Nonlinear relationships can lead to bias in biomass calculations and drift-foraging models when using summaries of invertebrate drift data

Abstract

Drift-foraging models offer a mechanistic description of how fish feed in flowing water and the application of drift-foraging bioenergetics models to answer both applied and theoretical questions in aquatic ecology is growing. These models typically include nonlinear descriptions of ecological processes and as a result may be sensitive to how model inputs are summarized because of a mathematical property of nonlinear equations known as Jensen’s inequality. In particular, we show that the way in which continuous size distributions of invertebrate prey are represented within foraging models can lead to biases within the modeling process. We begin by illustrating how different equations common to drift-foraging models are sensitive to invertebrate inputs. We then use two case studies to show how different representations of invertebrate prey can influence predictions of energy intake and lifetime growth. Greater emphasis should be placed on accurate characterizations of invertebrate drift, acknowledging that inferences from drift-foraging models may be influenced by how invertebrate prey are represented.

Appendices

Appendix A

We estimated fish energy intake for different summaries of invertebrate prey size using the foraging model developed by Rosenfeld and Taylor (2009) and applied by Hafs et al. (2014). The foraging model requires estimates of focal point velocities (position whereby fish hold, often near the bed) and velocity estimates of invertebrate drift in nearby faster velocity water. Hafs et al. (2014) used predicted velocities from a two-dimensional flow model to approximate velocities experienced by the fish. We used an equation based on Stewart (1980) to estimate focal point velocity, then a velocity differential based on Hayes and Jowett (1994) was applied to estimate the invertebrate drift velocity. The maximum swimming velocity was estimated using equations provided by Brett and Glass (1973). Rosenfeld and Taylor (2009) include a capture success function that adjusts the probability of intercepting prey to account for water velocity, fish size, and the distance of prey from the focal point. We did not use this capture success function and assumed that all prey items were captured (e.g. capture success =1). Other parameters used in the foraging model were taken from Hafs et al. (2014). The same set of prey biomass distributions (see, methods) were used as input for the foraging model.

Under the Rosenfeld and Taylor (2009) foraging model, energy intake varied with how the prey biomass was calculated and with how prey was distributed between size bins (Fig. 4). The highest energy intake was under a scenario with the distribution of prey increasing between the three size bins and lowest estimates under a scenario with decreasing distribution (Fig. 4). The scenario with only one size bin resulted in intermediate energy intake and was similar to the scenario with three size bins and a balanced prey size distribution. These patterns were consistent regardless of whether prey biomass estimates were based on a constant concentration and varying prey sizes (Fig. 4, panel a) or prey biomass held constant but distributed differently between sizes (Fig. 4, panel b). However, in the former scenario, differences in energy intake between prey distributions were larger. Overall, the results were largely similar to those of the foraging model used by Hayes et al. (2000) and presented in the text (see Fig. 1).

Fig. 4

Energy intake (kJ/day) for a range of fish weight (g) predicted from the foraging model used in Rosenfeld and Taylor (2009). Comparisons are made between representations of invertebrate prey differing in the number of prey sizes and how prey are distributed between sizes. All scenarios in panel a have the same starting prey concentration and prey biomass is estimated from the prey concentration and prey size using a length-mass relationship. All scenarios in panel b have the same starting prey biomass and differ in how this mass is distributed between prey sizes

Appendix B

We use a simulation to explore the influence of alternative invertebrate prey size binning strategies on biomass estimates from measured lengths given a nonlinear length-mass regression. The common length-mass regression of the form:

$$ {\mathrm{W}}_{\mathrm{i}}=\mathrm{a}\ast {{\mathrm{L}}_{\mathrm{i}}}^{\mathrm{b}} $$

was used with parameters a = 0.012 and b = 2.74 for Gammarus spp. (Benke et al. 1999). We used the beta distribution to simulate the size distribution of invertebrates with a right skew (shape parameters 2, 6), centered (shape parameter 4, 4), and left skew (shape parameters 6, 2). The beta distribution is bounded by 0 and 1, so we multiplied a constant (12) to simulate data over a range from 0 to 12 mm. We generated 10,000 random lengths and converted these lengths to mass, under each of the binning strategies and size distributions. This process was repeated 10,000 times and the mean and 95 % confidence intervals are presented for each scenario relative to the known true biomass estimate. This exercise was performed using the mean invertebrate size and five binning scenarios ranging from 0.1- to 3-mm size bins. The mid-points of the size bins were used when converting lenghts to mass (e.g., size bin 0 to 3 mm, mid-point 1.5 mm).

Using the mean invertebrate size consistently underestimated the biomass across the three prey size distributions we considered (Fig. 5). The magnitude of the bias when using the mean invertebrate size is dependent on how the invertebrate sizes are distributed. The right skewed example underestimating the biomass by greater than 40 % (Fig. 5, Panel a). For the right skewed and centered examples, the 2- and 3-mm scenarios overestimate biomass, while for the left skewed example the 95 % confidence intervals overlap 0. Overall, the 1-mm size bins and smaller produced estimates that were unbiased, regardless of how the sizes were distributed. Binning invertebrate drift collections at 1-mm intervals may be appropriate for many applications in order to balance precision in biomass estimates and efficiency of processing samples.

Fig. 5

Percent bias in estimates of invertebrate biomass for alternative binning strategies (mean size down to 0.1-mm size bins) interacting with different invertebrate size distributions, right skewed (panel a), centered (panel b), and left skewed (panel c)