Swarming Dynamics: Locust Marching Hopper Bands

This is a post from a series in my project.

I’m sure most of us are familiar with this biblical reference: the locust plague. As it turns out, there is quite a bit of interesting mathematics behind the dynamics of swarm formation. I know farmers and governments are probably the first ones to fund this kind of research; locust swarms cost billions of dollars in lost crops. Swarms can be huge – on the order of millions of individuals – and they can cover hundreds of square miles per day. If you aren’t familiar with locusts, the youtube video below should clarify some things:

Locusts are strange, little creatures.

• a cross between a cricket and a grasshopper
• two main behavioral states: solitary and sociable
• sparse environments, with low population densities: solitary bugs
• overabundant nutrients, with high population densities: sociable bugs
• behavior change induces bodily changes (ew, they get bigger)
• Bigger wings, legs, and eyes. All this, to fly long distances for food. In other words, a swarm is born.

Of course, there are many factors that cause this phase change, but generally, there’s a high correlation between population density and swarming. This research paper by a slew of researchers led by Chad Topaz from Macalester investigated the conditions required for an outbreak; that is, the research team investigated what triggers a solitary locust to become sociable. The researchers also posed some strategies for locust management policies. Here are some statements from that research paper:

• The condition for an outbreak is expressed as a ratio of known biological parameters. It is a threshold population density of locusts. Densities less than this point lead to solitary populations. Densities greater than this point lead to swarms.
• It is easier to prevent the formation of a swarm (e.g., reducing growing populations of locusts by killing a few suckers off) than to control an existing swarm (e.g., killing individuals within a swarm).

The researchers cover more detail about the specifics of the model, stability analysis, and numerical simulations of their results in the official research paper than I do, so this post should just be a simple overview of what’s happening. Below is the model the researchers came up with to predict how locusts change their behaviors from being solitary to sociable and vice versa. Another name for this model is a reaction-diffusion model. It really just tries to capture how the density of solitary locusts $s(x,t)$ and the density of sociable (also known as gregarious) locusts $g(x,t)$ change in any given area.

$\dot{s}+\nabla\cdot(\mathbf{v_s}s)=-f_2(\rho)s+f_1(\rho)g$
$\dot{g}+\nabla\cdot(\mathbf{v_g}g)=f_2(\rho)s-f_1(\rho)g$

The left hand side of this system of equations are derivatives in time and space that describe the rate the locusts change behavior from solitary to sociable. The right hand side is what we look at to analyze the system at equilibrium, which reveals the long-term behavior of the system. Together, they form the full model.

For you chemistry whackos out there, you can think of this system of equations as a system of chemicals. There is a predefined amount of chemicals you pour into the system (a finite number of solitary and sociable locusts), and then you wait for a while for the reaction to happen. When it’s all over, you look inside your reaction jar (or whatever you hold chemicals in) and see how the reaction progressed and what’s left over at the end. What’s the relationship between solitary and sociable locusts? They add up to a total number of locusts:

$\rho(x,t) = s(x,t) + g(x,t)$

How do we know how fast (or slow) locusts change their behaviors from one phase to the next? We model them with equations that increase (or decrease) with increasing density – $f_1(\rho), f_2(\rho)$. If you’re into the specifics, this is what the terms mean:

• $\dot{s}$ is the rate of change of density of solitary individuals
• $\dot{g}$ is the rate of change of density of sociable (gregarious) individuals
• $\nabla\cdot(\mathbf{v_s}s)$, $\nabla\cdot(\mathbf{v_g}g)$ is the change in velocity of the solitary (gregarious) individuals multiplied by the density of solitary (gregarious) individuals
• $f_{1,2}(\rho)$ represent conversion rates from one phase to the other, which depend on density

One of the main assumptions that the authors make for their model is that the rate that locusts change to being solitary ($f_1$) decreases with increasing population density and that the rate of locusts changing to being sociable ($f_2$) increases with increasing population density. In fact, these rates are monotonically decreasing and increasing, respectively. A plot of $f_{1,2}$ may help visualize these trends.

When you set the differential equations to zero, you can find the equilibrium points $\phi_{s,g}$ of the model. What I mean is: rearrange these equations to get an expression for $s, g$. We’ll call these special points $s_0, g_0$.

$0=-f_2(\rho)s+f_1(\rho)g$
$0=f_2(\rho)s-f_1(\rho)g$
$\rho(x,t) = s(x,t) + g(x,t)$

$s_0 = g_0\frac{f_1}{f_2}$

$g_0 = \frac{\rho}{\frac{f_1}{f_2} +1}$

In other words, at these points, the locust densities are stable in their behavioral phase (they settle in either solitary or sociable behaviors). A plot of this situation may clarify:

As you can see, when population density increases (the $x$ axis), the number of initially solitary locusts (the $y$ axis)decreases because they change their behavior to be sociable.

After a significant amount of algebra and transformations (irrelevant for us), the authors concluded that the conditions for an outbreak of a swarm are based on a ratio of known biological parameters.

$\phi_g + \phi_s = 1$
$\phi_g = \frac{g_0}{s_0+g_0}$
$\phi_g>\phi_g^*=\frac{R_sr_s^2}{R_sr_s^2-R_gr_g^2+A_ga_g^2}$

What this relation is trying to express is that once the density of sociable insects $\phi_g$ becomes greater than the equilibrium point $\phi_g^*$, the locusts begin to swarm. The researchers found a precise way to calculate this density threshold (above), where:

• $R_{s,g},A_g$ are the interaction amplitudes that determine the strengths of attraction to other insects
• $r_{s,g},a_g$ are the interaction length scales that represent typical distances over which one locust can sense and respond to another

The reason why the authors think that controlling locust populations before they swarm is easier than controlling an active swarm is because of the following plot. It’s a representation of how stability changes as the density of the locust population changes:

The solid red line represents stable solitary phase, and the dashed red line shows unstable solitary behavior. The green line is stable sociable behavior. The arrows indicate that as you increase locust density $\rho_0$, the population is stable in the solitary phase. However, once you move past the point marked with an asterisk, stability jumps from the solitary phase to the sociable phase. After this point, no matter how much you reduce the locust population, the individuals will continue to tend towards swarming behaviors. The technical name for this phenomenon is called hysteresis.

Numerical simulations

Numerical simulations of the model show some pretty neat results. They are meant to find the time elapsed until a swarm breaks out. You can see how the population is initially distributed between solitary and sociable insects, and as time goes one, how the distribution changes to favor swarming dynamics. For a certain set of parameters (chosen to illustrate a specific case), this is what they observed:

The dashed lines are solitary individuals and the solid lines are the sociable ones. You can see that in a matter of hours (about 3.45 hours), the dynamics change dramatically. At the end, you are just left with a travelling band of gregarious insects; no more solitary ones.

The findings of this paper have some neat implications for locust management that could save a big chunk of money for farmers and governments. Due to the sudden stability changes after a certain density threshold (hysteresis), it’s clear that the wise thing to do is to keep an eye on existing locust population densities so that they don’t pass that critical density. The plot of hysteresis shows that even if you were to reduce a population of millions of locusts in a swarm to close to zero individuals, these individuals would still want to swarm. That is a disheartening observation, but it’s possible to prevent this situation from happening if management techniques were focused on prevention rather than control of the swarms.

I’m always amazed at how advances in mathematics and new findings can lead us to unraveling some of nature’s mysteries. It makes things like ominous biblical plagues a lot less intimidating. To me, that’s the real value of math. It gives us the power and capability to begin to understand how seemingly impossibly complicated events happen.

(Moses from Charlton Heston’s Ten Commandments)