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Partitioning a reaction-diffusion ecological network for dynamic stability. (English) Zbl 1427.92099
Summary: The loss of dispersal connections between habitat patches may destabilize populations in a patched ecological network. This work studies the stability of populations when one or more communication links is removed. An example is finding the alignment of a highway through a patched forest containing a network of metapopulations in the patches. This problem is modelled as that of finding a stable cut of the graph induced by the metapopulations network, where nodes represent the habitat patches and the weighted edges model the dispersal between habitat patches. A reaction-diffusion system on the graph models the dynamics of the predator-prey system over the patched ecological network. The graph Laplacian’s Fiedler value, which indicates the well-connectedness of the graph, is shown to affect the stability of the metapopulations. We show that, when the Fiedler value is sufficiently large, the removal of edges without destabilizing the dynamics of the network is possible. We give an exhaustive graph partitioning procedure, which is suitable for smaller networks and uses the criterion for both the local and global stability of populations in partitioned networks. A heuristic graph bisection algorithm that preserves the preassigned lower bound for the Fiedler value is proposed for larger networks and is illustrated with examples.
Reviewer: Reviewer (Berlin)
92D40 Ecology
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35K57 Reaction-diffusion equations
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[1] Gilpin M. (2012) Metapopulation dynamics: empirical and theoretical investigations. New York, NY: Academic Press.
[2] Wilson DS. (1992) Complex interactions in metacommunities, with implications for biodiversity and higher levels of selection. Ecology 73, 1984-2000. (doi:10.2307/1941449)
[3] Tilman D, May RM, Lehman CL, Nowak MA. (1994) Habitat destruction and the extinction debt. Nature 371, 65-66. (doi:10.1038/371065a0)
[4] Reijnen R, van der Grift E, Van der Veen M, Pelk M, Lüchtenborg A, Bal D. (2000) De weg mét de minste weerstand; opgave ontsnippering. Technical report. Alterra, Wageningen, The Netherlands. See http://library.wur.nl/WebQuery/wurpubs/312298.
[5] Opdam P. (1991) Metapopulation theory and habitat fragmentation: a review of holarctic breeding bird studies. Landscape Ecol. 5, 93-106. (doi:10.1007/BF00124663)
[6] Hanski I. (1999) Metapopulation ecology. Oxford, UK: Oxford University Press.
[7] Levin SA. (1974) Dispersion and population interactions. Am. Nat. 108, 207-228. (doi:10.1086/282900)
[8] Chesson PL. (1981) Models for spatially distributed populations: the effect of within-patch variability. Theor. Popul. Biol. 19, 288-325. (doi:10.1016/0040-5809(81)90023-X) · Zbl 0472.92015
[9] Kareiva P. (1990) Population dynamics in spatially complex environments: theory and data. Phil. Trans. R. Soc. Lond. B 330, 175-190. (doi:10.1098/rstb.1990.0191)
[10] Amarasekare P. (1998) Interactions between local dynamics and dispersal: insights from single species models. Theor. Popul. Biol. 53, 44-59. (doi:10.1006/tpbi.1997.1340) · Zbl 0894.92029
[11] Bascompte J, Solé RV. (1994) Spatially induced bifurcations in single-species population dynamics. J. Anim. Ecol. 63, 256-264. (doi:10.2307/5544)
[12] Briggs CJ, Hoopes MF. (2004) Stabilizing effects in spatial parasitoid–host and predator–prey models: a review. Theor. Popul. Biol. 65, 299-315. (doi:10.1016/j.tpb.2003.11.001) · Zbl 1109.92047
[13] Amarasekare P. (2004) The role of density-dependent dispersal in source–sink dynamics. J. Theor. Biol. 226, 159-168. (doi:10.1016/j.jtbi.2003.08.007)
[14] Vance RR. (1984) The effect of dispersal on population stability in one-species, discrete-space population growth models. Am. Nat. 123, 230-254. (doi:10.1086/284199)
[15] Hassell M, Miramontes O, Rohani P, May R. (1995) Appropriate formulations for dispersal in spatially structured models: comments on Bascompte & Solé. J. Anim. Ecol. 64, 662-664. (doi:10.2307/5808)
[16] Rohani P, May RM, Hassell MP. (1996) Metapopulation and equilibrium stability: the effects of spatial structure. J. Theor. Biol. 181, 97-109. (doi:10.1006/jtbi.1996.0118)
[17] Jang SJ, Mitra AK. (2000) Equilibrium stability of single-species metapopulations. Bull. Math. Biol. 62, 155-161. (doi:10.1006/bulm.1999.0145) · Zbl 1323.92171
[18] Hanski I, Kuussaari M, Nieminen M. (1994) Metapopulation structure and migration in the butterfly Melitaea cinxia. Ecology 75, 747-762. (doi:10.2307/1941732)
[19] Turing AM. (1952) The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37-72. (doi:10.1098/rstb.1952.0012) · Zbl 1403.92034
[20] Levins R, Culver D. (1971) Regional coexistence of species and competition between rare species. Proc. Natl Acad. Sci. USA 68, 1246-1248. (doi:10.1073/pnas.68.6.1246) · Zbl 0217.57702
[21] Nakao H, Mikhailov AS. (2010) Turing patterns in network-organized activator–inhibitor systems. Nat. Phys. 6, 544. (doi:10.1038/nphys1651)
[22] Li MY, Shuai Z. (2010) Global-stability problem for coupled systems of differential equations on networks. J. Differ. Equ. 248, 1-20. (doi:10.1016/j.jde.2009.09.003) · Zbl 1190.34063
[23] Ide Y, Izuhara H, Machida T. (2016) Turing instability in reaction–diffusion models on complex networks. Physica A 457, 331-347. (doi:10.1016/j.physa.2016.03.055) · Zbl 1400.92060
[24] Manyombe MM, Tsanou B, Mbang J, Bowong S. (2017) A metapopulation model for the population dynamics of anopheles mosquito. Appl. Math. Comput. 307, 71-91. (doi:10.1016/j.amc.2017.02.039) · Zbl 1411.92280
[25] Caswell H. (1978) Predator-mediated coexistence: a nonequilibrium model. Am. Nat. 112, 127-154. (doi:10.1086/283257)
[26] McCauley E, Wilson WG, de Roos AM. (1993) Dynamics of age-structured and spatially structured predator-prey interactions: individual-based models and population-level formulations. Am. Nat. 142, 412-442. (doi:10.1086/285547)
[27] Crowley PH. (1981) Dispersal and the stability of predator-prey interactions. Am. Nat. 118, 673-701. (doi:10.1086/283861)
[28] Artzy-Randrup Y, Stone L. (2010) Connectivity, cycles, and persistence thresholds in metapopulation networks. PLoS Compt. Biol. 6, e1000876. (doi:10.1371/journal.pcbi.1000876)
[29] Urban D, Keitt T. (2001) Landscape connectivity: a graph-theoretic perspective. Ecology 82, 1205-1218. (doi:10.1890/0012-9658(2001)082[1205:LCAGTP]2.0.CO;2)
[30] Pecora LM, Carroll TL. (1998) Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80, 2109. (doi:10.1103/PhysRevLett.80.2109)
[31] Tanner HG, Jadbabaie A, Pappas GJ. (2007) Flocking in fixed and switching networks. IEEE Trans. Autom. Control 52, 863-868. (doi:10.1109/TAC.2007.895948) · Zbl 1366.93414
[32] Fax JA, Murray RM. (2004) Information flow and cooperative control of vehicle formations. IEEE Trans. Autom. Control 49, 1465-1476. (doi:10.1109/TAC.2004.834433) · Zbl 1365.90056
[33] Merris R. (1994) Laplacian matrices of graphs: a survey. Linear Algebra Appl. 197, 143-176. (doi:10.1016/0024-3795(94)90486-3) · Zbl 0802.05053
[34] Mohar B, Alavi Y, Chartrand G, Oellermann O. (1991) The Laplacian spectrum of graphs. Graph Theory Comb. Appl. 2, 12.
[35] Kernighan BW, Lin S. (1970) An efficient heuristic procedure for partitioning graphs. Bell Syst. Tech. J. 49, 291-307. (doi:10.1002/bltj.1970.49.issue-2) · Zbl 0333.05001
[36] Garey MR, Johnson DS. (1979) Computers and intractability: a guide to the theory of NP-completeness. New York, NY: W.H. Freeman.
[37] Merris R. (1998) Laplacian graph eigenvectors. Linear Algebra Appl. 278, 221-236. (doi:10.1016/S0024-3795(97)10080-5) · Zbl 0932.05057
[38] Kot M. (2001) Elements of mathematical ecology. Cambridge, UK: Cambridge University Press.
[39] Bhatia R. (1997) Matrix analysis. Berlin, Germany: Springer.
[40] Anderson WN, Morley TD. (1985) Eigenvalues of the Laplacian of a graph. Linear Multilinear A 18, 141-145. (doi:10.1080/03081088508817681) · Zbl 0594.05046
[41] Hahn G, Sabidussi G. (2013) Graph symmetry: algebraic methods and applications, vol. 497. Montreal, Canada: Springer Science & Business Media.
[42] Cvetkovic D, Simic S, Rowlinson P. (2009) An introduction to the theory of graph spectra. Cambridge, UK: Cambridge University Press.
[43] Ding J, Zhou A. (2007) Eigenvalues of rank-one updated matrices with some applications. Appl. Math. Lett. 20, 1223-1226. (doi:10.1016/j.aml.2006.11.016) · Zbl 1139.15003
[44] Ding J, Yao G. (2007) The eigenvalue problem of a specially updated matrix. Appl. Math. Comput. 185, 415-420. (doi:10.1016/j.amc.2006.07.040) · Zbl 1113.15010
[45] Lohmiller W, Slotine JJE. (1998) On contraction analysis for non-linear systems. Automatica 34, 683-696. (doi:10.1016/S0005-1098(98)00019-3) · Zbl 0934.93034
[46] Horn RA, Horn RA, Johnson CR. (1990) Matrix analysis. Cambridge, UK: Cambridge University Press.
[47] Deo N. (2017) Graph theory with applications to engineering and computer science. Mineola, NY: Courier Dover Publications.
[48] Schulz C. (2013) High quality graph partitioning. PhD thesis, Karlsruhe Institute of Technology, Karlsruhe, Germany. · Zbl 1269.68115
[49] Karypis G. (2011) METIS and ParMETIS. In Encyclopedia of parallel computing (ed. D Padua), pp. 1117-1124. Berlin, Germany: Springer.
[50] Bellen A, Zennaro M. (2013) Numerical methods for delay differential equations. Oxford, UK: Oxford University Press. · Zbl 0749.65042
[51] Karimi HR, Maass P. (2009) Delay-range-dependent exponential H∞ synchronization of a class of delayed neural networks. Chaos Solitons Fractals 41, 1125-1135. (doi:10.1016/j.chaos.2008.04.051) · Zbl 1198.93179
[52] Karimi HR, Gao H. (2010) New delay-dependent exponential H∞ synchronization for uncertain neural networks with mixed time delays. IEEE Trans. Syst. Man Cybern. B (Cybern.) 40, 173-185. (doi:10.1109/TSMCB.2009.2024408)
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