ISSN 1749-3889 (print), 1749-3897 (online)
International Journal of Nonlinear Science
Vol.14(2012) No.2,pp.131-141
Role of Vapor and Cloud Droplets on the Removal of Primary Pollutants Forming
Secondary Species from the Atmosphere: A Modeling Study
Shyam Sundar1 , Ram Naresh2 ∗
1
Department of Mathematics, P.S. Institute of Technology, Bhaunti, Kanpur-208020, India
2
Department of Mathematics, H. B. Technological Institute, Kanpur-208002, India
(Received 8 February 2012, accepted 29 March 2012)
Abstract: In this paper, a nonlinear mathematical model is proposed to study the removal of primary gaseous
pollutants forming secondary species, from the atmosphere by precipitation due to rain. In the modeling process, it is assumed that the growth of raindrops is directly proportional to the density of cloud droplets, which
are formed due to presence of vapor phase in the atmosphere. The proposed model is analyzed using stability
theory of differential equations and numerical simulations. It is observed that, if vapors do not condense
to form clouds then due to non-occurrence of rain pollutants would not be removed from the atmosphere.
Similar situation arises, when rain formation does not take place due to unfavorable atmospheric conditions
even if clouds are present. It is shown that the cumulative concentrations of primary pollutants, secondary
pollutants and the concentration of primary pollutants absorbed in raindrops decrease as the growth rate of
cloud droplets forming raindrops increases. The remaining equilibrium amount of pollutants would, however, depend upon the intensity of rain due to clouds, rate of emission of primary pollutants, and other removal
parameters. The numerical simulations support the qualitative results obtained.
Keywords: Vapor droplets; cloud droplets; raindrops; primary gaseous pollutants; secondary pollutants;
nonlinear interaction; stability; simulation
1
Introduction
In recent years, due to fast population growth and associated human activities, our environment has been continuously
stressed by different kinds of air pollutants affecting human beings in various ways[7, 12, 25, 27]. The question of
removal of these pollutants from the atmosphere is, therefore, very important. One of the most important mechanism for
removal of pollutants from the atmosphere is precipitation scavenging in which atmospheric gases/particulate matters are
absorbed/trapped in raindrops falling on the ground. It helps in cleaning the atmosphere from pollutants emitted from
various sources. This mechanism is subdivided into rain out (the process within the cloud where plume expands freely
into the cloud layer and material is removed by getting dissolved in water or if it consists of particles, by getting impacted
or entrapped by water drops) and washout (the process taking place below the cloud base for removal of pollutants by
rain).
In some experimental studies, it has been shown that the pollutants are significantly removed from the atmosphere by
precipitation scavenging[4, 8, 19, 21]. Various modeling studies have been made to understand the role of precipitation
scavenging by rain on the removal of pollutants from the atmosphere[1, 5, 6, 9, 10, 11, 13, 14, 26 ]. In particular, Hales
[13] presented some fundamentals of the theory of gas scavenging by rain. Garland [11] presented a simplified model to
study the deposition and wet removal of SO2 and particulate sulfate. Adeuwyi and Carmichael [1] have proposed a model
to investigate the gaseous absorption by water droplets from trace gas mixtures. Chen [5] studied the cloud dynamics of
sulfur dioxide in a raindrop using fully numerical simulation method (FNSM) considering gas and liquid phase.
The pollutants which are emitted directly into the atmosphere from various sources such as industrial stacks, household
discharges and vehicular exhausts etc. are called primary pollutants. Secondary pollutants are those which are formed in
the atmosphere by means of chemical conversion of primary pollutants e.g. sulfur dioxide is converted to sulfur trioxide
∗ Corresponding
author.
E-mail address: [email protected]
c
Copyright⃝World
Academic Press, World Academic Union
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132
International Journal of Nonlinear Science, Vol.14(2012), No.2, pp. 131-141
which combines with moisture and forms sulfuric acid. Some investigations related to removal of primary gaseous pollutants forming secondary species from the atmosphere due to precipitation scavenging by rain, have also been made using
mathematical models[2, 3, 18, 20, 22]. In particular, Alam and Seinfeld [2] have studied the dispersion of sulfur dioxide
and sulfate from a point source by taking into account the dry and wet deposition of both the species and obtained analytical solution of steady state, three dimensional atmospheric diffusion equation incorporating the first order conversion and
removal. A mathematical model to investigate the unsteady state dispersion of air pollutants from a time dependent point
source forming secondary species is also presented by taking into account the dry and wet deposition on the ground[22].
It may be noted that the above studies have been made using linear models. However, in real situations during
precipitation, the densities of cloud droplets and raindrops change, which affect the interaction of various phases such as
absorption of pollutants by raindrops and their removal due to falling of raindrops, making the phenomenon nonlinear.
This aspect should, therefore, be considered in the modeling process[15-17, 23]. For example, Naresh et. al. [15]
presented a nonlinear mathematical model for the removal of primary and secondary pollutants from the atmosphere by
rain but did not consider the effect of density of cloud droplets. In [23], the authors have studied the removal of gaseous
pollutants and particulate matters by rain taking into account the density of cloud droplets formed at a constant rate but
the role of vapor phase in the process was not considered.
It is pointed out here that the growth of raindrops is very much dependent on the vapor droplets and subsequent
formation of cloud droplets in the atmosphere[24]. In view of the above, in this paper, a nonlinear mathematical model
is proposed and analyzed to study the removal of primary gaseous pollutants forming secondary species by taking into
account the interaction processes involving the phases of vapor droplets, cloud droplets, raindrops, primary and secondary
pollutants and the phase of primary pollutants absorbed in raindrops.
2
Mathematical model
Consider the stable atmosphere of a polluted region where rain is taking place. We model the phenomenon of removal of
gaseous pollutants emitted in the environment and secondary species formed due to chemical conversion by rain below
the cloud base. When rainfall occurs, the raindrops interact with both the gaseous pollutants and the secondary species
and remove them by the process of absorption or by impaction caused by raindrops. Thus, we assume the following six
interacting phases in the atmosphere,
i. The vapor phase, which is formed naturally in the atmosphere.
ii. The cloud droplets phase, which is formed from vapor phase by cooling of vapors.
iii. The raindrops phase, which occurs due to precipitation caused by cloud droplets.
iv. The primary pollutants phase, which is formed due to emission of pollutants in the atmosphere.
v. The secondary pollutants phase, which is formed due to chemical conversion of primary pollutants in the atmosphere.
vi. Phase of primary pollutants absorbed in the raindrops.
To model the phenomena, following assumptions have been made,
i. The vapors are formed continuously at a rate q in the atmosphere.
ii. The growth rate of cloud droplets is in direct proportion to the density of vapors and further the growth of clouds is
enhanced by condensation of vapors on tiny water droplets present in the clouds.
iii. The growth rate of raindrops is in the direct proportion to the density of cloud droplets.
iv. The atmosphere is assumed to be calm and therefore the effects of convection and diffusion in the atmosphere have
not been considered in modeling the phenomenon.
v. The absorbed phase of secondary pollutants has not been considered in the modelling process due to the fact that it
may be in the form of aerosol particles, removed by the process of impaction.
Let Cv (t), Cd (t) and Cr (t) be the densities of vapors, cloud droplets and raindrops respectively in the atmosphere,
C(t) and Cs (t) be the cumulative concentrations of primary gaseous pollutants and secondary pollutants, Ca (t) be the
concentration of primary pollutants in absorbed phase. Let Q be the cumulative emission rate of primary pollutants with
natural depletion rate δ C. It is assumed that the secondary pollutants are formed at a rate δ0 with their natural depletion
rate δs Cs . The constant αs is the removal rate coefficient of secondary pollutants due to impaction by raindrops. The
absorption of primary pollutants by raindrops is assumed to be proportional to the cumulative concentration of primary
pollutants as well as the number density of raindrops (i.e. α C Cr ) with natural depletion rate k Ca and the remaining
amount is removed due to falling raindrops on the ground (i.e. ν Ca Cr ).
In view of the above assumptions and considerations, the system dynamics is assumed to be governed by the following
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133
nonlinear ordinary differential equations,
dCv
= q − µ0 Cv − µ1 Cv Cd
dt
(1)
dCd
= λCv + π1 µ1 Cv Cd − λ0 Cd
dt
(2)
dCr
= r Cd − r0 Cr − r1 Cr C
dt
(3)
dC
= Q − (δ + δ0 ) C − α CCr
dt
(4)
dCs
= δ0 C − δs Cs − αs Cs Cr
(5)
dt
dCa
= α CCr − k Ca − ν Ca Cr
(6)
dt
with Cv (0) ≥ 0, Cd (0) ≥ 0, Cr (0) ≥ 0, Cs (0) ≥ 0, Ca (0) ≥ 0
Here q is the constant growth rate of vapors with its natural depletion rate µ0 Cv , µ1 is the interaction rate coefficient
of vapors and cloud droplets, λ is the growth rate of cloud droplets with natural depletion rateλ0 Cd . Further, the growth
of cloud droplets is assumed to be enhanced by a rate π1 due to interaction of vapors and cloud droplets, (0≤ π1 ≤1).
The growth of raindrops is assumed to be directly proportional to the density of cloud droplets (i.e. r Cd ) with its natural
depletion rate r0 Cr , r1 is the depletion rate coefficient of raindrops due to interaction with primary pollutants. All the
constants in the model are assumed to be non-negative.
d
Remark 1 It may be noted here that if λ0 is very large due to unfavorable atmospheric conditions, dC
dt may become negative and pollutants would not be removed from the atmosphere due to non-occurrence of rain. Further, if the interaction
r
rate coefficient r1 is very large (i.e. if pollutants are hot enough), dC
dt may become negative, rain may not be formed and
the possibility of removal of pollutants from the atmosphere would be vanished.
Now to describe the bounds of dependent variables, we state the region of attraction of the model (1)-(6) in the form
of following lemma (without proof).
Lemma 1 The set
{
}
q
Q
Ω = (Cv , Cd , Cr , C, Cs , Ca ) : 0 ≤ Cv + Cd + Cr ≤
, 0 ≤ C + Cs + Ca ≤
,
λm
δm
attracts all solutions initiating in the interior of the positive octant, where λm = min {µ0 − λ, λ0 − r, r0 } and δm =
min {δ, δs , k}.
3
Equilibrium and stability analysis
The model has only one equilibrium point namely E ∗ (Cv∗ , Cd∗ , Cr∗ , C ∗ , Cs∗ , Ca∗ ) where Cv∗ , Cd∗ , Cr∗ , C ∗ , Cs∗ and Ca∗
are the positive solutions of the following equations,
Cv =
q
µ0 + µ1 Cd
(7)
λCv + π1 µ1 Cv Cd − λ0 Cd = 0
(8)
r Cd − r0 Cr − r1 Cr C = 0
(9)
C=
Q
δ + δ0 + α C r
(10)
δ0 C
δs + αs Cr
(11)
Cs =
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134
International Journal of Nonlinear Science, Vol.14(2012), No.2, pp. 131-141
Ca =
αCCr
k + vCr
(12)
From Eqs. (7) and (8), we get
µ1 λ0 Cd2 + (µ0 λ0 − π1 µ1 q)Cd − λ q = 0
This implies,
Cd =
−(µ0 λ0 − π1 µ1 q) +
Thus, there exists a unique root Cd∗ in 0 ≤ Cd ≤
from Eq.(7).
From Eqs. (9) and (10), we get
√
(µ0 λ0 − π1 µ1 q)2 + 4µ1 λ0 λ q
= Cd∗ (say)
2µ1 λ0
q
λm
(13)
without any condition. Using Cd∗ we can calculate the values of Cv∗
r0 α Cr2 + {r0 (δ + δ0 ) + r1 Q − r α Cd }Cr − r(δ + δ0 )Cd = 0
This implies,
Cr =
−{r0 (δ + δ0 ) + r1 Q − r α Cd } +
√
{r0 (δ + δ0 ) + r1 Q − r α Cd }2 + 4r0 α r(δ + δ0 )Cd
2r0 α
(14a)
Using Cd∗ from Eq.(13) in Eq.(14a), we get
√
−{r0 (δ + δ0 ) + r1 Q − r α Cd∗ } + {r0 (δ + δ0 ) + r1 Q − r α Cd∗ }2 + 4r0 α r(δ + δ0 )Cd∗
Cr =
= Cr∗ (say) (14b)
2r0 α
Hence, there exists a unique root Cr∗ in 0 ≤ Cr ≤ λqm without any condition. Using Cr∗ we can calculate the values
of C ∗ , Cs∗ and Ca∗ from the equations (10), (11) and (12) respectively.
From Eqs. (10), (11) and (12), we note that as the number density of raindrops increases, the concentrations C, Cs and
Ca decrease and for larger value of raindrops density, these may even tend to zero. Further, from Eq. (13), it is noted that
if the growth rate of density of vapors is zero (i.e. q = 0), the density of cloud droplets (Cd ) will be zero and therefore
pollutants would not be removed from the atmosphere.
In the following, we check the characteristics of various phases with respect to relevant parameters analytically.
3.1
Variation of Cr with λ
Differentiating Eq. (13), with respect to ‘λ’ we get
dCd
q
>0
=√
dλ
2
(µ0 λ0 − π1 µ1 q) + 4µ1 λ0 λ q
Differentiating Eq.(14a) with respect to ‘Cd ’ we get
[
]
r
r α Cd + r0 (δ + δ0 ) − r1 Q
dCr
=
1+ √
>0
dCd
2r0
{r0 (δ + δ0 ) + r1 Q − r α Cd }2 + 4r0 α r(δ + δ0 )Cd
dCr dCd
r
Therefore, dC
dλ = dCd dλ > 0.
Therefore Cr increases as λ increases. This implies that the number density of raindrops increases with increase in the
density of cloud droplets through cooling of vapor droplets.
3.2
Variation of C with λ
From Eq.(10), we note that
dC
dCr
α
= − (δ+δ0Q+α
Cr )2 < 0.
r
Now,
=
< 0 since dC
dλ > 0.
Therefore C decreases as λ increases. This implies that the cumulative concentration of primary gaseous pollutants
decreases with increase in the density of cloud droplets. This is due to the fact that the number density of raindrops
increases with increase in the density of cloud droplets as above.
dCs
dCs
dCa
dC
a
Similarly, we can also show that dC
dλ < 0, dλ < 0, dr < 0,
dr < 0, dr < 0, etc.
dC
dλ
dC dCr
dCr dλ
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Above analysis shows that the removal of primary pollutants increases as the density of cloud droplets or raindrops
increases. Further,
r
(1) If the coefficient r1 is very large, dC
dt may become negative and pollutants would not be removed from the
atmosphere.
dCs
(2) We also note that dC
dt < 0 and dt < 0 for large values of α and αs respectively. This implies that if removal rate
coefficients are large enough, all the pollutants would be removed from the atmosphere.
a
(3) For very large values of coefficients k and ν, dC
dt < 0 and the formation of absorbed phase is very transient and it
may not exist.
(4) If the growth rate of raindrops r due to clouds is very high, the number density of raindrops will be large and then
dCs
dCa
dC
dt < 0, dt < 0, dt < 0 and pollutants would be removed from the atmosphere emphasizing the role of clouds on
pollutant removal.
To see the stability behavior of E ∗ , we state the following theorems.
Theorem 1 Let the following inequalities hold,
{(λ + π1 µ1 Cd∗ ) − µ1 Cv∗ }2 < 2(µ0 + µ1 Cd∗ )(λ0 − π1 µ1 Cd∗ )
1
(δ + δ0 + α Cr∗ )(δs + αs Cr∗ )
2
1
(α Cr∗ )2 < (δ + δ0 + α Cr∗ )(k + ν Cr∗ )
2
then E ∗ is locally stable, (See Appendix-A for proof).
δ0 2 <
(15)
(16)
(17)
Theorem 2 Let the following inequalities hold in Ω,
(
q
λ + (π1 + 1)µ1
λm
)2
< 2(µ0 + µ1 Cd∗ )(λ0 − π1 µ1 Cv∗ )
(18)
1
(δ + δ0 + α Cr∗ )(δs + αs Cr∗ )
(19)
2
1
(α Cr∗ )2 < (δ + δ0 + α Cr∗ )(k + ν Cr∗ )
(20)
2
then E ∗ is globally asymptotically stable with respect to all solutions initiating in the interior of the positive octant, (See
Appendix-B for proof).
δ0 2 <
The above theorems imply that under certain conditions, the primary gaseous pollutants and secondary pollutants
would be removed from the atmosphere and removal rate increases as densities of cloud droplets and raindrops increase.
Remark 2 If µ1 , δ0 , α and λ are zero, the inequalities (15)-(20) are satisfied automatically. In such a case secondary
species and absorbed phase of primary pollutant will not be formed. It shows that primary gaseous pollutants would be
washed out completely from the atmosphere by natural factors such as gravity.
If the growth rate of cloud droplets is large then the possibility of satisfying the inequalities (15)-(20) is more plausible.
4
Numerical simulation
In this section, we integrate the system (1)-(6) numerically with the help of MAPLE 7.0 to study the behavior of the
system for different values of parameters. For that we consider the following set of parameter values,
q = 80, µ0 = 0.90, π1 = 0.7, µ1 = 0.001, λ = 0.7, λ0 = 0.8, r = 0.6, r0 = 0.45, r1 = 0.001,
Q = 2, δ = 0.50, δ0 = 0.15, α = 0.25, αs = 0.20, δs = 0.40, k = 0.40, ν = 0.55
The equilibrium E ∗ is calculated as,
Cv∗ = 81.869607, Cd∗ = 77.163590, Cr∗ = 102.867448,
C ∗ = 0.075852, Cs∗ = 0.000219, Ca∗ = 0.034236
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136
International Journal of Nonlinear Science, Vol.14(2012), No.2, pp. 131-141
Eigenvalues corresponding to E ∗ are obtained as,
−51.833724, −56.977096, −0.450000, −26.366937,
−0.859927 + 0.219058 i, −0.859927 − 0.219058 i
Since all the eigenvalues corresponding to E ∗ are either negative or have negative real part, therefore E ∗ is locally
asymptotically stable.
The global stability behavior of E ∗ in Cv − Cd and Cd − Cr plane is shown in the figure1 and figure 2 respectively.
It has also been checked that the local and global stability conditions are satisfied by this set of parameter values. In Figs.
3 and 4, variation of number density of raindrops Cr and cumulative concentration of primary gaseous pollutants C with
time ′ t′ is shown for different values of growth rate of vapors q (i.e. at q = 0, 40, 80) respectively. From these figures,
it is seen that if the growth rate of vapors is zero i.e. there is no vapor formation, the number density of raindrops will
be zero, (see Fig. 3) and the cumulative concentration of primary gaseous pollutants would increase continuously, (see
Fig. 4). Further, the number density of raindrops increases but the cumulative concentration of primary gaseous pollutants
decreases as the growth rate of vapors increases. In Figs. 5-8, the variation of number density of raindrops Cr , cumulative
concentration of primary gaseous pollutants C, the concentration of secondary pollutants Cs and the concentration of
primary pollutants absorbed in raindrops Ca with time ′ t′ is shown for different values of growth rate of cloud droplets
λ (i.e. at λ = 0, 0.5, 0.7) respectively. From these figures, it is seen that if the growth rate of cloud droplets is zero i.e.
there is no cloud formation leading to non- occurrence of rain and hence the number density of raindrops will be zero, (see
Fig. 5). The cumulative concentration of primary gaseous pollutants and the concentration of secondary species increase
continuously, while absorbed phase may not exist. Further, with increase in the growth rate of cloud droplets, the number
density of raindrops increases whereas concentrations of pollutants in the atmosphere decreases.
Figure 1: Global stability in Cv − Cd plane.
Figure 2: Global stability in Cd − Cr plane.
Figure 3: Variation of Cr with time ′ t′ for different
Figure 4: Variation of Cwith time ′ t′ for different
values of q.
values of q.
Figs. 9-11 show the variation of cumulative concentration of primary pollutants, concentration of secondary pollutants
and the concentration of primary pollutants absorbed in raindrops with time ′ t′ for different values of growth rate of
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137
raindrops i.e. at r = 0, 0.3, 0.6 respectively. It is seen that when r = 0 i.e. raindrops formation does not take place, the
concentrations of primary and secondary pollutants continue to increase while the absorbed phase is not formed. Further,
as the number density of raindrops increases, the concentration of these pollutants decreases with time.
Figure 5: Variation of Cr with time ′ t′ for different
Figure 6: Variation of C with time ′ t′ for different
values of λ.
values of λ.
Figure 7: Variation of Cs with time ′ t′ for different
Figure 8: Variation of Ca with time ′ t′ for differ-
values of λ.
ent values of λ.
Figure 9: Variation of C with time ′ t′ for different
Figure 10: Variation of Cs with time ′ t′ for differ-
values of r.
ent values of r.
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138
International Journal of Nonlinear Science, Vol.14(2012), No.2, pp. 131-141
From the above analysis, it is concluded that if the vapors are continuously formed in the atmosphere and clouds
are developed, then the resulting occurrence of rainfall due to favorable atmospheric conditions will significantly remove
pollutants from the atmosphere.
Figure 11: Variation of Ca with time ′ t′ for different values of r.
5
Conclusions
A nonlinear mathematical model is proposed to study the removal of primary gaseous pollutants forming secondary
species from the atmosphere by precipitation scavenging considering the effects of vapor and cloud droplets. The model
is studied qualitatively using stability theory of differential equations and numerical simulations. It is shown analytically
that the growth rate of raindrops increases while the concentrations of primary gaseous pollutants, secondary pollutants
and the concentration of primary pollutants absorbed in raindrops decreases as the growth rate of cloud droplets increases.
If due to unfavourable atmospheric conditions clouds are not formed, rain may not occur and pollutants would not be
removed from the atmosphere. It has also been shown numerically that if vapor phase does not lead to formation of
cloud droplets, the cumulative concentrations of primary gaseous pollutants and secondary species increases continuously
but decrease as the growth rate of cloud droplets increases. This decrease in the concentrations of pollutants is due to
increased level of raindrops formation. Further, the magnitude of primary and secondary pollutants removed in the rain
system depends upon the number density of raindrops, emission rate of primary pollutants, the rate of falling raindrops on
the ground and other removal parameters.
Acknowledgements
Authors are thankful to the reviewers for constructive comments and suggestions which helped us in finalizing the paper.
The financial support received from University Grants Commission, New Delhi, India through project F.No. 39-33/2010
(SR) for this research is gratefully acknowledged.
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International Journal of Nonlinear Science, Vol.14(2012), No.2, pp. 131-141
Appendix A
Proof. Proof of Theorem 1: Using the following positive definite function in the linearized system of model (1) - (6),
V =
1
2
2
2
2
2
(k1 Cv1
+ k2 Cd1
+ k3 Cr1
+ k4 C12 + k5 Cs1
+ k6 Ca1
)
2
(A1)
where ki (i = 1 to 6) are constants, to be chosen appropriately and Cv1 , Cd1 , Cr1 , C1 , Cs1 , Ca1 are small perturbations
from E ∗ , as follows
Cv = Cv∗ + Cv1 , Cd = Cd∗ + Cd1 , Cr = Cr∗ + Cr1 , C = C ∗ + C1 ,
Cs = Cs∗ + Cs1 , Ca = Ca∗ + Ca1
Differentiating (A1) with respect to ′ t′ we get, in the linearized system corresponding to E ∗ ,
2
2
2
V̇ = −k1 (µ0 + µ1 Cd∗ )Cv1
− k2 (λ0 − π1 µ1 Cd∗ )Cd1
− k3 (r0 + r1 C ∗ )Cr1
2
2
−k4 (δ + δ0 + α Cr∗ )C12 − k5 (δs + αs Cr∗ )Cs1
− k6 (k + νCr∗ )Ca1
+{k2 (λ + π1 µ1 Cd∗ ) − k1 µ1 Cv∗ }Cv1 Cd1 + k3 r Cd1 Cr1 − k3 r1 Cr∗ Cr1 C1 − k4 α C ∗ Cr1 C1
−k5 αs Cs∗ Cr1 Cs1 + k6 (α C ∗ − ν Ca∗ ) Cr1 Ca1 − k5 δ0 C1 Cs1 + k6 α Cr∗ C1 Ca1
Now V̇ will be negative definite under the following conditions,
{k2 (λ + π1 µ1 Cd∗ ) − k1 µ1 Cv∗ }2 < 2k1 k2 (µ0 + µ1 Cd∗ )(λ0 − π1 µ1 Cd∗ )
k3 r 2 <
2
k2 (λ0 − π1 µ1 Cd∗ )(r0 + r1 C ∗ )
5
(A2)
(A3)
k3 ( r1 Cr∗ )2 <
1
k4 (r0 + r1 C ∗ ) (δ + δ0 + α Cr∗ )
5
(A4)
k4 (α C ∗ )2 <
1
k3 (r0 + r1 C ∗ ) (δ + δ0 + α Cr∗ )
5
(A5)
2
k3 (r0 + r1 C ∗ )(δs + αs Cr∗ )
5
(A6)
k5 (αs Cs∗ )2 <
k6 (α C ∗ − ν Ca∗ )2 <
k5 δ0 2 <
2
k3 (r0 + r1 C ∗ )(k + ν Cr∗ )
5
(A7)
1
k4 (δ + δ0 + α Cr∗ )(δs + αs Cr∗ )
2
k6 (α Cr∗ )2 <
(A8)
1
k4 (δ + δ0 + α Cr∗ )(k + ν Cr∗ )
2
(A9)
Now choosing k1 = k2 = k4 = k5 = k6 = 1 and
<
1
5 (r0
+
{
(αs Cs∗ )2
(α C ∗ −ν Ca∗ )2
2(α C ∗ )2
∗ ,
∗ ,
∗
(δ+δ
(k+ν C
s Cr )
}r )
{0 +α Cr ) (δs +α
2(λ0 −π1 µ1 Cv∗ ) (δ+δ0 +α Cr∗ )
∗
, (r1 C ∗ )2
r1 C ) min
r2
r
5
2 (r0 +r1 C ∗ )
}
max
< k3
we get the stability conditions as given in the statement of the theorem. Hence V̇ will be negative definite provided the
conditions (15)- (17) are satisfied, proving the theorem.
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141
Appendix-B
Proof. Proof of Theorem 2: Using the following positive definite function,
U=
1
[m1 (Cv − Cv∗ )2 + m2 (Cd − Cd∗ )2 + m3 (Cr − Cr∗ )2 + m4 (C − C ∗ )2 + m5 (Cs − Cs∗ )2 + m6 (Ca − Ca∗ )2 ] (B1)
2
where the constants mi (i = 1 to 6) can be chosen appropriately.
Differentiating (B1) with respect to ‘t’ we get,
U̇ = −m1 (µ0 + µ1 Cd∗ )(Cv − Cv∗ )2 − m2 (λ0 − π1 µ1 Cv∗ )(Cd − Cd∗ )2 − m3 (r0 + r1 C ∗ )(Cr − Cr∗ )2
−m4 (δ + δ0 + α Cr∗ )(C − C ∗ )2 − m5 (δs + αs Cr∗ )(Cs − Cs∗ )2 − m6 (k + ν Cr∗ )(Ca − Ca∗ )2
+{m2 (λ + π1 µ1 Cd ) − m1 µ1 Cv }(Cv − Cv∗ )(Cd − Cd∗ ) + m3 r (Cd − Cd∗ )(Cr − Cr∗ )
−m3 r1 Cr (Cr − Cr∗ )(C − C ∗ ) − m4 α C (Cr − Cr∗ )(C − C ∗ )
−m5 αs Cs (Cr − Cr∗ )(Cs − Cs∗ ) + m6 (α C − ν Ca ) (Cr − Cr∗ )(Ca − Ca∗ )
−m5 δ0 (C − C ∗ )(Cs − Cs∗ ) + m6 α Cr∗ (C − C ∗ )(Ca − Ca∗ )
Sufficient conditions for U̇ to be negative definite are that the following inequalities hold:
{m2 (λ + π1 µ1 Cd ) − m1 µ1 Cv }2 < 2m1 m2 (µ0 + µ1 Cd∗ )(λ0 − π1 µ1 Cv∗ )
m3 r 2 <
2
m2 (λ0 − π1 µ1 Cv∗ )(r0 + r1 C ∗ )
5
m4 (α C)2 <
(B4)
1
m3 (r0 + r1 C ∗ ) (δ + δ0 + α Cr∗ )
5
(B5)
2
m3 (r0 + r1 C ∗ )(δs + αs Cr∗ )
5
(B6)
m5 (αs Cs )2 <
m6 (α C − ν Ca )2 <
m5 δ0 2 <
(B3)
1
m4 (r0 + r1 C ∗ ) (δ + δ0 + α Cr∗ )
5
m3 ( r1 Cr )2 <
(B2)
2
m3 (r0 + r1 C ∗ )(k + ν Cr∗ )
5
(B7)
1
m4 (δ + δ0 + α Cr∗ )(δs + αs Cr∗ )
2
(B8)
1
m4 (δ + δ0 + α Cr∗ )(k + ν Cr∗ )
(B9)
2
After maximizing LHS, minimizing RHS, and choosing the constants such that m1 = m2 = m4 = m5 = m6 = 1
m6 (α Cr∗ )2 <
and
5
2 (r0 +r1 C ∗ )
<
∗
(
(r0 +r1 C )
5
Q
δm
)2
min
{
max
{
α2s
(α +ν )2
2α2
∗ ,
∗
(δ+δ0 +α Cr∗ ) , (δs +αs C}
r ) (k+ν Cr )
2 (λ0 −π1 µ1 Cv∗ )
r2
,
}
< m3
(δ+δ0 +α Cr∗ )
2
r1 λqm
(
)
we get the conditions as stated in the theorem. Hence U̇ will be negative definite provided conditions (18) – (20) are
satisfied inside the region of attraction Ω, proving the theorem.
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