Modeling the Simultaneous Effect of Two Toxicants Causing Deformity in a Subclass of Biological Population

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. Khan1
, A.K.Agrawal2
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 Lyapunov’s
function. Numerical simulation is given to illustrate the qualitative behavior of model.
Keywords: Biological species, Deformity, Mathematical model, Stability, Two toxicants.
AMS Classification – 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 𝑁(𝑡) in the environment and
simultaneously affected by two different types of toxicants with environment concentrations 𝑇1 𝑡 and 𝑇2(𝑡)
(both toxicants are constantly emitted in the environment at the rates 𝑄1 and 𝑄2 respectively, from some

Modeling the Simultaneous Effect of Two Toxicants Causing Deformity in a Subclass of Biological…
external sources). These toxicants are correspondingly uptaken by the biological population at different
concentration rates 𝑈1 𝑡 and 𝑈2 𝑡 . These toxicants decrease the growth rate of biological population as well
as they also adversely affect a subclass of biological population with density 𝑁 𝐷(𝑡) and decay the capability of
reproduction. Here, 𝑁𝐴 𝑡 is the density of biological population which is capable in reproduction. Keeping
these views in mind, we have proposed the following model:
𝑑𝑁𝐴
𝑑𝑡
= 𝑏 − 𝑑 𝑁𝐴 − 𝑟1 𝑈1 + 𝑟2 𝑈2 𝑁𝐴 −
𝑟𝑁𝐴 𝑁
𝐾 𝑇1, 𝑇2
𝑑𝑁 𝐷
𝑑𝑡
= 𝑟1 𝑈1 + 𝑟2 𝑈2 𝑁𝐴 −
𝑟𝑁 𝐷 𝑁
𝐾 𝑇1, 𝑇2
− 𝛼 + 𝑑 𝑁 𝐷
𝑑𝑇1
𝑑𝑡
= 𝑄1 − 𝛿1 𝑇1 − 𝛾1 𝑇1 𝑁 + 𝜋1 𝜈1 𝑁𝑈1 2.1
𝑑𝑇2
𝑑𝑡
= 𝑄2 − 𝛿2 𝑇2 − 𝛾2 𝑇2 𝑁 + 𝜋2 𝜈2 𝑁𝑈2
𝑑𝑈1
𝑑𝑡
= 𝛾1 𝑇1 𝑁 − 𝛽1 𝑈1 − 𝜈1 𝑁𝑈1
𝑑𝑈2
𝑑𝑡
= 𝛾2 𝑇2 𝑁 − 𝛽2 𝑈2 − 𝜈2 𝑁𝑈2
𝑁𝐴 0 , 𝑁 𝐷 0 ≥ 0, 𝑇𝑖 0 ≥ 0, 𝑈𝑖 0 ≥ 𝑐𝑖 𝑁 0 , 𝑐𝑖 > 0, 0 < 𝜋𝑖 < 1 for 𝑖 = 1,2
All the parameters used in the model (2.1) are positive and defined as follows:
 𝑏 − the birth rate of logistically growing biological population,
 𝑑 − the death rate of logistically growing biological population,
 𝑟 − the growth rate of biological population in toxicants free environment, i.e. 𝑟 = (𝑏 − 𝑑)
 𝛼 − the decay rate of the deformed population due to high toxicity,
 𝑟1 & 𝑟2 − the decreasing rates of the growth rate associated with the uptakes of environmental
concentration of toxicants 𝑇1 and 𝑇2 respectively,
 𝛿1 & 𝛿2 − the natural depletion rate coefficients of 𝑇1 and 𝑇2 respectively,
 𝛽1 & 𝛽2 − the natural depletion rate coefficients of 𝑈1 and 𝑈2 respectively,
 𝛾1 & 𝛾2 − the depletion rate coefficients due to uptake by the population respectively,
(𝑖. 𝑒. 𝛾1 𝑇1 𝑁 & 𝛾2 𝑇2 𝑁)
 𝜈1 & 𝜈2 − the depletion rate coefficients of 𝑈1 and 𝑈2 respectively due to decay of some members of 𝑁,
(𝑖. 𝑒. 𝜈1 𝑁𝑈1 & 𝜈2 𝑁𝑈2)
 𝜋1 & 𝜋2 − the fractions of the depletion of 𝑈1 and 𝑈2 respectively due to decay of some members of
𝑁 which may reenter into the environment, 𝑖. 𝑒. 𝜋1 𝜈1 𝑁𝑈1 & 𝜋2 𝜈2 𝑁𝑈2
In the above model (2.1), total density of logistically growing biological population 𝑁 is equal to the
sum of density of biological population without deformity 𝑁𝐴 and with deformity 𝑁 𝐷, 𝑖. 𝑒. 𝑁 = 𝑁𝐴 + 𝑁 𝐷 .
So, the above system can be written in terms of 𝑁, 𝑁 𝐷, 𝑇1, 𝑇2, 𝑈1and 𝑈2 as follows:
𝑑𝑁
𝑑𝑡
= 𝑟𝑁 −
𝑟𝑁2
𝐾 𝑇1, 𝑇2
− 𝛼 + 𝑏 𝑁 𝐷
𝑑𝑁 𝐷
𝑑𝑡
= 𝑟1 𝑈1 + 𝑟2 𝑈2 (𝑁 − 𝑁 𝐷) −
𝑟𝑁 𝐷 𝑁
𝐾 𝑇1, 𝑇2
− 𝛼 + 𝑑 𝑁 𝐷
𝑑𝑇1
𝑑𝑡
= 𝑄1 − 𝛿1 𝑇1 − 𝛾1 𝑇1 𝑁 + 𝜋1 𝜈1 𝑁𝑈1 2.2
𝑑𝑇2
𝑑𝑡
= 𝑄2 − 𝛿2 𝑇2 − 𝛾2 𝑇2 𝑁 + 𝜋2 𝜈2 𝑁𝑈2
𝑑𝑈1
𝑑𝑡
= 𝛾1 𝑇1 𝑁 − 𝛽1 𝑈1 − 𝜈1 𝑁𝑈1
𝑑𝑈2
𝑑𝑡
= 𝛾2 𝑇2 𝑁 − 𝛽2 𝑈2 − 𝜈2 𝑁𝑈2

𝑁 0 ≥ 0, 𝑁 𝐷 0 ≥ 0, 𝑇𝑖 0 ≥ 0, 𝑈𝑖 0 ≥ 𝑐𝑖 𝑁 0 , 0 ≤ 𝜋𝑖 ≤ 1, for 𝑖 = 1,2
where 𝑐1, 𝑐2 > 0 are constants relating to the initial uptake concentration 𝑈𝑖 0 with the initial density
of biological population 𝑁(0).
In the model (2.2), the function 𝐾 𝑇1, 𝑇2 > 0 (for all values of 𝑇1 & 𝑇2) denotes the carrying capacity
of the environment for the biological population 𝑁 and it decreases when 𝑇1 or 𝑇2 or both increase.
we have,
initial carrying capacity, 𝐾0 = 𝐾 0, 0 and
𝜕𝐾
𝜕 𝑇 𝑖
< 0 for 𝑇𝑖 > 0, 𝑖 = 1,2 2.3
III. Equilibrium points and stability analysis
The model (2.2) has two non – negative equilibrium points 𝐸1 = 0, 0,
𝑄1
𝛿1
,
𝑄2
𝛿2
, 0, 0 and 𝐸2 =
(𝑁∗
, 𝑁 𝐷
∗
, 𝑇1
∗
, 𝑇2
∗
, 𝑈1
∗
, 𝑈2
∗
). It is obvious that equilibria 𝐸1 exist, hence existence of 𝐸1 is not discussed.
Existence of 𝑬 𝟐: The value of 𝑁∗
, 𝑁 𝐷
∗
, 𝑇1
∗
, 𝑇2
∗
, 𝑈1
∗
and 𝑈2
∗
are the positive solutions of the following system of
equations:
𝑁 =
1
𝑟
𝑟 − 𝑟1 𝑈1 − 𝑟2 𝑈2 𝐾 𝑇1, 𝑇2 3.1
𝑁 𝐷 =
𝑟1 𝑈1 + 𝑟2 𝑈2 𝑁𝐾 𝑇1, 𝑇2
𝑟𝑁 + 𝑟1 𝑈1 + 𝑟2 𝑈2 + 𝛼 + 𝑑 𝐾 𝑇1, 𝑇2
3.2
𝑇1 =
𝑄1 𝛽1 + 𝜈1 𝑁
𝑓1 𝑁
= 𝑔1 𝑁 3.3
𝑇2 =
𝑄2 𝛽2 + 𝜈2 𝑁
𝑓2 𝑁
= 𝑔2 𝑁 3.4
𝑈1 =
𝑄1 𝛾1 𝑁
𝑓1 𝑁
= 𝑕1 𝑁 3.5
𝑈2 =
𝑄2 𝛾2 𝑁
𝑓2 𝑁
= 𝑕2 𝑁 3.6
where, 𝑓1 𝑁 = 𝛿1 𝛽1 + 𝛾1 𝛽1 + 𝛿1 𝜈1 𝑁 + 𝛾1 𝜈1 1 − 𝜋1 𝑁2
3.7
𝑓2 𝑁 = 𝛿2 𝛽2 + 𝛾2 𝛽2 + 𝛿2 𝜈2 𝑁 + 𝛾2 𝜈2 1 − 𝜋2 𝑁2
(3.8)
Using equations (3.1-3.8), we can assume a function
𝐹 𝑁 = 𝑟𝑁 − 𝑟 − 𝑟1 𝑕1 𝑁 − 𝑟2 𝑕2 𝑁 𝐾 𝑔1 𝑁 , 𝑔2 𝑁 (3.9)
From (3.9), we can say that
𝐹 0 < 0 and 𝐹 𝐾0 > 0
this implies there must exist a root between 0 and 𝐾0 for the equation 𝐹 𝑁 = 0, says 𝑁∗
.
Uniqueness of 𝑬 𝟐:
For 𝑁∗
to be unique root of 𝐹 𝑁 = 0, we must have
𝑑𝐹
𝑑𝑁
= 𝑟 + 𝐾 𝑔1 𝑁 , 𝑔2 𝑁 𝑟1
𝑑𝑕1
𝑑𝑁
+ 𝑟2
𝑑𝑕2
𝑑𝑁
− 𝑟 − 𝑟1 𝑕1 𝑁 − 𝑟2 𝑕2 𝑁
𝜕𝐾
𝜕𝑇1
𝑑𝑔1
𝑑𝑁
+
𝜕𝐾
𝜕𝑇2
𝑑𝑔2
𝑑𝑁
> 0
where
𝑑𝑕1
𝑑𝑁
=
𝑄1 𝛾1
𝑓1
2
𝑁
𝛿1 𝛽1 − 𝛾1 𝜈1 1 − 𝜋1 𝑁2
(3.10)
𝑑𝑕2
𝑑𝑁
=
𝑄2 𝛾2
𝑓2
2
𝑁
𝛿2 𝛽2 − 𝛾2 𝜈2 1 − 𝜋2 𝑁2
(3.11)

𝑑𝑔1
𝑑𝑁
= −
𝑄1 𝛾1
𝑓1
2
𝑁
𝛽1
2
+ 2𝛽1 𝜈1 1 − 𝜋1 𝑁 + 𝜈1
2
1 − 𝜋1 𝑁2
< 0 (3.12)
𝑑𝑔2
𝑑𝑁
= −
𝑄2 𝛾2
𝑓2
2
𝑁
𝛽2
2
+ 2𝛽2 𝜈2 1 − 𝜋2 𝑁 + 𝜈2
2
1 − 𝜋2 𝑁2
< 0 (3.13)
Since,
𝜕𝐾
𝜕𝑇1
,
𝜕𝐾
𝜕𝑇2
< 0 (from eq. (2.3)) and
𝑑𝑔1
𝑑𝑁
,
𝑑𝑔2
𝑑𝑁
< 0 (from eq. (3.12-3.13)), this implies that:
𝑟 − 𝑟1 𝑕1 𝑁 − 𝑟2 𝑕2 𝑁
𝜕𝐾
𝜕𝑇1
𝑑𝑔1
𝑑𝑁
+
𝜕𝐾
𝜕𝑇2
𝑑𝑔2
𝑑𝑁
> 0
then
𝑑𝐹
𝑑𝑁
> 0, only when
𝑟 + 𝐾 𝑔1 𝑁 , 𝑔2 𝑁 𝑟1
𝑑𝑕1
𝑑𝑁
+ 𝑟2
𝑑𝑕2
𝑑𝑁
> 𝑟 − 𝑟1 𝑕1 𝑁 − 𝑟2 𝑕2 𝑁
𝜕𝐾
𝜕𝑇1
𝑑𝑔1
𝑑𝑁
+
𝜕𝐾
𝜕𝑇2
𝑑𝑔2
𝑑𝑁
(3.14)
Hence, if the conditions (3.14) is satisfied, the root 𝑁∗
of 𝐹 𝑁 = 0 is unique and lower than the carrying
capacity of the environment.
After that, we can compute the value of 𝑁 𝐷
∗
, 𝑇1
∗
, 𝑇2
∗
, 𝑈1
∗
and 𝑈2
∗
with the help of 𝑁∗
and equations (3.2-3.8).
3.1 Local stability analysis
To study the local stability behavior of the equilibrium points 𝐸1 = 0, 0,
𝑄1
𝛿1
,
𝑄2
𝛿2
, 0, 0 and 𝐸2 =
(𝑁∗
, 𝑁 𝐷
∗
, 𝑇1
∗
, 𝑇2
∗
, 𝑈1
∗
, 𝑈2
∗
), we compute the variational matrices 𝑀1 and 𝑀2 corresponding to the equilibrium points
𝐸1 and 𝐸2 such as:
𝑀1 =
𝑟 −(𝛼 + 𝑏) 0 0 0 0
0 −(𝛼 + 𝑑) 0 0 0 0
−
𝛾1 𝑄1
𝛿1
0 −𝛿1 0 0 0
−
𝛾2 𝑄2
𝛿2
0 0 −𝛿2 0 0
𝛾1 𝑄1
𝛿1
0 0 0 −𝛽1 0
𝛾2 𝑄2
𝛿2
0 0 0 0 −𝛽2
From 𝑀1, it is obvious that 𝐸1 is a saddle point unstable locally only in the 𝑁 − direction and with
stable manifold locally in the 𝑁 𝐷 − 𝑇1 − 𝑇2 − 𝑈1 − 𝑈2 space.
And
𝑀2 =
−𝑟
2𝑁∗
𝐾 𝑇1
∗
, 𝑇2
∗ − 1 − 𝛼 + 𝑏 𝑟𝑁∗2
𝐾1 𝑇1
∗
, 𝑇2
∗
𝑟𝑁∗2
𝐾2 𝑇1
∗
, 𝑇2
∗
0 0
𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
−
𝑟𝑁 𝐷
∗
𝐾 𝑇1
∗
, 𝑇2
∗ − 𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
𝑁∗
𝑁 𝐷
∗ 𝑟𝑁∗
𝑁 𝐷
∗
𝐾1 𝑇1
∗
, 𝑇2
∗
𝑟𝑁∗
𝑁 𝐷
∗
𝐾2 𝑇1
∗
, 𝑇2
∗
𝑟1 𝑁∗
− 𝑁 𝐷
∗
𝑟2 𝑁∗
− 𝑁 𝐷
∗
−𝛾1 𝑇1
∗
+ 𝜋1 𝜈1 𝑈1
∗
0 − 𝛿1 + 𝛾1 𝑁∗
0 𝜋1 𝜈1 𝑁∗
0
−𝛾2 𝑇2
∗
+ 𝜋2 𝜈2 𝑈2
∗
0 0 −(𝛿2 + 𝛾2 𝑁∗
) 0 𝜋2 𝜈2 𝑁∗
𝛾1 𝑇1
∗
− 𝜈1 𝑈1
∗
0 𝛾1 𝑁∗
0 −(𝛽1 + 𝜈1 𝑁∗
) 0
𝛾2 𝑇2
∗
− 𝜈2 𝑈2
∗
0 0 𝛾2 𝑁∗
0 −(𝛽2 + 𝜈2 𝑁∗
)
Here,
𝐾1 𝑇1
∗
, 𝑇2
∗
=
1
𝐾2 𝑇1
∗
, 𝑇2
∗ .
𝜕𝐾
𝜕𝑇1 𝑇1
∗,𝑇2
∗
< 0 and 𝐾2 𝑇1
∗
, 𝑇2
∗
=
1
𝐾2 𝑇1
∗
, 𝑇2
∗ .
𝜕𝐾
𝜕𝑇2 𝑇1
∗,𝑇2
∗
< 0

According to the Gershgorin’s disc, all the eigenvalues of variational matrix 𝑀2 are negative or having
negative real parts if
𝐾 𝑇1
∗
, 𝑇2
∗
< 2𝑁∗
(3.15)
𝛼 + 𝑏 + 𝑟𝑁∗2
𝐾1 𝑇1
∗
, 𝑇2
∗
+ 𝑟𝑁∗2
𝐾2 𝑇1
∗
, 𝑇2
∗
< 𝑟
2𝑁∗
𝐾 𝑇1
∗
, 𝑇2
∗ − 1 (3.16)
𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
−
𝑟𝑁 𝐷
∗
𝐾 𝑇1
∗
, 𝑇2
∗ + 𝑟𝑁∗
𝑁 𝐷
∗
𝐾1 𝑇1
∗
, 𝑇2
∗
+ 𝑟𝑁∗
𝑁 𝐷
∗
𝐾2 𝑇1
∗
, 𝑇2
∗
+ 𝑟1 𝑁∗
− 𝑁 𝐷
∗
+ 𝑟2 𝑁∗
− 𝑁 𝐷
∗
< 𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
𝑁∗
𝑁 𝐷
∗ (3.17)
−𝛾1 𝑇1
∗
+ 𝜋1 𝜈1 𝑈1
∗
+ 𝜋1 𝜈1 𝑁∗
< 𝛿1 + 𝛾1 𝑁∗
(3.18)
−𝛾2 𝑇2
∗
+ 𝜋2 𝜈2 𝑈2
∗
+ 𝜋2 𝜈2 𝑁∗
< 𝛿2 + 𝛾2 𝑁∗
(3.19)
𝛾1 𝑇1
∗
− 𝜈1 𝑈1
∗
+ 𝛾1 𝑁∗
< (𝛽1 + 𝜈1 𝑁∗
) (3.20)
𝛾2 𝑇2
∗
− 𝜈2
∗
𝑈2
∗
+ 𝛾2 𝑁∗
< 𝛽2 + 𝜈2 𝑁∗
(3.21)
Hence, we can state the following theorem.
Theorem 1: The equilibrium point 𝐸2 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 𝐸2,
we need a lemma which establishes the region of attraction of 𝐸2.
Lemma 1: The region
Ω = 𝑁, 𝑁 𝐷, 𝑇1, 𝑇2, 𝑈1, 𝑈2 : 0 ≤ 𝑁 ≤ 𝐾0, 0 ≤ 𝑁 𝐷 ≤
𝑟1 + 𝑟2 𝑄1 + 𝑄2 𝐾0
𝑟1 + 𝑟2 𝑄1 + 𝑄2 + 𝛿 𝑚 𝛼 + 𝑑
,
0 ≤ 𝑇1 + 𝑇2 + 𝑈1 + 𝑈2 ≤
𝑄1 + 𝑄2
𝛿 𝑚
where 𝛿 𝑚 = min 𝛿1, 𝛿2, 𝛽1, 𝛽2
attracts all solution initiating in the interior of the positive orthant.
Proof: From the first equation of model (2.2),
we have,
𝑑𝑁
𝑑𝑡
≤ 𝑟𝑁 −
𝑟𝑁2
𝐾0
= 𝑟 1 −
𝑁
𝐾0
𝑁
Thus, lim⁡sup𝑡→∞ 𝑁 𝑡 ≤ 𝐾0.
From the last four equations of model (2.2),
we have,
𝑑𝑇1
𝑑𝑡
+
𝑑𝑇2
𝑑𝑡
+
𝑑𝑈1
𝑑𝑡
+
𝑑𝑈2
𝑑𝑡
= 𝑄1 + 𝑄2 − 𝛿1 𝑇1 + 𝛿2 𝑇2 + 𝛽1 𝑈1 + 𝛽2 𝑈2 − 1 − 𝜋1 𝜈1 𝑁𝑈1 − 1 − 𝜋2 𝜈2 𝑁𝑈2
≤ 𝑄1 + 𝑄2 − 𝛿 𝑚 𝑇1 + 𝑇2 + 𝑈1 + 𝑈2
where 𝛿 𝑚 = min⁡(δ1, δ2, β1
, β2
)
Thus, lim sup
𝑡→∞
𝑇1 + 𝑇2 + 𝑈1 + 𝑈2 ≤
𝑄1 + 𝑄2
𝛿 𝑚
From the second equation of model (2.2),
we have,
𝑑𝑁 𝐷
𝑑𝑡
= 𝑟1 𝑈1 + 𝑟2 𝑈2 𝑁 − 𝑁 𝐷 −
𝑟𝑁 𝐷 𝑁
𝐾 𝑇1, 𝑇2
− 𝛼 + 𝑑 𝑁 𝐷
≤
𝑟1 + 𝑟2 𝑄1 + 𝑄2
𝛿 𝑚
𝐾0 − 𝑁 𝐷 − 𝛼 + 𝑑 𝑁 𝐷
Thus, lim sup𝑡→∞ 𝑁 𝐷 𝑡 ≤
𝑟1 + 𝑟2 𝑄1 + 𝑄2 𝐾0
𝑟1 + 𝑟2 𝑄1 + 𝑄2 + 𝛿 𝑚 𝛼 + 𝑑

proving the lemma.□
The following theorem establishes global asymptotic stability conditions for the equilibrium point 𝐸2.
Theorem 2: Let 𝐾 𝑇 satisfies the following inequalities in Ω with the assumptions in equation (2.3):
𝐾 𝑚 ≤ 𝐾 𝑇 ≤ 𝐾0, 0 ≤ −
𝜕𝐾
𝜕𝑇1
𝑇1, 𝑇2 ≤ 𝜅1, 0 ≤ −
𝜕𝐾
𝜕𝑇2
𝑇1, 𝑇2 ≤ 𝜅2
where 𝐾 𝑚 , 𝜅1 & 𝜅2 are positive constants.
Then 𝐸2 is globally asymptotically stable with respect to all solutions initiating in the interior of the
positive orthant, if the following conditions hold in Ω:
𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
− 𝛼 + 𝑏 +
𝑟𝐾0 𝑟1 + 𝑟2 (𝑄1 + 𝑄2)
𝐾 𝑇1
∗
, 𝑇2
∗
𝑟1 + 𝑟2 𝑄1 + 𝑄2 + 𝛿 𝑚 𝛼 + 𝑑
2
<
4𝑟
25
(𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
)
𝑁∗
𝑁 𝐷
∗
2𝑁∗
𝐾 𝑇1
∗
, 𝑇2
∗ − 1 (3.22)
𝛾1 + 𝜋1 𝜈1
𝑄1 + 𝑄2
𝛿 𝑚
+
𝑟𝐾0
2
𝜅1
𝐾 𝑚
2
2
<
4𝑟
15
𝛿1 + 𝛾1 𝑁∗
2𝑁∗
𝐾 𝑇1
∗
, 𝑇2
∗ − 1 (3.23)
𝛾2 + 𝜋2 𝜈2
𝑄1 + 𝑄2
𝛿 𝑚
+
𝑟𝐾0
2
𝜅2
𝐾 𝑚
2
2
<
4𝑟
15
𝛿2 + 𝛾2 𝑁∗
2𝑁∗
𝐾 𝑇1
∗
, 𝑇2
∗ − 1 (3.24)
𝛾1 + 𝜈1
𝑄1 + 𝑄2
𝛿 𝑚
2
<
4𝑟
15
𝛽1 + 𝜈1 𝑁∗
2𝑁∗
𝐾 𝑇1
∗
, 𝑇2
∗ − 1 (3.25)
𝛾2 + 𝜈2
𝑄1 + 𝑄2
𝛿 𝑚
2
<
4𝑟
15
𝛽2 + 𝜈2 𝑁∗
2𝑁∗
𝐾 𝑇1
∗
, 𝑇2
∗ − 1 (3.26)
𝑟𝐾0 𝜅1 𝑟1 + 𝑟2 𝑄1 + 𝑄2
𝐾 𝑚
2 𝑟1 + 𝑟2 𝑄1 + 𝑄2 + 𝛼 + 𝑑 𝛿 𝑚
2
<
4
15
𝛿1 + 𝛾1 𝑁∗
𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
𝑁∗
𝑁 𝐷
∗ (3.27)
𝑟𝐾0 𝜅2 𝑟1 + 𝑟2 𝑄1 + 𝑄2
𝐾 𝑚
2 𝑟1 + 𝑟2 𝑄1 + 𝑄2 + 𝛼 + 𝑑 𝛿 𝑚
2
<
4
15
𝛿2 + 𝛾2 𝑁∗
𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
𝑁∗
𝑁 𝐷
∗ (3.28)
𝑟1 𝐾0
2
<
4
15
𝛽1 + 𝜈1 𝑁∗
𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
𝑁∗
𝑁 𝐷
∗ (3.29)
𝑟2 𝐾0
2
<
4
15
𝛽2 + 𝜈2 𝑁∗
𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
𝑁∗
𝑁 𝐷
∗ (3.30)
𝛾1 + 𝜋1 𝜈1 𝑁∗ 2
<
4
9
𝛿1 + 𝛾1 𝑁∗
𝛽1 + 𝜈1 𝑁∗
(3.31)
𝛾2 + 𝜋2 𝜈2 𝑁∗ 2
<
4
9
𝛿2 + 𝛾2 𝑁∗
𝛽2 + 𝜈2 𝑁∗
(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:
𝐾 𝑇1, 𝑇2 = 𝐾0 −
𝑏11 𝑇1
1 + 𝑏12 𝑇1
−
𝑏21 𝑇2
1 + 𝑏22 𝑇2
(4.1)
and assuming a set of parameters
𝑏 = 0.005, 𝑑 = 0.00001, 𝑟1 = 0.0007, 𝑟2 = 0.0005, 𝑄1 = 0.001, 𝑄2 = 0.0004
𝛿1 = 0.004, 𝛿2 = 0.001, 𝛾1 = 0.0005, 𝛾2 = 0.0003, 𝜋1 = 0.0004, 𝜋2 = 0.0006
𝜈1 = 0.005, 𝜈2 = 0.003, 𝛽1 = 0.006, 𝛽2 = 0.004, 𝐾0 = 10.0, 𝑏11 = 0.0002,
𝑏12 = 1.0, 𝑏21 = 0.0001, 𝑏22 = 2.0, 𝜅1 = 0.001, 𝜅2 = 0.001, 𝐾 𝑚 = 3.0
(4.2)
For the above function and set of values of parameters (4.1-4.2), we have obtained equilibrium point
𝐸2(𝑁∗
, 𝑁 𝐷
∗
, 𝑇1
∗
, 𝑇2
∗
, 𝑈1
∗
, 𝑈2
∗
) with values 𝑁∗
= 9.7771, 𝑁𝑑
∗
= 0.0227, 𝑇1
∗
= 0.1113, 𝑇2
∗
= 0.1002, 𝑈1
∗
= 0.0099

and 𝑈2
∗
= 0.0088. Here, condition (3.14) satisfies which shows that the values 𝑁∗
, 𝑁 𝐷
∗
, 𝑇1
∗
, 𝑇2
∗
, 𝑈1
∗
and 𝑈2
∗
are
unique in the region Ω. The eigenvalues of variational matrix 𝑀2 corresponding to the equilibrium point 𝐸2 for
the model (2.2) are obtained as −0.0559, −0.0339, −0.0090, −0.0039, −0.0050 + 0.0002𝑖 and
−0.0050 − 0.0002𝑖. We note that four eigenvalues of variational matrix are negative and remaining two
eigenvalues have negative real parts which show that equilibrium point 𝐸2 is locally asymptotically stable. Also,
the equilibrium point 𝐸2 satisfies all the conditions of global asymptotic stability (3.22-3.32). (see Fig.1)
Fig.1: Nonlinear stability of (𝑵∗, 𝑵 𝑫
∗
) in 𝑵 − 𝑵 𝑫 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 𝑄1 and 𝑄2 respectively. Here, we take all
the parameters same as eq. (4.2) except 𝑄1 and 𝑄2. In both figures, we can see that when emission rate of
toxicant 𝑄1 as well as emission rate of toxicant 𝑄2 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 𝑵 𝑫 with time for different values of 𝑸 𝟏

Fig.3: Variation of deformed population 𝑵 𝑫 with time for different values of 𝑸 𝟐
In Fig.4 & Fig.5, we have represented the variation in the density of deformed population for different
values of the uptake rate coefficients 𝛾1 and 𝛾2 (all the parameters same as eq. (4.2) except 𝛾1 and 𝛾2
respectively). Here figures are showing that when the uptake rates of toxicants increase, density of deformed
population increases.
Fig.4: Variation of deformed population 𝑵 𝑫 with time for different values of 𝜸 𝟏

Fig.5: Variation of deformed population 𝑵 𝑫 with time for different values of 𝜸 𝟐
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 𝛼 (all the parameters same as eq. (4.2) except 𝛼). 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 𝑵 𝑫 with time for different values of 𝜶
Fig.7: 𝑵 and 𝑵 𝑫 for large emission rate of toxicant 𝑻 𝟏 in the environment

Fig.8: 𝑵 and 𝑵 𝑫 for large emission rate of toxicant 𝑻 𝟐 in the environment
In Fig.7 & Fig.8, we have represented the variation in the densities of Total population (𝑁) and
Deformed population 𝑁 𝐷 for large emission rate of toxicants 𝑄1 and 𝑄2. 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 𝐸1 and 𝐸2 in which 𝐸1 is
saddle point and 𝐸2 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 𝐸2
𝑊 𝑁, 𝑁 𝐷, 𝑇1, 𝑇2, 𝑈1, 𝑈2
=
1
2
𝑁 − 𝑁∗ 2
+
1
2
𝑁 𝐷 − 𝑁 𝐷
∗ 2
+
1
2
𝑇1 − 𝑇1
∗ 2
+
1
2
𝑇2 − 𝑇2
∗ 2
+
1
2
𝑈1 − 𝑈1
∗ 2
+
1
2
𝑈2 − 𝑈2
∗ 2
Differentiating 𝑊 with respect to 𝑡 along the solution of (2.2), we get
𝑑𝑊
𝑑𝑡
= 𝑁 − 𝑁∗
𝑟𝑁 −
𝑟𝑁2
𝐾 𝑇1, 𝑇2
− 𝛼 + 𝑏 𝑁 𝐷
+ 𝑁 𝐷 − 𝑁 𝐷
∗
𝑟1 𝑈1 + 𝑟2 𝑈2 𝑁 − 𝑁 𝐷 −
𝑟𝑁 𝐷 𝑁
𝐾 𝑇1, 𝑇2
− 𝛼 + 𝑑 𝑁 𝐷
+ 𝑇1 − 𝑇1
∗
𝑄1 − 𝛿1 𝑇1 − 𝛾1 𝑇1 𝑁 + 𝜋1 𝜈1 𝑁𝑈1 + 𝑇2 − 𝑇2
∗
𝑄2 − 𝛿2 𝑇2 − 𝛾2 𝑇2 𝑁 + 𝜋2 𝜈2 𝑁𝑈2
+ 𝑈1 − 𝑈1
∗
𝛾1 𝑇1 𝑁 − 𝛽1 𝑈1 − 𝜈1 𝑁𝑈1 + 𝑈2 − 𝑈2
∗
𝛾2 𝑇2 𝑁 − 𝛽2 𝑈2 − 𝜈2 𝑁𝑈2

using (3.1-3.8), we get after some calculation
𝑑𝑊
𝑑𝑡
= − 𝑟
2𝑁∗
𝐾 𝑇1
∗
, 𝑇2
∗ − 1 𝑁 − 𝑁∗ 2
− (𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
)
𝑁∗
𝑁 𝐷
∗ 𝑁 𝐷 − 𝑁 𝐷
∗ 2
− 𝛿1 + 𝛾1 𝑁∗
𝑇1 − 𝑇1
∗ 2
− 𝛿2 + 𝛾2 𝑁∗
𝑇2 − 𝑇2
∗ 2
− 𝛽1 + 𝜈1 𝑁∗
𝑈1 − 𝑈1
∗ 2
− 𝛽2 + 𝜈2 𝑁∗
𝑈2 − 𝑈2
∗ 2
+ − 𝛼 + 𝑏 + 𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
−
𝑟𝑁 𝐷
𝐾 𝑇1
∗
, 𝑇2
∗ 𝑁 − 𝑁∗
∗
+ 𝜋1 𝜈1 𝑈1 − 𝛾1 𝑇1 − 𝑟𝑁2
𝜂1 𝑇1, 𝑇2 𝑁 − 𝑁∗
𝑇1 − 𝑇1
∗
+ 𝜋2 𝜈2 𝑈2 − 𝛾2 𝑇2 − 𝑟𝑁2
𝜂2 𝑇1
∗
, 𝑇2 𝑁 − 𝑁∗
𝑇2 − 𝑇2
∗
+ 𝛾1 𝑇1 − 𝜈1 𝑈1 𝑁 − 𝑁∗
𝑈1 − 𝑈1
∗
+ 𝛾2 𝑇2 − 𝜈2 𝑈2 𝑁 − 𝑁∗
𝑈2 − 𝑈2
∗
− 𝑟𝑁𝑁 𝐷 𝜂1 𝑇1, 𝑇2 𝑁 𝐷 − 𝑁 𝐷
∗
𝑇1 − 𝑇1
∗
− 𝑟𝑁𝑁 𝐷 𝜂2 𝑇1
∗
, 𝑇2 𝑁 𝐷 − 𝑁 𝐷
∗
𝑇2 − 𝑇2
∗
+ 𝑟1 𝑁 − 𝑁 𝐷 𝑁 𝐷 − 𝑁 𝐷
∗
𝑈1 − 𝑈1
∗
+ 𝑟2 𝑁 − 𝑁 𝐷 𝑁𝐷 − 𝑁 𝐷
∗
𝑈2 − 𝑈2
∗
+ 𝜋1 𝜈1 𝑁∗
+ 𝛾1 𝑁∗
𝑇1 − 𝑇1
∗
(𝑈1 − 𝑈1
∗
) + 𝜋2 𝜈2 𝑁∗
+ 𝛾2 𝑁∗
𝑇2 − 𝑇2
∗
(𝑈2 − 𝑈2
∗
)
where,
𝜂1 𝑇1, 𝑇2 =
1
𝐾 𝑇1,𝑇2
−
1
𝐾 𝑇1
∗,𝑇2
𝑇1 − 𝑇1
∗ , 𝑇1 ≠ 𝑇1
∗
−
1
𝐾2 𝑇1
∗
, 𝑇2
𝜕𝐾
𝜕𝑇1
𝑇1
∗
, 𝑇2 , 𝑇1 = 𝑇1
∗
,
𝜂2 𝑇1
∗
, 𝑇2 =
1
𝐾 𝑇1
∗,𝑇2
−
1
𝐾 𝑇1
∗,𝑇2
∗
𝑇2 − 𝑇2
∗ , 𝑇2 ≠ 𝑇2
∗
−
1
𝐾2 𝑇1
∗
, 𝑇2
∗
𝜕𝐾
𝜕𝑇2
𝑇1
∗
, 𝑇2
∗
, 𝑇2 = 𝑇2
∗
Thus,
𝑑𝑤
𝑑𝑡
can be written as sum of the quadratics,
𝑑𝑤
𝑑𝑡
= −
1
2
𝑏11 𝑁 − 𝑁∗ 2
+ 𝑏12 𝑁 − 𝑁∗
∗
−
1
2
𝑏22 𝑁 𝐷 − 𝑁 𝐷
∗ 2
+ −
1
2
𝑏11 𝑁 − 𝑁∗ 2
+ 𝑏13 𝑁 − 𝑁∗
𝑇1 − 𝑇1
∗
−
1
2
𝑏33 𝑇1 − 𝑇1
∗ 2
+ −
1
2
𝑏11 𝑁 − 𝑁∗ 2
+ 𝑏14 𝑁 − 𝑁∗
𝑇2 − 𝑇2
∗
−
1
2
𝑏44 𝑇2 − 𝑇2
∗ 2
+ −
1
2
𝑏11 𝑁 − 𝑁∗ 2
+ 𝑏15 𝑁 − 𝑁∗
𝑈1 − 𝑈1
∗
−
1
2
𝑏55 𝑈1 − 𝑈1
∗ 2
+ −
1
2
𝑏11 𝑁 − 𝑁∗ 2
+ 𝑏16 𝑁 − 𝑁∗
𝑈2 − 𝑈2
∗
−
1
2
𝑏66 𝑈2 − 𝑈2
∗ 2
+ −
1
2
𝑏22 𝑁 𝐷 − 𝑁 𝐷
∗ 2
+ 𝑏23 𝑁 𝐷 − 𝑁 𝐷
∗
𝑇1 − 𝑇1
∗
−
1
2
𝑏33 𝑇1 − 𝑇1
∗ 2
+ −
1
2
𝑏22 𝑁 𝐷 − 𝑁 𝐷
∗ 2
+ 𝑏24 𝑁 𝐷 − 𝑁 𝐷
∗
𝑇2 − 𝑇2
∗
−
1
2
𝑏44 𝑇2 − 𝑇2
∗ 2
+ −
1
2
𝑏22 𝑁 𝐷 − 𝑁 𝐷
∗ 2
+ 𝑏25 𝑁 𝐷 − 𝑁 𝐷
∗
𝑈1 − 𝑈1
∗
−
1
2
𝑏55 𝑈1 − 𝑈1
∗ 2
+ −
1
2
𝑏22 𝑁 𝐷 − 𝑁 𝐷
∗ 2
+ 𝑏26 𝑁 𝐷 − 𝑁 𝐷
∗
𝑈2 − 𝑈2
∗
−
1
2
𝑏66 𝑈2 − 𝑈2
∗ 2
+ −
1
2
𝑏33 𝑇1 − 𝑇1
∗ 2
+ 𝑏35 𝑇1 − 𝑇1
∗
𝑈1 − 𝑈1
∗
−
1
2
𝑏55 𝑈1 − 𝑈1
∗ 2
+ −
1
2
𝑏44 𝑇2 − 𝑇2
∗ 2
+ 𝑏46 𝑇2 − 𝑇2
∗
𝑈2 − 𝑈2
∗
−
1
2
𝑏66 𝑈2 − 𝑈2
∗ 2
where,
𝑏11 =
2
5
𝑟
2𝑁∗
𝐾 𝑇1
∗
, 𝑇2
∗ − 1 , 𝑏22 =
2
5
(𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
)
𝑁∗
𝑁 𝐷
∗ , 𝑏33 =
2
3
𝛿1 + 𝛾1 𝑁∗
𝑏44 =
2
3
𝛿2 + 𝛾2 𝑁∗
, 𝑏55 =
2
3
𝛽1 + 𝜈1 𝑁∗
, 𝑏66 =
2
3
𝛽2 + 𝜈2 𝑁∗

𝑏12 = − 𝛼 + 𝑏 + 𝑟1 𝑈1
∗
+ 𝑟2 𝑈2
∗
−
𝑟𝑁 𝐷
𝐾 𝑇1
∗
, 𝑇2
∗ , 𝑏13 = 𝜋1 𝜈1 𝑈1 − 𝛾1 𝑇1 − 𝑟𝑁2
𝜂1 𝑇1, 𝑇2
𝑏14 = 𝜋2 𝜈2 𝑈2 − 𝛾2 𝑇2 − 𝑟𝑁2
𝜂2 𝑇1
∗
, 𝑇2 , 𝑏15 = 𝛾1 𝑇1 − 𝜈1 𝑈1 , 𝑏16 = 𝛾2 𝑇2 − 𝜈2 𝑈2
𝑏23 = −𝑟𝑁𝑁 𝐷 𝜂1 𝑇1, 𝑇2 , 𝑏24 = −𝑟𝑁𝑁 𝐷 𝜂2 𝑇1
∗
, 𝑇2 , 𝑏25 = 𝑟1 𝑁 − 𝑁 𝐷 , 𝑏26 = 𝑟2 𝑁 − 𝑁 𝐷
𝑏35 = 𝜋1 𝜈1 𝑁∗
+ 𝛾1 𝑁∗
, 𝑏46 = 𝜋2 𝜈2 𝑁∗
+ 𝛾2 𝑁∗
Thus,
dW
dt
will be negative definite provided
𝑏12
2
< 𝑏11 𝑏22 (3.33)
𝑏13
2
< 𝑏11 𝑏33 (3.34)
𝑏14
2
< 𝑏11 𝑏44 (3.35)
𝑏15
2
< 𝑏11 𝑏55 (3.36)
𝑏16
2
< 𝑏11 𝑏66 3.37
𝑏23
2
< 𝑏22 𝑏33 (3.38)
𝑏24
2
< 𝑏22 𝑏44 (3.39)
𝑏25
2
< 𝑏22 𝑏55 (3.40)
𝑏26
2
< 𝑏22 𝑏66 (3.41)
𝑏35
2
< 𝑏33 𝑏55 (3.42)
𝑏46
2
< 𝑏44 𝑏66 (3.43)
We note that (3.33-3.43) ⇒ (3.22-3.32) respectively. So, W is a Lyapunov’s function with respect to
the equilibrium 𝐸2 and therefore 𝐸2 is globally asymptotically stable under the conditions (3.22-3.32). Hence the
theorem. □
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