The generation mechanism of Turing-pattern in a Tree-grass competition model with cross diffusion and time delay

Rina Su; Chunrui Zhang; Rina Su; Chunrui Zhang

doi:10.3934/mbe.2022562

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 12: 12073-12103. doi: 10.3934/mbe.2022562

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The generation mechanism of Turing-pattern in a Tree-grass competition model with cross diffusion and time delay

Rina Su ^1,2,
Chunrui Zhang ^{3
,
,}

1.
College of Mechanical and Electrical Engineering, Northeast Forestry University, Harbin, 150040, China
2.
College of Mathematics and Physics, Inner Mongolia Minzu University, Tongliao, 028043, China
3.
Department of Mathematics, Northeast Forestry University, Harbin, 150040, China

Received: 10 November 2021 Revised: 27 June 2022 Accepted: 13 July 2022 Published: 18 August 2022

In this paper, we study the general mechanism of Turing-pattern in a tree-grass competition model with cross diffusion and time delay. The properties of four equilibrium points, the existence of Hopf bifurcation and the sufficient conditions for Turing instability caused by cross-diffusion are analyzed, respectively. The amplitude equation of tree-grass competition model is derived by using multi-scale analysis method, and its nonlinear stability is studied. The sensitivity analysis also verified that fire frequency plays a key role in tree-grass coexistence equilibrium. Finally, the Turing pattern of tree-grass model obtained by numerical simulation is consistent with the spatial structure of tree-grass density distribution observed in Hulunbuir grassland, China.
- tree-grass model,
- Turing instability,
- cross-diffusion,
- amplitude equation,
- pattern
Citation: Rina Su, Chunrui Zhang. The generation mechanism of Turing-pattern in a Tree-grass competition model with cross diffusion and time delay[J]. Mathematical Biosciences and Engineering, 2022, 19(12): 12073-12103. doi: 10.3934/mbe.2022562

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Abstract

In this paper, we study the general mechanism of Turing-pattern in a tree-grass competition model with cross diffusion and time delay. The properties of four equilibrium points, the existence of Hopf bifurcation and the sufficient conditions for Turing instability caused by cross-diffusion are analyzed, respectively. The amplitude equation of tree-grass competition model is derived by using multi-scale analysis method, and its nonlinear stability is studied. The sensitivity analysis also verified that fire frequency plays a key role in tree-grass coexistence equilibrium. Finally, the Turing pattern of tree-grass model obtained by numerical simulation is consistent with the spatial structure of tree-grass density distribution observed in Hulunbuir grassland, China.

References

[1]	J. I. House, S. Archer, D. D. Breshears, R. J. Scholes, Conundrums in mixed woody-herbaceous plant systems, J. Biogeogr., 30 (2003), 1763–1777. https://doi.org/10.1046/j.1365-2699.2003.00873.x doi: 10.1046/j.1365-2699.2003.00873.x
[2]	M. Sankaran, J. Ratnam, N. P. Hanan, Tree-grass co-existence in savannas revisited-insights from an examination of assumptions and mechanisms invoked in existing models, Ecol. Lett., 7 (2004), 480–490. https://doi.org/10.1111/j.1461-0248.2004.00596.x doi: 10.1111/j.1461-0248.2004.00596.x
[3]	F. S. Gilliam, W. J. Platt, R. K. Peet, Natural disturbances and the physiognomy of pine savannas: a phenomenological model, Appl. Veg. Sci., 9 (2006), 83–96. https://doi.org/10.1111/j.1654-109X.2006.tb00658.x doi: 10.1111/j.1654-109X.2006.tb00658.x
[4]	P. D'Odorico, L. Francesco, L. Ridolfi, A probabilistic analysis of fire-induced tree-grass coexistence in savannas, Am. Nat., 167 (2006), E79–E87. https://doi.org/10.1086/500617 doi: 10.1086/500617
[5]	W. J. Bond, F. Woodward, G. Midgley, The global distribution of ecosystems in a world without fire, New Phytol., 165 (2005), 525–538. https://doi.org/10.1111/j.1469-8137.2004.01252.x doi: 10.1111/j.1469-8137.2004.01252.x
[6]	B. Beckage, L. J. Gross, Grass feedbacks on fire stabilize Savannas, Ecol. Modell., 222 (2011), 2227–2233. https://doi.org/10.1016/j.ecolmodel.2011.01.015 doi: 10.1016/j.ecolmodel.2011.01.015
[7]	A. M. Gill, Adaptive responses of Australian vascular plant species to fires, Aust. Acad. Sci., (1981), 243–272.
[8]	R. E. Keane, K. C. Ryan, T. T. Veblen, Cascading effects of fire exclusion in rocky mountain ecosystems: a literature review, USDA Forest Serv. General Tech. Rep., 91 (2002), 1–31. https://doi.org/10.2737/RMRS-GTR-91 doi: 10.2737/RMRS-GTR-91
[9]	R. A. Bradstock, M. A. Gill, R. J. Williams, Flammable Australia: their roles in understanding and predicting biotic responses to fire regimes from individuals to landscapes, CSIRO Publishing, Colingwood Australia, 2012. https://doi.org/10.1007/978-1-4612-0873-0
[10]	W. A. Hoffmann, Fire and population dynamics of woody plants in a neotropical savanna: matrix model projections, Ecology, 80 (1999), 1354–1369. https://doi.org/10.2307/177080 doi: 10.2307/177080
[11]	B. Beckage, L. J. Gross, W. J. Platt, Modelling responses of pine Savannas to climate change and large-scale disturbance, Appl. Veg. Sci., 9 (2006), 75–82. https://doi.org/10.1111/j.1654-109X.2006.tb00657.x doi: 10.1111/j.1654-109X.2006.tb00657.x
[12]	W. Beckage, W. Platt, L. J. Gross, Vegetation, fire, and feedbacks: a disturbance-mediated model of Savannas, Am. Nat., 174 (2009), 805–818. https://doi.org/10.2307/27735896 doi: 10.2307/27735896
[13]	T. D. Johan, Toit, R. Kevin, C. B. Harry, The kruger experience: ecology and management of savanna heterogeneity, Island Press, Washington, 2003.
[14]	M. Mermoz, T. Kitzberger, T. T. Veblen, Landscape influences on occurrence and spread of wildfires in patagonian forests and shrublands, Ecology, 86 (2005), 2705–2715.
[15]	J. B. William, What limits trees in $C_4$ grasslands and Savannas, Annu. Rev. Ecol. Evol. Syst., 39 (2008), 641–659. https://doi.org/10.1146/annurev.ecolsys.39.110707.173411 doi: 10.1146/annurev.ecolsys.39.110707.173411
[16]	E. R. Caroline, A. T. Michael, M. Sankaran, S. I. Higgins, Savanna vegetation-fire-climate relationships differ among continents, Science, 343 (2014), 548–552. https://doi.org/10.1126/science.1247355 doi: 10.1126/science.1247355
[17]	Q. Ouyang, Nonlinear Science and the Pattern Dynamics Introduction, Peking University Press, Beijing, 2010.
[18]	C. Zhang, B. Zheng, R. Su, Realizability of the normal forms for the non-semisimple $1:1$ resonant Hopf bifurcation in a vector field, Commun. Nonlinear Sci. Numer. Simul., 91 (2020), 105407. https://doi.org/10.1016/j.cnsns.2020.105407 doi: 10.1016/j.cnsns.2020.105407
[19]	C. Zhang, B. Zheng, Steady state bifurcation and patterns of reaction-diffusion equations, Int. J. Bifurcat. Chaos, 30 (2020), 2050215. https://doi.org/10.1142/S0218127420502156 doi: 10.1142/S0218127420502156
[20]	W. Jiang, Q. An, J. Shi, Formulation of the normal form of Turing-Hopf bifurcation in partial functional differential equations, J. Differ. Equ., 268 (2020), 6067–6102. https://doi.org/10.1016/j.jde.2019.11.039 doi: 10.1016/j.jde.2019.11.039
[21]	K. Yun, K. Sourav, Sasmal, M. Komi, A two-patch prey-predator model with predator dispersal driven by the predation strength, Math. Biosci. Eng., 14 (2017), 843–880. https://doi.org/10.3934/mbe.2017046 doi: 10.3934/mbe.2017046
[22]	K. Kim, W. Choi, Local dynamics and coexistence of predator-prey model with directional dispersal of predator, Math. Biosci. Eng., 17 (2020), 6737–6755. https://doi.org/10.3934/mbe.2020351 doi: 10.3934/mbe.2020351
[23]	Q. Din, M. S. Shabbir, M. A. Khan, K. Ahmad, Bifurcation analysis and chaos control for a plant-herbivore model with weak predator functional response, J. Biol. Dyn., 13 (2019), 481–501. https://doi.org/10.1080/17513758.2019.1638976 doi: 10.1080/17513758.2019.1638976
[24]	A. M. Mahmoud, I. A. Ismail, A. A. Farah, H. Mohd, Codimension one and two bifurcations of a discrete-time fractional-order SEIR measles epidemic model with constant vaccination, Chaos Soliton. Fract., 140 (2020), 110104. https://doi.org/10.1016/j.chaos.2020.110104 doi: 10.1016/j.chaos.2020.110104
[25]	H. Huo, P. Yang, H. Xiang, Dynamics for an SIRS epidemic model with infection age and relapse on a scale-free network, J. Franklin Inst., 356 (2019), 7411–7443. https://doi.org/10.1016/j.jfranklin.2019.03.034 doi: 10.1016/j.jfranklin.2019.03.034
[26]	X. Yang, F. Liu, Q. Wang, H. Wang, Dynamics analysis for a discrete dynamic competition model, Adv. Differ. Equ., 2019 (2019), 1–17. https://doi.org/10.1186/s13662-019-2149-6 doi: 10.1186/s13662-019-2149-6
[27]	H. Kato, T. Takada, Stability and bifurcation analysis of a ratio-dependent community dynamics model on Batesian mimicry, J. Math. Biol., 79 (2019), 329–368. https://doi.org/10.1007/s00285-019-01359-y doi: 10.1007/s00285-019-01359-y
[28]	Y. Song, Hopf bifurcation and spatio-temporal patterns indelay-coupled van der Pol oscillators, Nonlinear Dyn., 63 (2011), 223–237. https://doi.org/10.1007/s11071-010-9799-y doi: 10.1007/s11071-010-9799-y
[29]	J. Lin, R. Xu, L. Li, Turing-Hopf bifurcation of reaction-diffusion neural networks with leakage delay, Commun. Nonlinear Sci. Numer. Simul., 85 (2020), 105241. https://doi.org/10.1016/j.cnsns.2020.105241 doi: 10.1016/j.cnsns.2020.105241
[30]	R. J. Scholes, S. R. Archer, Tree-grass interactions in Savannas, Ann. Rev. Ecol. Syst., 28 (1997), 517–544. https://doi.org/10.1146/annurev.ecolsys.28.1.517 doi: 10.1146/annurev.ecolsys.28.1.517
[31]	S. I. Higgins, W. J. Bond, W. Trollope, Fire, resprouting and variability: a recipe for grass-tree coexistence in Savanna, J. Ecol., 88 (2000), 213–229. https://doi.org/10.1046/j.1365-2745.2000.00435.x doi: 10.1046/j.1365-2745.2000.00435.x
[32]	J. Shi, Z. Xie, K. Little, Cross-diffusion induced instability and stability in reaction-diffusion systems, J. Appl. Anal. Comput., 1 (2011), 95–119. https://doi.org/10.2337/diab.31.7.585 doi: 10.2337/diab.31.7.585
[33]	Q. Li, Z. Liu, S. Yuan, Cross-diffusion induced Turing instability for a competition model with saturation effect, J. Math. Comput., 347 (2019), 64–77. https://doi.org/10.1016/j.amc.2018.10.071 doi: 10.1016/j.amc.2018.10.071
[34]	S. Chen, J. Shi, J. Wei, Time delay-induced instabilities and Hopf bifurcations in general reaction–diffusion systems, J. Nonlinear Sci., 23 (2013), 1–38. https://doi.org/10.1007/s00332-012-9138-1 doi: 10.1007/s00332-012-9138-1
[35]	V. Yatat, P. Couteron, J. J. Tewa, An impulsive modelling framework of fire occurrence in a size structured model of tree-grass interactions for savanna ecosystems, J. Math. Biol., 74 (2017), 1425–1482. https://doi.org/10.1007/s00285-016-1060-y doi: 10.1007/s00285-016-1060-y
[36]	Y. Su, J. Wei, J. Shi, Hopf bifurcations in a reaction-diffusion population model with delay effect, J. Differ. Equ., 247 (2009), 1156–1184. https://doi.org/10.1016/j.jde.2009.04.017 doi: 10.1016/j.jde.2009.04.017
[37]	F. Wei, Q. Fu, Hopf bifurcation and stability for predator-prey systems with Beddington-DeAngelis type functional response and stage structure for prey incorporating refuge, Appl. Math. Model., 40 (2016), 126–134. https://doi.org/10.1016/j.apm.2015.04.042 doi: 10.1016/j.apm.2015.04.042
[38]	F. Accatino, C. De Michele, R. Vezzoli, D. Donzelli, R. J. Scholes, Tree-grass co-existence in Savanna: interactions of rain and fire, J. Theor. Biol., 267 (2010), 235–242. https://doi.org/10.1016/j.jtbi.2010.08.012 doi: 10.1016/j.jtbi.2010.08.012
[39]	M. Garvie, Finite difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB, Bull. Math. Biol., 69 (2007), 931–956. https://doi.org/10.1007/s11538-006-9062-3 doi: 10.1007/s11538-006-9062-3

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