Mathematical Models of Hopper Bands for the Australian Plague Locust
By Jillian Kunze
Locusts are both an ancient pest and a contemporary menace that, when swarming, can completely destroy farmers’ crops. As large locust swarms move across a terrain and eat everything in their path, they develop certain structures based on the distribution of plants in the area. For example, bands of locusts develop dense fronts in lowcover areas that contain a lot of food. During a minisymposium presentation at the 2020 SIAM Annual Meeting, Jasper Weinburd of Harvey Mudd College presented a mathematical model for the structure of locust aggregations based on their resource consumption. Previous modelling efforts were based on locusts’ social interactions, so Weinburd’s model was the first that attempted to use food as the driving factor for the swarm’s collective structure.
Photos showing different structures of locust swarms.
The study focused on hopper bands, which are groups of juvenile locust nymphs that cannot fly. Instead, they exhibit a kind of jerky motion as they move across the ground in one of three ways—crawling, hopping, or stopping—while all heading in generally the same direction. This is the stage of locust life at which pesticide treatments are the most effective, so modelling the movement patterns of juvenile locust nymphs is of great interest.
Weinburd and his collaborators developed an agentbased model that simulates the movements of individual locusts and discovers how they impact the system at large. Because the juvenile locusts do not fly, the researchers considered them as a twodimensional system; they then took a onemeter cross section of the hopper band so that it became a onedimensional system. The model employed a large number of locusts that could at any time be in one of two possible states: stationary or moving. The practice of switching between those two states was governed by a discretetime Markov process, and the probability of a locust being in either state depended on the amount of food present. Stationary locusts caused the amount of resources near them to decrease.
As expected based on locusts’ real activity, the model produced aggregate collective motion that took the form of a travelling wave of locusts with a characteristic asymmetric profile. The model could also output other observables, such as the distance of individual locusts from the center and the average speed of the band (which tended to equilibrate to a constant value).
Output from the agentbased model of locust aggregation, which shows a travelling wave of locusts.
There were 10 parameter inputs to the model for the individual behaviors of locusts. Some were easy to quantify, such as the individual marching speed, but others like the individual foraging rate lacked welldefined values because they were more difficult to measure in field experiments. The researchers’ goal was to adjust the lesserknown parameters so that the model produced the expected values for observable quantities based on real field observations. However, it was computationally taxing to conduct multiple simulations that tested different parameter values, as the model required roughly 10 billion calculations per run.
Weinburd and his team tried an alternative method for moving from the input parameters to the observable quantities: meanfield partial differential equations (PDEs) in an advectionreaction model. Given a certain initial amount of food and total mass, they found that the PDE approach could predict the speed, amount of leftover food, and band shape. They could thus determine the ranges for the less wellconstrained input parameters without performing an excessive number of calculations — simply by identifying a combination of biologically reasonable inputs and outputs to the advectionreaction model.
Using Sobol indices, Weinburd’s group found that foraging rate was the input parameter with the biggest impact on the model’s output. The team also compared the results from the model to field data and discovered quantitative agreement; the model reproduced field observations within observed scatter. Moving forward, Weinburd and his colleagues are also working on a twodimensional, agentbased model of locust swarms with social interactions, which shows different aggregation patterns. In addition, they are currently collaborating with ecologists to track locusts in video data, which they can use to deduce locust interactions and further their modelling efforts.

Jillian Kunze is the associate editor of SIAM News. 