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doi: 10.3934/dcdsb.2021189
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Stability and dynamic transition of vegetation model for flat arid terrains

Lan Jia and  Liang Li ,

College of Mathematics, Sichuan University, Chengdu, Sichuan 610065, China

* Corresponding author: Liang Li

Received  October 2020 Revised  July 2021 Early access July 2021

Fund Project: This research was supported by the NSFC, Grant No.11901408 and 11771306

In this paper, we aim to investigate the dynamic transition of the Klausmeier-Gray-Scott (KGS) model in a rectangular domain or a square domain. Our research tool is the dynamic transition theory for the dissipative system. Firstly, we verify the principle of exchange of stability (PES) by analyzing the spectrum of the linear part of the model. Secondly, by utilizing the method of center manifold reduction, we show that the model undergoes a continuous transition or a jump transition. For the model in a rectangular domain, we discuss the transitions of the model from a real simple eigenvalue and a pair of simple complex eigenvalues. our results imply that the model bifurcates to exactly two new steady state solutions or a periodic solution, whose stability is determined by a non-dimensional coefficient. For the model in a square domain, we only focus on the transition from a real eigenvalue with algebraic multiplicity 2. The result shows that the model may bifurcate to an $ S^{1} $ attractor with 8 non-degenerate singular points. In addition, a saddle-node bifurcation is also possible. At the end of the article, some numerical results are performed to illustrate our conclusions.

Keywords: Klausmeier-Gray-Scott model, dynamic transition, Hopf bifurcation, the center manifold.
Mathematics Subject Classification: Primary: 35B32, 35B35; Secondary: 35B09, 35K40.
Citation: Lan Jia, Liang Li. Stability and dynamic transition of vegetation model for flat arid terrains. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021189
References:
[1]

J. J. R. Bennett and J. A. Sherratt, Long-distance seed dispersal affects the resilience of banded vegetation patterns in semi-deserts, Journal of Theoretical Biology, 481 (2019), 151-161.  doi: 10.1016/j.jtbi.2018.10.002.  Google Scholar

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[19]

Y. Mao, Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain, Discrete and Continuous Dynamical Systems-Series B, 23 (2018), 3935-3947.  doi: 10.3934/dcdsb.2018118.  Google Scholar

[20]

Y. Mao, D. Yan and C. Lu, Dynamic transitions and stability for the acetabularia whorl formation, Discrete and Continuous Dynamical Systems-Series B, 24 (2019), 5989-6004.  doi: 10.3934/dcdsb.2019117.  Google Scholar

[21]

Z. Pan, T. Sengul and Q. Wang, On the viscous instabilities and transitions of two-layer model with a layered topography, Communications in Nonlinear Science and Numerical Simulation, 80 (2020), 104978. doi: 10.1016/j.cnsns.2019.104978.  Google Scholar

[22]

J. A. Sherratt, Pattern solutions of the klausmeier model for banded vegetation in semi-arid environments Ⅰ, Nonlinearity, 23 (2010), 2657-2675.  doi: 10.1088/0951-7715/23/10/016.  Google Scholar

[23]

J. A. Sherratt, Pattern solutions of the klausmeier model for banded vegetation in semi-arid environments Ⅱ: Patterns with the largest possible propagation speeds, Proceedings of the Royal Society A Math. Phys. Eng. Sci., 467 (2011), 3272-3294.  doi: 10.1098/rspa.2011.0194.  Google Scholar

[24]

J. A. Sherratt, Pattern solutions of the klausmeier model for banded vegetation in semi-arid environments Ⅲ: The transition between homoclinic solutions, Physica D: Nonlinear Phenomena, 242 (2013), 30-41.  doi: 10.1016/j.physd.2012.08.014.  Google Scholar

[25]

J. A. Sherratt, Pattern solutions of the klausmeier model for banded vegetation in semiarid environments IV: Slowly moving patterns and their stability, SIAM Journal on Applied Mathematics, 73 (2013), 330-350.  doi: 10.1137/120862648.  Google Scholar

[26]

J. A. Sherratt, Pattern solutions of the klausmeier model for banded vegetation in semi-arid environments Ⅴ: The transition from patterns to desert, SIAM Journal on Applied Mathematics, 73 (2013), 1347-1367.  doi: 10.1137/120899510.  Google Scholar

[27]

J. A. Sherratt, When does colonisation of a semi-arid hillslope generate vegetation patterns?, Journal of Mathematical Biology, 73 (2016), 199-226.  doi: 10.1007/s00285-015-0942-8.  Google Scholar

[28]

J. A. Sherratt and G. J. Lord, Nonlinear dynamics and pattern bifurcations in a model for vegetation stripes in semi-arid environments, Theoretical Population Biology, 71 (2007), 1-11.  doi: 10.1016/j.tpb.2006.07.009.  Google Scholar

[29]

G.-Q. Sun, C.-H. Wang, L.-L. Chang, Y.-P. Wu, L. Li and Z. Jin, Effects of feedback regulation on vegetation patterns in semi-arid environments, Applied Mathematical Modelling, 61 (2018), 200-215.  doi: 10.1016/j.apm.2018.04.010.  Google Scholar

[30]

C. E. Tarnita, J. A. Bonachela, E. Sheffer, J. A. Guyton, T. C. Coverdale, R. A. Long and R. M. Pringle, A theoretical foundation for multi-scale regular vegetation patterns, Nature, 541 (2017), 398-401.  doi: 10.1038/nature20801.  Google Scholar

[31]

C. Valentin, J. M. d'Herbes and J. Poesen, Soil and water components of banded vegetation patterns, Catena, 37 (1999), 1-24.  doi: 10.1016/S0341-8162(99)00053-3.  Google Scholar

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S. van der Stelt, A. Doelman, G. Hek and J. D. M. Rademacher, Rise and fall of periodic patterns for a generalized klausmeier-gray-scott model, Journal of Nonlinear Science, 23 (2013), 39-95.  doi: 10.1007/s00332-012-9139-0.  Google Scholar

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X. Wang, W. Wang and G. Zhang, Vegetation pattern formation of a water-biomass model, Communications in Nonlinear Science and Numerical Simulation, 42 (2017), 571-584.  doi: 10.1016/j.cnsns.2016.06.008.  Google Scholar

[34]

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[35]

D. Zhang and R. Liu, Dynamical transition for S-K-T biological competing model with cross-diffusion, Mathematical Methods in the Applied Sciences, 41 (2018), 4641-4658.  doi: 10.1002/mma.4919.  Google Scholar

show all references

References:
[1]

J. J. R. Bennett and J. A. Sherratt, Long-distance seed dispersal affects the resilience of banded vegetation patterns in semi-deserts, Journal of Theoretical Biology, 481 (2019), 151-161.  doi: 10.1016/j.jtbi.2018.10.002.  Google Scholar

[2]

G. Consolo, C. Currò and G. Valenti, Supercritical and subcritical turing pattern formation in a hyperbolic vegetation model for flat arid environments, Physica D: Nonlinear Phenomena, 398 (2019), 141-163.  doi: 10.1016/j.physd.2019.03.006.  Google Scholar

[3]

V. Deblauwe, N. Barbier, P. Couteron, O. Lejeune and J. Bogaert, The global biogeography of semi-arid periodic vegetation patterns, Global Ecology and Biogeography, 17 (2008), 715-723.  doi: 10.1111/j.1466-8238.2008.00413.x.  Google Scholar

[4]

S. Getzin, H. Yizhaq and B. Bell, Discovery of fairy circles in australia supports self-organization theory, Proceedings of the National Academy of Sciences of the United States of America, 113 (2016), 3551-3556.  doi: 10.1073/pnas.1522130113.  Google Scholar

[5]

K. Gowda, H. Riecke and M. Silber, Transitions between patterned states in vegetation models for semiarid ecosystems, Physical Review E, 89 (2014), 022701. doi: 10.1103/PhysRevE.89.022701.  Google Scholar

[6]

D. Han, M. Hernandez and Q. Wang, Dynamical transitions of a low-dimensional model for rayleigh-benard convection under a vertical magnetic field, Chaos, Solitons and Fractals, 114 (2018), 370-380.  doi: 10.1016/j.chaos.2018.06.027.  Google Scholar

[7]

D. Han, M. Salvalaglio and Q. Wang, On the instabilities and transitions of the Western boundary current, Communications in Computational Physics, 26 (2019), 35-56.  doi: 10.4208/cicp.OA-2018-0066.  Google Scholar

[8]

D. Han, M. Hernandez and Q. Wang, Dynamic transitions and bifurcations for a class of axisymmetric geophysical fluid flow, SIAM Journal on Applied Dynamical Systems, 20 (2021), 38-64.  doi: 10.1137/20M1321139.  Google Scholar

[9]

D. Han, Q. Wang and X. Wang, Dynamic transitions and bifurcations for thermal convection in the superposed free flow and porous media, Physica D: Nonlinear Phenomena, 414 (2020), 132687. doi: 10.1016/j.physd.2020.132687.  Google Scholar

[10]

R. HilleRisLambers, M. Rietkerk, F. van den Bosch, H. H. T. Prins and H. de Kroon, Vegetation pattern formation in semi-arid grazing systems, Ecology, 82 (2001), 50-61.   Google Scholar

[11]

C.-H. Hsia, C.-S. Lin, T. Ma and S. Wang, Tropical atmospheric circulations with humidity effects, Proceedings of the Royal Society A, 471 (2015), 20140353. doi: 10.1098/rspa.2014.0353.  Google Scholar

[12]

C.-H. Hsia, T. Ma and S. Wang, Rotating boussinesq equations: Dynamic stability and transition, Discrete and Continuous Dynamical Systems, 28 (2010), 99-130.  doi: 10.3934/dcds.2010.28.99.  Google Scholar

[13]

B. J. Kealy and D. J. Wollkind, A nonlinear stability analysis of vegetative turing pattern formation for an interaction-diffusion plant-surface water model system in an arid flat environment, Journal of Theoretical Biology, 74 (2012), 803-833.  doi: 10.1007/s11538-011-9688-7.  Google Scholar

[14]

C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828.  doi: 10.1126/science.284.5421.1826.  Google Scholar

[15]

C. Lu, Y. Mao, T. Sengul and Q. Wang, On the spectral instability and bifurcation of the 2d-quasi-geostrophic potential vorticity equation with a generalized kolmogorov forcing, Physica D: Nonlinear Phenomena, 403 (2020), 132296. doi: 10.1016/j.physd.2019.132296.  Google Scholar

[16]

T. Ma and S. Wang, Bifurcation Theory and Application, Series A: Monographs and Treatises, 53. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. doi: 10.1142/5798.  Google Scholar

[17]

T. Ma and S. Wang, Dynamic transition and pattern formation for chemotaction system, Discrete and Continuous Dynamical Systems-Series B, 19 (2014), 2809-2835.  doi: 10.3934/dcdsb.2014.19.2809.  Google Scholar

[18]

T. Ma and S. Wang, Phase Transition Dynamics, 2$^nd$ edition, Springer, New York, 2019. doi: 10.1007/978-3-030-29260-7.  Google Scholar

[19]

Y. Mao, Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain, Discrete and Continuous Dynamical Systems-Series B, 23 (2018), 3935-3947.  doi: 10.3934/dcdsb.2018118.  Google Scholar

[20]

Y. Mao, D. Yan and C. Lu, Dynamic transitions and stability for the acetabularia whorl formation, Discrete and Continuous Dynamical Systems-Series B, 24 (2019), 5989-6004.  doi: 10.3934/dcdsb.2019117.  Google Scholar

[21]

Z. Pan, T. Sengul and Q. Wang, On the viscous instabilities and transitions of two-layer model with a layered topography, Communications in Nonlinear Science and Numerical Simulation, 80 (2020), 104978. doi: 10.1016/j.cnsns.2019.104978.  Google Scholar

[22]

J. A. Sherratt, Pattern solutions of the klausmeier model for banded vegetation in semi-arid environments Ⅰ, Nonlinearity, 23 (2010), 2657-2675.  doi: 10.1088/0951-7715/23/10/016.  Google Scholar

[23]

J. A. Sherratt, Pattern solutions of the klausmeier model for banded vegetation in semi-arid environments Ⅱ: Patterns with the largest possible propagation speeds, Proceedings of the Royal Society A Math. Phys. Eng. Sci., 467 (2011), 3272-3294.  doi: 10.1098/rspa.2011.0194.  Google Scholar

[24]

J. A. Sherratt, Pattern solutions of the klausmeier model for banded vegetation in semi-arid environments Ⅲ: The transition between homoclinic solutions, Physica D: Nonlinear Phenomena, 242 (2013), 30-41.  doi: 10.1016/j.physd.2012.08.014.  Google Scholar

[25]

J. A. Sherratt, Pattern solutions of the klausmeier model for banded vegetation in semiarid environments IV: Slowly moving patterns and their stability, SIAM Journal on Applied Mathematics, 73 (2013), 330-350.  doi: 10.1137/120862648.  Google Scholar

[26]

J. A. Sherratt, Pattern solutions of the klausmeier model for banded vegetation in semi-arid environments Ⅴ: The transition from patterns to desert, SIAM Journal on Applied Mathematics, 73 (2013), 1347-1367.  doi: 10.1137/120899510.  Google Scholar

[27]

J. A. Sherratt, When does colonisation of a semi-arid hillslope generate vegetation patterns?, Journal of Mathematical Biology, 73 (2016), 199-226.  doi: 10.1007/s00285-015-0942-8.  Google Scholar

[28]

J. A. Sherratt and G. J. Lord, Nonlinear dynamics and pattern bifurcations in a model for vegetation stripes in semi-arid environments, Theoretical Population Biology, 71 (2007), 1-11.  doi: 10.1016/j.tpb.2006.07.009.  Google Scholar

[29]

G.-Q. Sun, C.-H. Wang, L.-L. Chang, Y.-P. Wu, L. Li and Z. Jin, Effects of feedback regulation on vegetation patterns in semi-arid environments, Applied Mathematical Modelling, 61 (2018), 200-215.  doi: 10.1016/j.apm.2018.04.010.  Google Scholar

[30]

C. E. Tarnita, J. A. Bonachela, E. Sheffer, J. A. Guyton, T. C. Coverdale, R. A. Long and R. M. Pringle, A theoretical foundation for multi-scale regular vegetation patterns, Nature, 541 (2017), 398-401.  doi: 10.1038/nature20801.  Google Scholar

[31]

C. Valentin, J. M. d'Herbes and J. Poesen, Soil and water components of banded vegetation patterns, Catena, 37 (1999), 1-24.  doi: 10.1016/S0341-8162(99)00053-3.  Google Scholar

[32]

S. van der Stelt, A. Doelman, G. Hek and J. D. M. Rademacher, Rise and fall of periodic patterns for a generalized klausmeier-gray-scott model, Journal of Nonlinear Science, 23 (2013), 39-95.  doi: 10.1007/s00332-012-9139-0.  Google Scholar

[33]

X. Wang, W. Wang and G. Zhang, Vegetation pattern formation of a water-biomass model, Communications in Nonlinear Science and Numerical Simulation, 42 (2017), 571-584.  doi: 10.1016/j.cnsns.2016.06.008.  Google Scholar

[34]

Y. R. Zelnik, P. Gandhi, E. Knobloch and E. Meron, Implications of tristability in pattern-forming ecosystems, \emphChaos, 28 (2018), 033609. doi: 10.1063/1.5018925.  Google Scholar

[35]

D. Zhang and R. Liu, Dynamical transition for S-K-T biological competing model with cross-diffusion, Mathematical Methods in the Applied Sciences, 41 (2018), 4641-4658.  doi: 10.1002/mma.4919.  Google Scholar

Figure 1.  the topological structure of phase portrait of continuous transition as control parameter $ \lambda >\Lambda_{c} $.
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Figure 2.  the topological structure of phase portrait of jump transition as control parameter $ \lambda<\Lambda_{c}. $
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Figure 3.  the topological structure of phase portrait of jump transition as control parameter λ < Λc
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Figure 4.  the topological structure of phase portrait of jump transition as control parameter λ > Λc
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Figure 5.  8 nondegenerate singular points
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Figure 6.   
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Figure 7.  Plot of the numerical solution $ b $ at $ T = 50 $ (left) and plot of bifurcated solution $ \widetilde{b} $ (right) of the system (6)-(10), where the parameters are $ m = 0.5, d = 6, a = 1.00057. $
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Figure 8.  Plot of the numerical solution $ b $ and $ w $ at $ T = 600 $ of the system (6)-(10), where the parameters are $ m = 0.5, d = 6, a = 1.0004957. $
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Figure 9.  Plot of the numerical solution $ b $(left) at $ T = 600 $ and plot of bifurcated solution $ \widetilde{b} $ (right) of the system (6)-(10), where the parameters are $ m = 0.5, d = 6, a = 16.704876. $
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