Modelling locust foraging: How and why food affects hopper band formation

Locust swarms are a major threat to agriculture, affecting every continent except Antarctica and impacting the lives of 1 in 10 people. 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 formation of hopper bands by the juvenile wingless locust nymphs. It is thus important to understand the hopper band formation process to control locust outbreaks. On longer time-scales, environmental conditions such as rain events synchronize locust lifecycles and can lead to repeated outbreaks. On shorter time-scales, changes in resource distributions at both small and large spatial scales have an effect on locust gregarisation. It is these short time-scale locust-resource relationships and their effect on hopper band formation that are of interest. In this paper we investigate not only the effect of food on both the formation and characteristics of locust hopper bands but also a possible evolutionary explanation for gregarisation in this context. We do this by deriving a multi-population aggregation equation that includes non-local inter-individual interactions and local inter-individual and food interactions. By performing a series of numerical experiments we find that there exists an optimal food width for locust hopper band formation, and by looking at foraging efficiency within the model framework we uncover a possible evolutionary reason for gregarisation. Author summary Locusts are short horned grass hoppers that live in two diametrically opposed behavioural states. In the first, solitarious, they will actively avoid other locusts, whereas the second, gregarious, they will actively seek them out. It is in this gregarious state that locusts form the recognisable and destructive flying adult swarms. However, prior to swarm formation juvenile flightless locusts will form marching hopper bands and make their way from food source to food source. Predicting where these hopper bands might form is key to controlling locust outbreaks. Research has shown that changes in food distributions can affect the transition from solitarious to gregarious. In this paper we construct a mathematical model of locust-locust and locust-food interactions to investigate how and why isolated food distributions affect hopper band formation. Our findings suggest that there is an optimal food width for hopper band formation and that being gregarious increases a locusts ability to forage when food width decreases.

Having plagued mankind for millennia, locust swarms affect every continent except behavioural characteristics from gregarious locusts [2]. For instance using these 48 techniques Buhl and colleagues have found the critical density for the onset of collective 49 movement [23], the interaction range of locusts (13.5cm), and the way that hopper band 50 directional changes are affected by locust density [17]. One downside of SPP models is 51 that there are few analytical tools available to study their behaviour. In contrast, 52 continuum models can be analysed using an array of tools from the theory of partial giving a representation of the average behaviour of the group. The latter (continuum) 56 approach is adopted in this paper.

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The non-local aggregation equation, first proposed by Mogilner and 58 Edelstein-Keshet [25], is a common continuum PDE analogue of SPP models [26,27]. It 59 based a conservative mass system of the form 60 ∂ρ ∂t where Q is defined as some social interaction potential, ρ is the density of species in 61 question, and is the convolution operation. For this type of model the existence and 62 stability of swarms has been proven [25], and both travelling wave solutions [25] and 63 analytic expressions for the steady states [28] have been found. This model has been 64 further extended to include non-linear local repulsion which leads to compact and 65 bounded solutions [29]. While usually used for single populations, the model has been 66 further adapted to consider multiple interacting species [30]. 67 In a 2012 paper, Topaz et. al. [31] used a multispecies aggregation equation to model 68 locusts as two distinct behavioural sub-populations, solitarious and gregarious. By 69 considering the locust-locust interactions and the transition between the two states, 70 they were able to deduce both the critical density ratio of gregarious locusts that would 71 cause a hopper band to occur and visualised the rapid transition once this density ratio 72 had been reached [31]. For simplicity the model focused on inter-locust interactions and 73 ignored interactions between locusts and the environment. While there exists some 74 continuum models of locust food interactions to investigate the effect of food on peak 75 locust density [32] or to consider hopper band movement [33], we are not aware of any 76 studies that consider locust-locust and locust-food interactions as well as gregarisation 77 in a continuum setting. local density defined as ρ(x, t) = s(x, t) + g(x, t). For later convenience we will also 98 define the local gregarious mass fraction as and the global gregarious mass fraction as It is possible to write s and g in terms of Eq (5) as We assume that the time-scale of gregarisation is shorter than the life cycle of locusts, ignoring births and deaths and thus conserving the total number of locusts. We also allow for a transition from solitarious to gregarious and vice-versa depending on the September 16, 2020 6/32 local population density. Hence, conservation laws give equations of the form ∂g ∂t + ∇ · (J g local + J g non-local ) = K(s, g), (3a) where J (s,g) local is the flux due to local interactions, J (s,g) non-local is the flux due to 102 non-local interactions, and K(s, g) represents the transition between the solitarious and 103 gregarious states.

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In addition to locusts we include food resources, let c(x, t) denote the food density 105 (mass of edible material per unit area). We assume that locust food consumption 106 follows the law of mass action and on the time-scale of hopper band formation food 107 production is negligible, giving where κ is the locust's food consumption rate.

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Finally, for later convenience we will also define the local gregarious mass fraction as 110 and the global gregarious mass fraction as It is possible to write s and g in terms of Eq (5) as Local interactions. We now turn to specifying the local interaction terms in Eq (3a) 113 and Eq (3a). These are captured by taking the continuum limit of a lattice model, we 114 do this by following the work of Painter and Sherratt [34]. We begin by considering 115 solitarious locust movement on a one-dimensional lattice (we assume that local 116 gregarious locust behaviour is the same resulting in a similar derivation). Let s t i be the 117 September 16, 2020 7/32 number of solitarious locusts at site i at time t, and let g t i ,ρ t i , and c t i be similarly defined. 118 We assume that the transition rate for a locust at the i th site depends on the food 119 density at that site, and the relative population density between the current site and 120 neighbouring sites. If we let T ± i be the rate at which locusts at site i move to the right, 121 +, and left, −, during a timestep, then our transition probabilities are where F is a function of food density, τ is a function related to the local locust density, 123 and α and β represent probabilities of movement. If nutrients are abundant at the 124 current site, then we assume locusts are less likely to move to a neighbouring site, which 125 implies F is a decreasing function. We set, where c 0 is related to how long a locust remains stationary while feeding. We further 127 assume that as the locust population density rises at neighbouring sites relative to the 128 population density of the current site, the probability of moving to those sites decreases 129 proportional to the number of collisions between individuals that would occur. Using Thus, our transition probabilities are Then the number of individuals in cell i at time t + ∆t is given by From this, we can derive the continuum limit for both solitarious and gregarious locust September 16, 2020 8/32 densities, and find our local flux as where D and γ are continuum constants related to α and β respectively. In higher dimensions, the expressions for fluxes are: Non-local interactions. For our non-local interactions we adopt the fluxes used by Topaz et. al. [31]. By considering each locust subpopulation, solitarious and gregarious, as having different social potentials, we obtain the following expressions for the non-local flux We also adopt the functional forms if the social potentials used by Topaz et. al. [31], as 134 they are used extensively in modelling collective behaviour and are well studied [28].

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They are based on the assumption that solitarious locusts have a long range repulsive 136 social potential and gregarious locusts have a long range attractive and a shorter range 137 repulsive social potential. The social potentials are given by, where, R s and r s are the solitarious repulsion strength and sensing distance respectively. 139 Similarly, R g and r g are the gregarious repulsion strength and sensing distance. Finally 140 A g and a g are the gregarious attraction strength and sensing distance.

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Gregarisation dynamics. For the rates at which locusts become gregarious (or 142 solitarious) we again follow the work of Topaz et. al. [31]. We assume that solitarious locusts transition to gregarious is a function of the local locust density (and vice versa). 144 This gives our equations for kinetics as, where, f 1 (ρ) and f 2 (ρ) are positive functions representing density dependant transition rates. To make our results more directly comparable we again use the same functional forms as Topaz et. al. [31] , where δ 1,2 are maximal phase transition rates and k 1,2 are the locust densities at which 146 half this maximal transition rate occurs.

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A system of equations for locust gregarisation including food interactions.
By substituting our flux expressions, (8a) through to (9b), and kinetics term (10), into our conservation equations, (3a) and (3b), and rearranging the equation into a advection diffusion system, we obtain the following system of equations Then dropping the bar notation the dimensionless governing equations are Note that we have introduced the following dimensionless parameters, For notational simplicity we drop the · * notation in the rest of the paper.
. Finally, while the support of g is 169 infinite (due to the linear diffusion) the bulk of the mass is contained as a series of 170 aggregations, we will approximate the support of a single aggregation as Ω. Using these 171 assumptions we can rewrite Eq (13a) as a gradient flow of the form, with the minimisers satisfying Next, we follow the work of [29,32,35] and with a series of simplifying assumptions we 175 consider both the large and small mass limit in turn. First, define the mass of locusts as 176 To find the large mass limit, we begin with Eq (14) and assume that g(x) is 177 approximately rectangular and for a single aggregation that the support is far larger with support, The accuracy of this approximation is illustrated by The max value and support are labelled ||g|| ∞ and Ω respectively. For both the simulation and calculations D = 0.01, γ = 60, R g = 0.25, r g = 0.5, A g = 1, and c = 0 and 1. As the mass M is increased the gregarious locust shape g becomes increasingly rectangular as the maximum locust density does not depend on the total mass. In addition as the amount of food is increased from c = 0 on the left to c = 1 on the right, the maximum density for the gregarious locusts increases.
For the small mass limit, we begin with Eq (14) and approximate the social 188 interaction potential using a Taylor expansion, e −|x| r In addition, we ignore the effect of linear diffusion 190 within Ω. These assumptions give Eq (14) as Following [32], gives an estimate of the maximum gregarious locust density, ||g|| ∞ , as with support, where B is the β-function (for definition see [36], page 207).

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The results of these approximations can be seen in Fig 2. While these 195 approximations are less accurate than those of the large mass limit, they illustrate that 196 as the amount of food increases so too does the maximum hopper band density.

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However this effect is less pronounced than in the large mass limit. It also demonstrates 198 how the maximum hopper band density and support both increase with an increase in 199 locust mass. in terms of the global gregarious mass fraction Eq (6) and the total density as whereQ s andQ g are the Fourier transform of Q s and Q g respectively.

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From this, it can be seen that asρ increases the gregarious fraction required for 214 hopper band formation increases suggesting an upper locust density for hopper band amount of available food increases.

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For our specific functions Q g = R g e − |x| rg − A g e −|x| and Q s = R s e − |x| rs , taking the one 218 dimensional Fourier transforms of Q s and Q g using the following definition, gives the following relationship, From this we can find the maximum homogenous density,ρ, that locust aggregations 221 can still form. So taking Eq (21) and substitutingφ g = 1 we can solve forρ as, where ||g|| ∞ is maximum density for the large mass limit given by Eq (15).

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Finally, we calculate if it is possible for a particular homogeneous density of locusts to become unstable (and thus form a hopper band). By calculating the homogeneous steady state gregarious mass fraction as, then by combining with (21) we obtain an implicit condition for hopper band formation 224 In Eq (22), if the values on the left are not greater than those on the right then it is not 226 possible for a great enough fraction of locusts to become gregarious and for instabilities 227 to occur. As the value of the right hand side decreases as the amount of food increases, 228 we can deduce that the presence of food lowers the required density for hopper band 229 formation.

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Time until hopper band formation with homogeneous locust densities. We 231 also estimate time until hopper band formation with homogeneous locust densities and 232 September 16, 2020 16/32 a constant c. By assuming that s and g are homogeneous we can ignore the spatial 233 components of Eq (12a) and Eq (12b). We again denote the combined homogeneous 234 locust density asρ however nowρ = s(t) + g(t). Finally, assuming that g(0) = 0, we 235 find the homogeneous density of gregarious locusts as a function of time is given by Which we then solve for t * such that g(t * ) =φ gρ , whereφ g is given by Eq (20). This

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gives an estimation for time of hopper band formation (i.e. the time required for the 238 homogeneous densities to become unstable) as, Thus, as increasing food decreases the gregarious mass fraction required for hopper band 240 formation it follows that it also decreases the time required for hopper band formation. 241 Center of mass. Another property of the model is how the center of mass for the 242 locusts behaves. For a single population with diffusive terms it has been shown that the 243 center of mass does not move [29]. Using a similar method we look at how the total mass fraction (see Eq (5)), which gives We now consider the behaviour of the center of mass. For notational simplicity we let Then the position of the center of mass, C, of ρ, is given by The motion of the center of mass is then given by, where Eq (25) is obtained using integration by parts and our boundary conditions.

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Parameter selection and initial conditions. The bulk of the parameters, 269 R s , r s , R g , r g , A g , k, and δ, have been adapted from [31] to our non-dimensionalised 270 system of equations. We explore two parameter sets that we will term symmetric and solitarization. This is the default parameter set from Topaz et. al. [31] with an adjusted 275 k 1 term that is calculated using Eq (22) and the upper range for the onset of collective 276 behaviour as ≈ 65 locusts/m 2 [31,38]. This behaviour is characteristic of the Desert 277 locust (Schistocerca gregaria) [10].
magnitude longer than gregarisation, and the density of locusts for half the maximal 280 transition rate is lower for solitarization. This is the alternative set from Topaz et. 281 al. [31]. The Australia plague locust (Chortoicetes terminifera) potentially follows this 282 behaviour taking as little as 6 hours to gregarise but up to 72 hours to solitarise [37,39]. 283 The complete selection of parameters can be seen in Table 1. At the densities we are investigating we will assume that the majority of movement 285 will be due to locust-locust interactions rather than random motion, so we set our s(x, 0) = ρ amb 16.6 (16.6 + µ) and g(x, 0) = 0, (29) where ρ amb is a ambient locust density and µ is some normally distributed noise, 302 µ ∼ N (0, 1). To ensure that simulations were comparable, we set-up three locust initial 303 condition and rescaled them for each given ambient locust density. Finally, the initial 304 condition for food is given by a smoothed step function of the form,

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with α = 7, x 0 = L/2, F M being the food mass and ζ being the initial food footprint. 306 We will also introduce ω = 100ζ/L which expresses the food footprint as a percentage of 307 the domain.

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The effect of food on hopper band formation. To investigate the effect that two food masses were tested, F M = 1.5 and 3. As a control we also performed 315 simulations with both no food present and a homogeneous food source, represented by 316 ω = 0 and ω = 100 respectively, for each ambient locust density. 317 We varied the ambient locust density ranging from ρ amb = 0.8 to ρ amb = 1.4 for the 318 symmetric parameters. This range was selected based on Eq (22) so that in the absence 319 of food hopper band formation would not occur. We also ran three simulations for each 320 combination of ρ amb , ω, and F M with varied initial noise and took the maximum peak 321 density across the three simulations, as we found in certain cases the initial noise had an 322 effect on whether a hopper band would form.

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For the asymmetric variables we varied ρ amb from ρ amb = 0.3 to ρ amb = 0.55, to test 324 the effect food had on the time frame of hopper band formation. From Eq (22) there Eq (23) this will only occur outside or right at the end of our simulated time frame. We 327 ran a single simulations for each combination of ρ amb , ω, and F M .

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The results for the symmetric parameter experiments are displayed in Fig 4. The 329 plots show the final peak gregarious density for the varying food footprint sizes and 330 ambient locust densities. In the blue regions there was no hopper band formation and in 331 the green regions there was successful hopper band formation. It can be seen in the 332 plots that as the food mass is increased the minimum required locust density for hopper 333 band formation decreases. This effect is more pronounced within an optimal food width 334 and this optimal width increases as as the amount of food increases.
By looking at the instantaneous relative advantage vs the global gregarious mass 358 fraction prior to hopper band formation in Fig 7,  with this effect being more pronounced at some optimal food width.

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Analytical investigations of our model shows that a spatially uniform food source has 376 a variety of effects has on locust behaviour. Firstly, by considering a purely gregarious 377 population we found that the maximum locust density is affected by the amount of food 378 present, in that increasing food leads to increased maximum density. Then, by 379 performing a linear stability analysis we found the gregarious mass fraction required for 380 hopper band formation depends on both the ambient locust density and the amount of 381 food present, with increasing food decreasing the required gregarious mass fraction.

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Using this relationship we then found that our model also has a theoretical maximum

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While it has been shown that highly clumped food sources lead to a greater 390 likelihood of gregarisation [20], using numerical simulations we have shown that there 391 exists an optimal width for these food clumps for hopper band formation. This effect 392 was shown to lower the required density for hopper band formation via the symmetric 393 parameters and the required time via the asymmetric parameters. This optimal width is 394 dependent on the amount of food present relative to the locust population. This effect 395 appears to be brought about by the depletion of the food source, if the food source is 396 not sufficiently depleted, then a gregarious hopper band will fail to form because a 397 portion of the gregarious population will remain on the food.

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In 1957 Ellis and Ashall [41] found that dense but patchy vegetation promoted the 399 aggregation of hoppers and that sparse uniform plant cover promoted their dispersal.
found that as the gregarious mass fraction increases so too does the foraging advantage 402 of being gregarious. This effect is increased by the mass of food present but is increasingly easy to quantify during field surveys, and aerial surveys including drones 418 and satellite imagery [21,37]. Future research should focus on developing decision 419 support systems integrating predictive gregarisation models and GIS data from surveys. 420