\documentclass[twoside]{article}
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\markboth{\hfil Asymptotic and transient analysis \hfil}%
{\hfil Thomas C. Gard \hfil}
\begin{document}
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Fourth Mississippi State Conference on Differential Equations and\newline
Computational Simulations,
Electronic Journal of Differential Equations, \newline
Conference 03, 1999, pp 51-62. \newline
URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
\vspace{\bigskipamount} \\
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Asymptotic and transient analysis of stochastic core ecosystem models
\thanks{ {\em Mathematics Subject Classifications:} 92D25, 60H10.
\hfil\break\indent
{\em Key words:} stochastic plankton-fish models, stochastic differential
equations, \hfil\break\indent
asymptotic properties, ultimate boundedness, transient behavior,
exit probabilities.
\hfil\break\indent
\copyright 2000 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Published July 10, 2000. } }
\date{}
\author{ Thomas C. Gard }
\maketitle
\begin{abstract}
General results on ultimate boundedness and exit probability
estimates for stochastic differential equations are applied to
investigate asymptotic and transient properties of models of
plankton-fish dynamics in uncertain environments
\end{abstract}
\newtheorem{thm}{Theorem}
\section{Introduction}
Opposing general points of view on whether or not populations ultimately
survive are succinctly expressed recently by Halley and
Iwasa ([10]) and Jansen and Sigmund ([12]). Is
extinction certain if random variability is taken into account? Or if
parameters are restricted to realistic ranges and the mitigating effects
of
population communities are explicitly considered, will persistence occur
possibly after some initial risk period? Answers to such questions for
real systems are more than pedagogical niceties. They can, for example,
lead to proposed strategies for maintaining large scale bio-physico-chemical
systems such as the highly utilized natural systems constituting watershed
ecosystems. Central to dynamical models for watershed ecosystems are
what might be called core lake ecosystem models, such as plankton-fish
models
which describe the dynamics of a limiting nutrient $P$, and algae $A$,
zooplankton $Z$, and small fish $F$ populations
\begin{eqnarray}
\frac{dP}{dt} &=& \delta(P_I(t) - P) + g_P(t, P, A, Z, F)\nonumber\\
\frac{dA}{dt} &=&g_A(t,P,A,Z)-\delta A\nonumber\\
\frac{dZ}{dt}&=&g_Z(t,A,Z,F)-\delta Z \\
\frac{dF}{dt}&=&g_F(t,Z,F)+F_I(t)\nonumber
\end{eqnarray}
recently discussed in the literature.(See, for example Doveri
et.\,al.\,[3] where specific
functional forms for the interaction portions $g_P$, $g_A$, $g_Z$, and
$g_F$ of the net growth
rates are given.) In (1) $P_I$ denotes the nutrient input rate, and
$F_I$ the small
fish recruitment rate from large fish. Simplified submodels of (1) have
been
discussed; the $PA$ submodel is a resource-consumer model with similar
dynamics to the simple chemostat model ([18]), the
$PAZ$ submodel has been discussed by Ruan ([16],[17]) and others, and the
$AZF$ model is a
three-species food chain.
The relative novelty of (1) is the explicit inclusion of small
fish dynamics - the timing and size of large annual recruitment peaks
simultaneously effecting and being determined by $PAZ$ levels.
Temperature
and other seasonality time variations of parameters together with cyclic
nonlinearities in the model can lead to chaotic regimes ([15]).
All models of real biological systems account for uncertainty in
parameters and structure in one way or another. There is always a
variability ansatz,
although in many cases such assumptions are implicit. An explicit
approach on the other hand is to formulate models with well-defined stochastic
features in order to account for random variability. The class of
stochastic differential equation models for interacting populations is
such a
class which can take into account environmental randomness: the SDE model
analogous to (1) has the general form
\begin{eqnarray}
dP&=&[\delta(P_I-P)+g_P] dt+\sigma_P \,dW_P\nonumber\\
dA&=&[g_A-\delta
A] dt+\sigma_A \,dW_A\nonumber\\
dZ&=&[g_Z-\delta Z] dt+\sigma_Z \,dW_Z \\
dF&=&[g_F+F_I] dt+\sigma_F \,dW_F\,,\nonumber
\end{eqnarray}
where $W_P$, $W_A$, $W_Z$, and $W_F$ are standard Brownian motions and
$\sigma_P$, $\sigma_A$, $\sigma_Z$, and $\sigma_F$ denote the
corresponding
intensities of the noise fluctuations; the $\sigma$'s may be functions
of
the state variables and time. A specific example of (2),
motivated by a
stochastic model of two competitors in a chemostat suggested by
Stephanopoulos, Aris, and Fredrickson [19], is given by
\begin{eqnarray}
dP&=&[\delta_0(P_I-P)+g_P] dt+\delta_1(P_I-P)\, dW\nonumber\\
dA&=&[g_A-\delta_0 A] dt+\delta_1A\, dW\nonumber\\
dZ&=&[g_Z-\delta_0 Z] dt+\delta_1Z \,dW\\
dF&=&[g_F+F_I] \, dt\,.\nonumber
\end{eqnarray}
System (3) arises when the dilution or washout rate $\delta$ is viewed as
the sum of an average value $\delta_0$ plus a random noise fluctuation
with
intensity $\delta_1$ about the average:
\begin{equation}
\delta=\delta_0+\delta_1N\,.
\end{equation}
In (4) $N$ represents standard white noise - in a generalized sense
\begin{equation}
N=\frac{dW}{dt}
\end{equation}
with $W$ a standard Brownian motion. In the next section we give a
result
which obtains asymptotic estimates for the average values of the state
variables in (2) which is analogous to uniform persistence for
the corresponding deterministic model (1). Application to
specific
$PAZF$ models is incomplete at this time and the subject of ongoing
work. Transient behavior of the model may be important whether or not
the former
result applies. The third section contains a result which gives
estimates
for first exit location probabilities from certain bounded sets in the
feasible region which may indicate initial survival or extinction
tendencies
of populations. We show that this result can be applied to models of the
form (3).
\section{Persistence in the mean}
Permanence (uniform persistence together
with dissipativity) is the most basic general qualitative feature to
verify for
interacting population models ([11],[20]); it is
the model analog of mutual survival and non-explosion of the populations
represented in the model. Permanence means that there are positive
constants $K$ and $L$ such that for any component population $X(t)$ with
any positive initial value $X(0)$
\begin{equation}
K\leq \liminf_{t \to \infty} X(t)\leq\limsup_{t\to \infty}
X(t)\leq L.
\end{equation}
If
\begin{equation}
Y(t) = \ln X(t)
\end{equation}
or some other transformation of the ray $(0, \infty)$ to the line
$(-\infty, \infty)$, permanence of $X$ is equivalent to dissipativity or
ultimate boundedness of $Y$: there is a positive constant $M$ such that
for any initial value $Y(0)$
\begin{equation}
\limsup_{t\to\infty} |Y(t)|\leq M.
\end{equation}
There are well-known theorems in differential equations which give
ultimate
boundedness if a Liapunov function exists. In this section we will
apply an
analogous theorem of Miyahara ([14]) for stochastic
differential
equations. It is convenient to change notation here: let
\begin{eqnarray}
X&=&(X_{1,}X_2,X_3,X_4)=(P,A,Z,F)\,, \\
Y&=&\ln(X)\leftrightarrow Y_i=\ln(X_i),\; i =1,2,3,4.
\end{eqnarray}
Applying Ito's formula to (2) yields a transformed system of the
form
\begin{equation}
dY=H(t,Y) dt+\Gamma(t,Y) dW
\end{equation}
where here $W $= ($W_P$, $W_A$, $W_Z$, $W_F), H$ is a 4-d vector function,
and $\Gamma$ is a $4\times 4$ diagonal matrix function.
\begin{thm} {\rm (Miyahara [14])} Suppose there
exists a scalar function $V(t,y)$ which is ${\cal C}^1$ in $t$ and
${\cal C}^2$ in $y$ and a number $p\geq
1$ such that for some constants $a_1$ and $a_2$
and positive constants $c_1$ and $c_2$ and all $y$
\begin{enumerate}
\item $V(t, y) \geq-a_1+c_1\|y\|^p$
\item ${\cal L} V(t, y) \leq a_2-c_2 V(t, y)$
\end{enumerate}
\noindent where
${\cal L} V = V_t+H\cdot\nabla
V+\frac{1}{2} \mathop{\rm trace}(\Gamma\Gamma^T V_{yy})$
and $V_t$ is the partial derivative of $V$ with respect to $t,\ \nabla
V$ is the $y$-gradient of $V$ and $V_{yy}$ denotes
the matrix of second partial derivatives of $V$ with repect to $y$. %
Then for, any solution $Y(t)$ of (11),
\begin{equation}
\limsup_{t\to\infty} E \|Y(t)\|^p \leq \frac{a_1}{c_1}+\frac{a_2}{c_1
c_2}%
\end{equation}
where $E(\cdot)$ denotes the expected value or mean.
\end{thm}
Equations (6) - (8) suggest that the conclusion (12) of Theorem 1
could be called persistence (or permanence) in the mean for $X$.
Applying Theorem 1 requires
a candidate for the Liapunov function $V$. It has been shown by the
author ([8]) that
functions of the form
\begin{equation}
V(y)=\exp \{U(y)\}
\end{equation}
with
\begin{equation}
U(y)=\sum_{i = 1}^n\alpha_i[e^{y_i}-y_i-1]
\end{equation}
for some positive constants $\alpha_i$ can be applied to predator-prey
models. The function $U$ in (14) is Volterra's Liapunov function
transformed from the positive cone $R_{+}^n$ to all of $R^n$ by
(10). Utilizing the $1$-norm
$$
\|y\|=\sum_{i = 1}^n|y_i|
$$
we obtain
\begin{equation}
U(y) \geq-\beta+\alpha \|y\|
\end{equation}
where
$$
\beta=\sum_{i = 1}^n\alpha _i{\rm \ \ and \ \ }\alpha={\rm min\ }
\alpha _i.
$$
From (15) and (13) it follows that
$$
V(y) \geq exp\{-\beta+\alpha \|y\|\}\geq e^{-\beta}(1+\alpha \|y\|).
$$
Condition 1 of Theorem 1 is verified for $p=1$ with
\begin{equation}
a_1=0{\rm \ \ and \ \ } c_1=\alpha e^{-\beta}.
\end{equation}
Obtaining Condition 2 is more difficult. Generally one expects
\begin{equation}
a_2=a_2(\Gamma), c_2=c_2(\Gamma)%
\end{equation}
if the function $V$ has negative definite
derivative $\dot{V}$ along trajectories of the
corresponding deterministic system. The following example shows that
Condition 2 can be obtained for a single population dynamics model with a
Liapunov function of the form (13) even when $\Gamma$ is not
necessarily small. Consider the stochastic logistic model
\begin{equation}
dX=X(1-X) dt+\frac{1}{\sqrt{2}}X dW
\end{equation}
which transforms to
\begin{equation}
dY=\Bigl(\frac{3}{4}-e^Y\Bigr) dt+\frac{1}{\sqrt{2}} dW
\end{equation}
under $Y=\ln (X)$. The Liapunov function (13) here is
\begin{equation}
V(y)=\exp \{e^y-y-1\}.
\end{equation}
It is easy to see that
$$
V(y)\geq|y|
$$
and so we can actually do a little better than (16): we can take
\begin{equation}
a_1=0{\rm \ \ and\ \ } c_1=1.
\end{equation}
To get Condition 2 we need to estimate
\begin{eqnarray}
{\cal L} V(y)&=&\Bigl(1-\frac{1}{4}-e^y\Bigr)V'(y)+\frac{1}{2}\Bigl(%
\frac{1}{2}\Bigr)V''(y)\nonumber\\
&=&\Bigl(1-\frac{1}{4}-e^y\Bigr)(e^y-1)V(y)+%
\frac{1}{4}[(e^y-1)^2+e^y]V(y).
\end{eqnarray}
A brief calculation gives
\begin{equation}
{\cal L} V(y)=\frac{1}{4}[1-3(e^y-1)^2]V(y)\leq\frac{1}{2}-%
\frac{1}{4}V(y),
\end{equation}
i.e., we have Condition 2 satisfied with
\begin{equation}
a_2=\frac{1}{2}{\rm \ \ and\ \ } c_2=\frac{1}{4}.
\end{equation}
Using (21) and (24), the conclusion of the theorem yields
\begin{equation}
\limsup_{t\to\infty} E \|\ln (X(t))\| \leq\frac{1/2}{1/4}=2.
\end{equation}
Volterra's Liapunov function does not seem to work for resource-consumer
models; the counterpart to inequality (23) does not hold. So there
remain some problems in applying Miyahara's result to stochastic $PAZF$
models. We conclude this section by summarizing the conditions which
would
have to be met in order to get a specific result here. For clarity in
stating the conditions, we will consider only the following simplified
version of (2)
\begin{eqnarray}
dP&=&[\delta(P_I-P)+Pf_P(P,A)] dt+(P_I-P)\mu_P(P) \,dW_P\nonumber\\
dA&=&[Af_A(P,A,Z)-\delta A] dt+A\mu_A(A) \,dW_A\nonumber\\
dZ&=&[Zf_Z(A,Z,F)-\delta Z] dt+Z\mu_Z(Z) \,dW_Z\\
dF&=&[Ff_F(Z,F)+F_I] dt+F\mu_F(F)\, dW_F\,.\nonumber
\end{eqnarray}
In particular, we are ignoring nutrient recycling and we are assuming that
parameters are constants. Under the log transformation:
\begin{equation}
Y_1=\ln (P/P_I),Y_2=\ln (A),Y_3=\ln (Z),Y_4=\ln (F)
\end{equation}
the system becomes
\begin{eqnarray}
dY_1&=&[\delta(e^{-Y_1}-1)-\frac{\mu_P^2}{2}(e^{-Y_1}-1)^2+f_P]
dt+(e^{-Y_1}-1)\mu_P
\,dW_P\nonumber\\ > dY_2&=&[f_A-\delta-\frac{\mu_A^2}{2}] dt+\mu_A
\,dW_A\nonumber\\ > dY_3&=&[f_Z-\delta-\frac{\mu_Z^2}{2}] dt+\mu_Z \,dW_Z\\
dY_4&=&[f_F+F_Ie^{-Y_4}-\frac{\mu_F^2}{2}] dt+\mu_F \,dW_F\,,\nonumber
\end{eqnarray}
where in (28)
$f_P=f_P(P_Ie^{Y_1},e^{Y_2})$, $\mu_P=\mu_P(P_Ie^{Y_1})$,\dots.
If we choose a ${\cal C}^2$ function $V$ which satisfies Condition 1
of Miyahara's Theorem: for some number $p\geq 1$ there is a
constant $a_1$ and a positive constant $c_1$ such that
\begin{equation}
V(y) \geq-a_1+c_1\|y\|^p
\end{equation}
we need to verify Condition 2. Condition 2 in Miyahara's Theorem is, for
some positive constant $c$
\begin{eqnarray}
{\cal L} V+cV &=&\frac{\partial V}{\partial
y_1}[\delta(e^{-y_1}-1)+f_P]+\frac{\partial V}{\partial
y_2}[f_A-\delta]\nonumber\\
&&+\frac{\partial V}{\partial y_3}[f_Z-\delta]+\frac{\partial
V}{\partial y_4}[f_F+F_Ie^{-y_4}]\nonumber\\
&&+\frac{1}{2}\Bigl\{\mu_P^2\Bigl(%
\frac{\partial^2V}{\partial y^2_1}-\frac{\partial V}{\partial
y_1}\Bigr) (e^{-y_1}-1)^2+\mu_A^2\Bigl(\frac{\partial^2V}{\partial
y^2_2}-\frac{\partial V}{\partial y_2} \Bigr)\\
&&+\mu_Z^2\Bigl(\frac{\partial^2V}{\partial y^2_3}-\frac{\partial
V}{\partial
y_3} \Bigr)+\mu_F^2\Bigl(\frac{\partial^2V}{\partial y^2_4}-\frac{\partial
V}{\partial y_4} \Bigr)\Bigr\}+cV\nonumber
\end{eqnarray}
is bounded. The result then is
\begin{thm} Suppose there exists a ${\cal C}^2$ function
$V$ defined on $R^4$ which satisfies (29) and (30). Then, for
any solution $Y(t)$ of (28),
\begin{equation}
\limsup_{t\to \infty}
E \|Y(t)\|^p \leq \frac{a_1}{c_1}+\frac{b}{c_1 c}
\end{equation}
where
$b$ is a bound for ${\cal L} V+cV$ i. e., for any solution
$(P(t)$,$A(t),Z(t),F(t))$ of (26) and any positive number $\epsilon$,
\begin{equation}
E \|(\ln (P(t)/P_I),\ln (A(t)),\ln (Z(t)),\ln (F(t)))\|^p \leq
\frac{a_1}{c_1}+%
\frac{b}{c_1 c}+ \epsilon
\end{equation}
for all sufficiently large $t$.
\end{thm}
\section { Exit probabilities}
Even when some form of stability can be verified for
a model, transient behavior may still be important to investigate. If
trajectories enter a region of state space where one or more model
components
are small, features neglected in the model could lead to collapse before a
predicted recovery can occur. For models which attempt to account for
random effects, the situation is particularly critical. Estimating
certain
exit statistics is a natural first approach to deal with this
problem ([6],[7],[9],[13]). Suppose
\begin{equation}
X=(X_{1,}X_2,\ldots,X_n)
\end{equation}
represents the $n$ components of a stochastic dynamical population model
taking
values in the usual positive cone
\begin{equation}
R^n_{+}=\{x=(x_{1,}x_2,\ldots,x_n):x_i>0,i=1,2,\ldots,n\}
\end{equation}
in $n$-dimensional space, and $ B\subseteq R_{+}^n$ is a bounded set.
Then for any fixed $ x\in B$,
we can consider the realization
$$
X=X(t,x), t\geq 0
$$
of the model with $ X(0,x)=x\in B$,
and the corresponding first exit time of $X$,
\begin{equation}
\tau=\tau_x(B)=inf\bigl\{t:X(t,x)\notin B\bigr\}
\end{equation}
from $B$. The first exit time $\tau$ or even its mean or expected value
\begin{equation}
u(x)=E(\tau_x)
\end{equation}
gives an indication of persistence of $X$ relative to the set $B$ ([13]).
For example, if
$\tau_x=\infty$
for all $x\in B$, then $B$ is positive invariant for $X$, and if also the
boundary
$\partial B$ of $B$ is contained in $R_{+}^n$, then the set $B$ is a
candidate for a practical persistence estimate for the model. If it
could
also be shown that each realization $X$ which begins at an $x$ outside $B$
hits $B$ in a finite time before hitting the boundary of $R^n_{+}$,
verification of practical persistence would be complete. If the model for
$X $takes the form of a
stochastic differential equation
\begin{equation}
dX=G(X) dt+\Lambda(X) dW
\end{equation}
as discussed in the previous section, then it is known that the expected
exit
time $u$ solves the boundary value problem
\begin{eqnarray}
{\cal L} u(x) = -1, \quad x\in B\\
u(x) = 0, \quad x\in\partial B\,,\nonumber
\end{eqnarray}
where, as in Theorem 1 above,
\begin{equation}
{\cal L} u =G\cdot\nabla
u+\frac{1}{2} \mathop{\rm trace}(\Lambda\Lambda^T u_{xx})\,.
\end{equation}
For example, for the simple scalar problem,
\begin{equation}
dX = \sqrt{\varepsilon}\ dW, X(0) = x \in B = (0,1)
\end{equation}
with $\epsilon$ any positive number, the boundary value problem for
$u(x)=E(\tau_x)$ is
\begin{equation}
-1 = {\cal L}U(x) = \frac{\varepsilon}{2} u''(x), u(0) = 0 = u(1).
\end{equation}
The solution is easily calculated:
\begin{equation}
u(x)=\frac{1}{\epsilon}(x-x^2).
\end{equation}
Note that $\tau_x$ is finite almost surely in this example. The unit
interval $B$ here is not an estimate for practical persistence; in fact
persistence fails in this example. Although the above example is not a
very
interesting population model, it does exhibit what has become anticipated
behavior of randomly perturbed deterministic models - loss of stability.
In
this situation the size of $\tau_x$ or $E(\tau_x)$ still can indicate
relative persistence. One can also try to determine other exit
statistics
such as exit point location probabilities
\begin{equation}
v(x)=P\{X(\tau_x)\in\partial_\eta B\}
\end{equation}
where $\partial_\eta B$ is some particular subset of the boundary of $B$.
This can be both
physically relevant and mathematically tractable if the set
$B$ and the boundary portion $\partial_\eta B$ are suitably chosen.
Suppose $V$ is a ${\cal
C}^2$ function, and
\begin{equation}
B\subseteq\{x:\eta\leq V(x)\leq\gamma\}
\end{equation}
and
\begin{equation}
\partial_\eta B=\partial B\cap\{x:V(x)=\eta\}.
\end{equation}
We have the following result (See also [8].)
\begin{thm} Let $p =v(x)$, and $q =u(x)=E(\tau_x)$. Suppose there is
a constant $c\geq 0$ such that
\begin{equation}
{\cal L} V(x)\geq c{\rm ,\ for\ all\ } x\in B
\end{equation}
where ${\cal L}$ is the operator given by (39). Then
\begin{equation}
p\leq[\gamma-V(x)-cq]/[\gamma-\eta].
\end{equation}
\end{thm}
\noindent{\bf Remark 1.} Note that, if, for example,
\begin{equation}
V(x) = \frac{\eta + \gamma}{2}
\end{equation}
then (47) becomes
\begin{equation}
p\leq\frac{1}{2}-c q/[\gamma-\eta]
\end{equation}
i.\ e., the term $-c q/[\gamma-\eta]$
gives an estimate of the net bias due to the drift $G(x)$ and diffusion
$\Lambda(x)$ terms in (37).
\bigskip
\noindent Proof of Theorem 3. By Dynkin's formula ([4]) (or by
Ito's formula and taking expected values - see [5], for example)
applied to the process $V(X(t))$ on the random interval $[0,\tau]$ we have
\begin{equation}
EV(X(\tau))-V(x)=E\int_0^\tau{\cal L} V(X(s)) ds
\end{equation}
and then (46) yields
\begin{equation}
E\int_0^\tau{\cal L} V(X(s)) ds\geq cE(\tau).
\end{equation}
From (50) and (51) then we have
\begin{equation}
EV(X(\tau))\geq V(x)+cE(\tau).
\end{equation}
On the other hand, taking into account (44) and (45) , we get
\begin{equation}
EV(X(\tau))\leq p \eta+(1-p) \gamma.\end{equation}
Inequalities (52) and (53) give
\begin{equation} p \eta+(1-p) \gamma\geq V(x)+cE(\tau).\end{equation}
or
\begin{equation} p\leq[\gamma-V(x)-cq]/[\gamma-\eta].
\end{equation} \hfill$\diamondsuit$\medskip
Returning to the example (40): $dX=\sqrt{\epsilon}\ dW, X(0)=x\in
B=(0,1)$ \linebreak for a simple application of Theorem 3, we take
$$
\eta=0,\gamma=1,\ {\rm and\ } V(x)=x^r,
$$
for any number $r$ satisfying $10$ and $\delta_1$
sufficiently large,
\begin{eqnarray}
{\cal L} V(x)&=&\{[\delta_0(P_I-x_1)+g_P+\frac{1}{2}(\delta_1%
(P_I-x_1))^2]/x_1\nonumber\\
&&+ r_2[g_A-\delta_0 x_2+\frac{1}{2}(\delta_1x_2)^2]\\
&&+ r_3[g_Z-\delta_0 x_3+\frac{1}{2}(\delta_1x_3)^2]\nonumber\\
&&+r_4g_F\}V(x)\ \geq \ c\nonumber
\end{eqnarray}
for some positive constant c and for all $x\in B$, since everything in
(63)
is bounded, and all of the terms involving
$\delta_1$ are positive. We remark finally that it should be
noted that both $c$ and $q$ in (47) generally will depend on
$\delta_1$.
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\end{thebibliography}
\noindent{\sc Thomas C. Gard} \\
Department of Mathematics \\
The University of Georgia \\
Athens, GA 30602, USA \\
e-mail: tgard@uga.edu
\end{document}