A Two Species Amensalism Model with a Linearly Varying Cover on the First Species

Dr. Sita Rambabu. B¹ , Dr. Lakshmi  Narayan K²

¹Department of Mathematics, VJIT, Hyderabad-500085, India

²Department of Mathematics, VITS, Hyderabad - 508284, India

*Corresponding author E-mail: sitarambabu.b@gmail.com

ABSTRACT:

The present paper is devoted to an analytical investigation of a two species ammensalism model with a   cover linearly varying with the population on  the first species  (x)  to protect from the attacks of the second species (y).All the equilibrium points are identified and the local stability is discussed and global stability is also discussed by constructing suitable Lyapunov’s function supported by the numerical simulation by using Matlab.

KEYWORDS: Ammensal, cover, Equilibrium Point, Global stability, Lyapunov function.

1. INTRODUCTION:

The present chapter, we consider the interaction between two species, one species is Ammensal x(t) and another species is Enemy species y(t). We incorporated a linearly varying cover on Ammensal to increase the survival rate of Ammensal from the attacks of Enemy species. The criteria of equilibrium states of the system have been derived the existence of local and global stability of equilibrium states discussed by constructing suitable Lyapunov’s function. Numerical illustrations for the various values of parameters of the model have been appended to support the analytical discussion.

2. BASIC EQUATIONS AND NOMENCLATURE:

The model for the interacting species is given by

(2.1)

Where  is the Ammensal population provided with a cover of safety with constant values for a and b, from the attacks of the Enemy species.

3. EQUILIBRIUM STATES:

The system under investigation has four equilibrium states. They are

I.E_1: The extinct state                         (3.1)

II.E_2: The state in which only the Enemy species survives and the Ammensal extinct

                                                         (3.2)

III. E₃: The state in which Ammensal survive and the Enemy species extinct.

                                                         (3.3)

IV. E_4: The state in which both Ammensal and Enemy species survive

(3.4)

This case exist only when

4. THE STABILITY OF THE EQUILIBRIUM STATES:

Let =                                                                                                                                       (4.1)

where U = is the perturbation over the equilibrium state. The basic equations (1.1) are linearized to obtain the equations for the perturbed state.

                                                                                                                                                              (4.2)

where A=

The characteristic equation for the system is                                                                     (4.3)

The roots of the characteristic equations are negative real numbers or complex conjugates with the negative real parts, then the equilibrium points are stable.

4.1 Stability of the equilibrium state I:

are small perturbations in the species at the equilibrium states, then the linear version  of the system is given by

                                                                                                                     (4.1.1)

with the Characteristic equation is

The roots of the equations are a₁ & a₂

Since   then system is unstable.

The trajectories for this state are 

                                                                                                                        (4.1.2)

where , are the initial strengths of  respectively

The numerical illustrations for the various values of parameters in the Fig. 4.1.1 to 4.1.2

Case1:a₁=0.2; a₂=0.3; a=0.1; b=0.2; u₁₀=15;u₂₀=20;

                                        Fig. 4.1.1                                                                                                   Fig. 4.1.2

Fig. 4.1.1 shows the variation of Ammensal and Enemy species with respect to time, the curves are raise from their initial population strengths. Hence the system becomes unstable.

In Fig. 4.1.2 the phase plot of Ammensal and Enemy species shows that the population raised from their initial population sizes.

Case2:a₁=0.2; a₂=0.1; a=0.2; b=0.1; x₁₂=0.5;u₁₀=15;u₂₀=20;

Fig. 4.1.3                                                                                                                     Fig. 4.1.4

In the Fig. 4.1.3 initially the Enemy species population dominates the Ammensal till t= 1.5, later on Enemy species dominates and both populations are increasing from initial populations leads the unstable system.

Fig. 4.1.4 shows the phase plot of Ammensal and Enemy species populations, in which both are increasing from their initial population sizes

4.2. Discussion of Local Stability of the equilibrium state II:

By substituting  in equation (6.A.2.1), then lineraized system of equations obtained as

                                                                                              (4.2.1)

with the characteristic equation

The roots of the equations are                        (4.2.2)

Hence clearly  is negative.

System is stable if                                                                                             (4.2.3)

The trajectories for this state are

                                                                                        (4.2.4)

(4.2.5)             

where s₁&s₂ are the roots of the equation

The solution curves are illustrated from the Fig. 4.2.1 to 4.2.4

Case1:a₁=0.2; a₂=3;a=0.1;b=0.2;x₁₂=0.3; x₂₂=0.3;u₁₀=15;u₂₀=20;

In this case

Fig. 4.2.1                                                                                                                              Fig. 4.2.2

The curves shown in Fig. 4.2.1 are converging to equilibrium point origin after t =3 with diminishing amplitude make the stable system.

In Fig. 4.2.2 the phase plot of Ammensal and Enemy species are clearly shows that the curve converging to origin makes stable system.

Case2:a₁=0.2; a₂=0.1;a=0.1;b=0.2;x₁₂=0.3; x₂₂=0.4;u₁₀=15;u₂₀=20;

                 In this case

   Fig. 4.2.3                                                                                                                             Fig. 4.2.4

In the Fig. 4.2.3 the Enemy species dominates in its initial population strength till t=1, later on Ammensal dominates and exponentially increasing make the system unstable. Fig. 4.2.4 shows the phase plane of Ammensal and Enemy species shows that the system is unstable.

4.3 Discussion of Local Stability of the equilibrium state III:

The Linearized system of equations

                                                                                                                                              (4.3.1)

With the Characteristic equation is                                                             (4.3.2)

The roots of the equations are

System is unstable since

The trajectories for this state are 

(4.3.3)

The solution curves are illustrated from the Fig. 4.3.1 to 4.3.4

Case1:a₁=0.2;a₂=0.2;a=0.2;b=0.1;x₁₂=0.5;x₁₁=0.3;x₂₂=0.4;u₁₀=15; u₂₀=20;

Fig. 4.3.1                                                                                                 Fig. 4.3.2

In the Fig. 4.3.1 Enemy species population is positive exponential, Ammensal population is negative exponential for the above mentioned parametric values .the system is unstable because of one population has an exponential growth rate. This is clearly shown in Fig. 4.3.2 phase plane of Ammensal and Enemy species.

Case2:a₁=0.3; a₂=0.2;a=0.2;b=0.1;x₁₂=0.5; x₁₁=0.3;x₂₂=0.4;u₁₀=15;u₂₀=20;

Fig. 4.3.3                                                                                                                            Fig. 4.3.4

In the Fig. 4.3.3 Enemy species population is positive exponential, Ammensal population is negative exponential makes the unstable system. This is clearly shown in Fig. 4.3.4 phase plane of Ammensal and Enemy species.

4.4 Discussion of Local Stability of the equilibrium state IV:

The Lineraized system of equations

                                                                                                                             (4.4.1)

With Characteristic Equation

(4.4.2)

The roots of the equation are

The System is stable if (4.4.3)

The trajectories for this state are

                                                                                                                          (4.4.4)

The solution curves are illustrated from the Fig. 4.4.1 to 4.4.10

Case1:a₁=0.3; a₂=0.2;a=0.2;b=0.1;x₁₂=0.5;x₁₁=0.3;x₂₂=0.4;u₁₀=15; u₂₀=20;

Fig. 4.4.1                                                                                                                Fig. 4.4.2

In the Fig. 4.4.1 the population curves are converging to the equilibrium point E(3.689,2.709).Hence the system is asymptotically stable.

Case2:a₁=3; a₂=1;a=0.2;b=0.1;x₁₂=0.5; x₁₁=0.3;x₂₂=0.2;u₁₀=15;u₂₀=20;

Fig. 4.4.3                                                                                                                 Fig. 4.4.4

Two populations of Ammensal and Enemy species are decreasing in amplitude and after t=6.5 both curves are coincide .the Ammensal population is extent at t = 1.5, later slightly increasing up to t =6, then coincide with Enemy species population for the above mentioned parametric values. The converging equilibrium point (origin) shown in Fig. 4.4.3   

Case3:a₁=1; a₂=1;a=0.2;b=0.1;x₁₂=0.5; x₁₁=0.3;x₂₂=0.2;u₁₀=15;u₂₀=20;

Fig. 4.4.5                                                                                                                           Fig. 4.4.6

In Fig. 4.4.5, initially Enemy species dominates Ammensal up to

t =0.1,later on Ammensal dominates, the Ammensal population is increasing from its initial strength where as Enemy species population is decreases i.e it shown in Fig. 4.4.5 is unstable system.

Case4:a₁=1; a₂=1;a=0.2;b=0.1;x₁₂=0.5; x₁₁=0.3;x₂₂=0.2;u₁₀=15;u₂₀=20;

Fig. 4.4.7                                                                                                                           Fig. 4.4.8

Two curves of Ammensal and Enemy species are converging to the equilibrium point after t = 4.5, both curves are coincide make the stable system.

In Fig. 4.4.8 the phase plane shows the converging equilibrium point E(0.0002,0.0009) almost converging to origin make stable system

Case5:a₁=0.3;a₂=0.2;a=0.2;b=0.1;x₁₂=0.5; x₁₁=0.3;x₂₂=0.2;u₁₀=15; u₂₀=20;

Fig. 4.4.9                                                                                                                           Fig. 4.4.10

For the above parametric values the solution curves of Ammensal and Enemy species are converging to the equilibrium points as shown in Fig. 6.A.21

Fig. 4.4.10 shows the phase plots of Ammensal & Enemy species is converging to the equilibrium point E (3.34, 2.799) shows that the system is globally asymptotically stable.

5. GLOBAL STABILITY:

Theorem: The positive equilibrium point (3.4) is globally asymptotically stable

Proof: Let the Lyapunov function be 

Differentiate with respect to t along the trajectories of the system, we have

                                                                                                                                              (5.1)

                                                                (5.2)

Substitute  

We have (5.3)

By the inequality we have

(5.4)

choose

, therefore is globally asymptotically stable.

6. NUMERICAL EXAMPLE:

Example 1: a₁=3; a₂=2; a=0.5; b=0.2;α₁₁=0.5; α₁₂=0.5; α₂₂=0.2;  x= 10; y= 15;

Fig. 6.1                                                                                                          Fig. 6.2

Fig. 6.1 shows the variation of Ammensal and Enemy species with respect to time

The solution curves are negative exponentials and converging to the equilibrium points, the system is globally asymptotically stable for the mentioned parametric values.

Fig. 6.2 shows the phase plot of Ammensal and Enemy species converging to the equilibrium point E(1.4495,10)

Example 2: a₁=1.5; a₂=1.2; a=0.5; b=0.2; α₁₁=0.5; α₁₂=0.42; α₂₂=0.5; x= 10; y= 15;

Fig. 6.3                                                                                                         Fig. 6.4

Fig. 6.3 shows the variation of Ammensal and Enemy species with respect to time. The solution curves are negative exponentials and converging to the equilibrium points, shows the system is globally asymptotically stable for the above mentioned parametric values.

Fig. 6.4 shows the phase plot of Ammensal and Enemy species converging to the equilibrium point E (1.913, 2.4)

7. CONCLUSIONS:

The present paper is devoted to an analytical investigation of a two species Ammensalism model with a   cover linearly varying with the population on the Ammensal(x) to protect from the attacks of the Enemy species (y).All the equilibrium points are identified and the local stability is discussed and global stability is also discussed by constructing suitable Lyapunov’s function and stability analysis is supported by the numerical simulation by using Matlab.

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. JNTUH.

2. Lakshmi Naryan K, and Pattabhi Ramacharyulu NCh. Some threshold theorems for prey-predator model with harvesting. Int. J. of Math.Sci.  and Engg. Appls 2008; 2: 23-3.

3. Leonardo of Pisa (Fibonacci).  Liberabaci (Book of counting board), 1202.

4. Lotka. AJ. Elements of physical biology, Williams and Wilkins, Baltimore, 1925.

5. Lucas WF Roberts SF and Thrall RM. Discreate and system models (vol (III) of modules in Applied Mathematics). Springer-verlag, Heidel berg, 1983.

6. Marchuk GI. Mathematical Models in immunology. optimization software Inc., New York, 1983.

7. Matsuda H and Abrams PA. Effects of predators-prey interaction and adaptive change on sustainable yield. Can. J. Fish. Aquat. Sci./J. Can. Sci. Halieut. Aquat. 2004; 61: 175-184.

8. May RM. Stability and complexity in model Eco-Systems. Princeton University press. Princeton, 1973.

9. Mesterton-Gibbons. M. A technique for finding optimal two species  harvesting policies. Ecol.model. 1996; 92: 235-244.

10. Meyer WJ. Concepts of Mathematical Modeling. McGraw –Hill, 1985.

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

12. Phanikumar N, Pattabhiramacharyulu NCh. A three species eco-system consisting of a prey predator and host commensal to the prey. International Journal of Open Problems Compt. Math, 2010; 3(1): 92-113.

13. Rish S and Boucher DH. 1976 what ecologist looks for. Bulletin of the Ecological Society of America 57: 8-9.

14. Varma VS. A note  on  Exact solutions for a Special prey-predator or competing Species System. Bull.Math.Biol. 1977; 39: 619-622.

15. Volterra V. Leconssen la Theorie  Mathematique  de  la  Leitte Pou Lavie.  Gauthier-Villars, Paris,1931.

16. Liao SJ. beyond perturbation: introduction to the homotopy analysis method.CRC Press. Boca Raton: Chapman & Hall. 2003.

17. 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. Pelagia Research Library. 2012; 3 (3): 1821-1826.

Received on 10.09.2017       Modified on 24.10.2017

Accepted on 30.11.2017      ©A&V Publications All right reserved

Research J. Science and Tech. 2017; 9(4): 511-520.

DOI: 10.5958/2349-2988.2017.00088.2