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The authors have declared that no competing interests exist.

Conceived and designed the experiments: JP RS ML CMT. Performed the experiments: JP RS ML. Analyzed the data: CN JP OW BM RS ML AJB CMT. Wrote the paper: CN JP OW AJB CMT.

From bird flocks to fish schools and ungulate herds to insect swarms, social biological aggregations are found across the natural world. An ongoing challenge in the mathematical modeling of aggregations is to strengthen the connection between models and biological data by quantifying the rules that individuals follow. We model aggregation of the pea aphid,

From bird flocks to fish schools and ungulate herds to insect swarms, nature abounds with examples of animal aggregations

A central question in the study of aggregations pertains to the relationship between individual-level and group-level behaviors, and it is crucial to distinguish between these. Individual-level behaviors might include an organism's tendency to move closer to conspecifics, or to align its movement with that of its neighbors. Group-level properties describe characteristics of many individuals, such as the shape of an aggregation, its spatial density distribution, and its velocity distribution. The connection between individual and group-level behaviors is highly nontrivial, as is typical for a complex system

An ongoing challenge in aggregation modeling is to construct individual-level rules that are quantitatively accurate and well-tied to experimental data. Sometimes, modelers may attempt to calibrate models and infer parameters based on published field observations or experimental results, for example, as with recent studies of locust swarms

Some foundational results on pea aphid movement appear in

In the absence of predators, some aphids move infrequently

Despite the aforementioned account of chemical signaling, and while it is well-known that aphids aggregate around food sources

The experiments of

Given the evidence for social aggregation in some aphid species, our goal at present is to assess and model aggregation of the pea aphid. More specifically, in order to deduce individual-level rules, we conduct experiments to track the motion of aphids walking in a featureless circular arena. We observe that each aphid transitions stochastically between a moving and a stationary state. Moving aphids follow a correlated random walk. The probabilities of stopping and starting, as well as the random walk parameters, depend strongly on distance to an aphid's nearest neighbor. For large nearest neighbor distances, when an aphid is essentially isolated, its motion is ballistic. Aphids move faster, turn less, and are less likely to stop. In contrast, for short nearest neighbor distances, aphids move more slowly, turn more, and are more likely to become stationary; this behavior constitutes an aggregation mechanism. From the experimental data, we estimate the state transition probabilities and correlated random walk parameters as a function of nearest neighbor distance. With the individual-level model established, we assess whether it reproduces the macroscopic patterns of movement at the group level. To do so, we consider three distributions, namely distance to nearest neighbor, angle to nearest neighbor, and percentage of population moving at any given time. For each of these three distributions we compare our experimental data to the output of numerical simulations of our nearest neighbor model, and of a control model in which aphids do not interact socially. Our social nearest neighbor model reproduces salient features of the experimental data that are not captured by the control.

To host aphid colonies, we grew fava bean plants,

We performed experiments on a vibration isolation table (IsoStation, Newport Corp., Irvine, CA) in a darkened lab in order to minimize effects of the ambient environment. The experimental arena consisted of a polypropylene circular ring, with a radius of 20

Aphids are dimorphic insects that may develop into winged or wingless forms, depending on a complicated interaction between genetics and environment

To prepare our data for motion tracking, we converted raw video footage in.mts format to.mp4 using Handbrake video processing software with sampling in grayscale at 5

(A) Trajectories of 28 aphids during approximately 15

To prepare our raw data set for modeling (see next section) we enhanced it with several elementary, derived pieces of data, namely motion state (stationary or moving), step length (distance traveled in one frame), heading, turning angle, and distance to nearest neighbor. An aphid's step length in a current frame was calculated as magnitude of the difference between its current and previous positions. We considered an aphid to be moving in a given frame if its step length was sufficiently large. For small steps, corresponding to speeds less than 4×10^{−2}

Based on the observation that aphids in the experimental trials transitioned between stationary and moving behavior, we propose a probabilistic two-state model to describe aphid movement and social interaction dynamics. Let _{MS}_{SM}

Moving aphids appear (naively) to follow a correlated random walk

We will now quantify our four model parameters: probability of a moving aphid stopping (_{MS}_{SM}

To estimate the transition probabilities _{MS}_{SM}_{MS}

We then form a scatterplot of the probability within each bin versus the midpoint of the bin, resulting in

(A) _{MS}^{2} = 0.92. To give a further sense of the efficacy of the fit, we display each point according to the standard error of the mean within the bin it represents. If the model curve passes within two standard errors of the estimated value, we show it as a green square; otherwise, it is a red dot. (B) Like (A), but for the probability _{SM}^{2} = 0.52; see text for discussion.

The probability _{MS}

The probability Here, _{MS}

To fit _{MS}_{MS}_{MS}_{MS}_{MS}

The probability _{SM}_{MS}_{SM}

The first (exponential) term models collision avoidanceThe first (exponential) term is repulsive, consistent with the notion that aphids avoid settling too close to others. The second (rational) term is a “loneliness” term, capturing that aphids move more when they are in isolation.is attractive, modeling the tendency of solitary aphids to move in order to aggregate. Together, these two terms specify a particular distance at which an aphid is most likely to be stationary (namely the value of _{SM}_{MS}_{MS}

We tried several functional forms (including linear combinations of exponentials) but choose

We now turn to the parameters governing moving aphids' correlated random walks.

Each data point represents the mean step length within a bin of 800 elements from our experimental data set, where the data are binned by ^{2} = 0.82. To give a further sense of the efficacy of the fit, we display data points according to the same scheme used in

According to this model, aphids with neighbors nearby take short steps, and the step length increases and saturates as _{MS}_{SM}

Finally, we model the spread of the distribution of turning angles

(A) Turning angle distribution parameter 0<^{2} = 0.99. Green circles (red dots) points represent data bins for which the model prediction falls within (outside) a 95% confidence interval around the experimentally measured

According to this model, aphids with nearby neighbors will turn more often at wider angles, resulting in motion that is less ballistic and more diffusive.

Fitting the model as described previously, we find

In summary, our model consists of just four quantities: _{MS}_{SM}

As alluded previously, one component ignored in the model is the arena's boundary. While it is quite likely that the presence of a boundary wall influences aphids' movement, the majority of our data set is composed of aphids far from the boundary.

The circular experimental arena has a radius of 0.2

With our model for individual-level behavior established, we will presently assess the degree to which it reproduces group-level behaviors. For comparison and contrast, we also consider a control model in which aphids do not interact at all. For this non-interaction model, we use the asymptotic (limit of large

We now shift our focus to group-level behaviors. We compare the experimental data (

We will compare three different group-level behaviors by studying their corresponding cumulative distribution functions as computed across each data set. A cumulative distribution tells, for any particular value of a data variable (horizontal axis) the percentage of data in the data set that is less than or equal to that value (vertical axis). It will be convenient to call our cumulative distributions

Comparison | |||

0.0046 | 0.1083 | 0.0835 | |

0.0159 | 0.3226 | 0.3873 | |

0.0113 | 0.2181 | 0.1668 |

Comparison | |||

0.0431 | 0.0085 | 0.0152 | |

0.0352 | 0.0128 | 0.0135 | |

0.0078 | 0.0057 | 0.0035 |

Comparison | |||

0.0774 | 0.3226 | 0.7789 | |

0.4711 | 0.8915 | 0.8373 | |

0.3938 | 0.8806 | 1.6649 |

The first group-level behavior we consider is the distribution of nearest neighbor distances

(A) Cumulative distributions

The second group-level behavior we consider is the distribution of angle to nearest neighbor,

Finally, we consider the third group-level behavior, the distribution of the fraction

Through experiment and modeling, we have investigated the movement, social behavior, and aggregation of the pea aphid. Motion-tracked experimental data gives rise to a two-state model in which aphids transition stochastically between stationary and moving states. Moving aphids follow a correlated random walk. The state transition probabilities _{MS}_{SM}

Our mathematical model is strikingly different from some previous data-driven aggregation models. The model of golden shiner fish in

The biological conclusions of our work are as follows. First, we have provided strong quantitative evidence that pea aphids display social behavior, in that an individual's movement in a featureless environment is influenced by its nearest neighbor.

Second, we have gained insight into the mechanism by which aphids aggregate. The probability of a stationary aphid starting to move decreases if a neighbor is nearby. The probability of a moving aphid stopping increases if a neighbor is nearby. These two behaviors promote aggregation. Further, aphids that are moving take shorter steps and turn more when in the vicinity of neighbors, promoting motion that is more diffusive and less ballistic (that is, less likely to move it away from the neighbor). This is reminiscent of the classic run-and-tumble model of bacteria

ThirdFinally, our model of individual-level behavior gives some feeling for the sensing range of the aphid. We recall the exponential length scales

As evidenced by the metrics in the previous section, our individual-based social model reproduces group-level featuresbehaviors muchbetter than a control model. There remain many avenues for further investigation. While we have demonstrated that pea aphid behavior promotes aggregation, we have not focused on quantifying the degree of aggregation (beyond measuring the distribution of distance to nearest neighbor). One could investigate the typical population size of an aggregation and the typical time scales of an aggregation's formation and existence. Furthermore, we have not captured all of the experimental complexity in our simple model. As mentioned throughout, we have ignored the effects of the boundary. It would be useful to quantify more precisely the rules an aphid obeys when it encounters an immovable obstacle such as a boundary. Additionally, our model is arguably the simplest possible social model, in which social effects depend on a single nearest neighbor. One could investigate the degree to which an aphid responds simultaneously to multiple neighbors, keeping in mind the limits of aphid cognition. Finally, it could be interesting to augment our work, which describes aphid aggregation the absence of environmental cues, with a consideration of external factors such as nutrition sources. Such an investigation might shed further light on the aphid's role as a destructive crop pest.

Ken Moffett of the Macalester College machine shop built the experimental arena. Matthew Beckman of Augsburg College provided advice on our experimental setup. Raibatak Das of the University of Colorado, Denver shared a template of M atlab code helpful in our image processing and tracking. We benefitted from statistical discussions with Alicia Johnson, Victor Addona, and Danny Kaplan. As part of his undergraduate research experience at Macalester College, Trevor McCalmont contributed to a prototype of the experiment and model during early stages of this work. We are grateful to Macalester College for laboratory space in the XMAC (eXperiment, Modeling, Analysis and Computation) laboratory.