Meta-food-chains as a many-layer epidemic process on networks

Notable recent works have focused on the multilayer properties of coevolving diseases. We point out that very similar systems play an important role in population ecology. Specifically we study a meta-food-web model that was recently proposed by Pillai et al. [Theor. Ecol. 3, 223 (2009)]. This model describes a network of species connected by feeding interactions, which spread over a network of spatial patches. Focusing on the essential case, where the network of feeding interactions is a chain, we develop an analytical approach for the computation of the degree distributions of colonized spatial patches for the different species in the chain. This framework allows us to address ecologically relevant questions. Considering configuration model ensembles of spatial networks, we find that there is an upper bound for the fraction of patches that a given species can occupy, which depends only on the networks mean degree. For a given mean degree there is then an optimal degree distribution that comes closest to the upper bound. Notably scale-free degree distributions perform worse than more homogeneous degree distributions if the mean degree is sufficiently high. Because species experience the underlying network differently the optimal degree distribution for one particular species is generally not the optimal distribution for the other species in the same food web. These results are of interest for conservation ecology, where, for instance, the task of selecting areas of old-growth forest to preserve in an agricultural landscape, amounts to the design of a patch network.

Figure 1

Example of the two-step process for finding the degree distribution of the effective network of species 2. We start with the patch network and its degree distribution, $p_k$, (a). The first step identifies the nodes colonized by the species, ${\chi }_k$, (b). The second step removes links from colonized nodes to uncolonized nodes from the distribution giving the effective network, $g_k$, (c). This is the network upon which species 2 disperses.

Figure 2

Comparison of colonized abundance of species 1 in patch networks with different degree distributions. Shown are analytical (lines) and simulated (markers) abundances for species on patch networks with Poisson (blue) and scale-free (green) degree distributions with the same mean degree and the limit as calculated from the configuration model. For low $e/c$ the Poisson degree distribution leads to greater abundance, but for high $e/c⪆5$ it is the scale-free distributions that allow greater abundance. The insert highlights the region at which the lines cross.

Figure 3

Effective degree distributions experienced by species on different trophic levels for Poisson patch networks with $〈k〉=20$. Shown are the degree distribution of a particular patch network and the effective network of species 4 and 8 food chain with which all species $e_i/c_i=0.5$ from simulation (markers) and analytically (lines). The sum of the distribution is reduced for higher trophic levels as the species inhabits fewer patches.

Figure 4

Effective degree distributions experienced by species on different trophic levels for scale-free patch networks with $〈k〉=20$. Shown are the degree distribution of a particular patch network and the effective network of species 4 and 8 food chain with which all species $e_i/c_i=0.5$ from simulation (markers) and analytically (lines). The sum of the distribution is reduced for higher trophic levels as the species inhabits fewer patches.

Figure 5

Colonized abundance for many trophic levels of a food chain. Shown are the fraction of patches inhabited by each species in food chains on patch networks with Poisson degree distributions (blue) and scale-free degree distributions (green). All species have $e/c=0.5$, and both patch networks have $〈k〉=10$. For species in the low trophic levels the networks with Poisson degree distribution have greater abundance, but for species higher in the food chain it is the scale-free distribution that has the greater abundance. Further the scale-free network can support more species than the Poisson network.