measured a growth rate of 22% d⁻¹ [57]. In any case, model skill was noticeably higher among the “12 to 18 h daylight” subset of data (R² = 65%, Table 6), which excluded the above outliers and one other (unproblematic) data point. Note that we did not “tune” the model equations or larval traits to optimally fit the validation dataset, and that we used almost entirely independent data for parameterization and validation (there was overlap with respect to one trait in two studies).

Growth.

The list of validation studies encompassed a wide range of different approaches (Table 4), each subject to potential bias and random sampling error. For example, many lab studies are biased towards low growth, because larvae rarely thrive as well in artificial environments as they might under ideal natural conditions. Field studies, on the other hand, are likely to overestimate growth. For example, among simultaneously occurring and equally sized larvae, faster-growing individuals have, so far, suffered a shorter period of exposure to predation [2], [58], and may additionally have experienced lower instantaneous predation rates [59], [60]. Random sampling of the survivors is thus not sufficient for an unbiased characterization of the original population. Attempting to correct for these and other sources of uncertainty was beyond the scope of the present study, as was the distinction among individual growth rates (e.g., RNA-DNA ratios), “vertical” or “horizontal” population growth rates (i.e., based on patterns among simultaneously collected larvae or time-series of samples, respectively). Conceptually, it should be clear that the presented parameterizations represent average individuals, and that Quirks was used to model individual growth rates.

Prey concentration.

Matching (i.e., simulating) empirical prey concentrations for the purpose of validation was also problematic. Given the complexity of plankton dynamics in the field, and the difficulty of maintaining nominal food levels in small-scale experiments, constant “matched” prey concentrations can only have roughly approximated the actual conditions experienced by larvae. For this reason we were surprised that Quirks performed equally well (in terms of R²) for the satiated and prey-limited subsets of validation data. The fact that 10 of 12 data points fell very near the 1∶1 line (Figure 1B) is encouraging, but should not be over-interpreted. Quirks underestimated growth in 9 of 12 cases (including 5 of 5 with empirically observed negative growth), which may indicate that our parameterization slightly overestimated larval respiration or slightly underestimated foraging capacity. In summary of model validation, a long list of uncertainties would have prevented even a perfect model from reproducing 100% of variability in the validation dataset. The observed R² of up to 65% (Table 6) is therefore a conservative (low) estimate of Quirks’ model skill.

Maximum growth.

As expected, growth potential and prey requirement for 5% growth were sensitive to different parameters, and model sensitivity/parameter influence varied by larval type. Of the factors influencing growth potential, only respiration (traits x_res and x_rQ10) and the environment (temperature and photoperiod shortly after peak spawning, Table 5) differed among species. All other traits were either generic or rendered inconsequential by the arbitrarily high food concentration. In all larvae, growth potential was more sensitive to y_dig and y_eff (given ad libitum feeding, both are simply coefficients of digestion), than to x_res and y_act (coefficients of total and active respiration, respectively). Young anchovy larvae were most sensitive to all of the above traits, due to their high mass-specific respiration. For example, anchovy required a mere 3.4% increase in y_dig to boost growth potential by 10% while the other species required 4.5% to 4.9%. Young cod and herring larvae were similar in their sensitivities, even though herring had ∼18% higher x_res and were exposed to ∼10°C warmer water. On the other hand, young sprat larvae were 5%–31% more sensitive than herring, despite effectively identical traits, because they experienced much longer photoperiods (>16 versus <12 h daylight). Relative temperature changes were most important for anchovy (at 15.0°C), followed by herring (at 14.7°C), sprat (at 8.9°C) and cod (at 5.1°C). In contrast, the parameter influence of photoperiod was independent of the absolute photoperiod reference values. These initially counterintuitive results highlight the importance of species-specific reference points (i.e., realistic environmental conditions).

Prey requirement.

Unlike growth potential, prey requirement (for 5% growth) was sensitive to all endogenous traits and exogenous environmental factors. The influence of several important traits was similar among species. Prey requirement was approximately proportional to x_body, y_eff, and y_swim, and approximately inversely proportional to y_vis², making y_vis the most influential trait in all cases. Perturbations of several other parameters affected prey requirement very differently. Notably, young sprat and anchovy were much more sensitive to s than cod and herring larvae. The same pattern was also apparent for the less influential traits x_ing and y_det. To generalize this relationship, parameters differentially affecting access to small prey sizes are expected to be particularly influential in species with a developmental morphology similar to sprat and anchovy, i.e. small size at yolk depletion and small mouth gape. The influence of x_res and y_act was greatest in young anchovy and least in young cod larvae, primarily due to their vastly different ecophysiology. To generalize, larvae with low temperature-specific respiration rates in combination with cold environments (e.g., cod) are expected to be less sensitive to perturbations in x_res and y_act than larvae displaying the opposite pattern (e.g., anchovy). The influence of x_len perturbations was related to body shape, such that the thinnest larvae (sprat) were most sensitive, the somewhat less thin herring and anchovy were only slightly less sensitive, but the dramatically thicker cod were only half as sensitive. Neither y_hand, nor the parameters related to turbulence (ε, y_dist and y_turb) had a strong influence on prey requirement (see detailed discussion below). Since larvae only used from 64.8% (herring) to 88.4% (cod) of their full digestive capacity to achieve 5% growth, and inhabited waters well below their upper thermal tolerance, small changes in y_dig or x_tol were entirely without consequence.

Uncertainty.

Uncertainty in many traits clearly exceeded the range affecting 10% changes in growth potential or prey requirement. However, our literature review uncovered too little data for a meaningful trait-by-trait assessment of uncertainty. Instead, we will highlight just a few traits and processes of particular interest. First, uncertainty in gut evacuation rate (used to parameterize y_dig) and assimilation efficiency (the main component of y_eff) primarily limited the reliability of growth potential estimates, thus better measurements of digestion would be invaluable. The importance of these factors for mechanistic larval fish IBMs is well known and has recently been reviewed in detail [5], [61]. Second, uncertainty in volumetric foraging rate (used to parameterize y_vis) overshadowed all other factors influencing prey requirements, indicating a need for further experimental research in this area. For example, Rosenthal and Hempel [62] determined that 11- to 13-mm herring could search 3.1 to 5.2 l h⁻¹ based on estimated swimming speeds and visual fields. For the same 11 to 13 mm length range, Munk and Kiørboe calculated 3.2 l h⁻¹ from feeding strike rates and 9.2 l h⁻¹ from attack posture rates [63], while additional feeding strike data by the same authors for ∼13 mm herring yielded rates from 1.6 to 3.1 l h⁻¹ [15], [64]. Reanalysis of four studies of 6- to 8-mm cod resulted in estimates of 0.4 to 5.2 l h⁻¹ [55], [65]–[67], while three estimates for 5.5-mm anchovy (based on A. mitchilli and E. mordax) ranged from 0.17 to 0.55 l h⁻¹ [68]–[70]. Prey used in the above studies varied from 0.04-mm protozoans to 0.8-mm copepods, but 0.12- to 0.23-mm copepod nauplii were most common. Our results for 13-mm herring, 7-mm cod (and sprat), and 5.5-mm anchovy feeding on 0.2-mm prey were 5.0, 0.78, and 0.38 l h⁻¹, respectively, and took into account prey swimming and detection of prey. Treating prey detection as a type 2 functional response to prey size (equation 19) was consistent with the above studies, in that it helped explain some of the variability among literature estimates. However, our estimate of y_det = 0.07 mm was based on a single study of 13- to 18-mm herring reacting to 0.1- to 0.8-mm copepods [15], thus the value is quite uncertain. Third, handling time estimates for anchovy and sprat provided a nice example of a highly uncertain parameter that was, nevertheless, fairly inconsequential for Quirks output. Mean handling time in seven studies of cod, herring, and closely-related species of anchovy (E. mordax, and A. mitchilli) varied from ∼0.5 to 3.5 s. It seems likely that the generic value of 1.5 s was perfectly adequate for anchovy and sprat, since any value below 8.9 s (anchovy) or 17.7 s (sprat) would have yielded very similar model output.

Natural selection.

Our results led to several hypotheses regarding the evolutionary ecology of marine fish larvae. Due to the strong links between larval growth and survival [2], [3], [14], [58]–[60], phenotypes resulting in better growth are almost certainly favored by natural selection, yet the traits most likely to affect growth (assuming phenotypic variation among individuals) differ, depending on food limitation. When food is abundant, selection is likely to favor phenotypes with a fast metabolism, since the benefits of increased digestive capacity exceed the costs of increased respiration. There should also be selection for gains in digestive capacity or efficiency even at the expense of losses in foraging capacity or efficiency. In contrast, phenotypes with high foraging capacity and high gross growth efficiency (growth per ingestion) should be favored under food limitation. This reversal would result in selection towards slow metabolism and improved foraging (e.g., increased vision in all species, mouth gape in young anchovy and sprat) at the expense of digestive capacity or even metabolic efficiency.

Taking this line of reasoning a step further, we hypothesize that body plans prioritizing jaw and eye development over gut development may indicate species adapted to environments characterized by relatively low prey concentrations. In a previous study we estimated by principal component analysis that 16% of variability in the body shape of marine fish larvae involves a morphological gradient from a small head, eye, and jaw with a thick trunk to a large head, eye, and jaw with a thin trunk [3]. We may be able to use those data to test the above hypothesis. Given the strong influence of initial L on prey requirement, early life history strategies adapted to low-prey environments may also involve larger first-feeding larvae. This mechanism is independent from and synergistic with the finding that larger (longer) larvae can generally survive greater periods of starvation [14]. Note that the influence of x_body (body shape) perturbations is difficult to interpret in an evolutionary context, because changes in L affected both vision and mouth gape, while changes in M did not. Modeled larvae of enormous L and miniscule M would have a lower prey requirement than any realistic parameterization, following the computer science principle of “garbage in, garbage out.”

Absolute prey concentration.

While it was easy to find species-specific temperature and photoperiod reference points for sensitivity analyses (Table 5), prey concentration would have been a more elusive variable. This is why we originally chose maximum growth potential as the first and prey requirement for fixed growth as the second sensitivity analysis output variable. Both metrics and analyses are, by design, independent of estimates of prey concentration. This does not change the fact that absolute prey concentrations below the satiation point had a strong negative influence on modeled growth (Table 5). In the range between starvation and satiation, the influence of prey concentration was roughly linear with 1 mg m⁻³ of plankton resulting in growth of 0.8% (cod), 3.5% (anchovy), 9.3% (sprat), and 4.7% (herring) d⁻¹. Model sensitivity aside, it is interesting to compare absolute prey requirements of the four species. One might expect the requirements to be highest for larvae spawned in summer, when zooplankton biomass is highest in the North Sea [71]. Our model estimates do not agree with this expectation, since young cod larvae required a higher prey concentration in winter than sprat in spring, anchovy in summer or herring in autumn. This may simply reflect that our parameterizations were imperfect. Alternatively, the thermal tolerance of adults, eggs, or larvae may be more important in determining spawning seasonality than larval prey limitation.

Optimal prey size.

Plankton at 67% of the maximum ingestible prey size constituted highly favorable prey for all four larval types. This pattern arose primarily from the trade-off between the decreasing probability of capture success and the increasing benefit of successful capture (Figure 3). Note that the chosen metric is independent of encounter rate, and therefore somewhat different from the expected benefit per nearby prey organism (e.g., in an aquarium). The latter can be slightly increased by prey movement or substantially reduced by low observation success (Figure 3). More variable optima have been reported in previous modeling studies, e.g.: 54% for 7-mm cod [26], 44% for 13-mm herring [23], and 85% for 7-mm sprat larvae [26]. Since those models applied different assumptions (equations) for each species, it is unclear whether this variability reflected differences among species or among models.

Empirical studies have rarely provided sufficient information to calculate the favorability of different prey sizes for young fish larvae. One exception was a series of laboratory experiments by Munk, where 13.5-mm herring larvae were fed low concentrations of copepods of various sizes [15]. In that study, the largest prey treatment (0.6 mm, ∼57% of maximum ingestible length) yielded the highest estimated ingestion per encounter (prey dry mass • capture success) as well as the highest ingestion rate (prey dry mass • capture success • attack rate). However, a much smaller prey treatment (0.2 mm, ∼19%) was considered optimal by the author, whose definition was based on the number of ingested copepods, as opposed to their mass. This logic permeates the literature, and is often quantified in terms of Chesson’s α, also known as Chesson’s (or Manley’s) measure of prey preference [72]. When applied to fish larvae, α should perhaps be called a measure of prey “sampling,” as the terms “preference” and “selection” are quite misleading. In fact, α neither reflects theoretical benefits (since prey mass is not considered), nor empirical behavior (since prey detection and capture success are assumed to be constant). Instead, it indicates sampling bias for different prey types, i.e. the relative probability of ingestion corrected by ambient numeric concentration (which is different from encounter rate). For comparison with Munk’s data, we calculated Chesson’s α for 13.5-mm herring larvae as simulated by Quirks: the prey length with maximal α was equivalent (0.21 mm, ∼21%). Note that the nutritional quality (e.g., fatty acid composition) of prey organisms also influences their benefit to larval fish predators [73]. This further level of detail was not considered in the present study, but could be modeled by “correcting” prey mass by a food quality coefficient in the determination of foraging capacity (equation 13).

Diet composition.

Quirks assumes optimal foraging behavior, meaning that larvae only pursue and attack prey organisms when this is more profitable than spending the same time searching for better prey instead [9], [43]. Interestingly, optimal foraging was neither necessary for the model runs performed in this study, nor did it influence the results. In all cases, larvae achieved ad libitum feeding at prey concentrations far below the level at which they began actively excluding any ingestible prey from their diet. In other words, hungry larvae pursued everything from the smallest modeled prey class to the largest one they might possibly capture, while satiated larvae sometimes engaged in more complex selective feeding, but gained no benefit from doing so (because ingestion was limited by digestion). At later stages of development this would not hold true, and the explicit assumption of optimal foraging could become quite important.

Prey fields.

Quirks’ representation of plankton size structure as normalized size spectra with independent biomass and slope s, presented an opportunity to determine optimal prey fields for each larval type (Figure 4). In all four cases, optimal prey fields were so “steep” that they consisted to >95% biomass of plankton smaller than the optimal prey size. This demonstrates that the integrated benefit of all prey sizes can be quite independent from the concentration of optimal prey. The optimal values of s for anchovy (−1.9), cod (−1.4), and herring (−1.4) fell within the range from 0 to −2, which is regularly observed in the nearby Bay of Biscay [37], while the optimum for sprat (−2.5) may be unrealistically steep [36]. In any case, there was relatively little change in prey requirements among such steep prey size spectra, while “flat” prey fields became increasingly unfavorable for young larvae (Figure 4).

Turbulence.

Since the recognition that turbulence increases the rate of encounters between fish larvae and their prey [55], [74] but decreases subsequent pursuit success [49], [75], it has become common to include turbulence effects in individual-based models of larval fish foraging [5], [76], [77]. The relationship between foraging and turbulence is often assumed to be dome-shaped, with an optimal intermediate turbulence level that enhances encounter rates more than it reduces pursuit success. We can reproduce this shape in Quirks, but only for larvae swimming much more slowly than those discussed here. At the parameterized effective speeds of >4 mm s⁻¹ the net influence of turbulence was always negative (Figure 5). In the laboratory, optimal intermediate turbulence has also been demonstrated only for small (poorly developed, slow) larvae [78]–[80]. In one study, small (<5 mm L, 7–9 d post hatch) Japanese flounder (Paralichthys olivaceus) larvae benefited significantly from turbulent kinetic energy dissipation rates between 10⁻⁷ and 10⁻⁵ W kg⁻¹, while slightly larger individuals (<7 mm, 12–14 d) did not [80]. A pattern of diminishing benefits of turbulence with increasing size (development, speed) also emerges from most models, with the pattern of decline depending on assumptions regarding visual field geometry, foraging behavior, pursuit success, and relative turbulent velocity w between predator and prey [67], [81]–[83]. As discussed, Quirks assumes a continuously but imperfectly scanned visual cylinder, and empirically validated Kolmogorov 1941 theory determines w, which effectively increases the visual cylinder (equation 15) and reduces pursuit success (equation 20). Our linear reduction in pursuit success to zero at w>y_turb (1 L s⁻¹) was more simplistic than previous approaches, but gave very similar results (e.g., there was a correlation of R² = 95% between equation 20 and Fiksen and MacKenzie’s preferred “Mod4” [83] in the w range from 0.1 to 10 mm s⁻¹). Note that in situ turbulence varies with depth, allowing larvae to influence their exposure by vertical swimming behavior. Since Quirks does not account for such behavior, it makes little sense to address turbulence in even greater detail. For the same reason, light intensity was simplified as either insufficient (nighttime) or sufficient (daytime) for visual feeding, even though it varies continuously with time and depth in the field.

Stochasticity.

Another limitation of Quirks (and many other IBMs) is that highly stochastic processes such as prey encounters are represented by simple averages. On the one hand, this allows for a fully deterministic model, in keeping with the design goal of transparency. On the other hand, this precludes Quirks (in its current form) from modeling stochastic variability in growth or foraging success. Further, the lack of a state variable tracking larval gut contents means that various behavioral and physiological mechanisms related to stochastic foraging success are not represented. For example, modeled larvae cannot fill their guts towards the beginning and then save energy towards the end of a modeled time step. They also cannot benefit from increased metabolic efficiency during times of reduced gut turnover [30], [61], [84], e.g. after dusk.

Thermal tolerance.

Larval tolerance of high temperatures does not emerge mechanistically from the model and must therefore be defined a priori via the x_tol trait. This causes modeled growth potential to increase with temperature until just below the specified upper limit, which is unrealistic [3], [85]. Instead, unfavorable enzyme kinetics or insufficient aerobic scope for metabolism should cause growth potential to diminish as temperature approaches an ecological limit somewhat below the physiologically lethal temperature. We are not aware of any larval fish model that successfully incorporates such proximate mechanisms. There is a pressing need for additional research into the mechanisms limiting larval fish thermal tolerance, so that models such as Quirks can become more useful for climate research. Until this is accomplished, extra care should be taken in the interpretation of model growth predictions for temperatures that have not been empirically validated.

North sea fish larvae.

The most distinguishing features of young exogenously feeding anchovy, cod, herring (autumn-spawned), and sprat larvae at environmental conditions they typically experience in the North Sea were as follows. Young anchovy larvae were shortest, had the fastest metabolism, required a high prey concentration, and had the potential to grow fast. Young cod larvae were heavy and had the slowest metabolism, benefited from large prey, required a high prey concentration, and achieved the lowest growth potential. Young herring larvae were both long and heavy, benefited from large prey, had a low prey requirement, and could grow fast. Young sprat larvae were lightest and required the lowest prey concentration despite being limited to feeding on the smallest prey.

Quirks.

We found that Quirks could mechanistically explain the majority of variability among growth measured by numerous approaches for larvae of four species, from <30 to >300 µg in dry mass, at temperatures from <3 to >24°C and at photoperiods from <10 h to continuous daylight. To the best of our knowledge, this level of model skill is unprecedented for a diverse set of fish larvae and environments. By releasing Quirks as free software (Source Code S1, https://sourceforge.net/projects/larvalfishquirks/), we hope to encourage others to build on our efforts.