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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, solitary and gregarious. Individuals are
repelled (attracted) by others depending on their solitary
(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 | solitary}

\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 kilometers per day, stripping crops and
vegetation in its desolate 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.

Between plagues, locusts are mainly solitary creatures who live in
arid regions and lay eggs in small breeding grounds lush with
vegetation. Occasionally, resources are abundant enough to support
numerous hatchings, leading to a high density of adults. Overcrowding
at resource sites promotes a transition to a 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. Within the newly formed swarm, individuals mantain their
cohesiveness via direct sensory communication, or via chemical and
vibrational signaling \cite{Bre1954,DOrChuCha2006, MogEdel1999}.
Newly settled feeding grounds are also inevitably depleted, leading to
several stop-and-go cycles of traveling locust bands. Outbreaks 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 may be induced by visual or olfactory cues, but a
most potent stimulus is tactile: the repetitive stroking of the femora
of hind legs \cite{SimMcCHag1999,SimDesHag2001,RogMatDes2003} is
believed to function as a crowding indicator.  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}.  In the laboratory, it
has been shown that the solitary to gregarious switch can be induced
by rubbing a locust's hind leg for 5s per minute during a
period of 4h \cite{SimDesHag2001}. Cessation of physical contact
leads to a transition back to the solitary state after 4h.

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 the previous studies however, have focused on how an initially disperse,
solitary locust population spontaneously transitions into 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...


\section{The model}
As described above, locusts in a group are subject to attractive
and/or repulsive forces based on combined sensory, chemical, and
mechanical cues that affect their motion. Here we assume that such sensing is directionally
isotropic, a simplification used in many similar
models~\cite{EftVriLew20070,ParMar2005}.
We consider a locust population
density $\rho(\vec{x},t)$ moving at velocity $\vec v(\vec x,t)$ rather than individual locusts.
This standard continuum approach provides us with
analytical tools that can be brought to bear on the problem of
characterizing the initiation and structure of a swarm. Our framework
is based on a common swarm model that includes a conservation law
for $\rho$ and a representation for $\vec v$ given by 
\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}

\noindent
where $\vec{x}= (x,y)$ (e.g., in 2D). Here, we model $\vec v$ as a
convolution between the density $\rho(\vec x,t)$ and the social
interaction field $Q(\vec x, \vec x')$ that describes the influence of
the locust population at location $\vec{x'}$ on that at location
$\vec{x}$. We use the notation $\vec v = - \nabla Q * \rho$ to denote
the convolution in Eqn.\,\ref{eq:social} and assume that $Q (|\vec x -
\vec x'|)$ is radially symmetric and depends only on the distance
between $\vec x$ and $\vec x'$.  Eqns.\,\eqref{eq:social} have been
extensively studied in one and two dimensions for many specific
interaction choices
\cite{TopBer2004,BerLau2007,BodVel2005,BodVel2006}. Solutions
generally include distinct regimes such as steady state swarms,
spreading, and blow-up \cite{BerLau2007, LevTopBer2009, BerTop2011}.

To adapt Eqns.\,\eqref{eq:social} to biphasic desert locust swarm
dynamics, we denote the density of solitary and gregarious locusts by
$s(\vec{x},t)$ and $g(\vec{x},t)$, respectively, and the total density
by $\rho = s+g$.  We also include a density-dependent rate
$f_1(\rho)$ to model the transition from the gregarious to the solitary
state and $f_2(\rho)$ for the opposite switch. Our model thus reads
%
\begin{subequations}
\label{eq:ge}
\begin{alignat}{4}
\dot{s} &+ \nabla \cdot (v_s s)  &=  -f_2(\rho)s &+ f_1(\rho)g, \\
\label{eq:ge2}
\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}

\noindent
These equations are complete once we specify the form of the solitary and 
gregarious social interaction fields $Q_s$ and
$Q_g$. Since solitary locusts are crowd-avoiding we model
$Q_s$ as purely repulsive. Gregarious locusts, on the other
hand, are assumed to be attracted to others, except for short distance
repulsions due to excluded volume effects. Hence, we
let $Q_s$ and $Q_g$ be described by the following Morse-type
interactions
%
\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}
%
\noindent
where $R_s, R_g, A_g$ are interaction amplitudes and $r_s, r_g$ and
$a_g$ are interaction length scales. For cohesiveness to occur, the
parameters in $Q_g$ must be chosen in the so called ``clumping
regime'', per the conditions stated in \cite{DOrChuCha2006,
  ChuDOrMar2007, 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
ones. We model $f_{i=1,2}(\rho)$ 
via Hill-type functions so that
%
\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}
% 
Here, $\delta_{i=1,2}$ are maximal rates and $k_{i=1,2}$ are
characteristic locust densities at which the transitions occur at half
of their maximal values.  These choices allow $f_1$ to be a decreasing
function of $\rho$ and $f_2$ to increase with $\rho$, saturating at
$\delta_2$. Our complete model thus is represented by
Eqs.\,\eqref{eq:ge}-\eqref{eq:rates}.  We analyze them in 1D and 2D,
with appropriate boundary conditions in a finite domain and with
initial conditions that specify $s(\vec{x},0)$ and $g(\vec{x},0)$.
%and hence both $\rho(\vec{x},0)$ and the total mean
%density $\rho_0=\int \rho(\vec{x},0)~ d\vec{x}$.

\section{Biological Parameter Values}

\noindent
In order to estimate the parameters in Eq.\,\eqref{eq:rates} we draw
upon the experimental results of \cite{SimDesHag2001}.  Since phase
changes between states take approximately four hours we estimate
$\delta_1 = \delta_2 = 0.25\ $h$^{-1}$. The critical density for
gregarization is reported to be about $50$ - $80$ locusts /m$^2$. We
assume that the solitarization process has the same critical density,
and set $k_1=k_2\approx 65$ locusts /m$^2$.  To estimate the social
interaction parameters in Eqs.\,\eqref{eq:Q}, we apply instead the
results of \cite{BuhSumCou2006,BazRomTho2011}, who identify the
``sensing range'' of a locust as $0.14$ m, and the ``repulsion range''
as $0.04$ m, of the same order of magnitude as the approximately
$0.08$ m body length of a mature individual.  For the gregarious phase
we thus set the repulsion length scale at $r_g = 0.04$ m and the
attractive one at $a_g = 0.14$ m corresponding to the experimental
sensing range. These choices are in agreement with field observations
where attraction typically occurs at longer length scales compared to
repulsion. We also assume that solitary locusts tend to repel others
at their sensing range so that $r_s = 0.14$ m. Due to these choices,
in the remainder of this paper we will often be working under the
assumption that $r_g < a_g = r_s$.

Finally, we estimate $R_s$, $R_g$, and $A_g$ from explicit velocity
computations. The speed of a locust when it is alone
varies between $72$--$216$ 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 a hypothetical semi-infinite density field $\rho(x)
= \rho_0 \Heaviside(x)$ where $\Heaviside(x)$ is the Heaviside
function and $\rho_0 = 65$ locusts /m$^2$, the approximate critical
density of a gregarious group. A solitary locust placed 
at the swarm's edge at the origin, should move to the left with maximal
velocity $v_s^{max} = -216$ m/hr. We use the velocity definition
Eq.\,\eqref{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, 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 at the attraction length scale $a_g = 0.14$ m
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(-0.14,0) = -\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{Homogeneous steady states}

For any set of initial conditions, the mean locust density $\rho_0$ is
known, and corresponds to the the total density at the homogeneous
steady state. Accordingly, there is a family of homogeneous steady
states parameterized by the total homogeneous density $\rho_0$. The
densities of solitary and gregarious locust populations arising
from the steady state solutions of Eqs.\,\ref{eq:ge} and \ref{eq:ge2} 
are 
%
\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} \label{eq:g0}.
\end{eqnarray}
\end{subequations}
% 
In the small $\rho_0$ limit these can be approximated as
\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,
%\quad \beta=\frac{\delta_2}{\delta_1 k_2^2},
\end{equation}
%
so that the low-density steady state is composed mostly of solitary
locusts. In the large $\rho_0$ limit on the other hand
%
\begin{equation}
s_0 \approx \frac{\delta_1 k_2^2}{\delta_2 \rho_0}, \quad g_0 \approx 
\rho_0 -\frac{\delta_1 k_2^2}{\delta_2 \rho_0},
\end{equation}
% 
so that the high-density steady state is composed mostly of
gregarious insects. These estimates already point to the
nonmonotonicity of $s_0$ with total density.  Further evidence of this
relationship is given in Fig.\,\ref{fig:steadystate} where $s_0, g_0$
are shown as functions of the total density $\rho_0$. To obtain the
three sets of curves, we took a uniform distribution around each $\{
\delta_1, \delta_2, k_1, k_2\}$ parameter, centered at our biological
estimates ($k_1=k_2= 65$ locusts /m$^2$ and $\delta_1 = \delta_2 =
0.25$ h$^{-1}$), with a range from 0.7 to 1.3 times these estimated
values. We drew 10,000 sample parameter sets and plotted three pairs of
curves, corresponding to 25, 50, and 75 percentile levels of the
steady state value of $s_0$ (in blue) and $g_0$ (in green).

As noted, $s_0$ at first increases with $\rho_0$, as both solitary and
gregarious locusts accumulate. At a critical total density $\rho_c$,
$s_0$ reaches a maximum, whereas $g_0$ keeps monotonically increasing.
From a biological point of view, this model prediction implies that
homogeneous solitary groups can only exist up to $\rho_c$, beyond
which gregarization dominates.  While for the chosen functions
$f_{i=1,2}$, closed form expressions for the maximum cannot easily
be obtained, for the biologically estimated
parameters of this work, $k_1 = k_2 = k$ and $\delta_1 = \delta_2 =
\delta$, we find that the maximum solitary density is attained at
$\rho_0 =k$ when $s_0 = g_0 = k/2$.


\section{Linear stability analysis}
In this section we study the stability of the steady state solution in
Eqs.\,\eqref{eq:s0} and \eqref{eq:g0}, to determine under what conditions
uniformly spread locust populations aggregate or disperse, creating
local clusters or vacancies. We thus consider small perturbations
$s_1, g_1$ about the $s_0, g_0$ values and 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}
\noindent
which leads to 
$\rho(\vec{x},t) =s_0+g_0+ s_1(\vec{x},t)+g_1(\vec{x},t)$. 
Substituting Eqn.\,\eqref{eq:pert1} into Eq.\,\eqref{eq:ge} under the assumptions $s_1 \ll s_0$ and $g_1 \ll g_0$ 
and expanding to first order in the perturbations we 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,
\label{eq:fourier}
\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}
and where $A,B > 0$ for all $\rho_0 \neq 0$ since $f_1$ is a
monotonically increasing function of $\rho_0$ and $f_2$ is a
monotonically decreasing one. To further analyze the linearized
equations, we Fourier expand the perturbations as
%
\begin{equation}
\label{eq:perturb}
s_1(\vec{x},t) = \sum_\vec{q} \mathcal{S}_\vec{q}(t) \exp{i \vec{q}
  \cdot \vec{x}}, 
\quad s_2(\vec{x},t) = \sum_\vec{q} \mathcal{G}_\vec{q} (t) \exp{i
  \vec{q} \cdot
\vec{x}}.
\end{equation}
%
For each perturbation wave number $\vec q$ the components $\{ q_x,
q_y\}$ may be physically interpreted as the inverse of typical length
scales on which the perturbation occurs. The most unstable of these
may be interpreted as the typical size of an emerging gregarious or
solitary locust patch.  We allow for an infinitely large domain so
that there are no restrictionds on $\vec q$. Thus, upon substituting
the Fourier expansion of Eqn.\,\eqref{eq:peturb} into
Eqn.\,\eqref{eq:linearized} we find a set of ordinary differential
equations for each Fourier mode amplitude.  These are conveniently
written in the following matrix form
\begin{subequations}
\begin{gather}
\frac{d}{dt}
\begin{pmatrix}
\mathcal{S}_q \\\mathcal{G}_q
\end{pmatrix}
=
\mat{L}(q)
\begin{pmatrix}
\mathcal{S}_q \\ \mathcal{G}_q
\end{pmatrix}, \\
\mat{L}(q) \equiv \begin{pmatrix}
-s_0 q^2 \wh{Q}_s(q) - A & -s_0 q^2 \wh{Q}_s(q)+B \\ -g_0 q^2  \wh{Q}_g(q) + A & -g_0 q^2  \wh{Q}_g(q) - B
\end{pmatrix}.
\end{gather}
\end{subequations}
%
\noindent
Here, $q = |\vec{q}|$ is the perturbation wave number, and
$\wh{Q}_{s,g}(a)$ are the Fourier transforms of the two dimensional
social interaction potentials, respectively
%
\begin{eqnarray}
\label{eq:qhats}
\widehat{Q}_s(q)& =&  \frac{2 \pi R_s r_s^2}{(1+r_s^2 q^2)^{3/2}}, \\
\widehat{Q}_g(q) & = &   \frac{2 \pi R_g r_g^2}{(1+(r_g^2q^2)^{3/2}}-
\frac{2 \pi A_g a_g^2}{(1+a_g^2 q^2)^{3/2}}.
\end{eqnarray}
%
\noindent
The eigenvalues $\lambda_{i=1,2}(q)$ of $\mat{L}(q)$ can be written as
%
\begin{equation}
\lambda_1(q) = -q^2 \left[s_0 \wh{Q}_s(q) + 
g_0 \wh{Q}_g(q)\right], \quad \lambda_2 = -(A+B).
\end{equation}
%
The eigenvalue $\lambda_2$ is $q$ independent and negative since
$A,B>0$. Thus, the constant density steady state is stable if
$\lambda_1(q)<0$ for all admissible $q$. If $\lambda_1(q) \geq 0$ for
some $q$, then the constant density steady state is unstable to
perturbations of those wave numbers.  The pivotal
eigenvalue $\lambda_1(q)$ embodies all physically relevant parameters
and is written in the most general form. 
In order to make analytical progress, we let
$k_1 = k_2 = k$, $\delta_1 = \delta_2 = \delta$ and $r_s=a_g$, which
follow from our biological assumptions. Later it will be useful to
recall that within our parameter estimates $r_g < a_g$.  Inserting
these simplifications, we rewrite $\lambda_1(q)$ as
%
\begin{equation}
\label{eq:lambda1mod}
\lambda_1(q) = \frac{2\pi R_g \rho_0^3 r_g^2 q^2}{(h^2+\rho_0^2)(1+q^2 a_g^2)^{3/2}} 
\left[H(\rho_0) - h(q)
\right] 
\end{equation}
where
\begin{equation}
H(\rho_0) = \frac{\rho_0^2 A_g a_g^2 - k^2 R_s a_g^2}{R_g \rho_0^2 r_g^2},\quad
h(q) = \frac {(1+q^2 a_g^2)^{3/2}}{(1+q^2 r_g^2)^{3/2}}.
\end{equation}
% 
Since $h(q) > 0$, for instability to occur we must require
$H(\rho_0)> 0$, otherwise $\lambda_1(q)$ will always be negative. This
necessary - but not sufficient - condition translates to $ \rho_0^2 >
k^2 R_s / A_g $.  Let us now look more carefully at the terms inside
the square brackets of Eqn.~\eqref{eq:lambda1mod}.  Since $r_g < a_g$,
the function $h(q)$ is monotonically increasing. Hence, for
$\lambda_1(q)$ to be positive, it is also necessary that $H(\rho_0)$ be
at least greater than the minimum value of $h(q)$, attained at
$q=0$. Thus, a more stringent condition for instability is
\begin{equation}
H(\rho_0) > h(q=0) = 1.
\end{equation}
Using the fact that $h(q \to \infty) = a_g^3/ r_g^3 > 1$
we can now distinguish three cases:
\begin{itemize}
\item if $H(\rho_0)< 1$ then $\lambda_1(q)<0$ for all $q$ and the homogeneous
  steady state is stable to perturbations of all wave numbers.
\item if $1 \leq H(\rho_0) \leq a_g^3/r_g^3$ then $\lambda_1(q)$ will have a
  single root at $q = q_*$ and instabilities in the homogeneous steady
  state will arise for perturbations with small wave numbers $q <
  q_*$, giving rise to large clusters of gregarious or solitary
  patches.
\item if $H(\rho_0)> a_g^3/r_g^3$ then $\lambda_1(q) > 0$ for all $q$ 
and the homogeneous steady state
is unstable to perturbations of any wave number.
\end{itemize}

\noindent The condition $H(\rho_0) > 1$ 
that guarantees instability can be rewritten as
\begin{equation}
\rho_0 > 
\left(\frac{R_s}{A_g - R_g (r_g/a_g)^2}\right)^{1/2} k.
\end{equation}
% 
The above inequality is one of the most important results of this
paper and states that for sufficiently large locust densities uniform
distributions are no longer stable and concentrated patches of locusts
of various sizes may form.  In this work we do not include
descriptions of vegetation or crops, but the instability condition
may be considered an indicator of the locust plague, where patches of
high density locust groups may arise in confined spaces
causing much more damage than
if uniform locust populations were spread over the entire available
domain. Note that in order for the instability to occur 
not only the densituy has to be large enough, but also
the ``clumping condition'' introduced earlier and that represents
long range locust attraction, $A_g r_g^2 - R_g r_g^2 > 0 $, must hold.
 
For the locust parameter choices specified in this work, Eq.\,ref{eq:}
homogeneous solution is unstability as long as $\rho_0 > 59.9$, 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. [CHAD IS CONSIDERING: WHY AREN'T THEY CLOSER?] 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 most unstable wave number $k_{max}$
as a function of $\rho_0$. This wavenumber would be characterize the
cluster diameter of the swarm as it is first initiated.  {\bf[BRIEFLY
    EXPLAIN HOW OBT'D]}. For low densities, the most unstable
wavenumber is 0, indicating that large wavelength aggregation zones
destabilize first. As $\rho_0$ increases, there is a sharp transition
region in which $k_{max}$ grows rapidly. Here, clusters of some finite
size would be seen. We further observe that $k_{max}$ levels to a
plateau value. (As before, the three curves in
Fig.~\ref{fig:stability} are the 25th, 50th, and 75th percentile
values as parameters are varied). At the 50th percentile value the
large $\rho_0$ asymptotic value of $k_{max}$ is $k_{max} =
8.9$. {\bf[GIVE UNITS AND INDICATE PREDICTED CLUSTER DIAMETERS.]}

\section{Typical locust patch size} 
% 
We can now estimate the typical locust patch size $d$
when the uniform population is unstable to perturbations
by finding the most unstable wavelength $q_{max}$ that maximizes 
$\lambda(q)$. We can thus identify $d \simeq 1/q_{max}$.
We thus calculate $\lambda'(q)$ under the instability condition
$H(\rho) > 1$ to find

\begin{eqnarray}
\label{eq:maxwave}
\lambda'(q) = \frac{2 \pi R_g \rho_0^3 r_g^2 q}{(h^2+\rho_0^2)} 
\left[H \frac{2 - a_g^2 q^2}{(1 + a_g^2 q^2)^{5/2}}- 
\frac{2 - r_g^2 q^2}{(1+r_g^2 q^2)^{5/2}} \right].
\end{eqnarray}

\noindent
To find $q_{max}$ we set the above expression to zero $\lambda(q_{max}=0)$. 
Note that at $q=0$ the term in parenthesis 
in Eqn.\,\eqref{eq:maxwave} is given by $2(H (\rho_0)-1) > 0$
due the instability condition. Furthermore, at $q = \sqrt 2 / a_g$ 
the same term is $2 a_g^3 (r_g^2 - a_g^2)/(a_g^2 + 2
r_g^2)^{5/2} < 0$ since $r_g < a_g$. These two limits 
imply that $\lambda'(q_{max}) =0$ for $q_{max} < \sqrt 2 / a_g$.

THIS IS A BIT IFFY:

Numerical estimates show that for the parameters at hand
the maximum is obtained in the vicinity of $\sqrt 2/a_g$
so we let $q_{max} \simeq \sqrt{2}/a_g - q_{sh}$
with $0 < q_{sh} \ll \sqrt{2} / a_g$. After
expanding Eqn.\,\eqref{eq:maxwave} to leading orders
and in the limit of large $\rho_0$ we find 
%
\begin{eqnarray}
q_{sh} \simeq \frac{9 \sqrt{3}}{\sqrt{2 r_s}} \frac{R_g}{A_g} 
\frac{r_g^2}{r_s^2}
\end{eqnarray}
%
\noindent
we thus expect typical clusters to extend over the range

\begin{eqnarray}
d \simeq \frac {\sqrt{r_s}} 
{2 \left[1 - \frac{9 \sqrt{3}}{2 \sqrt{2}} \frac{R_g}{A_g} 
\frac{r_g^2}{r_s^2}
\right]}
\end{eqnarray}

\noindent
which, for the parameters at hand is rougly $d \simeq 0.11$ m.

CHAD: NOTE THAT FOR SURE, THIS IS CLOSE TO
KMAX = 8.9 WHICH YOU ALSO HAD IN THE PAPER
AND D = 1/8.9 = 0.11,
BUT WHAT DOES THIS MEAN PHYSICALLY? OUR CLUSTERS ARE MADE OF 
A COUPLE OF LOCUSTS, SINCE THEY ARE TYPICALLY 0.08 METERS???

%\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 \eqref{eq:ge}, we observe mass-balanced states in
which gregarious and solitary locusts spatially segregate into regions
with disjoint support.  This means that in the given regions $\rho
\approx s$ or $\rho \approx g$.  We can approximate this behaviour
with the following ``bulk'' state model reduction. For convenience,
we define the total number of solitary and gregarious locusts, $S$
and $G$ respectively as
%
\begin{equation}
S =  \int_{\Omega} s\,d\Omega, \quad G =  \int_{\Omega} g\,d\Omega,
\label{WasEq29}
\end{equation}
%
where $\Omega$ is our spatial domain, 
the total population mass is $M = S+G$, and mass fractions
are given as
%
\begin{equation}
\label{eq:whsg}
\phi_s = S/M, \quad \phi_g = G/M, \quad \phi_s + \phi_g = 1.
\end{equation}
%
We assume that solitary locusts are spread throughout most of the
domain $\Omega$, covering an area denoted $\alpha_s$, whereas
gregarious locusts clumped in a region whose area we call
$\alpha_g$. (This area can be estimated from the gregarious potential;
see \eqref{eq:Q}.)  Then in these regions, local densities are
approximately
%
\begin{equation}
s = S/\alpha_s, \qquad g = G/\alpha_g.
\label{WasEq 31}
\end{equation}
%
\noindent
If we now integrate Eqs.\,\eqref{eq:ge} over the domain 
the spatial terms vanish, and using $\rho \approx s$ or $\rho \approx
g$ in the disjoint regions, we find 
%
\begin{equation}
\int \dot{s} = -\int f_2(s)s + \int f_1(g)g = - \int \dot{g} 
\end{equation}
% 
Further assuming that $s, g$ are approximately constant in the
regions of their support, the above equations can be rewritten as
%
\begin{eqnarray}
 \dot{S} &=  - f_2(S/\alpha_s)S +  f_1(G/\alpha_g)G = - \dot{G} 
\end{eqnarray}
%
We can write the above in terms of the mass fractions $\phi_s, \phi_g$ 
from \eqref{eq:whsg} 
%
\begin{equation}
 \dot{\phi_s} =  - f_2(M\phi_s/\alpha_s)\phi_s +
 f_1(M\phi_g/\alpha_g)\phi_g = - \dot{\phi_g}
\end{equation}
%
Finally, substituting the definitions of $f_1, f_2$ 
from \eqref{eq:rates} we obtain
%
\begin{equation}
\label{BulkModelEqs}
 \dot{\phi_s} =  - \frac{c_1\phi_s^3}{1+c_2\phi_s^2} +  \frac{c_3
   \phi_g}{1+c_4 \phi_g^2} = - \dot{\phi_g} 
\end{equation}
%
where the $c_i$ reduced parameters are given by
% 
\begin{equation}
c_1 = \frac{\delta_2 M^2}{\alpha_s^2 k_2^2} , \quad c_2 =
\frac{M^2}{\alpha_s^2 k_2^2}, \quad c_3 = \delta_1, \qquad c_4 =
\frac{M^2}{\alpha_g^2 k_1^2}.
\end{equation}
% 
The ODEs \eqref{BulkModelEqs} can be used to compute the steady
state fractions $\phi_s, \phi_g$ by setting LHS to zero in each
equation. Here we consider the dynamics of this system.  It is
possible to reduce this to a single equation using $\phi_s=1-\phi_g$,
though the result is is complicated. An easier approach is to consider
the large M limit, in which \eqref{BulkModelEqs} becomes (to leading
order)
%
\begin{equation}
\label{BulkModelLargeM}
 \dot{\phi_s} =  - \delta_2 \phi_s +  \frac{c_3 }{c_4 \phi_g} = 
- \dot{\phi_g} 
\end{equation}
%
Noting the values of the constants in \eqref{BulkModelLargeM}, and the
fact that $0 \le \phi_s, \phi_g \le 1$ are dimensionless, we observe
that the first term is of $O(1)$ whereas the second term is of
$O(1/M^2)$. This means that, to leading order, for large $M$, the mass
fraction of the solitary locusts decays exponentially in time. This is
based on the assumption of a segregated state, and would be expected
to occur once segregation is nearly complete.

In short, for large M, the entire population will eventually become gregarious. This implies that the level $\rho_0$ that leads to instability is crucial. If the state of the population is in the stable regime, mass gregarization can be avoided. As soon as the population shifts beyond the border of stability (at which $\phi_s \approx \phi_g$), there is no avoiding the gregarization of the entire group.

%%%%%%%%%%%%OLDER STUFF HERE%%%%%%%%%%%%
%For a segregated steady state at mass balance, the number flux of gregarious locusts becoming solitarized per unit time, $G f_1(G/\alpha_g)$, should equal the number flux of solitary locusts becoming gregarized,$S f_2(S/\alpha_s)$. Equating these and substituting from \eqref{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}
%Solving \eqref{eq:massbalance1} using \eqref{eq:whsg} and dividing 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  TEMPORARY.. TO BE WRITTEN AFTER RESULTS COLLECTED]} 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.) and an NSERC Discovery grant (to LEK). We are grateful to the Mathematical Biosciences Institute where portions of this research were performed.
\end{acknowledgments}

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\begin{figure}
%\resizebox{0.5\textwidth}{!}{\includegraphics{steadystate.eps}}
\caption{Uniform steady state (SS) levels of solitary locusts, $s_0$
  (blue) and gregarious locusts, $g_0$ (green) as functions of the
  mean locust density $\rho_0$ on a log-log plot.  The three curves
  represent the 25, 50 and 75 percentile of the SS value obtained
  using a uniform distribution centered at parameter values estimated
  from biological data, with a range $\pm 30\%$. The steady state is
  stable in the white region and unstable in the red region. Vertical
  dashed lines are 25th and 75th percentile values for onset of
  instability. At the 50th percentile, instability occurs before the
  value of $g_0$ overtakes $s_0$.}\label{fig:steadystate}
\end{figure}

\begin{figure}
%\resizebox{0.5\textwidth}{!}{\includegraphics{kmax.eps}}
\caption{Maximally unstable wavelength. At low densities, only the wavenumber $k=0$ is unstable. Near the critical density, $k_{max}$ increases rapidly to some constant value as the density increases. Three curves correspond to same percentile values as in Fig~\ref{fig:steadystate}}\label{fig:stability}.
\end{figure}

{\bf [PUT IN SUPPLEMENT?]}  
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 solitary 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 solitary 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 rejecting 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.



\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*}
%





