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\\
\title{Towards a General Framework in Food Webs Driven by Data at Multiple Biological Levels}\\
\tiny Carlos J. Meli\'an,\\
National Center for Ecological Analysis and Synthesis, University of California, Santa Barbara.\\
\institute{\tiny The Sven Lov\'en Centre for Marine Sciences, University of Gothenburg, Str\"omstad, Sweden,\\ June 5, 2009}
\vspace{0.4 in}
\date{}

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\frame{\frametitle{Thanks!}
\begin{itemize}
\item Computing-scientist staff at NCEAS
\item Microsoft Research Ltd., Cambridge, UK.
\item Drew Allen, Jennifer Dunne, Jonathan Davies, Stanley Harpole, Stephen Hubbell, Pablo Marquet, Brad McRae, Mark Urban, and Tommaso Zillio,
\end{itemize}
}

\section{Background}

\subsection{Impressive development in food webs}
\frame{\frametitle{Food webs as a focus for unifying ecological theory}
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\begin{columns}
\begin{column}{3cm}
\vspace{-2.25 in}
\hspace{-2 in} \includegraphics[width=9cm]{IndividualSizeSpectrum.pdf}
\end{column}
\begin{column}{7cm}
\vspace{-1.5 in}
\begin{row}
\includegraphics[width=7cm,angle=90]{IndividualRankConnectivity.pdf}
\end{row}
\vspace{-1.5 in}
\begin{row}
\hspace{4.5 in} \includegraphics[width=6cm]{DegreeAllSpecies.pdf}
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\subsection{Data accumulation}
\frame{\frametitle{Data accumulation in Networks of data at multiple levels (GeneBank, KNB)}
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\begin{itemize}
\includegraphics[width=6cm]{CumulativeData.pdf}
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\subsection{Data accumulation at different levels + lack theoretical analysis of interacting individuals (reproduction mode)}
\frame{\frametitle{Food webs as a focus for unifying ecological-evolutionary theory}
\setbeamercolor{uppercol}{fg=black,bg=white}
\setbeamercolor{lowercol}{fg=black,bg=white}
\begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{}
\begin{enumerate}
\item < 1-| alert@1 > {\Large Models with explicit DNA evolution and speciation (Kimura:1968,Higgs $\&$ Derrida:1992,Gavrilets:2004)}
\item < 2-| alert@2 > {\Large Aggregation: Link raw data with dynamical models of interacting individuals and populations}
\end{enumerate}
\end{beamerboxesrounded}
}

\subsection{Speciation}
\frame{\frametitle{1. Models with explicit DNA evolution and speciation}
\begin{itemize}
\item < 1-| alert@1 > {\Large Why are there so many species?}
\item < 2-| alert@2 > {\small Ecological views have focused on the mechanisms that enable or constraint species coexistence (Hutchinson:1959)}
\item < 3-| alert@3 > {\Large Why are there so few kinds of animals?}
\item < 4-| alert@4 > {\small Additional constraints on the process of speciation, constraints set by the genetics rather than ecology (Felsenstein:1981)}
\end{itemize}
}

\subsection{Speciation}
\frame{\frametitle{Two broad categories of speciation:}
\begin{itemize}
\item < 1-| alert@1 > Ecological speciation: Reproductive isolation (RI) evolves between populations by divergent (positive,neutral,negative) selection arising from differences between ecological environments and ecological processes (Schluter:2009).
\item < 2-| alert@2 > Genetic speciation: Evolution of RI by the fixation of different types of mutations experiencing similar environments (Lynch:2007).
\end{itemize}
}

\subsection{Speciation}
\frame{\frametitle{1. Models with explicit DNA evolution and speciation}
\vspace{-0.15 in}
\begin{columns}
\begin{column}{4cm}
\vspace{0.05 in}
\includegraphics[width=8cm]{speciation.pdf}
\end{column}
\begin{column}{9cm}
\vspace{-2.25 in}
\includegraphics[width=7cm,angle=90]{speciation1.pdf}
\vspace{-1 in}
\includegraphics[width=7cm,angle=90]{speciation2.pdf}
\end{column}
\end{columns}
}

\subsection{Collapse}
\frame{\frametitle{2. Aggregation}
\vspace{-0.5 in}
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\begin{itemize}
\item[] 
\includegraphics<1>[width=7cm]{tree1.pdf} 
\item[]
\includegraphics<2>[width=7cm]{tree2.pdf} 
\item[]
\includegraphics<3>[width=7cm]{tree3.pdf} 
\item[]
\includegraphics<4>[width=7cm]{tree4.pdf}
\item[]
\includegraphics<5>[width=7cm]{tree5.pdf}
\item[]
\vspace{-0.75 in}
\includegraphics<6>[width=7cm]{tree6.pdf}
\end{itemize}
\end{column}
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\vspace{-0.5 in}
\begin{row}
\hspace{-0.25}\includegraphics<6>[width=6.5cm]{Polis1991.pdf}
%Yield effort curve for first 100 species of prey captured by the scorpion Paruroctonus mesaensis in the Coachella Valley
%The number of prey species continues to increase with observation time. The 100th prey species was recorded
%on the 181st survey night;an asymptote was never reached in 5yr and more than 2000
%person hours of field time.
\end{row}
\vspace{-2 in}
\begin{row}
\hspace{-0.25 in}\includegraphics<6>[width=6.5cm]{CumulativenumberpreyAll.pdf}
\end{row}
\end{column}
\end{columns}
}

\subsection{Collapse 2}
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\includegraphics<1>[width=6cm]{TempSalConnInd.pdf}
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\includegraphics<1>[width=6cm]{Connectance.pdf}
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\subsection{Summary}
\frame{\frametitle{Summary}
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\begin{itemize}
\item < 1-| alert@1 > Networks of Data (at individual level) from {\Large ONE} (type, location, level)...
\item < 2-| alert@2 > to {\Large MULTIPLE} (types, locations, levels).
\item < 3-| alert@3 > From {\Large ONE} specific question...
\item < 4-| alert@4 > to {\Large SEVERAL} questions
\item < 5-| alert@5 > From models with {\Large ONE} Output
\item < 6-| alert@6 > to models with {\large SEVERAL} Outputs
\item < 7-| alert@7 > From methods to select {\Large one model} according to {\Large ONE} specific output
\item < 8-| alert@8 > to methods to select {\Large models} according to {\Large SEVERAL} outputs
\end{itemize}
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\section{Questions}

\subsection{Questions}
\frame{\frametitle{Questions}
\begin{itemize} 
\item < 1-| alert@1 > {\large What is the minimum information we need to model diversification and coexistence at molecular and ecological levels? It is possible to estimate a maximum likelihood for the general structure of the individual interactions in such a framework?}
\item < 2-| alert@2 > {\Large Does (neutral, positive, negative) selection and evolution at molecular and ecological levels predict individual rank in connectance and species rank in abundance?} 
\item < 3-| alert@3 > {Dream question: \small Does (neutral, positive, negative) selection predict size-spectrum and species level food web properties?}
\end{itemize}
}

\section{The data: An Example}

\subsection{The raw data}
\frame{\frametitle{\tiny Sampling individual diets, size and abundance at several s-t conditions}
{\headcol {\tiny $30520$ inds after 144 samplings/year, Fish ($5725$/$54$); Shrimps ($365$/$258$); Mysids ($18393$)\\
$688$ x $10^6$ inds/$10^{5}$$m^{3}$, Fish $9$ x $10^6$; Shrimps $4556$; Mysids $679$ x $10^6$}}
\begin{columns}
\begin{column}{8cm}
\vspace{-0.2 in}
\includegraphics[width=10cm]{Sampling.pdf}
%sampled features of individuals, species abundance (each s-t) and metacommunity abundance
\end{column}
\end{columns}
}

\section{The model}

\subsection{The model}
\frame{\frametitle{\tiny DNA evolution model with explicit speciation: steps}
\vspace{-0.25 in}
\begin{itemize}[<+->]
\item[] 
\includegraphics<1>[width=7cm]{evo1.pdf}
\item[]
\includegraphics<2>[width=7cm]{evo2.pdf}
\item[]
\includegraphics<3>[width=7cm]{evo3.pdf}
\item[]
\includegraphics<4>[width=7cm]{evo4.pdf}
\end{itemize}
}

\subsection{The Model}
\frame{\frametitle{\tiny DNA evolution model with explicit speciation: formal steps}
\vspace{-0.25 in}
\begin{itemize}
\item < 1-| alert@1 > Genetic similarity; $q^{ij}$  = $\frac{1}{L} \sum_{u=1}^{L} S_{u}^{i} S_{u}^{j}$, thus $Q$ = $[q^{ij}]$, where $q^{ij}$ $\in$ $[-1,1]$ and ${S_{1}^{i},S_{2}^{i},...,S_{L}^{i}}$ is the genome of individual $i$:
\item < 2-| alert@2 > $q^{ij}$ as the fraction of identical sites ($f^{ij}$), thus $q^{ij}$  = $\frac{1}{L}\left[L f^{ij} - L(1 - f^{ij})\right]$ = 2 $f^{ij}$ - 1, and $f^{ij$} = $\frac{1 + q^{ij}}{2}$
\item < 3-| alert@3 > Which is the expected frequency of nucleotides that a new offspring $k$
will share with each individual $j$ in the population ($E[f^{kj}]$)?
\item < 4-| alert@4 >  $q^{ki} = \frac{e^{-2\mu}}{2}\left(q^{G_1(k) i} + q^{G_2(k) i}\right)$ and $q^{kk}$ = 1.
\item < 5-| alert@5 > Matrix Q approaches a mean value $Q^{*}$ = $\frac{1}{4J\mu}$
\item < 6-| alert@6 > If $q^{ij} < q^{\mathrm{min}}$, then reproductive isolation occurs. Note that $q^\mathrm{min}$ implicitly captures  the ecological (i.e., prezygotic) and genetic  (i.e., postzygotic) mechanisms of speciation.
\item < 7-| alert@7 > If $q^{\mathrm{min}}$ $>$ $Q^{*}$ then speciation happens (Higgs $\&$ Derrida 1992)
\end{itemize}
}

\subsection{Expected values}
\frame{\frametitle{\tiny Nucleotide heterozygosity, Speciation rate and Relative species abundance}
\vspace{-1.65 in}
\begin{columns}
\begin{column}{4cm}
\vspace{2 in} 
\includegraphics[width=5cm]{mutualisticnetworks.pdf}
\vspace{-2 in}
\includegraphics[width=5cm]{mutualisticnetworks.pdf}
\end{column}
\vspace{0.5 in}
\begin{column}{8cm}
\vspace{-0.2 in} \includegraphics[width=10cm,angle=90]{MultilevelNetworks2.pdf}
\end{column}
\end{columns}
}

\section{Results}

\subsection{Expected speciation rate}
\frame{\frametitle{Can we predict speciation rate in the neutral case?}
{\headcol {What is the number of steps ($x_{n}$) at which $q^{min}$ $>$ $q^{A A_n}$?}}
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\begin{column}{4cm}
\vspace{-0.5 in}
\includegraphics[width=10cm]{mutation.pdf} 
\end{column}
\begin{column}{5cm}
\vspace{-3 in}
\begin{itemize}
\item < 1-| alert@1 > \tiny $E[q^{kj}]$ = $\frac{e^{-2 \mu}}{2}(q^{G_{1}(k) j} +  q^{G_{2}(k) j})$,
\item < 2-| alert@2 > \tiny $q^{A A_{1}} = x_{1} = e^{-2 \mu}\frac{(q^{AA} + q^{AB})}{2}$,
\item < 3-| alert@3 > \tiny $q^{A A_{2}} = x_{2} = e^{-2 \mu 2} \left(\frac{(1 + E[AB])}{2}\right)^2$,\\
 where $E[AB]$ = $X$ = $(1 + q^\mathrm{min})/2$
\item < 4-| alert@4 > \tiny $q^\mathrm{min}  > e^{-2 \mu n} \left(\frac{(1 + X)}{2}\right)^n,$
\item < 5-| alert@5 > \tiny and applying logarithms result in:\\ $n$ = $\frac{log(q^\mathrm{min})}{-2 \mu + log(\frac{1 + X}{2})}$,
\item < 6-| alert@6 > \tiny and the expected speciation rate is:\\ $\nu$ = $\frac{1}{n}$ = $\frac{-2 \mu + log(\frac{1 + X}{2})}{log(q^\mathrm{min})}$.
\end{itemize}
\end{column}
\end{columns}
\end{beamerboxesrounded}
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\subsection{Expected speciation rate}
\frame{\frametitle{This expression gives an accurate prediction of the speciation rate...}
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\begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{}
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\includegraphics[width=5.5cm]{Fig4Presentation.pdf} 
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\subsection{Expected species abundance}
\frame{\frametitle{Can we predict the expected relative species abundance at steady state?}
\setbeamercolor{uppercol}{fg=black,bg=white}
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\begin{itemize}
\item < 1-| alert@1 > $\nu$ = $\phi(1) N(0,1)$ 
\item < 2-| alert@2 > $\phi(1)$ $N(2,1)$ + $\phi(1)$ $N(0,1)$ = $\sum_{i=2}^{J}$ $\phi(i)$ $N(1,i)$\\
\item < 3-| alert@3 > 
.\\
.\\
.\\
.\\
.\\
\item < 3-| alert@3 > $\phi(j)$ $N(j + 1,j)$ + $\phi(j)$ $N(j - 1,j)$ = $\sum_{i=j+1}^{J}$ $\phi(i)$ $N(j,i)$
\end{itemize}
\end{beamerboxesrounded}
}

\subsection{Maximum likelihood individual interactions}
\frame{\frametitle{Maximum likelihood}
\includegraphics[width=8cm]{MLE.pdf} 
}

\subsection{Maximum likelihood estimation of individual diets using the expected species abundance}

\frame{\frametitle{What is the probability of obtaining exactly the given empirical network at individual level ($N_{ind}(L)$)?}
\begin{equation}
\begin{center}
\operatorname{P}(N_{ind}(L) | p_{12},p_{21},p_{11},p_{22}) =
\vspace{0.2 in}

p^{L_{12}}_{12} (1 - p_{12})^{N^{prey}_{1} - L_{12}} $\dot$\\
p^{L_{21}}_{21} (1 - p_{21})^{N^{prey}_{2} - L_{21}} $\dot$\\
p^{L_{11}}_{11} (1 - p_{11})^{N^{prey}_{1} - L_{11}} $\dot$\\
\hspace{1.65 in} p^{L_{22}}_{22} (1 - p_{22})^{N^{prey}_{2} - L_{22}},
\end{center}
\end{equation}
\vspace{-0.85 in}

\begin{eqnarray}
\begin{center}
\ell(\overrightarrow{p} | N_{ind}(L)) = \prod^{S}_{i = 1} \prod^{S}_{j = 1} p^{L_{ij}}_{ij} (1 - p_{ij})^{N^{prey}_{i} - L_{ij}},\vspace{0.15 in}

\hspace{0.25 in}\ell(\mu_{R},\mu_{P},q^{min}_{R},q^{min}_{P} | N_{ind}(L)) = \prod^{S}_{i = 1} \prod^{S}_{j = 1} p^{L_{ij}}_{ij} (1 - p_{ij})^{N^{prey}_{i} - L_{ij}}.
\end{center}
\end{eqnarray}
}

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\subsection{Individual rank in connectance}
\frame{\frametitle{Individual rank in connectance}
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\vspace{-0.5 in}
\hspace{3 in}\pgfimage[width=5cm]{Sampling}
\vspace{-2.85 in}
\includegraphics[width=7cm]{TempSalConnInd.pdf}
\end{beamerboxesrounded}
}

\subsection{Neutral predictions individual and species levels} 
\frame{\frametitle{Neutral predictions at individual and species levels}
\vspace{-0.25 in}
{\headcol {\tiny  (Left) $J_{P}$ = Fish ($5725$); $J_{R}$ = Shrimps($258$);\\(Center) (Fish) $J_{P_M}$ = $2.4$ x $10^6$ inds/$10^{5}$$m^{3}$, $\mu_{P}$ = $10^{-7}$; $q^{min}_{P}$ = $0.57$;\\ (Right) (Shrimps) $J_{R_M}$: $4556$ inds/$10^{5}$$m^{3}$; $\mu_{R}$ = $2.5$ x $10^{-4}$; $q^{min}_{R}$ = $0.45$;\\ log $\ell(\overrightarrow{p}$) = -1273.47}}
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\vspace{-0.5 in}
\hspace{3 in}\pgfimage[width=5cm]{Sampling}
\vspace{-2 in}
\includegraphics[width=7cm]{NeutralTest.pdf}
\end{beamerboxesrounded}
}

\section{Conclusions}

\subsection{Conclusions}
\frame{\frametitle{Conclusions}
\begin{enumerate}
\item < 1-| alert@1 > A sampling theory of food webs in space and time to test genetic and ecological drift with the observed individual rank in connectance and species rank in abundance.
\item < 2-| alert@2 > Those two distributions depart from neutral expectations in the Guadalquivir estuary food web in all the s-t conditions (sampling of $30520$ inds. and an estimated abundance of $688$ x $10^6$ inds/$10^{5}$$m^{3}$).
\end{enumerate}
}

\subsection{General conclusions}
\frame{\frametitle{General Conclusions}
\begin{enumerate}
\item < 1-| alert@1 > Theoretical framework to understand diversification and coexistence at multiple biological levels.
\item < 2-| alert@2 > Birth-death process with neutral genome evolution (genetic and ecological drift) and a molecular filter that regulates the production of fertile offspring allow approximations at steady state that can be tested using thousands of individuals simultaneously.
\item < 3-| alert@3 > Future extensions include MLE to test the expected individual interactions according to the their size and the abundance of each species.
\end{enumerate}
}
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