1. Introduction

Various kinds of pollutants/toxicants discharged into the environment very often affect the resources and the species dependent on them[1-14]. In particular, Freedman and Shukla[1] studied the effect of a toxicant 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 toxicants on a two species competitive system was studied by Chattopadhyaya[2]. Samanta and Matti[3] studied the effect of toxicant on a single species by considering the three cases of toxicant emission i.e. instantaneous spill, constant emission and rapidly fluctuating random emission of toxicant into the environment. It was shown that the toxicant concentration emittedinstantaneously 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 (rapidly fluctuating random emission), the stability (local) of the system would depend on the washout rate of toxicant from the environment.

In recent years, some mathematical models have also been proposed to study the existence and survival of resourcedependent species living in a polluted environment [15-21]. 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 industrialization pressure are made. Shukla et al.[16] studied a model for survival of resource dependent population to see the effect of toxicant emitted from 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 environmental toxicant 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 toxic product. However, in real situations the intrinsic growth of resource biomass is, very often affected by intermediate toxic product which is formed inside the resource biomass due to some metabolic reactions. This resource biomass affected by the intermediate toxic product then affects the species dependent on it. In this direction, Naresh et al.[22] presented a mathematical model to study the effect of intermediate toxic product formed by uptake of a toxicant on plant biomass. They have shown that, as the rate of emission of toxicant increases, the equilibrium label of plant biomass decreases. Since the toxicants emitted into the atmosphere are uptaken by the resource biomass, an intermediate toxic product is formed which then affects the growth of resource biomass and the species dependent on it. Hence, in the present investigation, our main purpose is to study the effect of intermediate toxic product on the survival of resource dependent species. Thus, we propose a nonlinear mathematical model to see the effect of intermediate toxic product on the resource biomass and on the survival of species dependent on it[22].The paper is organized as follows, in Section 2 we present the mathematical model. The equilibrium 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 simulations of the model and conclusions are provided in Section 6.

2. 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 growth rate of resource biomass decreases with increase in the concentration of intermediate toxic product.

3. The growth 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 environment. It is assumed that, when the amount of toxicants uptaken by resource biomass interacts with the bio fluid (sap) present inside the biomass, an intermediate toxic 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 cumulative concentration of toxicants in the environment with natural depletion rate and is the depletion rate coefficient of toxicants due to uptake by resource biomass. The environmental 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 α₁ 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 ordinary differential equations,

(1)

(2)

(3)

(4)

(5)

The uptaken concentration of toxicants and the concentration of intermediate toxic product are also assumed to be depleted by an amount and respectively due to falling of biomass on the ground. A fraction of the depleted amount may also re-enter the environment, 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 which decreases as the concentration of intermediate toxic product increases and hence, we assume that,

The function denotes the carrying capacity of resource biomass which decreases as the concentration of toxicant increases and hence,

The function denotes the intrinsic growth rate of resource dependent species which increases as the resource biomass density increases and hence,

The function denotes the carrying capacity of resource biomass which decreases as the concentration of toxicant increases and hence,

3. Equilibrium Analysis

The model (1) – (5) has the following four equilibria namely,

1.

2. , where and

3.

4.

The existence of and is obvious.

3.1. Existence and Uniqueness of

The positive solution of variables in equilibrium is given by the following equations which are obtained by putting the right hand sides of model equations (1) - (5) to zero

(6)

(7)

(8)

(9)

From eqs. (7) and (8) we have,

(10)

From eq. (8) we get,

(11)

From eq. (9) we get,

(12)

Using eq. (6) we assume that,

which gives

(13)

From eq.(13), it can be seen that

(14)

and

(15)

also

(16)

From 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 and can be found from eqs. (10), (11) and (12) respectively.

3.2. Existence and Uniqueness of

(17)

(18)

(19)

(20)

(21)

As before, let

From this, it can be seen that and showing the existence of a root of in and the root will be unique, provided,

Knowing the value of, we can find the values of and from eqs. (18), (19), (20) and (21) respectively.

In the following we analyse the stability behavior of above equilibria.

4. Stability Analysis

In this section, we describe stability analysis of different equilibria.

Theorem 1

(i) Equilibria, and are unstable.

(ii) If the following inequalities hold,

(22)

(23)

(24)

(25)

where

then is locally asymptotically stable (See Appendix-A for proof).

To establish the nonlinear asymptotic stability of, we need the bounds of different variables. For this we propose the following region of attraction, stated without proof Freedman and So[23].

Lemma 1 The set

attracts all solutions initiating in the interior of non-negative octant, where and.

Theorem 2.

Let, , , ,, , , , , satisfying in for some constants , then if the following inequalities hold in,

(26)

(27)

(28)

(29)

where

is nonlinearly asymptotically stable with respect to all solutions initiating in the interior of the first octant. (See Appendix-B for proof)

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.

2. If and, then the conditions (24), (25), (28) and (29) are satisfied automatically.

The above analysis imply that as the rate of introduction of toxicants in the environment increases, then under certain conditions the densities of resource biomass and resource dependent species decrease and settle down to their respective equilibrium levels. It is pointed here that the magnitude of equilibrium of species would mainly depend upon the resource biomass density affected by intermediate toxic product formed inside the biomass due to some metabolic changes. The density of resource biomass decreases as cumulative concentration of toxicants in the environment increases and it may even tend to zero for very high concentration of toxicants and then the species dependent on it may not survive.

5. Numerical Simulations

In this section, we analyse the model (1) – (5) numerically with the help of MAPLE 7.0 to study the behaviour of the system for different values of parameters. For this we assume that,

Now we consider the following set of parameter values,

Equilibrium values of different variable in are obtained as,

Eigen values of the matrix corresponding to are given by,

Since all the eigen values corresponding to are negative and therefore is locally asymptotically stable.

The nonlinear asymptotic stability behaviour of in and plane is shown in the figure 1 and figure 2 respectively. In these figures, it is shown that the trajectories started at any point in the region approaches the equilibrium point showing that the equilibrium is nonlinearly asymptotically stable. In figures 3, 4, 5, the variation of the densities of biomass (B), resource dependent species and the concentration of intermediate toxic product is shown with time at different values of rate of emission of toxicants in the environment i.e. at respectively. From these figures, it is observed that the densities of biomass and species dependent on it decrease as the rate of introduction of toxicants increases and settle down to their respective equilibrium levels, while the concentration of intermediate toxic product increases. In figures 6 and 7, the variation of the densities of biomass (B) and resource dependent species is shown with time at different values of rate of formation of intermediate toxic product i.e. at respectively. From these figures, it can be seen that the densities of biomass and species dependent on it decrease as the rate of formation of intermediate toxic product increases and settle down to their respective equilibrium levels. In figure 8, it is shown that if the growth of resource biomass increases, the density of species dependent on it also increases.

Figure 1. Nonlinear stability in plane

Figure 2. Nonlinear stability in plane

Figure 3. Variation of biomass density B with time for different values of

Figure 4. Variation of density of resource dependent species with time for different values of

Figure 5. Variation of concentration of intermediate toxic product with time for different values of

Figure 6. Variation of biomass density with time for different values of

Figure 7. Variation of density of resource dependent specieswith time for different values of

Figure 8. Variation of density of resource dependent species with time for different values of

6. Conclusions

In this paper, we have proposed a nonlinear mathematical model to study the effect of intermediate 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 toxic product formed inside the resource biomass due to some metabolic reactions when toxicants present in the environment are uptaken by resource biomass. This affected resource biomass by the intermediate toxic 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 biomass and species dependent on it decrease as the rate of introduction of toxicants increases in the environment and settle down to their respective equilibrium levels while the concentration of intermediate toxic product increases. Further, as the rate of formation of intermediate toxic 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 toxicant. If the rate of emission of toxicants in the environment is large enough, then under certain conditions the resource biomass may become extinct due to increased level of intermediate toxic product affecting the growth of biomass 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 growth of resource biomass increases. Thus, if the formation of intermediate toxic product 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.

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,

From which we note that is a saddle point.

Similarly, it can also be checked that equilibrium point is unstable in - direction.

(ii) Consider the following positive definite function about ,

(A1)

where,

Differentiating (A1) with respect to we get

(A2)

Now using the linearized system of the model (1) – (5) corresponding to in (A2), we have,

Now will be negative definite, if the following inequalities are satisfied,

(A3)

(A4)

(A5)

(A6)

(A7)

(A8)

(A9)

Under the assumptions ,

will be negative definite provided the conditions (22) – (25) are satisfied, showing that is a Liapunov function and hence is locally asymptotically stable.

APPENDIX-B

Proof of the Theorem 2.

Consider the following positive definite function about

(B1)

where the constants m_i (i = 0,1,2,3,4) can be chosen appropriately.

On differentiation we get,

Where

Now will be negative definite under the following sufficient conditions,

(B2)

(B3)

(B4)

(B5)

(B6)

(B7)

(B8)

Maximizing LHS and minimizing RHS and choosing,

will be negative definite provided the conditions (26) – (29) are satisfied and hence the theorem.