International Journal of Scientific & Technology Research

Home About Us Scope Editorial Board Blog/Latest News Contact Us
10th percentile
Powered by  Scopus
Scopus coverage:
Nov 2018 to May 2020


IJSTR >> Volume 8 - Issue 10, October 2019 Edition

International Journal of Scientific & Technology Research  
International Journal of Scientific & Technology Research

Website: http://www.ijstr.org

ISSN 2277-8616

The “Determinant Of The Jacobian Matrix” As A Purely Determining Factor Of A Vegetation Pattern Formation Under Turing Instability

[Full Text]



Peter K. Nyarko, Daniel E. Bentil, Isaac K. Dontwi, Samuel Y. Mensah, Christiana C. Nyarko



Determinant, homogeneous plant equilibrium, plant metabolism responses, Turing analysis, vegetation patterns



The mechanism for growth, spread and vegetation pattern formation is largely unknown and poorly understood. To improve understanding of this mechanism, a mathematical model consisting of two nonlinear partial differential equations for soil water balance (N) and plant biomass density variable (P) to investigate the dynamics of forest growth and vegetation pattern formation was developed. The methods used include Michaelis-Menten Kinetics for the rate of nutrients uptake by a cell or organism for growth and Continuous-Time Markov (CTM) method as a standardized methodology that describes plant metabolism responses to multiple resource inputs. This CTM technique was used to obtain a simple plant growth component by synthesizing the four resources (light, water and nutrients together with temperature). To linearize the nonlinear model formulated in order to explain the dynamics of the growth, spread and vegetation pattern formation of the forest, the Taylor Series Expansion method was applied. The linear stability analysis of homogeneous steady-state solutions provided a reliable predictor of the onset and nature of pattern formation in the reaction-diffusion systems. The results revealed that, stability conditions needed for pattern formation is possible provided that . Thus, the homogeneous plant equilibrium decreases with decreasing rainfall until plant become extinct. Based on this condition, the trace and determinant criteria for stability were obtained as and respectively. Again, as increases or decreases, also increases or decreases respectively irrespective of the values of the other parameters. This suggests that which is a surrogate for a dimensionless infiltration capacity prohibits pattern formation at high levels. Hence, one may therefore not expect vegetation patterns to exist in situations of high fertility level and rich water condition. However, this is not the case.



[1] H. Temesgen, and K.V. Gadow, “Generalized height–diameter models-an application for major tree species in complex stands of interior British Columbia”, European Journal of Forest Research, Vol. 123, No.1, pp.45-51, 2004.
[2] V. Deblauwe, P. Couteron, J. Bogaert, and N. Barbier, “Determinants and dynamics of banded vegetation pattern migration in arid climates”, Journal of Ecological Monograph, Vol. 82, No. 1, pp 3–21. 2012
[3] J. Müller, “Floristic and structural pattern and current distribution of tiger bush vegetation in Burkina Faso (West Africa), assessed by means of belt transects and spatial analysis”, Application Ecological Environment Res, Vol. 11, pp 53–171, 2013.
[4] S. S. Berg and D.L. Dunkerley, “Patterned mulga near Alice Springs, central Australia, and the potential threat of firewood collection on this vegetation community”, Journal of Arid Environment, Vol. 59, pp 313–350. 2004
[5] M. Moreno-de las Heras, P.M. Saco, G.R. Willgoose, and D.J. Tongway, “Variations in hydrological connectivity of Australian semiarid landscapes indicate abrupt changes in rainfall-use efficiency of vegetation”, Journal of Geophys Res Vol. 117, pp G03009, 2012.
[6] J.D. Pelletier, S.B. DeLong, C.A. Orem, P. Becerra, K. Compton, K. Gressett, J. Lyons-Baral, L.A. McGuire, J.L. Molaro, and J.C. Spinler, “How do vegetation bands form in dry lands? Insights from numerical modeling and field studies in southern Nevada. USA”, Journal of Geophys Res, Vol. 117, pp F04026, 2012.
[7] G.G. Penny, K.E. Daniels, and S.E. Thompson, “Local properties of patterned vegetation: quantifying endogenous and exogenous effects”, Philos Trans R Soc A, Vol. 371, pp 20120359, 2013.
[8] E. Buis, A. Veldkamp, B. Boeken, and N. Van Breemen, “Controls on plant functional surface cover types along a precipitation gradient in the Negev Desert of Israel”, Journal of Arid Environment, Vol. 73, pp 82–90. 2009
[9] E. Sheffer, J. von Hardenberg, H. Yizhaq, M. Shachak, and E. Meron, “Emerged or imposed: a theory on the role of physical templates and self‐organisation for vegetation patchiness”, Ecology letters, Vol. 16, No. 2, pp.127-139, 2013.
[10] H. Yizhaq, S. Sela, T. Svoray, S. Assouline, and G. Bel, “Effects of heterogeneous soil-water diffusivity on vegetation pattern formation”, Water Resource Res, Vol. 50, pp 5743–5758, 2014.
[11] Bel, G., Hagberg, A., Meron, E., (2012), “Gradual regime shifts in spatially extended ecosystems”, Journal of Theoretical Ecology, Vol. 5, pp 591–604
[12] D. Dralle, G. Boisrame, and S.E. Thompson, “Spatially variable water table recharge and the hillslope hydrologic response: Analytical solutions to the linearized hillslope Boussinesq equation”, Journal of Water Resources Research, Vol. 50, pp 8515–8530. 2014
[13] K. Kellner, and O. J. H. Bosch, “Influence of patch formation in determining the stocking rate for southern African grasslands”, Journal of Arid Environments, Vol. 22, pp.99-105. 1992
[14] J.M. Thiery, J.M. d'Herbes, and C. Valentin, “A model simulating the genesis of banded vegetation patterns in Niger”, Journal of Ecology, pp.497-507, 1995.
[15] M. Jeltsch, A. Kaipainen, V. Joukov, X. Meng, M. Lakso, H. Rauvala, M. Swartz, D. Fukumura, R.K. Jain, and K. Alitalo, “Hyperplasia of lymphatic vessels. 1997.
[16] J. Bromley, J. Brouwer, A.P. Barker, S.R. Gaze, and C. Valentine, “The role of surface water redistribution in an area of patterned vegetation in a semi-arid environment, south-west Niger”, Journal of Hydrology, Vol. 198, No. 1-4, pp.1-29. 1997
[17] R. Lefever, and O. Lejeune, “On the origin of Tiger bush”, Bulletin of Mathematical Biology, Vol. 59, pp 263–294. 1997.
[18] E. Meron, “Pattern-formation approach to modelling spatially extended ecosystems”, Ecological Model, Vol. 234, pp 70–82. 2012.
[19] K. Gowda, H. Riecke, and M. Silber, "Transitions between patterned states in vegetation models for semiarid ecosystems”, Phys Rev E, Vol. 89, 022701. 2014
[20] M. Rietkerk, S.C. Dekker, P.C. de Ruiter, J. van de Koppel, “Self-organized patchiness and catastrophic shifts in ecosystems”, Science, Vol. 305, pp 1926–1929, 2004.
[21] S. Kéfi, M. Rietkerk, M. van Baalen, M. Loreau, “Local facilitation, bistability and transitions in arid ecosystems”, Theoretical Population Biology Vol. 71, pp 367–379. 2007
[22] R. Corrado, A.M. Cherubini, and C. Pennetta, “Early warning signals of desertification transitions in semiarid ecosystems”, Journal of Physical Review Education, Vol. 90, pp. 062705-1 to 062705-11. 2014
[23] M. Rietkerk, F. van den Bosch, J.V. van de Koppel, “Site-specific properties and irreversible vegetation changes in semi-arid grazing systems”, Oikos, Vol. 80, No. 1, pp.241-252, 1997.
[24] C. A. Klausmeier, “Regular and irregular patterns in semiarid vegetation”, Science, Vol. 284, pp 1826–1828. 1999
[25] de Wit, C.T., (1958), “Transpiration and crop yields”, Unknown Publisher, No. 64.6.
[26] LI Zi-zhen, WANG Wan-xion, and XU Cai-lin, “Dynamic model of crop growth system and numerical simulation of crop growth process under the multi-environment external force action”, Applied Mathematics and Mechanics, Vol. 24, No. 6, pp 727-737, 2003
[27] Li, Y. R. (2003), “Contraction Integrated Semigroups and their Application to Continuous Time Markov Chains”, Acta Mathematica Sinica, Vol. 19, No. 3, pp. 605-618.