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\author{Thomas Petzoldt\\ Technische Universit\"at Dresden\\ Institut f\"ur Hydrobiologie\And
        Karsten Rinke \\ Helmholtz-Centre\\ for Environmental Research - UFZ}
\title{\pkg{simecol}: An Object-Oriented Framework for Ecological Modeling in \proglang{R}}
%\VignetteIndexEntry{Introduction to the simecol Package}

%% for pretty printing and a nice hypersummary also set:
\Plainauthor{Thomas Petzoldt, Karsten Rinke} %% comma-separated
\Plaintitle{simecol: An Object-Oriented Framework for Ecological Modeling in R} %% without formatting
\Shorttitle{Ecological Modeling with \pkg{simecol}} %% a short title (if necessary)

%% an abstract and keywords
\Abstract{
The \pkg{simecol} package provides an open structure
to implement, simulate and share ecological models. A generalized object-oriented
architecture improves readability and potential code re-use of models and
makes \pkg{simecol}-models freely extendable and simple to use.
The \pkg{simecol} package was implemented in the \proglang{S4} class system
of the programming language \proglang{R}.
Reference applications, e.g.\ predator-prey models or grid models are provided
which can be used as a starting point for own developments.
Compact example applications and the complete code of an individual-based
model of the water flea \textit{Daphnia} document the efficient usage of  \pkg{simecol} for
various purposes in ecological modeling,  e.g.\ scenario analysis, stochastic
simulations and individual based population dynamics.
Ecologists are encouraged to exploit the abilities of \pkg{simecol} to structure
their work and to use \proglang{R} and object-oriented programming
as a suitable medium for the distribution and share of ecological modeling code.

\textbf{Note:} A previous version of this document has been published as
\citet{Petzoldt2007} in the Journal of
Statistical Software, \url{https://www.jstatsoft.org/v22/i09}.
Please refer to the original publication when citing this work.
}
\Keywords{ecological modeling, individual-based model,
  object-oriented programming (OOP), code-sharing, \proglang{R}}
\Plainkeywords{ecological modeling, individual-based model,
  object-orientated programming (OOP), code-sharing, R} %% without formatting


%% The address of (at least) one author should be given
%% in the following format:
\Address{
  Thomas Petzoldt\\
  Institut f\"ur Hydrobiologie\\
  Technische Universit\"at Dresden\\
  01062 Dresden, Germany\\
  E-mail: \email{thomas.petzoldt@tu-dresden.de}\\
  URL: \url{https://tu-dresden.de/Members/thomas.petzoldt/}\\
  \\
  Karsten Rinke\\
  Helmholtz-Centre for Environmental Research - UFZ\\
  Department of Lake Research\\
  Br\"uckstr. 3a\\
  39114 Magdeburg, Germany\\
  E-mail: \email{karsten.rinke@ufz.de}\\
  URL: \url{https://www.ufz.de/index.php?en=19635}
}

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\begin{document}

<<init,echo=FALSE, include=FALSE, fig=FALSE>>=
library("simecol")
data(lv, package = "simecol")
options("width"=72)
options("prompt" = "R> ", "continue" = "+  ")
@

%% include your article here, just as usual
%% Note that you should use the \pkg{}, \proglang{} and \code{} commands.

\section[Introduction]{Introduction}
%% Note: If there is markup in \(sub)section, then it has to be escape as above.

The \proglang{R} system with the underlying \proglang{S} programming language is well
suited for the development, implementation and analysis of dynamic
models. It is, in addition to data analysis, increasingly used for model simulations
in many disciplines like pharmacology \citep{Torno2004a}, psychology \citep{Maas2006},
microbiology \citep{Jensen2006}, epidemiology \citep{Hartley2006}, ecology
\citep{Fussmann2003,Rinke2005,Levey2005} or econometrics
\citep{Cribari-Neto1999,Zeileis2005}. Existing
applications already cover a range from small conceptual process and
teaching models up to coupled hydrodynamic-ecological models \citep{Rolinski2005}.
Small models can be implemented easily in pure \proglang{R} \citep{Petzoldt2003a}
or by means of the \proglang{XML}-based Systems Biology Markup Language \proglang{SBML}
and the corresponding Bioconductor package \citep{Radivoyevitch2004}.
For more complex or computation intensive simulations
\proglang{R} is primarily used as an environment for data management,
simulation control and data analysis, while the model core may be
implemented in other languages like \proglang{C/C++}, \proglang{FORTRAN} or
\proglang{JAVA}.

This works perfectly at the extremes, but problems appear with medium-sized
models. While larger modeling projects usually start with an extensive
planning and design phase carried out by experienced people,
small models can be implemented ad-hoc without problems in \proglang{R}
from scratch or by modification of online help examples. On the other hand, medium-sized
applications often start by extension of small examples up to ever
increasing size and complexity. An adequate design period is often skipped
and at the end of a modeling project no time remains for re-design or
appropriate documentation. The resulting programs are necessarily ill-structured
in most cases or at least exhibit a very special, proprietary design.

The situation is even worse in ecological modeling, because this discipline is broad,
modeling strategies vary substantially and ecological modelers
are very creative. Different families of models (e.g.\ statistical, differential
equations, discrete event, individual-based, spatially explicit) are applied alone
or in mixtures to different ecological systems (terrestrial, limnetic, marine)
and scales (individuals, laboratory systems, lakes/rivers/forests, oceans,
biosphere).
Not enough that there is a Babel of programming languages and simulation systems,
there is also a Babel of approaches. There are often cases where it seems to
be necessary to understand the whole source code when one tries to modify only
one single parameter value or to introduce a new equation and it is not seldom
easier to re-write code from scratch than to reuse an existing one.

We aim to propose a possible way out of the dilemma, an open
structure to implement and simulate ecological
models.
The \proglang{R} package \pkg{simecol} is provided to demonstrate the
feasibility of this approach including a starter set of examples and utility
functions to work with such models.

After giving a description of the design goals and the specification of the
\code{simObj} class in Sections \ref{s:goals} and \ref{s:approach}
we demonstrate basic use of the package in Section \ref{s:examples}.
The different mechanisms available to implement and simulate
\code{simObj} models are explained in Section \ref{s:implementation}.
A complete individual-based model is given in Section \ref{s:ownmodels}
to elucidate how to use and extend \pkg{simecol} in the
modeling process. Finally, we discuss perspectives of \proglang{R}
and \pkg{simecol} in ecological modeling as well as relations to
other packages (Section \ref{s:discussion}).

\section{Design goals}\label{s:goals}

Our first goal is to provide a generalized architecture for the
implementation of ecological models. Such a unified style, which can be
considered as a template or prototype of model implementations,
provides manifold advantages for a scientific community.
The structured architecture will increase readability of the code
by separating model equations from other code elements,
e.g.\ for numerical techniques. This will enable ecological modellers to use
\proglang{R} as a communication medium and allows to distribute
model source code together with its documentation, e.g.\ as executable part of the
``standard protocol for describing individual-based and agent-based models''
suggested by \citet{Grimm2006}.

We want to stress that we don not intend to establish yet another
simulation system. Complete simulation systems are numerous and
well-established for specified applications, e.g.\
STELLA\footnote{\url{https://www.iseesystems.com/}},
Berkeley Madonna\footnote{\url{https://www.berkeleymadonna.com/}},
VENSIM\footnote{\url{https://www.vensim.com/}},
EcoBeaker\footnote{https://www.ecobeaker.com/} or the GNU open
source system ECOBAS \citep{Benz2001a}. A comprehensive overview about
software used in ecological modeling is given elsewhere e.g.\ at
\url{https://www.systemdynamics.org/} or Chapter 8 of \citet{Grimm2005}.
These simulation systems can be extremely effective for the specific
class of applications they are intended for, but, they often lack the
full power and flexibility of a programming language.
In such cases, model frameworks or simulation libraries are commonly
used to support one specific model family, e.g.\ \proglang{PASCAL}
templates for ordinary differential and delay-differential equations
\citep{Gurney1998}, an object-oriented \proglang{C++} framework like
OSIRIS \citep{Mooij1996} for individual-based models or the
\proglang{Objective C} framework
SWARM\footnote{\url{https://www.swarm.org/}} for agent-based simulations.

An alternative approach is the use of high-level programming environments and
matrix oriented languages like
MATLAB\footnote{\url{https://www.mathworks.com/}} or \proglang{R}
\citep{Rcore2006}. Such languages allow a more interactive development
cycle, compared to compiled languages, and outweigh their performance
handicap by efficient algorithms and compiled libraries for numerics
and data management. Both openness and interactivity have made the
\proglang{R} system a universal scripting interface for the free
combination of a large diversity of applications in statistics and
scientific computing.

The second design goal is to be as open as possible and to take advantage of
the open philosophy of \proglang{R}. Users should be allowed to
employ the full power of \proglang{R}'s graphical, statistical and
data management functions and to use arbitrary code written in
\proglang{R} or compiled  languages.
The complete code of \pkg{simecol}-models should be published under a license
that minimizes dependence from others and guarantees unrestricted use in science
and education, including the possibility to be modified by others.
Within this context, \pkg{simecol} is intended to provide a framework on the
meta-level identifying structure-components of ecological simulation models.

Our third design goal is ease of use and simplicity.
One of the main characteristics of programming languages like \proglang{S} and
\proglang{R} is that users become programmers \citep{Chambers2000}.
Unfortunately, ecologists are commonly not well-trained in programming, which
often hampers their application of models. Therefore, we aim to provide a
software layer that bridges this gap and helps ecologists to work with models.
In consequence, this means for \pkg{simecol} that simplicity of implementation
is more important than efficiency. The system should support a broad
level of user experience -- in our case ecological models covering the
whole range from teaching models to research applications.

From the perspective of a first time user it should be possible to run simulations
without knowing too much about \proglang{R} and implementation details. A simulation
of an ecological model should be as easy as fitting a linear model in
\proglang{R} (see example in Section \ref{s:simulation}).
A number of memorable ``commands'', i.e.\ a few essential but not overwhelmingly
extensive generics for simulation, printing, plotting and data access, and utility functions
accompany this package. Both the functions and also the simulation models should
have meaningful defaults to enable new users to get almost immediate success
and to enable experienced developers to structure their applications and to avoid
unnecessary copy and paste \citep{Maechler2004}.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Approach}\label{s:approach}

The approach follows directly from the design goals to provide (i)
a standardized structure, (ii) open and reusable code and (iii) ease of use of
``the model''. It is almost self-evident to apply an object-oriented design,
consisting of:

\begin{enumerate}
\item A general and extensible class description suitable for ecological
  simulation models that allows sub-classes for different model families
  and multiple instances (objects of class \code{simObj})
  which can be used simultaneously without interference,
\item Generic functions which work on objects of these classes and behave
  differently depending on the model family they work with.
\end{enumerate}

All equations, constants and data needed for one particular simulation
should be included in the model object, with the exception of general and widely
needed functions, e.g.\ numerical algorithms. In the following sections we first
analyse what is generally needed and then describe the particular approach.

\subsection{State space approach}\label{c:statespace}

Most ecological models can be formulated by means of a state space representation, known
from statistics and control theory (Figure \ref{f:statespace}).
This applies to dynamic (discrete resp. continuous) systems as
well as to static, time independent systems when postulating that the latter case
is a subset. A general description that is valid for both
linear and nonlinear systems can be given as:

\begin{align}
\mathbf{\dot{x}}(t) & = \mathbf{f}(t, \mathbf{x}(t), \mathbf{u}(t), \mathbf{p})\\
\mathbf{y}(t)       & = \mathbf{g}(t, \mathbf{x}(t), \mathbf{u}(t), \mathbf{p})
\end{align}

where $\mathbf{x}$ is the state of the system and $\dot{\mathbf{x}}$ its first
derivative, $t$ the time, $\mathbf{u}(t)$ is the input vector (or vector of
boundary conditions), and
$\mathbf{y}$ is the output vector.
The functions $\mathbf{f}$ and $\mathbf{g}$ are
the state transition function and the observation function, respectively, which rely on
a vector  $\mathbf{p}$ of constant parameters.

A simulation of a dynamic system is obtained by applying a suitable numerical
algorithm to the function $\mathbf{f}$. This algorithm can be a simple iteration or,
when $\mathbf{f}$ is a system of ordinary differential equations, an appropriate
ODE solver or a function giving an analytical solution.

Compared to the usual statistical models in \proglang{R}, ecological
models are more diverse in their structure and exhibit tight
relationships between procedural code (methods, equations) and data.
Non-trivial ecological models are based on more or less modular building
blocks (submodels), which are either base equations or complex models
themselves.

\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{statespace}
\caption{\label{f:statespace} State space diagram of a dynamic system ($\mathbf{x}(t)$:
   state vector of the system, $\mathbf{x}(t_0)$ initial state, $\mathbf{\dot{x}}$:
   first derivative of the state vector,
   $\mathbf{u}$: input matrix, $\mathbf{y}(t)$: model output,
   $\mathbf{f}$ state transition function, $\mathbf{g}$ observation function,
   $\mathbf{p}$ constant parameters of
  $\mathbf{f}$ and $\mathbf{g}$), figure redrawn after \protect\citet{Ludyk1995},
  see also \protect\citet{Bossel1992} and \protect\citet{Wolkenhauer2005}.}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=\textwidth]{classes}
\caption{\label{f:classes} Class diagram of \code{simObj} and related classes.
 The subclasses and example model objects are provided as reference for
 user-defined and future extensions.}
\end{figure}


\subsection{The simObj specification}

In essence, what do we need to implement a not too narrow
class of ecological models?  We need self-contained objects derived from
classes with suitable properties and methods (data slots and function slots)
resulting from the state space description: state variables, model equations
and algorithms, model parameters (constants),
input values, time steps, the name of an appropriate numerical algorithm (solver),
and an optional set of possibly nested sub-models (sub-equations). These parts
are implemented as slots of the \code{simObj} class from which subclasses for
different model families can inherit (Figure \ref{f:classes}).

%\pagebreak

A small set of supporting functions is provided to work with these objects,
namely:

\begin{itemize}
\item Generic functions for simulation, printing, plotting,
  slot manipulation (accessor functions) and object creation
  (\code{initialize} functions),
\item Utility functions, e.g.\ neighborhood relations for cellular automata.
\end{itemize}

\subsection{Generic functions}

In the \proglang{S4} class model of the \proglang{S} language methods are based on generic
functions. They specify the behavior of a particular function, depending on the
class of the function arguments \citep{Chambers2001}. All generic functions in
\pkg{simecol} are defined as default methods for the class \code{simObj} and
specific methods exist if necessary for subclasses. If new subclasses are
defined for additional model families by the user it may be necessary to create
new methods that work with these user-defined data types and provide
the required functionality.

\subsubsection{Simulation}\label{s:simulation}

The core function to work with \code{simObj}ects is the generic function
\code{sim(simObj, ...)}, which, for dynamic systems, simulates an initial value
problem using the initial state, boundary conditions, equations and parameters stored in one
particular \code{simObj} instance by calling the numerical algorithm referred by its name
in the \code{solver} slot of \code{simObj}.
Common for all versions of \code{sim} is the pass-back modification behavior,
i.e.\ a modified version of the original \code{simObj}
is returned with a newly added or updated slot \code{out} holding the simulation results:

<<out, echo=TRUE, eval=FALSE>>=
library("simecol")
data(lv, package = "simecol")
lv <- sim(lv)
plot(lv)
o1 <- out(lv)
@

The functionality of \code{sim} can vary for different subclasses
of \code{simObj} e.g.\ \code{odeModel}, \code{gridModel}, \code{rwalkModel}.
This behavior results mainly from a different data structure of the state variables
and the set of numerical algorithms that are adequate for a given family of
ecological models.
Whereas ODE models have a vector for state and a data frame for outputs, grid
models may have a grid matrix for state and a list of grids (one grid per time step)
as output, and finally, random walk models may have a list for the initial state
of the particles and a list of lists for the output.

The returned \code{simObj} can be printed and plotted directly with appropriate
functions, the simulation results can be extracted with \code{out} or the
resulting \code{simObj} can be used in another simulation with modified data
or functionality.

\subsubsection{Accessor functions}

Similar to the \code{out} function other accessor functions are available for
all slots with (in opposite to \code{out}) not only read but also write access.
These functions are used similarly like the base function \code{names} and work with the
appropriate data structures, see help files for details. The functions
allow to change either the whole content of the respective slot or
to change single elements, e.g.\ parameter values, time steps or equations.
For example, the following will change only the parameter value of \code{k1}:

<<parm1,echo=TRUE>>=
parms(lv)
parms(lv)["k1"] <- 0.4
@

An entirely new parameter is added to the parameter vector via:

<<parm2,echo=TRUE>>=
parms(lv)["a"] <- 1
parms(lv)
@

Elements can be deleted when a modified version of the parameter vector is assigned:


<<parms3,echo=TRUE>>=
parms(lv) <- parms(lv)[-4]
parms(lv)
@

The behavior is analogous for all other slots with the exception of
\code{out}, given that the correct data type for the respective slot
(vector, list or matrix) is used.

In addition to the command line accessor functions, graphical
\proglang{Tcl/Tk} versions exist (\code{editParms}, \code{editTimes},
\code{editInit})\footnote{These functions replace the deprecated
\code{fixFoo} functions, e.g. \code{fixParms}. Use \code{foo <-
editParms(foo)} instead of \code{fixParms(foo)}}, however, more
complex data types cannot be handled yet by these functions.

\subsubsection{Numerical algorithms}

In order to simulate ecological models of various types, the appropriate numerical algorithms can be plugged into
the \code{sim} function either by using an existing function, e.g.\ from this package,
by imported solvers of package \pkg{deSolve} or by user-defined algorithms.

The algorithm used for one particular \code{simObj} is stored as character string
in the \code{solver} slot of the object.
User-defined algorithms have to provide interfaces (parameter line, output structure) and
functionality (see below) that fit into the respective object specification and are
compatible to the data structures of the particular class.

\subsection{Utility functions}

A few utility functions are provided for overcoming frequently occuring problems.
However, it is not planned to overload \pkg{simecol} with numerous utilities as most of
them are application-specific. Additional supporting functions should be written
in the user workspace when they are needed or may be included in optional packages.


\subsubsection{Interpolation}

Dynamic systems often require interpolation of input data. This is particularly
important for ODE solvers with automatic step size adjustment and there are cases
where excessive interpolation outweighs the advantages of automatic step size
determination.

The performance of linear approximation is crucial and we found that the performance
of the respective functions from the \pkg{base} package can be increased if
\code{approxfun} is used instead of \code{approx}, if matrices are used instead
of data frames and if the number of data (nodes) in the inputs is limited
to the essential minimum.
In addition to this, two special versions \code{approxTime}
and \code{approxTime1} provided by \pkg{simecol} may be useful, see the
help file for details.

\subsubsection{Neighborhood functions}

The computation of neighborhood is time critical for cellular automata. Two
\proglang{C++} functions, \code{eightneighbors} (direct neighbors) and
\code{neighbors} (generalized neighborhood with arbitrary weight matrices)
are provided for rectangular grids. The implementation of these functions is
straightforward and may serve as a starting point for even more efficient
solutions or other grid types, for example hexagonal or 3D grids.

Neighborhood functions can also be used for spatially explicit models.
Models of this family commonly include both, an explicit spatial
representation of organisms (in most cases with real-valued locations) and a
grid-based representation of environmental factors \citep{Dunning1995, Grimm2005}.

\subsection{Example models}

A set of small ecological models is supplied with the package. These models are intended
as a starting point for testing the package and for own developments. The models
are provided in two versions, as binary objects in the data directory and
in full source code in the directory ``examples''. The number of example models is intentionally
limited and will grow only moderately in the future. In addition to this,
ecological models which follow the \code{simObj} specification are well suited
to be published and shared between scientists either as single code objects or
in domain specific packages.

\begin{figure}
\rule{\textwidth}{0.1mm}
\\
\begin{minipage}{0.58\textwidth}
\small
\begin{Code}
CA <- new("gridModel",
    main = function(time, init, parms) {
      z     <- init
      nb    <- eightneighbors(z)
      pgen  <- 1 - (1 - parms$pbirth)^nb
      zgen  <- ifelse(z == 0 &
                 runif(z) < pgen, 1, 0)
      zsurv <- ifelse(z >= 1 &
                 runif(z) < (1 - parms$pdeath),
                 z + 1, 0)
      zgen + zsurv
    },
    parms  = list(pbirth = 0.02, pdeath = 0.01),
    times  = c(from = 1, to = 50, by = 1),
    init   = matrix(0, nrow = 40, ncol = 40),
    solver = "iteration"
)
init(CA)[18:22,18:22] <- 1
\end{Code}
\end{minipage}
\begin{minipage}{0.4\textwidth}
\includegraphics[width=\textwidth]{ca}
\end{minipage}
\\
\rule{\textwidth}{0.1mm}
\caption{\label{f:stochCA} Stochastic cellular automaton, source code (left) and
after 50 iterations (right).}
\end{figure}

\pagebreak{4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Two introductory examples}\label{s:examples}

\subsection{Cellular automaton}

At the first level of experience, users can simply explore example models
supplied with the package or provided by other users without
carrying too much on implementation details. They can be loaded with
\code{source} from harddisk or the Internet,
for example the stochastic cellular automaton shown in Figure \ref{f:stochCA}:

<<CA,eval=FALSE, echo=TRUE>>=
#library("simecol")
data(CA, package="simecol")
CA <- sim(CA)
plot(CA)
@

Note, that the \code{sim} function uses pass-back modification, i.e.\ the result
is the complete \code{simObj} with the model outputs inserted.
The advantage is that the resulting \code{simObj} is consistent, i.e.\ the model
output corresponds to the equations, parameters and other settings of the \code{simObj}.
Now, the settings may be inspected and changed, e.g.\ the number of time steps:

<<CA,eval=FALSE, echo=TRUE>>=
times(CA)
times(CA) <- c(to=100)
CA <- sim(CA)
plot(CA)
@

\subsection{Predator-prey model}\label{s:predprey}

A second built-in demonstration example of \pkg{simecol},
is the elementary Lotka-Volterra predator-prey model,
which can be given by two ordinary differential equations:

\begin{align}
\frac{dX_1}{dt} &= k_1 X_1 - k_2 X_1 X_2   \label{e:lv1}\\
\frac{dX_2}{dt} &= - k_3 X_2 + k_2 X_1 X_2 \label{e:lv2}
\end{align}

In order to reproduce a schoolbook example two scenarios may be created by modifying
two copies (clones) of \code{lv}:

<<lv1,echo=TRUE>>=
#library("simecol")
data(lv, package="simecol")
lv1 <- lv2 <- lv
@

We now inspect default settings of initial values and parameters, modify them
as required for \code{lv2} and simulate both scenarios:

<<lv2,echo=TRUE>>=
init(lv1)
parms(lv1)
parms(lv2)["k3"] <- 0.1
lv1 <- sim(lv1)
lv2 <- sim(lv2)
@

The outputs of \code{lv1} and \code{lv2} can be compared visually using the
plotting method of the \code{odeModel} class (\code{plot(lv1)}) or
with regular plotting functions after extracting the outputs (Figure \ref{f:lv12}).
It is quite obvious that scenario 1 produces stable cycles and that scenario 2
is at equilibrium for the given initial values and parametrization, because of:

\begin{alignat}{2}
\frac{dX_1}{dt} &= 0.2 \cdot 0.5 - 0.2 \cdot 0.5 \cdot 1 &= 0 \\
\frac{dX_2}{dt} &= -0.1 \cdot 1  + 0.2 \cdot 0.5 \cdot 1 &= 0
\end{alignat}


<<lv12,echo=FALSE,fig=TRUE,width=10,height=5>>=
o1 <- out(lv1); o2 <- out(lv2)
par(mfrow=c(1,2), las=1)
plot(o1$time, o1$prey, ylab="Prey, Predator",
  xlab="Time", col="blue", type="l", ylim=c(0,2), main="Scenario 1")
lines(o1$time, o1$predator, col="red", lty="dashed")
plot(o2$time, o2$prey, ylab="Prey, Predator",
  xlab="Time", col="blue", type="l", ylim=c(0,2), main="Scenario 2")
lines(o2$time, o2$predator, col="red", lty="dashed")
legend(10, 2, c("Prey", "Predator"), col=c("blue", "red"),
  lty=c("solid", "dashed"), lwd=1)
@

\begin{figure}
\centering
\includegraphics[width=\textwidth]{tp-lv12}
\caption{\label{f:lv12} Two scenarios of a basic Lotka-Volterra model.
Scenario 1 (left) shows stable cycles, Scenario 2 (right) is at
equilibrium.}
\end{figure}


It is a particular advantage of \proglang{R}, that the complete set
of statistical functions is immediately available, e.g.\ to inspect
summary statistics like the range:

<<lv_range,echo=TRUE>>=
sapply(o1[c("predator", "prey")], range)
sapply(o2[c("predator", "prey")], range)
@

The identity of the lower and upper limits for scenario 2 confirm the equilibrium
state. Moreover, the period length of the cycles of scenario 1 can be analysed by
means of spectral analysis:

<<lv_spectrum,echo=TRUE, include=FALSE>>=
tlv <- times(lv1)
ots <- ts(o1[c("predator", "prey")], start=tlv["from"],
          end=tlv["to"], deltat=tlv["by"])
sp  <- spectrum(ots, spans=c(3,3), log="no")
1/sp$freq[sp$spec[,1] == max(sp$spec[,1])]
@

which yields an estimated period length of approximately
\Sexpr{round(1/sp$freq[sp$spec[,1] == max(sp$spec[,1])])} time units.



\begin{table}[tp]
\caption{\label{t:lvmodel} Implementation example of the elementary
Lotka-Volterra model}

\hrulefill
\begin{Code}
lv <- new("odeModel",
  main = function (time, init, parms, ...) {
    x <- init
    p <- parms
    dx1 <-   p["k1"] * x[1] - p["k2"] * x[1] * x[2]
    dx2 <- - p["k3"] * x[2] + p["k2"] * x[1] * x[2]
    list(c(dx1, dx2))
  },
  parms  = c(k1=0.2, k2=0.2, k3=0.2),
  times  = c(from=0, to=100, by=0.5),
  init   = c(prey=0.5, predator=1),
  solver = "rk4"
)
\end{Code}
\hrulefill
\end{table}


\begin{table}[t]
\caption{\label{t:upca} Uniform Period Chaotic Amplitude Model after
  \citet{Blasius1999}. Note that function \code{f1} is nested within \code{f2}.}

\hrulefill
\begin{Code}
upca <- new("odeModel",
  main = function(time, init, parms) {
    with(as.list(c(init, parms)), {
      du <-  a * u           - alpha1 * f(u, v, k1)
      dv <- -b * v           + alpha1 * f(u, v, k1) - alpha2 * f(v, w, k2)
      dw <- -c * (w - wstar) + alpha2 * f(v, w, k2)
      list(c(du, dv, dw))
   })
  },
  equations  = list(
    f1 = function(x, y, k){x * y},                    # Lotka-Volterra
    f2 = function(x, y, k){f1(x, y, k) / (1 + k * x)} # Holling II
  ),
  times  = c(from=0, to=100, by=0.1),
  parms  = c(a=1, b=1, c=10, alpha1=0.2, alpha2=1,
              k1=0.05, k2=0, wstar=0.006),
  init = c(u=10, v=5, w=0.1),
  solver = "lsoda"
)

equations(upca)$f <- equations(upca)$f1
\end{Code}
\hrulefill
\end{table}

\section{Implementation of simecol models}\label{s:implementation}

\subsection{Lotka-Volterra model}

The implementation of the Lotka-Volterra equations is straightforward and results
in a compact \proglang{S4} object (Table \ref{t:lvmodel}). The two equations
(Eq. \ref{e:lv1}, \ref{e:lv2}) can easily be put into the
main function and there is no need for sub-equations. The code can be made even
simpler without the two assignments at the beginning of main, but with respect
to more structured models we found it generally advantageous to keep the default
values of the names in the parameter line and on the other hand to use common
symbols in the equations.

\subsection{Models with nested subequations}

For large models with numerous equations or for models with alternative
(i.e.\ exchangeable) submodels it may be preferable to use a separate structure.
Although \pkg{simecol} principally allows implementing subroutines as local
functions of the \code{main} slot or even directly in the user workspace such
a strategy would not be in line with our design goals. Instead, the
\code{equation}-slot of the \code{simObj} class definition provides the structure
where relevant submodels and model equations are stored.
Consequent usage of the \code{equation} slot helps to increase the readability
of the \code{main} function, leads to more structurized code and complies with
the object-oriented paradigm. Moreover, the \code{equation} slot can be used to
store alternative submodels, see Table \ref{t:upca} for a small example.

In this example, two versions of the functional response can be enabled alternatively
by assigning one of \code{f1} or \code{f2} to \code{f} via \code{equations}
(last line of the Table \ref{t:upca}) and with the same mechanism it is possible
to introduce further functional response curves.

The example shows also several techniques for scoping: (i) the parameter vector
is ``unpacked'' using \code{with}\footnote{This requires conversion of the parameter
vector into a list. Parameter vectors are only used in the \code{odeModel} class
to ease implementation of teaching models, all other classes natively use lists.},
(ii) parameters are explicitly passed to subequations and (iii) slot functions of \code{equations}
can be used without explicit pass through, a functionality that is provided by the
\code{solver} functions. The result is, that
all elements of the \code{equations} slot are visible within \code{main} and
within all other functions of \code{equations}.

\subsection{Input data}

The simulation models presented so far are autonomous, i.e.\ they have no external
forcing data (matrix $\mathbf u$ in Figure \ref{f:statespace}). Such time dependent
data, e.g.\ food availability or meteorological conditions, which are required in
many practical cases can be provided in the \code{inputs} slot.  In order to give
a minimal example we may create a new \code{odeModel} by modifying a clone
\code{lv\_ef} of the elementary predator-prey model.
To enable external forcing a modified version of the \code{main} slot is introduced,
that simulates substrate ($S$) dependent growth of the prey population:

<<lv_ef1,echo=TRUE>>=
lv_ef <- lv

main(lv_ef) <-  function (time, init, parms, ...) {
  x <- init
  p <- parms
  S <- approxTime1(inputs, time, rule=2)["s.in"]
  dx1 <-   S * p["k1"] * x[1] - p["k2"] * x[1] * x[2]
  dx2 <- - p["k3"] * x[2] + p["k2"] * x[1] * x[2]
  list(c(dx1, dx2))
}
@

For linear interpolation the utility function \code{approxTime1} is used here
to read the input at correct time steps from the input matrix, which can be given as:

<<lv_ef2,echo=TRUE>>=
inputs(lv_ef) <-  as.matrix(data.frame(
  time = c(0, 30, 30.1, 100),
  s.in = c(0,  0,  .5,     .5)
))
@

Note, that inputs are converted into a matrix for performance reasons because otherwise
repeated conversions were performed by \code{approxTime1}, or similarly by
\code{approx}, which would be time consuming, especially for larger
input data sets.

The resulting model can then be easily simulated and plotted and results in
Figure \ref{f:lv_eff3}:

<<lv_eff3,fig=TRUE,width=8,height=4,echo=TRUE>>=
o <- out(sim(lv_ef))
matplot(o$time,o[2:3], xlab="Time",
  ylab="Substrate, Prey, Predator", type="l",
  lty=c("solid", "dashed"), col=c("blue", "red"), las=1)
inp <- as.data.frame(inputs(lv_ef))
lines(inp$time, inp$s.in, col="darkgreen", lwd=2, lty="11")
@

\begin{figure}
\centering
\includegraphics[width=\textwidth]{tp-lv_eff3}
@
\caption{\label{f:lv_eff3} Externally forced predator-prey model
(prey: blue, solid; predator: red, dashed) with resource (green dotted line).}
\end{figure}


\begin{table}
\caption{\label{t:lv_efr} Predator-prey simulation with stochastic input
  variables. The example is derived from the externally forced object \code{lv\_ef}.
  An initialisation function \code{initfunc} is provided which is called
  by \code{initialize} and returns a re-initialized \code{obj} with
  a new random sample of input values. The utility function \code{fromtoby}
  is used to expand the time vector from its compact form
  \code{c(from=, to=, by=)} into a sequence.}

\hrulefill
%% switch R> prompt temporarily off  (table style)
<<echo=FALSE,include=FALSE>>=
options("prompt" = " ", "continue" = " ")
@

<<lv_efr,echo=TRUE,fig=TRUE,width=10,height=6>>=
lv_efr <- lv_ef
tt     <- fromtoby(times(lv_efr))
o      <- matrix(0, nrow=length(tt), ncol=10)
initfunc(lv_efr) <- function(obj) {
  tt <- fromtoby(times(obj))
  inputs(obj) <- as.matrix(data.frame(
    time = tt,
    s.in = pmax(rnorm(tt, mean=1, sd=0.5), 0)
  ))
  obj
}
for (i in 1:10) {
  lv_efr <- initialize(lv_efr)
  lv_efr <- sim(lv_efr)
  o[,i] <- out(lv_efr)$prey
}
matplot(tt, o, xlab="Time", ylab="Prey", las=1, type="l")
@

%% switch R> prompt on again
<<echo=FALSE,include=FALSE>>=
options("prompt" = "R> ", "continue" = "+ ")
@
\hrulefill
\end{table}

\begin{figure}
\centering
\includegraphics[width=\textwidth]{tp-lv_efr}
\caption{\label{f:lvefr} Ten stochastic realizations of the model from Table
  \ref{t:lv_efr}.}
\end{figure}


\subsection{Initializing}

Sometimes, it may be required to perform computations while initializing a
\code{simObj}. This may be either required to ensure consistency between different
slots (e.g.\ parameters, inputs and initial values) to perform error checking or
to create non-deterministic variants. Initializing methods, which exist in \proglang{R}
as class methods of the generic \code{initialize}, are called either explicitly
or implicitly during object creation. The syntax allows,
in principal, two different modes of use. One can either provide all slots of
the object in creation as named arguments to \code{new} or one can provide an
existing \code{simObj} as the first un-named argument to \code{initialize}
in order to get a re-initialized clone.\footnote{Initialization is now done
automatically before each call to sim (introduced in version 0.6).}

In the case of \code{simObj} this mechanism is extended by
an optionally existing function slot \code{initfunc}, which is executed during the
object creation process. Object creation is then as follows: in the first step
an incomplete object is created internally via \code{new} according to the slots given
and in the second step this object in creation is passed to the \code{obj} argument of
\code{initfunc} which performs the final steps and returns the complete object.

It would, of course, also be possible to create a provisional object first and
to modify it afterwards with accessor functions, but \code{initfunc} provides a more efficient solution and
helps to ensure consistency, e.g.\ between parameters and inputs, between
\code{times} and \code{inputs} or between
different state variables.

In the example shown in Table \ref{t:lv_efr} new instances with
different stochastic realizations of the input variables are created and simulated
(see Figure \ref{f:lvefr}).
Note that \code{initfunc} is called automatically every time, when
new instances are created via \code{new} or \code{initialize}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Creating own models}\label{s:ownmodels}

\subsection{The modeling cycle}

The modeling process is an iterative cycle of tasks \cite[see][]{Grimm2005}.
It begins with the formulation of (i) questions and (ii) hypotheses, (iii) the
translation of these questions into a specific model structure, (iv) the
implementation of the model in form of computer software, (v) the analysis,
test and (in most cases) revision of the model, and (vi) communication of the
model and its results to the scientific community.
Another view is given by \citet{Fowler1997}, who with respect to software modeling
suggested to distiguish three different perspectives:

\begin{enumerate}
\item Conceptional perspective,
\item Specification perspective,
\item Implementation perspective.
\end{enumerate}

These perspectives are complementary to the tasks defined by \citet{Grimm2005} when
tasks (i)--(ii) are regarded as conceptional and task (iii) as specification.
In the following we concentrate on task (iv) to explain by means of a real,
but still simple example, how a specified model can be implemented using \proglang{R}
and the \pkg{simecol} software.

\subsection{Conceptional perspective: an individual-based model of Daphnia}

The scientific purpose of the \textit{Daphnia} model given here was
the analysis of demographic effects of \textit{Daphnia} (water flea) populations.
Two main hypotheses should be tested:

\begin{itemize}
\item Size-selective predation leads to an increased population mortality rate,
  compared to non-selective predation by fish \citep{Mehner1998}.
\item In comparison to predictions from the conventional Lotka-Volterra approach
  the inclusion of demographic effects results in a delayed but then inexpectedly
  rapid decline of abundance during periods of food limitation due to
  ageing effects \citep{Huelsmann2002}.
\end{itemize}

Due to a multiple number of important features, the genus \textit{Daphnia} (water flea)
is an outstanding model organism in limnology, toxicology and population ecology, so
results derived on this example may be of general interest to other areas as well.

The \textit{Daphnia} model consists of three general parts:

\begin{enumerate}
\item A semi-empirical model of temperature and food dependent somatic growth and
  egg production derived from laboratory experiments
  (TeFI = temperature-food-interaction model) according to \citet{Rinke2003},
\item An empirical function of egg development time after \citet{Bottrell1976},
\item A non-spatial individual-based simulation of population dynamics.
\end{enumerate}

Individual-based models (IMBs) are a popular technique in ecological modeling
\citep{DeAngelis1992a,Grimm1999,Grimm2005}. It is our aim to demonstrate
how such models can be implemented with \pkg{simecol}.

\subsection{Model specification}

The state of the system is defined as a sample of individuals, each having four states:
age, size, number of eggs and age of eggs. Population development is dependend on
two environmental variables, food (phytoplankton, given in $\rm mg\,L^{-1}$ carbon)
and temperature (in \textcelsius). The model is simulated in fixed time steps
(usually 0.1 day) over a period of several days up to a few months. The time scales
are selected with respect to the egg development time, which is about 4.4 days at
15\,\textcelsius \citep{Bottrell1976}.

The life cycle of \textit{Daphnia} starts with the release of neonate individuals of
a given size ($L_0$) from the brood chamber of the mother into the water.
Somatic growth follows the von Bertalanffy growth equation \citep{Bertalanffy1957},
depending on several empirical parameters. As soon as an individual reaches a fixed size
at maturity (SAM) a clutch of eggs is produced (spawned), whereby the clutch size
(number of eggs) is controlled by food availablility.
After a temperature dependend egg development time the
individuals from this clutch are released (hatched) and the cycle is started again
(parthenogenetic, i.e.\ asexual reproduction).

Mortality can be modelled with arbitrary deterministic or stochastic mortality functions,
e.g.\ size dependent mortality due to fish predation, but for the first simulation
a deterministic fixed maximum age is used.
All equations and parameters are given in detail in \citet{Rinke2003}
and although more elaborate bioenergetic \textit{Daphnia}
models are available in the meantime \citep{Rinke2005,Rinke2005a}, the relatively
simple model given here should be sufficient for the intended purpose.

\subsection{Model implementation}

\subsubsection{Definition of a user-defined subclass}

A subclass for non-spatial individual-based models was not available in past versions of
\pkg{simecol}, but could be easily derived from \code{simObj} as class with
appropriate data types, in particular with a \code{data.frame} for the
table of the individuals stored in \code{init}:

<<ibm_class,echo=TRUE,eval=FALSE>>=
setClass("indbasedModel",
  representation(
    parms  = "list",
    init   = "data.frame"
  ),
  contains = "simObj"
)
@


Since \pkg{simecol} version 0.8-4 class \code{indbasedModel} is a built-in class. The
code snippet above is left in this document as an example how to derive user-defined
subclasses.

\subsubsection{Implementation of the model equations}

The implementation which is provided in Table \ref{t:ibmdaphnia1} and \ref{t:ibmdaphnia2}
starts with the selection of an appropriate data structure
for the state variables. A table of individuals
with four columns: age, size, number of eggs and age of eggs (\code{eggage})
is realized as data frame with one row for each individual.
The data frame is initialized with an arbitrary number of start individuals
(e.g.\ one single juvenile animal in the \code{init} slot).

The \code{main} function simulates the life cycle of \textit{Daphnia} and calls the
sub-equations \code{live}, \code{survive} and \code{hatch} which implement
the following processes:

\begin{description}
  \item[live:] The age of all individuals and the egg-age for
    individuals with eggs is incremented by the actual
    time step \code{DELTAT}. Then, the empirical function \code{tefi} is called
    to estimate length and potential number of eggs as a function of age, food
    and temperature. The data frame of individuals is then updated and for all
    adult individuals (size > size at maturity, SAM) which actually have no eggs
    the appropriate number of eggs is initialized.
  \item[survive:] The survival function returns the subset of surviving
    individuals. Note that it is particularly easy in \proglang{R} to implement
    survival with the \code{subset} function by simply applying a logical rule to
    the table of individuals and returning only those rows which match the condition.
  \item[hatch:] In a first step the the actual egg age is compared with the
    egg development time. Then the total number of mature eggs is counted
    and a data frame (\code{newinds}) with an appropriate number of individuals is
    created (function \code{newdaphnia}. Population growth occurs by appending
    the data frame with newborns (\code{newinds}) to the data frame of the
    surviving (\code{inds}).
\end{description}

All functions of the life cycle receive the actual table of individuals
(\code{init}) as their first argument and return an updated table
\code{inds} which is then passed back to \code{init}.

The model is simulated by iteration over the time vector. Note that in contrast
to ODE models the \code{main} function explicitly returns the new state and
not derivatives. To account for this, the \code{iteration} algorithm is to be
used here and not one of the ODE solvers like \code{euler}, \code{rk4} or \code{lsoda}.

A number of constant parameters is needed by the empirical model
\citep[see][for details]{Rinke2003}, which are represented as list within
the \code{parms} slot.

\subsection{Model simulation}

With the \code{ibm\_Daphnia} object derived from the \code{indbasedModel} class
and given as complete source code in Tables \ref{t:ibmdaphnia1} and
\ref{t:ibmdaphnia2} it is now possible to perform a first simulation:

<<ibm_daphnia, echo=FALSE, include=FALSE>>=
source("ibm_daphnia.R")
@

<<sim_ibm_daphnia, echo=TRUE,eval=FALSE>>=
solver(ibm_daphnia) <- "iteration"
ibm_daphnia <- sim(ibm_daphnia)
@

This already works with the \code{iteration} method provided by the package,
but the default behavior may not be optimal for the concrete subclass.

One disadvantage here is the fact that the default
\code{iteration} algorithm stores the complete state data structure
(i.e.\ the complete data frame) for each time step as list in the \code{out} slot.
This behavior is rather memory consuming for individual-based simulations with
several hundred or thousand individuals. Moreover, no adequate plotting method is
currently available for such a list of data frames and therefore the default
\code{plot(simObj)} method simply returns a warning message.

\subsection{Class-specific functions and methods}

Depending on the complexity of the model it may be necessary to supply either
an own solver function or a complete \code{sim} method. The difference is that only one
\code{sim} method is available for one particular class, but several solver
functions may be provided as alternatives for one class, either as \proglang{S4} methods with
different names or as ordinary (non generic) functions.

In most cases it should be sufficient to write class specific solvers, but for
complicated data structures or hybrid model architectures it may be necessary to
provide class specific \code{sim} methods.
In case of the individual-based \textit{Daphnia} model, the solver should be of type
``iterator'' but with additional functionality to reduce the amount of data
stored in \code{out}. To do this, \code{mysolver} given in Table \ref{t:mysolver} has a
local function \code{observer}, which for each time step returns one line
of summary statistics for all individuals. Additionally, it would be also possible to
write data to logfiles, to print selected data to screen or to display animated
graphics during simulation.

The argument naming of the solver functions is compatible with the ODE solvers of the
\pkg{deSolve}-package with respect to the first four arguments. Moreover, some essential
functionality must be provided by all solvers:

\begin{enumerate}
\item Extraction of slots of the \code{simObj} (argument \code{y}) to local
  variables, expansion of \code{y@times} via \code{fromtoby}, \code{attach} and
  \code{detach} for the list of equations as given in the example,
\item Iteration loop or any other appropriate numeric algorithm,
\item Assignment of the special parameter DELTAT (optional, if needed by the model),
\item Accumulation of essential simulation results to the outputs (\code{out}-slot)
  and assignment of explanatory variable names, in this case done by \code{observer}.
\end{enumerate}

Depending on the data structure, it is also possible to write a class specific plot
function:

<<ibm_plot,eval=FALSE, echo=TRUE>>=
setMethod("plot", c("indbasedModel", "missing"), function(x, y, ...) {
  o <- out(x)
  par(mfrow=c(2, 2))
  plot(o$times, o$meanage,  type="l", xlab="Time", ylab="Mean age (d)")
  plot(o$times, o$meaneggs, type="l", xlab="Time", ylab="Eggs per individual")
  plot(o$times, o$number,   type="l", xlab="Time", ylab="Abundance")
  plot(o$times, o$number,   type="l", xlab="Time", ylab="Abundance", log="y")
})
@

\subsection{Model application}

Now, the model can be simulated and used for the intended application,
e.g.\ hypothesis testing, parameter estimation or scenario analysis:

<<daphnia1,eval=TRUE,echo=TRUE,include=FALSE,fig=TRUE,width=8,height=8>>=
solver(ibm_daphnia) <- "myiteration"
ibm_daphnia <- sim(ibm_daphnia)
plot(ibm_daphnia)
@

\begin{figure}[t]
\centering
\includegraphics[width=0.7\textwidth]{tp-daphnia1}
\caption{\label{f:daphnia1} Result of the \code{plot} method of the
 \textit{Daphnia} model, top: mean age of the population and number of eggs
   indicating synchronized population development; bottom: exponential
   population growth in linear resp.\ logarithmic scale.}
\end{figure}

It is beyond the scope of this paper to provide an overview over simulation techniques
or to answer domain specific questions about \textit{Daphnia} population dynamics;
however, the following example is intended to give an impression how \pkg{simecol}
models can be used in practice. The example deals with the effect of size-selective
predation, similar to the more extensive analysis of \citet{Mooij1997}. Four
scenarios will be compared:

\begin{description}
  \item [Sc0:] no mortality at all,
  \item [Sc1:] constant mortality (independent of body length),
  \item [Sc2:] small individuals preferred (typical for invertebrate predators like
    \textit{Chaoborus}, phantom midge larvae),
  \item [Sc3:] large individuals preferred (typical for adult fish).
\end{description}

At the first step we create one clone of the \code{daphnia\_ibm}-object, assign
settings common to all scenarios and an initial sample population:

<<ibm_init,echo=TRUE>>=
Sc0                                   <- ibm_daphnia
times(Sc0)                            <- c(from=0, to=30, by=0.2)
parms(Sc0)[c("temp", "food", "mort")] <- c(15, 0.4, 0.1)
init(Sc0) <- data.frame(age=rep(10, 50), size = rep(2.5, 50),
                        eggs=rep(5, 50), eggage=runif(50, 0, 4))
@

Then we replace the default \code{survive}-function with a
more general one which depends on a user-specified mortality function \code{fmort}:

<<ibm_eqations,echo=TRUE>>=
equations(Sc0)$survive = function(inds, parms) {
  abundance <- nrow(inds)
  rnd       <- runif(abundance)
  mort      <- fmort(parms$mort, inds$size) * parms$DELTAT
  subset(inds, rnd > mort)
}
@

Copies of object \code{Sc0} are created and modified according to the scenario
specification.
In the example below we have
two functions with constant mortality and two other functions where per capita
mortality is higher for the larger or smaller individuals, respectively:

<<ibm_mort,echo=TRUE>>=
Sc1 <- Sc2 <- Sc3 <- Sc0
equations(Sc0)$fmort <- function(mort, x) 0
equations(Sc1)$fmort <- function(mort, x) mort
equations(Sc2)$fmort <- function(mort, x){
  mort * 2 * rank(-x) / (length(x) + 1)
}
equations(Sc3)$fmort <- function(mort, x){
  mort * 2 * rank(x) / (length(x) + 1)
}
@

Finally, the scenarios can be simulated, either line by line as in Section
\ref{s:predprey} or listwise with \code{lapply}:

<<ibm_seed,echo=FALSE>>=
set.seed(123)
@

%% workaround to avoid line break here
<<ibm_sim_sc,echo=FALSE>>=
sc <- lapply(list(Sc0=Sc0, Sc1=Sc1, Sc2=Sc2, Sc3=Sc3), sim)
@

\begin{Code}
sc <- lapply(list(Sc0=Sc0, Sc1=Sc1, Sc2=Sc2, Sc3=Sc3), sim)
\end{Code}

\begin{figure}
\centering
<<daphnia2,eval=TRUE,fig=TRUE,include=TRUE,width=8,height=6>>=
growthrate <- function(obj) {
  o <- subset(out(obj), times > 10)
  m <- lm(log(o$number) ~ o$times)
  as.vector(coef(m)[2])
}

abundplot <- function(ref, sc1, sc2, sc3){
  par(las=1)
  plot(out(ref)[c("times", "number")],
      type="l", xlab="Time", ylab="Abundance",
      log="y", ylim=c(10, 5000),
      main="",
      font=2, font.lab=2, lwd=2
  )
  lines(out(sc1)[c("times", "number")], col="blue", lty="6222", lwd=2)
  lines(out(sc2)[c("times", "number")], col="darkgreen", lty="22", lwd=2)
  lines(out(sc3)[c("times", "number")], col="red", lty="66", lwd=2)
  legend("topleft",
    c(paste("Sc0: no mortality,     r =", round(growthrate(ref),2), collapse=" "),
      paste("Sc1: random mort.   r =", round(growthrate(sc1),2), collapse=" "),
      paste("Sc2: small pref.        r =", round(growthrate(sc2),2), collapse=" "),
      paste("Sc3: large pref.         r =", round(growthrate(sc3),2), collapse=" ")
      ),
    col=c("black","blue","darkgreen","red"), lty=c("solid", "6222", "22", "66"),
    bty="n", lwd=2
  )
}
abundplot(sc$Sc0, sc$Sc1, sc$Sc2, sc$Sc3)
@
\caption{\label{f:daphnia2} Time series of \textit{Daphnia} abundance for
different scenarios with non-selective resp.\ size selective mortality. Population
growth rates ($r$ in $\rm d^{-1}$) were approximated by log-linear regression for
all data points after an initial period of 10 days.}
\end{figure}

The result shows very clearly the influence of demography on population growth
(Figure \ref{f:daphnia2}).
Given that the population growth rate $r$ without any mortality (i.e.\ equal to the birth rate $b$)
is approximately \Sexpr{round(growthrate(sc$Sc0), 2)}$\mathrm d^{-1}$ in \code{Sc0} and the mortality
rate $d$ is set to $0.1$$\mathrm d^{-1}$, it is plausible that the population
growth rate in \code{Sc1} is:

\[
r = b-d \approx \Sexpr{round(growthrate(sc$Sc1), 2)} \mathrm d^{-1}
\]

In case of size-selective predation, demography has to be taken into
account in order to get realistic estimations of $r$. The simulation
shows an increased population loss in case of fish predation (\code{Sc3},
$r=\Sexpr{round(growthrate(sc$Sc3), 2)}$) and a lower effect in case of
\textit{Chaoborus} (\code{Sc2}, $r=\Sexpr{round(growthrate(sc$Sc2), 2)}$).
Please see \citet{Mooij1997,Mooij2003} for details and how the results may depend
on fecundity of the prey, the shape of the selection function and the dynamics
of predator and prey.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Discussion}\label{s:discussion}

The main contribution of the \pkg{simecol} package is the proposal
of a generalized, declarative structure to describe ecological models.
This structure was inspired by the state space representation used in control theory
and is intended as a pragmatic solution to unify the upcoming diversity of
\proglang{R} implementations of ecological models.
The object-oriented \code{simObj} structure may be useful also in other areas
and for other models like continuous-time Markov processes and stochastic differential
equations.

With the set of examples presented and some additional models developed in our workgroup,
the matrix-oriented \proglang{R} language was found to be well suited for model
development (rapid prototyping) and model evaluation. According to our experience,
a structured OOP style is more efficient compared to a purely
functional style, or even worse, ad-hoc programming.

The functional OOP system of \proglang{R} is different from languages like
\proglang{JAVA} or \proglang{C++} and the approach of generic functions for common
tasks seems to be more appropriate for statistical data analysis
than for ecological simulation models which have not only variable data but also
variable code.
Moreover, the lack of references and the invisibility of member variables
in slot functions of the same object was seemingly inconvenient and needed
re-thinking. However, the \proglang{R} language with its \proglang{Scheme}
heritage \citep{Ihaka1996} is a ``programmable programming language''. Lexical
scoping and local environments \citep{Gentleman2000} allow to change its
default behavior if needed. There were temptations to apply an alternative OOP
paradigm that allows for
references e.g.\ \pkg{R.oo} \citep{Bengtsson2003} or \pkg{proto} \citep{Kates2005},
but it was decided to stay with the default behavior as much as possible.
Similarly, we used only flat object hierarchies and abandoned delegation-based
approaches and instead suggest cloning (creation time sharing) as a standard
technique to create derived objects.

At a first look \proglang{R} seems to be less suited for large applications, e.g.\
turbulence models, where \proglang{C} and \proglang{FORTRAN} are standard
or for complex individual-based simulations with large numbers of interacting
individuals, where class-based OOP in the flavor of \proglang{C++} or \proglang{JAVA}
is regarded as more natural \citep{Grimm2005}.
However, even such applications can take advantage of \pkg{simecol}, either
because of vectorization in \proglang{R} (\code{subset} is in fact highly efficient)
or due to the possibility to embed compiled code as shared library.
For large applications or external simulation programs, \pkg{simecol} objects
can be constructed as an interface provided that the external program is open
enough to be linked or at least is callable in batch mode.

The package is designed to be open for local extensions and further
evolution of the package itself. A limited number of classes will follow,
e.g.\ for individual-based models similar to the \textit{Daphnia} example or for
purely statistical models like neural networks. An integrated parameter estimation
functionality may follow as well as an interface to quantitative and qualitative
model evaluation criteria \citep{Jachner2007}. Moreover, interfaces to other promising
approaches to solve simulation models in \proglang{R} may be worth to be established,
e.g.\ to the \proglang{XML} based description language of the
bioconductor package \pkg{SBMLR} \citep{Radivoyevitch2004}, or
to the nonlinear mixed effects modelling package \pkg{nlmeODE} of \citet{Torno2004a}
who independently developed a similar list-based object structure for a class
of ordinary differential equation models.

Another appealing approach is the stoichiometry-matrix based approach for
ODE models of aquatic systems (wastewater treatment, biofilm, rivers and lakes,
\url{https://www.eawag.ch/organisation/abteilungen/siam/lehre/Modaqecosys/}).
These \proglang{R} scripts, developed by
a prominent group in water modeling \citep[e.g.\ ][]{Reichert1994,Reichert2001},
are currently used for teaching of aquatic modeling together with model assessment like
sensitivity and uncertainty analysis, optimization, and frequentist or Bayesian
model tests.

Our work presented so far can serve as a starting point and demonstrates, that
\proglang{R} together with OOP is well suited as a medium for the development, distribution and
share of ecological modeling code. Reference applications and utility functions
will help ecologists to structure their work.
The open source license of \proglang{R} and its accompanying packages
should encourage own applications, which remain under complete control of the
developer. Moreover, the intentionally lightweight character of \pkg{simecol} and
the compact code of the solutions would enable the user to unhinge his model from
\pkg{simecol} whenever required and to port it to other systems. As a conclusion,
one can make nothing wrong when starting to model with \proglang{R} but it may
be possible that one stays with it for a long time.


\section*{Acknowledgments}\label{s:acknowledgments}

We wish to express our thanks to Ren\`{e} Sachse for suggestions while developing
his own \pkg{simecol} models and to the participants of the
``Modeling for Limnologists'' workshop for testing and feedback.
We are grateful to two anynymous reviewers for their constructive remarks which helped
to improve the manuscript.

\clearpage

\begin{table}[p]
\caption{\label{t:ibmdaphnia1} Individual-based \textit{Daphnia} model (part I,
  class definition, main equation, parameters, initial state, time steps, solver)}
\small
\hrulefill
\begin{Code}
library("simecol")
## class indbasedModel is built in since simecol version 0.8-4
# setClass("indbasedModel",
#   representation(parms  = "list", init   = "data.frame"), contains = "simObj"
# )

ibm_daphnia <- new("indbasedModel",
  main = function(time, init, parms) {
    init <- live(init, parms)
    init <- survive(init, parms)
    init <- hatch(init, parms)
    init
  },
  parms = list(
    # parameters of the somatic growth equation
    a1          = 1.167,    # (mm)
    a2          = 0.573,    # (mg L^-1)
    a3          = 1.420,    # (mm)
    a4          = 2.397,    # (d),
    b1          = 1.089e-2, # (d^-1)
    b2          = 0.122,    # ((deg. C)^-1)
    # parameters of the clutch size equation
    X_max_slope = 23.83,    # (eggs)
    K_s_slope   = 0.65,     # (mg L^-1)
    beta_min    = -29.28,   # (eggs)
    u_c         = 1,        # (L mg^-1) unit conversion factor
    # parameters of the individual-based model
    L_0_Hall    = 0.35,     # (mm) SON (size of neonanates) of Hall data
    L_0         = 0.65,     # (mm) SON
    SAM         = 1.50,     # (mm) SAM (size at maturity)
    maxage      = 60,       # (d)
    # constant environmental conditions
    temp        = 20,       # (deg C)
    food        = 0.5       # (mg L^-1)
  ),
  init      = data.frame(age=0, size=0.65, eggs=0, eggage=0),
  times     = c(from=0, to=60, by=1),
  solver    = "myiteration", # or default method: "iteration"
  equations = list()
)
\end{Code}
\hrulefill
\end{table}



\begin{table}[p]
\caption{\label{t:ibmdaphnia2} Individual-based \textit{Daphnia} model
  (part II, equations and algorithms)}
\small
\hrulefill
\begin{Code}
equations(ibm_daphnia) <- list(
  newdaphnia = function(SON, n) {
    if (n>0) {
      data.frame(age = rep(0, n), size = SON, eggs = 0, eggage = 0)
    } else {
      NULL
    }
  },
  bottrell = function(temp) {
    exp(3.3956 + 0.2193 * log(temp) - 0.3414 * log(temp)^2)
  },
  tefi = function(time, temp, food, parms){
    with(parms, {
      deltaL <- L_0 - L_0_Hall
      k      <- b1 * exp(b2 * temp)
      L_max  <- (a1 * food)/(a2 + food) + a3 - k * a4
      L      <- L_max - (L_max - L_0_Hall) * exp (-k * time) + deltaL
      E      <- (X_max_slope * food)/(K_s_slope + food) * L +
                  beta_min * (1 - exp(-u_c * food))
      as.data.frame(cbind(L, E))
  })},
  live = function(inds, parms){
    with(parms,{
      ninds       <- nrow(inds)
      inds$age    <- inds$age + DELTAT
      inds$eggage <- ifelse(inds$size > SAM & inds$eggs > 0,
                            inds$eggage + DELTAT, 0)
      tefi_out    <- tefi(inds$age, temp, food, parms)
      inds$size   <- tefi_out$L
      neweggs     <- round(tefi_out$E)
      inds$eggs   <- ifelse(inds$size > SAM & inds$eggage==0,
                            neweggs, inds$eggs)
      inds
  })},
  survive  = function(inds, parms) subset(inds, inds$age < parms$maxage),
  hatch = function(inds, parms) {
    newinds <- NULL
    with(parms, {
      edt       <- bottrell(temp)
      have.neo  <- which(inds$eggs > 0 & inds$eggage > edt)
      eggs      <- inds$eggs[have.neo]
      new.neo   <- sum(eggs)
      inds$eggs[have.neo]  <-  inds$eggage[have.neo] <- 0
      newinds <- newdaphnia(L_0, new.neo)
      rbind(inds, newinds)
    })
  }
)
\end{Code}
\hrulefill
\end{table}


\begin{table}[p]
\caption{\label{t:mysolver} Solver function and \code{plot} method for the
  \textit{Daphnia} model}

\hrulefill
\begin{Code}
## a more appropriate solver (note the observer function
myiteration <- function(y, times=NULL, func=NULL, parms=NULL,
                        animate=FALSE, ...) {
  observer <- function(res) {
    # eggs, size, age, eggage
    number   <- nrow(res)
    meansize <- mean(res$size)
    meanage  <- mean(res$age)
    meaneggs <- mean(res$eggs)
    c(number=number, meansize=meansize, meanage=meanage, meaneggs=meaneggs)
  }
  init              <- y@init
  times             <- fromtoby(y@times)
  func              <- y@main
  parms             <- y@parms
  inputs            <- y@inputs
  equations         <- y@equations
  equations         <- addtoenv(equations)
  environment(func) <- environment()
  parms$DELTAT <- 0
  res <- observer(init)
  out <- res
  for (i in 2:length(times)) {
    time <- times[i]
    parms$DELTAT <- times[i] - times[i-1]
    init <- func(time, init, parms)
    res  <- observer(init)
    out  <- rbind(out, res)
   }
  row.names(out) <- NULL
  out <- cbind(times, out)
  as.data.frame(out)
}
## a plotting function that matches the output structure of the observer
setMethod("plot", c("indbasedModel", "missing"), function(x, y, ...) {
  o <- out(x)
  par(mfrow=c(2, 2))
  plot(o$times, o$meanage,  type="l", xlab="Time", ylab="Mean age (d)")
  plot(o$times, o$meaneggs, type="l", xlab="Time", ylab="Eggs per indiv.")
  plot(o$times, o$number,   type="l", xlab="Time", ylab="Abundance")
  plot(o$times, o$number, type="l", xlab="Time", ylab="Abundance", log="y")
})
## RUN the MODEL ##
ibm_daphnia <- sim(ibm_daphnia)
plot(ibm_daphnia)
\end{Code}
\hrulefill
\end{table}

\clearpage

\phantomsection
\addcontentsline{toc}{section}{References}
%\bibliographystyle{jss}
\bibliography{simecol}



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