ANALYSIS OF PLANT GROWTH DYNAMICS UNDER THE EFFECT OF TOXICITY: A DELAY DIFFERENTIAL EQUATION MODEL
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Abstract
A mathematical model is designed to examine plant growth under stress in the presence of toxicity with a delay. It is observed that toxic substances change the soil's structure and activity, which has a negative impact on the concentration of nutrients there. The availability of soil nutrients is one of the most significant elements affecting biomass. Total biomass is seen to be adversely affected by both nutrition deficiency and toxicity. It has been noted that the presence of toxicity changes the physiology and growth of the plant, which ultimately reduces crop growth and production. This adverse effect of toxicity is not seen immediately, but after an incubation period. This effect is demonstrated by taking into account the delay in the state variable. Additionally, Hopf bifurcation is observed for the crucial value of the delay parameter. Utilizing explicit techniques, the direction and stability of bifurcating periodic solutions are found. Sensitivity analysis is used to determine the sensitivity of solutions of the model when values of parameters are varied. MATLAB is used for simulation.
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Licensee MJS, Universiti Malaya, Malaysia. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
References
A. Lacointe, "Carbon allocation among tree organs: A review of basic processes and representation infunctional-structural tree models," 2000.
A. Overman, "A memoir on mathematical models of crop growth and yeild: Effect of geographic location," Univ. Florida.
A. R. Watkinson, "Density-dependence in single-species populations of plants," J. Theor. Biol., vol. 83, no. 2, 1980, doi: 10.1016/0022-5193(80)90297-0.
B. Ingalls, M. Mincheva, and M. R. Roussel, "Parametric Sensitivity Analysis of Oscillatory Delay Systems with an Application to Gene Regulation," Bull. Math. Biol., vol. 79, no. 7, 2017, doi: 10.1007/s11538-017-0298-x.
C. Sun, M. Han, and Y. Lin, "Analysis of stability and Hopf bifurcation for a delayed logistic equation," Chaos, Solitons and Fractals, vol. 31, no. 3, 2007, doi: 10.1016/j.chaos.2005.10.019.
D. H. Hardy, J. Myers and C. Stokes, Heavy metals in North Carolina soils : occurrence & significance. N.C. Department of Agriculture and Consumer Services, Agronomic Division.
D. Kaur and P. Kumar, "Bifurcation induced by delay parameter in plant growth dynamics," J. Phys. Conf. Ser., vol. 2267, 2022.
D. Sharsma, S. Sundar and R. Naresh, "Modelling the effect of toxicant on plant biomass with time delay," Int. J. non linear Sci., vol. 17, pp. 254–267, 2014.
Dipesh and P. Kumar, "Effect of time delay on dynamic of plant competition under allelopathy," Math. Method Appl. Sci.,vol. 45, pp.9308–9321, 2022.
F. A. Rihan, "Sensitivity analysis for dynmics systems with time-lags," J. Comput. Appl. Math., vol. 151, no. 2, 2003, doi: 10.1016/S0377-0427(02)00659-3.
G. A. Bocharov and F. A. Rihan, "Numerical modelling in biosciences using delay differential equations," J. Comput. Appl. Math., vol. 125, no. 1–2, 2000, doi: 10.1016/S0377-0427(00)00468-4.
G. Deleo, L. Delfuria, and M. Gatto, "The interaction between soil acidity and forest dynamics: A simple-model exhibiting catastrophic behavior," Theor. Popul. Biol., vol. 43, no. 1, 1993, doi: 10.1006/tpbi.1993.1002.
G. Ladas and C. Qian, "Oscillation and global stability in a delay logistic equation," Dyn. Stab. Syst., vol. 9, no. 2, 1994, doi: 10.1080/02681119408806174.
H. T. Banks, D. Robbins, and K. L. Sutton, "Generalized sensitivity analysis for delay differential equations," in International Series of Numerical Mathematics, vol. 164, 2013. doi: 10.1007/978-3-0348-0631-2_2.
J. B. Shukla, A. Agarwal, P. Sinha and B. Dubey, "Modelling Effects of Primary and Secondary Toxicant on Renewable Resources.," Nat. Resour. Model., vol. 16, pp. 99–120, 2008.
K. R. Schneider, "Hassard, B. D. / Kazarinoff, N. D. / Wan, Y.-H., Theory and Applications of Hopf Bifurcation. London Mathematical Society Lecture Note Series 41. Cambridge, Cambridge University Press 1981. 320 S., £ 15.00 A P/B. ISBN 0-521-23158-2," ZAMM - Zeitschrift für Angew. Math. und Mech., vol. 62, no. 12, 1982, doi: 10.1002/zamm.19820621221.
O. P. Misra and P. Kalra, "Effect of toxic metal on the structural dry weight of a plant: A Model," Int. J. Biomath., vol. 6, no. 5, 2013, doi: 10.1142/S1793524513500289.
O. P. Misra and P. Kalra, "Modelling Effect of Toxic Metal on the Individual Plant Growth: A Two Compartment Model," Am. J. Comput. Appl. Math., vol. 2, no. 6, 2013, doi: 10.5923/j.ajcam.20120206.06.
S. Trvedi and L. Erdei, "Effects of cadmium and lead on the accumulation of Ca2+ and K+ and on the influx and translocation of K+ in wheat of low and high K+ status," Physiol. Plant., vol. 84, no. 1, 1992, doi: 10.1111/j.1399-3054.1992.tb08770.x.
S. Ruan and J. Wei, "IMA Journal of Mathematics Applications in Medical Biology," IMA J. Math. Appl. Med. Biol. 1841-52 (2001)., vol. 18, pp. 41–52, 2001.
S. H. Saker and I. Kubiaczyk, "Oscillation and stability in nonlinear delay differential equations of population dynamics," Math. Comput. Model., vol 35, pp. 298–301, 2002.
T. J. H. M, "Mathematical Models in Plant Physiology. A Quantitative Approach to Problems in Plant and Crop Physiology.," Acad. Press. London, 1976.