article mathematical analysis of the three dimensional

Copyright © 2021 by Modern Scientific Press Company, Florida, USA

International Journal of Modern Mathematical Sciences, 2021, 19(1): 45-56

International Journal of Modern Mathematical Sciences

Journal homepage:www.ModernScientificPress.com/Journals/ijmms.aspx

ISSN:2166-286X

Florida, USA

Article

Mathematical Analysis of the Three Dimensional Lotka -

Volterra Model

*V. Ananthaswamy1, P. Felicia Shirly2, M. Subha3

1Research Centre and PG Department of Mathematics, The Madura College, Madurai, Tamil Nadu,

India.

2Department of Mathematics, Lady Doak College, Madurai, Tamil Nadu, India

3Department of Mathematics, Fatima College, Madurai, Tamil Nadu, India

*Author to whom correspondence should be addressed; E-Mail: [email protected]

Article history: Received 6 December 2020, Revised 10 February 2021, Accepted 12 February 2021,

Published 18 February 2021.

Abstract: This paper reflects some research outcome denoting as to how Lotka–Volterra

prey predator model has been solved by using new Homotopy analysis method. In this paper,

a non-linear mathematical model is used to analyse the dynamical relationship between

predator and their prey. This paper presents an approximate analytical method to solve the

non-linear differential equations. A simple and closed form of analytical expressions for

three dimensional Lotka – Volterra model are obtained. Numerical simulations are carried

out to justify analytical results.

Keywords: Lotka – Volterra equation; Mathematical modelling; Non-Linear differential

equations; Homotopy analysis method; Numerical simulation.

Mathematics Subject Classification Code (2010): 34E, 35K20, 68U20

1. Introduction

The developments of the qualitative analysis of non-linear differential equations are derived to

study many real life problems in mathematical biology. The modeling for the population dynamics of a

prey-predator system is one of the important goals in mathematical biology, which has received wide

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Int. J. Modern Math. Sci. 2021, 19(1): 45-56


46

attention by several authors [1]. The Lotka –Volterra model describes the number of ecological

competitors (or predator–prey) model which is dynamic by nature .This model was framed and gradually

gained its popularity in the technological arena. The simple prey–predator model is among the most

popular models, being frequently used to demonstrate a simple non-linear control system[2]. One

mathematical model that is frequently examined is the Lotka – Volterra predator-prey model. This refers

to a system in which there are two populations known as the predator and the prey. The model states that

the prey will grow at a certain rate but will also be eaten at a certain rate because of predators. The

predators will die at a certain rate but will then grow by eating prey. The aim of this paper is to study the

predator-prey models with at least one predator and two preys and to investigate the relationship that

exists between these preys and the predator using Lotka-Volterra model and differential equations.

2. Mathematical Formulation of the Problem

Mathematical models have become important tools in analyzing the dynamical relationship

between predator and their prey. The three dimensional Lotka – Volterra predator prey system has been

proposed to describe the population dynamics of the interacting species of a predator and its prey. The

two preys and one predator model with time delay is represented by the following system of three non-

linear differential equations [3]:

cxzbxyaxdt

dx (1)

fyzmydyxdt

dy (2)

kzhzygzxdt

dz (3)

where )(tx , )(ty and )(tz stand for the number of susceptible predator, susceptible prey and infected

prey populations respectively. The parameters ‘a’ is the natural death of the healthy susceptible predator,

‘b’ is the number of contact between susceptible prey and healthy susceptible predator, ‘c’ is the number

of contact between healthy susceptible predator and infected prey, ‘d’ is the number of contact between

healthy susceptible predator and susceptible prey, ‘m’ is the per capita birth rate of susceptible prey (per

time), ‘f’ is the number of contact between healthy susceptible prey and infected prey, ‘g’ is the number

of contact between healthy susceptible predator and infected prey, ‘h’ is number of contact between

healthy susceptible prey and infected prey and ‘k’ is the harvesting rate of the infected prey [5]. The

initial conditions for the eqns.(1)-(3) are as follows:

oxx )0( ,oyy )0( and

ozz )0( (4)

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47

3. Approximate Analytical Expressions of the Prey Predator System Using the

Homotopy Analysis Method

Liao proposed a powerful analytical method for nonlinear problems, namely the Homotopy

analysis method. It is a non perturbative analytical method for obtaining series solutions to non-linear

equations and has been successfully applied to numerous problems in science and engineering. In

comparison with other perturbative and non-perturbative analytical methods, Homotopy analysis method

offers the ability to adjust and control the convergence of a solution via the so-called convergence-

control parameter h. Furthermore, the obtained result is of high accuracy ([5] - [18]). Using Homotopy

analysis method we obtained the approximate solutions of the eqns. (1) - (3) as follows:

taoo

o

tm

o

tk

o

o

ta

o em

yb

k

zcxh

m

eyb

k

ezcxhextx

)( (5)

tmoo

o

tk

o

ta

o

o

tm

o ek

zf

a

xdyh

k

ezf

a

exdyheyty

)( (6)

ktooo

tm

o

ta

oo

tk

o em

yh

a

xgzh

m

eyh

a

exgzheztz

)( (7)

4. Numerical Simulation

In order to investigate the accuracy of the approximate analytical expressions with a finite

number of terms, the system of differential eqns. is solved numerically. To show the efficiency of the

present method our results are compared with the numerical solution (MATLAB program). The function

ode45 (Range – Kutta method) in MATLAB software which is a function of solving the initial value

problems is used to solve eqns. (1) – (3). The analytical solutions are compared with the numerical

solution in Figs. (1) to (3). Upon comparison, it is evident that both results give satisfactory agreement

for all values of parameters. The MATLAB program is also given in (Appendix B). In the following

plots, analytical solutions are represented by dotted lines and numerical simulation results are

represented by solid lines.

5. Results and Discussion

The eqns.(5) - (7) provide the new and simple analytical expressions for the number of

susceptible predator, susceptible prey and infected prey respectively. In order to analyse the influence

of the parameters hgfmdcba ,,,,,,, and k over a Lotka – Volterra model, the analytical expressions

eqns.(5) - (7) are plotted in Figs.(1)-(3) by fixing constant values for other parameters given in Table 1.



48

Table 1: Values of parameters

Parameters a b c d m f g h k

Values 0.1 0.01 0.4 0.1 0.02 0.03 0.1 0.04 0.005

Fig.1 represents the profile of susceptible predator )(tx verses time .t From Figs.1(a) - 1(c), it

can be inferred that the rate of increase of )(tx increases with increase in the parameters ba, and .c

From the Fig.1, it can be seen that )(tx highly depends on the value of natural death of the healthy

susceptible predator. Smaller changes in a natural death of the healthy susceptible predator can be seen

to have a large effect on )(tx compared to the number of contact between susceptible prey and healthy

susceptible predator and the number of contact between healthy susceptible predator and infected prey.

(a)

(b)



49

(c)

Fig. 1: Number of susceptible predator )(tx versus time t is plotted for varying the values of the

parameters ba, and c , by fixing constant values for other parameters given in Table1 using the

eqn.(5).

Fig.2 represents the profile of susceptible prey )(ty verses time .t From Figs. 2(a) - 2(c) it can

be seen that the rate of increase of )(ty increases with increase in the parameters md, and .f

Comparing the Fig.2, it can be seen that the rate of increase of )(ty is highly sensitive to per capita birth

rate of susceptible prey when compared with other two parameters.

(a)



50

(b)

(c)

Fig. 2: Number of susceptible prey )(ty versus time t is plotted for varying the values of the

parameters md, and f , by fixing constant values for other parameters given in Table 1 using

eqn.(6).

Fig. 3 represents the profile of infected prey )(tz verses time .t From Figs. 3(a) - 3(c), it can be

seen that the rate of increase of )(tz increases with increase in the parameters hg, and .k



51

(a)

(b)

(c)

Fig.3: Number of infected prey )(tz versus time t is plotted for varying the values of the parameters

hg, and k by fixing constant values for other parameters given in Table1 using eqn.(7).



52

6. Conclusion

The time dependent system of non-linear differential eqns. in predator–prey models in Lotka

– Volterra system have been solved analytically and numerically. The approximate analytical

expressions for the prey and predator populations are derived by using the Homotopy analysis method.

The primary result of this work is simple approximate calculations of prey and predator populations for

all values of parameters. The Homotopy analysis method is an extremely simple method and it is

also a promising method to solve other non-linear equations. We analyzed a mathematical model

dealing with two species of prey-predator system

References

[1] R. K. Naji , S. J. Majeed. The Dynamical Analysis of a Prey-Predator Model with a Refuge-Stage

Structure Prey Population. International Journal of Differential Equations, (2016):1-10.

[2] S. Paul, S. P. Mondal ,P. Bhattacharya. Numerical solution of Lotka Volterra prey predator model

by using Runge–Kutta–Fehlberg method and Laplace Adomian decomposition method. Alexandria

Engineering Journal, 55(1)(2016):613-617.

[3] J. Vasundhara Devi, R. V. G. Ravi Kumar. Modeling the prey predator problem by a graph

differential equation. European Journal of Pure and Applied Mathematics, 7(1)(2014):37-44.

[4] R. Soni , U. Chouhan. A Dynamic Effect of Infectious Disease on prey predator system

andHarvesting Policy. Biosciences Research Communications, 11(2)(2018):231-237.

[5] S. J. Liao. The proposed Homotopy analysis technique for the solution of non linear problems.

Ph.D.Thesis, Shanghai Jiao Tong University, (1992).

[6] S. J. Liao. An approximate solution technique which does not depend upon small parameters: a

special example. Int. J. Non-Linear Mech., 30(1995):371-380.

[7] S. J. Liao. Beyond perturbation introduction to the Homotopy analysis method. 1st edn., Chapman

and Hall, CRC press, Boca Raton, (2003):336.

[8] S. J. Liao. On the Homotopy analysis method for non-linear problems. Appl. Math. Comput.,

147(2004):499-513.

[9] S. J. Liao. An optimal Homotopy-analysis approach for strongly non-linear differential equations.

Commun. Nonlinear Sci. Numer. Simulat., 15(2010):2003- 2016.

[10] S. J. Liao. The Homotopy analysis method in non-linear differential equations. Springer and Higher

education press, (2012).

[11] S. J. Liao. An explicit totally analytic approximation of blasius viscous flow problems. International

Journal of Nonlinear Mechanics, 34(1999):759-778.

https://www.sciencedirect.com/science/article/pii/S1110016815002197#!





53

[12] S.J. Liao. On the analytic solution of magnetohydrodynamic flows non-Newtonian fluids over a

stretching sheet, J Fluid Mech., 488(2003):189-212.

[13] S. J. Liao. A new branch of boundary layer flows over a permeable stretching plate. International

Journal of Nonlinear Mechanics, 42(2007):819-830.

[14] V. Ananthaswamy, S. Kala and L. Rajendran. Approximate analytical solution of non-linear initial

value problem for an autocatalysis in a continuous stirred tank reactor, Homotopy analysis method.

International Journal of Mathematical Archive, 5(4)(2014):1-12.

[15] V. Ananthaswamy, M. Suba , A. Mohamed Fathima. Approximate Analytical Expressions of Non-

Linear Boundary Value problem for a Boundary Layer Flow using the Homotopy Analysis Method.

Madridge Journal of Bioinformatics and Systems Biology, 1(2)(2019):34-39.

[16] V. Ananthaswamy, L. Sahanya Amalraj. Thermal stability analysis of reactive hydromagnetic third-

grade fluid using Homotopy analysis method. International Journal of Modern Mathematical

Sciences, 14(1)(2016):25-41.

[17] V. Ananthaswamy, T. Iswarya. Analytical expressions of mass transfer effects on unsteady flow

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[18] V. Ananthaswamy, T. Iswarya. Analytical expressions of the effect of radiation on free convection

flow of heat and mass transfer. Nonlinear Studies. 23(1)(2016):133-147.

Appendix: A

Approximate analytical expressions of the eqn.(1)-(4) using Homotopy analysis method.

In this Appendix, we indicate how the eqn.(5) is derived.

To find the solution of eqn.(1) , we first construct a Homotopy as follows:

cxzbxyax

dt

dxphax

dt

dxp1

(A.1)

The approximate analytical solution of the eqn.(A.1) is as follows:

.....)()()()( 2

2

10 txptpxtxtx

(A.2)

The initial approximations are as follows:

0)0( xxo , ...3,2,10)0( ixi (A.3)

Substituting eqn.(A.2) into an eqn. (A.1) and arranging the coefficients of like powers of p,

we can obtain the following eqns.

0)()(

: taxdt

tdxp o

oo

(A.4)

0)()()()()()(

: 111 tztcxtythbxtaxdt

tdxp oooo

(A.5)



54

Solving the eqns.(A.4) and (A.5) using the eqn.(A.3), we can obtain the following results

at

o extx 0)(

(A.6)

atooo

mt

o

kt

oo e

m

by

k

czhx

m

eby

k

eczhxtx

)(1

(A.7)

According to the HAM, letting 1p , we can conclude that

)()()( 1 txtxtx o

(A.8)


fyzdxymy

dt

dyphmy

dt

dyp1

(A.9)

The approximate analytical solution of the eqn.(A.9) is as follows:

.....)()()()( 2

2

10 typtpytyty

(A.10)


0)0( yyo , ...3,2,10)0( iyi (A.11)

Substituting the eqn. (A.10) into an eqn. (A.9) and arranging the coefficients of like powers of p , we

can obtain the following eqns.

0)()(

: tmydt

tdyp o

oo

(A.12)

0)]()()()([)()(

: 111 tztfytytdxhtmydt

tdyp oooo

(A.13)


tm

o eyty 0)(

(A.14)

tmooo

tk

o

ta

oo e

k

zf

a

xdyh

k

ezf

a

exdyhty

)(1

(A.15)


)()()( 1 tytyty o

(A.16)


yzhxzgzk

dt

dzphzk

dt

dzp1

(A.17)

The approximate analytical solution of the eqn.(A.17) is as follows: .....)()()()( 2

2

10 tzptpztztz

(A.18)


0)0( zzo , ...3,2,10)0( izi (A.19)



55

Substituting the eqn.(A.18) into an eqn.(A.17) and arranging the coefficients of like powers of p , we

can obtain the following differential eqns.

0)()(

: tkzdt

tdzp o

oo

(A.20)

0)]()()()([)()(

: 111 tzthytztgxhtkzdt

tdzp oooo

(A.21)


kt

oo eztz )( (A.22)tkoo

o

tm

o

ta

oo e

m

yh

a

xgzh

m

eyh

a

exgzhtz

)(1

(A.23)


)()()( 1 tztztz o

(A.24)

After substituting the eqns. (A.6) and (A.7) into an eqn. (A.8), and substituting the eqns. (A.14) and

(A.15) into an eqn. (A.16) and substituting the eqns.(A.22) and (A.23) into an eqn. (A.24), we obtain the

solutions in the text eqns. (5), (6) and (7) respectively.

Appendix: B

MATLAB program to find the solution of the eqns. (1) - (4)

function predator

options= odeset('RelTol',1e-6,'Stats','on');

% initial conditions

Xo = [.01;3;1];

tspan = [0 .5];

tic

[t, X] =ode45(@TestFunction,tspan,Xo,options);

toc

figure

hold on

% plot(t, X(:,1),'-')

% plot(t, X(:,2),'-')

plot(t, X(:,3),'-')

legend('x1','x2','x3')

ylabel('x')

xlabel('t')

return



56

function [dx_dt]= TestFunction(t, x)

a=.1,b=.01,c=.4,d=.1,e=.02,f=.03,g=.1,h=.04,k=1;

dx_dt(1) =a*x(1)+b*x(1)*x(2)+c*x(1)*x(3);

dx_dt(2) =e*x(2)+d*x(1)*x(2)+f*x(2)*x(3);

dx_dt(3) =k*x(3)+g*x(1)*x(3)+h*x(2)*x(3);

dx_dt = dx_dt';

return

Appendix: C

Nomenclature

Symbol Meaning

)(tx Number of susceptible predator

)(ty Number of susceptible prey

)(tz Number of infected prey

a Natural death of the healthy susceptible predator

b Number of contact between susceptible prey and healthy susceptible predator

c Number of contact between healthy susceptible predator and infected prey

d Number of contact between healthy susceptible predator and susceptible prey

m The per capita birth rate of susceptible prey

f Number of contact between healthy susceptible prey and infected prey

g Number of contact between healthy susceptible predator and infected prey

h number of contact between healthy susceptible prey and infected prey

k The harvesting rate of the infected prey

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