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

Previous Article Next Article

Research article Special Issues

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.

Keywords:

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

Related Papers:

[1]	Yifeng Wang . Journal summary from Editor in Chief. AIMS Environmental Science, 2024, 11(3): 380-380. doi: 10.3934/environsci.2024018
[2]	Yifeng Wang . Journal summary from Editor in Chief. AIMS Environmental Science, 2025, 12(2): 252-252. doi: 10.3934/environsci.2025011
[3]	Yifeng Wang . Journal summary from Editor in Chief. AIMS Environmental Science, 2021, 8(2): 100-100. doi: 10.3934/environsci.2021007
[4]	Yifeng Wang . Journal summary from Editor in Chief. AIMS Environmental Science, 2022, 9(2): 217-217. doi: 10.3934/environsci.2022014
[5]	Yifeng Wang . Journal summary from Editor in Chief. AIMS Environmental Science, 2023, 10(2): 245-245. doi: 10.3934/environsci.2023014
[6]	Yifeng Wang . Journal summary from Editor in Chief. AIMS Environmental Science, 2018, 5(1): 64-66. doi: 10.3934/environsci.2018.1.64
[7]	Yifeng Wang . Journal summary from Editor in Chief. AIMS Environmental Science, 2016, 3(1): 140-140. doi: 10.3934/environsci.2016.1.140
[8]	Yifeng Wang . Journal summary from Editor in Chief. AIMS Environmental Science, 2020, 7(2): 153-155. doi: 10.3934/environsci.2020009
[9]	Yifeng Wang . Journal summary from Editor in Chief. AIMS Environmental Science, 2017, 4(2): 287-288. doi: 10.3934/environsci.2017.2.287
[10]	Rukhsana Kokkadan, Resha Neznin, Praseeja Cheruparambath, Jerisa Cabilao, Salma Albouchi . A Study of Infaunal Abundance, Diversity and Distribution in Chettuva Mangrove, Kerala, India. AIMS Environmental Science, 2023, 10(1): 82-92. doi: 10.3934/environsci.2023005

Abstract

Dear Editorial Board Members,

It is my pleasure to share with you the year-end report for AIMS Environmental Science. The journal went through a challenging year in the fifth year (2018). We have received 57 submissions with 27 published online (Figure 1). The most downloaded and cited papers are listed in Tables 1 and 2. The top read article received more than 4332 downloads.

Figure 1. Manuscript statistics.

DownLoad: Full-Size Img PowerPoint

Table 1. The top 10 articles with most pdf download: (By December 31th 2018).

Title	Usages
Quantifying the local-scale ecosystem services provided by urban treed streetscapes in Bolzano, Italy	4332
Low temperature selective catalytic reduction of NOover Mn-based catalyst: A review	1417
Remote sensing of agricultural drought monitoring: A state of art review	1307
Feasibility study of a solar photovoltaic water pumping system for rural Ethiopia	1278
Biophilic architecture: a review of the rationale and outcomes	1234
Nitrate pollution of groundwater by pit latrines in developing countries	1177
Urban agriculture in the transition to low carbon cities through urban greening	1136
Climate change and land management impact rangeland condition and sage-grouse habitat in southeastern Oregon	1097
A state-and-transition simulation modeling approach for estimating the historical range of variability	1080
Effects of urban green areas on air temperature in a medium-sized Argentinian city	1033

| Show Table

DownLoad: CSV

Table 2. The top 10 articles with most cited: (By December 31th 2018).

Title	Number
Traffic-related air pollution and brain development	14
Biophilic architecture: a review of the rationale and outcomes	11
An integrated approach to modeling changes in land use, land cover, and disturbance and their impact on ecosystem carbon dynamics: a case study in the Sierra Nevada Mountains of California	9
Nitrate pollution of groundwater by pit latrines in developing countries	9
The mechanism of kaolin clay flocculation by a cation-independent bioflocculant produced by Chryseobacterium daeguense W6	9
Enhancing water flux of thin-film nanocomposite (TFN) membrane by incorporation of bimodal silica nanoparticles	9
Linking state-and-transition simulation and timber supply models for forest biomass production scenarios	8
Climate change and land management impact rangeland condition and sage-grouse habitat in southeastern Oregon	8
Quantifying the local-scale ecosystem services provided by urban treed streetscapes in Bolzano, Italy	8
Combining state-and-transition simulations and species distribution models to anticipate the effects of climate change	8

| Show Table

DownLoad: CSV

I would like to thank all the board members for serving on the Editorial Board and their dedication and contribution to the journal AIMS Environmental Science. The goal in 2019 is to solicit more papers and increase paper citations. We will try our best to reduce the processing time and supply with a better experience for publication. To recognize the contribution of the Editorial Board members and authors during the years, we will offer that (1) for authors invited, the article processing charge (APC) is automatically waived; (2) each editorial board member is entitled for some waivers. I am looking forward to continuing working with you to make the AIMS Environmental Science a sustainable and impactful journal. Please don't hesitate to send me e-mails if you have new ideas and suggestions to help us to achieve this goal.

Yifeng Wang, Ph.D.

Editor in Chief, AIMS Environmental Science

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

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)