A Mathematical Study of Two Species Commensalism Model

 

Goverdhan Reddy. J,  Sita Rambabu. B

Vidya Jyothi Institute of Technology, Hyderabad

*Corresponding Author E-mail: goverdhanreddy.javini@gmail.com, sitarambabu.b@gmail.com

 

ABSTRACT

In this present paper we discussed two species commensalism model. Here first species (x) is commensal and the second species (y) is host. Commensalism is Ecological model interaction between two organisms. One organism benefits from other without harmed by the organism. Here we governed two non linear differential equations with natural resources and the model is represented by coupled non linear ordinary differential equations. All the equilibrium points are identified and discussed the local stability in each stage and also discussed the global stability by constructing suitable Lyapunov function and supported by numerical simulations using Mat Lab.

 

KEYWORDS: Commensal, Host, Equilibrium point, Global Stability, Lyapunov function.

 

1.INTRODUCTION:

In this paper we proposed a commensalism model of two species. Commensalism is Ecological model interaction between two organisms. One organism benefits from other without harmed by the organism. Here we governed two non linear differential equations with natural resources and the model is represented by coupled non linear ordinary differential equations. All the equilibrium points are identified and discussed the local stability in each stage and also discussed the global stability by constructing suitable Lyapunov function and supported by numerical simulations using  Mat Lab.

 

2. BASIC EQUATIONS:

The model equations for two species commensalism model are given by the following system of first order non linear ordinary differential equations employing the following notations.

 

                    

                              (2.1)

Where

x (t)        :  The population of the  commensal

y(t) : The population of the host

   : The natural growth rate of commensal

  : The natural growth rate of species2

        : The rate of decrease of the commensal due to insufficient food and inter species competition

: The rate of increase of the commensal due to successful attacks on the second species

: The rate of decrease of the species2 due to insufficient food and inter species competition

 

3. EQUILIBRIUM STATES:

The system under investigation has four equilibrium states. They are

: The extinct state

                                                   (3.1)

: The state in which only the first species extinct and the second species is survive.

                 (3.2)

:The state in which the  first  species exist and the second species extinct.

                                         (3.3)

: The state in which both the species exist.

                    (3.4)

 

4. THE STABILITY OF THE EQUILIBRIUM STATES:

To examine the local stability at the equilibrium states, linearzed the equations (2.1) by introducing the small perturbations  over  respectively, so that

The linearized systemof equations for the peturbuted state.

                                                                                                                      (4.1)

                                                               (4.2)

This is known as variational matrix.

The characteristic equation for the system is.                                                                      (4.3)

The equilibrium state is stable, if the roots of the equation (4.3) are negative in case they are real or the roots have negative real part in case they are complex.

 

4.1. STABILITY OF FULLY WASHED OUT STATE ():

The systems of linearized equations of the system (2.1) are

                                                                                                                                 (4.1.1)

The variational matrix of the system (2.1) at

                                                                                                                                               (4.1.2)

With the characteristic equation

       

                                                                                                                                                      (4.1.3)

Since all the roots are positive therefore the system is unstable.

 

TRAJECTORIES:

Where are initial strength.

 

4.2. STABILITY ANALYSIS OF FIRST SPECIES WASHED OUT STATE ():

The systems of linearized equations of the system (2.1) are

              

 

                                                                                                                                       (4.2.1)
The variational matrix of the system (2.1) at

 

                                                                                                                       (4.2.2)

With the characteristic equation

                                                               (4.2.3)

The system is unstable since one of the roots is positive.

 

Trajectories:

                                                                                                                                            (4.2.4)

 

4.3. STABILITY ANALYSIS OF SECOND SPECIES WASHED OUT STATE ():

The systems of linearized equations of the system (2.1) are

                                                                                                                                (4.3.1)

The variational matrix of the system (2.1) at

                                                                                                                 (4.3.2)

With the characteristic equation

                                                                               (4.3.3)

The system is unstable since one of the roots is negative.

Trajectories:

                                          Where                                                      (4.3.4)

 

 

 

4.4. STABILITY ANALYSIS OF CO-EXISTING STATE OR NORMAL STEADY STATE ():

The systems of linearized equations of the system (2.1) are

                                                                                                                       (4.4.1)

The variational matrix of the system (2.1) at

                                                                                                                  (4.4.2)

With the characteristic equation

       

By Routh-Hurwitz criteria

       

                                                                                                         (4.4.3)

Therefore the real parts of the roots of this equation are negative if

Therefore the system is stable.

 

Trajectories:

                    (4.4.4)

 

5. GLOBAL STABILITY:

Theorem: The Equilibrium point ( ) is globally asymptotically stable.

Proof:    Let the Lyapunov function for the case  is

                                                                               (5.1)

                                                                                               (5.2)

                                                                                                                            

                                                                                                                           (5.3)

                                                                          

                                                                                  (5.4)

  Substitute                                                                                            (5.5)

                                                   (5.6)

                                                                                             (5.7)

       The interior equilibrium point   is globally asymptotically stable                 

 

6. NUMERICAL EXAMPLE:

Case1:a1=0.2,a2=0.3,u10=10, u20=15


Fig 1.1.1                                                                  Fig1.2.1

 

The figure 1.1.1 shows the variation of species1 and species2 with respect to time, the curves rises from their initial population strengths. Hence the system becomes unstable.

The figure 1.1.2 shows the phase plot of species1 and species2. The population rises from their initial strengths.

 

CASE 2:  a1=0.2,a2=0.3,u10=10,u20=15,x12=0.2,x22=0.5

 

Fig 1.2.1                                                                                                                            Fig 1.2.2

 

The figure 1.2.1 shows the variation of species1 and species2 with respect to time, the curves rises from their initial population strengths. Hence the system becomes unstable.

The figure 1.2.2 shows the phase plot of species1 and species2. The population rises from their initial strengths.

 

CASE 3:  a1=2, a2=3, x11=0.5, x12=0.3, u10=10, u20=20

 


                    Fig 1.3.1                                                                                                                            Fig 1.3.2

The figure 1.3.1 shows the variation of species1 and species2 with respect to time, the curves rises from their initial population strengths. Hence the system becomes unstable.

 

The figure 1.3.2 shows the phase plot of species1 and species2. The population rises from their initial strengths.

 

CASE 4: a1=2, a2=3, x11=0.5, x12=0.9, x22=0.8, u10=10, u20=20


Fig 1.4.1                                                                                                                            Fig 1.4.2

When the initial strength of commensal increases, the above figures shows commensalism effect due to interaction with species2. The phase space trajectory also shows the same.

 

7. CONCLUSION:

In this present paper identified all the equilibrium points and supported by numerical simulation by using mat lab and observed that in case-IV the commensalism effect is on species one due to interaction with species2. i.e., the population strength of commensal is increased due to interaction with species2.and also the system is asymptotically stable it was observed by construting suitable Lyapunov function

 

8. REFERENCES:

1.     Lakshmi Narayan. K.: A Mathematical study of Prey-Predator Ecological Models with a partial covers for the prey and alternative food for the predator, Ph.D thesis, 2004, J.N.T. University.

2.     Lakshmi Naryan. K., and Pattabhi Ramacharyulu. N.ch: Some threshold theorems for prey-predator model with harvesteng”, Int. J. of Math.Sci.  and Engg. Appls,vol (2), No.2, 2008, PP-23-3.

3.     Lotka. A.J.: Elements of physical biology, Williams and Wilkins, Baltimore, 1925.

4.     Matsuda.H and Abrams.P.A.: Effects of predators-prey interaction and adaptive change on sustainable yield, Can. J. Fish. Aquat. Sci./J. Can. Sci. Halieut. Aquat., 61 (2004) 175-184.

5.     May, R.M.: Stability and complexity in model Eco-Systems”, Princeton University press, Princeton, 1973.

6.     Mesterton-Gibbons.M.: A technique for finding optimal two species harvesting policies, Ecol.model., 92 (1996) 235-244.

7.     Meyer, W.J.: Concepts of Mathematical Modeling, McGraw –Hill, 1985.

8.     Paul Colinvaux.: Ecology, John Wiley and Sons Inc., New York, 1986.

9.     Phanikumar.N.,Pattabhiramacharyulu.N.Ch.,: “A three species eco-system consisting of a prey  predator and host commensal to the prey” International journal of open problems compt.math, 3(1),(2010).92-113.

10.   Rish, S. and D.H. Boucher.: 1976 what ecologist looks for. Bulletin of the Ecological Society of America 57:8-9.

11.   Varma, V.S.: A note on Exact solutions for a Special prey-predator or competing Species System, Bull.Math.Biol. vol.39, 1977, PP 619-622.

12.   Volterra, V: Leconssen la theorie  mathematique  de  la  leitte pou lavie,  Gauthier-Villars, Paris,1931.

13.   Sita Rambabu.B.,Lakshmi Narayan.K., and Shahanaz Bathul.: A Mathematical study of Two Species Amensalism Model With a Cover for the first Species by Homotopy Analysis Method, Advances in Applied Science Research, 2012, 3 (3):pp: 1821-1826 Pelagia Research Library.

14.   A Two Species Amensalism Model 4with Harvesting By Homotopy Analysis    Method”, B. Sita Rambabu,K. L. Narayan and Shahanaz Bathul,Bulletin of   Society for mathematical services & standards (B SO MAS S),Vol. I No. 1     (2012), pp. 1-10.

 

 

 

 

 

 

 

Received on 11.06.2017       Modified on 21.07.2017

Accepted on 18.08.2017      ©A&V Publications All right reserved

Research J. Science and Tech. 2017; 9(3):385-391.

DOI: 10.5958/2349-2988.2017.00067.5