Modelling locust foraging: How and why food affects group formation

Locusts are short horned grasshoppers that exhibit two behaviour types depending on their local population density. These are: solitarious, where they will actively avoid other locusts, and gregarious where they will seek them out. It is in this gregarious state that locusts can form massive and destructive flying swarms or plagues. However, these swarms are usually preceded by the aggregation of juvenile wingless locust nymphs. In this paper we attempt to understand how the distribution of food resources affect the group formation process. We do this by introducing a multi-population partial differential equation model that includes non-local locust interactions, local locust and food interactions, and gregarisation. Our results suggest that, food acts to increase the maximum density of locust groups, lowers the percentage of the population that needs to be gregarious for group formation, and decreases both the required density of locusts and time for group formation around an optimal food width. Finally, by looking at foraging efficiency within the numerical experiments we find that there exists a foraging advantage to being gregarious.

where with our specific functions given by Finally, we give the measures of foraging efficiency defined in [1].The per capita 5 contact with food for solitarious and gregarious locusts, respectively are given by where M is given by ( 2).The instantaneous relative advantage at time t is given by 1 Foraging efficiency 8 Many measurements of foraging efficiency/advantage rely on an individuals ability to 9 extract energy from a food source.The measure we look at, known as foraging 10 efficiency, is the ratio of energy gained to energy spent and is given mathematically by 11 Laguë et al. [2] as where F (t) is the total energy gained by foraging for time t, p 1 is the energy cost per 13 unit time during foraging, p 2 is the energy lost per unit time by travel between food 14 patches for time t * .Note that the marginal value theorem [2,3] is a simpler version of

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(5) given by 16 where R(t) is the rate of energy gain, F (t) is the total energy gained by foraging a patch of food in time t, and t * is the time to travel between patches.
18 However, (5) is not spatially explicit, we thus need to convert our spatially explicit subscript • g denote gregarious locusts and • s denote solitarious.Then, let f g (c, s, g) be a 22 function describing the energy gain for gregarious locusts per unit area per unit time

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(the derivation for solitarious is similar and thus omitted here).We can then calculate 24 the total instantaneous gregarious energy gain, I g at time t as 25 This allows us to calculate the average gregarious individual's instantaneous energy gain by dividing I g (t) by the total number of gregarious locusts, where M and φ g are given by ( 2) and ( 3) respectively.We can then calculate the average gregarious individuals total energy gain over a timer the time interval, [0, t], by 27 integrating, giving rise to By substituting (1c) into (7) and taking into account only the gregarious contribution to ρ we obtain this is the definition of cumulative relative advantage from Tania et al. [1].
Finally, as the proportion of the population that is gregarious is changing in time B(t) becomes difficult to interpret.We instead assume that φ g (t) is constant over the short interval [t, t + ∆t] for some ∆t 1, we then let b(t) be the cumulative relative advantage over this interval, which we will term the instantaneous relative advantage to be in line with Tania et al. [1].We find which is the measure we use in the main text.We thus find that an instantaneous relative advantage would lead to a cumulative relative advantage for a fixed gregarious 37 mass fraction, and this in turn would imply a foraging advantage to being gregarious.

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We note that the results when using (10) instead of (11) give a similar impression.Cumulative relative advantage of gregarious locusts vs time at various food footprints and food masses.In these simulations ρ amb = 0.95 and κ = 0.09, with the symmetric parameter set.The homogeneous food source is labelled ω = 100%.Similar to the results presented in Figure 7 in the main text, as time increases so too does the cumulative relative foraging advantage of being gregarious.This effect is increased by the mass of food present but is diminished by the size of the food footprint.

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Fig 1.  Cumulative relative advantage of gregarious locusts vs time at various food footprints and food masses.In these simulations ρ amb = 0.95 and κ = 0.09, with the symmetric parameter set.The homogeneous food source is labelled ω = 100%.Similar to the results presented in Figure7in the main text, as time increases so too does the cumulative relative foraging advantage of being gregarious.This effect is increased by the mass of food present but is diminished by the size of the food footprint.
dτ, , p 1 and p 2 are equal for both solitarious and gregarious.If they are 33 unequal we would end up with some scalar multiple of B(t).It should be noted that 30E s (t) = t 0 κη s (τ ) dτ p 1 t + p 2 t * .*