Effect of Intermediate Toxic Product on the Survival of a Resource Dependent Species: A Modeling Study

A nonlinear model is proposed and analysed to study the effect of intermediate toxic product on the survival of a resource dependent species in a polluted environment. It is assumed that when resource biomass uptakes pollutants/toxicants, a liquid (sap) present in the body of b iomass reacts with such toxicants and as such intermediate toxic product is formed. Th is toxic substance then affects the biomass and the species dependent on it. The analysis of the model shows that with increase in the cumulative emission rate of toxicants in the atmosphere, the densities of resource biomass and the species dependent on it decrease and attain their lowest equilibrium. If the rate of emission of toxicants is large enough, the resource biomass may become ext inct under certain conditions and the species dependent on it may not survive. The model analysis also suggests that if the fo rmation of intermediate toxic p roduct is restricted by way of controlling the emission of toxicants in the environment, the resulting growth of resource biomass would lead to survival of species dependent on it.


Introduction
Various kinds of pollutants/toxicants discharged into the environment very often affect the resources and the species dependent on them [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. In particular, Freed man and Shukla [1] studied the effect of a to xicant on single species and predator-prey systems. They have assumed that the intrinsic growth rate of species decreases with increase in uptaken concentration of toxicant whereas its carrying capacity decreases with the environmental concentration of toxicant. The effect of to xicants on a two species competitive system was studied by Chattopadhyaya [2]. Samanta and Matti [3] studied the effect of to xicant on a single species by considering the three cases of toxicant emission i.e. instantaneous spill, constant emission and rapidly fluctuating random emission of to xicant into the environment. It was shown that the toxicant concentration emitted instantaneous ly would not be sufficient to kill the population whereas the constant emission makes the population to settle down to steady state. In the third case (rapid ly fluctuating random emission), the stability (local) of the system would depend on the washout rate of toxicant fro m the environ ment.
In recent years, some mathemat ical models have also been proposed to study the existence and survival of resource dependent species liv ing in a polluted environ ment [15][16][17][18][19][20][21]. Dubey and Hussain [15] proposed models for the survival of two competing species dependent on a resource in an industrial environment. They assumed competing species to be partially dependent, wholly dependent or predating on the resource biomass. They concluded that an appropriate level of resource biomass can be maintained to ensure the survival of species if suitable efforts to conserve the resource biomass and to control the undesired label of industrializat ion pressure are made. Shukla et al. [16] studied a model for survival of resource dependent population to see the effect of toxicant emitted fro m external sources as well as formed by its precursors. They have shown that the densities of resource and the population decrease as the cumulative emission rate of environ mental to xicant increases.
It is pointed out here that in the above studies, the effect of toxicant on resource biomass and species dependent on it is considered but without incorporating the process of formation of intermediate to xic p roduct. However, in real situations the intrinsic g rowth of resource bio mass is, very often affected by intermediate to xic product which is formed inside the resource biomass due to some metabolic reactions. This resource biomass affected by the intermediate to xic product then affects the species dependent on it. In this direction, Naresh et al. [22] presented a mathemat ical model to study the effect of intermediate to xic p roduct formed by uptake of a toxicant on plant biomass. They have shown that, as the rate of emission of to xicant increases, the equilibriu m label of plant bio mass decreases. Since the toxicants emitted into the atmosphere are uptaken by the resource biomass, an intermediate to xic product is formed wh ich then affects the growth of resource bio mass and the species dependent on it. Hence, in the present investigation, our main purpose is to study the effect of intermediate to xic product on the survival of resource dependent species. Thus, we propose a nonlinear mathematical model to see the effect of intermed iate toxic product on the resource bio mass and on the survival of species dependent on it [22]. The paper is organized as fo llo ws, in Section 2 we present the mathematical model. The equilibriu m analysis is carried out in Section 3 and in Section 4, the stability analysis of the model is presented. In Section 5, we present the numerical simu lations of the model and conclusions are provided in Section 6.

Mathematical Model
The following assumptions have been made in the modeling process, 1. The densities of resource and species dependent on it are governed by logistic models.
2. The gro wth rate of resource bio mass decreases with increase in the concentration of intermediate to xic product.
3. The gro wth rate of resource dependent species increases with increase in the density of resource biomass.
4. The carrying capacities of resource biomass as well as that of species dependent on it decrease with environmental concentration of toxicants.
Let be the resource biomass density, affected by toxicants emitted at a constant rate in the environ ment. It is assumed that, when the amount of toxicants uptaken by resource biomass interacts with the bio flu id (sap) present inside the biomass, an intermed iate to xic product is formed which affects the growth of resource biomass. Let be the density of resource dependent species, its growth rate is enhanced by the resource biomass density. Let be the cumulat ive concentration of toxicants in the environment with natural depletion rate and is the depletion rate coefficient of to xicants due to uptake by resource biomass. The environ mental concentration of toxicants affect the carrying capacities and of the resource biomass and the resource dependent species respectively. It is assumed that the uptake of the toxicants by the resource biomass is directly proportional to the density of resource biomass and the concentration of toxicants. Let be the concentration of toxicants uptaken by the resource biomass with as natural depletion rate coefficient and be the concentration of intermediate toxic product formed with a rate α 1 and as its natural depletion rate coefficient.
Keeping in view of the above assumptions and considerations, the system dynamics is assumed to be governed by the following nonlinear ord inary differential equations, The uptaken concentration of toxicants and the concentration of intermediate to xic product are also assumed to be depleted by an amount and respectively due to falling of bio mass on the ground. A fraction of the depleted amount may also re-enter the environ ment, thus increasing the growth of toxicants. The constants and are reversible rate coefficients. All the constants are assumed to be non-negative.
In the model, the function denotes the intrinsic growth rate of resource biomass wh ich decreases as the concentration of intermediate to xic product increases and hence, we assume that, , for The function denotes the carrying capacity of resource biomass which decreases as the concentration of toxicant increases and hence, , for The function denotes the intrinsic growth rate of resource dependent species which increases as the resource biomass density increases and hence, , for The function denotes the carrying capacity of resource biomass which decreases as the concentration of toxicant increases and hence, , for

4.
The existence of and is obvious.

Existence and Uni queness of E 2
The positive solution of variables in equilibriu m is given by the follo wing equations which are obtained by putting the right hand sides of model equations (1) -(5) to zero Fro m eqs. (7) and (8) we have, Fro m eq. (8) we get, Fro m eq. (9) we get, Using eq. (6) we assume that, which gives (13) Fro m eq.(13), it can be seen that (14) and (15) also (16) Fro m eqs. (14) and (15) it is clear that there exist a root of in . The root will be unique, provided, where, Knowing the value of , the values of an d can be found from eqs. (10), (11) and (12) respectively.

Existence and Uni queness of * E
The positive solution of variable in * E is given by the following algebraic equations, (20) As before, let  18), (19), (20) and (21) respectively.
In the following we analyse the stability behavior of above equilibria.

Stability Analysis
In this section, we describe stability analysis of different equilibria.
Theorem 1 (i) Equilibria , an d are unstable.
(ii) If the following inequalities hold, where then * E is locally asy mptotically stable (See Appendix-A for proof).
To establish the nonlinear asymptotic stability of * E , we need the bounds of different variab les. For this we propose δ ν δ δ ν θ αδ π αν satisfying in for some constants , then if the following inequalities hold in , where * E is nonlinearly asymptotically stable with respect to all solutions initiating in the interior of the first octant. (See Appendix-B for proof) Remarks: 1. If and are very small, then the possibility of satisfying conditions (22) -(29) is more plausible showing that these parameters have destabilizing effect on the system.
The above analysis imply that as the rate of introduction of toxicants in the environ ment increases, then under certain conditions the densities of resource bio mass and resource dependent species decrease and settle down to their respective equilib riu m levels. It is pointed here that the magnitude of equilibriu m of species would main ly depend upon the resource biomass density affected by intermediate toxic product formed inside the bio mass due to some metabolic changes. The density of resource bio mass decreases as cumulative concentration of toxicants in the environment increases and it may even tend to zero fo r very high concentration of toxicants and then the species dependent on it may not survive.

Numerical Simulations
In this section, we analyse the model (1)  i.e. at respectively. Fro m these figures, it can be seen that the densities of biomass and species dependent on it decrease as the rate of formation of intermediate to xic product increases and settle down to their respective equilib riu m levels. In figure 8, it is shown that if the growth of resource biomass increases, the density of species dependent on it also increases.

Conclusions
In this paper, we have proposed a nonlinear mathematical model to study the effect of intermed iate toxic product on the survival of a resource dependent species living in a polluted environment. It is assumed that the growth of resource biomass is affected by the intermediate to xic p roduct formed inside the resource biomass due to some metabolic reactions when toxicants present in the environment are uptaken by resource biomass. This affected resource bio mass by the intermediate to xic product then affects the resource dependent species. The model is analysed using stability theory of differential equations and computer simulations. It is shown that densities of resource bio mass and species dependent on it decrease as the rate of introduction of toxicants increases in the environment and settle down to their respective equilibriu m levels while the concentration of intermediate to xic product increases. Further, as the rate of formation of intermed iate to xic product increases, the densities of resource biomass and the species dependent on it also decrease and their magnitudes are less than their respective densities when they are not affected by to xicant. If the rate of emission of toxicants in the environment is large enough, then under certain conditions the resource biomass may become ext inct due to increased level of intermediate toxic p roduct affecting the gro wth of bio mass and as such the species dependent on it may not survive. It is also observed that the density of species dependent on resource biomass increases if the g rowth of resource bio mass increases. Thus, if the formation of intermediate to xic product is restricted by way of controlling the emission of toxicants in the environment, the resulting growth of resource bio mass would lead to survival of species dependent on it.

APPENDIX-A
Proof of the Theorem 1.
(i) The variational matrix corresponding to is given by, It can be seen that the two eigen values of are positive, therefore is a saddle point. The variational matrix corresponding to is given by, Fro m wh ich we note that is a saddle point.
Similarly, it can also be checked that equilibriu m point is unstable in -direction.   is locally asymptotically stable.