126 Kalyan Das and Nurul Huda Gazi

exposure. International standards for drinking water have been placed by organizations such as the World Health Organization (WHO), however local conditions determine the nature of the standards that are to be legislated by different countries, and thus fluoride limits in drinking water standards may differ from one country to another. This paper investigates the potential health risks involved with both lower and higher concentrations of fluoride in drinking water, as well as posing possible measures of mitigation to eliminate such harmful threats. It also provides a survey of fluoride content in several bottled water samples around the World.

The quality of groundwater is very important in evaluating its utility in various fields such as domestic public water supply and agriculture. The assessment of the dissolved constituents thus become very important for safe drinking water. Some ions dissolved in water and present in appropriate concentration are essential for human beings while higher concentration results in toxicity. Fluorine is the most electronegative of all elements and is physiologically more active than any other ion. Fluorine is never found free in nature. Fluorine in drinking water is totally in an ionic form and hence it rapidly, totally and passively passes through the intestinal mucosa and interferes with major metabolic activities of the living system. Fluorine is often called a two-edge sword. The occurrence of fluoride in ground water has attracted attention globally, since it has considerable impact on human health. The major sources of fluoride in groundwater are fluoride-bearing rocks such as fluorspar, cryolite, fluorapatite and hydroxylapatite [1]. The fluoride content in the groundwater is a function of many factors such as availability and solubility of fluoride minerals, velocity of flowing water, temperature, pH, concentration of calcium and bicarbonate ions in water, etc. Excess fluoride affects plants and animals also. The permissible limit for fluoride in drinking water is 1.5 mg/L (WHO, 1984).

fluoride exposure are dental fluorosis, teeth mottling, skeletal fluorosis and deformation of bones in children as well as in adults. An estimated 62 million people in India residing in 17 states are affected with fluorosis. According to a recent report, 3555 habitations have been identified as fluoride affected settlements spreading over the Vellore, Dharmapuri, Trichy, Karur, Salem, Namakkal, Erode, Coimbatore and Virudhunagar Districts of Tamil Nadu, India. Dental fluorosis occurs during tooth formation and becomes apparent upon eruption of the teeth. It ranges from very mild symmetrical whitish areas on the teeth to pitting of the enamel, frequently associated with brownish discoloration. Skeletal fluorosis, in turn, is caused by a complex dose-related action of fluoride on bone. Fluoride increases bone formation and simultaneously decreases bone resorption. As a result of these effects, bone changes observed in people heavily exposed to fluoride ranges from osteosclerosis to exostosis formation of varying degrees. Fluoride exists fairly abundantly in the earth’s crust and can enter groundwater by natural processes; the soil at the foot of mountains is particularly likely to be high in fluoride from the weathering and leaching of bedrock with a high fluoride content.

According to 1984 guidelines published by the World Health Organization (WHO), fluoride is an effective agent for preventing dental caries if taken in ‘optimal’ amounts. But a single ‘optimal’ level for daily intake cannot be agreed because the nutritional status of individuals, which varies greatly, influences the rate at which fluoride is absorbed by the body. A diet poor in calcium, for example, increases the body’s retention of fluoride. Water is a ma jor source of fluoride intake. The 1984 WHO guidelines suggested that in areas with a warm climate, the optimal fluoride concentration in drinking water should remain below 1 mg/litre (1ppm or part per million), while in cooler climates it could go up to 1.2 mg/litre. The differentiation derives from the fact that we perspire more in hot weather and consequently drink more water. The guideline value (permissible upper limit) for fluoride in

The ma jor health problems caused by
Нелинейные явления в сложных системах Т. 17, No 2, 2014

Role of Beneficial Bacteria on Contaminated Fluoride Drinking Water 127

drinking water was set at 1.5 mg/litre, considered a threshold where the benefit of resistance to tooth decay did not yet shade into a significant risk of dental fluorosis. The WHO guideline value for fluoride in water is not universal: India, for example, lowered its permissible upper limit from 1.5 ppm to 1.0 ppm in 1998. In many countries, fluoride is purposely added to the water supply, toothpaste and sometimes other products to promote dental health. It should be noted that fluoride is also found in some foodstuffs and in the air (mostly from production of phosphate fertilizers or burning of fluoride- containing fuels), so the amount of fluoride people actually ingest may be higher than assumed. It has long been known that excessive fluoride intake carries serious toxic effects. But scientists are now debating whether fluoride confers any benefit at all.

2. The mathematical model

A simplified model is constructed for the effect of bacterial consumption of fluoride in the culture medium. Two dimensional differential equations are used for fluoride and bacterial dynamics. Michaelis-Menton kinetics is followed in the bacterial consumption of fluoride. The two dimensional non-linear model equations are as follows.

(2.1). To ensure the existence and uniqueness of

solutions of the model system (2.1), we seek the

solution in R+2 = {(x,y):x≥0,y≥0} so that

all the standard results on existence, uniqueness

and continuous dependence on initial conditions

are applicable since the growth rate functions

2
are sufficiently smooth in R+. The solutions

with the given initial conditions are non-negative with positive time follows from the integral formulation of (2.1) [6]. The system describing the evolution of X(t) = [x(t), y(t)] , is said to be bounded if all the solution trajectories of (2.1) are uniformly asymptotically bounded for t ≥ 0. In other words, there exists a constant M such that limt→∝sup∥X∥ ≤ M. The results regarding boundedness of solutions are given in the following lemma.

Lemma 1 All the solutions of equations (2.1) with non-negative initial conditions are positive.

Proof The proof is obvious. ?

Lemma 2 All the solutions of equations (2.1) that are initiated in R+2 one eventually uniformly bounded in the region A ⊂ R+2 , where A = {(x,y)∈R+2 :0≤x≤a,0≤x+y≤a}and hence the system (2.1) are dissipative.

Proof Dissipation of a ecological system means the breaking up and scattering of the components and move away from each other by dispersion, but from mathematical point of view a dynamical system in which the phase space volume contracts along the trajectory and this means that the generalized divergence is less than 0.

To prove the Lemma 2, we use the standard comparison theorem. We shall first show that

2 [x(t),y(t)] → A, for all (x0,y0) ∈ R+.

From the first equation of (2.1), it follows that dx ≤ a.

dt

̇ So,x(t)≤afort≥0.Similarly,S+θS(t)<a,

where θ = Min[1,c(1−d)], S(t) = x(t)+y(t). So,S(t)≤agivesx(t)+y(t)≤aforallt≥0. Hence,x(t)+y(t)isboundedandsoy(t)is, and one can write y(t) ≤ a. Also, the set A is positively invariant by Lemma 1.

dx cxy =a− ,

dt x+k

dcxy
withx(0)=x0 >0,y(0)=y0 >0andd<1,

where x = fluoride pool; a = initial amount of fluoride; c = fluoride consumption rate of bacteria; k = Half saturation constant of bacteria on fluoride; y = bacteria pool; d = growth rate of bacteria; μ = mortality rate of bacteria.

Analytical as well as numerical solutions of the above equation are to be solved to know the dynamics of the fluoride pool and bacteria. Now, we establish some basic properties of equations

dy
dt x+k

(2.1)

=

− μy,

Nonlinear Phenomena in Complex Systems Vol. 17, no. 2, 2014

128 Kalyan Das and Nurul Huda Gazi

So, for all (x0, y0) ∈ R+2 , [x (t) , y (t)] approaches the set A as t → ∞. Therefore, the system (2.1) is uniformly bounded. This completes the proof of the Lemma. ?

From the above Lemma, it is ensured that there exist Ω (x0, y0) ∈ A where Ω is the ω-limit

2 set of the orbits initiated in R+.

A stationary point or equilibrium point is

one which has associated with it a vector of zero

length. That is, if a system is at a stationary

point at one instant in time, it will remain at that

point at the next instant. The trivial equilibrium

E0 (0, 0) exist. The interior equilibrium E∗ (x∗ , y∗ )

of the system (2.1) are obtained by setting dx = dt

dy = 0 in the positive quadrant of the x−y plane, dt

where x∗ = kμ/(cd − μ), y∗ = ad/μ. Clearly, E∗ (x∗, y∗) is feasible provided μ < cd.

Let us consider a dynamical system dx = n dt

f (x) where x ∈ s ⊂ R . The steady state of the above dynamical system are those value(s) of x such that f(x) = 0. Steady state is usually denoted by x ̄. A steady state x ̄ is said to be locally stable if for |x ̄ − x0| < δ implies |x ̄ − x (t)| < ε, where t, δ are small (positive) numbers and δ depends on ε and x0 is initial (or, perturbed) position. If, however, |x ̄ − x0| < δ ⇒ limt→∝x (t) = x ̄, then x ̄ is said to be locally asymptotically stable. If for any initial point in the phase space, the orbit x (t) approaches to x ̄, then x ̄ is said to be globally asymptotically stable.

In order to investigate the local asymptotic stability of the steady state, we linearize the system (2.1) about E∗. Let u(t) and v(t) be the respective linearized variables of the system (2.1). Then, we have

To determine the nature of stability of the system (2.2), we require the sign of the real parts of the roots of equation (2.3). It follows that E0 (0, 0) is locally stable in x − y directions. Next, the steady state E∗ (x∗ , y∗ ) is locally asymptotically stable because P > 0, Q > 0. Therefore, we may conclude that the existence of an interior equilibrium implies the local asymptotic stability of the system (2.2) and we have the sufficient condition for local asymptotic stability of the steady state in the following theorem.

In the next result we will show that the conditions for the existence of E∗ implies the global asymptotic stability of the system (2.2).

Theorem 2.1 If μ > cd then the interior equilibrium point E∗ (x∗ , y∗ ) of the system (2.1) is

2 locally and globally asymptotically stable in R+.

Proof Let us define, h(x,y) = y1, clearly, h > 0 ify>0.

du dt

= −

kcy∗ (x∗ + k)2

u (t) − dv

cx∗

v (t) ,

In this section, we consider the global asymptotic stability of the system (2.1) by constructing a suitable Lyapunov function. We define a Lyapunov function

Since E∗ (x∗, y∗) is

locally

globally

cxy x+k

cdxy x+k

Now, f1 (x,y) = a−
Therefore,∇(x,y)= ∂ (f1h)+ ∂ (f2h)= ∂ (a−

, f2 (x,y) = ∂x ∂y

bx − cx ) + ∂ ( cdx − μ) = − ck < 0. y x+k ∂y x+k (x+k)2

−μy. ∂x y

asymptotically stable and the system (2.1) is

dissipative from the lemma, then from Bendixon-

Dulac criterion [15], we may conclude that

E∗ (x∗, y∗) is locally and globally asymptotically

2
stable in R+. This completes the proof of the

theorem.

3.

?

Global stability

The characteristic equation associated with system (2.2) is given by

V(x,y)=x−x∗−x∗ln x∗

whereP=

kc2 dx
∗3 >0.

(x∗ + k)

kcdy∗
dt = (x +k)2u(t). (2.2)

∗ (x)

D(ξ)=ξ2+Pξ+Q=0, (2.3)

kcy∗ (x∗ +k)

y

∗

3 >0,Q=
Нелинейные явления в сложных системах Т. 17, No 2, 2014

[

( y )]

(3.1)

y−y∗ −y∗ln
in subsequent steps. It can be easily verified that

+l1
where l1 is a suitable constant to be determined

(x∗ +k)

=

= −

−

x x x+k

+ l1(y − y∗)

x+k

− μy

Role of Beneficial Bacteria on Contaminated Fluoride Drinking Water 129

the function V is zero at the equilibrium point E∗(x∗,y∗) and is positive for all other positive values of x, y.

The time derivative of V along with the solutions of (2.1) is

with g(x) = a, p(x) = cx , h(x) = x+k

cdx , where x(t) and y(t) are the densities of x+k

fluoride and bacteria in contaminated drinking water respectively; g(x) is the growth rate of fluoride the in absence of bacteria; p(x) and h(x) represent respectively the functional and numerical response of the bacteria.

The specific standard assumptions on g(x), p(x) and h(x) are:

(A1)g(x):R+ →RisofclassC1,g(0)=afor allx∈R+
(A2)p:R+ →R+ isofclassC1,p(0)=0and p′(x)>0forallx∈R+

(A3)h:R+ →R+ isofclassC1,h(0)=0and h′(x)>0forallx∈R+
We also need the following more general assumptions:

dV x − x∗ dx y − y∗ dy dt= x dt+l1 y dt

x−x[acy] [dcx] ∗

c(x − x∗)
(x + k)(x∗ + k)

[x∗(y−y∗)+k(y−y∗)−l1dk]

−

(x−x∗) .

kcy∗
x(x + k)(x∗ + k)

2

Therefore,

(A4) There exists E∗ (x∗ , y∗ ) such that h(x∗ ) = μ and g(x∗)−y∗p(x∗) = 0 with 0 < x∗ < ∞ and 0 < y∗ < ∞

dV c

[x∗(x−x∗)2+x∗(y−y∗)2

[g(x)]′
(A5) p(x∗) p(x) |x=x∗ > 0.

dt

≤ −
+k(x − x∗)2 + k(y − y∗)2 − 2l1(x − x∗)dk],

2(x + k)(x∗ + k)

The above conditions (A1)-(A5) guarantee that has at least one critical point E∗(x∗,y∗) inside the population quadrant and that it is a repeller. We shall assume that E is unique.

For local structural stability analysis of the equilibrium point, we refer to [4, 7, 16]. In order to obtain a sufficient criterion for uniqueness of limit cycles, we define:

dV dk i.e., dt <0ifx<x∗ andtakingl1 = 2.

This shows that dV (< 0) is negative definite dt

and hence by Lyapunov’s theorem on stability, it follows that the positive equilibrium E∗(x∗,y∗) is globally asymptotically stable provided x < x∗ which implies x < μk/(cd − μ).

Thus, we find that the feasible region in which the interior equilibrium point E∗(x∗,y∗) is globally asymptotically stable, is left side of the line x = M, in the first quadrant of the xy phase- plane, where M = μk/(cd − μ).

4. Existence and uniqueness of limit cycle

In the present study we consider model system (2.1) which can be put in the following form

G1(x)ψ(x) G0 (x)

, (4.2)

dx = g(x) − yp(x), dt

dy = yh(x) − μy, (4.1) dt

ψ(0) > 0, G0(x) = −g(x)[h(x) − μ], G1(x) = []′ ′

g(x) p (x)
[h(x) − μ] + p(x) p(x) and G2(x) = p(x) . Our

main result can be stated as follows:

Theorem 4.1 Consider the system (4.1) and let (A1)–(A5) hold. Further we assume that
(A6) N(x)<0 for 0≤x<x∗
(A7) M(x) is non-increasing for 0 ≤ x < x∗ Then system has at most one limit cycle around

Nonlinear Phenomena in Complex Systems Vol. 17, no. 2, 2014

where

M(x) = −
N(x) = G1(x)ψ(x) + 2G0(x),

ψ2(x)
ψ(x)ψ′ (x) = Σ2j=1Gj (x)ψ′ (x), ψ(x)

(4.3) ̸= 0,

130 Kalyan Das and Nurul Huda Gazi

the unstable equilibrium point E∗ (x∗ , y∗ ) in R+2 \{E∗} and if it exists it is stable.

The following result can be viewed as a consequence of Theorem 4.1.

Theorem 4.2 Consider the system and let (A1)-

(A5) hold. Further assume that

Let μ = γ ̄ − 1, γ ̄ is a constant quantity and satisfy the hypothesis of Hopf–Theorem. Then, by the Jacobian matrix of the linearized system evaluated at E∗(x∗,y∗) can be written as

()

0

J(μ) =
We assume that the determinant of J(μ) is

det J(μ) = (γ ̄ − μ)g(x∗)h′ (x∗)

which is positive for all μ while Trace J(μ) = 0. For μ < 0, the equilibrium point E = (x∗,y∗) is stable and it is unstable for μ > 0. The eigenvalues λ1,2(μ) of J(μ) are complex for |μ| small enough, with Reλ1,2 (μ) = 12 Trace J (μ) satisfying the following conditions:

(A8)Q(x) = 1 + 1 p(x) g(x) g(x) h(x)−μ

[ ]′
g(x) p(x)

g(x∗)h′ (x∗) (γ ̄−μ) p(x∗)

−p(x∗ )
.

0

isnon- increasing for 0 ≤ x < x∗. Then system has at most one limit cycle around the unstable equilibrium point E∗(x∗, y∗) in R+2 \{E∗} and if

it exists it is stable.

We can combine Theorem 4.2 and Poincare ́– Bendixon Theorem to get the following ([8]):

Theorem 4.3 Suppose in the system [ ]′

(A9) g(x) >0
p(x) x=x∗ [ g(x) ]′

(A10) K (x) = p(x) p(x) h(x)−μ

are non-increasing functions on 0 ≤ x < x∗. Then the system has exactly one limit cycle which is globally asymptotically stable with respect to the set R+2 \{E∗}.

Remark 4.1. According to the above results, the following problem then becomes important to consider. Consider the generalized Gause system

1′

Reλ1,2(0) = 2Trace J(0), and Imλ1,2(0) =

[Reλ1,2(μ)] |μ=0 > 0

√

dx = g(x) − yp(x), dt

dy = γy[−q(y) + h(x)(]4.4) dt

The condition of the Hopf–Andronov bifurcation Theorem are satisfied and so a limit cycle exists for |μ| small enough, μ > 0.

Remark 4.2. The existence of a limit cycle follows also from the Poincare–Bendixon Theorem and the size of the region can also be readily determined by the Poincare–Bendixon approach.

4.2. Uniqueness of limit cycles

In this section we want to give an algebraic proof that the Gause system has all most one limit cycle. For this we transform (4.1) in Lienard’s equation and derive Theorem 4.1 and Theorem 4.2 respectively.

Lemma 4.1 A system of the form (4.1) can be transformed into the form of Lienard’s equation

′′ ′

where q(y) is a polynomial of degree n ≥ 3 with the properties that q(0) > 0 and q′ (0) > 0 for all y > 0. Derive criteria under which limit cycle are unique. For existence, uniqueness of limit cycle in Gauss Predator-prey system, we refer to [2] etc.

4.1. Existence of limit cycles

In this section we shall look for the Hopf– Andronov bifurcation point which occurs if the eigenvalues of the linearized system of the equilibrium point E∗(x∗, y∗) are purely imaginary.

X + f(X)X + q(X) = 0. (4.5) Нелинейные явления в сложных системах Т. 17, No 2, 2014

detJ(0) ̸= 0.

Role of Beneficial Bacteria on Contaminated Fluoride Drinking Water 131

Or, into the equivalent system

dX dV
dτ =−V−F(X), dτ =q(X),

Finally, the substitution ds = φ(X ) = [∫X ] dτ

exp 0 H2(ξ)dξ transform (4.10) into the Lienard’s equation (4.5), where

f(X) = H1(X)φ(X), q(X) = H0(X)φ2(X). (4.12)

F(X) = by a change of variables.

f(ξ)dξ

Proof First, we introduce the co-ordinates X = x−x∗, Y = y−y∗ in order to put the equilibrium point E∗(x∗, y∗) in the origin. Now equation (4.1) transforms into

dX = (X +x∗)g(X +x∗)−(Y +y∗)p(X +x∗), dt

dY
dt = (Y +y∗)[h(X +x∗)−μ].

(4.7)

It is not hard to see that if we substitute the regular mapping

X = X, Z = (X+x∗)g(X+x∗)−(Y +y∗)p(X+x∗) into (4.7) and develop in powers of Z, it follows

that

dXdZ 2 dt = Z, dt = G0(X) + G1(X)Z + G2(X)Z

(4.8) where the functions G0, G1 and G2 are defined in the previous section. Now we introduce the third

Thus the proof of the lemma is complete. ? Remark 4.3. As is readily seen, the mapping X = X, Z = Xg(X)−Yp(X) is regular:

change of variables dt = 1+U > 0, where ds ψ(X)

ψ(X)ψ′(X) = Σ2j=0Gj(X)ψ′(X), ψ(X)̸=0, ψ(0)>0,

which when substituted into (4.8) gives

(4.9)

(A12) F(0) = 0, f(0) < 0, where f(X) = F (X)

(A13) All limit cycles are contained in the region

a<X<b,−∞<V <∞wherea<0<band [ ]′

f(X) > 0 where X increases in a < X < b and q(X)

0<X<b.
From conditions (A11) and (A12) we know

that the origin is the only equilibrium point of the system (4.6), which is unstable; from condition (A13) we know that the system (4.6) can not have more than one limit cycle; if one exists, it must be a stable cycle. The condition that the integral of q tends to infinity if X does, may be changed to the curve V + F (X ) = 0 is defined for all X ∈ (−∞, ∞) (see [8, 17]).

First, we observe that (A11) is equivalent

>0, x̸=x ∗

(4.13)

dX dU
ds =U, ds =−H0(X)−H1(X)U−H2(X)U .

Here

2

(4.10)

(4.11)

∫ X 0

(4.6)

G0(X) H0(X) = − ,

(10)
det ′ ′ =−p(X)̸=

[Xg(X)] − Y p (X) −p(X) 0 and one-to-one.

[X(X,Y ),Z(X,Y )] = [X(X∗,Y ∗),Z(X∗,Y ∗)] implies(X,Y)=(X∗,Y∗).

Thus, an additional hypothesis is that t(s, U ) and Z(s,U) are one-one and bi-continuous.

Proof of Theorem 4.1. From Lemma 1, the

system (4.1) can be transformed into Lienard’s

form (4.6) (same as (4.5)) for which we have a

uniqueness result of the limit cycles due to [17].

Thus, from Zhang’s Theorem, we need only to

verify that the following three conditions hold for

the system (4.6), when f and q are as in (4.12)

(A11)Xq(X) > 0forX ̸= 0;G(−∞) =

G(+∞) = ∞, where G(X) = ∫ X q(ξ)dξ 0′

ψ2(X) G1(X)ψ(X) + 3G0(X)

,

(x − x∗)G0(x) ∗ 0 ψ2(x)

H1(X) = −
H2(X) = − .

(x−x )H (x)=−
where G0(x) = −g(x)[h(x) − μ].

ψ2(X) G1(X)ψ(X) + 2G0(X) ψ2(X)

Nonlinear Phenomena in Complex Systems Vol. 17, no. 2, 2014

132 Kalyan Das and Nurul Huda Gazi

By the usual isocline analysis, we have, G0(x) > 0 for 0 < x < x∗ which implies (4.13) and respectively (A11). We now observe that (A12) is equivalent to

average value. So, the equilibrium population distribution fluctuate randomly about their average value. The governing equation of the system with environmental fluctuation is stochastic differential equations. It is not possible to find a time independent equilibrium point of such stochastic system. There is a continuous spectrum of disturbances generated by the environmental fluctuation and the system is in tension between two countervailing tendencies. The random fluctuation are responsible to spread the population cloud, on the other hand the dynamics of stabilizing population interactions tend to restore the population to their mean value, in order to compact the cloud. This type of compact cloud population distribution is called stochastically stable system. There are two types of stochasticity: the demographic stochasticity and the environmental stochasticity in population biology.

To study the stochastic fluctuation of the environment for the model system as described in (2.1), we incorporate stochastic perturbation term to the deterministic model system (2.1). We assume that stochastic perturbations are of white noise type which are directly proportional to the distances x(t), y(t) from the equilibrium point E∗ whose coordinates are x∗, y∗ influence on x ̇(t), ẏ(t) respectively. By this way the system (3.1) takes the following form

(4.14)

p(x)
(A5) and the fact that ψ(0) > 0, we obtain (4.14) and consequently (A12). Finally, using the assumption (A6) that

[ ]′ 1

H1(0) < 0 where φ(0)H1 (0) =

=

−G1 (0)
−p(x∗) g(x) |x=x∗. Obviously, by virtue of

[ ]′

= N(x) > 0 φ(x)

φ(x)
for0<x<x∗ .Weobservethat(A13)is

equivalent to

[H1(x)]′ [G1(x)ψ(x)]′
H0(x) = G0(x) >0 (4.15)

for 0 < x < x∗.
Now, since G1 (x)φ(x) = −M (x) and by (A7)

H0 (x)
M (x) is non-increasing for 0 < x < x∗ , it follows

that (4.15) holds and respectively (A13). Thus, conditions (A11)–(A13) of Zhang’s Theorem are satisfied and Theorem 4.1 is proved. ?

So, the above theorems ensure the existence and the uniqueness of limit cycle which is stable.

5. The model under random noise of the environment

We want to study the effect of environmental fluctuation on the model system and stochastic stability of the endemic equilibrium point E∗. The natural phenomena do not follow the deterministic law rather oscillate randomly about some average value. Thus the deterministic equilibrium point is not a fixed state. Due to environmental fluctuation, the birth rate, carrying capacity, competition coefficients, inter- specific interaction coefficients, death rates and the other parameters involved with the system exhibit random fluctuation around their

[ cxy]
dx(t) = a − dt + σ1(x − x∗)dξt1,

Нелинейные явления в сложных системах Т. 17, No 2, 2014

dy(t) = y Here σ1 ,

[ x+k] −μ + dcx

dt + σ2(y − y∗)dξt2(.5.1)

x+k σ2 are real

positive constants, ξti (i = 1,2) are the standard Wiener process [6] independent of each other. The model system (5.1) of the stochastic differential equation is said to have multiplicative random white noise. We take the model system as Ito stochastic differential system. We wonder whether the dynamical behaviour of the model is robust with environmental fluctuation by investigating asymptotic stability behaviour of the equilibrium point E∗ for the system (5.1). The above

where

xy , g(t,X )

Xt = ( cxy )

[a −

Role of Beneficial Bacteria on Contaminated Fluoride Drinking Water

133

stochastic differential model may be put into the form

(σ1u1 0) 0 σ2u2

̄

(ξt1) 2 .

dXt = f(t, Xt) + g(t, Xt)dξt ()

equivalent to the stability of system (5.3) around

the origin (0, 0). Now we define the mean square

stability of system (5.3) around (0,0). We state

here the theorem of means square stability in

probability for the stochastic differential system

∂V T ∂V (t, U) LV(t,V)= ∂t +f (U(t)) ∂U

]
( x+k ) ()

y −μ+ dcx

x+k

,

t σ1(x−x∗) 0 ̄ ξt1

0 σ2(y − y∗) and ξt = ξt2

.

(t≥t0×R,t0∈R andV ∈C2(R)isa function such that V (t, U ) has twice continuous derivatives with respect to U and one bounded derivative with respect to t for almost all t ≥ 0. For function from R the generating operator L of equation (5.3) is defined as

The solution of system (5.2) with some

initial condition is an Ito process [6]. The vector

function f(t,Xt) is defined as drift coefficient

which varies slowly and the components are

continuous. g(t, Xt) is the rapidly varying ̄

continuous random component. ξt is a multi- dimensional stochastic Wiener process and defined as diffusion matrix. g(t,Xt) depends on the solution Xt of the stochastic differential equation (5.1) and it has effect on the dynamics of the model system. The diagonal form of g(t,Xt) indicates that the system is effected by diagonal noise.

1[ ∂2V ] +2Tr gT(t,U)∂U2g(t,U)

̄

f (t, Xt )

(5.2)

= =

g(U(t)) =
The stability of system (5.2) around E∗ is

(5.3). Let R be the set defined as R = [ m T]

and ξt =

ξt

5.1. Effect of randomness on the equilibrium state

We have a model system (5.2) whose governing equations are stochastic differential. So we are in a position to find the stochastic stability of the model system (5.2) in mean square sense around E∗ (x∗ , y∗ ). We note that E∗ is also an equilibrium point of the model system (5.2). It is very difficult to find the stochastic stability criteria of the nonlinear system (5.2). So we linearize the system around E∗ with the following transformationu1 =x−x∗,u2 =y−y∗.Thuswe have the linearized system

̄

V(t,U)= 1[wu2 +(u+v)2] u2 2

( kcy∗ −

u (t) −

cx∗ (x∗ +k)

)

v (t)

(5.3) Here U(t) = u1 , f(U(t)) =

dU (t) = f (U (t)dt + g(U (t))dξ(t). ()

(5.4) where =col(,),2 =

∂V
()∂U ∂u1∂u2 ∂U

∂2V ,i,j = 1,2. ∂ui∂uj 2×2

Theorem: Suppose there exist a function V (t, U ) ∈ C20 (R) satisfying the inequality

where ki, i = 1,2,3, p > 0 is some suitable constant. Then the trial solution of (5.3) is exponentially p-stable with t ≥ 0. If p = 2, then the trivial solution of (5.3) is exponentially mean square stable. Furthermore, the solution is globally stable in probability.

We will use the above theory to the system (5.3) to study the stability. So, our problem remains in constructing a Lyapunov function V(t,U) allowing to obtain mean square stability conditions in terms of system parameters. Let us choose a function

pp

k1|U| ≤V(t,U)≤k2|U| , LV (t, U ) ≤ −k3 |U |p ,

∂V∂V ∂2V

(5.5)

(x∗ +k)2
(x∗ +k)2

, with V(t,0) = 0, w > 0 is constant. We remark here that the Lyapunov function may be

kcdy∗

u (t)
Nonlinear Phenomena in Complex Systems Vol. 17, no. 2, 2014

134 Kalyan Das and Nurul Huda Gazi

constructed in several ways. So, the Lyapunov function we have constructed here is not unique.

Theorem 5.1 If
2 4kcy∗(w∗ − d) 2 2kcdy∗

are satisfied then the equilibrium point E∗ is mean square stable.

Proof To prove the theorem we calculate the terms of (5.4) with Lyapunov function as in (5.5). Here

σ1 < (x + k)2 w , σ2 < (x + k)2 ∗∗∗

∂V ∂t

+

cx kcy kcdy ]

[

= −(1+w)

kcy kcdy ] [ ∗∗2∗∗∗

fT(U(t))col

= −w

∗

+

∗ + (x∗ +k)

(x∗ +k)2

(x∗ +k)2

u + −(1+w)

− + uv (x∗ +k)2 (x∗ +k)2

(∂V) [

kcy (x∗ +k)2

kcdy ] ∗

cx

kcdy ]
∗ uv,

cx∗ 2 −v,

(x∗ +k)

[

u2+ −w

(x∗ + k)

∂U

(x∗ +k)2

(x∗ +k)2
Thus assembling the above expressions in

expression (5.4) and arranging, we get

and

1 [ ∂2V
2Tr gT(t,U)∂U2g(t,U) = 2[wσ12u2 +σ2v2].

[

− (2w+1)

]

1

LV(t,V)=−

2

cx kcy kcdy ]

1 ]
uv− − σ2 v <−

[2kcy(w−d)

[2kcy∗(w − d) (x∗ + k)2

1 ]
− wσ12 u2−

kcy∗ (x∗ + k)2

u2 ∗∗∗∗22∗

[ cx
(x∗ +k) 2 (x∗ +k)2

(x∗ +k)

+

(x∗ +k)2

−2

(x∗ +k)2

1 ] [ cx kcy kcdy ] [ cx 1 ] 22∗∗∗∗22

− wσ1 u − (2w+1) + 2 (x∗ +k)

(x∗ +k)2

−2 uv− (x∗ +k)2

(x∗ +k)

− σ2 v . 2

Let us now choose w = w∗ = where

2kcdy∗ −kcy∗ −cx∗ (x∗ +k) cx∗ (x∗ +k)

x∗(x∗ + k) then

provided

(2d

−

1)ky∗ >

 

 

2kcy∗ (w∗ −d) (x∗+k)2

0

1 2

P =

−wσ 0 2 1

kcdy∗ (x∗+k)2

− 1σ2 2 2

.

LV (U (t), t) ≤ −U T P U
Нелинейные явления в сложных системах Т. 17, No 2, 2014

(5.6)

(5.7) If all the principal minors of P are positive