The Mathematics of a Locust Swarm

AAccounts of locust plagues reach back further than almost any other agricultural record, and the picture they hand down is remarkably uniform: a purposeful army advancing on the nearest green field. Modern reporting has inherited the metaphor, describing swarms that hunt, besiege and choose their targets, as though several hundred million insects shared a single plan. The animal itself offers little support for such language. A solitary locust is a drab, slow grasshopper that avoids others of its kind, and its eyes resolve little beyond a few centimetres, so no individual can be said to know where the crops of a region lie. What changes is not the insect's knowledge but its company, since close contact alters its colour, its physiology and its temperament within a matter of hours. The question that has occupied researchers since the middle of the last century is therefore mathematical rather than zoological, and it is easily stated: how can a rule that each animal obeys locally, using only what it can see and touch, produce a formation many kilometres across? The answers now on offer disagree about something more basic than mechanism, namely whether the formation is moving towards anything at all.

BThe account that has attracted most attention over the past two decades turns the familiar picture inside out. Helena Thorbjorn, an entomologist at the Vallance Institute of Population Biology, maintains that a marching band is not an advance but a retreat, since each insect moves forward chiefly because the insect immediately behind it is a potential predator. Hoppers short of protein will eat one another, and the abdomen of the animal in front is the nearest available supply. In arena trials conducted between 1996 and 2003, her group filmed bands whose rear-facing sensory nerves had been silenced; those individuals stopped marching within eleven minutes, while untreated controls in the same enclosure were still moving three hours later. Across the larger trials, 27 per cent of marked hoppers had been partly eaten by the end of a single day. From these figures Thorbjorn draws a conclusion that is deliberately austere: the heading a band takes carries no information whatever about where food lies, and the alignment that observers admire is simply what results when many thousands of animals are each escaping the one behind. Vegetation, on this reading, is where a band stops rather than what it is seeking.

CField entomologists have not accepted the picture, and their objection is more than sentimental. Idris Okonkwo, who has directed survey work for the Sahel Locust Monitoring Unit since 2011, contends that an enclosure of a square metre or two cannot stand in for terrain where hunger and greenery differ from one slope to the next. Tracking 46 bands across southern Mauritania in 2014, his teams found that a band's heading swung towards a patch of vegetation whenever such a patch lay within roughly 200 metres, a regularity that the escape model does not predict. The correlation is not easy to explain. Okonkwo suspects that wind-borne chemical cues are responsible, although he has not demonstrated this, and he concedes that his own evidence is thin in ways that matter. The bands were followed during a single rainy season, in one district, and only between mid-morning and dusk, so that any behaviour peculiar to the cooler hours of the day went entirely unrecorded. What the survey establishes, then, is that the escape model is incomplete rather than that it is wrong.

DBetween the two positions lies a quantity that both camps now treat as decisive, and it is not a property of any individual insect. Bands do not align at every density. In a simulation of 1,200 hoppers published in 2021 by Ana Ferreira, a former student of Thorbjorn's, movement remained essentially random below about twenty animals per square metre; between twenty and sixty the proportion travelling in a common direction climbed steeply; above sixty, alignment was all but complete and the modelled band ceased to respond to the food gradients built into its landscape. The transition, in other words, occupies a range of densities only a few insects wide. Arena work is necessarily carried out above that transition, since a confined space holding several hundred hoppers cannot be anything else, which suggests that Thorbjorn's rule is not so much wrong as permanently observed under a single condition. Okonkwo's field bands, thinner and free to spread, spent much of their time below it. Ferreira's model reproduces both sets of observations without amendment, provided that density is allowed to vary as it does outdoors.

EFor those who must actually deal with an upsurge, the dispute matters far less than either side supposes. Control is overwhelmingly a matter of timing. A hopper band occupying a hectare of ground can be treated by a single vehicle in a morning; once the insects have moulted into flying adults and merged into a swarm covering 80 square kilometres, the same operation requires aircraft, agreement between governments and, by one estimate from 2004, some forty times the expenditure for each tonne of vegetation saved. Both researchers therefore urge intervention before the density transition rather than after it, and they do so for opposite reasons: Thorbjorn because alignment cannot begin until crowding is severe, Okonkwo because thin bands are still steered by what surrounds them and can be found wherever the vegetation is greenest. The writer's own view is that both mechanisms are real and that their relative weight shifts with crowding, an answer that will satisfy neither party. It has, at least, the merit of being testable. What it does not offer is the reassuring image with which this account began, of an army that knows where it is going.