# American Institute of Mathematical Sciences

June  2022, 27(6): 3375-3398. doi: 10.3934/dcdsb.2021189

## Stability and dynamic transition of vegetation model for flat arid terrains

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

* Corresponding author: Liang Li

Received  October 2020 Revised  June 2021 Published  June 2022 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.

Citation: Lan Jia, Liang Li. Stability and dynamic transition of vegetation model for flat arid terrains. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3375-3398. 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [14] C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828.  doi: 10.1126/science.284.5421.1826. [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. [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. [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. [18] T. Ma and S. Wang, Phase Transition Dynamics, 2$^nd$ edition, Springer, New York, 2019. doi: 10.1007/978-3-030-29260-7. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [14] C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828.  doi: 10.1126/science.284.5421.1826. [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. [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. [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. [18] T. Ma and S. Wang, Phase Transition Dynamics, 2$^nd$ edition, Springer, New York, 2019. doi: 10.1007/978-3-030-29260-7. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
the topological structure of phase portrait of continuous transition as control parameter $\lambda >\Lambda_{c}$.
the topological structure of phase portrait of jump transition as control parameter $\lambda<\Lambda_{c}.$
the topological structure of phase portrait of jump transition as control parameter λ < Λc
the topological structure of phase portrait of jump transition as control parameter λ > Λc
8 nondegenerate singular points
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 $m = 0.5, d = 6, a = 1.00057.$
Plot of the numerical solution $b$ and $w$ at $T = 600$ of the system (6)-(10), where the parameters are $m = 0.5, d = 6, a = 1.0004957.$
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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