We try to compare the solutions by some numerical techniques when we apply the methods on some mathematical biology problems. The RungeKuttaFehlberg (RKF) method is a promising method to give an approximate solution of nonlinear ordinary differential equation systems, such as a model for insect population, onespecies LotkaVolterra model. The technique is described and illustrated by numerical examples. We modify the population models by taking the Holling type III functional response and intraspecific competition term and hence we solve it by this numerical technique and show that RKF method gives good results. We try to compare this method with the Laplace Adomian Decomposition Method (LADM) and with the exact solutions.
Mathematical models of population growth have been formed to provide a significant angle of the real ecological situation. The meaning and importance of each parameter in the models have been defined biologically [
For getting exact population Holling type III functional response plays an important role in population dynamics. Holling type III functional response should be taken into the predatorprey interactions, which is proposed by Holling [
In the field of science and technology, numerous significant physical phenomena are frequently modeled by nonlinear differential equations. Such equations are often very difficult to solve analytically. Yet, analytical approximate methods are very important for obtaining the accurate solutions which have gained much significance in recent years. There are various methods, undertaken to find out approximate solutions to nonlinear problems. Homotopy Perturbation Method (HPM), Homotopy Analysis Method (HAM), Differential Transform Method (DTM), Variational Iteration Method (VIM), Adomian Decomposition Method (ADM), Laplace Adomian Decomposition Method (LADM), and RungeKuttaFehlberg (RKF) method are some very popular methods. The purpose of this paper is to bring out the numerical solution of various population models by using the approach, namely, RungeKuttaFehlberg (RKF) method.
Recently, different scientists use the numerical method in their different problems. Here we try to give some references showing the importance of quasinumerical methods techniques in present time: Kumar and Baskar [
The accurate solutions of population growth models may become a difficult task if the equations are highly nonlinear. To overcome the situation there we take the numerical simulation, until there are no particular numerical methods for solving such problems. So to fill up the gap, here we find the approximated solutions of some population models by applying such some reliable, efficient, and more comfortable numerical technique (e.g., LADM, RKF) and try to conclude which one is the best.
The principal aim of this paper is to perform systematic analysis of the comparisons among exact solution and some reliable numerical techniques on the dynamics of the population model. The said numerical methods shall be more acceptable and reliable for solving such kind of problem. The issues which are addressed in the paper are described in the following:
Adding Holling type III functional response in insect population model and LotkaVolterra model for better formulation
Applying two numerical methods LADM and RKF for solution of the models and find the solution
Analysis of the comparisons among exact solution, Laplace Adomian Decomposition Method (LADM), and RungeKuttaFehlberg (RKF) method on the models
The solution of the models by different numerical techniques being illustrated numerically and graphically
The necessary algorithm for numerical techniques given
The paper is organized as follows. The basic literature survey on population growth model, functional response, and numerical techniques is addressed in Section
The Laplace Adomian Decomposition Method (LADM) [
Consider the following nonlinear differential equation:
Taking Laplace transform on both sides of the above equation, we get
For numerical computation, we get the expression as
Split the given equation into two parts. The first part is
Apply the Laplace transform to second part, determine the coefficient of
Calculate the Adomian polynomials for the function
Divide the first part to the coefficient of
Construct the solution using inverse Laplace transform to
End
One of the most popular methods with a constant step size is the fourthorder RungeKutta (RK4) method. Reasonably the RungeKutta method can [
Consider the initial value problem
The problem is to solve the initial value problem in the above equation by means of RungeKutta methods [
First we need some definitions:
Read
For
Consider
Repeat for better approximation.
End the programme.
Otherwise,
Repeat Steps
Suppose that an insect population
We now solve (
Consider the initial value problem
First we define
Now we modify the above model by introducing Holling type III functional response which is defined in the following:
The feeding rate saturates at the maximal feeding rate
We now solve (
Now applying LADM, we have the following recursive algorithm:
Consider the initial value problem
First we define
Here we consider interspecific competition term in the insect population model. Interspecific competition is the most important aspect in the population growth dynamics model [
Solving (
Consider the initial value problem
First we define
We define the LotkaVolterra model into different ways as follows.
Mathematical models of population growth have been formed to provide an inconceivable significant angle of true ecological situation. The meaning of each parameter in the models has been defined biologically. For
In case of one species, (
Solving (
Consider the initial value problem
First we define
Now we modify the LotkaVolterra model by introducing Holling type III functional response which is defined as
Solving (
Consider the initial value problem
The numerical solutions obtained by using the RKF, LADM method and are compared with the exact solution for different population models.
From Table
Numerical comparison when initial condition is

Exact solution  Solution by RKF method  Solution by LADM 
ERKF  ELADM 

0  1000.0000000  1000.0000000  1000.00000000 


0.1  1217.4076457  1217.4447223  1216.07381402 


0.2  1453.7920776  1453.8275070  1444.17894316 


0.3  1673.5897680  1673.6084290  1647.56834151 


0.4  1831.8331881  1831.8125036  1788.52160650 


0.5  1889.5969624  1889.5159617  1838.84276708 


0.6  1831.3776006  1831.2908389  1788.12230677 


0.7  1672.7981525  1672.7809816  1646.85165910 


0.8  1452.8457426  1452.9182247  1443.28432644 


0.9  1216.4762073  1216.5807034  1215.15811427 


1  999.1956876  999.2604905  999.19564898 


ELADM
ERKF
Comparison among the exact solution and the solutions obtained by using RKF method and LADM for model I.
In Table
Numerical comparison of solutions when

Exact solution  Solution by RKF method  Solution by LADM (three iterates)  ERKF  ELADM 

0  100.00000000  100.0000000  100.0000000 


0.05  110.44538789  110.1789856  110.4297547 


0.1  121.68557418  121.108106  121.5572303 


0.15  133.42386083  132.497513  133.0020711 


0.2  145.25872028  143.9584501  144.3172756 


0.25  156.69395306  155.0138903  155.0163121 


0.3  167.16677455  165.1252092  164.6053933 


0.35  176.09213446  173.7346800  172.6181278 


0.4  182.92083874  180.3198596  178.6494928 


0.45  187.20173364  184.4522441  182.3860869 


0.5  188.63838966  185.8501193  183.6299329 


0.55  187.1284265  184.4153954  182.3136691 


0.6  182.7773689  180.2468100  178.5057434 


0.65  175.8845511  173.6266764  172.4051363 


0.7  166.9030599  164.9840041  164.3260955 


0.75  156.3834523  154.8416109  154.6742722 


0.8  144.9109052  143.7574377  143.9164278 


0.85  133.047794  132.2701488  132.5464437 


0.9  121.2890397  120.8566363  121.0506779 


0.95  110.0347113  109.90535847  109.8757207 


1  99.57968623  99.70572240  99.4013312 


Evaluation among the exact solution and the solutions obtained by using LADM and RKF method for model II.
In Table
Numerical comparison when

Exact solution  Solution by RKF method  Solution by LADM (three iterates) 
ERKF  ELADM 

0  100.00000  100.00000  100.0000000 


0.05  110.43960  110.43957  110.4247550 


0.1  121.67210  121.67914  121.5472313 


0.15  133.40060  133.41682  132.9870734 


0.2  145.22320  145.25102  144.2972796 


0.25  156.64390  156.68567  154.9913184 


0.3  167.09980  167.15791  164.5754023 


0.35  176.00650  176.08282  172.5831401 


0.4  182.81540  182.91116  178.6095088 


0.45  187.07650  187.19184  182.3411071 


0.5  188.49450  188.62843  183.5799579 


0.55  186.96800  187.11853  182.2586994 


0.6  182.60380  182.76773  178.4457794 


0.65  175.70170  175.87525  172.3401786 


0.7  166.71530  166.89424  164.2561445 


0.75  156.19490  156.37518  154.5993285 


0.8  144.72520  144.90324  143.8364918 


0.85  132.86810  133.04075  132.4615159 


0.9  121.11750  121.28263  120.9607589 


0.95  109.87270  110.02889  109.7808110 


1  99.42780  99.57443  99.3014312 


Evaluation among the exact solution and the solutions obtained by using LADM and RKF method for model III.
From Table
Numerical comparison when

Exact solution  Solution by RKF method  Solution by LADM method (three iterates)  ERKF  ELADM 

0  0.10000000  0.10000000  0.10000000 


0.1  0.10713679  0.10713679  0.10714000 


0.2  0.11453291  0.11453291  0.11456000 


0.3  0.12216385  0.12216385  0.12226000 


0.4  0.13000114  0.13000114  0.13024000 


0.5  0.13801261  0.13801261  0.13850000 


0.6  0.14616290  0.14616290  0.14704000 


0.7  0.15441399  0.15441399  0.15586000 


0.8  0.16272591  0.16272591  0.16496000 


0.9  0.17105750  0.17105750  0.17434000 


1  0.17936718  0.17936718  0.17936718 


1.1  0.18761383  0.18761383  0.19394000 


1.2  0.19575756  0.19575756  0.20416000 


1.3  0.20376050  0.20376050  0.21466000 


1.4  0.21158743  0.21158743  0.22544000 


1.5  0.21920638  0.21920638  0.23650000 


1.6  0.22658907  0.22658907  0.24784000 


1.7  0.23371122  0.23371122  0.25946000 


1.8  0.24055276  0.24055276  0.27136000 


1.9  0.24709782  0.24709782  0.28354000 


2  0.25333471  0.25333471  0.29600000 


Evaluation between the exact solution and the solutions obtained by using LADM and RKF method for model IV.
From Table
Numerical comparison when

Exact solution  Solution by RKF method  Solution by LADM (two iterates)  ERKF  ELADM 

0  0.1000000000  0.1000000000  0.1000000000 


0.1  0.1065991220  0.1065991220  0.1065049505 


0.2  0.1133693593  0.1133693590  0.1130099010 


0.3  0.1202826611  0.1202826611  0.1195148515 


0.4  0.1273083819  0.1273083815  0.1260198020 


0.5  0.1344137893  0.1344137893  0.1325247525 


0.6  0.1415646647  0.1415646643  0.1390297030 


0.7  0.1487259514  0.1487259514  0.1455346535 


0.8  0.1558624471  0.1558624468  0.1520396040 


0.9  0.1629394898  0.1629394898  0.1585445545 


1  0.1699236292  0.1699236290  0.1650495050 


1.1  0.1767832382  0.1767832382  0.1715544554 


1.2  0.1834890574  0.1834890572  0.1780594059 


1.3  0.1900146405  0.1900146404  0.1845643564 


1.4  0.1963367003  0.1963367001  0.1910693069 


1.5  0.2024353397  0.2024353396  0.1975742574 


1.6  0.2082941773  0.2082941768  0.2040792079 


1.7  0.2139003623  0.2139003624  0.2105841584 


1.8  0.2192445053  0.2192445047  0.2170891089 


1.9  0.2243205144  0.2243205144  0.2235940594 


2  0.2291253839  0.2291253840  0.2300990099 


Evaluation between the exact solution and the solutions obtained by using LADM and RKF method for model V.
In this paper, we describe the method for finding numerical solution of insect population model and LotkaVolterra model. Here we apply two numerical methods called RKF and LADM for solutions of the said models. Here the numerical solutions obtained by using the RKF show high accuracy and these are compared with the LADM solution. So we can say that these numerical results show that the RKF method is an acceptable and reliable numerical technique for the solution of linear and nonlinear differential equation models on population models. It can be seen clearly from the graphical representations that RKF gives quite good results after a certain considerable time intervals. This is a very useful method, which will be undoubtedly found applicable in broad applications. The advantage of the RKF over the LADM is that there is no need for the evaluations of the Adomian polynomials and the advantage of RKF over RK4 is that it has a good accuracy using variable step size. Hence it provides an efficient numerical solution.
The authors declare that they have no competing interests.