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

#### Abstract

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.

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DOI: http://dx.doi.org/10.5614%2Fj.math.fund.sci.2018.50.3.2

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