Endogenous spatial heterogeneity in a multi-patch predator-prey system: insights from a field-parameterized model

Abstract

The causes and consequences of spatial heterogeneity in population dynamics are of both theoretical and practical interest. Previously, we described (Kummel et al., Oikos 122:896–906, 2013) a field system in which predation by ladybugs drives the development of strong spatial heterogeneity among aphid populations living on nearby plants. In this paper, we investigate the detailed mechanisms responsible for this phenomenon. We develop a detailed mathematical model of the system, parameterized by an extensive experimental work showing that ladybugs tend to remain on plants with high aphid numbers and are attracted to plants on which ladybugs are actively feeding. The model reproduces important aspects of the field system and allows us to explore how various behavioral features contribute to these dynamics. The results indicate that spatial heterogeneity results from the random aspect of ladybug foraging that causes some large aphid populations to be under-exploited. For parameter values that are unrealistic for our system, the model displays other types of complex dynamics, including predator swarming and chaos. Our study illustrates how a realistic, carefully parameterized model can connect individual behavior to larger scale spatiotemporal dynamics.

Appendices

Appendix 1. Ladybug arrival and departure study (study A)

This study allowed us to parameterize the departure rate of ladybugs as a function of the number of aphids present on the yucca raceme at the time of departure. The study also allowed us to examine what drives arrival rates of ladybugs. Counter to our flight tracing study described in the body of the article, this study showed that for convergent ladybugs, ladybug arrival rates were strongly determined by the number of conspecific ladybugs present on the plant. However, given that all of the ladybugs were actively feeding during the study, it is still possible that the attraction was to feeding/fed ladybugs as opposed to conspecifics per se. The study also shows that the conditions on the nearest neighbor influence arrival rates, departure rates, and turnover rates of ladybugs on the focal plant.

This study was conducted in a short grass step field site near Colorado Springs along an approximately 100-m section of an abandoned railroad grade. In our experimental section, we identified all 71 flowering racemes of Yucca glauca and counted the number of arrivals and departures of ladybugs in a 20-min period, together with a count of aphids and both species of ladybugs at the beginning of the 20-min observation period. We also counted the number of ladybugs and aphids on the nearest neighbor raceme and measured the distance to the nearest neighbor raceme. For the initial 22 of the observation periods, we did not distinguish whether the arrival or departure was from the convergent or seven-spotted ladybug, and for the remaining 59 observation periods, we did distinguish between the two species. The results of the departure rate of ladybugs are present in the body of the article.

The dependence of arrival rates of H. convergens on conspecifics

H. convergens arrived with greater frequency on plants with high numbers of conspecific coccinellids, indicating significant self-attraction. The relationship was linear and therefore each additional coccinellid increased the attractiveness of the group by the same degree regardless of group size. Interestingly, the number of aphids on the focal plant did not affect H. convergens arrival rates (R² = 0.028, n = 59, p = 0.205). Contrary to our flight tracing study, the arrival rate did not depend on the ladybug × aphid term (R² = 0.021, n = 59, p = 0.274).

Impacts of conditions on nearest neighbor

H. convergens also arrived with greater frequency on plants whose nearest neighbor had large numbers of H. convergens. In a multiple regression (R² = 0.552, n = 59, p < 0.0005), arrival rates increased with increasing number of H. convergens on nearest neighbor (slope = 0.233, p < 0.0005) and with increasing number of H. convergens on the focal plant (slope = 0.289, p < 0.0005). Because H. convergens numbers on the focal plant did not correlate with its numbers on the nearest neighbor (r = − 0.048, n = 59, p = 0.717) and both independent variables share the same units, the slopes indicate the true contribution of each independent variable and the slopes are directly comparable.

Interestingly, total departure rates were high on plants with large numbers of H. convergens on the nearest neighbor plant. This conclusion is based on a multiple linear regression (R² = 0.576, n = 59, p < 0.0005), which showed that total departure rates (individuals per time) increased both with increasing numbers of H. convergens on the nearest neighbor (slope = 0.289 p = 0.002) and with increasing numbers of H. convergens on the focal plant (slope = 0.717, p < 0.0005). A similar relationship holds also for per-capita departure rates.

Given that large H. convergens numbers on the nearest neighbor plant increased arrival and departure rates, this should increase turnover rates on the focal plant. Turnover rates can be calculated either as input rate divided by the reservoir size (arrival based) or as output rate divided by the reservoir size (departure based). Both arrival-based and departure-based turnover rates increased with increasing difference between nearest neighbor NN and focal (F) ladybug population sizes: when NN > F departure turnover rates were high and when NN < F turnover rates were low (departure-based R² = 0.148, n = 59, p = 0.003; arrival-based R² = 0.201, n = 59, p < 0.0005).

Appendix 2. The olfactometer experiment (study B)

This study provided the evidence that convergent ladybugs (1) are attracted to the scent of other convergent ladybugs feeding on aphids, (2) are not attracted or repelled by the scent of aphids alone, and (3) are repelled by the scent of starving convergent ladybugs.

Ladybug attraction to different scents was measured through their movement in an olfactometer where different scents were introduced at opposing ends of the long axis of a diamond-shaped arena (32 cm length, 10 cm width, 4 cm height).

The scents were produced in half-gallon mason jars and were drawn through 40-cm silicone tubing into the slightly de-pressurized arena. The slightly lower pressure within the arena was created by removing air from the center of the arena by an aquarium pump, with the speed of air removal adjusted by a clamp. The scents were produced as follows. The scent of aphids was produced by closing a yucca raceme with approximately 5000 aphids into the mason jar; the scent of feeding ladybugs was produced by enclosing a group of 30 field-harvested Hippodamia convergens feeding on a raceme with approximately 5000 aphids; the scent of starved ladybugs was produced by enclosing a group of 30 field-harvested H. convergens starved for 24 h and a yucca raceme free of aphids; the control scent was produced by enclosing a yucca raceme free of aphids. During the experiment, the control scent of the empty raceme was paired with one of the experimental scents.

Convergent ladybugs used in this experiment were field collected daily approximately an hour before the onset of the experiment and were kept in a glass jar at room temperature.

To measure ladybug attraction/avoidance of the experimental scent, we placed a single ladybug into the center of the arena and recorded its position along the long axis of the olfactometer every 10 s for the total of 10 min (60 readings). If the insect did not move for 3 min or more, the data of the trial was discarded. To aid the recording of the ladybug positions, the arena was lined with paper and divided into 2-cm-wide strips running perpendicular to the long axis of the arena. The center of the arena was assigned the value of zero, and strips progressively closer to the port with the experimental scent were assigned positive values corresponding to the distance from the center. Strips closer to control scent port on the other side of the arena were assigned negative values corresponding to the distance from the center. The attraction/avoidance score was calculated as the average of the 60 readings, a positive number indicating attraction and a negative number indicating avoidance. The absolute value of the score indicated the magnitude of the attraction/avoidance. We tested a total of 24 field harvested H. convergens (8 per condition). We did not determine the sex of the beetles. The paper lining the arena was changed between the trials.

Appendix 3. Mesh bag field experiment (study C)

This study provided evidence that ladybugs in the field prefer to aggregate in locations where ladybugs are actively feeding and avoid locations with starved ladybugs. We were not able to distinguish whether this was because they arrived at the locations with feeding ladybugs more often or whether they remained there longer. The aim of the experiment was to count the number of free field ladybugs that were present on the outside of mesh bags which enclosed our experimental conditions. The experimental conditions comprised a design where the presence and absence of aphids was fully crossed with the presence and absence of convergent ladybugs.

The study was conducted in a 20 × 30 m experimental area in a short grass steppe location near Colorado Springs on a steep hill densely populated with flowering Yucca glauca. To account for the potential effect of the hill, such as varying moisture conditions, we deployed a block design. The hill was divided into five horizontal strips, each 6 m wide. In each strip, four yucca that had at least 3000 aphids on a raceme were randomly chosen among the flowering yucca plants present and these were randomly assigned into one of the following four experimental conditions: (1) empty raceme, (2) aphids only, (3) ladybugs only, (4) aphids and ladybugs. Racemes in conditions 1 and 3 were stripped off their aphids using a toothbrush. Racemes were then enclosed in bridal veil mesh bags. Bags in conditions 3 and 4 were supplied with 15 field-caught convergent ladybugs. We visited the experimental area 24 h later and counted all the ladybugs that were present on the outside of the mesh bags.