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ORIGINAL ARTICLE

Inverse problem of the Holling-Tanner model
and its solution

Adejimi Adesola Adeniji, Igor Fedotov, Michael Y. Shatalov
Department of Mathematics and Statistics, Tshwane University of Technology

adejimi.adeniji@gmail.com, fedoptovi@tut.ac.za, shatalovm@tut.ac.za

Received: 8 August 2018, accepted: 5 December 2018, published: 19 December 2018

Abstract—In this paper we undertake to consider
the inverse problem of parameter identification of
nonlinear system of ordinary differential equations
for a specific case of complete information about
solution of the Holling-Tanner model for finite
number of points for the finite time interval. In this
model the equations are nonlinearly dependent on
the unknown parameters. By means of the proposed
transformation the obtained equations become lin-
early dependent on new parameters functionally
dependent on the original ones. This simplification
is achieved by the fact that the new set of pa-
rameters becomes dependent and the corresponding
constraint between the parameters is nonlinear. If
the conventional approach based on introduction of
the Lagrange multiplier is used this circumstance
will result in a nonlinear system of equations. A
novel algorithm of the problem solution is proposed
in which only one nonlinear equation instead of
the system of six nonlinear equations has to be
solved. Differentiation and integration methods of
the problem solution are implemented and it is
shown that the integration method produces more
accurate results and uses less number of points on
the given time interval.

Keywords-Parameter estimation, Goal function,
Absolute error curves, Inverse method, Holling-

Tanner model, Least square method, Differentiation
method, Integration method

I. INTRODUCTION

The numerical evaluation of known coefficient
of a dynamical system i.e. the problem of dy-
namical system identification, is one of the most
important problem of the mathematical biology
[1], ecology [2], [3], [4], etc. Usually, to identify
a dynamics of a system, it is necessary to have
certain statistical information for time values about
the unknown functions of this system. In the
present paper we consider the inverse problem
of parameter identification of the Holling-Tanner
predator-prey model [5], [6]. This model is widely
used in mathematical biology, for example, in the
study of transmissible disease [7]. Several investi-
gations have been done by various researchers on
the mite-spider-mite, lynx-hare and sparrow-hawk-
sparrow competition [8], [9], [10]. In [11], the
authors proposed a method consisting in the direct
integration of a given dynamical system with the
subsequent application of quadrature rules and the
least square method [12], [13] provided that there

Copyright: c© 2018 Adeniji et al. This article is distributed under the terms of the Creative Commons Attribution License
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source are credited.
Citation: Adejimi Adesola Adeniji, Igor Fedotov, Michael Y. Shatalov, Inverse problem of the
Holling-Tanner model and its solution, Biomath 7 (2018), 1812057,
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A.A. Adeniji, I. Fedotov, M.Y. Shatalov, Inverse problem of the Holling-Tanner model and its solution

is complete statistical information about the un-
known function. In this paper, we assume that the
complete information about the competing species
is available and the two methods of solution, dif-
ferentiation and integration methods, are proposed.
The problem of the Holling-Tanner model iden-
tification has its specifics, because it nonlinearly
depends on the unknown parameters. It is possible
to transform this model to a new form where the
equations of the system linearly depends on the set
of new parameters. These new parameters are not
independent and we need to consider the constraint
between the parameters, which are nonlinear. The
Holling-Tanner model has only one constraint and
hence, can be simply treated by a novel method
developed by the authors. The theoretical consid-
erations are accompanied by numerical examples
where the developed algorithm is tested for both
differentiation and integration methods of solution.
It is shown that the integration methods is more
accurate than the differentiation one and needs less
amount of experimental information.

II. MAIN RESULTS

In our paper we consider the Holling-Tanner
model [9], [14] described by the following system
of equations:



ẋ = b1x− b2x2 − b3 x·yb4+x,
ẏ = b5y − b6 y

2

x
,

t = 0, x(0) = x0, y(0) = y0

(1)

where x = x(t), y = y(t), ẋ = dx(t)
dt

, ẏ = dy(t)
dt

,
t is time and b1, · · ·b6 are positive constant pa-
rameters [15]. Initial conditions for this system
are formulated so that at t = 0 : x(t = 0) =
x0 > 0 and y(t = 0) = y0 > 0. The main results
relating to solution of this initial value problem
were obtained in [10], [16], [17], [18] as

• Solution of the initial value problem (1)
{x(t),y(t)} with positive initial conditions is
positive, i.e. x(t) > 0 and y(t) > 0 for t ≥ 0.

• Initial value problem (1) has the positive
steady-state solution [15] (x̃, ỹ) which cor-

responds to either stable focus or stable node
critical point depending on b1, · · ·b6 so that:

x̃=
b1b6−b3b5−b2b4b6 +

√
∆

2b2b6
> 0,

(2)
ỹ =

b5(b1b6−b3b5−b2b4b6 +
√

∆)

2b2b
2
6

> 0.

where

∆ = (b1b6−b3b5− b2b4b6)2 +4b1b2b4b26.

• Initial value problem (1) has unstable steady-
state solution

(˜̃x, ˜̃y) = (
b1
b2
, 0),

which corresponds to the saddle critical point.

III. ON SOLVABILITY OF IDENTIFICATION
PROBLEM

Assume that solution of initial problem (1),
x(t) and y(t) is given on the finite time interval
t ∈ [0,T] with initial t = 0 and terminal t = T
time instants in N + 1 equispaced time instants
ti =

T
N
i ∈ [0,T]:

xi = x(ti), yi = y(ti) (i = 0, · · · ,N) (3)

Lets us formulate the identification problem for
parameters b1, · · ·b6 from the known solution (3)
This problem can be solved if the conditions of
the following theorem are satisfied:

Theorem 1. Parameters b1,b2,b3,b4 of model
(1) can be identified by the least squares
method if (N + 1) × 1-vector columns
[xi] ,

[
x2i
]
,
[
x3i
]
, [ẋi] , [xi,yi] are linearly

independent. Parameters b5 and b6 of the
above mentioned model can be identified by the
mentioned method if (N + 1) × 1-vector columns
[yi] and [

y2i
x1

] are linearly independent.

Proof: By multiplying the first equation of
system (1) by (b4 + x) and grouping the resulting
terms we obtain

C1(−x3(t)) + C2(−x(t)y(t)) + C3(−ẋ(t))
+C4(x(t)) + C5(x

2(t)) + (−x(t)ẋ(t)) = 0, (4)

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A.A. Adeniji, I. Fedotov, M.Y. Shatalov, Inverse problem of the Holling-Tanner model and its solution

where C1 = b2, C2 = b3, C3 = b4, C4 = b1b4,
C5 = b1−b2b4 are new unknown parameters. It is
easy to check that the parameters C1,C3,C4,C5
satisfy the following constrains: C1C23 + C3C5 −
C4. Considering x(t) and y(t) in time instants t =
ti we obtain the following overdetermined system
of N + 1 linear algebraic equations:

C1 ~f1 +C2 ~f2 +C3 ~f3 +C4 ~f4 +C5 ~f5−~f6 = 0, (5)

where

~f1 = [f1i] =
[
−x3i

]
, ~f2 = [f2i] = [−xiyi] ,

~f3 = [f3i] = [−ẋi] , ~f4 = [f4i] = [xi] ,
~f5 = [f5i] =

[
x2i
]
, and ~f6 = [f6i] = [xiẋi]

are (N + 1) × 1-vector columns. Hence, the un-
known parameters C1, C2, C3, C4 and C5 can be
found by, for example, the least squares method
[19] by means of the constrained minimization of
function G1:

G1 = G1(C1,C2,C3,C4,C5,λ) =

=
1

2
(C1 ~f1+C2 ~f2+C3 ~f3+C4 ~f4+C5 ~f5−~f6)T

(C1 ~f1+C2 ~f2+C3 ~f3+C4 ~f4+C5 ~f5−~f6)
+ λ(C1C

2
3 + C3C5 −C4) −→ min (6)

This problem can be solved providing that vectors
~f1, · · · , ~f5 are linearly independent in (6). The
last term contains the Lagrange multiplier λ and
the constraint between coefficients C1, · · · ,C5.
Moreover, the second equation of system (1) can
be rewritten in time instants t = ti as the following
overdetermined system of N + 1 linear algebraic
equations:

C6 ~f7 + C7 ~f8 − ~f9 = 0, (7)

where

~f7 = [f7i] = [yi] , ~f8 = [f8i] =

[
−y2i
xi

]
,

~f9 = [f9i] = [ẏi] ,C6 = b5,C7 = b6.

That is why coefficients C6,C7 can be found by
application of the least square method by means

of minimization of function G2

G2 =G2(C6,C7) =
1

2

(
C6 ~f7 + C7 ~f8 + ~f9

)T
(C6 ~f7 + C7 ~f8 + ~f9) −→ min (8)

This problem can be solved providing that vectors
~f7 and ~f8 are linearly independent of (8).

Remark 2. In vectors ~f3, ~f6 the component ẋi,
and in vector ~f9 the components ẏi are calculated
by means of numerical differentiation of xi,yi with
respect to time t and that is why the proposed
method is called the differential method of identi-
fication.

Corollary 3. Parameters b1,b2,b3,b4 of the model
(1) can be identified by the least square method
[19] if (N + 1) × 1-vector columns[∫ ti

0
x(τ)dτ

]
,

[∫ ti
0
x2(τ)dτ

]
,

[∫ ti
0
x3(τ)dτ

]
,

[xi −x0] ,
[∫ ti

0
x(τ)y(τ)dτ

]
are linearly dependent. Parameters b5 and b6 of
the abovementioned model can be identified by
the abovementioned method if (N + 1)×1-vector

columns
[∫ ti

0
y(τ)dτ

]
and

[∫ ti
0

y2(τ)

x(τ)
dτ

]
are

linearly dependent.

Proof: Integrating expression (4) with respect
to time t ∈ [0,T] we obtain

C1

(
−
∫ t
0
x3(τ)dτ

)
+ C2

(
−
∫ t
0
x(τ)y(τ)dτ

)
+ C3 (x0 −x(t)) + C4

(∫ t
0
x(τ)dτ

)
+ C5

(∫ t
0
x2(τ)dτ

)
−
(

1

2
(x2(t) −x20)

)
= 0.

(9)

Integrating second equation of system 5 with re-
spect to time t ∈ [0,T] we have

C6

(∫ t
0
y(τ)dτ

)
+ C7

(
−
∫ t
0

y2(τ)

x(τ)
dτ

)
−C3 (y(t) −y0) = 0. (10)

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A.A. Adeniji, I. Fedotov, M.Y. Shatalov, Inverse problem of the Holling-Tanner model and its solution

Performing all integrations in (9) and (10) from 0
to tj ∈ [0,T] we obtain the following overdeter-
mined systems of N + 1 linear algebraic equations

C1 ~g1 + C2 ~g2 + C3 ~g3 + C4 ~g4 + C5 ~g5 − ~g6 = 0,
C6 ~g7 + C7 ~g8 − ~g9 = 0, (11)

where

~g1 =

[
−
∫ ti
0
x3(τ)dτ

]
, ~g2 =

[
−
∫ ti
0
x(τ)y(τ)dτ

]
,

~g3 = [x0 −xi] , ~g4 =
[∫ ti

0
x(τ)dτ

]
,

~g5 =

[∫ ti
0
x2(τ)dτ

]
, ~g6 =

[
1

2
(x2i −x

2
0)

]
,

~g7 =

[∫ ti
0
y(τ)dτ

]
, ~g8 =

[
−
∫ ti
0

y2(τ)

x(τ)
dτ

]
,

~g9 = [yi −y0]

are the (N + 1)×1-vector columns. Now applying
the method used in Theorem 1 we prove the
Corollary.

Remark 4. In vector ~g1, ~g2, ~g4, ~g5, ~g7, ~g8 the inte-
grals are calculated by means of numerical inte-
gration of xi,yi and their combinations with re-
spect to time t and that is why the proposed method
is called the integration method of identification.

Remark 5. Note that expressions (5), (7) and
(11) are linear with respect to unknown constants
C1, · · · ,C7. Direct use of the constraint minimiza-
tion using the Lagrange multiplier with constraint:

C1C
2
3 + C3C5 −C4 = 0 (12)

produces nonlinear system of equations for de-
termination of six unknowns C1,C2,C3,C4,C5,λ.
Thus the search is performed in six-dimensional
space of parameters and hence this method sub-
stantially complexifies the solution procedure. De-
termination of parameters and C6 and C7 needs
solution of linear system of two algebraic equa-
tions. In the next section we describe an original
problem solution algorithm reducing the search
space dimension to one and using only linear
matrix manipulations in the process of solution,

which substantially simplifies and accelerates the
problem solution.

IV. SOLUTION OF THE PARAMETER
IDENTIFICATION PROBLEM

There are four original independent parameters
(b1,b2,b3,b4) in the first equation of (1). First
four C- parameters (C1,C2,C3,C4) depend on b-
parameters so that there is one-to-one correspon-
dence between them. The parameter C5 depends
on the first four C-parameter as follows:

C5 =
C4
C3
−C1C23. (13)

Hence, it is possible to consider (C1,C2,C3,C4)
as independent parameters and introduce new
name for the dependent parameter C5 = −λ. The
novel algorithm will be considered in detail for
the differentiation method of solution, i.e. with
~f1,··· ,9 - vector columns(see expression (5) and
(7). The integration method of solution uses the
same algorithm in which ~f1,··· ,9 - vector columns
are changed to ~g1,··· ,9 -ones (see (11)). Param-
eter λ will be selected from the given interval
λ ∈ [λmin,λmax] and substituted in goal function
G3 which is composed as follows

G3 = G3(C1,C2,C3,C4,λ)

=
1

2

(
C1 ~f1+C2 ~f2+C3 ~f3+C4 ~f4−(λ~f5+ ~f6)

)T
(
C1 ~f1+C2 ~f2+C3 ~f3+C4 ~f4−(λ~f5+ ~f6)

)
(14)

and subjected to minimization. In expression (14),
parameter λ is considered as constant at every
minimization and minimization itself is performed
with respect to parameters C1,C2,C3,C4. Solu-
tion of this problem is given by the following
formula

C(λ) = [C1(λ), C2(λ), C3(λ), C4(λ)]
T

=
(
(LT1 L1)

−1LT1
)
R(λ), (15)

where

L1 =
[
~f1 ~f2 ~f3 ~f4

]T
, R(λ) = λ~f5 + ~f6. (16)

In expression (15) it is possible to calculate 1 ×
(N +1)- vector row

(
(LT1 L1)

−1LT1
)

only once and

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A.A. Adeniji, I. Fedotov, M.Y. Shatalov, Inverse problem of the Holling-Tanner model and its solution

after that perform its multiplication by (N +1)×1-
vector row R(λ), which is very fast operation.
Components of vector C(λ) and C5 = −λ are
substituted in the constraint (12) to obtain the
following nonlinear scalar equation

C1(λ)C
2
3 (λ) −λC3(λ) −C4(λ) = 0, (17)

which is solved with respect to λ. All roots of
Equation (17) are found (sometimes to find all
the roots it is necessary to expand the interval
λ ∈ [λmin,λmax] to the left or to the right or
to both sides). After finding a particular root λ
the corresponding b-parameters are calculated as
follows:

b1(λ) =
C4(λ)

C3(λ)
, b2(λ) = C1(λ),

b3(λ) = C2(λ), b4(λ) = C3(λ). (18)

(See (4)). The estimations of b-parameters are
obtained from the proper selection of root λ = λ̄:

b̄1 = b1(λ̄), b̄2 = b2(λ̄),

b̄3 = b3(λ̄), b̄4 = b4(λ̄) (19)

(one of the criteria of the correct choice of λ̄ must
be positiveness of all estimated b̄ parameters, see
Numerical Examples). Parameters b5 and b6 are
estimated by means of minimization of the goal
function of Equation 8. Solution of this problem
is given by the formulas:[

b̄5

b̄6

]
= (LT2 L2)

−1LT2
~f9 (20)

where L2 =
[
~f7 ~f8

]
is (N + 1)×2- matrix, (See

(7)).
Expression (15)-(20) give solution to the identi-
fication problem by means of the differentiation
method. To find solution of the problem by the
integral method it is necessary to consider vectors
~g1,··· ,9 (See expression 11) instead of ~f1,··· ,9. In the
next section you will find more information about
application of the differentiation and integration
methods.

V. NUMERICAL EXAMPLES

Let us solve the initial problem of Equation 1
with the following parameters:

b1 = 0.2 b2 = 0.01 b3 = 0.05

b4 = 1 b5 = 0.062 b6 = 0.0223 (21)

and initial conditions: [x0 y0]
T

= [10 5]
T . The

stable critical point has coordinates (x̃, ỹ) ≈
(7.77064, 21.4066) (see Equation 2) and it is the
stable focus (eigenvalues of the linearized system
in the vicinity of the critical point are ν1,2 ≈
−0.0138 ± 0.0735i, where i2=−1). The unstable
saddle has coordinates (˜̃x, ˜̃y)=(20,0). Numerical
solution x = x(t) on the time interval t ∈ [0,T =
150] in N + 1 = 25 points is shown in Figure
1 and solution y = y(t) is shown in Figure 2.
Performing solution by means of the differential

Fig. 1. Graph of solution x = x(t)

Fig. 2. Graph of solution y = y(t)

method in accordance with the described algorithm
we obtain nonlinear Equation (12) from which
the parameters are calculated: λ1 ≈ −0.3282,
λ2 ≈ −0.1091 and λ3 ≈ 0.2087. As we see,
only λ3 parameter can be selected from three

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A.A. Adeniji, I. Fedotov, M.Y. Shatalov, Inverse problem of the Holling-Tanner model and its solution

TABLE I
VALUES OF b-PARAMETERS, CORRESPONDING TO DIFFERENT ROOTS OF EQUATION (12) FOR N + 1 = 25

(DIFFERENTIATION METHOD)

Original Values λ1 ≈ −0.3282 λ2 ≈ −0.1091 λ3 ≈ 0.2087

b1 = 0.2000 −0.0580 −0.0621 0.2129

b2 = 0.0100 −0.0150 −0.0045 0.0108

b3 = 0.0500 −0.0295 −0.0026 0.0491

b4 = 1.0000 −18.0380 −10.5185 0.3906

roots, because λ1 and λ2 generate the negative
values of b-parameters. The relative error of the
b-parameters corresponding to λ3-parameter are as
follows:

ERROR%(b1) ≈ 6.437%
ERROR%(b2) ≈ 7.725%
ERROR%(b3) ≈ 1.832%
ERROR%(b4) ≈ 60.943% (22)

Estimation of parameters b5 and b6 gives coinci-
dence with the original values of the parameters in
four decimals with the following relative errors:

ERROR%(b5) ≈ 0.029%
ERROR%(b6) ≈ 0.028% (23)

Comparison of original graphs with graphs ob-
tained by numberical solution of initial problem
(1) with the same initial conditions but with esti-
mated parameters is shown in Figure 3 and Figure
4.

As we see the estimated parameters gives quite
good estimation of the process dynamics. The
estimated values of the steady states are as follows
(˜̃x, ˜̃y) ≈ (7.7143, 21.4282) with relative errors:

ERROR%(x̃) ≈ 0.102%
ERROR%(ỹ) ≈ 0.101% (24)

Estimation of the parameters with N + 1 = 49
points gives λ1 ≈ −0.2585, λ2 ≈ −0.0878, λ3 ≈
0.1914 and the following values of parameters (see
Table 2)

Fig. 3. Graph of original solution x = x(t) (dots) and
solution with estimated parameters (solid line)

Fig. 4. Graph of original solution y = y(t) (dots) and
solution with estimated parameters (solid line)

As we see, only λ3 parameter can be selected
from the three roots, because λ1 and λ2 generate
the negative values of b-parameters. One can see
the substantial improvement of the parameters
estimations. The relative errors of the b-parameters

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A.A. Adeniji, I. Fedotov, M.Y. Shatalov, Inverse problem of the Holling-Tanner model and its solution

TABLE II
VALUES OF b-PARAMETERS, CORRESPONDING TO DIFFERENT ROOTS OF EQUATION (12) FOR N + 1 = 49

(DIFFERENTIATION METHOD)

Original Values λ1 ≈ −0.2585 λ2 ≈ −0.0878 λ3 ≈ 0.1914

b1 = 0.2000 −0.0405 −0.0485 0.2010

b2 = 0.0100 −0.0119 −0.0036 0.0101

b3 = 0.0500 −0.0271 −0.0022 0.0500

b4 = 1.0000 −18.3099 −10.9975 0.9553

corresponding to λ3- parameter are as follows:

ERROR%(b1) ≈ 0.496%
ERROR%(b2) ≈ 0.583%
ERROR%(b3) ≈ 0.087%
ERROR%(b4) ≈ 4.469% (25)

Estimation of parameters b5 and b6 gives coinci-
dence with the original ones in four decimals with
the following relative errors:

ERROR%(b5) ≈ 0.002%
ERROR%(b6) ≈ 0.002% (26)

Comparison of original graphs with graphs ob-
tained by numerical solution of initial problem (1)
with the same initial conditions but with estimated
parameters is shown in Figure 5 and Figure 6.

Fig. 5. Graph of original solution x = x(t) (dots) and
solution with estimated parameters (solid line)

As we see the estimated parameters give very
good estimation of the process dynamics. The

Fig. 6. Graph of original solution y = y(t) (dots) and
solution with estimated parameters (solid line)

estimated values of the steady states are as follows
(˜̃x, ˜̃y) ≈ (7.7077, 21.4102) with relative errors:

ERROR%(x̃) ≈ 0.017%
ERROR%(ỹ) ≈ 0.017% (27)

Absolute errors in calculation of x = x(t) and
y = y(t) in the differentiation method for N +
1 = 25 and N + 1 = 49 points are shown in
Figure 7 and Figure 8. Performing solution by
means of the integration method in accordance
with the described algorithm we obtain three roots
of nonlinear equation (12): λ1 ≈ −0.2391, λ2 ≈
−0.0725, λ3 ≈ 0.1899.

As we see, only λ3 parameter can be selected
from the three roots, because λ1 and λ2 generate
the negative values of b-parameters. The relative
errors of the b-parameters corresponding to λ3-

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TABLE III
VALUES OF b-PARAMETERS, CORRESPONDING DIFFERENT ROOTS OF EQUATION (12) FOR N + 1 = 25 (INTEGRATION

METHOD)

Original Values λ1 ≈ −0.2391 λ2 ≈ −0.0725 λ3 ≈ 0.1899

b1 = 0.2000 −0.0302 −0.0386 0.1999

b2 = 0.0100 −0.0115 −0.0032 0.0100

b3 = 0.0500 −0.0282 −0.0022 0.0500

b4 = 1.0000 −18.1074 −10.6869 0.9997

Fig. 7. Absolute Errors of Calculation for N +1 = 25 points
(Differentiation method)

Fig. 8. Absolute Errors of Calculation for N +1 = 49 points
(Differentiation method)

parameter are as follows:

ERROR%(b1) ≈ 0.052%
ERROR%(b2) ≈ 0.044%
ERROR%(b3) ≈ 0.059%
ERROR%(b4) ≈ 0.033% (28)

Fig. 9. Absolute errors of calculation for N +1 = 25 points
(Integration method) .

Estimation of parameters b5 and b6 gives coinci-
dence with the original values of b-parameter in
four decimals with the following relative errors:

ERROR%(b5) ≈ 0.008%
ERROR%(b6) ≈ 0.007% (29)

Comparison of original graphs with graphs ob-
tained by numerical solution of initial problem (1)
with the same initial conditions but with estimated
parameters are visually indistinguishable from Fig-
ure 5 and Figure 6. Absolute errors in calculation
of x = x(t) and y = y(t) in the integration method
for N + 1 = 25 points are shown in Figure 9.

The parameters are estimated with very high
accuracy at N + 1 = 25 points. The estimated
values of the steady states are as follows (˜̃x, ˜̃y) ≈
(7.7062, 21.4061) with relative errors:

ERROR%(x̃) ≈ 0.002%
ERROR%(ỹ) ≈ 0.002% (30)

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http://dx.doi.org/10.11145/j.biomath.2018.12.057


A.A. Adeniji, I. Fedotov, M.Y. Shatalov, Inverse problem of the Holling-Tanner model and its solution

VI. CONCLUSION

Two methods of solution of the inverse problem
on parameter identification of the Holling-Tanner
model with complete information are discussed.
These are the differentiation and integration meth-
ods of solution. The conditions are indicated at
which all parameters of the model can be iden-
tified. The main disadvantage of the conventional
method of constraint minimization by means of
the Lagrange multipliers is that the method gen-
erates a system of six nonlinear equations with
unknown initial guess values. Proposed is the
novel method of the problem solution in which
the six dimensional space of search is reduced
to one dimensional space and the procedure of
the initial guess value is performed by fast vector
multiplication. Numerical examples of the pro-
posed algorithm implementation are demonstrated
for the differentiation and integration methods. It
is shown that the integration method generates
more accurate results than the differentiation one.
The integration method also needs less number
of points on the fixed time interval to produce
accurate results than the differentiation method.

VII. ACKNOWLEDGMENT

The authors acknowledge the department of
Mathematics and Statistics of the Tshwane Uni-
versity of Technology towards the research.

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http://dx.doi.org/10.11145/j.biomath.2018.12.057

	Introduction
	Main results
	On solvability of identification problem
	Solution of the parameter identification problem
	Numerical examples
	Conclusion
	Acknowledgment
	References