Structural stability of invasion graphs for Lotka-Volterra systems. (English) Zbl 07839903
Summary: In this paper, we study in detail the structure of the global attractor for the Lotka-Volterra system with a Volterra-Lyapunov stable structural matrix. We consider the invasion graph as recently introduced in [J. Hofbauer and S. J. Schreiber, J. Math. Biol. 85, No. 5, Paper No. 54, 22 p. (2022; Zbl 1501.92113)] and prove that its edges represent all the heteroclinic connections between the equilibria of the system. We also study the stability of this structure with respect to the perturbation of the problem parameters. This allows us to introduce a definition of structural stability in ecology in coherence with the classical mathematical concept where there exists a detailed geometrical structure, robust under perturbation, that governs the transient and asymptotic dynamics.
MSC:
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
| 37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |
| 34D45 | Attractors of solutions to ordinary differential equations |
| 92D25 | Population dynamics (general) |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34D20 | Stability of solutions to ordinary differential equations |
| 34D23 | Global stability of solutions to ordinary differential equations |
| 05C90 | Applications of graph theory |
References:
| [1] | Afraimovich, V.; Tristan, I.; Huerta, R.; Rabinovich, MI, Winnerless competition principle and prediction of the transient dynamics in a Lotka-Volterra model, Chaos Interdiscip J Nonlinear Sci, 18, 4, 2008 · Zbl 1309.92064 · doi:10.1063/1.2991108 |
| [2] | Aragão-Costa, ER; Caraballo, T.; Carvalho, AN; Langa, JA, Stability of gradient semigroups under perturbation, Nonlinearity, 24, 2099-2117, 2011 · Zbl 1225.37018 · doi:10.1088/0951-7715/24/7/010 |
| [3] | Arrow, KJ; McManus, M., A note on dynamic stability, Econometrica, 26, 448-454, 1958 · Zbl 0107.37201 · doi:10.2307/1907624 |
| [4] | Barabás, GE; D’Andrea, R.; Stump, SM, Chesson’s coexistence theory, Ecol Monogr, 88, 277-303, 2018 · doi:10.1002/ecm.1302 |
| [5] | Bascompte, J.; Jordano, P., Mutualistic networks, 2014, Princeton: Princeton University Press, Princeton |
| [6] | Bortolan, MC; Carvalho, AN; Langa, JA; Raugel, G., Nonautonomous perturbations of Morse-Smale semigroups: stability of the phase diagram, J Dyn Differ Equ, 34, 2681-2747, 2022 · Zbl 1508.37042 · doi:10.1007/s10884-021-10066-6 |
| [7] | Chesson, P., Multispecies competition in variable environments, Theor Popul Biol, 45, 227-276, 1994 · Zbl 0804.92025 · doi:10.1006/tpbi.1994.1013 |
| [8] | Chi, C-W; Hsu, S-B; Wu, L-I, On the asymmetric May-Leonard model of three competing species, SIAM J Appl Math, 58, 211-226, 1998 · Zbl 0919.92036 · doi:10.1137/S0036139994272060 |
| [9] | Cross, GW, Three types of matrix stability, Linear Algebra Appl., 20, 253-263, 1978 · Zbl 0376.15007 · doi:10.1016/0024-3795(78)90021-6 |
| [10] | Esteban, FJ; Galadí, JA; Langa, JA; Portillo, JR; Soler-Toscano, F., Informational structures: a dynamical system approach for integrated information, PLoS Comput Biol, 14, 9, 2018 · doi:10.1371/journal.pcbi.1006154 |
| [11] | Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, 2013, New York: Springer, New York |
| [12] | Guerrero, G.; Langa, JA; Suarez, A., Architecture of attractor determines dynamics on mutualistic complex networks, Nonlinear Anal Real World Appl., 34, 17-40, 2017 · Zbl 1370.34109 · doi:10.1016/j.nonrwa.2016.07.009 |
| [13] | Hale, JK, Asymptotic Behavior of Dissipative Systems, 1988, Providence: American Mathematical Society, Providence · Zbl 0642.58013 |
| [14] | Hang-Kwang, L.; Pimm, SL, The assembly of ecological communities: a minimalist approach, J Anim Ecol, 62, 4, 749-765, 1993 · doi:10.2307/5394 |
| [15] | Hastings, A.; Abbott, KC; Cuddington, K.; Francis, TB; Gellner, G.; Lai, Y-C; Morozov, A.; Petrovskii, S.; Scranton, K.; Zeeman, ML, Transient phenomena in ecology, Science, 361, 6406, eaat6412, 2018 · doi:10.1126/science.aat6412 |
| [16] | Hastings, A.; Abbott, KC; Cuddington, K.; Francis, TB; Lai, Y-C; Morozov, A.; Petrovskii, S.; Zeeman, ML, Effects of stochasticity on the length and behaviour of ecological transients, J R Soc Interface, 18, 180, 20210257, 2021 · doi:10.1098/rsif.2021.0257 |
| [17] | Hofbauer, J.; Sigmund, K., The theory of evolution and dynamical systems, 1988, Cambridge: Cambridge University Press, Cambridge · Zbl 0678.92010 |
| [18] | Hofbauer, J.; Sigmund, K., Evolutionary games and population dynamics, 1998, Cambridge: Cambridge University Press, Cambridge · Zbl 0914.90287 · doi:10.1017/CBO9781139173179 |
| [19] | Hofbauer, J.; Schreiber, SJ, Permanence via invasion graphs: incorporating community assembly into modern coexistence theory, J Math Biol, 85, 54, 2022 · Zbl 1501.92113 · doi:10.1007/s00285-022-01815-2 |
| [20] | Johnson, CR, Hadamard products of matrices, Linear Multilinear Algebra, 1, 295-307, 1974 · Zbl 0281.15013 · doi:10.1080/03081087408817030 |
| [21] | Kalita, P.; Langa, JA; Soler-Toscano, F., Informational structures and informational fields as a prototype for the description of postulates of the integrated information theory, Entropy, 21, 5, 493, 2019 · doi:10.3390/e21050493 |
| [22] | Kelley, A., The stable, center-stable, center, center-unstable, unstable manifolds, J Differ Equ, 3, 546-570, 1967 · Zbl 0173.11001 · doi:10.1016/0022-0396(67)90016-2 |
| [23] | Kraaijevanger, JFBM, A characterization of Lyapunov diagonal stability using Hadamard products, Linear Algebra Appl, 151, 245-254, 1991 · Zbl 0724.15014 · doi:10.1016/0024-3795(91)90366-5 |
| [24] | Labarca, M.; Pacifico, MJ, Stability of Morse-Smale vector fields on manifolds with boundary, Topology, 29, 57-81, 1990 · Zbl 0717.58032 · doi:10.1016/0040-9383(90)90025-F |
| [25] | Lischke, H.; Löffler, T., Finding all multiple stable fixpoints of n-species Lotka-Volterra competition models, Theor Popul Biol, 115, 24-34, 2017 · Zbl 1381.92084 · doi:10.1016/j.tpb.2017.02.001 |
| [26] | Logofet, DO, Matrices and graphs: stability problems in mathematical ecology, 1993, Boca Raton: CRC Press, Boca Raton |
| [27] | MacArthur, RH, Species packing and what interspecies competition minimizes, Proc Natl Acad Sci USA, 64, 1369-1375, 1969 · doi:10.1073/pnas.64.4.1369 |
| [28] | May, RM, Stability and complexity in model ecosystems, 1973, Princeton: Princeton University Press, Princeton |
| [29] | May, RM; Leonard, WJ, Nonlinear aspects of competition between three species, SIAM J Appl Math, 29, 243-253, 1975 · Zbl 0314.92008 · doi:10.1137/0129022 |
| [30] | Moylan, PJ, Matrices with positive principal minors, Linear Algebra Appl, 17, 53-58, 1977 · Zbl 0356.15005 · doi:10.1016/0024-3795(77)90040-4 |
| [31] | Morton, RD; Law, R.; Pimm, SL; Drake, JA, On models for assembling ecological communities, Oikos, 75, 3, 493, 1996 · doi:10.2307/3545891 |
| [32] | Novak, M.; Yeakel, JD; Noble, AE; Doak, DF; Emmerson, M.; Estes, JA; Jacob, U.; Tinker, MT; Wootton, JT, Characterizing species interactions to understand press perturbations: What is the community matrix?, Ann Rev Ecol Evolut Syst, 47, 1, 409-432, 2016 · doi:10.1146/annurev-ecolsys-032416-010215 |
| [33] | Portillo, JR; Soler-Toscano, F.; Langa, JA, Global structural stability and the role of cooperation in mutualistic systems, PLoS ONE, 17, 4, 2022 · doi:10.1371/journal.pone.0267404 |
| [34] | Prishlyak, AO; Bilun, SV; Prus, AA, Morse flows with fixed points on the boundary of 3-manifolds, J Math Sci, 274, 881-897, 2023 · Zbl 1536.37029 · doi:10.1007/s10958-023-06651-3 |
| [35] | Robinson, C., Structural stability on manifolds with boundary, J Differ Equ, 37, 1-11, 1980 · Zbl 0441.58010 · doi:10.1016/0022-0396(80)90083-2 |
| [36] | Robinson, JC, Infinite-dimensional dynamical systems, 2001, Cambridge: Cambridge University Press, Cambridge · Zbl 0980.35001 · doi:10.1007/978-94-010-0732-0 |
| [37] | Rohr, RP; Saavedra, S.; Bascompte, J., On the structural stability of mutualistic systems, Science, 345, 1253497, 2014 · doi:10.1126/science.1253497 |
| [38] | Serván, CA; Allesina, S., Tractable models of ecological assembly, Ecol Lett, 24, 1029-1037, 2021 · doi:10.1111/ele.13702 |
| [39] | Takeuchi, Y., Global dynamical properties of Lotka-Volterra systems, 1996, Singapore: World Scientific Publishing Co. Pte. Ltd., Singapore · Zbl 0844.34006 · doi:10.1142/2942 |
| [40] | Zeeman, EC, Stability of dynamical systems, Nonlinearity, 1, 1, 115-155, 1988 · Zbl 0643.58005 · doi:10.1088/0951-7715/1/1/005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
