# American Institute of Mathematical Sciences

July 2018, 23(5): 1895-1915. doi: 10.3934/dcdsb.2018186

## Persistence of complex food webs in metacommunities

 1 Fraunhofer Institute for Solar Energy Systems, Freiburg, Germany 2 Department of Engineering Mathematics, University of Bristol, Bristol, UK

* Corresponding author: Thilo Gross

Received  April 2017 Revised  August 2017 Published  May 2018

Meta-foodweb theory is considered a promising approach for explaining species diversity and foodweb complexity. Recently Pillai et al. (2010) proposed a simple modeling framework for the dynamics of food webs at the metacommunity level. Here, we employ this framework to compute general conditions for the persistence of complex meta-foodwebs. The persistence conditions found depend on the connectivity of the resource patches and the structure of the assembled food web, thus linking the underlying spatial patch-network and the species interaction network. We find that the persistence of omnivores is more likely when it is feeding on (a) prey on low trophic levels, and (b) prey on similar trophic levels.

Citation: Gesa A. Benndorf, Thilo Gross. Persistence of complex food webs in metacommunities. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 1895-1915. doi: 10.3934/dcdsb.2018186
##### References:
 [1] Y. Ahn, H. Jeong, N. Masuda and J. Noh, Epidemic dynamics of two species of interacting particles on scale-free networks, Physical Review E, 74 (2006), 066113. doi: 10.1103/PhysRevE.74.066113. [2] E. Barter and T. Gross, Meta-foodchains as a many-layer epidemic process on networks, Physical Review E, 93 (2016), 022303. [3] G. A. Böhme and T. Gross, Analytical calculation of fragmentation transitions in adaptive networks, Physical Review E, 83 (2011), 035101. [4] L. Bolchoun, B. Drossel and K. Allhoff, Spatial topologies affect local food web structure and diversity in evolutionary metacommunities, Scientific Reports, 7 (2017), 1818. doi: 10.1038/s41598-017-01921-y. [5] A. Brechtel, P. Gramlich, D. Ritterskamp, B. Drossel and T. Gross, Master stiability functions reveal diffusion-driven instabilities is multilayer networks, arXiv: 1610.07635 (2016). [6] U. Brose, R. Williams and N. Martinez, Allometric scaling enhances stability in complex food webs, Ecology Letters, 9 (2006), 1228-1236. doi: 10.1111/j.1461-0248.2006.00978.x. [7] O. Diekmann, J. Heesterbeek and M. Roberts, The construction of next-generation matrices for compartmental epidemic models, Journal of The Royal Society Interface, 7 (2010), 873-885. doi: 10.1098/rsif.2009.0386. [8] P. Gramlich, S. Plitzko, L. Rudolf, B. Drossel and T. Gross, The influence of dispersal on a predator-prey system with two habitats, Journal of Theoretical Biology, 398 (2016), 150-161. doi: 10.1016/j.jtbi.2016.03.015. [9] D. Gravel, F. Massol and M. Leibold, Stability and complexity in model meta-ecosystems, Nature Communications, 7 (2016), 12457. doi: 10.1038/ncomms12457. [10] T. Gross, L. Rudolf, S. Levin and U. Dieckmann, Generalized models reveal stabilizing factors in food webs, Science, 325 (2009), 747-750. doi: 10.1126/science.1173536. [11] I. Hanski, Metapopulation dynamics, Nature, 396 (1998), 41-49. [12] G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297. doi: 10.1126/science.131.3409.1292. [13] G. Hutchinson, Homage to Santa Rosalia or why are there so many kinds of animals?, American Naturalist, 43 (1959), 145-159. doi: 10.1086/282070. [14] J. Kamgang and G. Sallet, Computation of threshold conditions for epidemiological models and global stability of the disease-free equilibrium (dfe), Mathematical Biosciences, 213 (2008), 1-12. doi: 10.1016/j.mbs.2008.02.005. [15] B. Karrer and M. Newman, Competing epidemics on complex networks, Physical Review E, 84 (2011), 036106. doi: 10.1103/PhysRevE.84.036106. [16] R. Levins, Some demographic and genetic consequences of environmental heterogeneity for biological control, Bulletin of the Ecological Society of America, 15 (1969), 237-240. doi: 10.1093/besa/15.3.237. [17] N. Masuda and N. Konno, Multi-state epidemic processes on complex networks, Journal of Theoretical Biology, 243 (2006), 64-75. doi: 10.1016/j.jtbi.2006.06.010. [18] K. McCann, A. Hastings and G. Huxel, Weak trophic interactions and the balance of nature, Nature, 395 (1998), 794-798. doi: 10.1038/27427. [19] A. Mougi, Spatial complexity enhances predictability in food webs, Scientific Reports, 7 (2017), 43440. doi: 10.1038/srep43440. [20] A. Mougi and M. Kondoh, Food-web complexity, meta-community complexity and community stability, Scientific Reports, 6 (2016), 24478. doi: 10.1038/srep24478. [21] A. Muneepeerakul, J. Weitz, S. Levin, A. Rinaldo and I. Rodriguez-Iturbe, A neutral metapopulation model of biodiversity in river networks, Journal of Theoretical Biology, 245 (2007), 351-363. doi: 10.1016/j.jtbi.2006.10.005. [22] M. Newman, Threshold effects for two pathogens spreading on a network, Physical Review Letters, 95 (2005), 108701. doi: 10.1103/PhysRevLett.95.108701. [23] P. Pillai, M. Loreau and A. Gonzalez, A patch-dynamic framework for food web metacommunities, Theoretical Ecology, 3 (2010), 223-237. doi: 10.1007/s12080-009-0065-1. [24] P. Pillai, A. Gonzalez and M. Loreau, Metacommunity theory explains the emergence of food web complexity, Proceedings of the National Academy of Sciences, 108 (2011), 19293-19298. doi: 10.1073/pnas.1106235108. [25] P. Pillai, A. Gonzalez and M. Loreau, Evolution of dispersal in a predator-prey metacommunity, The American Naturalist, 179 (2012), 204-216. [26] N. Tromeur, L. Rudolf and T. Gross, Impact of dispersal on the stability of metapopulations, Journal of Theoretical Biology, 392 (2016), 1-11. doi: 10.1016/j.jtbi.2015.11.029. [27] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [28] J. Vandermeer, Omnivory and the stability of food webs, Journal of Theoretical Biology, 238 (2006), 497-504. doi: 10.1016/j.jtbi.2005.06.006.

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##### References:
 [1] Y. Ahn, H. Jeong, N. Masuda and J. Noh, Epidemic dynamics of two species of interacting particles on scale-free networks, Physical Review E, 74 (2006), 066113. doi: 10.1103/PhysRevE.74.066113. [2] E. Barter and T. Gross, Meta-foodchains as a many-layer epidemic process on networks, Physical Review E, 93 (2016), 022303. [3] G. A. Böhme and T. Gross, Analytical calculation of fragmentation transitions in adaptive networks, Physical Review E, 83 (2011), 035101. [4] L. Bolchoun, B. Drossel and K. Allhoff, Spatial topologies affect local food web structure and diversity in evolutionary metacommunities, Scientific Reports, 7 (2017), 1818. doi: 10.1038/s41598-017-01921-y. [5] A. Brechtel, P. Gramlich, D. Ritterskamp, B. Drossel and T. Gross, Master stiability functions reveal diffusion-driven instabilities is multilayer networks, arXiv: 1610.07635 (2016). [6] U. Brose, R. Williams and N. Martinez, Allometric scaling enhances stability in complex food webs, Ecology Letters, 9 (2006), 1228-1236. doi: 10.1111/j.1461-0248.2006.00978.x. [7] O. Diekmann, J. Heesterbeek and M. Roberts, The construction of next-generation matrices for compartmental epidemic models, Journal of The Royal Society Interface, 7 (2010), 873-885. doi: 10.1098/rsif.2009.0386. [8] P. Gramlich, S. Plitzko, L. Rudolf, B. Drossel and T. Gross, The influence of dispersal on a predator-prey system with two habitats, Journal of Theoretical Biology, 398 (2016), 150-161. doi: 10.1016/j.jtbi.2016.03.015. [9] D. Gravel, F. Massol and M. Leibold, Stability and complexity in model meta-ecosystems, Nature Communications, 7 (2016), 12457. doi: 10.1038/ncomms12457. [10] T. Gross, L. Rudolf, S. Levin and U. Dieckmann, Generalized models reveal stabilizing factors in food webs, Science, 325 (2009), 747-750. doi: 10.1126/science.1173536. [11] I. Hanski, Metapopulation dynamics, Nature, 396 (1998), 41-49. [12] G. Hardin, The competitive exclusion principle, Science, 131 (1960), 1292-1297. doi: 10.1126/science.131.3409.1292. [13] G. Hutchinson, Homage to Santa Rosalia or why are there so many kinds of animals?, American Naturalist, 43 (1959), 145-159. doi: 10.1086/282070. [14] J. Kamgang and G. Sallet, Computation of threshold conditions for epidemiological models and global stability of the disease-free equilibrium (dfe), Mathematical Biosciences, 213 (2008), 1-12. doi: 10.1016/j.mbs.2008.02.005. [15] B. Karrer and M. Newman, Competing epidemics on complex networks, Physical Review E, 84 (2011), 036106. doi: 10.1103/PhysRevE.84.036106. [16] R. Levins, Some demographic and genetic consequences of environmental heterogeneity for biological control, Bulletin of the Ecological Society of America, 15 (1969), 237-240. doi: 10.1093/besa/15.3.237. [17] N. Masuda and N. Konno, Multi-state epidemic processes on complex networks, Journal of Theoretical Biology, 243 (2006), 64-75. doi: 10.1016/j.jtbi.2006.06.010. [18] K. McCann, A. Hastings and G. Huxel, Weak trophic interactions and the balance of nature, Nature, 395 (1998), 794-798. doi: 10.1038/27427. [19] A. Mougi, Spatial complexity enhances predictability in food webs, Scientific Reports, 7 (2017), 43440. doi: 10.1038/srep43440. [20] A. Mougi and M. Kondoh, Food-web complexity, meta-community complexity and community stability, Scientific Reports, 6 (2016), 24478. doi: 10.1038/srep24478. [21] A. Muneepeerakul, J. Weitz, S. Levin, A. Rinaldo and I. Rodriguez-Iturbe, A neutral metapopulation model of biodiversity in river networks, Journal of Theoretical Biology, 245 (2007), 351-363. doi: 10.1016/j.jtbi.2006.10.005. [22] M. Newman, Threshold effects for two pathogens spreading on a network, Physical Review Letters, 95 (2005), 108701. doi: 10.1103/PhysRevLett.95.108701. [23] P. Pillai, M. Loreau and A. Gonzalez, A patch-dynamic framework for food web metacommunities, Theoretical Ecology, 3 (2010), 223-237. doi: 10.1007/s12080-009-0065-1. [24] P. Pillai, A. Gonzalez and M. Loreau, Metacommunity theory explains the emergence of food web complexity, Proceedings of the National Academy of Sciences, 108 (2011), 19293-19298. doi: 10.1073/pnas.1106235108. [25] P. Pillai, A. Gonzalez and M. Loreau, Evolution of dispersal in a predator-prey metacommunity, The American Naturalist, 179 (2012), 204-216. [26] N. Tromeur, L. Rudolf and T. Gross, Impact of dispersal on the stability of metapopulations, Journal of Theoretical Biology, 392 (2016), 1-11. doi: 10.1016/j.jtbi.2015.11.029. [27] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [28] J. Vandermeer, Omnivory and the stability of food webs, Journal of Theoretical Biology, 238 (2006), 497-504. doi: 10.1016/j.jtbi.2005.06.006.
Schematic illustration of the emergence of complex meta-foodwebs. Local food webs (bubbles, arrows) exist in geographical patches embedded in a geographical network (grey). It is assumed that competitive exclusion restricts the local dynamics such that in every particular patch only a simple food chain can be realized. However, because different food chains are realized in different patches, aggregating all observations of species and interactions into a single network still leads to a complex food web
Example of food webs emerging in the metacommunity. Specialist species are depicted by black nodes, while the omnivore is shown as a grey node. Specialists are numbered according to the trophic level they belong to. Arrows indicate feeding links. Shown are the spatially aggregated food webs that can be observed depending on the value of the normalized extinction rate $z = e/c$ and the mean connectivity $\langle k \rangle$. The omnivore $x$ becomes extinct due to competitive exclusion by specialist 3 for $z<\langle k \rangle/(6+3\sqrt{2})$ (a), coexists with the specialist 3 for $z<\langle k \rangle/6$ (b), and persists in a system where the specialist 3 cannot survive for $z<\langle k \rangle/3$ (c). These parameter ranges are not drawn to scale
Recipe for the calculation of persistence thresholds
Illustration of the feeding interactions of omnivores. We study two types of omnivores leading to structurally different persistence conditions: Ⅰ) omnivores feeding on consecutive species, and Ⅱ) omnivores feeding on non-consecutive species. The figure illustrates the notation where prey species are labeled as $i-1$ and $j-1$ respectively
Dependence of the coexistence range for an omnivore on the trophic levels of its prey. Shown is the range of the normalized extinction rate ($\bar{z}$) for which an omnivore can coexist with a specialist food chain (shaded, see Eqs. (18), (19)). Each omnivore is assumed to be able to feed on two specialist species on the tropic levels $i-1$ and $j-1$. The figure shows that the ranges of coexistence grow when $i$ and $j$ are closer together and become maximal if the omnivore feeds on two species which are on consecutive levels. Furthermore coexistance ranges are larger for omnivores feeding lower in the food chain. The dashed region marks the parameter range where all omnivores with a given values of $i$ can coexists with the specialist chain
Equilibrium densities of three different omnivores from numerical simulations (symbols) and theoretical predictions (lines). Shown is the proportion of patches occupied by the respective species (patch-density) over the normalized extinction rate of all species $\bar{z}$. Dashed lines indicate parameter values where the respective specialist competitor goes extinct in the metacommunity. The equilibrium patch densities observed in simulations display a close to linear behavior and are well described by the analytical formula in Eq. (21). The parameters for the simulation are the same as in Fig. 10
Comparison of the predicted ranges with simulation results. Plotted are the equilibrium patch-densities $p_3^*$ (●), $p_4^*$ ($\circ$) and $p_x^*$($\times$), obtained from network simulations. Eeach point corresponds to an average over 10 independent simulation runs. We used $N = 10000, \langle k \rangle = 20, c = 0.05$ and varied $\bar z = e$ within the indicated range. Dashed lines correspond to the persistence thresholds predicted by our analytical calculations. Coexistence range for omnivore $x$: $2/(13+\sqrt{97})<\bar z<1/(5+\sqrt{13})$, persistence threshold for specialist 3: $\bar z = 1/6$ and persistence threshold for specialist 4: $\bar z = 1/10$
General food web configuration consisting of a chain of specialists and a generalist $g$. The generalist feeds on two prey species in two independent chains arising from two types of habitat. In A), the trophic level of both prey species is the same ($i = j$), in B) the trophic levels differ at least by one ($i>j$). The corresponding coexistence ranges for both configurations are derived in the text
Comparison between the coexistence ranges for omnivores and generalists. For a system of two types of habitat and 10 trophic levels (20 specialists in two branches), we plot the coexistence range for an omnivore $x$, feeding upon prey species from exclusively one branch and the coexistence range for a generalist $g$, feeding upon two prey species from both branches (inset). Throughout the whole $j$-range, the coexistence range for the omnivore is larger than the coexistence range for the generalist
Example for a maximal food web emerging in the metacommunity. In contrast to Fig. 2, trophic levels of the omnivores' prey species differ by more than one. Symbols are the same as in Fig. 2. Shown are different food web configurations which arise in the given parameter ranges. The corresponding parameter ranges (not true to scale) are obtained by the mathematical formalism presented in the text
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