The Study of Effect of Toxic Metal on Plant Growth Dynamics with Time Lag: A Two-Compartment Model

Preety Kalra, Pankaj Kumar


A two-compartment mathematical model is proposed for the study of individual plant growth dynamics with time lag due to the presence of toxic metals in the soil. It is assumed in the model that nutrient uptake by the roots is hindered by the presence of the toxic metals. It is further assumed that there is less transfer of nutrients from the root compartment to the shoot compartment due to the toxic metals. However, the nutrient concentration decreases in the root compartment as well as in the shoot compartment, resulting in a decrease of the structural dry weight of the roots and shoots respectively. This effect was studied by considering time lag in the utilization coefficient of the nutrient concentration in the roots in the presence of toxic metals. It is further assumed in the model that the nutrient use efficiency is also affected by the presence of toxic metals, resulting in a decrease of the structural dry weight of the shoots. The inclusion of time lag results in the disturbance of the interior equilibrium stability and Hopf bifurcation occurs for a critical value of the delay parameter. This entire phenomenon was captured by numerical simulation.


Concentration of nutrients; equilibrium; Hopf-bifurcation; structural dry weight; time delay;

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Thornley, J.H.M, Mathematical Models in Plant Physiology, A Quantitative Approach to Problems in Plant and Crop Physiology, Academic Press, London, 1976.

Lacointe, A., Carbon Allocation Among Tree Organs: A Review of Basic Processes and Representation in Functional-Structural Tree Models, Annals of Forest Science, 57, pp. 521-533, 2000.

Godin, C., Hanan, J., Kurth, W., Lacointe, A. & Takenaka, A., Prusinkiewicz, P., Proceedings of the 4th International Workshop on Functional-Structural Plant Models, Montpellier, France, UMR AMAP, 2004.

Leo, G.D., Furia, L.D. & Gatto, M., The Interaction between Soil Acidity and Forest Dynamics, Theoretical Population Biology, 43, pp. 31-51, 1993.

Guala, S.D., Vega, F.A. & Covelo, E.F., The Dynamics of Heavy Metals in Plant-soil Interactions, Ecological Modelling, 221, pp. 1148-1152, 2010.

Guala, S.D., Vega, F.A. & Covelo, E.F., Modelling the Plant Soil Interaction in the Presence of Heavy Metal Pollution and Acid Variations, Environ. Monit. Assess., 185, pp. 73-80, 2013.

Misra, O.P. & Kalra, P., Modelling Effect of Toxic Metal on the Individual Plant Growth: A Two Compartment Model, American Journal of Computational and Applied Mathematics, 2(6), pp. 276-289, 2012.

Misra, O.P. & Kalra, P., Effect of Toxic Metal on the Structural Dry Weight of a Plant: A Model, International Journal of Biomathematics, 6(5), pp. 1350028 (27 Pages), 2013.

Dieudonne, J., Foundations of Modern Analysis, New York Academic Press, 1960.

Ruan, S. & Wei, J., On the Zeros of a Third-Degree Exponential Polynomial with Applications to A Delayed Model for The Control of Testosterone Secretion, IMA J. Math. Appl. Medic. Biol., 18, pp. 41-52, 2001.

Kubiaczyk, I. & Saker, S.H., Oscillation and Stability in Nonlinear Delay Differential Equations of Population Dynamics, Mathematical and Computer Modelling, 35, pp. 295-301, 2002.

Ruan, S. & Wei, J., On Zeros of a Transcendental Function with Applications to Stability of Delay Differential Equations with Two Delays, Dyn. Continuous Discr. Impuls Syst. Ser. A: Math. Anal., 10, pp. 863-874, 2003.

Naresh, R., Sharma, D. & Sundar, S., Modeling the Effect of Toxicant on Plant Biomass with Time Delay, International Journal of Nonlinear Science,17, pp. 254-267, 2014.

Shukla, A., Dubey, B. & Shukla, J.B., Effect of Environmentally Degraded Soil on Crop Yield: The Role of Conservation, Ecological Modelling, 86, pp. 235-239, 1996.

Sikarwar, C.S. & Mishra, O.P., Effect of Time Delay on the Dynamics of A Multi Team Prey Predator System, Ph. D. Thesis, Jiwaji University, Gwalior, 2012.

Naresh, R., Sundar, S. & Shukla, J.B., Modelling the Effect of An Intermediate Toxic Product Formed by Uptake of a Toxicant on Plant Biomass, Applied Mathematics and Computation, 182, pp. 151-160, 2006.

Huang, G., Liu, A. & Fory, U., Global Stability Analysis of Some Nonlinear Delay Differential Equations in Population Dynamics, J. Nonlinear Sci., 26, pp. 27-41, 2016.

Zhang, T., Jiang, H. & Teng, Z., On the Distribution of the Roots of a Fifth-Degree Exponential Polynomial with Applications to A Delayed Neural Network Model, Neurocomputing, 72, pp. 1098-1104, 2009.



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