How to count your bugs?

Ability to estimate local population density of an insect is critical in many fields, from pest management to conservation. No method currently exists that reliably connects trap catch with the insect population density, including the corresponding uncertainty. Here we report a simple and universal predictive relationship for a probability of catching an insect located a given distance away from the trap. This relationship allows to estimate, from a single catch, the most likely population density along with its statistical upper and lower bounds. To test the generality of this equation we used 10 distinct trapping data sets collected on insects from 5 different orders and major trapping methods: chemical-baited and light. For all of the datasets the equation faithfully describes the relationship between location of an insect and probability to catch it. The ability to estimate absolute population density from a single trap catch will significantly improve our understanding of insect population dynamics and allow for more effective research, management, and conservation programs.


Introduction
The question posed in the title of the paper may seem trivial: you catch your bugs with a net and count them. However, that procedure will tell you how many bugs are in the net and nothing abut what you really want to know -how many bugs are in the area around you.
Any quantitative analysis of trap catches must include a crucial step -directly relating trap catches to the absolute population density of an insect. The absence of a reliable and universal (applicable to all insects) procedure for this crucial step hinders conservation, management and research programs making it difficult to interpret catches, provide recommendations, develop management tactics and evaluate treatment efficacies. The availability of statistically reliable estimates of the absolute population density would significantly improve existing conservation and management programs by allowing them to optimize efforts based on the goal, available resources and the efficacy of the previous efforts. In research programs, this would significantly improve interpretation of results and facilitate optimization of existing tactics and development of new ones.
In a recent study we analyzed a wealth of trap catch data collected for European gypsy moth Lymantria dispar dispar (L.) and derived a simple mathematical relationship between catch probability and distance to a pheromone-baited trap, which, in turn, allowed us to connect the actual trap catch with the most probably population density, along with statistical bounds on absolute gypsy moth population density (Onufrieva et al 2020). However, the key question remained unanswered: is there a general relationship of this type that might apply to all insects and trap types? In this paper we demonstrate the generality of this mathematical relationship using several species of insects from various orders, and two major trapping methods: chemical baited and light attraction.
Insects are the most diverse group of organisms and it is highly improbable that their behavior with respect to attractants could be described by a universal law. However, if such a law were to be found it could have significant impact on the entire field of entomology. This work is about finding and validating such a law.

Analysis
We used the predictive relationship for a probability of catching an insect (spT fer(r)) located at a distance r from the trap that was developed for gypsy moth (Onufrieva et al. 2020) to investigate if it could be applied to other insects.
For gypsy moth, a wealth of data points is available (Onufrieva et al., 2020), which allowed us to come up with the most robust protocol for fitting Equation 1 (see Results). Specifically, for gypsy moth, males were released at distances 0,15,25,30,45,50,60,75,80,100,150,200,250,300,500,600,900,1000,1200, and 1500 m from pheromone-baited traps, therefore, short and long distances were balanced and had equal weight in determination of D50. The gypsy moth data set is also unique in that 12 distinct points are available for large values of r (r > 75 m). The availability of multiple data points at long distances had previously allowed us (Onufrieva et al., 2020) to come up with what we believe is the most accurate estimate of D50 = 26 ± 3 m, which was based on a log-log fit for long distance data points only. However, data available for the other insects studied here does not include spTfer(0) and experimental design is often unbalanced, including either mostly short or long distances, and not very many of them. To overcome this limitation, we developed a 2-step protocol for fitting Equation 1 to data that is missing spTfer(0).
Step 1: Use untransformed data to estimate spTfer (0)  ). This 2-step procedure ensures that the catches at large distances are given equal weight to the catches at short distances. For insect data that includes experimentally measured spTfer(0), only step 2 should be used.
Using the actual spTfer(0) in untransformed and log-transformed model yielded D50 = 21.7 ± 3 m and D50 = 27.3 ± 3 m, respectively. The latter value is closest to the one obtained previously, which supports the use of the 2-step fitting procedure including the log-transformed 2 nd step. The estimate of D50 obtained using the 2-step protocol proposed for the datasets missing spTfer(0) is higher than the estimates obtained using the other two methods, but nevertheless Equation 1 with their respective parameter sets approximates the experimental data reasonably well ( Fig. 1) in all three cases. Both probability of catch in the immediate proximity to the trap spTfer(0) and D50 are crucial for establishing a relationship between catch probability and distance to a baited trap, deriving bounds on absolute population density, and estimating the most probable population density of an insect (Fig. 3).
To derive bounds on average population density ̅ , we use the procedure described in (Onufrieva et al 2020). Once spTfer (0), D50 and Rmax are estimated, we define: Ln(spT fer (r ))

Equation 3
With that, the lower and upper bounds on the average density ̅ where M is the number of insects caught, p is the confidence level (p = 0.95 here) and 2 ( , ) is the quantile function (corresponding to a lower tail area q) of the χ 2 distribution with n degrees of freedom ( Table 2).
The most probable average male density in the trapping area is: To convert the male density to number of males per ha, and assuming D50 and Rmax are given in meters, μ in Equations (4) and (5) needs to be multiplied by 10,000.
We note that the probability of catching an insect located in the immediate proximity to the trap spTfer(0) provides a reference point for the rest of the trap catches, which is why it is important to measure spTfer(0) empirically, since as we saw in the example with gypsy moth data, estimating spTfer(0) by fitting Equation 1 to the experimental data is possible, but may not always match the experimentally obtained spTfer(0), which, in turn, may lead to an over-or underestimated D50. In western corn rootworm (see Results), our estimated D50 = 11 m agrees with results reported by Wamsley et al. (Wamsley et al., 2006), who observed significant drop of trap catches beyond 30 m away from the trap. However, trap catch collected at the distance of 16 m away from the trap is also significantly lower compared to the catch in a trap located 3 m away (Fig. 2E). This, once again, demonstrates the importance of measuring spTfer (0) empirically rather than estimating it by fitting Equation 1 to an incomplete experimental dataset. In Douglas fir beetle, D. pseudotsugae, previous studies reported that traps attracted beetles from at least 200 m (Dodds & Ross, 2002), but beyond this distance the recapture rate drops, which agrees with our estimate of D50 =184 ± 33 m (see Results).

Results
The main result of this work is the universal predictive relationship for a probability of catching an insect (spTfer(r)) located at a distance r from the trap.
where spTfer(0) (Miller et al., 2010) is the probability of catching an insect located in the immediate proximity to a baited trap and D50 is the distance from a baited trap at which the probability to catch an insect is ½ of the probability to catch an insect in the immediate proximity to the trap (spTfer (0)).
Results of the analysis conducted to estimate spTfer(0) and D50 for the studied insects are shown in Table 1 and Fig. 2. In all cases, the estimates of D50 obtained using untransformed and log-transformed data were very similar, within standard error of the mean (Table 1) (0)) and D50 for various insects in orders Lepidoptera, Coleoptera, Hymenoptera, Diptera, and Hemiptera. Experimental Rmax is listed, except for Sphingidae (marked *), which was estimated using method described in Miller et al. (2015).

Discussion
The key result of this work is the demonstration that a universal equation exists that faithfully describes the relationship between the probability to catch an insect and how far it is from the trap. The relationship is a simple formula with only 2 key parameters: spTfer(0), which is a probability to catch an insect released in the immediate proximity to the trap and D50 which we define as the distance from a baited trap at which the probability to catch an insect is ½ of the probability to catch an insect released in the immediate proximity to the trap (spTfer (0)). The strength of this definition is threefold: (1) it directly corresponds to what can be measured in field experiments and, (2) the concept of D50 can be easily illustrated on the graph of spTfer(r) vs. r, from which D50 value can be immediately estimated, at least approximately, as the value of r at which spTfer(r) = 1/2spTfer(0) (Fig. 2), and (3) the definition applies to any trap type.
To understand the biological meaning of D50, and its possible relationship to insect physiology, we compared D50 values derived from the trapping experiments with direct measurements of insect physiological response to appropriate attractant, where available. In gypsy moth, we estimated D50 = 26 ± 3m (Table 1), while Elkinton et al. (Elkinton & Cardé, 1984) observed wing fanning starting at a distance of 20 m from the pheromone source. Our estimate of D50 for European sawfly (D50 = 250 ± 21 m) agrees with the results of behavioral studies reported by Östrand et al. (2000), who observed response in N. sertifer to pheromone sources located 200 m away. Based on the agreement of our results with physiological studies, we suggest that qualitative biological meaning of D50 is effective attractive distance at which the probability that the lure elicits a response from is substantial. To formulate a more quantitative relationship will require more detailed physiological experiments than currently available.
In gypsy moth, numerical values of D50 and plume reach described by Miller et al. (2015) happen to be similar, but the match is purely coincidental and does not hold for most insects studied here. For most insects the values of plume reach (a pheromone-specific concept), and D50 (a universal characteristic of any trap) differ significantly.
One of the most striking results of this study is that the same number of insects caught in a trap may translate into order of magnitude different population densities in the field (Fig. 3). The qualitative explanation is that the population density is sensitive to parameters of the trap-insect system, particularly D50, and spTfer(0). Without knowing these key characteristics, based on the trap catch alone, one cannot make any quantitative assessment of what the actual insect population might be. The meaning of "catch zero" and "catch one" become clear only in light of the established relationship with the statistical bounds on the population density. When no insects are caught in the trap, we can conclude that, even though the insects might still be present in the field, their population density cannot exceed the specific threshold (upper bound, 95% confidence, Fig. 3). Likewise, if only a single insect has met its sad end in the trap, one can conclude that the actual population density cannot, with 95% confidence, be lower than the appropriate lower bound (Fig. 3).
It is remarkable that the simple Equation 1 works so well (average R = 0.91) across 5 orders of insects collected using very different attractants, such as chemical and light, selected randomly from the literature based on the available data despite the fact that parameters of analyzed trap-insect models vary widely: D50 ranged 6.5 -250 m and the estimated probability of catch in the immediate proximity to the trap spTfer(0) ranged 0.02 -0.7 (Table 1). This universality is the consequence of the universal set of principles that we applied to trapping of all insects: two-dimensional active movement space (insects following the terrain), finite active life span, and converged trap catches (collection time is long enough) used in well-designed trapping experiments.
Importance of conservation and pest management programs cannot be overstated as climate change, loss of biodiversity, and biological invasions remain the most serious environmental problems facing society. Inability to interpret insect trap catch data quantitatively, which includes directly relating trap catches to the absolute population density of an insect, hinders conservation, management, and research programs by making it difficult to provide recommendations, develop management tactics and evaluate treatment efficacies. The universal method reported here fills a key knowledge gap: it allows rigorous estimation of the most likely insect population density, along with the corresponding upper and lower bounds, from the number of insects caught by a single trap. The method is universal, in that it can be used for any trap-insect system. We believe this method will help develop technologies for improved insect population detection and management, but most importantly, will help drive future basic and applied research in multiple areas of entomology and ecology.