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.

Bifurcation induced by delay parameter in plant growth dynamics

“…The delay models about its positive equilibrium point been asymptotically stable under some necessary and sufficient conditions for all positive solutions obtained by Gopalsamy [16] and Ladas [17]. The effect of time lag in plant growth is studied using a two-compartment mathematical model by Kalra and Kumar [18].…”

Delay differential equation model of forest biomass and competition between wood‐based industries and synthetic‐based industries

“…It has been shown that the birthrate has reduced in the leading districts [11]. Various mathematical models proposed to understand the effect of time-lag in plant growth under the effect of toxic metals [12][13][14]. The nature of the exponentially characterizing equation's zeros and stability of non-zero equilibrium point have been studied in brief, which are helpful in the study of bifurcation and complex dynamics [15,16].…”

Effect of time delay on dynamic of plant competition under allelopathy

“…So, in the complexities, the delay concept comes into effect. The influence of the delay parameter on toxicant‐effected plant growth dynamics was analyzed using a nonlinear delay differential equation system 16–20 . Mukhopadhyay et al 13 have altered the Maynard‐Simth's 10 method to a delay differential equation model.…”

“…The influence of the delay parameter on toxicant-effected plant growth dynamics was analyzed using a nonlinear delay differential equation system. [16][17][18][19][20] Mukhopadhyay et al 13 have altered the Maynard-Simth's 10 method to a delay differential equation model. They found the distinct delay necessary for the development of the harmful substance to be generated by the organisms.…”