×

zbMATH — the first resource for mathematics

Model of pattern formation in marsh ecosystems with nonlocal interactions. (English) Zbl 1435.92095
In the present work the authors propose a phenomenological model to describe the dynamics of the marsh edge in terms of two-way interactions between marsh grass Spartina alternifora and sedimentation.
The aim of this paper lies in understanding whether the well-known scale-dependent (nonlocal) feedback between marsh vegetation and sedimentation can lead to spatially variable shoreline configurations. They propose a system of reaction-diffusion equations with an additional integral term with a Mexican-hat kernel function that describes the nature of this scale-dependent feedback. The proposed system is highly cooperative; as cooperative systems in which lack the classic activator-inhibitor mechanism necessary for pattern formation, it becomes of interest how and under what conditions spatial patterns may develop. It is performed a biharmonic approximation of our system and carry out analysis on the simpler biharmonic system that expresses the kernel function as separate short-range and long-range diffusion terms. Using the more mathematically tractable biharmonic system, we are then able to derive general condition for the formation of spatial patterns in a cooperative system such as in the introduced model. Further, using numerical simulations, the authors establish that the biharmonic model, while an approximation, is consistent with the original model, and therefore we can apply the theoretical results from the biharmonic system to help gain insight into the formation of patterns in the original system. After parameterizing the kernel function using a set of reasonable parameters from literature and it is established that spatial patterns can develop, given that the scale-dependent interactions between marsh vegetation and sediment dynamics are strong enough.
MSC:
92D40 Ecology
92D25 Population dynamics (general)
35K57 Reaction-diffusion equations
35B36 Pattern formations in context of PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
Software:
MATCONT
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adams, J.; Grobler, A.; Rowe, C.; Riddin, T.; Bornman, T.; Ayres, D., Plant traits and spread of the invasive salt marsh grass, Spartina alterniflora Loisel., in the Great Brak estuary, South Africa, Afr J Mar Sci, 34, 3, 313-322 (2012)
[2] Altieri, Ah; Silliman, Br; Bertness, Md, Hierarchical organization via a facilitation cascade in intertidal cordgrass bed communities, Am Nat, 169, 2, 195-206 (2007)
[3] Amari, S-I, Dynamics of pattern formation in lateral-inhibition type neural fields, Biol Cybern, 27, 2, 77-87 (1977) · Zbl 0367.92005
[4] Balke, T.; Klaassen, Pc; Garbutt, A.; Van Der Wal, D.; Herman, Pmj; Bouma, Tj, Conditional outcome of ecosystem engineering: a case study on tussocks of the salt marsh pioneer Spartina anglica, Geomorphology, 153, 232-238 (2012)
[5] Banerjee, M.; Volpert, V., Prey-predator model with a nonlocal consumption of prey, Chaos Interdiscip J Nonlinear Sci, 26, 8, 083120 (2016) · Zbl 1378.92080
[6] Barbier, N.; Couteron, P.; Lefever, R.; Deblauwe, V.; Lejeune, O., Spatial decoupling of facilitation and competition at the origin of gapped vegetation patterns, Ecology, 89, 6, 1521-1531 (2008)
[7] Bates, Pw; Ren, X., Transition layer solutions of a higher order equation in an infinite tube, Commun Partial Differ Equ, 21, 1-2, 109-145 (1996)
[8] Bates, Pw; Ren, X., Heteroclinic orbits for a higher order phase transition problem, Eur J Appl Math, 8, 2, 149-163 (1997) · Zbl 0873.35099
[9] Bayliss, A.; Volpert, V., Patterns for competing populations with species specific nonlocal coupling, Math Model Nat Phenom, 10, 6, 30-47 (2015) · Zbl 1338.35237
[10] Bertness, Md, Ribbed mussels and Spartina alterniflora production in a New England salt marsh, Ecology, 65, 1794-1807 (1984)
[11] Bertness, Md; Callaway, R., Positive interactions in communities, Trends Ecol Evol, 9, 5, 191-193 (1994)
[12] Bertness, Md; Grosholz, E., Population dynamics of the ribbed mussel, Geukensia demissa: the costs and benefits of an aggregated distribution, Oecologia, 67, 2, 192-204 (1985)
[13] Billingham, J., Dynamics of a strongly nonlocal reaction-diffusion population model, Nonlinearity, 17, 1, 313 (2003) · Zbl 1050.35038
[14] Borgogno, F.; D’Odorico, P.; Laio, F.; Ridolfi, L., Mathematical models of vegetation pattern formation in ecohydrology, Rev Geophys, 47, 1, RG1005 (2009)
[15] Bouma, T.; Van Duren, L.; Temmerman, S.; Claverie, T.; Blanco-Garcia, A.; Ysebaert, T.; Herman, Pmj, Spatial flow and sedimentation patterns within patches of epibenthic structures: combining field, flume and modelling experiments, Cont Shelf Res, 27, 8, 1020-1045 (2007)
[16] Bouma, T.; Friedrichs, M.; Van Wesenbeeck, B.; Temmerman, S.; Graf, G.; Herman, Pmj, Density-dependent linkage of scale-dependent feedbacks: a flume study on the intertidal macrophyte Spartina anglica, Oikos, 118, 2, 260-268 (2009)
[17] Bouma, T.; Temmerman, S.; Van Duren, L.; Martini, E.; Vandenbruwaene, W.; Callaghan, D.; Balke, T.; Biermans, G.; Klaassen, P.; Van Steeg, P., Organism traits determine the strength of scale-dependent bio-geomorphic feedbacks: a flume study on three intertidal plant species, Geomorphology, 180, 57-65 (2013)
[18] Britton, N., Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J Appl Math, 50, 6, 1663-1688 (1990) · Zbl 0723.92019
[19] Castets, V.; Dulos, E.; Boissonade, J.; De Kepper, P., Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern, Phys Rev Lett, 64, 24, 2953 (1990)
[20] Chen, Y.; Kolokolnikov, T.; Tzou, J.; Gai, C., Patterned vegetation, tipping points, and the rate of climate change, Eur J Appl Math, 26, 6, 945-958 (2015) · Zbl 1375.92078
[21] Couteron, P.; Lejeune, O., Periodic spotted patterns in semi-arid vegetation explained by a propagation-inhibition model, J Ecol, 89, 4, 616-628 (2001)
[22] Dakos, V.; Kéfi, S.; Rietkerk, M.; Van Nes, Eh; Scheffer, M., Slowing down in spatially patterned ecosystems at the brink of collapse, Am Nat, 177, 6, E153-E166 (2011)
[23] De Jager, M.; Weissing, Fj; Van De Koppel, J., Why mussels stick together: spatial self-organization affects the evolution of cooperation, Evol Ecol, 31, 1-12 (2017)
[24] Deegan, La; Johnson, Ds; Warren, Rs; Peterson, Bj; Fleeger, Jw; Fagherazzi, S.; Wollheim, Wm, Coastal eutrophication as a driver of salt marsh loss, Nature, 490, 7420, 388-392 (2012)
[25] Dhooge, A.; Govaerts, W.; Kuznetsov, Y.; Meijer, H.; Sautois, B., New features of the software Matcont for bifurcation analysis of dynamical systems, Math Comput Model Dyn Syst, 14, 2, 147-175 (2008) · Zbl 1158.34302
[26] Dibner, R.; Doak, D.; Lombardi, E., An ecological engineer maintains consistent spatial patterning, with implications for community-wide effects, Ecosphere, 6, 9, 1-17 (2015)
[27] D’Odorico, P.; Laio, F.; Ridolfi, L., Patterns as indicators of productivity enhancement by facilitation and competition in dryland vegetation, J Geophys Res Biogeosci, 111, G3, 1-7 (2006)
[28] Fagherazzi, S., Coastal processes: storm-proofing with marshes, Nat Geosci, 7, 10, 701-702 (2014)
[29] Fagherazzi, S.; Kirwan, Ml; Mudd, Sm; Guntenspergen, Gr; Temmerman, S.; D’Alpaos, A.; Van De Koppel, J.; Rybczyk, Jm; Reyes, E.; Craft, C., Numerical models of salt marsh evolution: ecological, geomorphic, and climatic factors, Rev Geophys, 50, 1, RG1002 (2012)
[30] Fagherazzi, S.; Mariotti, G.; Wiberg, P.; Mcglathery, K., Marsh collapse does not require sea level rise, Oceanography, 26, 3, 70-77 (2013)
[31] Fuentes, M.; Kuperman, M.; Kenkre, V., Nonlocal interaction effects on pattern formation in population dynamics, Phys Rev Lett, 91, 15, 158104 (2003)
[32] Gedan, Kb; Kirwan, Ml; Wolanski, E.; Barbier, Eb; Silliman, Br, The present and future role of coastal wetland vegetation in protecting shorelines: answering recent challenges to the paradigm, Clim Change, 106, 1, 7-29 (2011)
[33] Gierer, A.; Meinhardt, H., A theory of biological pattern formation, Biol Cybern, 12, 1, 30-39 (1972)
[34] Gleason, Ml; Elmer, Da; Pien, Nc; Fisher, Js, Effects of stem density upon sediment retention by salt marsh cord grass, Spartina alterniflora loisel, Estuaries, 2, 4, 271-273 (1979)
[35] Goodman, Je; Wood, Me; Gehrels, Wr, A 17-year record of sediment accretion in the salt marshes of Maine (USA), Mar Geol, 242, 1-3, 109-121 (2007)
[36] Gourley, S.; Chaplain, Ma; Davidson, F., Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation, Dyn Syst Int J, 16, 2, 173-192 (2001) · Zbl 0988.35082
[37] Green, Jb; Sharpe, J., Positional information and reaction-diffusion: two big ideas in developmental biology combine, Development, 142, 7, 1203-1211 (2015)
[38] Halpern, Bs; Silliman, Br; Olden, Jd; Bruno, Jp; Bertness, Md, Incorporating positive interactions in aquatic restoration and conservation, Front Ecol Environ, 5, 3, 153-160 (2007)
[39] Hardaway CS Jr, Byrne RJ (1999) Shoreline management in Chesapeake Bay. Special report in Applied Marine Science and Ocean Engineering No. 356, Virginia Institute of Marine Science, William & Mary, VA, USA
[40] He, Q.; Bertness, Md; Altieri, Ah, Global shifts towards positive species interactions with increasing environmental stress, Ecol Lett, 16, 5, 695-706 (2013)
[41] Hiscock, Tw; Megason, Sg, Mathematically guided approaches to distinguish models of periodic patterning, Development, 142, 3, 409-419 (2015)
[42] Kéfi, S.; Guttal, V.; Brock, Wa; Carpenter, Sr; Ellison, Am; Livina, Vn; Seekell, Da; Scheffer, M.; Van Nes, Eh; Dakos, V., Early warning signals of ecological transitions: methods for spatial patterns, PLoS ONE, 9, 3, e92097 (2014)
[43] Kéfi, S.; Holmgren, M.; Scheffer, M., When can positive interactions cause alternative stable states in ecosystems?, Funct Ecol, 30, 1, 88-97 (2016)
[44] Klausmeier, Ca, Regular and irregular patterns in semiarid vegetation, Science, 284, 5421, 1826-1828 (1999)
[45] Liu, Q-X; Weerman, Ej; Herman, Pmj; Olff, H.; Van De Koppel, J., Alternative mechanisms alter the emergent properties of self-organization in mussel beds, Proc R Soc Lond B Biol Sci, 279, rspb20120157 (2012)
[46] Liu, Q-X; Herman, Pmj; Mooij, Wm; Huisman, J.; Scheffer, M.; Olff, H.; Van De Koppel, J., Pattern formation at multiple spatial scales drives the resilience of mussel bed ecosystems, Nat Commun, 5, 5234 (2014)
[47] Madzvamuse, A.; Ndakwo, Hs; Barreira, R., Cross-diffusion-driven instability for reaction-diffusion systems: analysis and simulations, J Math Biol, 70, 4, 709-743 (2015) · Zbl 1335.92035
[48] Mariotti, G.; Fagherazzi, S., A numerical model for the coupled long-term evolution of salt marshes and tidal flats, J Geophys Res Earth Surf, 115, F1, F01004 (2010)
[49] Martínez-García R, Lopez C (2018) From scale-dependent feedbacks to long-range competition alone: a short review on pattern-forming mechanisms in arid ecosystems. Preprint arXiv:1801.01399
[50] Martínez-García, R.; Calabrese, Jm; Hernández-García, E.; López, C., Vegetation pattern formation in semiarid systems without facilitative mechanisms, Geophys Res Lett, 40, 23, 6143-6147 (2013)
[51] Martínez-García, R.; Calabrese, Jm; Hernández-García, E.; López, C., Minimal mechanisms for vegetation patterns in semiarid regions, Philos Trans R Soc A Math Phys Eng Sci, 372, 2027, 20140068 (2014)
[52] Merchant, Sm; Nagata, W., Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition, Theor Popul Biol, 80, 4, 289-297 (2011) · Zbl 1323.92177
[53] Möller, I.; Kudella, M.; Rupprecht, F.; Spencer, T.; Paul, M.; Van Wesenbeeck, Bk; Wolters, G.; Jensen, K.; Bouma, Tj; Miranda-Lange, M., Wave attenuation over coastal salt marshes under storm surge conditions, Nat Geosci, 7, 10, 727 (2014)
[54] Murray, Jd, Mathematical biology. II spatial models and biomedical applications (Interdisciplinary applied mathematics) (2001), New York: Springer, New York
[55] Nakamasu, A.; Takahashi, G.; Kanbe, A.; Kondo, S., Interactions between zebrafish pigment cells responsible for the generation of Turing patterns, Proc Natl Acad Sci, 106, 21, 8429-8434 (2009)
[56] Ninomiya, H.; Tanaka, Y.; Yamamoto, H., Reaction, diffusion and non-local interaction, J Math Biol, 75, 5, 1203-1233 (2017) · Zbl 1386.35209
[57] Nyman, Ja; Delaune, Rd; Roberts, Hh; Patrick, W. Jr, Relationship between vegetation and soil formation in a rapidly submerging coastal marsh, Mar Ecol Prog Ser, 96, 269-279 (1993)
[58] Parshad, Rd; Kumari, N.; Kasimov, Ar; Abderrahmane, Ha, Turing patterns and long-time behavior in a three-species food-chain model, Math Biosci, 254, 83-102 (2014) · Zbl 1323.92182
[59] Perry, Je; Atkinson, Rb, York river tidal marshes, J Coast Res, 57, 40-49 (2009)
[60] Priestas, Am; Mariotti, G.; Leonardi, N.; Fagherazzi, S., Coupled wave energy and erosion dynamics along a salt marsh boundary, Hog Island Bay, Virginia, USA, J Mar Sci Eng, 3, 3, 1041-1065 (2015)
[61] Pringle, Rm; Tarnita, C., Spatial self-organization of ecosystems: integrating multiple mechanisms of regular-pattern formation, Ann Rev Entomol, 62, 1, 359-377 (2017)
[62] Raspopovic, J.; Marcon, L.; Russo, L.; Sharpe, J., Digit patterning is controlled by a Bmp-Sox9-Wnt Turing network modulated by morphogen gradients, Science, 345, 6196, 566-570 (2014)
[63] Rietkerk, M.; Van De Koppel, J., Regular pattern formation in real ecosystems, Trends Ecol Evol, 23, 3, 169-175 (2008)
[64] Rosen, Ps, Erosion susceptibility of the Virginia Chesapeake Bay shoreline, Mar Geol, 34, 1-2, 45-59 (1980)
[65] Rovinsky, Ab; Menzinger, M., Self-organization induced by the differential flow of activator and inhibitor, Phys Rev Lett, 70, 6, 778 (1993)
[66] Schile, Lm, Modeling tidal marsh distribution with sea-level rise: evaluating the role of vegetation, sediment, and upland habitat in marsh resiliency, PLoS ONE, 9, 2, e88760 (2014)
[67] Schwarz, C.; Bouma, T.; Zhang, L.; Temmerman, S.; Ysebaert, T.; Herman, Pmj, Interactions between plant traits and sediment characteristics influencing species establishment and scale-dependent feedbacks in salt marsh ecosystems, Geomorphology, 250, 298-307 (2015)
[68] Sheehan, Mr; Ellison, Jc, Tidal marsh erosion and accretion trends following invasive species removal, Tamar estuary, Tasmania, Estuar Coast Shelf Sci, 164, 46-55 (2015)
[69] Shi, J.; Xie, Z.; Little, K., Cross-diffusion induced instability and stability in reaction-diffusion systems, J Appl Anal Comput, 1, 1, 95-119 (2011) · Zbl 1304.35329
[70] Siebert, J.; Schöll, E., Front and Turing patterns induced by Mexican-hat-like nonlocal feedback, Europhys Lett (EPL), 109, 4, 40014 (2015)
[71] Siero, E.; Doelman, A.; Eppinga, M.; Rademacher, J. D.; Rietkerk, M.; Siteur, K., Striped pattern selection by advective reaction-diffusion systems: resilience of banded vegetation on slopes, Chaos Interdiscip J Nonlinear Sci, 25, 3, 036411 (2015) · Zbl 1374.92167
[72] Silliman, Br; Van De Koppel, J.; Mccoy, Mw; Diller, J.; Kasozi, Gn; Earl, K.; Adams, Pn; Zimmerman, Ar, Degradation and resilience in louisiana salt marshes after the BP-Deepwater Horizon oil spill, Proc Natl Acad Sci, 109, 28, 11234-11239 (2012)
[73] Silliman, Br; Schrack, E.; He, Q.; Cope, R.; Santoni, A.; Van Der Heide, T.; Jacobi, R.; Jacobi, M.; Van De Koppel, J., Facilitation shifts paradigms and can amplify coastal restoration efforts, Proc Natl Acad Sci, 112, 46, 14295-14300 (2015)
[74] Stumpf, Rp, The process of sedimentation on the surface of a salt marsh, Estuar Coast Shelf Sci, 17, 5, 495-508 (1983)
[75] Tonelli, M.; Fagherazzi, S.; Petti, M., Modeling wave impact on salt marsh boundaries, J Geophys Res Oceans (2010)
[76] Turing, Am, The chemical basis of morphogenesis, Philos Trans R Soc Lond B Biol Sci, 237, 641, 37-72 (1952) · Zbl 1403.92034
[77] Van De Koppel, J.; Herman, Pmj; Thoolen, P.; Heip, Ch, Do alternate stable states occur in natural ecosystems? Evidence from a tidal flat, Ecology, 82, 12, 3449-3461 (2001)
[78] Van De Koppel, J.; Rietkerk, M.; Dankers, N.; Herman, Pmj, Scale-dependent feedback and regular spatial patterns in young mussel beds, Am Nat, 165, 3, E66-E77 (2005)
[79] Van De Koppel, J.; Van Der Wal, D.; Bakker, Jp; Herman, Pmj, Self-organization and vegetation collapse in salt marsh ecosystems, Am Nat, 165, 1, E1-E12 (2005)
[80] Van Der Heide, T.; Eklöf, Js; Van Nes, Eh; Van Der Zee, Em; Donadi, S.; Weerman, Ej; Olff, H.; Eriksson, Bk, Ecosystem engineering by seagrasses interacts with grazing to shape an intertidal landscape, PLoS ONE, 7, 8, e42060 (2012)
[81] Van Hulzen, J.; Van Soelen, J.; Bouma, T., Morphological variation and habitat modification are strongly correlated for the autogenic ecosystem engineer Spartina anglica (common cordgrass), Estuar Coasts, 30, 1, 3-11 (2007)
[82] Van Wesenbeeck, Bk; Van De Koppel, J.; Herman, Pmj; Bouma, Tj, Does scale-dependent feedback explain spatial complexity in salt-marsh ecosystems?, Oikos, 117, 1, 152-159 (2008)
[83] Vandenbruwaene, W.; Temmerman, S.; Bouma, T.; Klaassen, P.; De Vries, M.; Callaghan, D.; Van Steeg, P.; Dekker, F.; Van Duren, L.; Martini, E., Flow interaction with dynamic vegetation patches: implications for biogeomorphic evolution of a tidal landscape, J Geophys Res Earth Surf (2011)
[84] Watt, C.; Garbary, Dj; Longtin, C., Population structure of the ribbed mussel Geukensia demissa in salt marshes in the southern gulf of St. Lawrence, Canada, Helgol Mar Res, 65, 3, 275 (2010)
[85] White, K., Spatial heterogeneity in three species, plant-parasite-hyperparasite, systems, Philos Trans R Soc B Biol Sci, 353, 1368, 543 (1998)
[86] Yang, W.; Wang, Q.; Pan, X.; Li, B., Estimation of the probability of long-distance dispersal: stratified diffusion of Spartina alterniflora in the Yangtze river estuary, Am J Plant Sci, 5, 24, 3642 (2014)
[87] Ysebaert, T.; Yang, S-L; Zhang, L.; He, Q.; Bouma, Tj; Herman, Pmj, Wave attenuation by two contrasting ecosystem engineering salt marsh macrophytes in the intertidal pioneer zone, Wetlands, 31, 6, 1043-1054 (2011)
[88] Zaytseva S, Shaw LB, Lipcius RN, Shi J, Kirwan ML (2018) Pattern formation in marsh ecosystems modeled through the interaction of marsh vegetation, mussels and sediment. Manuscript in preparation
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.