Modeling the Simultaneous Effect of Two Toxicants Causing Deformity in a Subclass of Biological...
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IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 6 Ver. IV (Nov. - Dec. 2015), PP 70-82
www.iosrjournals.org
DOI: 10.9790/5728-11647082 www.iosrjournals.org 70 | Page
Modeling the Simultaneous Effect of Two Toxicants Causing
Deformity in a Subclass of Biological Population
Anuj Kumar1, A.W. Khan
1, A.K.Agrawal
2
1(Department of Mathematics, Integral University, Lucknow, India) 2(Department of Mathematics, Amity Univesity, Lucknow, India)
Abstract: In this paper, we have proposed and analyzed a mathematical model to study the simultaneous effect
of two toxicants on a biological population, in which a subclass of biological population is severely affected and
exhibits abnormal symptoms like deformity, fecundity, necrosis, etc. On studying the qualitative behavior of
model, it is shown that the density of total population will settle down to an equilibrium level lower than the
carrying capacity of the environment. In the model, we have assumed that a subclass of biological population is
not capable in further reproduction and it is found that the density of this subclass increases as emission rates
of toxicants or uptake rates of toxicants increase. For large emission rates it may happen that the entire
population gets severely affected and is not capable in reproduction and after a time period all the population
may die out. The stability analysis of the model is determined by variational matrix and method of Lyapunovβs
function. Numerical simulation is given to illustrate the qualitative behavior of model.
Keywords: Biological species, Deformity, Mathematical model, Stability, Two toxicants.
AMS Classification β 93A30, 92D25, 34D20, 34C60
I. Introduction
The dynamics of effect of toxicants on biological species using mathematical models ([1], [2], [3], [4],
[5], [6], [7], [8], [9], [10], [11]) have been studied by many researchers. These studies have been carried out for
different cases such as: Rescigno ([9]) proposed a mathematical model to study the effect of a toxicant on a
biological species when toxicant is being produced by the species itself, Hallam et. al.([6], [7]) proposed and
analyzed a mathematical model to study the effect of a toxicant on the growth rate of biological species, Shukla
et. al. [11] proposed a model to study the simultaneous effect of two different toxicants, emitted from some
external sources, etc. In all of these studies, it is assumed that the toxicants affect each and every individual of
the biological species uniformly. But it is observed that some members of biological species get severely
affected by toxicants and show change in shape, size, deformity, etc. These changes are observed in the
biological species living in aquatic environment ([12], [13], [14], [15], [16], [17], [18], [19], [20]) and in
terrestrial environment, in plants ([21], [22]) and in animals ([23], [24], [25], [26]).
The study of such very important observable fact where a subclass of the biological species is
adversely affected by the toxicant and shows abnormal symptoms such as deformity, incapable in reproduction
etc. using mathematical models is very limited. Agrawal and Shukla [2] have studied the effect of a single
toxicant (emitted from some external sources) on a biological population in which a subclass of biological
population is severely affected and shows abnormal symptoms like deformity, fecundity, necrosis, etc. using
mathematical model. However, no study has been done for this phenomenon under the simultaneous effect of
two toxicants. Therefore, in this paper we have proposed a dynamical model to study the simultaneous effect of
two toxicants (both toxicants are constantly emitted from some external sources) on a biological species in
which a subclass of biological population is severely affected and shows abnormal symptoms like deformity,
fecundity, necrosis, etc.
II. Mathematical Model
We consider a logistically growing biological population with density π(π‘) in the environment and
simultaneously affected by two different types of toxicants with environment concentrations π1 π‘ and π2(π‘)
(both toxicants are constantly emitted in the environment at the rates π1 and π2 respectively, from some
Modeling the Simultaneous Effect of Two Toxicants Causing Deformity in a Subclass of Biologicalβ¦
DOI: 10.9790/5728-11647082 www.iosrjournals.org 71 | Page
external sources). These toxicants are correspondingly uptaken by the biological population at different
concentration rates π1 π‘ and π2 π‘ . These toxicants decrease the growth rate of biological population as well
as they also adversely affect a subclass of biological population with density ππ·(π‘) and decay the capability of
reproduction. Here, ππ΄ π‘ is the density of biological population which is capable in reproduction. Keeping
these views in mind, we have proposed the following model:
πππ΄
ππ‘= π β π ππ΄ β π1π1 + π2π2 ππ΄ β
πππ΄π
πΎ π1,π2
πππ·
ππ‘= π1π1 + π2π2 ππ΄ β
πππ·π
πΎ π1,π2 β πΌ + π ππ·
ππ1
ππ‘= π1 β πΏ1π1 β πΎ1π1π + π1π1ππ1 2.1
ππ2
ππ‘= π2 β πΏ2π2 β πΎ2π2π + π2π2ππ2
ππ1
ππ‘= πΎ1π1π β π½1π1 β π1ππ1
ππ2
ππ‘= πΎ2π2π β π½2π2 β π2ππ2
ππ΄ 0 ,ππ· 0 β₯ 0, ππ 0 β₯ 0, ππ 0 β₯ πππ 0 , ππ > 0, 0 < ππ < 1 for π = 1,2
All the parameters used in the model (2.1) are positive and defined as follows:
π β the birth rate of logistically growing biological population,
π β the death rate of logistically growing biological population,
π β the growth rate of biological population in toxicants free environment, i.e. π = (π β π)
πΌ β the decay rate of the deformed population due to high toxicity,
π1 & π2 β the decreasing rates of the growth rate associated with the uptakes of environmental
concentration of toxicants π1 and π2 respectively,
πΏ1 & πΏ2 β the natural depletion rate coefficients of π1 and π2 respectively,
π½1 & π½2 β the natural depletion rate coefficients of π1 and π2 respectively,
πΎ1 & πΎ2 β the depletion rate coefficients due to uptake by the population respectively,
(π. π. πΎ1π1π & πΎ2π2π)
π1 & π2 β the depletion rate coefficients of π1 and π2 respectively due to decay of some members of π,
(π. π. π1ππ1 & π2ππ2)
π1 & π2 β the fractions of the depletion of π1 and π2 respectively due to decay of some members of
π which may reenter into the environment, π. π. π1π1ππ1 & π2π2ππ2
In the above model (2.1), total density of logistically growing biological population π is equal to the
sum of density of biological population without deformity ππ΄ and with deformity ππ·, π. π. π = ππ΄ + ππ· .
So, the above system can be written in terms of π,ππ· ,π1,π2,π1and π2 as follows:
ππ
ππ‘= ππ β
ππ2
πΎ π1,π2 β πΌ + π ππ·
πππ·
ππ‘= π1π1 + π2π2 (π β ππ·) β
πππ·π
πΎ π1,π2 β πΌ + π ππ·
ππ1
ππ‘= π1 β πΏ1π1 β πΎ1π1π + π1π1ππ1 2.2
ππ2
ππ‘= π2 β πΏ2π2 β πΎ2π2π + π2π2ππ2
ππ1
ππ‘= πΎ1π1π β π½1π1 β π1ππ1
ππ2
ππ‘= πΎ2π2π β π½2π2 β π2ππ2
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π 0 β₯ 0, ππ· 0 β₯ 0, ππ 0 β₯ 0, ππ 0 β₯ πππ 0 , 0 β€ ππ β€ 1, for π = 1,2
where π1, π2 > 0 are constants relating to the initial uptake concentration ππ 0 with the initial density
of biological population π(0).
In the model (2.2), the function πΎ π1,π2 > 0 (for all values of π1 & π2) denotes the carrying capacity
of the environment for the biological population π and it decreases when π1 or π2 or both increase.
we have,
initial carrying capacity, πΎ0 = πΎ 0, 0 and ππΎ
πππ< 0 for ππ > 0, π = 1,2 2.3
III. Equilibrium points and stability analysis
The model (2.2) has two non β negative equilibrium points πΈ1 = 0, 0,π1
πΏ1,π2
πΏ2, 0, 0 and πΈ2 =
(πβ,ππ·β ,π1
β,π2β,π1
β,π2β). It is obvious that equilibria πΈ1 exist, hence existence of πΈ1 is not discussed.
Existence of π¬π: The value of πβ, ππ·β , π1
β, π2β, π1
β and π2β are the positive solutions of the following system of
equations:
π =1
π π β π1π1 β π2π2 πΎ π1,π2 3.1
ππ· = π1π1 + π2π2 ππΎ π1,π2
ππ + π1π1 + π2π2 + πΌ + π πΎ π1,π2 3.2
π1 =π1 π½1 + π1π
π1 π = π1 π 3.3
π2 =π2 π½2 + π2π
π2 π = π2 π 3.4
π1 =π1πΎ1π
π1 π = π1 π 3.5
π2 =π2πΎ2π
π2 π = π2 π 3.6
where, π1 π = πΏ1π½1 + πΎ1π½1 + πΏ1π1 π + πΎ1π1 1 β π1 π2 3.7
π2 π = πΏ2π½2 + πΎ2π½2 + πΏ2π2 π + πΎ2π2 1 β π2 π2 (3.8)
Using equations (3.1-3.8), we can assume a function
πΉ π = ππ β π β π1π1 π β π2π2 π πΎ π1 π ,π2 π (3.9)
From (3.9), we can say that
πΉ 0 < 0 and πΉ πΎ0 > 0
this implies there must exist a root between 0 and πΎ0 for the equation πΉ π = 0, says πβ.
Uniqueness of π¬π:
For πβ to be unique root of πΉ π = 0, we must have
ππΉ
ππ= π + πΎ π1 π ,π2 π π1
ππ1
ππ+ π2
ππ2
ππ β π β π1π1 π β π2π2 π
ππΎ
ππ1
ππ1
ππ+
ππΎ
ππ2
ππ2
ππ > 0
where
ππ1
ππ=
π1πΎ1
π12 π
πΏ1π½1 β πΎ1π1 1 β π1 π2 (3.10)
ππ2
ππ=
π2πΎ2
π22 π
πΏ2π½2 β πΎ2π2 1 β π2 π2 (3.11)
Modeling the Simultaneous Effect of Two Toxicants Causing Deformity in a Subclass of Biologicalβ¦
DOI: 10.9790/5728-11647082 www.iosrjournals.org 73 | Page
ππ1
ππ= β
π1πΎ1
π12 π
π½12 + 2π½1π1 1 β π1 π + π1
2 1 β π1 π2 < 0 (3.12)
ππ2
ππ= β
π2πΎ2
π22 π
π½22 + 2π½2π2 1 β π2 π + π2
2 1 β π2 π2 < 0 (3.13)
Since, ππΎ
ππ1,
ππΎ
ππ2< 0 (from eq. (2.3)) and
ππ1
ππ,
ππ2
ππ< 0 (from eq. (3.12-3.13)), this implies that:
π β π1π1 π β π2π2 π ππΎ
ππ1
ππ1
ππ+
ππΎ
ππ2
ππ2
ππ > 0
then ππΉ
ππ> 0, only when
π + πΎ π1 π ,π2 π π1
ππ1
ππ+ π2
ππ2
ππ > π β π1π1 π β π2π2 π
ππΎ
ππ1
ππ1
ππ+
ππΎ
ππ2
ππ2
ππ (3.14)
Hence, if the conditions (3.14) is satisfied, the root πβ of πΉ π = 0 is unique and lower than the carrying
capacity of the environment.
After that, we can compute the value of ππ·β , π1
β, π2β, π1
β and π2β with the help of πβ and equations (3.2-3.8).
3.1 Local stability analysis
To study the local stability behavior of the equilibrium points πΈ1 = 0, 0,π1
πΏ1,π2
πΏ2, 0, 0 and πΈ2 =
(πβ,ππ·β ,π1
β,π2β,π1
β,π2β), we compute the variational matrices π1 and π2 corresponding to the equilibrium points
πΈ1 and πΈ2 such as:
π1 =
π β(πΌ + π) 0 0 0 0
0 β(πΌ + π) 0 0 0 0
βπΎ1π1
πΏ10 βπΏ1 0 0 0
βπΎ2π2
πΏ20 0 βπΏ2 0 0
πΎ1π1
πΏ10 0 0 βπ½1 0
πΎ2π2
πΏ20 0 0 0 βπ½2
From π1, it is obvious that πΈ1 is a saddle point unstable locally only in the π β direction and with
stable manifold locally in the ππ· β π1 β π2 βπ1 βπ2 space.
And
π2 =
βπ
2πβ
πΎ π1β ,π2
β β 1 β πΌ + π ππβ2πΎ1 π1
β,π2β ππβ2πΎ2 π1
β,π2β 0 0
π1π1β + π2π2
β βπππ·
β
πΎ π1β ,π2
β β π1π1
β + π2π2β πβ
ππ·β ππβππ·
βπΎ1 π1β,π2
β ππβππ·βπΎ2 π1
β,π2β π1 π
β β ππ·β π2 π
β β ππ·β
βπΎ1π1β + π1π1π1
β 0 β πΏ1 + πΎ1πβ 0 π1π1π
β 0βπΎ2π2
β + π2π2π2β 0 0 β(πΏ2 + πΎ2π
β) 0 π2π2πβ
πΎ1π1β β π1π1
β 0 πΎ1πβ 0 β(π½1 + π1π
β) 0πΎ2π2
β β π2π2β 0 0 πΎ2π
β 0 β(π½2 + π2πβ)
Here,
πΎ1 π1β,π2
β =1
πΎ2 π1β,π2
β . ππΎ
ππ1 π1
β,π2β
< 0 and πΎ2 π1β,π2
β =1
πΎ2 π1β,π2
β . ππΎ
ππ2 π1
β,π2β
< 0
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According to the Gershgorinβs disc, all the eigenvalues of variational matrix π2 are negative or having
negative real parts if
πΎ π1β,π2
β < 2πβ (3.15)
πΌ + π + ππβ2πΎ1 π1β,π2
β + ππβ2πΎ2 π1β,π2
β < π 2πβ
πΎ π1β,π2
β β 1 (3.16)
π1π1β + π2π2
β βπππ·
β
πΎ π1β,π2
β + ππβππ·
βπΎ1 π1β,π2
β + ππβππ·βπΎ2 π1
β,π2β + π1 π
β βππ·β
+ π2 πβ β ππ·
β < π1π1β + π2π2
β πβ
ππ·β (3.17)
βπΎ1π1β + π1π1π1
β + π1π1πβ < πΏ1 + πΎ1π
β (3.18)
βπΎ2π2β + π2π2π2
β + π2π2πβ < πΏ2 + πΎ2π
β (3.19)
πΎ1π1β β π1π1
β + πΎ1πβ < (π½1 + π1π
β) (3.20)
πΎ2π2β β π2
βπ2β + πΎ2π
β < π½2 + π2πβ (3.21)
Hence, we can state the following theorem.
Theorem 1: The equilibrium point πΈ2 is locally asymptotically stable if the conditions (3.15-3.21) are satisfied.
3.2 Global stability analysis
To found a set of sufficient conditions for globally asymptotically stable behavior of the equilibria πΈ2,
we need a lemma which establishes the region of attraction of πΈ2.
Lemma 1: The region
Ξ© = π,ππ· ,π1,π2,π1,π2 : 0 β€ π β€ πΎ0 , 0 β€ ππ· β€ π1 + π2 π1 + π2 πΎ0
π1 + π2 π1 + π2 + πΏπ πΌ + π ,
0 β€ π1 + π2 + π1 + π2 β€ π1 + π2
πΏπ
where πΏπ = min πΏ1,πΏ2,π½1 ,π½2
attracts all solution initiating in the interior of the positive orthant.
Proof: From the first equation of model (2.2),
we have,ππ
ππ‘β€ ππ β
ππ2
πΎ0= π 1 β
π
πΎ0 π
Thus, limsupπ‘ββπ π‘ β€ πΎ0.
From the last four equations of model (2.2),
we have,ππ1
ππ‘+ππ2
ππ‘+ππ1
ππ‘+ππ2
ππ‘= π1 + π2 β πΏ1π1 + πΏ2π2 + π½1π1 + π½2π2 β 1 β π1 π1ππ1 β 1 β π2 π2ππ2
β€ π1 + π2 β πΏπ π1 + π2 + π1 + π2
where πΏπ = min(Ξ΄1, Ξ΄2, Ξ²1
, Ξ²2
)
Thus, lim supπ‘ββ
π1 + π2 + π1 + π2 β€π1 + π2
πΏπ
From the second equation of model (2.2),
we have,πππ·
ππ‘= π1π1 + π2π2 π β ππ· β
πππ·π
πΎ π1,π2 β πΌ + π ππ·
β€ π1 + π2 π1 + π2
πΏπ πΎ0 β ππ· β πΌ + π ππ·
Thus, lim supπ‘ββππ· π‘ β€ π1 + π2 π1 + π2 πΎ0
π1 + π2 π1 + π2 + πΏπ πΌ + π
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proving the lemma.β‘
The following theorem establishes global asymptotic stability conditions for the equilibrium point πΈ2 .
Theorem 2: Let πΎ π satisfies the following inequalities in Ξ© with the assumptions in equation (2.3):
πΎπ β€ πΎ π β€ πΎ0, 0 β€ βππΎ
ππ1
π1,π2 β€ π 1 , 0 β€ βππΎ
ππ2
π1,π2 β€ π 2
where πΎπ ,π 1 & π 2 are positive constants.
Then πΈ2 is globally asymptotically stable with respect to all solutions initiating in the interior of the
positive orthant, if the following conditions hold in Ξ©:
π1π1β + π2π2
β β πΌ + π +ππΎ0 π1 + π2 (π1 + π2)
πΎ π1β,π2
β π1 + π2 π1 + π2 + πΏπ πΌ + π
2
< 4π
25(π1π1
β + π2π2β)πβ
ππ·β
2πβ
πΎ π1β,π2
β β 1 (3.22)
πΎ1 + π1π1 π1 + π2
πΏπ+ππΎ0
2π 1
πΎπ2
2
<4π
15 πΏ1 + πΎ1π
β 2πβ
πΎ π1β,π2
β β 1 (3.23)
πΎ2 + π2π2 π1 + π2
πΏπ+ππΎ0
2π 2
πΎπ2
2
<4π
15 πΏ2 + πΎ2π
β 2πβ
πΎ π1β,π2
β β 1 (3.24)
πΎ1 + π1 π1 + π2
πΏπ
2
<4π
15 π½1 + π1π
β 2πβ
πΎ π1β,π2
β β 1 (3.25)
πΎ2 + π2 π1 + π2
πΏπ
2
<4π
15 π½2 + π2π
β 2πβ
πΎ π1β,π2
β β 1 (3.26)
ππΎ0π 1 π1 + π2 π1 + π2
πΎπ2 π1 + π2 π1 + π2 + πΌ + π πΏπ
2
<4
15 πΏ1 + πΎ1π
β π1π1β + π2π2
β πβ
ππ·β (3.27)
ππΎ0π 2 π1 + π2 π1 + π2
πΎπ2 π1 + π2 π1 + π2 + πΌ + π πΏπ
2
<4
15 πΏ2 + πΎ2π
β π1π1β + π2π2
β πβ
ππ·β (3.28)
π1πΎ0 2 <
4
15 π½1 + π1π
β π1π1β + π2π2
β πβ
ππ·β (3.29)
π2πΎ0 2 <
4
15 π½2 + π2π
β π1π1β + π2π2
β πβ
ππ·β (3.30)
πΎ1 + π1π1 πβ 2 <
4
9 πΏ1 + πΎ1π
β π½1 + π1πβ (3.31)
πΎ2 + π2π2 πβ 2 <
4
9 πΏ2 + πΎ2π
β π½2 + π2πβ (3.32)
The proof of Theorem 2 is given in Appendix A.
IV. Numerical simulation
To make the qualitative results more clear, we give here numerical simulation of model (2.2) by
defining the function:
πΎ π1,π2 = πΎ0 βπ11π1
1 + π12π1β
π21π2
1 + π22π2 (4.1)
and assuming a set of parameters
π = 0.005, π = 0.00001, π1 = 0.0007, π2 = 0.0005, π1 = 0.001, π2 = 0.0004πΏ1 = 0.004, πΏ2 = 0.001, πΎ1 = 0.0005, πΎ2 = 0.0003, π1 = 0.0004, π2 = 0.0006π1 = 0.005, π2 = 0.003, π½1 = 0.006, π½2 = 0.004, πΎ0 = 10.0, π11 = 0.0002,π12 = 1.0, π21 = 0.0001, π22 = 2.0, π 1 = 0.001, π 2 = 0.001, πΎπ = 3.0
(4.2)
For the above function and set of values of parameters (4.1-4.2), we have obtained equilibrium point
πΈ2(πβ,ππ·β ,π1
β,π2β,π1
β,π2β) with values πβ = 9.7771, ππ
β = 0.0227, π1β = 0.1113, π2
β = 0.1002, π1β = 0.0099
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and π2β = 0.0088. Here, condition (3.14) satisfies which shows that the values πβ,ππ·
β ,π1β,π2
β,π1β and π2
β are
unique in the region Ξ©. The eigenvalues of variational matrix π2 corresponding to the equilibrium point πΈ2 for
the model (2.2) are obtained as β0.0559, β0.0339, β0.0090, β0.0039, β0.0050 + 0.0002π and
β0.0050 β 0.0002π. We note that four eigenvalues of variational matrix are negative and remaining two
eigenvalues have negative real parts which show that equilibrium point πΈ2 is locally asymptotically stable. Also,
the equilibrium point πΈ2 satisfies all the conditions of global asymptotic stability (3.22-3.32). (see Fig.1)
Fig.1: Nonlinear stability of (π΅β,π΅π«
β ) in π΅βπ΅π« plane for different initial starts
In Fig.2 & Fig.3, we have shown the changes in density of deformed population with respect to time
for different values of emission rates of toxicant in the environment π1 and π2 respectively. Here, we take all
the parameters same as eq. (4.2) except π1 and π2. In both figures, we can see that when emission rate of
toxicant π1 as well as emission rate of toxicant π2 increases the density of the deformed population also
increases, which shows that more members of the population will get deformed if the rate of toxicant emission
increases.
Fig.2: Variation of deformed population π΅π« with time for different values of πΈπ
Modeling the Simultaneous Effect of Two Toxicants Causing Deformity in a Subclass of Biologicalβ¦
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Fig.3: Variation of deformed population π΅π« with time for different values of πΈπ
In Fig.4 & Fig.5, we have represented the variation in the density of deformed population for different
values of the uptake rate coefficients πΎ1 and πΎ2 (all the parameters same as eq. (4.2) except πΎ1 and πΎ2
respectively). Here figures are showing that when the uptake rates of toxicants increase, density of deformed
population increases.
Fig.4: Variation of deformed population π΅π« with time for different values of πΈπ
Modeling the Simultaneous Effect of Two Toxicants Causing Deformity in a Subclass of Biologicalβ¦
DOI: 10.9790/5728-11647082 www.iosrjournals.org 78 | Page
Fig.5: Variation of deformed population π΅π« with time for different values of πΈπ
In Fig.6, we have shown the variation in density of deformed population corresponding to the decay
rate of the deformed population due to high toxicity πΌ (all the parameters same as eq. (4.2) except πΌ). In this
figure, we can see that when the decay rate of deformed population increases density of deformed population
decreases.
Fig.6: Variation of deformed population π΅π« with time for different values of πΆ
Fig.7: π΅ and π΅π« for large emission rate of toxicant π»π in the environment
Modeling the Simultaneous Effect of Two Toxicants Causing Deformity in a Subclass of Biologicalβ¦
DOI: 10.9790/5728-11647082 www.iosrjournals.org 79 | Page
Fig.8: π΅ and π΅π« for large emission rate of toxicant π»π in the environment
In Fig.7 & Fig.8, we have represented the variation in the densities of Total population (π) and
Deformed population ππ· for large emission rate of toxicants π1 and π2. These figures show that density of
total population gets severely affected and is not capable in reproduction for large emission rates.
V. Conclusion
In this paper, we have proposed and analyzed a mathematical model to study the simultaneous effect of
two toxicants on a biological population, in which a subclass of biological population is severely affected and
exhibits abnormal symptoms like deformity, fecundity, necrosis, etc. Here, we assume that these two toxicants
are being emitted into the environment by some external sources such as industrial discharge, vehicular exhaust,
waste water discharge from cities, etc. The model (2.2) has two equilibrium points πΈ1 and πΈ2 in which πΈ1 is
saddle point and πΈ2 is locally and globally stable under some conditions. The qualitative behavior of model (2.2)
shows that the density of total population will settle down to an equilibrium level, lower than its initial carrying
capacity. It is assumed that a subclass of biological population is not capable in reproduction. Under this
assumption, it is found that the density of this subclass increases as emission rates of toxicants or uptake rates of
toxicants increase and when the decay rate of deformed population increases, density of deformed population
decreases. For large emission rates, it may happen that the entire population gets severely affected and is not
capable in reproduction and after a time period all the population may die out. So, we need to control the
emission of toxicants from industries, household and vehicular discharges in the environment to protect
biological species from deformity.
Appendix A. Proof of the Theorem 2.
Proof: we consider a positive definite function about πΈ2
π π,ππ· ,π1,π2,π1 ,π2
=1
2 π β πβ 2 +
1
2 ππ· βππ·
β 2 +1
2 π1 β π1
β 2 +1
2 π2 β π2
β 2 +1
2 π1 βπ1
β 2
+1
2 π2 β π2
β 2
Differentiating π with respect to π‘ along the solution of (2.2), we get
ππ
ππ‘= π β πβ ππ β
ππ2
πΎ π1,π2 β πΌ + π ππ·
+ ππ· βππ·β π1π1 + π2π2 π β ππ· β
πππ·π
πΎ π1,π2 β πΌ + π ππ·
+ π1 β π1β π1 β πΏ1π1 β πΎ1π1π + π1π1ππ1 + π2 β π2
β π2 β πΏ2π2 β πΎ2π2π + π2π2ππ2
+ π1 βπ1β πΎ1π1π β π½1π1 β π1ππ1 + π2 β π2
β πΎ2π2π β π½2π2 β π2ππ2
Modeling the Simultaneous Effect of Two Toxicants Causing Deformity in a Subclass of Biologicalβ¦
DOI: 10.9790/5728-11647082 www.iosrjournals.org 80 | Page
using (3.1-3.8), we get after some calculation
ππ
ππ‘= β π
2πβ
πΎ π1β,π2
β β 1 π β πβ 2 β (π1π1
β + π2π2β)πβ
ππ·β ππ· β ππ·
β 2 β πΏ1 + πΎ1πβ π1 β π1
β 2
β πΏ2 + πΎ2πβ π2 β π2
β 2 β π½1 + π1πβ π1 β π1
β 2 β π½2 + π2πβ π2 β π2
β 2
+ β πΌ + π + π1π1β + π2π2
β βπππ·
πΎ π1β,π2
β π β πβ ππ· βππ·
β
+ π1π1π1 β πΎ1π1 β ππ2π1 π1,π2 π β πβ π1 β π1β
+ π2π2π2 β πΎ2π2 β ππ2π2 π1β,π2 π β πβ π2 β π2
β
+ πΎ1π1 β π1π1 π β πβ π1 βπ1β + πΎ2π2 β π2π2 π β πβ π2 β π2
β
β ππππ·π1 π1,π2 ππ· βππ·β π1 β π1
β β ππππ·π2 π1β,π2 ππ· β ππ·
β π2 β π2β
+ π1 π β ππ· ππ· βππ·β π1 βπ1
β + π2 π β ππ· ππ· β ππ·β π2 β π2
β
+ π1π1πβ + πΎ1π
β π1 β π1β (π1 β π1
β) + π2π2πβ + πΎ2π
β π2 β π2β (π2 β π2
β)
where,
π1 π1,π2 =
1
πΎ π1 ,π2 β
1
πΎ π1β,π2
π1 β π1β , π1 β π1
β
β1
πΎ2 π1β,π2
ππΎ
ππ1
π1β,π2 , π1 = π1
β
,
π2 π1β,π2 =
1
πΎ π1β,π2
β1
πΎ π1β,π2
β
π2 β π2β , π2 β π2
β
β1
πΎ2 π1β,π2
β
ππΎ
ππ2
π1β,π2
β , π2 = π2β
Thus, ππ€
ππ‘ can be written as sum of the quadratics,
ππ€
ππ‘= β
1
2π11 π β πβ 2 + π12 π β πβ ππ· β ππ·
β β1
2π22 ππ· βππ·
β 2
+ β1
2π11 π β πβ 2 + π13 π β πβ π1 β π1
β β1
2π33 π1 β π1
β 2
+ β1
2π11 π β πβ 2 + π14 π β πβ π2 β π2
β β1
2π44 π2 β π2
β 2
+ β1
2π11 π β πβ 2 + π15 π β πβ π1 β π1
β β1
2π55 π1 βπ1
β 2
+ β1
2π11 π β πβ 2 + π16 π β πβ π2 β π2
β β1
2π66 π2 β π2
β 2
+ β1
2π22 ππ· β ππ·
β 2 + π23 ππ· βππ·β π1 β π1
β β1
2π33 π1 β π1
β 2
+ β1
2π22 ππ· β ππ·
β 2 + π24 ππ· βππ·β π2 β π2
β β1
2π44 π2 β π2
β 2
+ β1
2π22 ππ· β ππ·
β 2 + π25 ππ· βππ·β π1 βπ1
β β1
2π55 π1 βπ1
β 2
+ β1
2π22 ππ· β ππ·
β 2 + π26 ππ· βππ·β π2 β π2
β β1
2π66 π2 βπ2
β 2
+ β1
2π33 π1 β π1
β 2 + π35 π1 β π1β π1 β π1
β β1
2π55 π1 βπ1
β 2
+ β1
2π44 π2 β π2
β 2 + π46 π2 β π2β π2 βπ2
β β1
2π66 π2 β π2
β 2
where,
π11 =2
5 π
2πβ
πΎ π1β,π2
β β 1 , π22 =
2
5 (π1π1
β + π2π2β)πβ
ππ·β , π33 =
2
3 πΏ1 + πΎ1π
β
π44 =2
3 πΏ2 + πΎ2π
β , π55 =2
3 π½1 + π1π
β , π66 =2
3 π½2 + π2π
β
Modeling the Simultaneous Effect of Two Toxicants Causing Deformity in a Subclass of Biologicalβ¦
DOI: 10.9790/5728-11647082 www.iosrjournals.org 81 | Page
π12 = β πΌ + π + π1π1β + π2π2
β βπππ·
πΎ π1β,π2
β , π13 = π1π1π1 β πΎ1π1 β ππ2π1 π1,π2
π14 = π2π2π2 β πΎ2π2 β ππ2π2 π1β,π2 , π15 = πΎ1π1 β π1π1 , π16 = πΎ2π2 β π2π2
π23 = βππππ·π1 π1,π2 , π24 = βππππ·π2 π1β,π2 , π25 = π1 π β ππ· , π26 = π2 π β ππ·
π35 = π1π1πβ + πΎ1π
β , π46 = π2π2πβ + πΎ2π
β
Thus,dW
dt will be negative definite provided
π122 < π11π22 (3.33)
π132 < π11π33 (3.34)
π142 < π11π44 (3.35)
π152 < π11π55 (3.36)
π162 < π11π66 3.37
π232 < π22π33 (3.38)
π242 < π22π44 (3.39)
π252 < π22π55 (3.40)
π262 < π22π66 (3.41)
π352 < π33π55 (3.42)
π462 < π44π66 (3.43)
We note that (3.33-3.43) β (3.22-3.32) respectively. So, W is a Lyapunovβs function with respect to
the equilibrium πΈ2 and therefore πΈ2 is globally asymptotically stable under the conditions (3.22-3.32). Hence the
theorem. β‘
References [1]. Agrawal AK, Sinha P, Dubey B and Shukla JB, Effects of two or more toxicants on a biological species: A non-linear mathematical
model and its analysis, In Mathematical Analysis and Applications. A.P. Dwivedi (Ed), Narosa Publishing House, New Delhi,
INDIA, 2000, 97 β 113.
[2]. Agrawal AK, Shukla JB, Effect of a toxicant on a biological population causing severe symptoms on a subclass, South Pacific
Journal of Pure and Applied Mathematics, 1 (1), 2012, 12 β 27.
[3]. DeLuna JT and Hallam TG, Effect of toxicants on population: a qualitative approach IV. Resource - Consumer Toxicant models,
Ecol. Modelling 35, 1987, 249 β 273.
[4]. Freedman HI and Shukla JB, Models for the effect of toxicant in single species and predator-prey systems, J. Math. Biol. 30, 1991,
15 β 30.
[5]. Hallam TG and Clark CE, Nonautonomous logistic equation as models of population in a deteriorating environment, J. Theor. Biol.
93, 1982, 303 β 311.
[6]. Hallam TG, Clark CE and Jordan GS, Effects of toxicants on populations: a qualitative approach II. First order kinetics, J. Math.
Biol. 18, 1983, 25 β 37.
[7]. Hallam TG, Clark CE and Lassiter RR, Effects of toxicants on populations: a qualitative approach I. Equilibrium environmental
exposure, Ecol. Modelling 18, 1983, 291 β 304.
[8]. Hallam TG and Deluna JT, Effects of toxicants on populations: a qualitative approach III. Environmental and food chain pathways,
J. Theor. Biol. 109, 1984, 411 β 429.
[9]. Rescigno A, The struggle for lifeβV. one species living in a limited environment, Bulletin of Mathematical Biology, 39(4), 1977,
479-485.
[10]. Shukla JB and Agrawal AK, Some mathematical models in ecotoxicology; Effects of toxicants on biological species, Sadhana 24,
1999, 25 β 40.
[11]. Shukla JB and Dubey B, Simultaneous effects of two toxicants on biological species: A mathematical model, J. Biol. Systems 4,
1996, 109 β 130.
[12]. Cushman RE, Chironomid deformities as indicators of pollution from synthetic, coal - derived oil, Freshw. Biol. 14, 1984, 179 β
182.
[13]. Dickman MD, Yang JR and Brindle ID, Impacts of heavy metals on higher aquatic plant, diathom and benthic invertebrate
communities in the Niagara River watershed, near Welland, Ontario, Water Pollut. Res. J. Canada 25, 1990, 131 β 159.
[14]. Dickman M, Lan Q and Matthews B, Teratogens in the Niagara River watershed as reflected by chironomid (Diptera:
Chironomidae) labial deformities. Can. Assoc. Water Pollut. Res. Control 24, 1990, 47 β 79.
[15]. Dickman M, Brindle I and Benson M, Evidence of teratogens in sediments of the Niagara River watershed as reflected by
chironomid (Diptera: Chironomidae) deformities. J. Great Lakes Res. 18, 1992, 467 β 480.
[16]. Dickman M and Rygiel G, Chironomid larval deformity frequencies, mortality, and diversity in heavy - metal contaminated
sediments of a Canadian riverline wetland, Environ. International 22, 1996, 693 β 703.
[17]. Hamilton AL and Saether O, The occurrence of characteristic deformities in the chironomid larvae of several Canadian lakes, Can.
Ent. 103, 1971, 363 β 368.
Modeling the Simultaneous Effect of Two Toxicants Causing Deformity in a Subclass of Biologicalβ¦
DOI: 10.9790/5728-11647082 www.iosrjournals.org 82 | Page
[18]. Hartwell SI, Wright DA and Savitz JD, Relative sensitivity of survival, growth and reproduction of Eurytemora Affinis (Copepoda)
to assessing polluted estuaries, Water, Air and Soil Pollut. 71, 1993, 281 β 291.
[19]. Patil VK and David M, Behavioral and morphological endpoints: as an early response to sublethal malathion intoxication in the
freshwater fish, Labeo rohita, Drug and chemical toxico., 33 (2), 2010, 160 β 165.
[20]. Veeramachaneni DNR, Palmer JS and Amann RP, Long term effects on male reproduction of early exposure to common chemical
contaminants in drinking water, Human Reproduction, 16 (5), 2001, 979 β 987.
[21]. Kozlowski TT, {Responses of Plants to Air} Academic Press, New York, 1975.
[22]. Kozlowski TT, Impacts of air pollution on forest ecosystem, BioScience 30, 1980, 88 β 93.
[23]. Miguel BL, Donald JB, Ravinder SS, Fernando GF, Induction of Morphological Deformities and Moulting Alterations in
Litopenaeus vannamei (Boone) Juveniles Exposed to the Triazole - Derivative Fungicide Tilt, Arch. Environ. Contam. Toxicol. 51,
2006, 69 β78.
[24]. Ronit HK and Eldad BC, The effect of colchicine treatment on sperm production and function: a review, Human Reproduction, 13
(2), 1998, 360 β 362.
[25]. Saquib M, Ahmad A and Ansari K, Morphological and physiological responses of Croton Bonplandianum baill. to air pollution, 17,
2010, 35 β 41.
[26]. Sun PL, Hawkins WE, Overstreet RM and Brown-Peterson NJ, Morphological Deformities as Biomarkers in Fish from
Contaminated Rivers in Taiwan, Int. J. Environ. Res. Public Health, 6, 2009, 2307 β 2331.