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\vspace{-5 in}
\\
\title{Diversification and coexistence in multilevel biological networks}\\
\tiny Carlos J. Meli\'an \\
\institute{EcoLunch @NCEAS,\\ February 19, 2009}
\vspace{0.4 in}
\date{\tiny \hspace{-0.3 in} Meli\'an, C. J., Alonso, D., V\'azquez, D. P., Regetz, J., and Allesina, S.\\ {\em Unifying theories of molecular, community and network evolution, submitted}}

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\begin{document}

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%\section[Outline]{}	% this puts the outline before EACH section automatically & will highlight the section you're about to talk about
%\frame{\tableofcontents}

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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{Motivation \& Questions}

\subsection{Motivation}
%1%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{\frametitle{Motivation 1}
\setbeamercolor{uppercol}{fg=black,bg=white}
\setbeamercolor{lowercol}{fg=black,bg=white}
\begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{}
\begin{itemize}
\item < 1-| alert@1 > Networks of Data (at individual level) from {\Large ONE} (type, location, time, level)...
\item < 2-| alert@2 > to {\Large MULTIPLE} (types, locations, time, 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 one {\Large Output}
\item < 6-| alert@6 > to models with several {\Large Outputs}
\item < 7-| alert@7 > From methods to select {\Large one model} according to one specific output
\item < 8-| alert@8 > to methods to select {\Large models} according to several outputs
\end{itemize}
\end{beamerboxesrounded}
}
\subsection{Motivation}
%5%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{\frametitle{Motivation 2: Genome evolution-mating graphs-ecological graphs}
\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}
}
%2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Motivation}
%\frame{\frametitle{Motivation 2}
%{\headcol {What is the minimum information we need to model diversification and coexistence at molecular and ecological levels?}}
%\setbeamercolor{uppercol}{fg=black,bg=white}
%\setbeamercolor{lowercol}{fg=black,bg=white}
%\begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{}
%\begin{center}
%\includegraphics[width=10cm]{XXX.pdf} 
%\end{center}%
%\end{beamerboxesrounded}
%}
%\subsection{Motivation}
%3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Paradox of Diversity
%\frame{\frametitle{Motivation 3}
%\setbeamercolor{uppercol}{fg=black,bg=white}
%\setbeamercolor{lowercol}{fg=black,bg=white}
%\begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{}
%\begin{itemize}
%\item < 1-| alert@1 > {\Large Why are there so many species?}\\
%{\small Ecological views have focused on the mechanisms that enable or constraint species coexistence \citep{Hutchinson:1959}}
%\item < 2-| alert@2 > {\Large Why are there so few kinds of animals?}\\
%{\small Additional constraints on the process of speciation, constraints set by the genetics rather than ecology...\citep{Felsenstein:1981}}
%\end{itemize}
%\end{beamerboxesrounded}
%}%make clear to a broad audience the work I'll present is at the interface ecol-evol and that is rare and why it is cutting-edge and important-what insights do we gain beyond those if i had only focused on the ecological or evolutionary aspects alone...

%4%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Motivation}
\frame{\frametitle{Motivation 3}
\begin{itemize}
\item < 1-| alert@1 > Two broad categories of speciation \citep{Schluter:2009}:
\item < 2-| alert@2 > Ecological speciation: Reproductive isolation (RI) evolves between populations by divergent natural selection arising from differences between ecological environments.
\item < 3-| alert@3 > Genetic speciation: Evolution of RI by the fixation of different ``advantageous'' mutations experiencing similar environments.
\item < 4-| alert@4 > \Large Lack of statistical test and hypothesis about the ecological and evolutionary processes underlying speciation and diversity...\citep{Gavrilets2:2004}
\end{itemize}
}


%\frame{\frametitle{Example 2: Genome evo-Mating \& ecological networks}
%\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{0.25 in}
%\vspace{-0.2 in} \includegraphics[width=12cm,angle=90]{MultilevelNetworks2.pdf}
%\end{column}
%\end{columns}
%}
\subsection{Questions}
%6%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{\frametitle{Questions}
\begin{itemize}  
\item < 1-| alert@1 > {\small Can speciation (ecological and/or genetic) happen in homogeneous environments?}
\item < 2-| alert@2 > {\large How many speciation modes can happen?}
\item < 3-| alert@3 > {\Large Can negative frequency-dependent selection at molecular and ecological levels maximize speciation rate, coexistence and genetic--species diversity?}
\item < 4-| alert@4 > {\Large Does neutral evolution at molecular and ecological levels predict adaptive radiation in {\em Cichlids}?}
\end{itemize}
}


%3%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\frame{\frametitle{Example 3: Ecological interactions-spatial processes}
%{\headcol {Exploring $\alpha$, $\beta$ and $\gamma$ tritrophic diversity with neutral metawebs}}
%\begin{columns}
%\begin{column}{8cm}
%\includegraphics[width=6cm,angle=90]{distancegraphallequal.pdf}
%\end{column}
%\hspace{-1 in}
%\begin{column}{8cm}
%\includegraphics[width=6cm,angle=90]{distancegraphcommon.pdf}
%\end{column}
%\end{columns}
%}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Background %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Model}

%7%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Model 1}
\frame{\frametitle{Parameters}
{\headcol {What is the minimum information we need to model diversification and coexistence at molecular and ecological levels?}}
\pause
\begin{enumerate}
\item < 1-| alert@1 > First principles $\rightarrow$ birth-death process at individual level ($i = 1,2,...J$).
\pause
\item < 2-| alert@2 > Genome evolution in haploid individuals\footnote{\hyperlink{GenomeEvolution}{\beamergotobutton{GenomeEvolution}}}%->whole genome and how to collapse the genome in the neutral case
\pause
\item < 3-| alert@3 > Equivalence and symmetric interactions at molecular and ecological levels $\rightarrow$ Non-interactive models.
\pause
\item < 4-| alert@4 > Mutation rate ($\mu$)\footnote{\hyperlink{MutationRate}{\beamergotobutton{MutationRate}}}
\pause
\item < 5-| alert@5 > Evolution of molecular constraints ($q^\mathrm{min}$, prezygotic and postzygotic implicit mechanisms)\footnote{\hyperlink{MolecularConstraints}{\beamergotobutton{MolecularConstraints}}}
\end{enumerate}
}

%8%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Model 2}
\frame{\frametitle{Neutral and frequency-dependent speciation models}
\begin{enumerate}
\item < 1-| alert@1 > Kill one individual at random from a uniform distribution
\item < 2-| alert@2 > Select one parent $G_1(k)$ from a uniform distribution
\item < 3-| alert@3 > We chose at random a second parent $G_2(k)$ among the individuals compatible with the first parent (i.e., $q^{G_1(k) G_2(k)}$ $>$ $q^{min}$)
\item < 4-| alert@4 > Given the similarity
between the parent of the new individual $k$ and the individual $i$
already in the population we update the similarity matrix (Q) according to:
\begin{equation}
  \begin{cases}
    q^{ki} = \frac{e^{-2\mu}}{2}\left(q^{G_1(k) i} + q^{G_2(k) i}\right)\label{A1}\\
    q^{kk} = 1.
    \end{cases}
\end{equation}
\item < 5-| alert@5 > The frequency--dependent model is the same but with individuals with rare alleles increasing the probability to mate.\footnote{\hyperlink{FrequencyDependent}{\beamergotobutton{FrequencyDependent}}}
\end{enumerate}
}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Results and Summary %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Theoretical Results}
%9%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Results}
\frame{\frametitle{Question 1}
\begin{itemize}
\item < 1-| alert@1 > {\Large Can speciation (ecological and/or genetic) happen in homogeneous environments?}
\end{itemize}
}
\subsection{Results}
%10%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{\frametitle{Yes}
{\headcol {Evolutionary graphs and the genetic species concept}}
\setbeamercolor{uppercol}{fg=black,bg=white}
\setbeamercolor{lowercol}{fg=black,bg=white}
\begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{}
\includegraphics[width=10cm]{Fig1.pdf} 
\end{beamerboxesrounded}
}
%11
\subsection{Results}
\frame{\frametitle{Question 2}
\begin{itemize}  
\item < 1-| alert@1 > {\Large How many speciation modes can happen?}
\end{itemize}
}
%12
\subsection{Results}
\frame{\frametitle{Two: Fission and mutation-induced speciation}
\begin{itemize}
\item < 1-| alert@1 > Fission happens because the death of an individual splits a previously connected component.
\setbeamercolor{uppercol}{fg=black,bg=white}
\setbeamercolor{lowercol}{fg=black,bg=white}
\begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{}
\includegraphics[width=4cm]{Fig1.pdf} 
\end{beamerboxesrounded}
\item < 2-| alert@2 > {\small What is the minimum mutation rate ($\mu_\mathrm{min}$) for the mutation induced speciation mode to happen?}\\{\tiny Because we need  $q^{ki}$  = $\frac{e^{-2\mu}}{2}\left(q^{G_1(k) i} + q^{G_2(k) i}\right)$ $<$ $q^\mathrm{min}$, then
$\mu_\mathrm{min}$ = $- \left(\frac{log\left(\frac{2q^\mathrm{min}}{1 + q^\mathrm{min}}\right)}{2}\right)$.\\ For example, if $q^\mathrm{min} = 0.95$, the minimum mutation rate to have mutation-induced speciation is $\approx$ $0.006$.}% This value is even higher if $q^\mathrm{min}$ = $0.90$ (i.e., $0.013$) and $0.85$ (i.e., $0.02$). These values are biologically unrealistic. Indeed, they are larger than the ones explored in the simulations (i.e., in $[5 \cdot
%%10^{-5}, 10^{-3}]$).}
\end{itemize}
}

%13
\subsection{Results}
\frame{\frametitle{Question 3}
\begin{itemize}
\item < 1-| alert@1 > {\Large Can negative frequency-dependent selection at molecular and ecological levels maximize speciation rate, coexistence and genetic--species diversity?}
\end{itemize}
}

%14%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Results}
\frame{\frametitle{Can we predict speciation rate in the neutral case?}
{\headcol {What is the number of steps ($x_{n}$) at which $q^{min}$ $>$ $q^{A A_n}$?}}
\setbeamercolor{uppercol}{fg=black,bg=white}
\setbeamercolor{lowercol}{fg=black,bg=white}
\begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{}
\vspace{-0.5 in}
\includegraphics[width=10cm]{mutation.pdf} 
\end{beamerboxesrounded}
}

%15%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Results}
\frame{\frametitle{The expected speciation rate}
\begin{itemize}
\item < 1-| alert@1 > The number of steps ($x_{n}$) at which
$q^{min}$ $>$ $q^{A A_n}$ that is proportional to the speciation rate is:\\
\begin{equation}
\frac{1}{x_{n}} = \frac{- 2 \mu + log(\frac{1 + X}{2})}{log(q^{min})} \label{A3}
\end{equation}
\item < 2-| alert@2 > where we know the value of $X$ is in the range [1,$q^{min}$], thus the
expected value of $X$ is = $\frac{1 + q^{min}}{2}$.
\end{itemize}
}
%16
\subsection{Results}
\frame{\frametitle{We now can estimate the simulated speciation rate ($\nu$)} 
\setbeamercolor{uppercol}{fg=black,bg=white}
\setbeamercolor{lowercol}{fg=black,bg=white}
\begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{}
\vspace{-0.65 in}
\includegraphics[width=9cm]{Fig3Presentation.pdf} 
\end{beamerboxesrounded}
}


%17%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Results}
\frame{\frametitle{Thus, the expected speciation rate is}
\setbeamercolor{uppercol}{fg=black,bg=white}
\setbeamercolor{lowercol}{fg=black,bg=white}
\begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{}
\begin{equation}
\nu = \alpha + \beta \left[\frac{- 2 \mu + log(\frac{1 + X}{2})}{log(q^{min})}\right],\label{A5}
\end{equation}
\end{beamerboxesrounded}
\\
where $\alpha$ is equal to -0.34 and the slope $\beta$ is equal to
1.45.
}
%18
\subsection{Results}
\frame{\frametitle{This expression gives an accurate prediction of the speciation
rate...}
\setbeamercolor{uppercol}{fg=black,bg=white}
\setbeamercolor{lowercol}{fg=black,bg=white}
\begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{}
\vspace{-0.5 in}
\includegraphics[width=5.5cm]{Fig4Presentation.pdf} 
\vspace{-0.5 in}
\begin{equation}
v = \frac{\# Sp. events}{Generation}
\end{equation}
\end{beamerboxesrounded}
}

%19
\subsection{Results}
\frame{\frametitle{and the predicted values are higher in the neutral case...}
\setbeamercolor{uppercol}{fg=black,bg=white}
\setbeamercolor{lowercol}{fg=black,bg=white}
\begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{}
\vspace{-0.5 in}
\includegraphics[width=7cm]{SpeciationEventsPresentation.pdf} 
\end{beamerboxesrounded}
}
%20
\subsection{Results}
\frame{\frametitle{Does higher speciation rate in the neutral scenario imply higher genetic and species diversity?}
\setbeamercolor{uppercol}{fg=black,bg=white}
\setbeamercolor{lowercol}{fg=black,bg=white}
\begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{}
\vspace{-0.5 in}
\includegraphics[width=8.5cm]{Fig5Presentation.pdf} 
\end{beamerboxesrounded}
}
%21
\section{Predictions}
\subsection{Results}
\frame{\frametitle{Does neutral evolution at molecular and ecological levels predict adaptive radiation in {\em Cichlids}?}
\setbeamercolor{uppercol}{fg=white,bg=white}
\setbeamercolor{lowercol}{fg=white,bg=white}
\begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{}
\vspace{-0.5 in}
\includegraphics[width=4cm]{cichlids.jpg} 
\end{beamerboxesrounded}
\begin{itemize}
\item < 1-| alert@1 > 88 colonizations in the last 20my in 88 different lakes in eastern Africa. 
\item < 2-| alert@2 > 34 radiations!
\end{itemize}
}

\subsection{Results}
\frame{\frametitle{Species number through time}
%\setbeamercolor{uppercol}{fg=white,bg=white}
%\setbeamercolor{lowercol}{fg=white,bg=white}
%\begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{}
\vspace{-0.5 in}
\includegraphics[width=6cm]{LakeBarombi.pdf} 
\vspace{-0.4 in}
\begin{equation}
\ell \left(q^{min}, \mu | \frac{Extant(Sp)}{time}\right) \rightarrow \frac{\# Sp. events \times 0.01}{100} \rightarrow \frac{v \times Gens. \times 0.01}{100}
\end{equation}
\vspace{-0.2 in}
\begin{itemize}
\item < 1-| alert@1 > {\small Assuming 1 Gen/y, $q^{min}$ = 0.1 and $\mu$ = 10^{-10}  $\approx$ 10^{12} individuals/Gen}
\end{itemize}
%\end{beamerboxesrounded}
}

\section{Conclusions}
%21%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame{\frametitle{Summary}
\begin{enumerate}
\item < 1-| alert@1 > The expected speciation rate (i.e., fission and not mutation--induced) in the neutral case can be
estimated using an approximation that takes into account the mutation
rate, neutral genome evolution and a molecular filter to have fertile offspring.
\item < 2-| alert@2 > The dynamics of speciation is dramatically different
in both scenarios. While speciation events are predictable in the
neutral case, rapid series of speciation events happen in the
frequency-dependent scenario. Then the system reaches a plateau without
further speciation events. Rare species in the system increases their
abundance, decreasing the probability of extinction and increasing
genetic--species diversity and coexistence.
\item < 3-| alert@3 > Negative frequency--dependent scenario is a potent selection amplifier, suppress speciation rate and increases genetic--species diversity.
\end{enumerate}
}
%22
\frame{\frametitle{General Conclusion}
\begin{enumerate}
\item < 1-| alert@1 > $NTME$ and $NTB$ offer a theoretical framework to add explicit mechanisms of speciation and link the ecology and evolution of coexistence and diversity at multiple levels.
\item < 2-| alert@2 > Evolutionary graphs at multiple biological levels have fascinating applications. For example, from the approach presented here is it possible to develop a biodiversity number with explicit genome evolution and speciation. This approach can be tested assuming neutral genome evolution and ecological behavior using thousands of individuals simultaneously. 
\item < 3-| alert@3 > This approach represents a statistical test of neutral evolution at molecular and ecological levels to explore simultaneously diversity and coexistence (i.e., {\em Cichlid} radiation).
\end{enumerate}
}

%\frame{\frametitle{Novel questions that can be addressed (I)}
%\begin{enumerate}  
%\item < 1-| alert@1 > Toward a new biodiversity number with explicit genome evolution and multiple speciation modes.
%\item < 2-| alert@2 > Main goal: ``Explore the consequences of molecular and ecological processes on mutation-induced, fission, microallopatric and other speciation modes on contemporary biodiversity...''
%\item < 3-| alert@3 > Example
%\end{enumerate}
%}

\section{Appendix}
%23%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Reproductive Isolation 1}
\frame[label=GenomeEvolution]{\frametitle{Genome evolution}
\begin{itemize}
\item < 1-| alert@1 > Let us assume that the genome of each individual
  is represented by a sequence of $N$ loci. Each locus can assume
  values for two alleles ($+1/-1$). The genome of each individual $i$
  can be written as (${S_{1}^i,S_{2}^i,...,S_{N}^i}$), where $S_{u}^i$
  is the $u^{th}$ locus for the individual $i$
\item < 2-| alert@2 > Genetic similarity between individual $i$ and $j$ is defined as $q^{ij}$  = $\frac{1}{N} \sum_{u=1}^N S_{u}^i S_{u}^j$.
\item < 3-| alert@3 > The genetic similarity matrix is $Q$ = $[q_{ij}] = 1$ for all $q_{ij}$.
\item < 4-| alert@4 > We know the evolution of this matrix in the limit $N$
$\rightarrow$ $\infty$ in the infinite genome limit \citep{Higgs&Derrida:1992}, because each pair of alleles contributing the similarity between each pair of individuals $k$ and $i$ comes with equal probability from one of the two possible combinations of the parents of $k$ and individual $i$.
\item < 5-| alert@5 > The development of viable and fertile offspring is possible only between organisms having an overlap greater than $q^{min}$ loci responsible of reproductive isolation (postzygotic RI).
\end{itemize}
}

\subsection{Reproductive Isolation 2}
\frame[label=MutationRate]{\frametitle{Mutation rate}
\begin{itemize}
\item < 1-| alert@1 > Evolving individuals $\rightarrow$ $\mu_{S_{u}^i}$ is neutral, equal and independent for all units of sequence N $\rightarrow$ $\mu_{S_{u}^i} = \mu$
\end{itemize}
}

\subsection{Reproductive Isolation 3}
\frame[label=MolecularConstraints]{\frametitle{Molecular constraints}
\headcol{from \citep{Coyne:1992}}
\setbeamercolor{uppercol}{fg=black,bg=white}
\setbeamercolor{lowercol}{fg=black,bg=white}
\begin{beamerboxesrounded}[upper=upperco,lower=lowercol,shadow=true]{}
\includegraphics[width=5cm]{SexualIsolation.pdf}
\end{beamerboxesrounded}
}

\subsection{Frequency-dependent model}
\frame[label=FrequencyDependent]{\frametitle{Frequency-dependent Model}
\begin{itemize}
\item < 1-| alert@1 > {\tiny Each individual $i$ of species $k$ is chosen for reproduction according to:
\begin{equation}
 P_{i,k} = {\cal{N}} F_{i,k},  
\end{equation}
where individual fitness is defined as:
\begin{equation}
 F_{i,k} = \frac{1}{\sum_{j=1}^{N_{k}} H(q^{ij}-q^{min})}   
\end{equation}
Thus we write:
\begin{equation}
 P_{i,k} = {\cal{N}} \frac{1}{\sum_{j=1}^{N_{k}} H(q^{ij}-q^{min})} 
\end{equation}
where $\cal{N}$ is a normalization factor, $N_{k}$ is the abundance of
species $k$, and $H(\alpha)$ is
\begin{equation*}
H(\alpha) =
\begin{cases}
1 & \text{if $\alpha > 0$}\\
0 & \text{otherwise} 
\end{cases}
\end{equation*}
We now calculate the normalization factor by using the normalization
requirement, i.e., by summing $P_{i,k}$ across all individuals 
and species 1 must be obtained:
\begin{equation}
{\cal{N}} \sum_{i=1}^{S} \sum_{j=1}^{N_{k}} F_{i,k} = 1, 
\end{equation}
where S is the number of species, then:
\begin{equation}
{\cal{N}} = \frac{1}{\sum_{i=1}^{S} \sum_{j=1}^{N_{k}} F_{i,k}}
\end{equation}
Therefore, the probability of birth for each $i$ individual is:
\begin{equation}
  P_{i,k} = \frac{F_{i,k}}{\sum_{i=1}^{S} \sum_{j=1}^{N_{k}} F_{i,k}}}
\end{equation}
\end{itemize}
}
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\bibliographystyle{ecology}
{\tiny \bibliography{EvolutionMultilevel}}
\end{document}