The effect of an external toxicant on a biological species in case of deformity: a model
Model. Earth Syst. Environ.
The effect of an external toxicant on a biological species in case of deformity: a model
Anuj Kumar Agarwal 0 1 2 3 4
A. W. Khan 0 1 2 3 4
A. K. Agrawal 0 1 2 3 4
0 & Anuj Kumar Agarwal
1 Mathematics Subject Classification 34 C60 92D25 93A30
2 Department of Mathematics, Amity University , Lucknow, Uttar Pradesh , India
3 Department of Mathematics, Integral University , Lucknow, Uttar Pradesh , India
4 A. K. Agrawal
In this paper, a mathematical model is proposed and analyzed to study the effect of an external toxicant on a biological species. Here, we have considered that the toxicant is constantly emitted in the environment form some external source and aftereffect of this external toxicant some members of biological species shows deformity as incapable in reproduction. The analytical results of model system are established by stability analysis and Hopfbifurcation theory. The model's results show, when emission of external toxicant increases, total population density decreases and density of deformed subclass increases. For highly emission of external toxicant, system become unstable and shows a supercritical Hopfbifurcation. To verify the analytical results, a numerical simulation is provided.
Mathematical model; Biological species; Toxicant; Deformity; Hopfbifurcation

37L10
Introduction
Mathematical models are used in largescale to predict the
various nature of real life problems; e.g. in ecology,
epidemiology, ecotoxicology and other many problems.
Many researchers also used mathematical models to
predict the growth of biological species in toxic environment.
They have proposed and analyzed mathematical models
by considering different cases, such as effect of a single
toxicant or more than one toxicant on biological species,
allelopathy case, deformity in a subclass of species, etc.
(Freedman and Shukla 1991; Shukla and Agrawal 1999;
Shukla et al. 2003; Agrawal and Shukla 2012; Kumar
et al. 2016)
, to provide important insights for the effect of
toxicants on biological species. In particular,
Freedman
and Shukla (1991)
have studied the effect a single
toxicant on a species with a consideration that toxicant
affected on the growth rate and decreasing the carrying
capacity of the environment.
Shukla and Agrawal (1999)
have proposed a model by considering a situation in
which toxicant emitted by a biological species and
decreased the density of other biological species (case of
allelopathy). As an interesting observable fact,
Agrawal
and Shukla (2012)
studied a model for the aftereffect of
a single toxicant a subclass of biological species shows
deformity as incapable in reproduction. Here, it is
assumed that toxicant is emitted in the environment from
some external source. Further, understand the case of
deformity aftereffect of a toxicant in more meaningful
manner,
Kumar et al. (2016)
proposed and analyzed a
model with an assumption that toxicant emitted by
biological species itself. They have shown, toxicant
decreased the total population density and a subclass of
species suffers from deformity. For higher emission rate,
the model system becomes unstable.
In this paper, we proposed and analyzed a mathematical
model to study the effect of an external toxicant on a
biological species a subclass of which is severely affected
and gets deformed. This case is similar to the case studied
by
Agrawal and Shukla (2012)
. But, the proposed model in
this study is a modified version of the model by
Agrawal
and Shukla (2012)
. The results obtained by modified model
are more closer to real life in case of deformity. In this
study, we also check the existence of hopfbifurcation and
the nature of bifurcating periodic solutions.
Mathematical model
We assume a biological species of population density N(t)
at time t, is logistically growing and surviving in a polluted
environment. This polluted environment having a toxicant
which is constantly emitted in the environment from some
external sources. The environmental concentration of this
toxicant is T(t) at time t. We assume that U(t) is the
concentration of toxicant T(t), taken up by the biological
species N(t) at time t. This toxicant is decreasing the
growth rate of species N(t) as well as a subclass of species
with population density NDðtÞ shows deformity as
incapability in reproduction. The remaining population density
which is free from deformity is assumed as NAðtÞ. Keeping
these facts in mind, we propose the following mathematical
model:
ddNtA ¼ ðb
dND
dt ¼ r1UNA
dT
dt ¼ Q
dU
dt ¼ cTN
dÞNA
r1UNA
rNDN
KðT Þ
rNAN
KðTÞ
All the parameters considered in the model are positive
constants. b and d are the natural birth and death rate of
biological species. r represents the intrinsic growth rate of
biological species. Q is the rate at which external toxicant
is constantly emitted in the environment. The external
toxicant T(t) is uptaken by the species at the rate c.
Aftereffect of this toxicant, the deformedfree population density
decreases at the rate r1. a is the mortality rate of deformed
population due to high toxicity in the environment. d and b
are naturally depletion rates of T(t) and U(t) respectively.
U(t) is depleted at the rate m due to die out of some
members of species and a fraction p of this depletion is
reentered into the environment. c [ 0 is a proportionality
ð2Þ
ð3Þ
ð4aÞ
ð4bÞ
ð4cÞ
ð4dÞ
The region of attraction X that attracts all the positive
solution of model system (3), is as follows:
X ¼ ðN; ND; T; UÞ : 0
N
K0; 0
ND
r1QK0
ðr1Q þ dmða þ dÞÞ
;
0 T þ U
Q
dm
where dm ¼ minðd; bÞ.
The proof of region of attraction X is given in Appendix
Equilibrium points
The model system (3) have two nonnegative equilibrium
points E1ð0; 0; Qd ; 0Þ and E2ðN ; ND; T ; U Þ. Here, the
existence of E1 is understandable, hence omitted. The
existence of E2 is as follows:
The values N ; ND; T and U of the equilibrium point E2
are the positive solutions of the following nonlinear equations,
constant used to calculate the initial uptake concentration
of toxicant. K(T) is a decreasing function of T to measure
the carrying capacity of the environment.
i.e.
ð1Þ
N ¼ ðr
r1UÞKðT Þ
r
r1UNKðTÞ
ND ¼ rN þ ðr1U þ a þ dÞKðTÞ
Qðb þ mNÞ
T ¼
f ðNÞ
QcN
U ¼ f ðNÞ ¼ hðNÞ ðsayÞ
¼ gðNÞ ðsayÞ
where
f ðNÞ ¼ db þ ðcb þ dmÞN þ cmð1
pÞN2
ð4eÞ
The Eq. (4c) shows that T is directly proportional to the
parameter Q and from the Eq. (2), carrying capacity
K(T) decreases as T increase. Hence, the carrying capacity
of the environment decreases when the emission rate of
external toxicant Q increases.
Let
pÞN þ m2ð1
pÞN2g\0
FðNÞ ¼ rN
at N ¼ 0,
Fð0Þ ¼
rK
at N ¼ K0
ðr
Q
d
r1hðNÞÞKðgðNÞÞ
\0
FðK0Þ ¼ rK0
ðr
r1hðK0ÞÞKðgðK0ÞÞ [ 0
Eqs. (6) and (7) show that FðNÞ ¼ 0 has a solution in the
interval ½0; K0 .
Also, The root N of FðNÞ ¼ 0 is unique, if
ddNF ¼ r þ r1KðgðNÞÞ ddNh ðr r1hðNÞÞ ddKT ddNg [ 0
ð5Þ
ð6Þ
ð7Þ
ð8Þ
ð9aÞ
ð9bÞ
ð10Þ
E2.
X_ ¼ M2X þ N2
where,
2 n 3
X ¼ 6646 nsd 7577;
u
Moreover,
m11 ¼
r
m21 ¼ r1U
from, Eqs. (4c) and (4d)
dg
dN ¼
Qc 2
f 2ðNÞ fb þ 2bmð1
dh Qc
dN ¼ f 2ðNÞ fdb
cmð1
pÞN2g
since, ddKT \0 (from (2)) and ddNg (from (9a))
so,
ðr
dK dg
r1hðNÞÞ dT dN
The equation FðNÞ ¼ 0 has a unique root N , only when
dh
r þ r1KðgðNÞÞ dN
Dynamical behavior corresponding to E1
The Jacobian matrix M1 corresponding to the equilibrium
points E1 is as follows
2 r ða þ bÞ 0 0 3
M1 ¼ 6666664 c0QcdQ ða00þ dÞ 00d 00b 7777757
d
Since, one of the eigenvalue of Jacobian matrix M1 is r [ 0
and all the other eigenvalues are ða þ dÞ; d; b\0,
which confirm that E1 is a saddle point locally
unstable manifold in N direction and locally stable manifold in
ND T U space.
Dynamical behavior corresponding to E2
To study the dynamical behavior, we linearize the model
system (3) corresponding to the equilibrium point E2 ¼
ðN ; ND; T ; U Þ by taking the following transformation:
N ¼ N þ n; ND ¼ ND þ nd; T ¼ T þ s; U ¼ U
þ u:
here, n; nd; s and u are taken as small perturbations around
So, the model system (3) can be written in the terms of
n; nd; s and u as follows:
ð11Þ
M2 is a Jacobian matrix corresponding to the equilibrium
point E2. Thus, the characteristic equation of M2 can be
written as:
pðxÞ ¼ x4 þ c1x3 þ c2x2 þ c3x þ c4
½cj Q¼Q [ 0 for ðj ¼ 1; . . .; 4Þ
½H2 Q¼Q ¼ ½c1c2
Hence, we can state the following theorem to set up the
local asymptotically stablility corresponding to the
equilibrium point E2.
Theorem 1 The equilibrium point E2 of model system (3)
is locally asymptotically stable under the conditions (13).
Existence of Hopfbifurcation
The model system (3), has a possibility of Hopfbifurcation
(Hassard et al. 1981; Kuznetsov 2004; Seydel 2009)
corresponding to the equilibrium point E2. By treating Q (i.e. the
emission rate of external toxicant) as a bifurcation parameter,
we check the existence of Hopfbifurcation. It is obvious that a
Hopfbifurcation may exist if all the eigenvalues of Jacobian
matrix are having negative real parts except a purely
imaginary complex conjugate pair. In this case, the Jacobian matrix
M2 having four eigenvalues xj ¼ Rj þ iIj ðj ¼ 1; . . .; 4Þ (say).
So, the Hopfbifurcation exist only when R1; R2 ¼ 0; I1 ¼
I2 ¼6 0 & R3; R4\0 at the critical value Q ¼ Q (say).
According to the Liu’s criterion
(Liu 1994)
, the model
system (3) undergoes a Hopfbifurcation at the critical
value Q ¼ Q [ 0, if
ð14aÞ
ð14bÞ
ð14cÞ
ð14dÞ
dRj
dQ Q¼Q
6¼0; for j ¼ 1; 2
A New Detecting Method For Conditions of Existence of
Hopfbifurcation
(Jiaqi and Zhujun 1995)
describe the last
condition (14d) in the terms of coefficients of characteristic
Eq. (12) as follows:
dR
dQ Q¼Q
¼
"ddQ ðc1c2c3
2c1ð4c4
20 v 0 0 3
6 v 0 0
J ¼ P 1M2P ¼ 660 0 J1 00 7577 and
4
0 0 0 J2
2F1ðy1;y2;y3;y4Þ3
f ¼ 6664ff23ððyy11;;yy22;;yy33;;yy44ÞÞ7757
f4ðy1;y2;y3;y4Þ
Here,
4 4 4 4
n ¼ XP1jyj; nd ¼ XP2jyj; s ¼ XP3jyj; u ¼ XP4jyj
j¼1 j¼1 j¼1 j¼1
Now, we evaluated the following quantities at critical value
of parameter Q ¼ Q and ðy1;y2;y3;y4Þ ¼ ð0;0;0;0Þ.
l2 ¼
Hence, the following theorem express the nature of
bifurcating periodic solutions.
Theorem 3 If l2 [ 0 (or l2\0), the model system (3)
shows a supercritical (or subcritical) Hopfbifurcation and
the bifurcating periodic solutions exist for Q [ Q (or
Q\Q ), if b2\0 (or b2 [ 0), the bifurcating periodic
solutions are stable (or unstable), if s2 [ 0 (or s2\0), the
period of bifurcating solutions increases (or decreases).
Numerical simulation
We provide numerical simulation to back up our analytical
results for the model system (3). A matlab package
MATCONT
(Dhooge et al. 2003)
is used for the graphical
representation of model system (3).
We assume, the carrying capacity function as
b1T
1 þ b2T
KðT Þ ¼ K0
and a set of parameters as:
b ¼ 0:55;
a ¼ 0:0002;
p ¼ 0:02;
b1 ¼ 0:02;
d ¼ 0:0006;
d ¼ 0:08;
m ¼ 0:0002;
b2 ¼ 1:0
Fig. 1 Timeseries graph of
total and deformed population
corresponding to the parameter
Q
ð18Þ
ð19aÞ
ð19bÞ
The equilibrium point E2 contains the value
N ¼ 9:5989; ND ¼ 0:3834; T
The condition (10) holds and ddNF ¼ 0:5495 [ 0, which
show that N is unique, in addition E2 is unique. The local
stability conditions (13) corresponding to E2 are also
satisfied.
Figure 1 shows the total density and density of deformed
subclass of biological species corresponding to the
parameter Q (the remaining parameters are same as (19b)).
The Fig. 1 shows that when the emission rate of external
toxicant increases, the total density N decrease and the
density of deformed subclass ND firstly increase then
decrease with N. For large emission rate Qð¼ 0:950Þ both
densities are oscillating.
Figure 2 shows the real and imaginary parts of
eigenvalues of Jacobian matrix M2 corresponding to the
parameter Q. The real parts of all eigenvalues (i.e. Ri\0,
i ¼ 1; ; 4) are negative for Q\Q ð¼ 0:83648Þ. At Q ¼
Q two eigenvalues become purely imaginary (i.e. R1 ¼
R2 ¼ 0 and I1 ¼ I2 6¼ 0), which confirms that a model
system (3) undergoes a Hopfbifurcation at Q ¼ 0:83648.
Figure 3 shows the densities of both populations N and
ND with respect to the emission rate of external toxicant Q.
Both densities N and ND become stable at equilibrium level
for Q\Q . After crossing the critical value
Q ð¼ 0:83648Þ, the equilibrium point losses its stability
and a supercritical Hopfbifurcation occurs (since
l2 ¼ 2:2456 10 4 [ 0). Both densities start oscillating
around their equilibrium level with stable bifurcating
periodic orbits (since b2 ¼ 3:5116 10 7\0).
Conclusion
A mathematical model is proposed to examine the growth of
biological species in the case a subclass of species shows
deformity, when an external toxicant is constantly emitted in
the environment. The analytical results of model show, as
emission rate of external toxicant increases, total population
density decreases and density of deformed subclass
population firstly increase then decrease with total population
density. If emission rate crosses the critical value, the model
system shows a supercritical Hopfbifurcation and all the
bifurcating periodic solutions of model system are stable.
Appendix A: Proof of the region of attraction X
Proof From the model system (3), we have
dN
dt
rN
rN2
K0
¼ r 1
Thus, lim supt!1 NðtÞ
Also, we have
N
K0
K0
N
dT dU
dt þ dt ¼ Q
Q
dT
bU
dmðT þ UÞ
ð1
pÞmNU
where dm ¼ minðd; bÞ:
Thus, lim supt!1ðTðtÞ þ UðtÞÞ dQm
From the second equation of model (3), we have
ða þ dÞND
Q
r1 dm ðK0
NDÞ
Thus, lim supt!1 NDðtÞ
r1QK0
½r1QþdmðaþdÞ
providing the region of attraction X:
h
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