International ejournals International ejournal of Mathematics and Engineering 91 (2010)

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1 Available online at International ejournals International ejournal of Mathematics and Engineering 9 ( ISS A TWO SPECIES MOA AMMESALISM -GLOBAL STABILITY AALYSIS K.V.L..ACHARYL Faculty of Mathematics Bapatla Engineering College Bapatla-50 kvlna@yahoo.com ABSTRACT The present paper intends to test the global stability of a two species Monad Ammensalism which is instituted by Liapunov s stability criteria. It is extracted by constructing a suitable Liapunov s function for appraising the global stability of the model in the case of normal steady state. AMS Classification: 9 5, 9 40 Key words: Equilibrium states, Stability, Liapunov s function for global stability Introduction: K.V.L..Acharyulu and and.ch.pattabhi Ramacharyulu [-5] analyzed the Local stability of an Ammensal- enemy eco-system on the quasi-linear basic balancing equations. Local stability analysis for an Ammensal- enemy eco-system with various resources in different cases has been also fulfilled in the author s earlier work. Several authors like Lotka[7], Kapur[6] etc. utilized this method in various situations for global stability. The present investigation is mainly focused on the establishment of the global stability of the coexistent equilibrium state of a two species Monad Ammensalism with limited resources by employing a property constructed by Liapunov s function with Liapunov s criteria for global stability Basic concepts: dx Consider an autonomous system F (, x y and dy F (, x y ( Assume that this system has an isolated initial point taken as (0, 0. Consider a function E(x,y possessing continuous partial derivatives along with the path of (.This path is represented by C= [(x (t, y (t] in the parametric form. E(x,y can be regarded as a function of t along C with rate of change If the total energy of physical system has a local minimum at a certain equilibrium point then the point is said to be stable.liapunov generalized this principle by constructing a function E(, whose rate of change is given by E E E E E.. F F ( t t t corresponding to the system.

2 K.V.L..ACHARYL International ejournal of Mathematics and Engineering 9 ( (iitheorem (A: If there exists a Liapunov s function E (x,y for the system (, then the critical point (0,0 is stable. Further, if this function has additional property that the function ( is negative definite, then the critical point (0, 0 is asymptotically stable. (3 The following theorem provides to ascertain the definiteness of a Liapunov s function. (iiitheorem (B: The function E(x,y = ax +bxy+cy is positive definite if a>0 and b 4ac<0 and negative definite if a<0, b 4ac<0. (4 otations adopted and are the populations of the Ammensal and enemy species with natural growth rates a and a respectively. a = The rate of decrease of the Ammensal due to insufficient food. a = The rate of increase of the Ammensal due to inhibition by the enemy. a = The rate of decrease of the enemy due to insufficient food. K i = ai aii are the carrying capacities of i, i =,,β = Monad type parameters. The state variables and as well as the model parameters a, a, a, a, K, and K are assumed to be non-negative constants. 3 Construction of Basic Equations of the model: The model equations for a two species Ammensal interaction with a monod typevariable coefficient of Ammensalism with limited resources is given by the following system of non-linear ordinary differential equations. (i Equation for the growth rate of Ammensal Species (S : d a K F (5 In equation (5 the function F ( is the characteristic of the Ammensalism with respect to the enemy with the properties: F ( is bounded and F ( a constant, as. The Ammensal characteristic model considered is a two parameter model of the monad type : F ( = (6 Here F 0 is a parameter characteristic of Ammensalism. Further ( 0 is another parameter signifying the strength of Ammensalism: 0 strong Ammensalism, 0 weak Ammensalism and 0, the interaction would be neutral. (ii Equation for the growth rate of enemy species (S : d a K (7 4 Equilibrium Points and Equilibrium states: The system has the following four equilibrium states E -E 4 resulting from d d = 0; = 0 (8 E : Fully washed out state : = 0; = 0 (9 E : The state in which only the : = 0; = K (0 enemy survives and the Ammensal is washed out E 3 : The state in which only the : = K ; = 0 ( Ammensal survives and the enemy is washed out

3 K.V.L..ACHARYL International ejournal of Mathematics and Engineering 9 ( K E 4 : Co-existent state : = Ka ; = K a K (Both Ammensal and enemy survive ( 5 Global stability of the model by Liapunov s function : The linearized basic equations are d a d = - a (4 The characteristic equation is a a 0 (3 (5 Equation (5 is of the form + p + q = 0 where p = a + a >0 (6 q = a a >0 (7 K K a a 4a a K K Therefore the conditions for Liapunov s function are satisfied ow we define E (, = ½ (a + b + c (8 where a = b = c = a a a a (9 (0 a a a ( ( = pq = ( a + a (a a From equations (6 and (7 it is clear that > 0 and a > 0 Also ( a aa ( a a a ( ac b - a (3

4 K.V.L..ACHARYL International ejournal of Mathematics and Engineering 9 ( a aa a aa = - a (4 (ac-b = a a + a 3 a 3 +a a a a 3 a a (5 (ac b > 0 ac b > 0 i.e., b ac < 0 (6 Therefore the function E (, at (8 is positive definite. E d E d a b a Further + b c a (7 By substituting values of a, b and c from equations (9, (0 and ( in (7 we get a aa E d E d a ( a a a a a ( K a a ( a a a a u ( a a a = a } (8 a a a a ( a a (

5 K.V.L..ACHARYL International ejournal of Mathematics and Engineering 9 ( ( ( a a 4 a a a ( a a a a = a a a aa a a a a a 4 aa ( 4 ( aa ( a a = - = = + a a a a a a a a a a aa ( ( a a ( a a a a a a a a a a a a a a = ( (9 E d E d ( u u (30 which is clearly negative definite. So E (, is a Liapunov s function for the linear system. ext we prove that E (, is also a Liapunov s function for the non-linear system also. If f and f are two functions in and defined by f (, = a a (3 f (, = a a (3 E E we now have to show that f f is negative definite. Putting = + and = + in (5 and (7 we get du ( ( = a a (

6 K.V.L..ACHARYL International ejournal of Mathematics and Engineering 9 ( a a ( ( ( = a a a ( a a du f (, = a F (, (33 where F(, = a du Also a a d f(, a G(, (34 where G (,u = - a E From (8 = a + b (35 du E = b + c (36 du ow E E f f a b a F(, d (b + cu [- a u + G (, ] (37 E E f f a b a d b c a + [(a + b F (, + (b +c G(, ] (38 From (9 E E f d f - ( + + (a +b F(, + (b +c G (, (39 By introducing polar co-ordinates = r cos, = r sin we can write the equation(39 as E E f d f - (r + r{[a cos + b sin ] F(, + [b cos + c sin ] G(, } (40 Let us denote largest of the numbers a, b, c by M. r Our assumptions imply that F (, < 6 M and G ( r, < for all sufficiently 6 M small r > 0. E E 4Kr r So f f r 0 d 6M 3 (4 Thus the function E (, is positive definite with the condition that

7 K.V.L..ACHARYL International ejournal of Mathematics and Engineering 9 ( E E f f is negative definite d The equilibrium state E 4 is asymptotically stable. Conclusion: The Global stability of a mathematical model of two species Monad Ammensalism with limited resources in the co-existent equilibrium state is explained and It is observed that the normal state is asymptotically stable. REFERECES []. Acharyulu. K.V.L.. & Pattabhi Ramacharyulu..Ch..; On The Stability of An Ammensal- Harvested Enemy Species Pair With Limited Resources in International journal of computational Intelligence Research (IJCIR, Vol.6, o.3; pp , June 00. []. Acharyulu. K.V.L.. & Pattabhi Ramacharyulu..Ch. An Ammensal-Enemy specie pairwith limited and unlimited resources respectively-a numerical approach, Int. J. Open Problems Compt. Math (IJOPCM., Vol. 3, o.,pp.73-9., March 00. [3]. Acharyulu. K.V.L.. & Pattabhi Ramacharyulu..Ch.;"On An Ammensal-Enemy Ecological Model With Variable Ammensal Coefficient is accepted for publication in International Journal of Computational Cognition( IJCC [4]. Acharyulu. K.V.L.. & Pattabhi Ramacharyulu..Ch In view of the reversal time of dominance in an Enemy-Ammensal species pair with unlimited and limited resources respectively for stability by numerical technique, International journal of Mathematical Sciences and Engineering Applications(IJMSEA ; Vol.4, o. II, June 00. [5]. Acharyulu. K.V.L.. & Pattabhi Ramacharyulu..Ch. On The Stability Of An Ammensal - EnemySpecies Pair With nlimited Resources. International e Journal Of Mathematics And Engineering (I.e.J.M.A.E., Volume-,Issue-II,pp-40-49;00. [6]. Kapur J.., Mathematical Modeling, Wiley Eser (985 [7]. Lotka A.J., Elements of Physical Biology, Williams & Wilking, Baltimore, 95.