Modelling Effect of Toxic Metal on the Individual Plant Growth: A Two Compartment Model

A two co mpart ment mathematical model for the ind ividual plant growth under the stress of toxic metal is studied. In the model it is assumed that the uptake of to xic metal adsorbed on the surface of soil by the plant is through root compart ment thereby decreasing the root dry weight and shoot dry weight due to decrease in nutrient concentration in each compart ment. In order to visualize the effect of to xic metal on p lant growth, we have studied two models that is, mod el for plant growth with no toxic effect and model fo r plant growth with toxic effect. Fro m the analysis of the models the criteria for plant growth with and without toxic effects are derived. The numerical simu lation is done using Matlab to support the analytical results.


Introduction
Soil normally contains a lo w concentrat ion of heavy metals such as copper (Cu) and zinc (Zn), wh ich are the essential macronutrients for the optimu m gro wth of the p lants . Met als su ch as cad miu m (Cd ), ars en ic (A r), chro miu m (Cr), lead (Pb ), nickel(Ni), mercu ry(Hg) and seleniu m (Se) to xic to plants are not usually found in agricultural soil [1]. Over the last few years, the level of heavy metals are increasing in the agricu ltural fields as a consequence of increasing environ mental pollution fro m industrial, agricu ltural, energy and mun icipal wastes. A reduction in p lant growth has been observed due to the presence of elevated levels of heavy metals like cadmiu m, arsenic, n ickel, lead and mercu ry [2]. Cad miu m (Cd ) is among the most widespread heavy metals found in the surface soil layer wh ich inhibits the uptake of nutrients by plants and as well as its growth [3]. The inhib ition of plant growth can be caused by the phytotoxic effect of cad miu m on different p rocesses in p lants, includ ing respirat ion, p hot osyn th es is , carboh yd rat e met ab o lis m an d wat er relat ion [4]. Cad miu m (Cd ) ia a to xic met al, caus ing phytotoxicity, and its uptake and accu mulation in plants causes reduction in photosynthesis, diminishes water and nutrient uptake [5]. Heavy metals interfere with the uptake and distribution of essential mineral nutrients in a plant, causing deficiencies and nutrient imbalance [6]. The to xic metals in the soil system could result in the leach ing of essential cation away fro m the rooted zone, decreasing plant the activities of several enzy mes, seed germinat ion and seedling growth [12][13][14][15]. Seed gemination inhibit ion by heavy metals has been reported by many researchers [16][17][18]. Agricultural research almost completely rely upon experimental and empirical works, co mbined with statistical analysis and very few mathematical modelling analysis has been carried out in this direction [3][4], [19][20]. Many of the mo dels that are currently used by agronomists and foresters to predict harvests and schedule fertilization, irrigation and pesticides application are of emp irical form. A major limitat ion in all these approaches is the unpredictability of the environmental inputs [21]. Thornley in itiated the work related to the mathematical modelling of ind ividual plant growth processes and mathemat ical models were applied to a wide variety of topics in p lant physiology [22]. The majority of these focuses on processes that are modelled independently such as photosynthesis, flu id transport, respiration, transpiration and stomatal response and the general goal of the models was to predict the effect of a variety of environmental factors, including radiat ion input, humid ity, wind, CO 2 concentration and temperature on these process rates. The soil-nutrient-plant interaction represents a good example of a relat ionship that operates at individual, population, and ecosystem levels. Nutrients influence individual plant growth, which has subsequent effect on population growth dynamics wh ich in turn influence production of standing crop. The models that have been developed to describe the growth of individual plants in crop has been classified by Benamin and Hard wick [23], according to the assumption that how resources are shared. A continuous-time model fo r the growth and reproduction of a perennial herb with discrete growing season is considered in [24] and optimal resource allocation in perennial plants has been determined and studied. In the paper [25], a transient three-dimensional model for soil water and solute transport with simultaneous root growth, root water and nutrient uptake is studied and discussed. In this paper, authors have presented a model to study the interactive relationships between changing soil-water and nutrient status and root activity. The authors in the paper [19] have studied the influence of acid deposition on forests by means of a mathematical model taking the state variables as forest dry weight, alu min iu m concentration in trees and soil, and proton concentration in soil. Referrence [3] have given a mathematical model to study the effect of cad miu m (Cd ) on nutrient uptake by crop such as Barley and have shown through their model that how the accumulation of Cd in plants inhibit its growth rate. Experimental and mathematical simu lation to study the effects of toxic metal; cadmiu m on the plant growth promoting rhizobacteria and plant interaction have been carried out by [4]. Nit rogen dynamics in soil, its availab ility to the crop and the effects of nitrogen deficiency on crop performance were studied in the model g iven by the researchers [26]. A non-spatial, sizestructured continuum model of p lant growth, without focusing on a particular species, but with emphasis on a dense tree-dominated forest is considered and studied by [27] and in this paper a closed form solution for the equilibriu m size density distribution is obtained along with the analytical conditions for co mmunities persistence. Crops and vegetables grown on polluted soil accumu late heavy metals that cause decrease in their yield, and in order to study the uptake of heavy metals and its accumulation by crops, mathemat ical models can be used. In paper [20], a study has been conducted through mathematical model to understand the cadmiu m uptake by radish, carrot, spinach and cabbage. In this paper a dynamic macroscopic numerical model for heavy metal transport and its uptake by vegetables in the root zone is considered and analysed numerically. A very few mathemat ical models to study the effects of toxic metal on plant growth exist [3][4], [19][20].
In view of the above, therefore in this paper, a two compart ment mathematical model for the plant growth under the stress of toxic metal is proposed and analyzed. For the modelling purpose, the plant is divided into root and shoot compartments in which the state variables considered are nutrient concentration and dry weight. In the model it is assumed that the uptake of to xic metal adsorbed on the surface of soil by the plant is through root compartment thereby decreasing the root dry weight and shoot dry weight due to decrease in nutrient concentration in each compart ment. In the model it is further assumed that the maximu m root dry weight and shoot dry weight decrease due to the presence of toxic metal in root co mpart ment. Fro m the analytical and nu merical analysis of the model the criteria for p lant growth under the stress of toxic metal are derived.

Mathematical Model
Model 1 (Model with no to xic effect): In this model the plant growth dynamics is studied by assuming that the plant is divided into root and shoot (stem, leaf, flower) co mpart ments in wh ich the state variables associated with the each compart ment are nutrient concentration and dry weight. Let r W and s W denote the root dry weight and shoot dry weight respectively. 0 S and 1 S denote the nutrient concentration in root and shoot respectively. With these notations, the mathematical model of the plant growth dynamics is given by the following system of nonlinear differential equations: with the in itial conditions as: In the present analysis we assume the fo llo wing forms for growth functions ) ( 0 S r and ) ( 1 S r [22], [28] : In absence of nutrient concentration plant will not grow and eventually they will die out.  [26]. In plant growth, it is considered that during the initial stage, i.e., during the lag phase, the rate of plant growth is slow. Rate of growth then increases rapidly during the exponential phase. After so me time the growth rate slowly decreases due to limitation of nutrient. This phase constitutes the stationary phase . The   (12) with the in itial conditions as: Here, we assume the follo wing forms for Along with the parameters of model 1, we have the following additional parameters in model 2 such as 1 , m k , f and h , which are described as follo ws: 0 Q is the input rate of heavy metals.  is the first order rate constant.  is the soil bulk density. k is the linear adsorption and absorption coefficient.
k is the Michaelis-Menten constant. f is the first order rate coefficient. h is the natural decay rate of C  due to soil depletion on account of natural process.  is natural decay rate of C . Here, all the parameters N K , 10 D , to be positive constants.

Analysis of Model 1
Now, we show that the solutions of the model g iven by (1) to (4) for all positive initial values Proof: By adding Eqs. (1) and (2), we get, and then by the usual comparison theorem we get as : Fro m Eqs. (20) and (21), we have where, * 1 10 20 1 2 10 20 2 0 Fro m the nature of the roots of the characteristic equation (23) we derive that the equilibriu m point * E is always locally asymptotically stable. Now, we d iscuss the global stability of the interior equilibriu m point * E of the system (1)-(4). The non-linear stability of the interio r positive equilib riu m is determined by the following theorem. Therorem 3.2: In addition to assumptions (6), let for some positive constants r K and s K less than 1.    V  is given by   (24)   V is a Liapunov function with respect to * E , whose domain contains 1 B , proving the theorem.
The above theorem shows, that provided inequalities (25) to (27) hold, the system settles down to a steady state solution.

Anal ysis of Model 2
Now, in the following we show that the solutions of model given by (7) to (12) Proof: By adding Eqs. (7) and (8), we get, and then by the usual comparison theorem we get as : and the positive value of 0 S and 1 S can be obtained by solving the following pair of equations: Fro m the nature of the roots of the characteristic equation Now, we d iscuss the global stability of the interior equilibriu m point Ẽ of the system (7)- (12). The non-linear stability of the interior positive equilibriu m state is determined by the follo wing theorem. Therorem 3.4: In addition to assumptions (6) and (13), for some positive constants  We consider a positive definite function about Then the derivatives along solutions, 2 V  is given by After some algebraic man ipulations, this can be written as V  can be written as the sum of three quadratic forms 33 2 22 V is a Liapunov function with respect to Ẽ , whose domain contains 2 B , proving the theorem.
The above theorem shows, that provided inequalities (47) to (50) hold, the system settles down to a steady state solution. which is asymptotically stable (see Figure 1). Further, to illustrate the global stability of interior equilibriu m of model 1 graphically, nu merical simu lation is performed for different init ial conditions (see Table 1 and 2) and results are shown in Figures 2 and 3 for phase plane and phase plane respectively. All the trajectories are starting from different initial condit ions and reach to interior equilibriu m .    . Phase plane graph for nutrient concentration in shoot S1 and shoot dry weight Ws at different initial conditions given in Table 1  Hence, Ẽ is asymptotically stable (see Figure 4).  Table 3 and 4) and results are shown in Figures 5 and 6 Table 4. Different initial conditions for S1 and Ws of model 2 S1(0) 0.1 10 16 2

Numerical Example
Ws(0) 1 0.1 16 18 Figure 5. Phase plane graph for nutrient concentration in root S0 and root dry weight Wr at different initial conditions given in Table 3 for model 2 (with toxic effect) showing the global stability behaviour Figure 6. Phase plane graph for nutrient concentration in shoot S1 and shoot dry weight Ws at different initial conditions given in Table 4

Conclusions
Equilibriu m of model 1 is shown to be asymptotically stable (see Fig. 1). The equilibria of model 2 is shown to be asymptotically stable (see Fig. 4). Fro m Figures 7(a) and 7(b), it may be noted that the equilibriu m levels of nutrient concentrations in each compart ment with no to xic effect are mo re than that of the equilibriu m levels of nutrient concentrations in respective compart ments when toxic effect is considered. Figure 7(a). Graph between nutrient concentration in root S0 and time t for model 1(with no toxic effect) and for model 2(with toxic effect) Figure 7(b). Graph between nutrient concentration in shoot S1 and time t for model 1(with no toxic effect) and for model 2(with toxic effect) Further, fro m Figures 8(a) and 8(b), it is observed that the equilibriu m levels of root dry weight and shoot dry weight with no to xic effect are more than that of equilibriu m levels of the root dry weight and shoot dry weight when toxic effect is being considered. Fro m the expressions (37) and (38) it may be noted that the root dry weight and shoot dry weight will decrease and may tend to zero with increasing C  . Fro m Eqs. (22) and (43), it is concluded that for large 1  , the nutrient concentration in shoot with to xic effect is less than that of nutrient concentration in shoot when no toxic effect is considered. The Figures 9(a)  with respect to C  . Fro m these figures it is observed that the toxicity of the metal will adversely effect the plant growth in its early stages resulting in loss of crop productivity [8], [29].