Exploring Locust Behavior: Flight Optimization, Population Dynamics, and Energy Efficiency 

Frederic Koran, Claire Leader, Alexandra Stoilova, Alina Weng

Abstract

Mathematics play a vital role in understanding locust physiology, behavior, and adaptability by linking flight efficiency, population dynamics, and energy metabolism. Locusts achieve flight efficiency through specialized wing structures, with finite element modeling (FEM) showing how pre-cambered curvature and tapered veins optimize aerodynamics. Advanced models like PARSEC parameterization and Nash genetic algorithms reveal a 77% boost in efficiency, conserving energy for migrations. 

Efficiency is also demonstrated by population dynamics that are driven by density-dependent behaviors and environmental factors. Sigmoidal functions link density to phase polyphenism, while logistic growth equations explain rapid population increases under favorable conditions. Advection-diffusion models predict migration patterns, and Lotka-Volterra equations reveal oscillating predator-prey trends. Energy metabolism adapts to diet, life stages, and behaviors. Scaling laws show a nonlinear link between body mass and energy use, with juveniles relying on anaerobic bursts and adults on sustained aerobic pathways. During gregarious phases, increased oxygen consumption and fat oxidation meet energy demands, while dehydration-induced oxidative stress highlights migration costs. These interconnected models reveal how locusts efficiently use energy through flight and metabolism and shows how they adapt to population pressures offering a cohesive view of their unique design solutions. 

Introduction

Locusts (Fig. 1) are highly adaptable insects known for their complex behavior and significant agricultural impact. They are species of grasshopper capable of forming massive swarms under specific conditions. Normally, locusts are solitary creatures, but they can be brought together against their natural predisposition to avoid one another, resulting in mass crowding (Simpson, 2005). Swarms containing up to 70 billion individuals have been observed in a single region, often referred to as “plagues” (Green 2022). During their gregarious or swarming phase locusts migrate rapidly.  

In 1988, a swarm incredibly crossed the Atlantic Ocean, traveling from Africa to South America in just ten days (Green, 2022). Locusts can travel such vast distances in very short periods due to the geometry of their wings and their specific flying mechanisms. By utilizing unique venation patterns, wing structure (including camber, and thickness), and flying dynamics (such as, degree of curvature, angle of incidence, and radius) locusts optimize their flight (Isakhani et al., 2021). These adaptations allow for increased aerodynamics and consequently enable them to conserve energy. Conserved energy means they can sustain longer flights with fewer rest periods, making even transoceanic flights possible!  

Between July and September 2003, locust swarms formed in Sahelian West Africa, migrating through Mauritius, Morocco, Algeria, Senegal, and Niger, devastating millions of hectares of agriculture, crops, and ecosystems (Issacs, 2004). A single locust can consume its body weight in vegetation daily, and a single swarm can devour up to 200 tons of vegetation in one day, moving relentlessly from one resource to the next once an area is depleted (Gifford, 2020). In the affected African countries, up to 80% of the workforce is employed in agriculture. The combined effects of widespread unemployment and food insecurity severely threatened the quality of life and economic stability in these regions (Issacs, 2004). 

Understanding and predicting locust population trends is essential for mitigating their destructive impacts. Mathematical modeling provides insights into how factors such as weather patterns and vegetation health influence locust populations directly and indirectly. By analyzing long-term trends and patterns, we can improve predictions of swarming events and develop proactive measures to protect ecosystems and agriculture.  

Locusts experience changes from solitary to gregarious phases—a phenomenon called phase polyphenism—driven by both biological and environmental factors. In arid and desert areas throughout Africa and India, the Desert Locust thrives despite harsh conditions. Typically, these regions cannot support large grasshopper populations, but after heavy monsoon rains, the soil stays wetter for longer, triggering mass breeding events. If a nymph hatches alone, it is likely to remain in the solitary phase. However, high-density hatching due to monsoon conditions leads to physical and behavioral transformations. Visual and olfactory cues encourage locusts to aggregate, creating a positive feedback loop where crowding reinforces swarming behavior (Green, 2022). Mathematical models can correlate population density with swarming probability, showing how increases in density trigger swarming (Collett et al., 1998). Physical contact between locusts, including touching of the hind legs, releases serotonin, which promotes aggregation (Green, 2022) and triggers changes in physical activity causing a shift in metabolic rate. Solitary locusts maintain energy-optimized metabolic systems, but gregarious phases need higher energy to sustain swarming behavior. In swarms, locusts are far more mobile compared to their sedentary solitary counterparts. They meet these energy demands by increasing their oxygen consumption, consequently causing oxidative stress (Du et al., 2022). This design solution highlights how the benefits of swarming outweigh its energetic cost.  

Changes to physical traits due to environmental shifts are called phenotypic plasticity, in this case it is density dependent (Pener & Simpson, 2009). The origin of such density-dependent transformations is said to be influenced by the shifting monsoon belt during the Sahara Desert’s formation. This varied the edges of the desert causing environmental instability that favored phenotypic plasticity and historically shaped the adaptability of locust populations (Green, 2022). These environmental pressures also influenced flight dynamics, critical for locust dispersal and survival in unstable habitats. 

Flight Modelling

Locusts have a set of forewings and hindwings that can be divided into different parts. The forewings can be divided into three parts: the anterior panels of the remigium, the posterior panels of the remigium and the clavus. The hindwings can be separated into two parts: the stiff anterior remigium and the vannus or vannal fan (Fig. 2) (Kumar & Sariyev, 2013). 

Fig. 2 Locust wing anatomy. Wing anatomical nomenclature: clavus (c.), claval furrow (c.f.), anterior panel of the remigium (a.p.r.), posterior panel of the remigium (p.p.r.), remigium (r.), median flexion line (m.f.l.), vannus (v.). [Adapted from Kumar & Sariyev, 2013]  

Locust hindwings generate camber during motion. This is called the umbrella effect.  During the upstroke of the hindwing, the broad and corrugated vannus curves creating bending in the radiating and unbranched veins of that section. These veins are the anal and jugal veins that are along the corrugations of the vannus. The membrane close to the border of the vannus is in tension and the inner membrane of the vannus remains loose and corrugated. Thus, the hindwings are like the curvature of an umbrella when opened (Wootton, 1995). The umbrella effect depends on the geometry of the wings. The length of the supporting veins and the acute angle α decreases as the insect abdomen is approached, starting from the remigium (Fig. 2b). Camber generation is proportional to the number of veins in the vannus region and the size of the acute angle (Isakhani et al., 2021). 

Logarithmic Spiral

To model the umbrella effect, the shape of the hindwing is studied. The outer margin of the hindwings is not shaped like a quarter circle but rather follows the shape of a truncated logarithmic spiral. The logarithmic spiral follows equation 1: 

(1)  

is the angle through which the radius revolves, is the proportionality constant, and is the radius length (Fig. 3a). According to this equation, the higher the constant , the tighter the spiral becomes because of an increase in angle. However, the hindwing does not spiral completely and only corresponds to a partial logarithmic spiral. The perimeter of the hindwing only spirals until less than 80º and a sharper turn is observed when the wing-base is approached (Wootton, 1995). However, this model does not consider the radiation of veins. Locust veins do not spread from a single point but rather a short bar.  

The margin of the hindwing is represented as a flexible cord in tension and the veins as a radiating series of rods. The cords and rods are separated by an angle ∅ and they form triangles (Fig. 3b). These similar triangles decrease in size as the insect abdomen is approached. By considering a triangle ABC, a relationship between the proportionality constant of the logarithmic spiral and the triangles can be deduced through mathematical manipulation seen in equation 2: 

(2)  

(Fig 3c) (Wootton, 1995).  

Fig. 3 (a) Logarithmic spiral. (b) Rods and cords form a series of similar triangles, see (c), diminishing in size posteromesally from the distal apex of the model vannus towards the site of the insect’s abdomen. [Adapted from Wootton, 1995] 

The umbrella effect only occurs if the membrane between two veins does form a flat triangle, but a panel with convex sides. Thus, compression can be seen on the rods to imitate this component. When the model is fully extended, three forces can be observed at the apex of the rods (Fig. 4). These three forces are related by the following equation: 

As the insect body is approached, the compression in the rods increases since F₁ of the new triangle corresponds to F₂ of the previous one. The compression of the rod increases by a ratio of (Wootton, 1995). This logarithmic spiral shape of the vannus helps the folding efficiency of the locust wing (Wootton et al., 2000). 

Fig. 4 The forces F₁, F₂, F₃, operating at the apex of a rod separating two triangles (Wootton, 1995). 

Finite Element Model 

The logarithmic spiral model is not entirely accurate since the wing membrane, cross-veins and intercalary veins were not considered in the model. The perimeter and the supporting veins were also modelled as several straight cords and straight rods respectively. The rigidity was assumed to be constant along the vein and the same for all veins. Veins taper and radiate from a cuticular bar. Both the veins and the wing margins are typically curved rather than straight. A finite element model was created for a more accurate take on locust hindwings. The finite element model divides a complex structure into simpler constituent elements. These elements go through a process called meshing where they are connected at common nodes. Nodes are interpolated by a field quantity. Since the nodes are meshed, the field quantity is interpolated throughout the structure. Thus, the finite element model can represent structures of any shape with any support conditions that are subjected to complex and numerous loads.  

The umbrella effect was modelled with this approach. The model consisted of the membrane and the dorsal longitudinal veins of the vannus of the wings, disregarding small cross-veins and ventral veins. The constituent elements of the models were membrane elements and beams representing the main supporting veins. Meshing of 1669 elements of these constituents, 220 beams and 1449 membrane elements, was found to show promising results (Fig. 5). The finite element model was subjected to field quantities such as stress and displacement during the application of loads and torque to the models (Herbert et al., 2000). 

Fig. 5 The final mesh of the ‘comprehensive’ model consisting of 220 beam (red) and 1449 membrane (blue) elements (Herbert et al., 2000). 

Five models were created from this approach. The ‘comprehensive’ model was the most realistic and accurate model of the locust vannus. The model includes pre-camber curvature of the vannus and tapering of the veins. Tapering veins were represented by dividing the model into three and assigning different flexural rigidity to the beam parts by changing the Young’s modulus value. The ‘no-camber’ model differed in the initial curvature, where pre-camber curvature was disregarded. The ‘straight-veined’ model included the pre-camber curvature but neglected the curvature of the veins. The ‘identical-vein’ model differed in vein tapering, where constant flexural rigidity was given throughout the veins. The ‘mock-geometry’ model disregarded the curved structure of the real cuticular bar where veins radiate from. Veins are joined at a straight line (Fig. 6) (Herbert et al., 2000).   

Fig.6 The geometry, boundary and load conditions of the five finite element analyses. (A) The ‘comprehensive’ model; (B) the ‘no-camber’ model; (C) the ‘straight-veined model; (D) the ‘identical-vein’ model and (E) the ‘mock-geometry’ model. The line below indicates the profile of each model, through points x-y. [Adapted from Herbert et al., 2000] 

Several experiments, such as force and torque application, were performed on these different models. By measuring the torque required to induce deformations in the vannus, it was determined that the ‘comprehensive’ model produced the most similar deformations to the real locust wing, at a torque of 3.5 x 10^-6 Nm, while the other models failed to do so (Herbert et al., 2000). Thus, wing shape as well as properties of veins and membrane are structurally significant for the locust wing. The ‘mock-geometry’ demonstrated that the shape of the wing and the arrangement of veins are significant. The ‘straight-veined’ model suggested that pre-cambering is diminished due to straight veins. Vein tapering was shown to be a natural adaptation to structural loading. Bending decreases from the cuticular bar to the wing border during aerodynamic loading. Vein tapering helps minimize the mass and moments of inertia needed for appropriate rigidity to resist this force. Camber was also demonstrated to be significant in the transition from an unloaded and loaded state (Herbert et al., 2000). Thus, vein tapering is used as a design solution to resist stress and acceleration applied to the wings during flight. 

Optimized Model Using Nash Genetic Algorithm 

Locust flight is optimized through gliding as it maximizes energy conservation by minimizing wing articulation. To analyze and model this natural adaptation, locust wing geometry is parameterized into a mathematical form through PARSEC parameterization. PARSEC parameterization represents the wing structure as a linear combination of base functions with eleven parameters seen in figure 7. A hybrid optimization technique is used to enhance the model and increase resemblance with the real wing (Isakhani et al., 2021). This technique is the Nash genetic algorithm formed from the Nash strategy and the bioinspired genetic algorithms (Giocoli, 2004).  

The Nash strategy is a non-cooperative and multi-objective optimization technique seen in game theory. This strategy consists of using players as distinct objectives for the problem where no players can have an individual gain by changing their strategy. The equilibrium N represents the optimized state of the problem. It is achieved when the enhancement of the objectives is no longer possible by the players. Bioinspired genetic algorithms are based on natural selection and genetics. The generation can solely evolve if its players have satisfied certain conditions such as maximizing the objective or minimizing cost. This algorithm permits the use of a wide range of variables with complex functions representing the conditions to meet.  

The wing model is considered as a two-player game, each representing the drag and lift coefficients. The lift-to-drag ratio must be optimized to obtain efficient gliding and thus, optimized flight. These coefficients depend on different design variables, such as the leading-edge angle of incidence (α_HW) and the leading-edge radius (r_e) for the drag coefficient as well as the trailing-edge angle of incidence (β_HW) and the crest curvatures (c_b, c_t) for the lift coefficient, but also similar design variables like the horizontal (u_t, u_b) and vertical (v_t, v_b) position variables (Fig. 7). Sixteen strategy combinations were found to obtain equilibrium N, where the lift coefficient is relatively high and the drag coefficient relatively low, thus maximizing the lift-to-drag ratio. The optimized model showed 77% improvement in gliding performance compared to the unoptimized model. Thus, locust flight is optimized through gliding where an efficient lift-to-drag ratio is maintained. The optimal ratio requires a balance between the coefficients, since drag cannot be fully disregarded since it is required for aerodynamic stability and control. 

Fig. 7 Illustration of the PARSEC parameterization and Nash GA hybridization: a) PARSEC parameterization adapted for a locust-inspired hind foil and b) depiction of design variables and their level of dependency on each player (C_L or C_D) (Isakhani et al., 2021). 

Table 1 List and description of the PARSEC parameters corresponding to the bioinspired airfoil’s defining attributes (Isakhani et al., 2021) 

Population Dynamics

Locusts possess the remarkable ability to rapidly change their population size and activity levels in response to environmental conditions. While their population dynamics are complex, they break down to interactions of biological processes, environmental factors, and now, human activities.  With a proper understanding of these dynamics and processes, we may be able to reduce the negative impacts and apply the population trends. 

Swarming Behavior: Phase Transformation and Density Dependent Function 

Locusts have a key feature to transition between solitary and gregarious phases, two phases which determine whether they act more as individuals, or seek other locusts and swarming, respectively. Their ability to transition between phases can be triggered by increased population density and results in behavioral and physiological changes (Pener & Simpson, 2009). This can determine whether locusts’ function in gregarious phase, which would cause them to find and form cohesive swarms which travel distances and consume large quantities of vegetation. 

Fig. 8 As the population density increases, regardless of dispersed or clumped food, locusts tend to move from the solitary to gregarious phase. (Sword, Lecoq, & Simpson, 2010)  

Mathematically, there is a probability modeled by P(D) – function of probability based on input density, where , where D is the locust density and K is the half saturation constant (the density at which there is 50% chance of phase transformation). This function produces a sigmoidal curve – an s shaped curve where the rate of change accelerates after a certain point, then slows down as it approaches a maximum value – reflecting the likelihood of locusts in gregarious phase increases with higher density (Collett et al., 1998). At low density, P(D) is small, and locusts remain solitary, as D approaches K, P(D) increases, showing the higher probability of gregarious phase and consequentially, swarming.  

Fig. 9 Sigmoidal modeling in locusts in solidary (separate) vs gregarious (consolidated) SNR refers to the signal to noise ratio, which is an indicator in swarming. (Johnson & Romero, 2021)

For example, if K = 100 locusts per square mile, and the currently density D = 150, probability of gregarization:

, or 60% chance that locusts will enter gregarious phase. In 2020, in East Africa, a period of unusually heavy rainfall led to increased vegetation and breeding sites, which increased locust densities to surpass the critical threshold (FAO, 2020). Using the density-dependent function, they were able to predict the likelihood of swarming and implement early control measures. 

There will always be environmental variability, so stochastic elements – which include randomness – can be incorporated into the model. For instance, Monte Carlo simulations, which take repeated random sampling, consider random fluctuations in factors like rainfall and temperature and how that affects swarming (Bazazi et al., 2011). 

Rapid Population Growth: Logistic Growth and Density Dependent Control 

The locust population grows rapidly and exponentially under favorable conditions without limiting factors (Uvarov, 1977).  This means that initially, the population growth can be modeled using N(t) = N₀e^rt, where N(t) models the population size at time t, N₀ is the initial population and r is the growth rate. While this equation models the free growth of the population at first, there are inevitable environmental limitations. These limitations, called density dependent factors, are due to limiting factors such as scarce sources. To take this into account, the logistic growth model has K, the carrying capacity of the environment, representing the maximum population size. The represents the reduction in growth rate as population approaches the carrying capacity. Using logistic growth models, we may accurately predict when populations will reach critical levels and use controlling measures to prepare (Hunter, 2004). 

Fig. 10 Shows the difference between exponential and logistic growth, where logistic slows down and maxes out at the carrying capacity of environment, also giving it a point of maximum growth. (Wilcox & Jessop, 2012) 

Predator-Prey Dynamics and Age-Structured Models: 

Locust populations and their interactions with predators can be modeled by Lotka-Volterra equations. These equations describe cyclical oscillations in populations: an increase in predator population leads to a decline in prey population. This in turn reduces the food available for predators, causing their population to decrease, which subsequently allows prey population to recover and increase, and so on. These two equations, and (L is locust population and P is predator population), where a is locust growth rate, b is predation rate coefficient, c is predator mortality rate, and d is the efficiency of converting consumed prey into predator offspring model an everlasting cycle where the populations constantly shift from too high or too low (Wilps & Diop, 1997).  

Fig. 11 General Lotka-Volterra equations show graphically the relationship between prey and predator population. As one increases, consequently, the other decreases in an endless loop (Wikipedia contributors, n.d.). 

There are also age dependent models that divide locusts into age classes and use an equation to represent the dynamics: where N_i is the number of individuals in age class i, m_i is the mortality rate in age class i, and f_i-1 is the fecundity (to produce offspring) rate from previous age class. We would essentially write out a different equation for each age class, then use those rates to compare. While this model is not applicable for swarming or immediate threats, using values of mortality or fecundity can simulate a population over time. Using that data, we may identify critical stages where interventions are most effective (Retkute et al., 2023). 

Energy Metabolism 

Energy metabolism in locusts is a highly active process influenced by diet, life cycle, and behavioral phases. Being herbivorous, locusts depend upon plant material for supplying their metabolic needs, where nutritional content in the food plays a major role in their growth and development, energy production. The quality and quantity of food available to the locust will determine the physiological status and behavioral traits it exhibits at any given stage, an example being transformation between solitary and gregarious phases. Locusts, in their life cycles, exhibit metabolic changes that characterize the various growth stages ranging from the embryonic to mature adult insect. The locusts also have very different metabolic needs between their solitary and gregarious phases. In the solitary phase, locusts have a more energetic economic metabolism, while in the gregarious phase, metabolic activity is turned up to supply energy for the highly energetic swarming behavior. These changes are not simply reactions to environmental triggers but are under rigid control by both genetic and physiological mechanisms. The locust diet is a foundation of energy metabolism and changes in metabolic processes during life cycles. Changes in metabolic processes can be seen between the solitary and gregarious phase. Understanding such factors provides insight into adaptive strategies that enable locusts to thrive in diverse and often challenging environments. 

Influences via diet 

The greatest influence on energy metabolism is the intake of that energy via an organism’s diet. For an organism to exist most efficiently in its environment, it requires a certain amount of nutrients, thus creating a very complex relationship with its surroundings. Ideally, locusts will ingest both carbohydrates and protein at a 1:1 ratio to sustain their daily living (Raubenheimer & Simpson, 1993), which can be referred to as an “intake target”. To achieve this, they ingest leaves, any tender tissues of plants, or scavenged dry plant matter, and they must eat approximately their weight in plant matter each day (National Geographic, 2015). When their bodies are deprived from one or several nutrients, locusts adjust their feeding behavior in such a way to restore nutrient imbalances to get as close to their intake target as possible. In a geometrical framework, nutrient-deprived locusts do not merely consume more food but, rather, shift the relative proportions of protein to carbohydrate in their diet to equalize their nutrient intake. When they shift these proportions, they restrict themselves to a rail (Fig. 12) (Raubenheimer & Simpson, 1993), thus guiding their feeding behavior to ensure that their intake of protein and carbohydrate remains within a viable range for survival and reproduction. 

Fig. 12 The x-axis represents the amount of carbohydrates consumed (in equal, but undefined units) and the y-axis represents the amount of protein consumed (in equal, but undefined units) by different groups of locusts. The diagonal lines represent the rails that confine the locusts’ diet depending on how much of each they eat. Each x on the lines represents where the locust groups stopped eating to get as close to the intake target as possible, which is boxed. (Raubenheimer & Simpson, 1993) 

The study found that, although the groups of locusts closest to their intake target exhibited the most stable growth, locusts with an extremely unbalanced diet were able to thrive. From an evolutionary perspective, compensatory feeding confers locusts with a key advantage in unpredictable environments. Natural habitats often offer foods with unbalanced nutritional profiles, and the ability to dynamically adjust feeding behavior ensures locusts can still thrive in such conditions (Raubenheimer & Simpson, 1993). This flexibility increases their chances of survival, especially when food is in short supply or when new or fluctuating food sources are encountered. This behavior likely evolved over time as an adaptive response to the challenges of varied and often suboptimal food availability, helping locusts maximize the efficiency of their feeding strategy and adapt to diverse ecological niches. 

Metabolism through life cycle 

Locusts’ metabolism encourages the use of mathematical models to understand how metabolic rates scale with age, size, and activity type. Research on metabolic physiology in the migratory locust, Locusta migratoria, and American locust, Schistocerca americana, has been developed under the framework of scaling laws and energetical models to quantify the relationship among body size, metabolic rate, and energetic demands from different behaviors, like jumping. These studies enable a deeper understanding of how locusts manage energy from the start to the end of their life cycle, shifting from anaerobic to aerobic metabolism. 

Through the implementation of scaling laws, the alteration in locusts’ metabolic rate through growth can be found (Snelling et al., 2011). The major measures are M_RO2, or resting metabolic rate, and M_MO2, which is the maximum hopping metabolic rate. For this reason, the present study applies a power-law scaling approach, modeling metabolic rate as a function of body mass: M_RO2 ∝ M^0.97 and M_MO2 ∝ M^1.02, where exponents are derived from experimental data (Fig. 13).  

Fig. 13 On the graph to the left, the unfilled circles represent the relationship between body mass (M_b) and resting metabolic rate (M_RO2) adult locusts; the filled circles represent the relationship between body mass and juvenile maximum metabolic rate during hopping (M_MO2,juv); the M_MO2 of adult locusts are also presented as filled triangles. On the graph to the right, the filled circles represent the relationship between body mass (M_b) and the maximum metabolic rate of juvenile hopping muscle (M_MO2,juv,hop); the M_MO2,hop of adult locusts is also presented as filled triangles. (Snelling et al., 2011) 

These scaling laws imply that both M_RO2 and M_MO2 increase nonlinearly with body mass. Despite their overall smaller body size, juvenile locusts have proportionally higher energy expenditure per unit mass in high power activities like hopping. This suggests there is an ontogenetic shift in metabolic efficiency as they mature. 

Anaerobic versus aerobic production of energy for high power activity is integrated within the same mathematical framework. Consequently, during maximum hopping efforts, juvenile locusts are heavily dependent on anaerobic metabolism, which is generally less energy-efficient than the relatively slower-acting aerobic metabolism, leading to the production of lactic acid (Kirkton et al., 2005) (Fig. 14). The anaerobic contribution is modeled here as a fraction of total energy expenditure that decreases with age as the locust’s aerobic metabolic capacity improves. It is seen as a shift in metabolic efficiency, where juvenile locusts rely more on oxidative phosphorylation and less on glycolysis to meet their energy demand. 

Fig. 14 Lactate levels increased with age and time spent jumping in the first 2 min (N=8–10 at each point). Values are means. With instars being the life stages of a locust (2nd is the most juvenile to 6th being the most adolescent before becoming an adult) (Kirkton et al., 2005). 

Similarly, the application of mathematical models to analyze the differential characteristics of energy pathways according to developmental stages of the S. americana. Quantification of oxygen consumption and lactate production during jumping and a model describing the relative contributions of both aerobic and anaerobic metabolism at different life stages are presented (Fig. 15). Combining respirometry with mathematical modeling and showing that the nymphs rely almost exclusively on anaerobic metabolism for energy supply of short jumping bursts, with very little consumption of oxygen uptake. As locusts mature, they develop a more efficient tracheal system, and their aerobic respiratory capacity increases accordingly. Adults can maintain jumping activities for a longer period using a greater proportion of more efficient aerobic pathways with fewer by-products (Kirkton et al., 2005). 

Fig. 15 Whole body carbon dioxide emission during jumping across different instars of S. americana. Values are means (Kirkton et al., 2005). 

Both studies incorporate metabolic scaling equations to quantify the shift in energy pathways and the scaling of metabolic rates with size. They emphasize that the ontogenetic modification of locust physiology is complemented by quantifiable changes in metabolic resource allocation, with bigger and older locusts being more metabolically efficient during sustained activity.  

Energy consumption during flight 

Energy consumption during flight is of paramount importance in the understanding of the metabolic demands and survival strategies of locusts, especially during long-distance migrations or swarming events. In general, locusts depend on fat as the main energy source during flight; therefore, fat metabolism is an important aspect of their ability to sustain long periods of activity. By analyzing the combustion of fat and other substances, researchers can quantify how locusts manage their energy reserves and optimize flight efficiency. This ability to efficiently store and burn fat allows locusts to endure long flights in search of food or favorable environmental conditions, a key adaptation for survival in variable environments. 

The energy consumption of flying locusts, if the combustion of fat and non-fat material (the remainder) occurs during flight, can be calculated (Weis-Fogh, 1952). Some of the key variables used in these calculations include L = total lipids (fat) in a group R = total remainder (non-fat dry matter like proteins and carbohydrates) Q = total quantity combusted, fat and remainder other variables used to derive the formula include d = deviation from the mean percentage n = number of individuals v = summation of volume indices. Also required is the weight of the group (w) and Δ, the estimate of error in the initial amounts of fat and remainder. 

Since the flying and control groups were alike before flight, data from the latter group were used to estimate the initial amounts of fat, L, and remainder, R, in the flying group. Initial fat and remainder in the flying group were estimated from equations 1 and 2, respectively: 

(1)  

(2)  

Here, L₁ and R₁ are the initial amounts in the control group, and L₂ and R₂ are the estimates for the flying group. The error terms Δ_L and Δ_R reflect estimation uncertainties. 

The combustion of fat and remainder during flight was then quantified using: 

(3)  

(4)  

These calculations determine how much fat (Q_L), and remainder (Q_R) were metabolized during flight. 

These estimates suggest that locusts use mainly fat as their energy source during flight (Weis-Fogh, 1952). This is particularly important in long-distance migration, especially in swarming when the insects must maintain flight over a very long period. Fat is more energetic than proteins or carbohydrates per unit weight and is thus a superior fuel for flying. This also highlights how the locusts’ metabolic systems are tuned for energy conservation and endurance, enabling them to fly for long distances without depleting their energy reserves too quickly. Such energy efficiency is a prime adaptation that enables locusts to migrate over long distances and survive in hostile environments, thus ensuring their survival during periods of food scarcity or migration. 

Conclusion

The extraordinary adaptability of locusts demonstrates the interplay between biology, environmental factors, and mathematical optimization. Their ability to form massive swarms and undertake long flights because of efficient flight mechanics and wing structure has been modelled using mathematical algorithms to highlight their unique abilities (Isakhani et al., 2021). These design solutions demonstrate the importance of energy-efficient flight to their survival. Population dynamics further illustrate the role of mathematical modeling in understanding locust behavior. Sigmoidal functions linking population density to swarming probability emphasize that density-dependent feedback loops drive phase polyphenism (Collett et al., 1998). Mathematical predictions can be made about locust aggregation based on environmental cues, like wind, rain, and resource availability (Reynolds & Reynolds, 2012). These predictions are important tools for anticipating swarming events to help mitigate their effects especially in poorer and more vulnerable countries. Energy metabolism is another area where the locust demonstrates its remarkable design solutions through mathematical frameworks. Scaling laws and models of metabolic rate reflect the transition from energy-efficient states to the high metabolic demands of swarming. The ability to balance the energetic costs of swarming with the benefits of strength in numbers is shaped by evolutionary pressures. Integrating mathematical models in flight optimization, population dynamics, and energy metabolism helps us gain a deeper understanding of the factors driving locust adaptability, also offering tools to mitigate their destructive impacts and develop more effective management strategies. 

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