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1 Available online at International ejournals ISSN International ejournal of Mathematics and Engineering 95 (00) GLOBAL STABILITY OF AN IMMIGRATING COMMENSAL OST MOEL WIT LIMITE RESOURCES N. Phani umar C.Srinivasa umar B. Ravindra Reddy 3 and N. Ch. Pattabhi Ramacharyulu 4. Faculty in Mathematics, epartmental umanities & Sciences Malla Reddy Engineering College, Secunderabad, India. phanikumar_nandanavanam@yahoo.com. Gopal Reddy College of Engineering & Technology, Pedakangarla (V), Patancheru (M), Medak (t) 5039, A.P, India, drcskumar4@gmail.com 3. JNTU College of Engineering, Nachupally (ondagattu), arimnagar50550, India. rbollareddy@gmail.com 4. Ex. Faculty, epartment of Mathematics & umanities, National Institute of Technology, Warangal, India., pattabhi933@yahoo.com ABSTRACT In this paper we establish the global stability of an immigrating commensal host model with limited resources, by constructing a suitable Liapunov s function in case of co-existent equilibrium state. AM Classification: 95, 940. eywords: Equilibrium state, Liapunov s function, Global stability.. INTROUCTION Phani umar and N.Ch.Pattabhi Ramacharyulu etc. [6] examined the local stability of an immigrating commensal host model with limited resources on the quasi-linear basic balancing equations. The present investigation is mainly devoted to establish the global stability of the co-existent equilibrium state of an immigrating commensal host model with limited resources,by employing a properly constructed Liapunov s function.. LIAPUNOV S STABILITY ANALYSIS Many approaches are available for the stability analysis of linear,time-invarient systems. owever for non-linear systems and/or time-varying systems, stability analysis may be extremely difficult or impossible. Liapunov Stability analysis is one method that may be applied for non-linear systems. In 89 A.M. Liapunov introduced the direct method to study the global stability of equilibrium states in case of linear and non-linear systems. is method is based on the chief characteristic of constructing a scalar function called Liapunov s function. That is by using the direct method of Liapunov,we can determine the stability of a system without solving the state. This is quite advantageous because solving non-linear and/or time-invarying state equation is very difficult. To day this method is widely recognized as an efficient tool in theory of control systems, dynamical systems, systems with

2 N. Phani umar, C.Srinivasa umar, B. Ravindra Reddy and N. Ch. Pattabhi Ramacharyulu International ejournal of Mathematics and Engineering 95 (00) time lag, power system analysis, and time varying non-linear feed back systems, multi species ecological systems and so on. The stability behaviour of solutions of linear and weakly non-linear system is done by using the techniques of variation of constants formulae and integral inequalities. So this analysis is confined to a small neighborhood of operating point i.e., local stability. Further, the techniques used there in require explicit knowledge of solutions of corresponding linear systems. The stability behavior of a physical system is discussed by several authors like apoor [], Lotka [3], Ogata [4] Bhaskara Rama Sarma and N.Ch.PattabhiRamacharyulu [], Lakshminarayan and N.Ch.Pattabhi Ramacharyulu [5] etc. If the total energy of a physical system has a local minimum at a certain equilibrium point, then that point is stable. This idea was generalized by Liapunov to study stability problems in a broader context... STABILITY BY LIAPUNOV S IRECT METO Consider an autonomous system dx F( x, y) dy G( x, y) () Assume that this system has an isolated critical point taken as (0, 0). Consider a function E(x, y) possessing continuous partial derivatives along 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 de E dx E dy x y de E E F x y x y, G x, y ().. EFINITIONS. E (x, y) is said to be positive definite if E (x, y)>0 (x,y) (0,0). E (x, y) is said to be positive semi-definite if E (x, y) > 0 & E (0, 0) =0 3. E (x, y) is said to be negative definite if E (x, y) < 0 4. E (x, y) is said to be negative semi-definite if E (x, y) < 0 & E (0, 0) =0 A Positive definite function E (x, y) with the property that () is negative semi-definite is called a Liapunov s function for the system (). The following theorem is the basic discovery. Theorem: If there exists a Liapunov s function E (x, y) for the system (), then the critical point (0,0) is stable. Further more, if this function has additional property that the function () is negative definite, then the critical point (0, 0) is asymptotically stable. 3. BASIC EQUATIONS OF TE MOEL The basic equations for the growth rate of a flourishing commensal and host species with limited resources are given by dn = a [ N N + C N N + ] dn = a [ N N + ] (3)

3 N. Phani umar, C.Srinivasa umar, B. Ravindra Reddy and N. Ch. Pattabhi Ramacharyulu International ejournal of Mathematics and Engineering 95 (00) TE EQULIBRIUM STATES dn The system has one equilibrium state resulting from Co-existence state E : dn = 0; E : N C ; N C ai where i=, i =, are the carrying capacities of N i. aii a C =, the commensal co-efficient. a 5. LOCAL STABILITY ANALYSIS The present authors [6] discussed the local stability of the above equilibrium state and which is stable. 6. LIAPUNOV S FUNCTION FOR GLOBAL STABILITY The linearized perturbed equations over the perturbations (u, u ) of the system (3) are du a N u Ca Nu (4) CN du a N u (5) The characteristic equation is a N a N 0 C N i.e., a N a N aa N N 0 CN CN This is in the form of p q 0 where p= a N a N >0 (6) C N q = aa N N >0 (7) C N Therefore the conditions for Liapunov s function are satisfied. Now define E (u, u ) = ½ (au + b u u +cu ) (8) where a N aa N N CN a (9) = 0

4 N. Phani umar, C.Srinivasa umar, B. Ravindra Reddy and N. Ch. Pattabhi Ramacharyulu International ejournal of Mathematics and Engineering 95 (00) b Ca a N N (0) CN CN a N C a N a a N N c () pq a N a N aa N N () CN CN From (6) and (7) it is clear that > 0 and a > 0. Also, (ac-b ) = a N a a N N C N a N C a N a a N N C N C N C a a N N (ac b ) > 0 b ac< 0 (3) The function E (u, u ) at (8) is positive definite. Further E du E du = u u au bu a N u Ca Nu bu cu a N u C N = a a N u a Ca N ba N ba N u u CN CN bca N ca N u (4) Substituting the values of a, b and c in (4) we get

5 N. Phani umar, C.Srinivasa umar, B. Ravindra Reddy and N. Ch. Pattabhi Ramacharyulu International ejournal of Mathematics and Engineering 95 (00) E du E du = u u a a N N a N a N aa N N a N a N C N C N u = - u u C N C N u = u u (5) (6) E du E du = u u u u which is clearly negative definite. So, E ( u, u ) is a Liapunov function for the Linear system. Next we prove that E (u, u ) is also a Liapunov function for the non-linear system If f and f are two functions in N and N defined by f (N, N ) = a N a N a N N a (7) f (N, N ) = an an a (8) E E Now we have to show that f f is negative definite u u By putting N = N + u and N = N + u in (7) and (8) we get du = a ( N +u ) a ( N +u ) + a ( N +u ) ( N +u ) + a = a N u Ca Nu F( u, u) C N where F(u, u ) = - a u + a u u du f (u, u ) = = - a N u Ca N u F( u, u) CN Similarly du a ( N u) a ( N u) au = a N u G( u, u) where G(u, u ) = - a u du f (u, u ) = a N u G( u, u) From (8) (9) (0)

6 N. Phani umar, C.Srinivasa umar, B. Ravindra Reddy and N. Ch. Pattabhi Ramacharyulu International ejournal of Mathematics and Engineering 95 (00) E = au +bu u () E = bu + cu u () E E Now f f ( au bu) a N u Ca N u F( u, u) u u CN + ( bu cu) a N u G( u, u) = ( au bu ) a N u Ca N u ( bu cu ) a N u CN ( au bu ) F ( u, u) + ( bu cu) G( u, u) (3) E E f f ( u u ) ( au bu ) F( u, u ) ( bu cu ) G( u, u) (4) du du By introducing polar coordinates (4) becomes E E f f r r[( a cos bsin ) F( u, u) ( bcos csin ) G( u, u) (5) du du Let us denote the largest of the numbers a, b, c by. r Our assumptions imply that F (u, u ) < 6 and G(u r, u ) <, for all sufficiently 6 small r > 0. E E 4r So f + f r = - r 0 (6) u u 6 3 Thus E (u, u ) is a positive definite function with the property that E E f f is negative definite. (7) u u The equilibrium point E is asymptotically stable. REFERENCES [] Bhaskara Rama Sarma B., N.Ch. Pattabhiramacharyulu, On Global Stability of Two-species competing Eco-system with decay and Replenishment for one of the species, International Journal of Scientific computing Vol () (July-ecember 008); PP [] apur J.N., Mathematical Modelling, Wiley Eser (985) [3] Lotka A.J., Elements of Physical Biology, Williams and willians, Baltimore (95). [4] Ogata.., Modern Control Engineering, Prentice-all India(000) [5] Lakshmi Narayan. A Mathematical study of a prey predator Ecological Model with a partial cover for the prey and alternative food for the predator, Ph.. Thesis, JNTU., (005) [6] Phanikumar N., Pattabhi Ramacharyulu N.Ch., On the Stability of a Commensal ost harvested (Immigration) Species pair with Limited resources accepted for publication in International Journal of Application and Applied mathematics.