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\title{Spatial structure and dynamics in a marine food web\footnote{In {\em Aquatic Food Webs: an Ecosystem Approach.} Belgrano, A., Scharler, U., Dunne, J., and Ulanowicz, R., editors.}}
\author{Carlos J. Meli\'an\footnote{Corresponding author;
phone:+34 954 232340; fax:+34 954 621125; e-mail:
cmelian@ebd.csic.es} , Jordi Bascompte, and Pedro Jordano
\\
\\
\\Integrative Ecology Group 
\\Estaci\'on Biol\'ogica de Do\~nana, CSIC
\\Apdo. 1056, E-41080, Sevilla, Spain}


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\begin{abstract}

Trophic interactions, as many other relevant ecological processes, are spatially extended. Previous work has pointed out the importance of the spatial domain to understand food webs. Here we built on this body of work by considering how space shapes the structure and dynamics of food webs.  Specifically, we show that trophic modules (simple tri-trophic food chains and omnivory chains) composed by the same set of species are over-represented in pair-wise comparisons between habitats in a large Caribbean marine food web. Using a spatially structured food web model, we show how local species abundance (i.e., within habitats) is determined by the interplay between the degree of dispersal among habitats and the structure of the trophic module. Understanding the dynamical consequences of food webs patterns in a set of interacting communities is relevant for assessing the role of regional versus local processes for the persistence of diversity.

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\small{\centerline{Keywords: metacommunity, space, food webs, dispersal,}
\small{\centerline{predation, competition, community wide-patterns.}
\end{abstract}


\newpage

\section{Introduction}

The role of space in population and community dynamics has been recently emphasized (e.g. Tilman and Kareiva 1997; Hanski and Gilpin 1997; Bascompte and Sol\'e 1998). 
Several models for the coexistence of interacting species in heterogeneous environments have been formulated. These include the energy and material transfer across ecosystem boundaries and its implication for succession and diversity (Margalef 1963; Polis et. al 1997), the geographic mosaic of coevolution (Thompson 1994), the regional coexistence of competitors via a competition-colonization trade-off (Tilman 1994), the randomly assembly of communities via recruitment limitation (Hubbell 1997), and metacommunities (Wilson 1992). As a general conclusion of these approaches, succession, dispersal, local interactions, and spatial heterogeneity have appeared strongly linked to the persistence of diversity. However, the underlying structure of ecological interactions in a spatially structured ecosystem and its implications for the persistence of biodiversity remains elusive by the lack of synthetic data (Loreau et. al 2003).

Introducing space and multiple species in a single framework is a complicated task. As Caswell and Cohen (1993) argued, it is difficult to analyze patch-occupancy models with a large number of species because the number of possible patch states increases exponentially with species richness. Therefore, most spatial studies have dealt with a few number of species (Hanski 1983), predator-prey systems (Kareiva 1987) or n-competing species (Caswell and Cohen 1993; Tilman 1994; Mouquet and Loreau 2003). On the other hand, the bulk of studies in food web structure and dynamics have dealt with both a large (but see Hori and Noda 2001) and a small (but see Caldarelli et al. 1998) number of species respectively, but make no explicit reference to space (Caswell and Cohen 1993; Holt 1996; Holt 1997). Only a few studies have explored the role of space on a small subset of trophic interacting species (Holt 1997; Meli\'an and Bascompte 2002).

The present study is an attempt to link structure and dynamics in a spatially structured large marine food web. We use data on the diet of $5526$ specimens belonging to $208$ fishes (Randall 1967) of Caribbean community in five different habitats (Opitz 1996; Meli\'an and Bascompte 2004; Bascompte et al., submitted). First, we detect structure by addressing how simple trophic modules, tri-trophic food chains ($FCs$), and chains with omnivory ($OMN$) with the same set of species are shared among the five habitats. Second, we extend a previous metacommunity model (Mouquet and Loreau 2002) by incorporating the dynamics of trophic modules in a set of connected communities. Specifically, the following questions are addressed:
(1) How are simple trophic modules compossed by the same set of species represented among habitats? 
(2) How does the interplay between dispersal and food web structure affect species dynamics at both local and regional scales?

\newpage

\section{Food Web Data and Methods}


\subsection{Data Collection: Peculiarities and Limitations}

The Caribbean fish community here studied covers the geographic area of Puerto Rico-Virgin Islands. Data were obtained in an area over more than 1000 $km^2$ covering the US Virgin Islands of St. Thomas, St. John and St. Croix (200 $km^2$), the British Virgin Islands (343 $km^2$) and Puerto Rico (554 $km^2$). The fish species analyzed provide mainly from the study by Randall (1967) synthesized by Opitz (1996).  

Spatially explicit presence/absence community matrices were created by considering the presence of each species in a specific habitat only when that particular species was recorded foraging or breeding in that area (Opitz 1996; Froese and Pauly 2003). Community matrices include both the trophic links and the spatial distribution of $208$ fish species identified to the species level. Randall's list of shark species was completed by Opitz (1996), which included more sharks with affinities to coral reefs of the Puerto Rico-Virgin islands area, based on accounts in Fischer (1978). The final spatially explicit community matrix has $3138$ interactions representing five food webs in five habitat types. Specifically, the habitat types here studied are mangrove/estuaries (m; $40$ species and $94$ interactions), coral reefs (c; $170$ species and $1569$ interactions), seagrass beds/algal mats (a; $98$ species and $651$ interactions), sand (s; $89$ species and $750$ interactions ), and offshore reefs (r; $22$ species and $74$ interactions). 

\newpage 
\subsection{Null model of food web structure}


We consider tri-trophic food chains (Fig. 1a) and food chains with omnivory (Fig. 1c). We count the number and species composition of such trophic modules within the food web at each community. We then make pair-wise comparisons among communities ($n=10$ pair-wise comparisons) and count the number of chains (with identical species at all trophic levels) shared by each pair of communities. To asses whether this shared number is higher or lower than expected by chance we develop a null model. 
This algorithm randomizes the empirical data yet strictly conserves ingoing and outgoing links for each species. In this algorithm, a pair of directed links $A-B$ and $C-D$ are randomly selected. They are rewired in such a way that $A$ becomes connected to $D$, and $C$ to $B$, provided that none of these links already existed in the network, in which case the rewiring stops, and a new pair of links is selected.

The model was tested against empirical data using each food web habitat to generate $200$ replicates. For each replicate we compare pair-wise habitats, and count the shared number of simple tri-trophic food chains and chains with omnivory containing exactly the same set of species. For each pair of habitats, we estimate the probability that a random replicate has a shared number of identical species chains equal or higher than the observed value. Recent algorithm analysis suggest that this null model represents a conservative test for presence-absence matrices (Mikl\'os and Podani 2004).

\subsection{Dynamic Metacommunity-food web model}

In order to assess the local and regional dynamics of the structure studied, we extend a previous metacommunity model (Mouquet and Loreau 2002; 2003) by incorporating trophic modules (tri-trophic food chains and food chains with omnivory) in a set of interacting communities. The model follows the formalism of previous metapopulation models (Levins 1969) applied to the scale of the individual (Hasting 1980; Tilman 1994). At the local scale (within communities), we consider a collection of identical discrete sites given that no site is ever occupied by more than one individual. The regional dynamics is modeled as in mainland-island models with immigration (Gotelli 1991), but with an explicit origin of immigration that is a function of emigration from other communities in the metacommunity (Mouquet and Loreau 2003). Therefore, the model includes three hierarchical levels (individual, community, and metacommunity). The model reads as follows:
\begin{eqnarray}
{dP_i_k \over dt}= \theta I_i_k V_k+(1-d) c_i_k P_i_k V_k-m_i_k P_i_k+R_i_k P_i_k-C_i_k P_i_k.\label{eq:1}
\end{eqnarray}  
At the local scale, $P_i_k$ is the proportion of sites occupied by species $i$ in community $k$. Each community consists of $S$ species that indirectly compete within each trophic level for a limited proportion of vacant sites, $V_k$, defined as;
\begin{eqnarray}
{V_k}= 1-\sum_{j=1}^S P_j_k,\label{eq:2}
\end{eqnarray} 
{\em where $P_j_k$ represents the proportion of sites occupied by species $j$ within the same trophic level in community $k$}. The metacommunity is constituted by $N$ communities. For each species in the community, we considered an explicit immigration function $I_i_k$. Emigrants were combined in a regional pool of dispersers that was equally redistributed to all other communities, except that no individual returned to the community it came from (Mouquet and Loreau 2003). After immigration, individuals were associated to the parameters corresponding to the community they immigrated to. Dispersal success, $\theta$, is the probability that a migrant will find a new community. $I_i_k$ reads as;
\begin{eqnarray}
{I_i_k}= \frac{d}{N-1}\sum_{l\neq k}^N c_i_l P_i_l,\label{eq:3}
\end{eqnarray} 
where $d$ is the fraction of individuals dispersing to other habitats. $c_i_k$, is the local reproductive rate of species $i$ in community $k$, and $m_i_k$ is the mortality rate of species $i$ in community $k$. $R_i_k$ represents the amount of resources available to species $i$ in community $k$;
\begin{eqnarray}
{R_i_k}=\sum_{j=1}^S a_i_j_k P_j_k,\label{eq:4}
\end{eqnarray} 
where $a_{ijk}$ is the predation rate of species $i$ on species $j$ in community $k$, and the sum is for all prey species. Similarly, $C_i_k$ represents the amount of consumption exerted on species $i$ by all its predators in community $k$, and can be written as follows:
\begin{eqnarray}
{C_i_k}=\sum_{j=1}^S a_j_i_k P_j_k,\label{eq:5}
\end{eqnarray} 
where $a_{jik}$ is the predation rate of species $j$ on species $i$ in community $k$, and the sum is for all predator species. 

We have numerically simulated a metacommunity consisting of $6$ species in $6$ communities. In each community, the six species can form either two simple tri-trophic food chains, or two omnivory chains.  The two trophic modules within each community are linked only by indirect competition between species within the same trophic level. {\em This is the simplest case. Future simulations could introduce interferences between larger number of trophic modules}. We assumed a species was locally extinct when its proportion of occupied sites was lower than $0.01$. Mortality rates ($m_i_k$) are constant and equal for all species. Dispersal success ($\theta$), was set to $1$.

We considered potential reproductive rates to fit the constraint of strict regional similarity, $SRS$ (Mouquet and Loreau 2003). That is, species within each trophic level have the same regional basic reproductive rates, but these change locally among communities. Under $SRS$, each species within each trophic level is the best competitor in one community. Similarly, we introduce the constraint of regional trophic similarity ($SRTS$). That is, each consumer has the same set of local energy requirements but distributed differently among communities. Additionally, we assumed a direct relationship between the resource's local reproductive rate and the intensity it is predated with (Jennings and Mackinson 2003).

The results here presented were obtained with a single set of species parameters. Under the $SRS$ and $SRTS$ scenarios, results are qualitatively robust to deviations from these parameter values.

\newpage
\section{Results}


We calculated the number of tri-trophic food chains, and omnivory chains common to all pairs of communities, and compared this number with the predicted by our null model (Fig 1b and 1d). The coral reef habitat shares with other habitats a number of $FCs$ and $OMN$ larger than expected by chance ($P<0.0001$), except for Mangrove in $FCs$ (P$<$0.002) and $OMN$ ($P<0.01$). Similarly, seagrass beds/algal mats and sand share a significant number of $FCs$ and $OMN$ ($P<0.0001$). 

Globally, from the $10$ possible inter-community comparisons, five share a number of modules higher than expected by chance (Fig. 1a and 1c where arrow thickness is proportional to statistical significance). This suggests that habitats with a significant proportion of common trophic modules are mainly composed by a regional pool of individuals. 

The average fraction of shared $FCs$ and $OMN$ between habitat pairs is $38\% \pm24.5\%$ and $41\% \pm25\%$ respectively, which still leaves more than $50\%$ of different trophic modules between habitats. However, it is interesting to note that $15$ species are embedded in more than $75\%$ of trophic modules, which suggests that a small number of species are playing an important role in connecting through dispersal the local dynamics of communities.

{\em Under $SRS$ and $SRTS$ scenarios assumed, regional species abundance and inter-community variance are equal for each two species within the same trophic level. Regional abundance in $OMN$ is higher, equal and lower for top, intermediate, and basal species respectively. Local abundances differ significantly between the two modules explored}. Specifically, when there is no dispersal ($d=0$) there is local exclusion by the competitively superior species (Mouquet and Loreau 2002). This occurs for the basal and top species in the simple trophic chain. The variance in the abundance of the basal and top species between local communities is thus higher without dispersal for tri-trophic food chains (Fig. 2a).  However, the situation is completely different for omnivory. Now, the inter-community variance is very low for both the basal and top species in the absence of dispersal, and dramatically increased with $d$ in the case of the top species.  When communities are extremely linked, the top species disappears from two communities and is extremely abundant in the remaining communities.

Finally, we can see in Fig. 2b (as compared with Fig. 2a) that the inter-community variance for high $d$-values is higher in a metacommunity with omnivory. Thus, the interplay between dispersal among spatially structured communities and food web structure greatly affect local species abundances. 
  
\newpage

\section{Summary and Discussion}

It is well known that local communities can be structured by both local and regional interactions (Ricklefs 1987). However, it still remains unknown what trophic structures are shared by a set of interacting communities and its dynamical implications for the persistence of biodiversity. The present study is an attempt to link local and regional food web structure and dynamics in a spatially structured marine food web.

Communities in five habitats of the Caribbean have shown significantly similar trophic structures which suggest that these communities are open to immigration (Karlson and Cornell 2002). It has been recently shown that mangroves in the Caribbean strongly influence the local community structure of fish on neighbouring coral reef (Mumby et al. 2004). Additionally, recent empirical studies have shown that dispersal among habitats and local species interactions are key factors for metacommunity structure (Shurin  2001; Cottenie et al. 2003; Kneitel and Miller 2003; Cottenie and De Meester 2004), and the persistence of local and regional diversity (Mouquet and Loreau 2003). However, it still remain unclear how the interplay between dispersal and more complex trophic structures constraint or allow the persistence of more species in local communities (Carr et al. 2002; Kneitel and Miller 2003).

In the present work, closed metacommunities ($d=0$) with tri-trophic food chains showed an extreme variation in local abundances for both the basal and top species (Fig. 2a). However, the reverse happens in open communities with omnivory ($d=1$). The top species becomes unstable, and goes extinct in two local communities (Fig. 2b). Further data synthesis and theoretical work is needed to integrate the functional links between habitats and the dynamical consequences of the structural pattern reported.  

In summary, similar trophic modules over-represented in a set of different habitats suggest links between communities. The level of connectivity among these communities and the type of trophic modules alter local abundance of species and promotes local changes in diversity. The present chapter is only a first attempt to link empirical data on the structure of spatially heterogeneous food webs and its dynamical consequences. It still remains unexplored how the results here presented change by the introduction of a larger number of interacting modules in a set of interacting communities.
 

\section{Acknowledgments}

We thank the editors of this book for inviting us to contribute this chapter. Funding was provided by the Spanish Ministry of Science and Technology (Grants REN2003-04774 to JB and REN2003-XXXXX to PJ, and Ph.D. Fellowship FP2000-6137 to CJM).   

\newpage
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\section{Figure Legends}

$\bullet$ [Fig. 1] Food web modules studied here are (a) tri-trophic food chains ($FCs$), and (c) omnivory chains ($OMN$). Circles represent the five different habitats. The thickness of the link connecting two habitats is proportional to the statistical significance in the degree of similarity between these two habitats. Similarity is measured as the number of common trophic modules. (b) and (d) represent the frequency of common tri-trophic food chains and omnivory chains respectively in all pair-wise community comparisons. Black and white histograms represent the observed and the expected values, respectively. Habitat types are mangrove/estuaries (m), coral reefs (c), seagrass beds/algal mats (a), sand (s), and offshore reefs (r). {\em As noted, coral reefs (c), shares with the rest of the habitats a number of FCs and OMN larger than expected by chance, which suggest a high degree of connectance promoted by dispersal}.


$\bullet$ [Fig. 2] Inter-community variance in local species abundance for the basal (continuous line), intermediate (dotted line), and top (circles) species as a function of the proportion of dispersal between communities ($d$). (a) represents tri-trophic food chains and (b) omnivory chains. {\em Parameter values are $m_i_k=0.2$, $c_i_k$ for basal species is $3$ in the first community, $2.8$ in the second, $2.6$, $2.4$, $2.2$, and $2$ in the sixth community. For intermediate species is $1.5$ in the first community, $1.4$, $1.3$, $1.2$, $1.1$, and $1$. Top species reproductive values are $0.8$, $0.75$, $0.7$, $0.65$, $0.6$, and $0.55$. Predation rate of intermediate and top species $j$ on species $i$ in community $k$ are $0.6$ in the first community, $0.5$, $0.4$, $0.3$, $0.2$, and $0.1$. The initial proportion of sites occupied by species $i$ in community $k$, ($P_i_k$) is set to $0.01$. As noted, in closed metacommunities tri-trophic food chains showed an extreme variation in local abundances for both the basal and top species ($P_i_k<0.01$) in two and three communities respectively, and the reverse happens in open communities with omnivory. The top species becomes unstable, and goes extinct in two local communities ($P_i_k<0.01$).}

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\section{Figures}
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\begin{center}
\includegraphics[width=14cm]{spacecaribe.eps}
\centerline{\caption{Fig. 1}}
\end{center}

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\begin{center}
\includegraphics[width=8cm]{varlocal.eps}
\centerline{\caption{Fig. 2}}
\end{center}

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