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\begin{document}

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

\author{Roberta Graff\affil{1}{University of Cambridge, Cambridge,
United Kingdom},
Javier de Ruiz Garcia\affil{2}{Universidad de Murcia, Bioquimica y Biologia
Molecular, Murcia, Spain},
\and
Franklin Sonnery\affil{2}{}}

\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. Here we model the
spatio-temporal dynamics and treansitions between these two
phases. [ABSTRACT TO BE DONE LATER].
\end{abstract}

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

\abbreviations{}


\noindent
Outbreaks of the desert locust \emph{Schistocerca gregaria} have
afflicted vast areas across northern Africa, the Middle East and
south-west Asia for centuries. Depending on climate and vegetation
conditions, billions of these voracious insects may aggregate into
destructive swarms, typically covering areas of a thousand square
kilometers, traveling a few hundred kilometers each day, and leaving
behind land mostly outstripped of vegetation
\cite{Ken1951,Alb1967,Uva1977,Rai1989}.  The latest locust plague
occurred in West Africa between 2003--2005 and severely disrupted
subsistence and export agriculture by destroying \$2.5 billion in
crops. Despite \$400 million spent in control efforts, loss rates
escalated to 50\% in certain regions \cite{Bel2005,BraDjiFay2006}.
These numbers alone attest to the urgency of finding optimal ways to
control, manage and possibly predict outbreaks.
%Outbreaks are difficult to predict and
%depend on several environmental, geographical and climatic factors.

In their quiet periods, 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 to
irrigation. Occasionally, resources are
abundant enough to support numerous hatchings, leading to a
large number of adult locusts who, due to crowding, acquire gregarious
tendencies. 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
each other. Behavioral phase is tremendously plastic
\cite{Uva1977,SimMcCHag1999,PenSim2009} and strongly dependent on
local population density, so that in sparse surroundings a gregarious
locust turns into a solitary one \cite{SimMcCHag1999} and viceversa in
crowded environments. It is this phenotype switch that takes
place when large numbers of locusts gather at the same nutritional
site, allowing individuals to transform from solitary to
gregarious and swarm together \cite{Des2000,ColDesSim1998}.

At a more mechanistic level, locust gregorization 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
Refs.\,\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
Ref.\,\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 Ref.\,\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}

\noindent
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 $\vec{Q}$ has the appropriate symmetry (DO WE
NEED THIS LINE?) Eq.\,\ref{eq:social} has been studied in
Refs.\,\cite{BerLau2007,BodVel2005,BodVel2006} for specific choices of
$Q$, and in a two-dimensional setting in \cite{TopBer2004}.  Generally,
the dynamics arising from Eq.\,\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
Ref.\,\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}

\noindent
where the velocities are given by

\begin{equation}
\label{eq:v2d}
v_s = -\nabla(Q_s^{2D} * \rho), \quad v_g  -\nabla(Q_g^{2D} * \rho).
\end{equation}

\noindent
in accordance to Eq.\,\ref{eq:social}.

STOP FOR TODAY

\vspace{5cm}
\noindent
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 $Q_s^{2D}$ and the gregarious potential
$Q_g^{2D}$ are
\begin{equation}
\label{eq:Q}
Q_s^{2D}(\vec{x}) = R_s \exp{-|\vec{x}|/r_s}, \quad Q_g^{2D}(\vec{x}) = R_g \exp{-|\vec{x}|/r_g} - A_g \exp{-|\vec{x}|/a_g}.
\end{equation}

Assume that variation in the swarm is negligible in one direction,
which we take to be the $y$-direction. Then (\ref{eq:ge}) reduces to
one dimension with interaction potentials that we refer to as
quasi-two-dimensional, as in \cite{BerTop2011}. The potentials are
\begin{equation}
\label{eq:Q}
Q_{s,g}(x) =  \int_{-\infty}^{\infty} Q_{s,g}(x,y)\,dy.
\end{equation}
and the velocity is
\begin{equation}
\label{eq:v}
v_s = -\partial_x (Q_s * \rho), \quad v_g  -\partial_x(Q_g * \rho).
\end{equation}



The parameters in $Q_g$ must be chosen in the clumping regime, per the
conditions stated in \cite{LevTopBer2009}. [Not the right condition
  anymore since we are using quasi-2d.]


Now consider the density dependent rates of phase change. The
solitarization rate $f_1$ should be high when local density is low,
and should approach zero when density is high. Conversely, the
gregarization rate $f_2$ should approach zero when local density low,
and should be significant when density is high. Adopt the functional
forms

\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}



%This model has ten parameters, $\delta_{1,2}$, $k_{1,2}$, $R_{s,g}$, $r_{s,g}$, $A_g$, and $a_g$. To reduce the number of parameters, nondimensionalize. Let

%\begin{equation}

%\rhot = \rho/k_1, \qquad \wt{M} = M/k_1, \qquad \gt = g/k_1, \qquad \st = s/k_1, \qquad \xt = x/r_s, \qquad \teet = t\delta_1.

%\end{equation}

%Then define new, dimensionless parameters

%\begin{alignat}{8}

%\wt{\delta}_2 &= \delta_2/\delta_1,  &\quad& \wt{k}_2 = k_2/k_1, &\quad& \wt{r}_g &= r_g/r_s, &\quad& \wt{a}_g &= a_g/r_s, &\quad& \wt{R}_{s,g} &= R_{s,g}/\beta, &\quad& \wt{A}_g = A_g/\beta,

%\end{alignat}

%where

%\begin{equation}

%\beta = \frac{r_s^2 \delta_1}{k_1}.

%\end{equation}

%

%Substitute the nondimensionalization into (\ref{eq:dimensioned_ge}) through (\ref{eq:dimensioned_fluxes}) and drop hats on variables and parameters to obtain

%\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

%\begin{equation}

%v_s = -\nabla(Q_s * \rho), \quad v_g = -\nabla(Q_g * \rho),

%\end{equation}

%with

%\begin{equation}

%\label{eq:Q}

%Q_s = R_s \exp{-|x|}, \quad Q_g = R_g \exp{-|x|/r_g} - A_g \exp{-|x|/a_g},

%\end{equation}

%and

%\begin{equation}

%\label{eq:rates}

%f_1(\rho) =  \frac{1}{1+ \rho^2}, \quad f_2(\rho) =  \frac{\delta_2 \left( \rho/k_2 \right)^2}{1+ \left( \rho/k_2 \right)^2}.

%\end{equation}

\section{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. 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 (\ref{eq:v}) to write
\begin{equation}
v_s(0) = -\partial_x \left\{Q_s * \rho_0 \Heaviside(x)\right\}|_{x=0} = v_s^{max},
\end{equation}
which we solve to find $R_s = 41.54$ 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) = -\partial_x\left\{Q_g * \rho_0 \Heaviside(x)\right\}|_{x=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\}|_{x=-0.14} = 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 $\rho_0$. For any given $\rho_0$, the homogeneous steady state is

\begin{equation}
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}, \quad g_0 =  \frac{\delta_2 \rho_0^3 (k_1^2+\rho^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{equation}

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.

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 (\ref{eq:pert1}) and (\ref{eq:pert2}) into (\ref{eq:dimensioned_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 (\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{equation}
\frac{d}{dt}
\begin{pmatrix}
\mathcal{S} \\\mathcal{G}
\end{pmatrix}
=
\mat{L}(k)
\begin{pmatrix}
\mathcal{S} \\ \mathcal{G}
\end{pmatrix}, \quad \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{equation}

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{equation}
\label{eq:qhats}
\widehat{Q}_s(k) = 2\frac{R_s r_s}{1+(r_s k)^2}, \quad \widehat{Q}_g(k) =2\left[ \frac{R_gr_g}{1+(r_gk)^2}-\frac{A_gr_a}{1+(r_ak)^2}\right].
\end{equation}

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 condition does not depend on the functions $f_{1,2}$ which describe behavioral phase change. It only depends on the social interaction potentials ad on the relative amounts of solitary and gregarious locusts.





\section{Bulk theory for segregated states}



In simulations of (\ref{eq:dimensioned_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{thebibliography}{10}


\bibitem{Ken1951}
Kennedy JS
\newblock (1951) The migration of the desert locust (\emph{{S}chistocerca gregaria {F}orsk}).
\newblock \emph{Proc Roy Soc Lond B} 235:163--290.

\bibitem{Alb1967}
Albrecht FO
\newblock (1967) Les grands probl{\`{e}}mes de la biologie, 
in \emph{Polymorphisme phasaire et biologie des acridiens migrateurs}
\newblock (Masson, Paris).

\bibitem{Uva1977}
Uvarov B
\newblock (1977) \emph{Grasshoppers and locusts}
\newblock (Cambridge Univ Press, London) Vol.{}~2.

\bibitem{Rai1989}
Rainey RC
\newblock (1989) \emph{Migration and Meteorology: Flight Behavior and the
  Atmospheric Environment of Locusts and other Migrant Pests}, Oxford Science
  Publications
\newblock (Clarendon Press, Oxford).

\bibitem{Bel2005}
Bell M
\newblock (2005) The 2004 desert locust outbreak.
\newblock \emph{Bull Am Meteor Soc} 86:S60.

\bibitem{BraDjiFay2006}
Brader L, {et~al.}
\newblock (2006) Towards a more effective response to desert locusts and their
  impacts on food security, livelihoods and poverty: Multilateral evaluation of
  the 2003--05 desert locust campaign, (United Nations Food and Agriculture
  Organization), Technical report.

\bibitem{AppHei1999}
Applebaum SW, Heifetz Y
\newblock (1999) Density-dependent physiological phase in insects.
\newblock \emph{Ann Rev Entomol} 44:317--341.

\bibitem{PenSim2009}
Pener MP, Simpson SJ
\newblock (2009) Locust phase polyphenism: An update.
\newblock \emph{Adv Insect Physiol} 36:1 -- 272.

\bibitem{Dir1953} 
Dirsh VM
\newblock (1953) Morphometric studies on phases
  of the desert locust (\textit{Schistocerca gregaria Forskal}. 
\newblock \emph {Antilocust  Bull} 16:1--34.

\bibitem{IslRoeSim1994} 
Islam MS, Roessingh P, Simpson S J and McCaffery AR 
\newblock(1994b) Parental effects on the behavior and
  coloration of nymphs of the desert locust Schistocerca
  gregaria. \newblock \emph{J Insect Physiol} 40:173--181.

\bibitem{SchAlb1999} 
Schmidt GH and Albutz R
\newblock (1999) Identification of solitary and gregarious populations of
  the desert locust, Schistocerca gregaria, by experimental breeding. 
\newblock \emph{Entomol Gen} 24:161--175.

\bibitem{SimMcCHag1999}
Simpson SJ, McCaffery AR, H{\"{a}}gele BF
\newblock (1999) A behavioral analysis of phase change in the desert locust.
\newblock \emph{Biol Rev} 74:461--480.

\bibitem{RogMatDes2003}
Rogers S, {et~al.}
\newblock (2003) Mechanosensory-induced behavioural gregarization in the desert
  locust schistocerca gregaria.
\newblock \emph{J Exp Bio} 206:3991--4002.

\bibitem{Des2000}
%\bibitem{despland2000}
Despland E, Collett M, Simpson S
\newblock (2000) Small-scale processes in desert locust swarm formation: how
  vegetation patterns influence gregarization.
\newblock \emph{Oikos} 88:652--662.

\bibitem{ColDesSim1998}
%\bibitem{collett1998}
Collett M, Despland E, Simpson S, Krakauer D
\newblock (1998) Spatial scales of desert locust gregarization.
\newblock \emph{Proc Natl Acad Sci} 95:13052.

\bibitem{SimDesHag2001}
Simpson SJ, Despland E, H{\"{a}}gele BF, Dodgson T
\newblock (2001) Gregarious behavior in desert locusts is evoked by touching
  their back legs.
\newblock \emph{Proc Natl Acad Sci} 98:3895--3897.

\bibitem{AnsRogOtt2009}
Anstey ML, Rogers SM, Ott SR, Burrows M, Simpson SJ
\newblock (2009) Serotonin mediates behavioral gregarization underlying swarm
  formation in desert locusts.
\newblock \emph{Science} 323:627--630.


\bibitem{BazBuhHal2008}
%\bibitem{bazazi2008}
Bazazi S, Buhl J, Hale JJ, Anstey ML, Sword GA,
Simpson SJ, Couzin ID
%Bazazi S, {et~al.}
\newblock (2008) Collective motion and cannibalism in locust migratory bands.
\newblock \emph{Curr Biol} 18:735--739.

\bibitem{EdeWatGru1998}
Edelstein-Keshet L, Watmough J, Gr{\"{u}}nbaum D
\newblock (1998) Do travelling band solutions describe cohesive swarms? {A}n
  investigation for migratory locusts.
\newblock \emph{J Math Bio} 36:515--549.

\bibitem{TopBerLog2008}
Topaz CM, Bernoff AJ, Logan S, Toolson W
\newblock (2008) A model for rolling swarms of locusts.
\newblock \emph{Eur Phys J Spec Top} 157:93--109.

\bibitem{BuhSumCou2006}
%\bibitem{buhl2006}
%Buhl J, {et~al.}
Buhl J, Sumpter DJT, Couzin ID,
Hale JJ, Despland E, Miller ER, Simpson SJ
\newblock (2006) From disorder to order in marching locusts.
\newblock \emph{Science} 312:1402.


\bibitem{VicCziBen1995}
Czir{\'{o}}k, A {et~ al}
%Ben-Jacob, E. and Cohen, I. and Vicsek, T.
\newblock (1995) Novel type of phase transition in a system
or self-driven particles.
\newblock \emph{Phys Rev Lett} 75:1226--1229.


\bibitem{HolChe1996}
Holt J, Cheke R
\newblock (1996) Models of desert locust phase changes.
\newblock \emph{Ecol Model} 91:131--137.

\bibitem{LevTopBer2009}
Leverentz AJ,  Topaz CM and Bernoff AJ
\newblock(2009)	Asymptotic dynamics of attractive-repulsive swarms.
\newblock \emph{SIAM J Appl Dyn Sys} 8:880--908.

\bibitem{BerTop2011}
Bernoff AJ, Topaz CM
\newblock (2011) A primer of swarm equilibria.
\newblock \emph{SIAM J Appl Dyn Sys} 10:212--250.

\bibitem{Bre1954} 
Breder CM 
\newblock (1954) Equations descriptive of fish schools and
other animal aggregations.
\newblock \emph{Ecol} 35:361--370.

\bibitem{DOrChuCha2006} D'Orsogna MR, Chuang YL, Bertozzi AL and
  Chayes L 
\newblock (2006) Self-propelled particles with soft-core
  interactions: patterns, stability and collapse.  
\newblock \emph{Phys Rev Lett} 96:104302.

\bibitem{MogEdel1999}
%\bibitem{mogilner1999}
Mogilner A, Edelstein-Keshet L
\newblock (1999) A non-local model for a swarm.
\newblock \emph{J Math Biol} 38:534--570.

\bibitem{EftVriLew20070}
Eftimie R, de Vries G, Lewis M A, Lutscher F
\newblock (2007) Modeling group formation and activity patterns in 
self-organizing collectives of individuals.
\newblock \emph {Bull Math Biol} 69:1537--1565.

\bibitem{ParMar2005}
Partan SR, Marler P,
\newblock (2005) 
Issues in the classification of multimodal communication signals.
\newblock \emph {Am Nat} 166:231--245.

\bibitem{BerLau2007}
Bertozzi AL, Laurent T,
\newblock (2009) The behavior of solutions of multidimensional
aggregation equations with mildly singular interaction kernels.
\newblock \emph {Chi Ann Math Ser B} 30:463--482.

\bibitem{BodVel2005}
Bodnar M, Velasquez JJL,
\newblock (2005) Derivation of macroscopic equations for individual 
cell-based models: a formal approach.
\newblock \emph{Math Meth Appl Sci} 28:1757--1779.

\bibitem{BodVel2006}
Bodnar M, Velasquez JJL,
\newblock (2006) An integro-differential equation arising as a limit
of individual cell-based models.
\newblock \emph{J Diff Eq} 222:341--380.

\bibitem{TopBer2004}
Topaz CM, Bertozzi AL,
\newblock(2004) Swarming patterns in a two-dimensional kinematic model
for biological groups.
\newblock \emph{SIAM J Appl Math} 65:152--174.

%\bibitem{bazazi2011}
%Bazazi S, {et~al.}
\bibitem{BazRomTho2011}
Bazazi S, Romanczuk P, Thomas S, Schimansky-Geier L, Hale JJ, 
Miller GA {et~al.}
%Sword GA, Simpson SJ, Couzin ID
\newblock (2011) Nutritional state and collective motion: from individuals to
  mass migration.
\newblock \emph{Proc Roy Soc B}
  278:356-363

\bibitem{SchWhi2011}
%\bibitem{schellinck2011}
Schellinck J, White T
\newblock (2011) A review of attraction and repulsion models of aggregation:
  Methods, findings and a discussion of model validation.
\newblock \emph{Ecol Mod} 222:1897--1911

\bibitem{RomCouSch2009}
Romanczuk P, Couzin I, Schimansky-Geier L
\newblock (2009) Collective motion due to individual escape and pursuit
  response.
\newblock \emph{Phys Rev Lett} 102:10602.

\bibitem{YatErbEsc2009}
%\bibitem{yates2009}
%Yates C, {et~al.}
Yates AC, Erban R, Escudero C, Couzin ID, Buhl J, 
Kevrekidis IG {et~al}
%Maini PK, Sumpter DJT
\newblock (2009) Inherent noise can facilitate coherence in collective swarm
  motion.
\newblock \emph{Proc Natl Acad Sci} 106:5464--5469.


\end{thebibliography}

\end{article}
\end{document}










%\bibitem{rogers2003}
%Rogers S, {et~al.}
%\newblock (2003) Mechanosensory-induced behavioural gregarization in the desert
%  locust schistocerca gregaria.
%\newblock \emph{Journal of experimental biology} 206:3991.

\bibitem{roessingh1993}
Roessingh P, Simpson S, James S
\newblock (1993) Analysis of phase-related changes in behaviour of desert
  locust nymphs.
\newblock \emph{Proceedings: Biological Sciences} pp 43--49.

\bibitem{simpson1999}
Simpson S, McCaffery A, H{\"a}gele B
\newblock (1999) A behavioural analysis of phase change in the desert locust.
\newblock \emph{Biological Reviews} 74:461--480.

\bibitem{sumpter2008}
Sumpter D, Buhl J, Biro D, Couzin I
\newblock (2008) Information transfer in moving animal groups.
\newblock \emph{Theory in Biosciences} 127:177--186.

\bibitem{sumpter2006}
Sumpter D
\newblock (2006) The principles of collective animal behaviour.
\newblock \emph{Philosophical Transactions of the Royal Society B: Biological
  Sciences} 361:5.


\bibitem{WalRai1951}
Waloff Z, Rainey RC
\newblock (1951) Field studies on factors affecting the displacements of desert
  locust swarms in eastern {A}frica.
\newblock \emph{Anti-Locust Bull.} pp 1--50.

\bibitem{Ell1953}
Ellis PE
\newblock (1953) The gregarious behavior of marching \emph{{L}ocusta
  {M}igratoria {M}igratorioides} ({R} \& {F}.) hoppers.
\newblock \emph{J. Exp. Bio.} 30:214--234.

\bibitem{Uva1966}
Uvarov B
\newblock (1966) \emph{Grasshoppers and Locusts}
\newblock (Cambridge University Press, London, UK) Vol.{}~1.

\bibitem{Wal1972}
Waloff Z
\newblock (1972) Observations on the airspeeds of freely flying locusts.
\newblock \emph{Anim. Behav.} 20:367--372.

\bibitem{SkaPopRof1990}
Skaf R, Popov GB, Roffey J
\newblock (1990) The desert locust - an international challenge.
\newblock \emph{Philosophical Transactions of the Royal Society of London
  Series B-Biological Sciences} 328:525--538.

\bibitem{RoeSim1994}
Roessingh P, Simpson SJ
\newblock (1994) The time-course of behavioral phase-change in nymphs of the
  desert locust, schistocerca-gregaria.
\newblock \emph{Physiol. Entomol.} 19:191--197.



\bibitem{Jof1997}
Joffe SR
\newblock (1997) Economic and policy issues in desert locust management: A
  preliminary analysis., (United Nations Food and Agriculture Organization),
  Desert Locust Technical Series AGP/DL/TS/27.


\bibitem{ColDesSim1998}
Collett M, Despland E, Simpson SJ, Krakauer DC
\newblock (1998) Spatial scales of desert locust gregarization.
\newblock \emph{Proc. Natl. Acad. Sci.} 95:13052--13055.


\bibitem{DesSim2000}
Despland E, Simpson S
\newblock (2000) Small-scale vegetation patterns in the parental environment
  influence the phase state of hatchlings of the desert locust.
\newblock \emph{Physiol. Entomol.} 25:74--81.

\bibitem{DesSim2000b}
Despland E, Simpson S
\newblock (2000) The role of food distribution and nutritional quality in
  behavioural phase change in the desert locust.
\newblock \emph{Animal Behaviour} 59:643--652.

\bibitem{DesColSim2000}
Despland E, Collett M, Simpson S
\newblock (2000) Small-scale processes in desert locust swarm formation: how
  vegetation patterns influence gregarization.
\newblock \emph{Oikos} 88:652--662.


\bibitem{WilThoBla2002}
Wilson K, {et~al.}
\newblock (2002) Coping with crowds: Density-dependent disease resistance in
  desert locusts.
\newblock \emph{Proc. Natl. Acad. Sci.} 99:5471--5475.








\bibitem{ReySwoSim2009}
Reynolds AM, Sword GA, Simpson SJ, Reynolds DR
\newblock (2009) Predator percolation, insect outbreaks, and phase polyphenism.
\newblock \emph{Current Biology} 19:20--24.


\bibitem{EscYatBuh2010}
Escudero C, {et~al.}
\newblock (2010) Ergodic directional switching in mobile insect groups.









\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*}






