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Spatiotemporal dynamics of a vegetation model with nonlocal delay in semi-arid environment

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Abstract

Vegetation patterns can reflect the spatial distribution of vegetation in both space and time. In semi-arid regions, the absorption of water by vegetation is a nonlocal process meaning that its roots can absorb water from themselves throughout the region. However, the effects of the nonlocal interaction on the distribution of vegetation pattern are not clear. In this paper, a dynamical model of vegetation pattern with nonlocal delay is investigated. Through the analysis of Turing instability, we obtain the conditions for the generation of stationary patterns. By numerical simulations, various spatial distribution of vegetation in semi-arid areas are qualitatively depicted. It is found that the stripe intervals in pattern decrease with the increase in the intensity of nonlocal delay effect, and then a dot-line mixed pattern appears, which eventually evolves into a high-density dot pattern. This indicates that vegetation has evolved from low-density stripe distribution to high-density point distribution. The results show that the nonlocal delay effect enhances vegetation biomass. Therefore, we can take measures to increase the intensity of nonlocal delay effect to increase vegetation density, which theoretically provides new guidance for vegetation protection and desertification control.

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References

  1. Jones, C.G., Lawton, J.H., Shachak, M.: Organisms as ecosystem engineers. Oikos 69, 373–386 (1994)

    Google Scholar 

  2. Lemordant, L., Gentine, P., et al.: Critical impact of vegetation physiology on the continental hydrologic cycle in response to increasing CO\(_{2}\). Proc. Natl. Acad. Sci. 115, 4093–4098 (2018)

    Google Scholar 

  3. Gallagher, R.V., Allen, S., Wright, I.J.: Safety margins and adaptive capacity of vegetation to climate change. Sci. Rep. 9, 8241 (2019)

    Google Scholar 

  4. Danielsen, F.: The Asian Tsunami: a protective role for coastal vegetation. Science 310, 643 (2005)

    Google Scholar 

  5. Hillerislambers, R., Rietkerk, M., Frank, V.D.B., et al.: Vegetation pattern formation in semi-arid grazing systems. Ecology 82, 50–61 (2001)

    Google Scholar 

  6. Couteron, P., Lejeune, O.: Periodic spotted patterns in semi “rid vegetation explained by a propagation” inhibition model. J. Ecol. 89, 616–628 (2001)

    Google Scholar 

  7. Sun, G.-Q.: Mathematical modeling of population dynamics with Allee effect. Nonlinear Dyn. 85, 1–12 (2016)

    MathSciNet  Google Scholar 

  8. Von, J.H., Meron, E., Shachak, M., et al.: Diversity of vegetation patterns and desertification. Phys. Rev. Lett. 87, 198101 (2001)

    Google Scholar 

  9. Shnerb, N.M., Sara, P., Lavee, H., et al.: Reactive glass and vegetation patterns. Phys. Rev. Lett. 90, 038101 (2002)

    Google Scholar 

  10. Barbier, N., Couteron, P., Lejoly, J., et al.: Self-organized vegetation patterning as a fingerprint of climate and human impact on semi-arid ecosystems. J. Ecol. 94, 537–547 (2006)

    Google Scholar 

  11. Ursino, N., Rulli, M.C.: Combined effect of fire and water scarcity on vegetation patterns in arid lands. Ecol. Model. 221, 2353–2362 (2010)

    Google Scholar 

  12. Rietkerk, M., Boerlijst, M.C., et al.: Self-organization of vegetation in arid ecosystems. Am. Nat. 160, 524–530 (2002)

    Google Scholar 

  13. Rietkerk, M.: Self-organized patchiness and catastrophic shifts in ecosystems. Science 305, 1926–1929 (2004)

    Google Scholar 

  14. Herrmann, H.-J.: Pattern formation of dunes. Nonlinear Dyn. 44, 315–317 (2006)

    MATH  Google Scholar 

  15. Marinov, K., Wang, T., Yang, Y.: On a vegetation pattern formation model governed by a nonlinear parabolic system. Nonlinear Anal. Real World Appl. 14, 507–525 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Evaristo, J., Jasechko, S., Mcdonnell, J.J.: Global separation of plant transpiration from groundwater and streamflow. Nature 525, 91–94 (2015)

    Google Scholar 

  17. Zemp, D.C., Schleussner, C.F., et al.: Self-amplified Amazon forest loss due to vegetation-atmosphere feedbacks. Nat. Commun. 8, 14681 (2017)

    Google Scholar 

  18. Brandt, M., Hiernaux, P., Rasmussen, K., et al.: Changes in rainfall distribution promote woody foliage production in the Sahel. Commu. Biol. 2, 133 (2019)

    Google Scholar 

  19. Meron, E., Gilad, E., et al.: Vegetation patterns along a rainfall gradient. Chaos Soliton. Fract. 19, 367–376 (2004)

    MATH  Google Scholar 

  20. Maestre, F.T., et al.: Plant species richness and ecosystem multifunctionality in global drylands. Science 335, 214–218 (2012)

    Google Scholar 

  21. Liu, L., Zhang, Y., Wu, S., et al.: Water memory effects and their impacts on global vegetation productivity and resilience. Sci. Rep. 8, 2962 (2018)

    Google Scholar 

  22. Rees, M., et al.: Long-term studies of vegetation dynamics. Science 293, 650–655 (2001)

    Google Scholar 

  23. Klausmeier, C.A.: Regular and irregular patterns in semiarid vegetation. Science 284, 1826–1828 (1999)

    Google Scholar 

  24. Borgogno, F., D’Odorico, P., Laio, F., Ridolfi, L.: Mathematical models of vegetation pattern formation in ecohydrology. Rev. Geophys. 47, RG1005 (2009)

    Google Scholar 

  25. Sun, G.-Q., et al.: Effects of feedback regulation on vegetation patterns in semi-arid environments. Appl. Math. Model. 61, 200–215 (2018)

    MathSciNet  Google Scholar 

  26. Sherratt, J.A., Synodinos, A.D.: Vegetation patterns and desertification waves in semi-arid environments: mathematical models based on local facilitation in plants. Discrete Cont. Dyn. B 17, 2815–2827 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Hillerislambers, R., Rietkerk, M., et al.: Vegetation pattern formation in semi-arid grazing systems. Ecology 82, 50–61 (2001)

    Google Scholar 

  28. Handa, I., Harmsen, R., Jefferies, R.: Patterns of vegetation change and the recovery potential of degraded areas in a coastal marsh system of the Hudson Bay lowlands. J. Ecol. 90, 86–99 (2002)

    Google Scholar 

  29. Rietkerk, M., Bosch, Fvd, Koppel, Jvd: Site-specific properties and irreversible vegetation changes in semi-arid grazing systems. Oikos 80, 241–252 (1997)

    Google Scholar 

  30. Rietkerk, M., Ketner, P., Burger, J., Hoorens, B., Olff, H.: Multiscale soil and vegetation patchiness along a gradient of herbivore impact in a semi-arid grazing system in West Africa. Plant Ecol. 148, 207–224 (2000)

    Google Scholar 

  31. Van de Koppel, J., Rietkerk, M., van Langevelde, F., et al.: Spatial heterogeneity and irreversible vegetation change in semiarid grazing systems. Am. Nat. 159, 209–218 (2002)

    Google Scholar 

  32. Wilson, A., et al.: Positive-feedback switches in plant communities. Adv. Ecol. Res. 23, 263–336 (1992)

    Google Scholar 

  33. Deblauwe, V., Couteron, P., Bogaert, J., et al.: Determinants and dynamics of banded vegetation pattern migration in arid climates. Ecol. Monogr. 82, 3–21 (2012)

    Google Scholar 

  34. Van de Koppel, J., Rietkerk, M.: Spatial interactions and resilience in arid ecosystems. Am. Nat. 163, 113–121 (2004)

    Google Scholar 

  35. Kefi, S., Rietkerk, M., Alados, C.L., Pueyo, Y., Papanastasis, V.P., ElAich, A., De Ruiter, P.C.: Spatial vegetation patterns and imminent desertification in Mediterranean arid ecosystems. Nature 449, 213–217 (2007)

    Google Scholar 

  36. Bastiaansen, R., Jaibi, O., Deblauwe, V., et al.: Multistability of model and real dryland ecosystems through spatial self-organization. Proc. Natl. Acad. Sci. 115, 11256–11261 (2018)

    Google Scholar 

  37. Seddon, A.W.R., Macias-Fauria, M., et al.: Sensitivity of global terrestrial ecosystems to climate variability. Nature 531, 229–232 (2016)

    Google Scholar 

  38. Braswell, B.H., et al.: The response of global terrestrial ecosystems to interannual temperature variability. Science 278, 870–873 (1997)

    Google Scholar 

  39. Forzieri, G., Alkama, R., et al.: Satellites reveal contrasting responses of regional climate to the widespread greening of Earth. Science 356, 1180–1184 (2017)

    Google Scholar 

  40. Kutzbach, J.E., Bonan, G.B., Foley, J.A., et al.: Vegetation and soils feedbacks on the response of the African monsoon response to orbital forcing in early to middle holocene. Nature 384, 19–26 (1996)

    Google Scholar 

  41. Forzieri, G., Alkama, R., et al.: Satellites reveal contrasting responses of regional climate to the widespread greening of Earth. Science 360, 022701 (2018)

    Google Scholar 

  42. Gowda, K., Riecke, H., Silber, M.: Transitions between patterned states in vegetation models for semiarid ecosystems. Phys. Rev. E 89, 022701 (2014)

    Google Scholar 

  43. Gowda, K., et al.: Assessing the robustness of spatial pattern sequences in a dryland vegetation model. Proc. R. Soc. A 472, 20150893 (2016)

    MathSciNet  MATH  Google Scholar 

  44. Lejeune, O., Tlidi, M., et al.: Vegetation spots and stripes: Dissipative structures in arid landscapes. Int. J. Quantum. Chem. 98, 261–271 (2004)

    Google Scholar 

  45. Zelnik, Y.R., Kinast, S., et al.: Regime shifts in models of dryland vegetation. Phil. Trans. R. Soc. A 371, 20120358 (2013)

    MathSciNet  MATH  Google Scholar 

  46. Bel, G., Hagberg, A., Meron, E.: Gradual regime shifts in spatially extended ecosystems. Theor. Ecol. 5, 591–604 (2012)

    Google Scholar 

  47. Martone, M., et al.: High-resolution forest mapping from Tandem-X interferometric data exploiting nonlocal filtering. Remote Sens. 10, 1477 (2018)

    Google Scholar 

  48. Biswas, A., et al.: Identifying effects of local and nonlocal factors of soil water storage using cyclical correlation analysis. Hydrol. Process 26, 3669–3677 (2012)

    Google Scholar 

  49. Thompson, S., Katul, G., Terborgh, J., et al.: Spatial organization of vegetation arising from non-local excitation with local inhibition in tropical rainforests. Physica D 238, 1061–1067 (2009)

    MATH  Google Scholar 

  50. Chen, S.-S., et al.: Threshold dynamics of a diffusive nonlocal phytoplankton model with age structure. Nonlinear Anal. Real World Appl. 50, 55–56 (2019)

    MathSciNet  MATH  Google Scholar 

  51. Guo, S.-J.: Stability and bifurcation in a reaction–diffusion model with nonlocal delay effect. J. Differ. Equ. 259, 1409–1448 (2015)

    MathSciNet  MATH  Google Scholar 

  52. Winckler, J., Lejeune, Q., et al.: Nonlocal effects dominate the global mean surface temperature response to the biogeophysical effects of deforestation. Geophys. Res. Lett. 46, 745–755 (2019)

    Google Scholar 

  53. Boushaba, K., Ruan, S.-G.: Instability in diffusive ecological models with nonlocal delay effects. J. Math. Anal. Appl. 258, 269–286 (2001)

    MathSciNet  MATH  Google Scholar 

  54. Sun, G.-Q., Wang, S.-L., Ren, Q., Jin, Z., Wu, Y.-P.: Effects of time delay and space on herbivore dynamics: linking inducible defenses of plants to herbivore outbreak. Sci. Rep. 5, 11246 (2015)

    Google Scholar 

  55. Bao, X.-M., Tian, C.: Delay driven vegetation patterns of a plankton system on a network. Physica A 521, 74–88 (2019)

    MathSciNet  Google Scholar 

  56. Wang, K.-K., Ye, H., Wang, Y.-J., et al.: Time-delay-induced dynamical behaviors for an ecological vegetation growth system driven by cross-correlated multiplicative and additive noises. Eur. Phys. J. E 41, 60 (2018)

    Google Scholar 

  57. Zeng, C., Han, Q., Yang, T., et al.: Noise- and delay-induced regime shifts in an ecological system of vegetation. J. Stat. Mech. Theory E 10, P10017 (2013)

    MathSciNet  Google Scholar 

  58. Li, L., Jin, Z., Li, J.: Periodic solutions in a herbivore-plant system with time delay and spatial diffusion. Appl. Math. Model. 40, 4765–4777 (2016)

    MathSciNet  MATH  Google Scholar 

  59. Wu, T., Fu, H., Feng, F., et al.: A new approach to predict normalized difference vegetation index using time-delay neural network in the arid and semi-arid grassland. Int. J. Remote Sens. 40, 1–14 (2019)

    Google Scholar 

  60. Tian, C.-R., Ling, Z., Zhang, L.: Delay-driven spatial patterns in a network-organized semiarid vegetation model. Appl. Math. Comput. (in Press) (2020)

  61. Han, Q., Yang, T., Zeng, C., et al.: Impact of time delays on stochastic resonance in an ecological system describing vegetation. Physica A 408, 96–105 (2014)

    MathSciNet  MATH  Google Scholar 

  62. Britton, N.F.: Aggregation and the competitive exclusion principle. J. Theor. Biol. 136, 57–66 (1989)

    MathSciNet  Google Scholar 

  63. Guo, Z.-G., Song, L.-P., Sun, G.-Q., Li, C., Jin, Z.: Pattern dynamics of an SIS epidemic model with nonlocal delay. Int. J. Bifurc. Chaos 29, 1950027 (2019)

    MathSciNet  MATH  Google Scholar 

  64. Sun, G.-Q., Wang, C.-H., Wu, Z.Y.: Pattern dynamics of a Gierer–Meinhardt model with spatial effects. Nonlinear Dyn. 88, 1385–1396 (2017)

    Google Scholar 

  65. Ma, J., Tang, J.: A review for dynamics in neuron and neuronal network. Nonlinear Dyn. 89, 1569–1578 (2017)

    MathSciNet  Google Scholar 

  66. Wang, C., Ma, J.: A review and guidance for pattern selection in spatiotemporal system. Int. J. Mod. Phys. B 32, 1830003 (2018)

    MathSciNet  MATH  Google Scholar 

  67. Ma, J., Qin, H., Song, X., et al.: Pattern selection in neuronal network driven by electric autapses with diversity in time delays. Int. J. Mod. Phys. B 29, 1450239 (2015)

    Google Scholar 

  68. Gourley, S.A., So, W.H.: Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain. J. Math. Biol. 44, 49–78 (2002)

    MathSciNet  MATH  Google Scholar 

  69. Koster, R.D., et al.: Regions of strong coupling between soil moisture and precipitation. Science 305, 1138–1140 (2004)

    Google Scholar 

  70. Sun, G.-Q., Jusup, M., Jin, Z., Wang, Y., Wang, Z.: Pattern transitions in spatial epidemics: mechanisms and emergent properties. Phys. Life Rev. 19, 43–73 (2016)

    Google Scholar 

  71. Wang, T.: Pattern dynamics of an epidemic model with nonlinear incidence rate. Nonlinear Dyn. 77, 31–40 (2014)

    MathSciNet  MATH  Google Scholar 

  72. Wang, Y., Cao, J., Li, X., Alsaedi, A.: Edge-based epidemic dynamics with multiple routes of transmission on random networks. Nonlinear Dyn. 91, 403–420 (2018)

    MathSciNet  Google Scholar 

  73. Wang, W., Liu, S., Liu, Z.: Spatiotemporal dynamics near the Turing–Hopf bifurcation in a toxic-phytoplankton–zooplankton model with cross-diffusion. Nonlinear Dyn. 98, 27–37 (2019)

    MATH  Google Scholar 

  74. Sun, G.-Q., Zhang, J., Song, L.-P., Jin, Z., Li, B.-L.: Pattern formation of a spatial predator–prey system. Appl. Math. Comput. 218, 11151–11162 (2012)

    MathSciNet  MATH  Google Scholar 

  75. Ghorai, S., Poria, S.: Pattern formation in a system involving prey–predation, competition and commensalism. Nonlinear Dyn. 89, 1309–1326 (2017)

    Google Scholar 

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Acknowledgements

The project is funded by the National Key Research and Development Program of China (Grant No. 2018YFE0109600), National Natural Science Foundation of China under Grant Nos. 11671241 and 41875097, Outstanding Young Talents Support Plan of Shanxi province, Selective Support for Scientific and Technological Activities of Overseas Scholars of Shanxi province, and High-Level Talent Project of Jiangsu Province (Six Talent Peaks and Grant No. JNHB-071).

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Authors and Affiliations

  1. Department of Computer Science and Technology, North University of China, Taiyuan, 030051, Shanxi, China

    Qiang Xue & Zun-Guang Guo

  2. Data Science and Technology, North University of China, Taiyuan, 030051, Shanxi, China

    Qiang Xue & Zun-Guang Guo

  3. Department of Mathematics, North University of China, Taiyuan, 030051, Shanxi, China

    Qiang Xue, Gui-Quan Sun & Zhen Jin

  4. Complex Systems Research Center, Shanxi University, Taiyuan, 030006, Shanxi, China

    Qiang Xue & Gui-Quan Sun

  5. Center for Ecology and Environmental Sciences, Northwestern Polytechnical University, Xi’an, 710072, China

    Chen Liu

  6. College of Physics Science and Technology, Yangzhou University, Yangzhou, 225002, Jiangsu Province, China

    Yong-Ping Wu & Guo-Lin Feng

  7. National Climate Center, China Meteorological Administration, Beijing, 100081, China

    Guo-Lin Feng

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  1. Qiang Xue
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Xue, Q., Sun, GQ., Liu, C. et al. Spatiotemporal dynamics of a vegetation model with nonlocal delay in semi-arid environment. Nonlinear Dyn 99, 3407–3420 (2020). https://doi.org/10.1007/s11071-020-05486-w

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