Locust and locust density-dependent phase polyphenism

Locusts are short-horned grasshoppers that exhibit density-dependent phase-polymorphism or polyphenism (the phenomenon in which two or more distinct phenotypes are produced by the same genotype). They appear in two forms or phases [6]: crowding induces the gregarious phase, most notorious for its tendency to aggregate and form massive swarms; while isolation leads to the solitary phase, in which individuals actively avoid other locusts. The differences between the gregarious and solitary phases are collectively termed phase characteristics, extending from behavior and ecology, through morphology and anatomy, to physiology and biochemistry (see recent reviews in [7,8]). Locust phase polyphenism is continuous, and many intermediate forms can be found between the extreme gregarious and solitary phases. The phase transformation, i.e., the response to changes in population density, is reversible and not developmental-stage- or age-specific. Some phase characteristics, especially behavioral patterns, respond rapidly within the same instar (larval stage). Other characteristics, for example, color changes, only show an overt response in the next or subsequent instars. Parental density affects the color of the hatchlings and may have some behavioral and further effects on the next generation ([9–12], and see [13] for a review). Full-scale phase transformation takes several generations, and presumably occurs only in the field. In the laboratory, locusts kept either under crowding or under isolation, respectively approach the characteristics of the gregarious and the solitary phases [8].

Some 15 locust species, belonging to several, sometimes distant, phylogenetic groups, are considered true locusts, i.e., they express density-dependent polyphenism, swarming, and migration (see [6,8,14] and references within). Most notorious (and, thus, most studied) is the desert locust (Schistocerca gregaria), which inhabits dry grasslands and deserts and has threatened agricultural production in Africa, the Middle East, and Asia for centuries. Hereon, unless explicitly noted, we will mostly refer to the desert locust, as this is the most investigated species and a major part of our knowledge is based on it. The range of the migratory locust (Locusta migratoria) is wider than that of any other. It is found in grasslands throughout Africa, major parts of Eurasia, the East Indies, tropical Australia, and New Zealand. The Australian plague locust (Chortoicetes terminifera) is the most important pest species of locust in Australia, due to the large areas infested, the frequency of outbreaks, and its ability to produce several generations in a year. These, as well as other, less prominent locust species, may differ in their ability to express the phase phenomenon in its entirety, in the dynamics and magnitude of their response to changes in population density [8,14,15]. This important point has been largely overlooked by biologists and even more by theoreticians, tempted to generalize from a single-species study to all locusts. As our knowledge of diverse species increases, we may be introduced to key species-specific differences in the dynamics of the phase phenomenon [6,8–10]. The intricate phylogeny of locusts also offers a challenge to any attempt at explaining the evolution of the locust phenomenon (see the section “Evolution of the Locust Phenomenon” below).

As mentioned, locust behavior is a prominent phase characteristic. The importance of the change in behavior is that it precedes and facilitates all other phase changes. The major behavioral characteristic of locusts in the gregarious phase is their strong attraction to their conspecifics, which translates to active aggregation behavior (Fig 1A) [6,16,17]. Gregarious locusts are also generally more active, including their strong propensity to march in huge bands of hoppers or to form flying swarms as adults. In contrast, solitary-reared locusts actively avoid contact with other locusts [18–20]. They are more sedentary and cryptic in behavior, do not march, and fly less. The behavioral phase transformation is a positive-feedback process. Solitary-reared locusts have been reported to acquire most of the behavioral characteristics of the gregarious phase within 4–8 hours of crowding, including the propensity to aggregate, and thus reinforce the gregarizing stimuli provided by other locusts [19]. Recently, it has been shown that even a 30-minute exposure of a solitary-reared nymph to a small crowd of locusts is sufficient for the induced change in behavior to persist, even after re-isolation of the locust for 24–72 hours [20]. Details of the dynamics of phase transformation and its interactions with the environment are still far from being fully resolved [8].

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Fig 1.

(A) A major behavioral characteristic of locusts in the gregarious phase is their strong propensity to form mass aggregations, as demonstrated by these desert locust nymphs. (B) Both swarming and marching start very early, already a few days after hatching, as demonstrated by these first larval instar desert locusts. (C) The endless marching bands of locust nymphs exemplify extreme coordination in their movement vectors during collective motion. (D) Road kills (nymphs hit by passing cars while the swarm crosses a road) will be immediately cannibalized by others (arrows). Locusts stopped in the midst of the swarm to feed on the cadavers will be totally ignored by others. Continuous and careful monitoring of individuals within a crowd in controlled lab experiments (E and F) have enhanced the development of quantitative analysis and theoretical modeling of individual dynamics and interactions, leading to various models of coordinated collective behavior.

https://doi.org/10.1371/journal.pcbi.1004522.g001

Locust swarming and coordinated movement

As mentioned, swarming is a hallmark of locust behavior. Swarming per se is a result of the attraction and aggregation tendency of locusts in the gregarious phase. Locust-coordinated mass movement includes the flight of adult swarms, but even more conspicuous is the marching of nymphs or hopper bands. It should be noted, however, that the formation of dense bands of hoppers is independent of the marching behavior: the multitude of different behavioral patterns of the bands include other, relatively quiescent, states, such as basking, perching, sheltering, and, of course, feeding. Hopper densities in quiescent bands are extremely high (Fig 1A), and the area that a quiescent band covers can be 2–4-fold smaller than that covered by the same band when marching, as reported for desert locusts by Ashall and Ellis [21,22], with even greater differences reported for other locusts ([23] and references therein).

Both swarming and marching start very early on, at the first larval instar, a few days after hatching (Fig 1B). Desert locust nymphs (as well as other locusts) have a diurnal pattern of behavior (e.g., [24,25]). The locust bands will normally have one period of marching in the forenoon and another in the afternoon, strongly dependent on weather conditions such as temperature, but also on clouds obscuring the sun, as well as on rain or cold wind [23]. In our own observations of the desert locust (Fig 1; Negev desert, Israel, spring of 2013; Ayali unpublished), marching usually took place from just before noon to two hours before sunset. The marching hopper bands can cover very different areas, from a few hundred square meters up to several kilometers. As reported for different locust species, their shape can also vary, ranging from columnar to frontal structures [23,25]. During marching of the hoppers, bands may meet, merge, or fuse, and the coordinated movement will thereby further enhance the swarming (this is also true for the flying swarms).

Undisturbed nymphs within marching bands will advance predominantly by very consistent walking rather than by jumping or hopping (the latter are aversive reactions, most commonly demonstrated in response to danger or threat [26–29]). This behavior is most typical to the desert and migratory locusts and may be less so in other species. Regarding the speed of marching, Uvarov [23] had already noted that “the actual speed of individual hoppers bears only a remote and uncertain relation (by a factor of 3–4) to the marching rate of the bands.” The reason for this is that the proportion of hoppers marching at any given moment can be as low as 10%. This has been confirmed by others (cited therein, and see also [24]) and by our own observations: individual marching locusts very often stop for periods of various durations. Of special interest is the behavior of hoppers at the edges of the marching band. Again, according to Uvarov ([23], and references within), hoppers advancing beyond the edge of a marching band frequently stop or even turn back to rejoin it. Similar observations were reported by Ellis and Ashall [24]. In their detailed description of marching bands of the Australian plague locust, Buhl et al. [30] refer specifically to the front edge of the band, finding that locusts at the front will slow down or simply turn more often, reducing their net displacement in comparison to those behind (as also reported for fish schools [31,32]). These behaviors will serve to maintain the integrity of the marching bands. They often result in bands comprising an extremely dense front followed by an exponential decay of density, with a consequent loss of cohesion toward the back [30]. What the major factors are that affect the direction of marching is still an open question.

The cannibalistic propensity of locusts has been recently associated with marching behaviour and was even suggested as a major driving force for the formation of collective motion in locust nymphs [33–35]. The controlled experiments and intriguing results of Bazazi et al. (e.g., [33]) indicate reduced marching of nymphs in an experimental arena following manipulation of sensory inputs from behind (visual and/or tactile). These reports are a good example of experimental results directly motivating a modeling approach, as discussed in the section “Escape and pursuit” below. They also underlie a putative theory for the evolution of phase polyphenism (see “Evolution of the Locust Phenomenon”). Cannibalism is very well documented in locusts (and other grasshoppers), e.g., [22,36–38]. However, it is accepted by most authors that cannibalism is mainly directed toward hatchlings [22,39], freshly molted [40], injured, or dead animals [37], and is rarely directed toward active, healthy individuals. Moreover, the documented dynamics within the marching band as described above (pauses of varied length, behavior at the band edges, etc.) are not consistent with the nymphs escaping predation or trying to cannibalize on their marching companions. The results of Bazazi et al. [33] may also be interpreted as merely supporting an instrumental role of visual inputs in the local interactions among locusts (see also 3.2 below), as well as confirming the cannibalistic tendency toward unhealthy individuals (surgically manipulated in this case).

While Uvarov [3] goes as far as to categorically reject any connection between ground movement and hunger (page 83 therein), it is well established that the nutritional state of the hoppers does have an important impact on the amount of marching, its speed, and the distance travelled [34,41]. Hence, it may be suggested that in addition to possible species-specific differences, different interactions among locust nymphs may occur at different behavioral contexts, and maybe also at different times of the day (e.g., some effect of a cannibalistic drive during the onset of marching in the morning, but non-cannibalistic interactions when marching is established or during high-density aggregation in vegetation during the hottest period of the day, etc.).

Flying swarms of adult locusts will migrate predominantly downwind. There are very few studies focusing on local dynamics within the flying swarm. Within a given section of the swarm, locusts were observed to generally fly with their bodies parallel, separated by as little as 10 cm [42–45]. Camhi et al. [46] even reported wing-beat coupling among pairs of locusts flying in tandem. Although aligned with their neighbors, locust groups within the body of the swarm head in random directions [43]. The integrity of the swarm is maintained by the fact that, similar to the marching hoppers, upon reaching the edges of the swarm locusts will turn back to re-join it. Hence, in the flying swarms, it is not clear whether and how local interactions and dynamics translate to coordinated swarm locomotion.

Experimental approaches to the study of locust swarming

Traditionally, studies of locust behavior have been limited to field observations during locust outbreaks in the affected African regions. Such reports include the pioneering work of Uvarov [6,23,47], Kennedy [48,49], and others. These efforts have been aided by the development of techniques that enabled the investigation of particular aspects of the behavior of individual locust hoppers in bands (e.g., the photographic techniques of Stower [50]). A crucial step towards a more advanced understanding of coordinated behavioral mechanism in locusts was made by the first laboratory experiments, including the generation of marching behavior in closed circular arenas and the experimental manipulation of locust density [16,17,51–53]. These early reports already identified the strong attraction of gregarious locusts to conspecifics, which translates into active aggregation behavior; and, furthermore, already suggested that locust collective movement is highly dependent on the density of animals in the group.

Behavioral experiments were supplemented by extensive literature on locust physiology and neurobiology (e.g., [54], and references within). Most significant were advances in describing neurobiological correlates to locust phase polyphenism (starting with [55,56]), as well as attempts at identifying the neurobiological-related mechanisms of the phase change (reviewed in [57]). These have disclosed relevant sensory modalities that may be important in the complex interactions between individuals within a locust swarm.

Using modern computer-vision tracking methods (Fig 1E and 1F), locust marching behavior in the controlled environment of the lab has lent itself easily to continuous and careful monitoring of the crowd and the individuals, and to quantitative analysis. In the experiments by Buhl et al. [58], groups of up to 120 insects were placed in a ring-shaped arena with an external diameter of 80 cm. Locusts were allowed to march freely for several hours, monitored by a video camera. The films were later analyzed to produce individual trajectories, allowing a detailed quantitative analysis of the swarm dynamics. Recently, Ariel et al. [59] used similar experiments to investigate the animal–animal interaction in order to reveal some of the microscopic details of locust motion within a crowd—when does it walk, pause, or turn? A surge in such quantitative studies of locust coordinated behavior in the past decade or two (see below) have kept locust studies up-to-date and relevant and ensured their place at the forefront of the interdisciplinary efforts to unravel the secrets behind collective animal behavior.

Locusts as self-propelled particles

The experiments by Buhl et al. [58] of locust marching in a circular arena motivated increased theoretical interest in the field. One of the main conclusions of this work was that the “swarm” synchronizes in either clockwise (CW) or counter-clockwise (CCW) movement. Furthermore, the time spent in an ordered state (a synchronized direction) increased with the number of animals, as switching from CW to CCW and vice-versa become rare events. The ring-shaped arena and directional switching motivated Buhl et al. to suggest a one-dimensional (1D) model (referred to hereafter as the Buhl model) adapted from Czirók et al. ([72]; referred to hereafter as the Czirók model). The applicability of a relatively simple 1D model to a real-life experiment quickly attracted interest by the theoretical community.

The Czirók model is essentially a 1D version of the well-known 2D Scalar Noise Model (SNM) of Viscek et al. [74]. We begin here with a description of the model that is continuous in time. Consider N particles with positions x(t) = (x₁(t),…,x_N(t)) and dimensionless velocities u(t) = (u₁(t),…,u_N(t)) moving along a line of length L with periodic boundaries. In order to facilitate comparison between the model and its many variants, we write the dynamics as a system of Stochastic Differential Equations (SDEs), (1) where, is white noise (independent for each i), v parameterizes the speed of particles, and σ is the standard deviation of external noise. The interaction of a focal particle i with its conspecifics is assumed to depend only on the local average direction within an interaction distance Δ, where A_i(t) denotes the set of neighbors of particle i at time t up to an interaction distance Δ, A_i(t) = {j:|x_i(t)-x_j(t)|≤Δ} and n_i(t) = |A_i(t)| is the number of neighbors. In the Czirók model, as in most later variations of this model, the function G(·) is taken to be an odd, piece-wise continuous function,

Discretizing (Eq 1) using Euler-Maruyama with step-size Δt, (2) where ξ_i(t) are independent random variables with zero mean and variance σ². Taking uniformly distributed noise, ξ_i(t)~U[-η/2,η/2], yields the model described by Yates et al. [75]. Taking Δt = 1 yields the discrete Czirók model,

In order to analyze and quantify order in the system, consider an order parameter ϕ(t) defined as

Fig 2A depicts two sample trajectories of ϕ(t) at different concentrations of ρ = N / L. In the following, simulation results are presented with the parameters N = 100, η = 2, Δ = 1, and v = 0.1, unless specified otherwise. At low concentrations, the system is disordered and ϕ(t) fluctuates around zero. However, at higher concentrations, ϕ(t) fluctuates around one of two metastable states located at +1 and -1, with occasional rapid transitions between them. This suggests that, when order is high, the system admits a coarse-grained description as a two-state continuous time Markov chain (CTMC). S1A Fig shows the distribution of waiting times between transitions, which is approximately exponential, as expected for a CTMC. In simulations, metastable states were defined using a cutoff for ϕ(t), A₊ = {ϕ(t)>p}, and A_- = {ϕ(t)< -p}, where p = 0.7. Fig 2C shows the average transition rate between the two metastable states for fixed noise η = 2 and varying ρ (see also S2 Fig). Interestingly, the switching rate shows a plateau for a wide range of concentrations. The length of the plateau seems to grow with N, suggesting a limiting equilibrium rate at the thermodynamic limit (N→∞ at constant ρ and η), which is the biologically relevant limit for natural large locust swarms.

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Fig 2.

(A) Sample trajectories for the Czirók model. Blue curve: ρ = 0.1. Red curve: ρ = 3.2. (B) Sample trajectories with individual-choice (I-C) interactions. Blue curve: ρ = 0.2. Red curve: ρ = 3.2. Note that the time scale between two models differs by a factor of ten. (C) Mean transition rates as a function of ρ. (D) The average order parameter as a function of ρ.

https://doi.org/10.1371/journal.pcbi.1004522.g002

The main significance of the Czirók model and its 2D predecessor [74] is the understanding that the two types of behaviors—fluctuating around ϕ = 0 or ϕ = ±1—can be thought of as phases in the sense of statistical physics. Consider the average order parameter where brackets denote averaging over all time steps and initial conditions. Fig 2D shows the average order parameter in the Czirók model (blue lines) as a function of ρ for fixed noise η. The figure suggests that for large N, the curve becomes singular at a critical density ρ_c. In statistical physics, this type of behavior is called a phase transition (not to be confused with the solitarious-gregarious transition discussed above) [62,64,65,74]. If the function ϕ(ρ) is discontinuous at ρ_c, then the transition is a first order transition, similar to the solid/liquid/gas transition in solid state physics. However, if ϕ(ρ) is continuous at ρ_c, but not differentiable, then the transition is called a second order transition, similar to the transition from a paramagnet to a ferromagnet in iron (a ferromagnet can remain magnetized after an external magnetic field is switched off, while a paramagnet cannot). Similar behavior is observed at constant ρ and variable η, indicating that η is analogues to temperature. See [76] for a discussion of temperature in Viscek-type models. The type of transition, critical concentrations, critical noise, and exponents in SPP models are studied numerically [65,72,74,77].

The significance of the observation that the Czirók model undergoes a phase transition is in understanding that order and disorder can be characterized as two distinct phases. Switching between the phases requires some external change in the system parameters, such as concentration or the amount of noise.

Individual pause-and-go movement

Following their own experimental observations of locusts marching in a ring-shaped arena, Ariel et al. [59] suggested a modified model, based on the biological paradigm that animal decisions constitute a sequence of discrete events—when to stop, walk, or turn, rather than a continuous process. The main observation underlying the model is that the locusts follow an intermittent motion pattern that is composed of interchanging walking and standing periods (see also [23,24]; details in the above section, “Locust swarming and coordinated movement”). Pause-and-go motion was previously observed by [83] in an arena housing a single locust. In the Ariel et al. pause-and-go model [59], a particle is either standing (v = 0) or moving. Standing particles "do not participate in the game," i.e., they are ignored by moving particles and are not taken into account in calculating local averages. Similarly, the local average velocity and the order parameter are defined in terms of only moving particles, for example, and

Ariel et al. [59] studied a pause-and-go model based on first-principle interactions as obtained from experiments. Here, we present a modified version of the model, which is easier to compare with the models discussed above:

1. A walking particle moves with a fixed velocity vu_i for a duration that is exponentially distributed with rate k_walk.
2. A standing particle starts moving with rate k_stand that depends on the number of walking neighbors. The function k_stand(n_i,moving) is increasing in n_i,moving, i.e., local movement increases the probability of standing animals to start walking. For simplicity, we take k_stand(n_i,moving(t)) to be a step function that jumps from a low rate k_stand,0 to a higher one k_stand,1 if n_i,moving(t) is above a threshold n_c,moving.

When a particle starts walking (i.e., changes from a standing to a walking state) it has a probability α to align with the local order among moving individuals, plus the usual noise term, . However, the probability α depends on the local order 〈u(t)〉_i,moving. For simplicity, we assume that \alpha(x) is a linear, increasing function. This dependence implies that animals have a higher tendency to align with ordered crowds than with disordered ones.

S4 Fig shows the global order parameter as a function of concentration for the pause-and-go model described above. The model suggests some qualitative predictions on the global properties or coarse-grained dynamics of the swarm, as discussed in the section “Coarse-Graining and Macroscopic Observables” below.

Three-zone models

One of the first modeling approaches for collective motion, proposed in the early 1980s by Akoi [84] and independently by Reynolds [85], is the widely used three-zone, or Avoidance–Alignment–Attraction (AAA), model. The model, popular with biologists for its intuitive assumptions, but less so with theoreticians for its difficult analysis, assumes that the interaction between conspecifics can be divided into the zones defined by three characteristic lengths, as follows.

• Individuals are repelled from conspecifics that are closer than a distance r₁.
• If no conspecifics are present up to distance r₁, then an individual aligns its movement direction with conspecifics that are closer than a distance r₂>r₁.
• If no conspecifics are present up to distance r₂, then an individual is attracted to conspecifics that are closer than a distance Δ>r₂.

A 1D version of the model was studied in the context of locust swarming by Bode et al. [80]. The algorithmic implementation of their model, which assumes asynchronous dynamics, is as follows. For every time step Δt repeat N times:

1. Choose an individual i at random.
2. If i has neighbors (up to distance Δ), choose a neighbor k at random and update u_i according to
3. Advance: x_i = x_i+vu_iΔt.

Note that the model does not include avoidance interactions, which, as far as locusts in a circular arena are concerned, do not make sense in 1D. Bode et al. [80] analyzed the dynamics under their model using coarse-grained variables, as described in the section “Coarse-Graining and Macroscopic Observables” below.

Escape and pursuit

Recently, a new paradigm that strives to give a comprehensive answer to several aspects of the locust phenomenon focused on cannibalism as a dominant feature of locust behavior and interactions ([33–35], and see 2.2 above). Building on this idea, Romanczuk et al. [63] suggested a modeling approach termed “escape and pursuit” (E&P). In this model, animals are continuously escaping from conspecifics behind them in order to avoid being eaten, while at the same time pursuing other animals in front, attempting to eat them.

The E&P model is a 2D SPP model. As an extension of the approaches described in the section “Locusts as self-propelled particles,” above, neighbors of a focal particle are categorized into four types: either at the front or behind, and either approaching or receding. Here, conspecifics that are either approaching from the front or receding from the back are ignored. Each of the two remaining types is averaged and weighted separately.

To be precise, the model assumes that individuals follow Langevin dynamics of the form where γ and D are the friction and diffusion coefficients, respectively, and describes the effective social force on particle i, which is given by

Here, k = e,p is the effective escape (e) and pursuit (p) response. The heaviside function is denoted by H(·), and is the relative dimensionless velocity of particle j with respect to i, where u_ji = u_j-u_i, r_ji = r_j-r_i, and . In addition, χ_k are interaction strengths, and N_k the number of particles i is pursuing or escaping. See [63] for details.

Unfortunately, a 1D version of the E&P model is problematic because, depending only on the initial condition, animals split into CW and CCW moving groups that do not interact. This observation is not consistent with our own experimental observations [59] indicating that one of the principle sources of synchronization in marching nymphs is that of animals turning to join a crowd moving in the opposite direction.

While offering a very attractive mechanism, the E&P model is somewhat inconsistent with various biological observations. As noted above, individuals in a marching band will very often stop for periods of various durations (see also [23,24]); when encountering a standing hopper, the marching ones will ignore it (Fig 1D). Furthermore, as noted in the section “Locust swarming and coordinated movement” above, locust marching behavior is mostly very different from escape or aversive reactions [27], which usually involve escape jumps [26,28]. Undisturbed marching desert locusts are very rarely seen hopping or jumping [29]. Also inconsistent with E&P is the behavior of hoppers at the front and side edges (e.g., the tendency of leading locusts to slow down or even turn around to join the bulk of the swarm [23,24,27,30]; detailed in the section “Locust swarming and coordinated movement”). According to the E&P model, locusts will chase receding conspecifics in front of them, escape conspecifics approaching from behind, and ignore others (approaching from the front or receding at the back). Therefore, under the rules of E&P, a nymph moving beyond the front of the swarm will not turn around but, rather, keep escaping the approaching swarm at its back.

Buhl et al. [86] reported their findings in a study of the Australian plague locust specifically aimed at analysing the spatial distribution of locusts within naturally occurring locust bands and inferring on the interactions within the band. These authors reported a tendency for locusts to interact with neighbours all around them, rather than a bias toward pursuing individuals ahead or escaping from the ones following behind. The E&P model may account for the collective behavior of locust nymphs under certain conditions and circumstances, in accordance with the context- and time-dependent role of cannibalism suggested in the section “Locust swarming and coordinated movement.”

Interim summary

Most of the models presented above share a common assumption regarding the animal’s calculation of a local average (or weighted average) that serves as the main influence of conspecifics on the individual. From a biological perspective, this assumption is not realistic, as insects and other animals are often more sensitive to extremal statistics, such as abrupt or unexpected changes (in the behavior of others or in the environment). In this sense, the pause-and-go model, in which the probability of an animal to start marching depends on the visual flux around it, more precisely on the number of walking neighbors exceeding a threshold, may be more realistic.

The next section further compares the different models in terms of the macroscopic dynamics of the entire swarm.

Modeling adult locust flying swarms

The model of Edelstein-Keshet et al. [87] studies the swarming behavior of locusts, focusing on the long-distance migratory swarms of flying adults. The main objective of this work was to investigate the issue of maintaining swarm cohesion; specifically, to find traveling wave solutions, which are particular solutions that have a fixed shape and move at constant speed. The model is given in terms of partial differential equations (PDEs) describing the time evolution of the continuous density of flying locusts, f(x,t), and stationary locusts (those on the ground), s(x,t). In this respect, the swarm exhibits intermittent motion reminiscent of the pause-and-go model. However, the latter includes a mechanism for turning and choice of direction that is missing here. In addition, there are no periodic boundary conditions. For simplicity, the swarm is considered as 1D. The dynamics is given by where, R(s,t) is the take-off rate, G(s,f) the landing rate, D describes random motion (when flying), and U describes the flying velocity (wind speed and self-propulsion). More elaborate models that take into account additional effects, such as dependence of D and U on the local densities, slow motion on the ground, and density-dependent turning, are also considered. The main conclusion of the paper is that traveling waves, representing cohesively moving swarms, cannot exist unless non-local interactions, i.e., global interaction between animals within the entire swarm, are allowed. Even then, “traveling wave solutions seem to bear some unrealistic properties.” See [87] for details.

Modeling locust phase transformation

Feistel and Feistel [88] approach the question of locust phase polyphenism and the concentration-dependent transition from the point of view of statistical physics. Assuming a homogenous, well-mixed population, they consider a biologically realistic mean-field model that describes the time evolution of four parameters: (i) External mechanical stress that increases production of (ii) a stress hormone. The latter affects individual behavior by reducing the (iii) average local concentration, increasing (iv) the rate of collective interactions. Increased collective interaction increases the external mechanical stress, thus creating a positive feedback loop. It is shown that, depending on model parameters, the dynamics undergoes a bifurcation, and the population can be classified into three phases: solitary, gregarious, and unstable. Transitions are analogous to either first- or second- order phase transitions in statistical physical, depending on parameters, such as the stress hormone production rate.

The model of Topaz et al. [89] studies the phenomenon of locust outbreak, which is characterized by rapid large-scale transitions between scattered populations of solitary animals to densely-packed gregarious swarms (see also [90]). The Topaz model is given in terms of integro-differential equations that describe the time evolution of a two-dimensional density of solitary animals, s(x,t), moving at velocity v_s(x,t), and the density of gregarious animals, g(x,t), moving at velocity v_g(x,t). Densities satisfy the continuity equation (conservation of mass) where k_s→g(ρ) and k_g→s(ρ) are the rate of transition from solitary to gregarious and vice versa, at a total local concentration ρ = s+g. It is assumed that gregarious animals are attracted to conspecifics while solitary animals are repelled. The strength of the interaction decreases exponentially with distance, where * denotes convolution and R_k±,r_k± are parameters, k = g or s. Numerical solutions as well as stability analysis reveal conditions for outbreaks.

It should be noted that our current understanding of questions related to solitary locust ecology and population dynamics, such as competition over resources, sexual selection and mate finding, habitat range, displacement, and more, is very limited, as most research focus has naturally been placed on the gregarious phase.

The work of Topaz et al. [89] strongly suggests the need to integrate the complex effects of density, as well as the dynamics of phase transformation, in the models of marching behavior (“Modeling Marching Locust Nymphs,” above). This model stands out in making an attempt to bridge different scales, from the large scale dynamics accompanying the emergence of swarms of gregarious locusts from scattered solitary populations, to the more local scale of the mechanisms inducing coordinated motion of the swarms.