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\begin{document}

\title{Desert locust dynamics: Behavioral phase change and swarming}

\author{Chad Topaz\affil{1}{Macalester College, St. Paul, U.S.A.},
Maria D'Orsogna\affil{2}{California State University, Northridge, U.S.A.},
Leah Edelstein-Keshet\affil{3}{University of British Columbia, Vancouver, Canada},
\and
Andrew Bernoff\affil{4}{Harvey Mudd College, Claremont, U.S.A.}}

\contributor{Submitted to Proceedings of the National Academy of Sciences
of the United States of America}

\maketitle

\begin{article}
\begin{abstract}
The African Locust \emph{Schistocerca gregaria} is known to have two
interconvertible phases, solitarious and gregarious. Individuals are 
repelled (attracted) by others depending on their solitarious (gregarious)
state, and crowding tends
to bias conversion towards the gregarious form. Here we model the
spatio-temporal interactions and transitions between the two
phases using nonlocal (integro-partial) differential equations. We
use steady state and linear stability analysis to characterize the 
conditions for onset of a locust ``plague'', characterized by mass transition to the 
gregarious form. Model reduction to approximate ``bulk'' theory allows us
to quantify the size and density of the emergent gregarious clusters, and
numerical simulations provide qualitative descriptions
of the swarm structure.
\end{abstract}

\keywords{locust | phase change | swarm | gregarious | solitarious}

\abbreviations{}

\dropcap{O}utbreaks of the desert locust \emph{Schistocerca gregaria} 
regularly afflict vast areas of northern Africa, the Middle East and
south-west Asia. Depending on climate and vegetation
conditions, billions of voracious locusts aggregate into
destructive swarms that span areas up to a thousand square
kilometers. A locust swarm can travel a few hundred km per day, 
stripping crops and vegetation in its destructive path.
\cite{Ken1951,Alb1967,Uva1977,Rai1989}.  The latest locust plague
in West Africa (2003--2005) severely disrupted 
agriculture, destroying \$2.5 billion in crops destined for both 
subsistence and export. Despite control efforts totalling \$400 million,
 loss rates escalated to 50\% in certain regions \cite{Bel2005,BraDjiFay2006}.
These numbers alone attest to the urgency of finding better ways to
predict, manage, and control locust plague outbreaks.
%Outbreaks are difficult to predict and
%depend on several environmental, geographical and climatic factors.

Between plagues, locusts are mainly solitary creatures who live
in arid regions and lay eggs in small breeding grounds lush with
vegetation. 
%, usually after seasonal rains or due toirrigation. 
Occasionally, resources are
abundant enough to support numerous hatchings, leading to a
high density of adults. Overcrowding at resource sites promotes
a transition to the gregarious phase, in a self-reinforcing process.
The available supply of vegetation and water at the
breeding ground is eventually exhausted, and locusts migrate
\textit{en masse} to other locations in search for nourishment. Newly
settled feeding grounds are also inevitably depleted, leading to
several stop-and-go cycles of traveling locust bands. This
phenomenon may be exacerbated in periods of drought, when large
numbers of locusts congregate on the same breeding or feeding
grounds.

Desert locusts are \emph{phase polyphenic:} while sharing the
same genotype, individuals may display different phenotypes
\cite{AppHei1999,PenSim2009} such as morphology \cite{Dir1953},
coloration \cite{IslRoeSim1994}, reproductive features
\cite{SchAlb1999} and most significantly behavior \cite{SimMcCHag1999,
  RogMatDes2003}. The latter can change from \emph{solitary} where
locusts seek isolation to \emph{gregarious} where they attract
one another. Behavioral state is plastic
\cite{Uva1977,SimMcCHag1999,PenSim2009} and strongly dependent on
local population density: in sparse surroundings, a gregarious
locust  transforms into the solitary state \cite{SimMcCHag1999} and vice versa in
crowded environments. It is this phenotype switch that takes
place when large numbers of locusts gather at a
site, promoting a massive change from solitary to
gregarious form and initiating the swarm plague\cite{Des2000,ColDesSim1998}.

Locust gregarization is induced by
tactile, visual and olfactory cues. One of the most potent stimuli has
been shown to be the regular touching of an individual's hind legs by
another \cite{SimMcCHag1999,SimDesHag2001,RogMatDes2003}. Although
other areas of the body such as mouth, face, and abdomen may be
self-activated while performing common tasks such as walking, feeding
or grooming \cite{SimDesHag2001}, the hind femora are touched only by
other insects in the field so that intense leg rubbing may function as
an indicator of crowding conditions.  The mechanosensory stimulation
of leg nerves leads to subsequent serotonin cascades in the brain, and
in turn to the onset of gregarious behavior
\cite{SimDesHag2001,RogMatDes2003,AnsRogOtt2009}. Furthermore, contact
with individuals coming from behind enhances the tendency of a
stationary locust to move \cite{BazBuhHal2008}.  It has been shown
that the solitary to gregarious switch can be induced by rubbing a
locust's hind leg for 5 seconds per minute during a period of four
hours \cite{SimDesHag2001}. Similarly, cessation of physical contact
leads to a transition back to the solitary state after four hours.

There have been relatively few mathematical studies of locust
behavior, especially of the switch between solitary and gregarious
phases in a spatio-temporal framework. Both
\cite{EdeWatGru1998} and \cite{TopBerLog2008} studied the
dynamics of rolling patterns formed by migrating locust groups. Data
driven models include the self-propelled particle model of
\cite{BuhSumCou2006} where the observed transition between
disordered to ordered locust movement was described via well-known
physics paradigms \cite{VicCziBen1995}. A logistic map was introduced
in \cite{HolChe1996} to describe population switching via a
birth rate and a carrying capacity dependent on population density
modulated by stochastic effects.

None of these studies have focused on how an initially disperse,
solitary locust population transitions to an aggregated, gregarious
one. The goal of this paper is to provide a mathematical description
of such behavioral phase changes by taking into account the most
relevant biological findings and by including relevant spatial features
that cause locusts to clump or disperse. OUR MAIN FINDING IS...
The paper is organized as
follows/our main results/conclusions.

\section{The model}

Individuals in a swarm are subject to attractive or repulsive forces
resulting either from direct sensory communication or via the indirect
signaling provided by chemical or vibrational sources
\cite{Bre1954,DOrChuCha2006, MogEdel1999}. While communications may be
unidirectional (\emph{e.g.}, narrow sight) or omnidirectional
(\emph{e.g.}  hearing), many organisms process a combination of
inputs, effectively resulting in omnidirectional
sensing~\cite{EftVriLew20070,ParMar2005}. Social forces are typically
cast in the form of superimposed pairwise interactions, although
many-body choices are also possible. In this work we will consider a
traveling swarm that is well-described by a continuum population
density $\rho(\vec{x},t)$ subject to social interactions. The
continuum assumption holds well in the context of locust gregorization
due to the large population numbers involved, both in the gregarious
and solitary phases (IS THIS TRUE?).  A simple model commonly used 
is

\begin{eqnarray}
\label{eq:social}
\rho_t + \nabla \cdot (\rho \vec{v}) = 0, \quad \vec{v} = -
\int_\Omega {\nabla \vec{Q}}(\vec{x}-\vec{x}')  \rho (\vec{x}',t) \,
d\vec{x}' 
%-\nabla Q*\rho.
\end{eqnarray}


where the velocity $\vec{v}$ is determined via a convolution with the
social interaction field $Q(\vec x - \vec x')$ that describes the
influence of the population at location $\vec{x'}$ on that at location
$\vec{x}$. For notational simplicity we denote the convolution via an
asterisk, so that $\vec v = - \nabla Q * \rho$.  If sensing is
omnidirectional then $\vec{Q}$ is radially symmetric. Eq. \ref{eq:social} has been studied in
\cite{BerLau2007,BodVel2005,BodVel2006} for specific choices of
$Q$, and in a two-dimensional setting in \cite{TopBer2004}.  [NOTE: THIS PREV. SENTENCE IS NOT SO WELL WRITTEN, AND ALSO, THERE ARE A LOT OF OTHER PAPERS, ALBEIT THEORETICAL, THAT STUDY THIS EQ.] Generally,
the dynamics arising from \ref{eq:social}
include at least three different asymptotic behaviors:
spreading, steady state, and blow-up \cite{LevTopBer2009}.  The
compactly supported steady states, which may include jump
discontinuities at the swarm's edges, are studied in depth in
\cite{BerTop2011} in one spatial dimension.

In the context of biphasic desert locust swarms we denote 
the density of solitary and gregarious locusts 
by $s(\vec{x},t)$ and $v(\vec{x},t)$ respectively, where 
$\vec{x}= (x,y)$. We also denote the total density by
$\rho = s +g$.  Extending Eq. \ref{eq:social} to include switching transitions between the
solitary and gregarious phases we write

\begin{subequations}
\label{eq:ge}
\begin{alignat}{4}
\dot{s} &+ \nabla \cdot (v_s s)  &=  -f_2(\rho)s &+ f_1(\rho)g, \\
\dot{g} &+ \nabla \cdot (v_g g) &=  \phantom{-}f_2(\rho)s &-
f_1(\rho)g,  
\end{alignat}
\end{subequations}
where the velocities are given by
\begin{equation}
\label{eq:v}
v_s = -\nabla(Q_s * \rho), \quad v_g  -\nabla(Q_g * \rho).
\end{equation}

Since solitarious locusts avoid crowds, they should display social
repulsion. Gregarious locusts should display attraction coupled with
short range repulsion, in order to form finite aggregations. The
solitarious potential $Q_s$ and the gregarious potential
$Q_g$ are
\begin{equation}
\label{eq:Q}
Q_s(\vec{x}) = R_s \exp{-|\vec{x}|/r_s}, \quad Q_g(\vec{x}) = R_g \exp{-|\vec{x}|/r_g} - A_g \exp{-|\vec{x}|/a_g}.
\end{equation}
The parameters in $Q_g$ must be chosen in the clumping regime, per the
conditions stated in \cite{LevTopBer2009}. Specifically, we require $R_g a_g - A_g r_g > 0$ so that repulsion
dominates at short length scales, and $A_g a_g^2 - R_g r_g^2>0$ so that attraction dominates at longer length scales.

Now consider the density dependent rates of phase change. Thesolitarization rate $f_1$ should be high when local density is low,and should approach zero when density is high. Conversely, thegregarization rate $f_2$ should approach zero when local density low,and should be significant when density is high. Adopt the functionalforms

\begin{equation}
\label{eq:rates}
f_1(\rho) =  \frac{\delta_1}{1+ \left( \rho/k_1 \right)^2}, \quad f_2(\rho) =  \frac{\delta_2 \left( \rho/k_2 \right)^2}{1+ \left( \rho/k_2 \right)^2}.
\end{equation}

\section{Biological Parameter Values}

We draw upon the experimental results of \cite{SimDesHag2001} to estimate phase change parameters in (\ref{eq:rates}). Phase change from the gregarious to solitarious state takes approximately four hours, and vice versa. Thus, $\delta_1 = \delta_2 = 1/4 = 0.25\ h^{-1}$. The critical density for gregarization is $50$ - $80$ locusts /$m^2$. We assume that the solitarization process has the same critical density, and we set the characteristic density transition scales to be $k_1=k_2= 65$ locusts /$m^2$.

To estimate social interaction range parameters in (\ref{eq:Q}), we apply results of \cite{BuhSumCou2006,BazRomTho2011}, which identify the ``sensing range'' of a locust as $0.14$ $m$, and the ``repulsion range'' as $0.04$ $m$, on order with the approximately $0.08$ $m$ body length of a mature individual. We set our repulsion length scales $r_s = r_g = 0.04$ $m$, corresponding to the experimental repulsion range. Since attraction typically operates over longer scales, we take $a_g = 0.14$ $m$, corresponding to the experimental sensing range.

We find $R_s$, $R_g$, and $A_g$ from explicit computations of velocity under our model. The speed of a locust when it is alone varies from $72$ - $216$ $m/hr$, depending on diet \cite{BazRomTho2011}. At the upper end, this is roughly 1 body length per second. When it is moving in a group, the speed varies in a tighter range of $144$ - $216$ $m/hr$ \cite{BazRomTho2011}. To estimate $R_s$ we imagine the hypothetical semi-infinite density field $\rho(x) = \rho_0 \Heaviside(x)$. Here, $\Heaviside(x)$ is the Heaviside function and $\rho_0 = 65$ locusts /$m^2$, the approximate critical density of a gregarious group. A solitarious locust placed at the origin, at the swarm's edge, should move to the left with maximal velocity $v_s^{max} = -216$ $m/hr$. We use the velocity definition Eq. \ref{eq:v} to write
\begin{equation}
v_s(0,0) = -\partial_x \left\{Q_s * \rho_0 \Heaviside(x)\right\}\big|_{(0,0)} = v_s^{max},
\end{equation}
which we solve to find $R_s = 11.87$ locusts$\cdot m^2/hr$. Similarly, a gregarious locust at the origin should move to the right with maximal velocity $v_g^{max} = 216$ $m/hr$, and so
\begin{equation}
v_g(0,0) = -\partial_x\left\{Q_g * \rho_0 \Heaviside(x)\right\}\big|_{(0,0)} = v_g^{max}.
\end{equation}
A gregarious locust placed a short distance away from the swarm equal to the attraction length scale should also move to the right, but with a slower velocity which we take to be the minimal velocity in a crowd, $v_g^{min}=144$ $m/hr$. Thus
\begin{equation}
v_g(-0.14) = -\partial_x\left\{Q_g * \rho_0 \Heaviside(x)\right\}\big|_{(-0.14,0)} = v_g^{min}.
\end{equation}
Together, these two conditions determine $R_g = 5.13$ locusts$\cdot m^2/hr$ and $A_g = 13.33$ locusts$\cdot m^2/hr$.

\section{Steady states and linear stability}

There is a family of homogeneous steady states parameterized by the total homogeneous density $\rho(x,t) = \rho_0$. For any given $\rho_0$, the homogeneous steady state is

\begin{subequations}
\label{eq:steadystate}
\begin{eqnarray}
s_0 & = & \frac{\rho_0 \delta_1 k_1^2 (k_2^2+\rho_0^2)}{\delta_1 k_1^2 k_2^2 + \delta_1 k_1^2 \rho_0^2 + \delta_2 k_1^2 \rho_0^2 +\delta_2 \rho_0^4} \label{eq:s0} \\
g_0 & = & \frac{\delta_2 \rho_0^3 (k_1^2+\rho_0^2)}{\delta_1 k_1^2 k_2^2 + \delta_1 k_1^2 \rho_0^2 + \delta_2 k_1^2 \rho_0^2 +\delta_2 \rho_0^4}.
\end{eqnarray}
\end{subequations}

In the small $\rho_0$ limit,

\begin{equation}
s_0 \approx  \rho_0 - \frac{\delta_2}{\delta_1 k_2^2} \rho_0^3, \quad g_0 \approx   \frac{\delta_2}{\delta_1 k_2^2} \rho_0^,
\end{equation}

and thus the low-density steady state is composed mostly of solitarious locusts. In the large $\rho$ limit,

\begin{equation}
s_0 \approx \frac{\delta_1 k_1^2}{\delta_2} \frac{1}{\rho_0}, \quad  g_0 \approx \rho_0 - \frac{\delta_1 k_1^2}{\delta_2} \frac{1}{\rho_0},
\end{equation}

and thus the high-density steady state is composed mostly of gregarious locusts.

We now calculate the linear stability of the homogeneous steady state.

The steady states are shown in Fig. \ref{fig:steadystate}. We imagine a a uniform distribution around each parameter. The distribution is centered at the value for each that we have estimated from biological data. The range of the uniform distribution goes from 0.7 to 1.3 times this mean value. We draw 10,000 parameter sample parameter sets. On each plot, you see three sets of blue curves, and these correspond to the 25, 50, and 75 percentile levels of the steady state value of $s$. The three green curves are similar, but for $g$.

One interesting thing to note is that$s_0$ is not monotonically increasing with $\rho_0$. Rather, there is a global maximum. From a biological point of view, I think this is a very nice feature of the model. It means that under our model, nature only supports up to a particular density of homogeneous solitary groups. Because of the form of $f_{1,2}$, there is no closed form solution for the coordinates of the maximum. However, there is an easy closed form solution if $k_1 = k_2 = k$ and $\delta_1 = \delta_2 = \delta$, as is the case for our biologically estimated parameters. This solution is $(\rho_0,s_0)=(k,k/2)$.

We can look for an approximate solution when $k_{1,2}$ and $\delta_{1,2}$ are each slightly detuned from equality. The basic idea is to expand everything in a power series in a small parameter $\epsilon$. Take the formula for $s_0$ from Eq. \ref{eq:s0}, differentiate it, and set it equal to zero to look for the critical point. This is our governing equation. Then expand everything in a power series. Without loss of generality, we can do this as 
\begin{subequations}
\begin{gather}
k_1 = k + \epsilon K, \quad k_2 = k - \epsilon K, \\
\delta_1 = \delta + \epsilon \Delta, \quad \delta_2 = \delta - \epsilon \Delta, \\
\rho_0 = \rho_{00} + \epsilon \rho_{01}.
\end{gather}
\end{subequations}
Substituting the power series into the equation for $s^\prime$ and solving at $\mathcal{O}(1)$ and $\mathcal{O}(\epsilon)$ yields
\begin{equation}
\rho_{00} = k, \quad \rho_{01} = K + \frac{\Delta k}{\delta}.
\end{equation}
Thus, the maximum solitarious density occurs at $\rho_0 \approx k + K + \Delta k/\delta$. Substituting back into the original formula for $s_0$ and keeping through $\mathcal{O}(\epsilon)$ gives us the maximum steady state solitarious density, which is $s_{max} \approx k/2 + \Delta k/(2\delta)$.

I've checked how good this approximation is vis-a-vis the parameter sensitivities. That is, I've set $k = 65$ and $\delta = 0.25$. I've let the deviation $K$ be as large as $0.3 k$ and the deviation $\Delta$ be as large as $0.3 \delta$ -- that is, I've considered up to 30\% deviation from the mean values. Comparing the exact (numerical) values for the critical point's coordinates to the approximate values, you get up to 20\% error for the $\rho_0$ coordinate and 15\% error for the $s$ coordinate.

Another interesting feature of the graph of $s_0$ and $g_0$ is that there is a point of equality between the two curves. By setting $s_0 = g_0$ in Eq. \ref{eq:steadystate} and throwing away the trivial case $\rho_0=0$, one finds one positive solution, which is that $s_0=g_0$ when
\begin{equation}
\rho_0 = \frac{k_1}{2\delta_2} \Bigl( \delta_1-\delta_2  + \sqrt{(\delta_1 - \delta_2)^2 + 4 \delta_1 \delta_2 (k_2/k_1)^2} \Bigr).
\end{equation}
This solution is exact (no assumptions or approximations are necessary). In the white and grey region of Fig. \ref{eq:steadystate}, $s_0 > g_0$ at the 50th percentile, and the reverse is true in the pink region. The solid, vertical purple lines lines show the 25th and 75th percentile of the equality point. 

We can also calculate the linear stability of the steady state solution. Let
\begin{equation}
\label{eq:pert1}
s(\vec{x},t) = s_0+s_1(\vec{x},t), \quad g(\vec{x},t) = g_0+g_1(\vec{x},t).
\end{equation}
Here, $s_1$ and $g_1$ are perturbations to the steady state. Also define
\begin{equation}
\label{eq:pert2}
\rho(\vec{x},t)=\rho_0+\rho_1(\vec{x},t) =s_0+g_0+ s_1(\vec{x},t)+g_1(\vec{x},t).
\end{equation}
Substitute Eq. \ref{eq:pert1} and Eq. \ref{eq:pert2} into Eq. \ref{eq:ge}, assume the perturbations $s_1$ and $g_1$ are small, \emph{i.e.}, ${\cal O}(\epsilon)$, and Taylor expand through ${\cal O}(\epsilon)$ to obtain the linearized equations
\begin{subequations}
\label{eq:linearized}
\begin{eqnarray}
\dot{s}_1 & = & s_0 Q_s * \nabla^2 (s_1+g_1)  - A s_1 + B g_1, \\
\dot{g}_1 & = & g_0 Q_g * \nabla^2 (s_1+g_1) + A s_1 - B g_1
\end{eqnarray}
\end{subequations}
where
\begin{subequations}
\begin{eqnarray}
A & = & f_2(\rho_0)+f_2^\prime(\rho_0)s_0-f_1^\prime(\rho_0)g_0\\
B & = & f_1(\rho_0)+f_1^\prime(\rho_0)g_0-f_2^\prime(\rho_0)s_0.
\end{eqnarray}
\end{subequations}
Since $f_1$ is monotonically decreasing and $f_2$ is monotonically increasing, $f_1^\prime(\rho_0) < 0$ and $f_2^\prime(\rho_0) > 0$ (excluding the trivial possibility $\rho_0$=0$)$. Hence, $A,B>0$.

To further analyze the equations, Fourier expand the perturbations as
\begin{equation}
s_1(\vec{x},t) = \sum_\vec{k} \mathcal{S}_\vec{k} (t) \exp{i \vec{k} \cdot \vec{x}}, \quad s_2(\vec{x},t) = \sum_\vec{k} \mathcal{G}_\vec{k} (t) \exp{i \vec{k} \cdot \vec{x}}.
\end{equation}
The admissible wave vectors $\vec{k}$ depend on the chosen domain $\Omega$. In the event that $\Omega = \mathbb{R}^n$ then all $\vec{k}$ are admissible (per the Fourier transform). Substituting into Eq. \ref{eq:linearized} yields a set of ordinary differential equations for each Fourier mode amplitude. These are conveniently written in matrix form. Dropping the subscripts on $\mathcal{S}$ and $\mathcal{G}$ for simplification of notation, we have
\begin{subequations}
\begin{gather}
\frac{d}{dt}
\begin{pmatrix}
\mathcal{S} \\\mathcal{G}
\end{pmatrix}
=
\mat{L}(k)
\begin{pmatrix}
\mathcal{S} \\ \mathcal{G}
\end{pmatrix}, \\
\mat{L}(k) \equiv \begin{pmatrix}
-s_0 k^2 \wh{Q}_s(k) - A & -s_0 k^2 \wh{Q}_s(k)+B \\ -g_0 k^2  \wh{Q}_g(k) + A & -g_0 k^2  \wh{Q}_g(k) - B
\end{pmatrix}.
\end{gather}
\end{subequations}
Here, $k = |\vec{k}|$ is the perturbation wave number, and $\wh{Q}_{s,g}(k)$ are the Fourier transforms of the social interaction potentials, that is, 
\begin{subequations}
\label{eq:qhats}
\begin{eqnarray}
\widehat{Q}_s(k)& =&  2\pi \frac{R_s r_s^2}{(1+r_s^2 k^2)^{3/2}}, \\
\widehat{Q}_g(k) & = & 2\pi \left[ \frac{R_g r_g^2}{(1+(r_g^2k^2)^{3/2}}-\frac{A_g a_g^2}{(1+a_g^2 k^2)^{3/2}}\right].
\end{eqnarray}
\end{subequations}

The eigenvalues of $\mat{L}(k)$ are
\begin{equation}
\lambda_1(k) = -k^2 \left[s_0 \wh{Q}_s(k) + g_0 \wh{Q}_g(k)\right], \quad \lambda_2 = -(A+B).
\end{equation}
The eigenvalue $\lambda_2$ is negative since $A,B>0$. Thus, the constant density steady state is stable if $\lambda_1(k)<0$ for all admissible $k$. If $\lambda_1(k)>0$ for some $k$, then the constant density steady state is unstable to perturbations of those wave numbers. %Factoring out a power of $-k^2$, the instability condition, then, is \begin{equation} s_0 \wh{Q}_s(k) + g_0 \wh{Q}_g(k) < 0 . \end{equation}
Crucially, this pivotal eigenvalue $\lambda_1(k)$ does not depend on the functions $f_{1,2}$ which describe behavioral phase change. It only depends on the social interaction potentials and on the relative amounts of solitary and gregarious locusts.

We can further analyze this eigenvalue. The possible unstable wave numbers $k$ are those satisfying $\lambda_1(k)>0$, where
\begin{eqnarray}
\label{eq:lambda1orig}
\lefteqn{\lambda_1(k)  =  - 2 \pi k^2 \times }\\
& & \left[\frac{s_0 R_s r_s^2}{(1+k^2 r_s^2)^{3/2}} +
\frac{g_0 R_g r_g^2}{(1+k^2 r_g^2)^{3/2}} -
\frac{g_0 A_g a_g^2}{(1+k^2 a_g^2)^{3/2}}
\right] \nonumber
\end{eqnarray}
where we have substituted for $\widehat{Q}_g(k)$ and $\widehat{Q}_s(k)$.

In order to make analytical progress we set $k_1 = k_2 = h$ and
$\delta_1 = \delta_2 = \delta$. The use of $h$ is to avoid confusion
with the wavelength $k$. We also use the fact that $r_s=a_g$. These equalities all hold for our biologically estimated parameters. Later
it will be useful to recall that within our model $r_g < r_s$.
Inserting these simplifications, we find
\begin{equation}
\label{eq:lambda1mod}
\lambda_1(k) = \frac{2\pi R_g \rho_0^3 r_g^2 k^2}{(h^2+\rho_0^2)(1+k^2 r_s^2)^{3/2}} 
\left[M - m(k)
\right] 
\end{equation}
where
\begin{equation}
M = \frac{\rho_0^2 A_g r_s^2 - h^2 R_s r_s^2}{R_g \rho_0^2 r_g^2}
\end{equation}
and
\begin{equation}
m(k) = \frac {(1+k^2 r_s^2)^{3/2}}{(1+k^2 r_g^2)^{3/2}}.
\end{equation}

Since $m(k) > 0$, for instability to occur we must
require $M > 0$, otherwise $\lambda_1(k)$ will always be negative. This
is not a sufficient condition, but it is a necessary one and translates
to
\begin{equation}
\rho_0 > \sqrt{\frac{R_s}{A_g}} h
\end{equation}
which in our case for $R_s = 11.87, A_g= 13.3$ and $h=65$ implies that
if $\rho_0 < = 61.4$ the system is stable to perturbations of all wave numbers.

Let us now look
at the terms inside the square brackets of Eq.~\ref{eq:lambda1mod}.  We know
that $r_g < r_s$. This implies that the function $m(k)$ is
monotonically increasing. Hence, for $\lambda_1(k)$ to be positive, it is
also necessary that $M$ be greater that the minimum value of $m(k)$
which is attained at $k=0$. Thus, a more stringent condition is that
\begin{equation}
M > m(k=0) = 1.
\end{equation}
Using the fact that $m(k \to \infty) = r_s^3/ r_g^3 > 1$
we can now distinguish three cases:
\begin{itemize}
\item if $M < 1$ then $\lambda(k)<0$ and the homogeneous steady state is stable to perturbations of all wave numbers.
\item if $1 \leq M \leq r_s^3/r_g^3$ then
instabilities will arise for perturbations with wave numbers in a band extending from $k=0$ to some finite $k$.
\item if $M > r_s^3/r_g^3$ then the system is unstable to perturbations of \emph{any} wave number.
\end{itemize}

The condition $M > 1$ to guarantee instability can be rewritten as
\begin{equation}
\rho_0 > \sqrt{
\frac{R_s}{A_g - R_g \frac{r_g^2}{r_s^2}}} h
\end{equation}
so that, we are now guaranteed instability as long as $\rho_0 >
61.4$, close to the 50th percentile value of $\rho_0=57.5$ in Fig. \ref{fig:steadystate}, which is the left border of the grey region. [NOTE FROM CHAD: WHY AREN'T THEY CLOSER??? I NEED TO FIGURE OUT...] The vertical black dashed lines indicate the 25th and 75th percentile values for the onset of instability. Note that at the 50th percentile level, instability occurs before $g_0$ overtakes $s_0$ in value.

Fig. \ref{fig:stability} shows the maximally unstable wave number $k_{max}$ as a function of $\rho_0$. Interestingly, there is a sharp transition region in which $k_{max}$ grows rapidly, but it doesn't grow without bound. Rather, it levels off to a plateau. The three curves in the figure are the 25th, 50th, and 75th percentile values. At the 50th percentile value the large $\rho_0$ asymptotic value of $k_{max}$ is $k_{max} = 8.9$.

In the source file, there is text below this line that has been commented out, but that contains a discussion of the existence of a maximally unstable wave number (though we cannot actually compute that number analytically because of the algebraic form of things. I wasn't sure whether or not we want to include this, so I commented it out, but we can change this.


%\section{Maximum Instability}
%\noindent
%Finally, if we are interested in the most unstable wavelength, in the
%case where $M > 1$, we can calculate $\lambda'(k)$ to find
%
%\begin{eqnarray}
%\lambda'(k) = \frac{2 \pi R_g \rho_0^3 r_g^2 k}{h^2+\rho_0^2} 
%\left[M \frac{2 - r_s^2 k^2}{(1 + r_s^2 k^2)^{5/2}}- 
%\frac{2 - r_g^2 k^2}{(1+r_g^2 k^2)^{5/2}} \right].
%\end{eqnarray}
%
%\noindent
%
%Note that the two terms in the parenthesis contain the function
%
%\begin{eqnarray}
%g(y) = \frac{2 - y^2}{(1 + y^2)^{5/2}}
%\end{eqnarray}
%
%\noindent
%so that finding the maximum corresponds
%to setting the term in parenthesis to zero and
%finding values of $k=k^*$ so that
%
%\begin{eqnarray}
%\label{tosolve}
%M g(r_s k^*) = g(r_g k^*)
%\end{eqnarray}
%
%\noindent
%The $g(y)$ function is such that $g(0) = 2$, $g(y \to \infty) = 0$, it
%intersects the $y$ axis at $y = \sqrt{2}$ initially decreases and
%attains a minimum at $y = 2$.  We note that this means, specifically,
%that at $k=0$ the function $M g(r_s k=0) > g(r_g k=0)$ since $M > 1$
%in this case.  Hence, $omega'(k)$ starts out positive for small $k$.
%On the other hand, the function $M g(r_s k)$ will be zero at the
%location $k_s =\sqrt{2}/r_s$, whereas the function $g(r_g k)$ will be
%zero at the location $k_g = \sqrt{2}/r_g$. We also know that $k_g >
%k_s$, due to our choice $r_s< r_g$. In particular, this means that for
%$k=k_s$ $M g(r_s k =r_s k_s) = 0$ but $g(r_g k=r_g k_s)$ is still
%positive, because it will become zero for larger $k=k_g$.
%
%These facts lead us to conclude that $M g(r_s k = r_sk_s) = 0 
%< g(r_g k = k_s)$, and that there must exist a point
%$0 < k^*< k_s$  where the two terms are equal. In particular,
%for the choice of $r_s = 0.14$ this leads to $k^* < 10$.
%The value of $k^*$ is the value of $k$ that leads to the fastest
%growing perturbation. There is another solution to
%Eq.\,\ref{tosolve},but that will represent a minimum.
%
%
%\begin{figure}[t]
%\label{Fig1}
%\begin{center}
%\includegraphics[width=5.3in, angle=0]{Chad.pdf}
%\caption{The magenta curve is the function $g(r_g k)$
%while the blue curve is the $M g(r_s k)$ curve, where $M > 1$.
%Here, we set $M=5$, $r_g = 0.04$ and $r_s = 0.14$. Note that
%as predicted, the two meet at a value $k^* < \sqrt{2}/r_s$
%which is the point that makes the curve $M g(r_s k) = 0$.
%In this case, $k^* \simeq 6.02$}
%\end{center}
%\end{figure}




\section{Bulk theory for segregated states}



In simulations of (\ref{eq:ge}), we observe mass-balanced states in which gregarious and solitarious locusts spatially segregate. We attempt a rough calculation of such solutions. The solitarious locusts are spread throughout most of the domain $\Omega$, covering an area we refer to as $\alpha_s$. The gregarious locusts are concentrated in a clump whose area we call $\alpha_g$, which may be estimated from the gregarious potential; see (\ref{eq:Q}). For convenience, define the total number of solitary locusts and gregarious locusts,

\begin{equation}
S =  \int_{\Omega} s\,d\Omega, \quad G =  \int_{\Omega} g\,d\Omega,
\end{equation}

and the total population mass $M = S+G$.

The local densities that solitarious and gregarious locusts sense in their respective patches are

\begin{equation}
s = S/\alpha_s, \qquad g = G/\alpha_g.
\end{equation}



For a segregated state at mass balance, the number flux of gregarious locusts becoming solitarized per unit time is $G f_1(G/\alpha_g)$. Similarly, the number flux of solitarious locusts becoming gregarized is $S f_2(S/\alpha_s)$. Equating these expressions and substituting from (\ref{eq:rates}) yields

\begin{equation}
\label{eq:massbalance1}
\frac{\delta_2 S^3}{\alpha_s^2 k_2^2+S^2} = \frac{\delta_1 k_1^2 \alpha_g^2 G}{\alpha_g^2 k_1^2+G^2}.
\end{equation}



To find the mass-balanced states, we must solve (\ref{eq:massbalance1}). To simplify this calculation, we define the soliatious and gregarious mass fractions

\begin{equation}
\label{eq:whsg}
\phi_s = S/M, \qquad \phi_g = G/M,
\end{equation}

so that

\begin{equation}
\phi_s + \phi_g = 1.
\end{equation}

Substituting (\ref{eq:whsg}) into (\ref{eq:massbalance1}) and dividing through by $M$ yields

\begin{equation}
\frac{c_1 \phi_s^3}{1+c_2 \phi_s^2} = \frac{c_3 \phi_g}{1+c_4 \phi_g^2}.
\end{equation}

where

\begin{equation}
c_1 = \frac{\delta_2 M^2}{\alpha_s^2 k_2^2} , \qquad c_2 = \frac{M^2}{\alpha_s^2 k_2^2}, \qquad c_3 = \delta_1, \qquad c_4 = \frac{M^2}{\alpha_g^2 k_1^2}.
\end{equation}




\section{Discussion}

{\bf THIS MATERIAL FROM LEK IS TEMPORARY AND NEEDS TO BE EDITED,
  RERGANIZED, ETC]} A recent general review of models for aggregation
  based on attraction and repulsion is provided by
  \cite{SchWhi2011}.  Purely theoretical models for swarming
  include integro-differential equations
  \cite{MogEdel1999}. Application of such ideas to flying locust
  swarms include \cite{EdeWatGru1998} and \cite{TopBerLog2008}.

As far as marching locusts, there have been a number of studies in
which data collected in the laboratory and theoretical models have
been combined.  Most models concerned with alignment of locusts moving
in a group \cite{RomCouSch2009,BuhSumCou2006,YatErbEsc2009} as well as
proportion of locusts moving at a given time
\cite{BazBuhHal2008,BazRomTho2011} depending on treatments such as diet and
denervation. The motivation in many of these models is to explore the
transition between a disordered and a coherent marching group capable
of great destructive force.  \cite{RomCouSch2009} formulated an
abstract model of collective motion, with repulsion and attraction
that was then modified by \cite{BazRomTho2011} for locusts. The authors
describe an individual-based model with locust in 2 states (stopped,
moving) with stochastic transitions. They consider that locusts sense
others in a spatial range and that this leads to an escape-dominated
response with a parameter $\chi$ that reflects the strength of social
interactions.  They take a repulsive range of 2 cm and assume that the
strength of the repulsion is $\chi_r$=10cm/s$^2$.  They write a
Langevin equation for the speed and orientation of each locust. The
main output of the model is proportion of individuals moving and mean
group speed as a function of the group density. (The mean group speed
varies sigmoidally (over the range 0-7 cm/sec) with density in the
range of 0-100 locusts /m$^2$.



\begin{acknowledgments}
This work was supported by National Science Foundation Grant DMS-1009633 (to C.T.). We are grateful to the Mathematical Biosciences Institute where portions of this research were performed.
\end{acknowledgments}

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\end{article}

\begin{figure}
\resizebox{\textwidth}{!}{\includegraphics{steadystate.eps}}
\caption{Steady states and stuff.}\label{fig:steadystate}
\end{figure}

\begin{figure}
\resizebox{\textwidth}{!}{\includegraphics{kmax.eps}}
\caption{Maximally unstable wavelength.}\label{fig:stability}
\end{figure}


\end{document}










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%
%
%
%\begin{figure}
%\centerline{\includegraphics[width=.4\textwidth]{figsamp.eps}}
%\caption{LKB1 phosphorylates Thr-172 of AMPK$\alpha$ \textit{in vitro}
%and activates its kinase activity.}\label{afoto}
%\end{figure}
%
%\begin{figure*}[ht]
%\begin{center}
%\centerline{\includegraphics[width=.7\textwidth]{figsamp.eps}}
%\caption{LKB1 phosphorylates Thr-172 of AMPK$\alpha$ \textit{in vitro}
%and activates its kinase activity.}\label{afoto2}
%\end{center}
%\end{figure*}
%
%\begin{table}[h]
%\caption{Repeat length of longer allele by age of onset class.
%This is what happens when the text continues.}
%\begin{tabular}{@{\vrule height 10.5pt depth4pt  width0pt}lrcccc}
%&\multicolumn5c{Repeat length}\\
%\noalign{\vskip-11pt}
%Age of onset,\\
%\cline{2-6}
%\vrule depth 6pt width 0pt years&\multicolumn1c{\it n}&Mean&SD&Range&Median\\
%\hline
%Juvenile, 2$-$20&40&60.15& 9.32&43$-$86&60\\
%Typical, 21$-$50&377&45.72&2.97&40$-$58&45\\
%Late, $>$50&26&41.85&1.56&40$-$45&42\tablenote{The no. of wells for all samples was 384. Genotypes were
%determined by mass spectrometric assay. The $m_t$ value indicates the
%average number of wells positive for the over represented allele.}
%\\
%\hline
%\end{tabular}
%\end{table}
%
%
%\begin{table*}[ht]
%\caption{Summary of the experimental results}
%\begin{tabular*}{\hsize}
%{@{\extracolsep{\fill}}rrrrrrrrrrrrr}
%\multicolumn{3}{l}{Parameters}&
%\multicolumn{5}{c}{Averaged Results}&
%\multicolumn{5}{c}{Comparisons}\cr
%\hline
%\multicolumn1c{$n$}&\multicolumn1c{$S^*_{MAX}$}&
%\multicolumn1c{$t_1$}&\multicolumn1c{\ $r_1$}&
%\multicolumn1c{\ $m_1$}&\multicolumn1c{$t_2$}&
%\multicolumn1c{$r_2$}&\multicolumn1c{$m_2$}
%&\multicolumn1c{$t_{lb}$}&\multicolumn1c{\ \ $t_1/t_2$}&
%$r_1/r_2$&$m_1/m_2$&
%$t_1/t_{lb}$\cr
%\hline
%10\tablenote{Stanford Synchrotron Radiation Laboratory (Stanford University,
%Stanford, CA)}&1\quad &4&.0007&4&4&.0020&4&4&1.000&.333&1.000&1.000\cr
%10\tablenote{$R_{\rm FREE}=R$ factor for the $\sim 5$\% of the randomly
%chosen unique ref\/lections not used in the ref\/inement.}&5\quad &50&.0008&8&50&.0020&12&49&.999&.417&.698&1.020\cr
%100\tablenote{Calculated for all observed data}&20\quad &2840975&.0423&95&2871117&.1083&521&---&
%.990&.390&.182&---\ \ \cr
%\hline
%\end{tabular*}
%\end{table*}
%





