A patch-dynamic framework for food web metacommunities

Abstract

The metacommunity concept has proved to be a valuable tool for studying how space can affect the properties and assembly of competitive communities. However, the concept has not been as extensively applied to the study of food webs or trophically structured communities. Here, we demonstrate how to develop a modelling framework that permits food webs to be considered from a spatial perspective. We do this by broadening the classic metapopulation patch-dynamic framework so that it can also account for trophic interactions between many species and patches. Unlike previous metacommunity models, we argue that this requires a system of equations to track the changing patch occupancy of the various species interactions, not the patch occupancy of individual species. We then suggest how this general theoretical framework can be used to study complex and spatially extended food web metacommunities.

Appendices

Appendix A

We can appreciate the motivation underlying Eq. 4 by considering the portion of a hypothetical food web shown in Fig. 5. The subgraph S _4,9 of the food web graph in Fig. 5 represents all the species and feeding links lying on a directed path between species 4 and species 9 (solid black lines). From Eq. 3, we know that f (S _4,9) gives the fraction of patches occupied by species 9 that are also occupied by species 4. In this case

$$ f\left( {{S_{4,9}}} \right) = \left( {{\rho_{\left( {4,5} \right)}} \cdot {\rho_{\left( {5,6} \right)}} \cdot {\rho_{\left( {6,8} \right)}} \cdot {\rho_{\left( {8,9} \right)}}} \right) + \left( {{\rho_{\left( {4,5} \right)}} \cdot {\rho_{\left( {5,7} \right)}} \cdot {\rho_{\left( {7,8} \right)}} \cdot {\rho_{\left( {8,9} \right)}}} \right) + {\rho_{\left( {4,9} \right)}}. $$

(7)

However, what we are actually interested in is the fraction of species 9-occupied patches where species 9 will be affected by extinctions of species 4. Looking at Fig. 5, we see that extinction of species 4 in patches containing the food chain sequence 4, 5, 7, 8, and 9 will not lead to the extinction of species 9, since an intermediate species 7 can switch predation to 4’s resource species 3, as indicated by the (3, 7) link (dashed line). As a result, a patch with the food chain sequence 3, 4, 5, 7, 8 and 9 will reassemble, after 4 goes extinct, into 3, 7, 8 and 9.

Fig. 5

An example of a subgraph of a hypothetical food web network

In order to know how the regional abundance of a species i is affected by the extinction rate of any particular species k below it, we need to know what fraction of patches containing species i also contains some intermediate species m capable of switching consumption to one of k’s resources upon k’s extinction. This is given by the second term in Eq. 4, $f\left( {\bigcup\limits_{m \in C(l)} {\left( {{S_{k,m}} \cup {S_{m,i}}} \right)} } \right)$. For the example given here,

$$ f\left( {\bigcup\limits_{m \in C(l)} {\left( {{S_{k,m}} \cup {S_{m,i}}} \right)} } \right) = {\rho_{\left( {4,5} \right)}} \cdot {\rho_{\left( {5,7} \right)}} \cdot {\rho_{\left( {7,8} \right)}} \cdot {\rho_{\left( {8,9} \right)}}. $$

(8)

Subtracting Eq. 8 from Eq. 7, as per Eq. 4 gives us the net fraction of patches occupied by 9, where 9 will go extinct upon 4 going extinct whenever 4 is consuming resource 3: $\Phi_{4,9}^3 = \left( {{\rho_{\left( {4,5} \right)}} \cdot {\rho_{\left( {5,6} \right)}} \cdot {\rho_{\left( {6,8} \right)}} \cdot {\rho_{\left( {8,9} \right)}}} \right) + {\rho_{\left( {4,9} \right)}}$.

Appendix B

We demonstrate here the application of Eq. 2 for the top two pairwise interactions in Fig. 2c (we assume v _5,4 > v _6,4 and v _3,1 > v _6,1).

For the (6, 7) interaction, the first term in Eq. 4 becomes $A = {c_{7,6}}{p_{\left( {6,7} \right)}}\left( {{p_{\left( {1,6} \right)}} + {p_{\left( {4,6} \right)}} - {p_{\left( {6,7} \right)}}} \right)$. Here, the growth in the fractions of patches with the (6, 7) interactions is due to the total number of species 7 colonisers (c _7,6 p _(6,7)) landing on available resource 6 patches, itself given by the total fractions of resource 6 patches (p _(1,6) + p _(4,6)), minus those already occupied by consumer 7 (p _(6,7)).

The second term given by $B = \left( {{e_{7,6}} + {\mu_{7,6}}} \right){p_{\left( {6,7} \right)}}$ simply gives the extinction rate of the (6, 7) interaction due to the extinction rate of consumer 7 (e _7,6) and the rate that resource 6 is driven extinct due to predation by consumer 7, (μ _7,6).

Since there are no competitors for resource 6 and no alternative resources for consumer 7 aside from 6, terms C, D and E are all equal to 0. However, a decrease in the (6, 7) interaction can occur due to extinctions of species further down the food web. The effect that species extinctions further down the food web has on the (6, 7) interaction will be determined by the degree of patch overlap between the (6, 7) interaction and the rate of extinction for each species below 6. In the F term, the extinction rates of all species interactions below (6, 7), multiplied by the fraction of the 7-occupied patches that will be affected by that extinction are summed (see Appendix A for an example). When this sum is multiplied by the density of (6, 7) patches, p _(6,7), we get $F = {p_{\left( {6,7} \right)}}\left( {\left( {{e_{6,1}}{\rho_{\left( {1,6} \right)}} + {\rho_{\left( {4,6} \right)}}} \right.\left( {\left( {{e_{6,4}} + {\mu_{6,4}}} \right) + \left( {{e_{4,3}} + {\mu_{4,3}}} \right) + {\rho_{\left( {2,3} \right)}}\left( {{e_{3,2}} + {\mu_{3,2}}} \right) + {\rho_{\left( {1,3} \right)}}\left( {{e_{3,1}} + {\mu_{3,1}}} \right)} \right)} \right)$.

The (6, 7) interaction can also decrease due to competitive displacement of species further down the food web—in this case, species 5 displacing 6 from resource 4 patches. Thus, here, G is equal to the number of species 5 colonisers, c _5,4 p _(4,5), that successfully land on and displace consumer species 6 from resource 4 in food chains with the (6, 7) interaction, ρ _(6,4) p _(6,7), giving us G = p _(6,7) (c _5,4 p _(4,5) ρ _(6,4)). Finally, H = 0, since there is no predator species that can drive 7 extinct from top–down effects. The overall equation for the (6, 7) interaction is then

$$ \begin{array}{*{20}{c}} {\frac{{{\text{d}}{p_{\left( {6,7} \right)}}}}{{{\text{d}}t}} = {c_{7,6}}{p_{\left( {6,7} \right)}}\left( {{p_{\left( {1,6} \right)}} + {p_{\left( {4,6} \right)}} - {p_{\left( {6,7} \right)}}} \right) - \left( {{e_{7,6}} + {\mu_{7,6}}} \right){p_{\left( {6,7} \right)}}} \\ { - {p_{\left( {6,7} \right)}}\left( {\left( {{e_{6,1}}{\rho_{\left( {1,6} \right)}} + {\rho_{\left( {4,6} \right)}}} \right.\left( {\left( {{e_{6,4}} + {\mu_{6,4}}} \right) + \left( {{e_{4,3}} + {\mu_{4,3}}} \right) + {\rho_{\left( {2,3} \right)}}\left( {{e_{3,2}} + {\mu_{3,2}}} \right) + {\rho_{\left( {1,3} \right)}}\left( {{e_{3,1}} + {\mu_{3,1}}} \right)} \right)} \right)} \\ { - {p_{\left( {6,7} \right)}}\left( {{c_{5,4}}{p_{\left( {4,5} \right)}}{\rho_{\left( {6,4} \right)}}} \right).} \\ \end{array} $$

Similarly for the (4, 6) interaction, A gives growth of the interaction due to colonisation of available resource 4 patches by consumer 6. However, now, in order to determine the total number of species 6 colonisers produced, one must sum over all the resource patches occupied by 6 (c _6,4 p _(4,6) + c _6,1 p _(1,6)), giving us $A = \left( {{c_{6,4}}{p_{\left( {4,6} \right)}} + {c_{6,1}}{p_{\left( {1,6} \right)}}} \right)\left( {{p_{\left( {3,4} \right)}} - {p_{\left( {4,5} \right)}} - {p_{\left( {4,6} \right)}}} \right)$. B is defined similarly to the previous example.

The (4, 6) interaction, unlike (6, 7), can decrease due to competitive displacement of the consumer species; in this case, 6 can be displaced from resource 4 by 5 giving us C = c _5,4 p _(4,5) (p _(4,6)). Since there is no way 6 can directly switch from some alternative resource onto 4, both D and E are equal to 0. The F and H terms are determined similarly to the previous example, while G = 0, since there are no species below the interaction that can be displaced by superior competitors. The overall equation then for the (4, 6) interaction is

$$ \begin{gathered} \frac{{{\text{d}}{p_{\left( {4,6} \right)}}}}{{{\text{d}}t}} = \left( {{c_{6,4}}{p_{\left( {4,6} \right)}} + {c_{6,1}}{p_{\left( {1,6} \right)}}} \right)\left( {{p_{\left( {3,4} \right)}} - {p_{\left( {4,5} \right)}} - {p_{\left( {4,6} \right)}}} \right) - \left( {{e_{6,4}} + {\mu_{6,4}}} \right){p_{\left( {4,6} \right)}} \\ - {c_{5,4}}{p_{\left( {4,5} \right)}}\left( {{p_{\left( {4,6} \right)}}} \right) - {p_{\left( {4,6} \right)}}\left( {\left( {{e_{4,3}} + {\mu_{4,3}}} \right) + {\rho_{\left( {2,3} \right)}}\left( {{e_{3,2}} + {\mu_{3,2}}} \right) + {\rho_{\left( {1,3} \right)}}\left( {{e_{3,1}} + {\mu_{3,1}}} \right)} \right) \\ - {\mu_{7,6}}{p_{\left( {4,6} \right)}}. \\ \end{gathered} $$

Appendix C

Algorithm for transforming a food web graph

Below, we outline an algorithm that can transform the food web graph so that no directed path in the graph will represent an a priori impossible food chain configuration. The algorithm required is relatively straightforward: (1) move up the vertex set until you come to a generalist species, t, with more than one incoming edge, at least one of which is a bypass link, which directly connects the consumer to a resource further down one of its other food chains. (2) Start at the resource vertex in one of the bypass links that has the lowest index value, and from there, start moving up the vertex set one index number at a time, checking all the incoming edges for each vertex while doing so. If the current vertex, k, has more than one incoming edge, at least one of which is from a directed path rooted in one of the bypass resources and at least one of which is from a directed path not rooted in one of the bypass resources, and if at least one of vertex k’s outgoing edges is on a directed path towards t, then split vertex k into two parallel vertices, each with the same outgoing edge as before, but no incoming edges. (3) Attach to the original vertex, k, any incoming edges from directed paths that were not rooted in one of the bypass resources, and to the new vertex, k’, attach the incoming edges from paths that are rooted in the one of the bypass resources. (4) Continue this until you arrive at the consumer vertex t and then disconnect all incoming edges from t that are part of a directed path from one of the bypass species. (5) Start again at (1) moving up the food web looking for the next generalist predator with bypass loops in order to establish a new t, and then repeat (2)–(4).

This algorithm can be applied to the network before the system of differential equations is defined. A simple example of transformation can be observed for the graph in Fig. 2c, which becomes, by the above method, the graph shown in Fig. 6. In Fig. 6, the distribution of each consumer’s feeding links among its prey can be considered independently of how the prey’s own feeding links are distributed among its own resources and so on. As a result, the frequency of patch overlap between any two species in a directed path or food chain can be determined by multiplying the ρ values of the pairwise interactions along the path between them.

Fig. 6

The food web in Fig. 2c after being transformed by the algorithm described in Appendix C to ensure that all directed paths depicted in the food web graph are not a priori impossible food chains