CHAPTER 2

**EFFECT OF TOXICANT AND PRECURSOR IN A STAGE – STRUCTUREDPOPULATION: A TIME DELAY **

MODELLING STUDY

INTRODUCTION

In the natural world, many species have a life history that takes their individual members through two stages: immature and mature with a time lag. In particular, we have in mind a mammalian population, which exhibits these two distinct stages (Aiello and Freedman, [1990]). From the general knowledge of biology, we know that immature individuals’ viability is more frangible than the mature individuals. Theeffect of environmental influences (like temperature, presence of toxicants, precursors etc.) on the immature individuals and mature individuals are always different due to varied activities of species. In human being, there are several excellent studies (Bruckner, [1981]) exhibiting that children are more sensitive to a toxicant than adults. A population is therefore divided into immature and mature individuals for a fixed period of time.

Many investigators have proposed and analyzed various stage – structured models to study the time – delay effect in population growth (Cui and Song, [2004]; Gourley and Kuang, [2003]; Gui and Ge, [2005]; Gurney et. al., [1986]; Nisbet et. al., [1989]; Wang et. al., [2001]). Aiello and Freedman (1990) on a single species growth model with stage – structure set up the following stage – structured model

where and are the densities of immature and mature population, respectively. represent the birth rate, is the immature death rate, is the mature death rate and it is of logistic nature, is the maturation time. Recently, Agarwal M. and Sapna, [2009] considered the time delay model for the effect of toxicant in a single species growth with stage – structure.

In real situation, however, toxicants are emitted into the environment by various human activities (e.g. Industrial, household, vehicular discharges, etc.) either directly or formed by its precursors which are population density dependent (Shukla et al.,[2009]). A precursor is a substance such as gases, particulate matter etc. that, when revealed into the atmosphere, undergoes chemical reaction or transformation such that the reaction product is a pollutant. But generally, the term ‘precursor’ is used to represent an intermediate product produced by human actions which is converted into a toxicant in the environment, harmful to itself as well as to other species living in the same habitat.

Keeping all the above points in mind, in this chapter,we generalize Agarwal M. and Sapna,[2009] paper on a single species stage-structured model with two life stages: immature and mature with a constant maturation time delay and study the effect of toxicant emitted directly into the environment by natural sources as well as formed by precursor of population. This situation is modelled by the system of four ordinary differential equations. Stability theory of nonlinear differential equation and fourth order Runge - Kutta method are used to analyze and predict the behavior of the model.

**2.1 THE MATHEMATICAL MODEL**

We consider a single species stage – structured model affected by the toxicants emitted directly into the environment from various sources as well as formed by precursors of population and expresses it as the system of following differential equations

(2.1.1)

Here, and arethe densities of immature and mature populations, respectively. is the cumulative concentration of precursors produced by the population forming the toxicant. is the concentration of toxicant in the environment at time.The constant is the rate of emission of toxicant into the environment from various external sources.is the concentration of augmentation of toxicant into the environment by the precursor of population. is the growth rate coefficient of the cumulative concentration of precursors, is its depletion rate coefficient due to natural factors, whereas is the depletion rate coefficient caused by its transformation into the toxicants of concentration .The constant is the natural washout rate coefficient of toxicant present in the environment, and are the depletion rate coefficients of toxicant concentration in the environment due to its uptake by the immature and mature populations, respectively.

Model (2.1.1) is derived under the following assumptions:

(H1):denotes the length of time from birth to maturity. The immature those born at time andthat survive to time exit from the immature stage and enter into the mature stage.

(H2): The birth rate of the immature population at any time decreases as the toxicant increases, where we choose it to be of the form The death rate of immature population increases with an increase in toxicant, where we choose it to be of the formThe term that appears in the first and second equation of model (2.1.1) represents the immature population born at time (i.e. and surviving at the time (with the immature death rate and therefore represents the transformation from immature to mature stage. The term represents the death rate of mature population and increases withan increase in the concentration of toxicant.

(H3): is given initial mature population and is the initial concentration of toxicant.

Now for the continuity of initial conditions, we require

(2.1.2)

the total surviving immature population from the observed births on

Taking from (2.1.2) and assuming that and are continuous (for mathematical reasons) and nonnegative (for biological reasons), then solutions of system (2.1.1) exist and are unique for all

The functions and satisfy following conditions:

for

for (2.1.3)

for

For ecological reasons, we make the following assumptions

and on (2.1.4)

**2.2 POSITIVITY AND BOUNDEDNESS OF SOLUTIONS **

Lemma (2.2.1):If the assumption (2.1.4) holds, then the solutions of system (2.1.1) with given initial conditions are positive for all

Proof:First we show that for all Assume that, there exists Evaluating the second equation of model (2.1.1) at time we obtain

Then but by the definition of this is a contradiction. Hence for all

Now, consider the equation

(2.2.1)

By positivity of we have on Solving equation (2.2.1), and using equation (2.1.2) in it, we get

This implies that

(2.2.2)

If one changes variables in the second interval by one obtains Hence > 0, and since is strictly decreasing, then on so on

So by induction, we can show that, for all

Now from the third equation of model (2.1.1), we have

This implies that,

(2.2.3)

Integration of (2.2.3) gives that

for all

Now, from the fourth equation of model (2.1.1), we have

This implies that,

This again implies that,

(2.2.4)

where is the maximum value of the function

Integration of (2.2.4) gives that

for all

This completes the proof of the lemma (2.2.1).

Lemma (2.2.2):All solutions of system (2.1.1) which initiate in are bounded with ultimate bound

where, and

Proof:We define a function

By positivity of solutions, Then calculating the derivative of along solutions of system (2.1.1), we get

For any positive constant we have

If we take such that then we obtain

(2.2.5)

where is the maximum value of the function

From (2.2.5), we have

this implies that

Moreover, we have

This proves the boundedness of and

From the third equation of model (2.1.1), we have

This implies that,

This proves the boundedness of

From the fourth equation of model (2.1.1), we have

This implies that,

This proves the boundedness of

Hencethe proof of the lemma (2.2.2) is complete.

**2.3BOUNDARY EQUILIBRIA AND STABILITY**

Model (2.1.1) has two nonnegative equilibria:

and

Existence of is obvious.

exists, if the system of equations

(2.3.1)

has a positive solution

Second equation of (2.3.1) gives

say. (2.3.2)

Using (2.3.2) inthe first equation of (2.3.1), we get

say. (2.3.3)

Using (2.3.2) and (2.3.3) in third equation of (2.3.1), we get

say. (2.3.4)

Now, from equations (2.3.2), (2.3.3) and (2.3.4) it is noted that are function of only. To show the existence of we define a function as follows

(2.3.5)

we have,

where

and

Thus there will be a in the interval for which

The sufficient condition for to be uniqueiswhere

>0. (2.3.6)

whereand

Let the equilibrium has variational matrixgiven by

For which three of the eigenvalues are negative and one is positive, giving a saddle point which is stable in the space and unstable in the –direction.

The variational matrix about is given by

where

The characteristic equation corresponding to is given by

(2.3.7)

where

Here, ' denotes the differentiation with respect to

For For the characteristic equation (2.3.7) takes the form

the characteristic equation (2.3.7) takes the form

where

According to Routh–Hurwitz criterion, equilibrium point is locally asymptotically stable in the absence of delay provided the following conditions are satisfied

and

Now, when stability of system can change only if there exists at least one root of equation (2.3.7) such that Let be one such root. Substituting this in equation (2.3.7) and equating real and imaginary parts, we have

(2.3.8)

(2.3.9) Squaring and adding (2.3.8) and (2.3.9), we get

(2.3.10)

Substituting in equation (2.3.10), we get

(2.3.11)

Let where

Then equation (2.3.11), takes the form

(2.3.12)

If we assume that

(i) (2.3.13)

(ii) (2.3.14)

thenequation (2.3.12) has no positive real root.

Thus if conditions(2.3.13) and (2.3.14)are satisfied, then there is no such that is an eigenvalue of the characteristic equation (2.3.7). That is to say that will never be purely imaginary root of equation (2.3.7). Therefore, the real parts of all eigenvalues of (2.3.7) are negative for all If either of any conditions (2.3.13), and (2.3.14) are not satisfied then critical value of time delay for which stability change occur is given by

(2.3.15)

**2.4 NUMERICAL SIMULATION AND DISCUSSION**

To facilitate the interpretation of our analytical findings by numerical simulation, we assume the following particular forms of the functions in model (2.1.1)

Where shows the concentration of toxicant present in the mature population due to which the birth rate of immature population decreases, whereas are toxicant concentration present in mature and immature population due to which their death rate increases.

Now,consider the set of parameter values as

(2.4.1)

For the above set of parameter values, the equilibrium is given by

Here, we note that all conditions for to be locally asymptotically stable are satisfied.Now graphs are plotted for different in order to conclude and confirm some important points.

In Figure(1), the immature and mature populations are plotted against time for different rates of emission of toxicants, From this plot, we can infer that as the rate of emission of toxicants increases, equilibrium densities of both immature and mature population decrease as expected.From this figure, it is also noted that for a given value of initial start as the time passes both immature and mature populations tend to their corresponding value of the equilibrium pointand hence coexist in the form of stable steady state assuring the local stability of This figure also shows that the qualitative behaviors of both immature and mature populations are same whereas the quantitative behaviors of immature and mature populations are different as the concentration of toxicants in the environment increases. Comparing the behavior of immature population and mature population from this figure, it is noted that the equilibrium density of immature population is more prone to toxicant effect than the mature population which is biologically reasonable. However, if the population is unstructured then we cannot see this distinct quantitative change in the equilibrium densities of immature and mature population. **Such distinct dynamical outcomes highlight the importance of the structured population generalization.**

Figure 1, Graph of and versus for different values of and

othervalues of parameters are same as in equation (2.1.1).

From the existence and stability criteria are recognized to be the important parameters. Therefore, we outline the effects of these parameters on the system dynamics in figures(2)–(9), respectively. In figures (2) – (5), the variations of immature and mature populations are shown for differentrespectively. It is concluded that with the increase of these parameters, equilibrium densities of both immature and mature populations decrease. From these figures, it is also noted that theimmature populations is more prone to increase in parametersthan mature population.Here it is also observed that the equilibrium densities of immature and mature population are more sensitive to the parameter in comparison to

Figure2, Graph of versus for different values of and other

values ofparameters are same as in equation (2.1.1).

Figure 3, Graph of versus for different values of and other

values ofparameters are same as in equation (2.1.1).

Figure 4, Graph of versus for different values of and other

values ofparameters are same as in equation (2.1.1).

Figure5, Graph of and versus for different values of andother

values ofparameters are same as in equation (2.1.1).

In figures (6) and(7), the immature and mature populations are plotted against time for different From these figures, we can infer that the equilibrium densities of both immature and mature population decrease with increase in the value of in the interval [0,6.62) and then it become extinct at Thus, when the maturation time the positive steady state disappears and population dies out. This shows the sensitivity of the model dynamics on the maturation time delay.

Figure6, Graph of versus for different values of and other

values of parameters are same as in equation (2.1.1).

Figure 7, Graph of versus for different values of and other

values of parameters are same as in equation (2.1.1).

The effect of parameter on immature and mature population is shown in figures(8) and (9). From these figures, we conclude that with increase in coefficient of augmentation of toxicant by precursor of populationthe endemic level ofboth immature and mature populationdecreases. It is due to the reason that as the rate of transformation of precursor into toxicant increases, the endemic level of concentration of toxicant increases into the environment, due to which the equilibrium density of both immature and mature population decreases.From these figures, it can be seen that the qualitative behavior of both immature and mature population are same whereas the quantitative behavior of both population are different for the increase in parameter Here, we observed that immature population is more prone to increase in the density of toxicant by precursorthan mature population. The variation of immature and mature population in the presence and absence of precursor is plotted in figure(10). From this plot we can infer that presence of precursors decreases the endemic level of both populations.

Figure8, Graph of versus for different values of and other

values ofparameters are same as in equation (2.1.1).

Figure9, Graph of versus for different values of and other

values ofparameters are same as in equation (2.1.1).

Figure 10, Graph of versus for presence and absence of Precursor and

other values ofparameters are same as in equation (2.1.1).

Moreover, the variation of equilibrium values with toxicants and precursors is also shown explicitly in the following table.

Equilibrium Values / Absence of ToxicantPrecursors both / Presenceof

Toxicant only / Presence of Toxicants & Precursors both

/ 1.7980 / 1.1236 / 1.0275

/ 0.1832 / 0.1250 / 0.1162

From above table, we note that when immature and mature population are affected by the toxicants but no precursors are formed, the equilibrium values of population are greater than their values when precursors are formed for the same set of parameters. This implies that, while assessing the adverse effect of toxicants, one must be careful to assess not only direct emitted toxicants but also produced by precursor of population.

Figure 11, Graph of versus for different initial starts for the

set of parameter values given in equation (2.1.1).

In figure(11), the variation of the immature population with the mature population is plotted for different initial starts. It is depicted from the graph that solutions converge to the equilibrium point for different initial starts, indicating the global stability of

2.5 CONCLUSION

In this chapter, a nonlinear mathematical model for the effect of toxicant emitted into the environment from external sources as well as formed by precursor of populationon a stage – structured population with two life stages, immature and mature is proposed and analyzed. It is further assumed that the time to reach maturity i.e. maturation time is constant.

An equilibrium analysis is presented and is found, analytically as well as graphically that,the nontrivial equilibrium is locally as well as globally asymptotically stable under certain conditions. It has been concluded from the analysis that as the cumulative rate of emission of the toxicant from external sources as well as its formation from precursors of population increases, the equilibrium density of both immature and mature populations decreases. From the analysis, it can be seen that the viability of immature individuals is more prone to toxicant effect (emitted directly by natural sources as well as its formation by precursor of population) than mature individuals and hence supports the biological fact that children are more sensitive to toxicant than adults. Also, it is noted that in thepresence of toxicant, the death rate of immature individuals is more sensitive in comparison to growth rate anddeath rate of immature and mature individuals, respectively. So, we need to control the emission of toxicantwhose concentration increases more into the environment due to presence of precursor,for the survival of population

It is also obtained that if the maturation time if immature population to mature population increases, then densities of both population decreases. Also, it is predicted that after certain increment, steady state appears and population dies out. Again, it is observed that when both immature and mature population are affected by a toxicant but no precursor is formed, the equilibrium level of immature and mature population are greater, than their values when precursor is formed by the population.

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