QUALITATIVE STUDY OF A GENERALISED BRUSSELATOR TYPE EQUATION

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1 ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N ( ) 145 QUALITATIVE STUDY OF A GENERALISED BRUSSELATOR TYPE EQUATION B.S. Lakshmi Deartment of Mathematics JNTUH College of Engineering Kukatally Hyderabad bslakshmi2000@yahoo.com S.S. Phulsagar Research Scholar Deartment of Mathematics JNTUH College of Engineering Kukatally, Hyderabad ss.maths@yahoo.co.in M.A.S. Srinivas Deartment of Mathematics JNTUH College of Engineering Kukatally, Hyderabad massrinivas@gmail.com Abstract. In this aer we discuss a system of equations which is a generalisation of the Brusselator equations [13]. Such equations usually deal with some autocatalytic reactions. Some equations related to non-allosteric enzyme reactions which are similar to the Michaelis-Menten equations with regard to functional resonse term are also analysed. Keywords: Autocatalytic reactions, Poincare comactification, enzyme reactions. 1. Introduction and Motivation The classic tri-molecular Brusselator reaction model is known to dislay comlex behaviour [4],[3],[6],[7]. In this aer we consider a multimolecular Brusselator tye of reaction with a secial emhasis on the trimolecular reaction. The generalised Brusselator tye equations based on a multimolecular reaction could have a ossible form (1) dt = 1 ax x y q, dt = b(x y q y), where x, y 0, integers, q 0 and arameters a 0, b 0 [10]. Several authors have investigated the cases where a = 0 (see for examle [11]).. Corresonding author

2 B.S. LAKSHMI, S.S. PHULSAGAR, M.A.S. SRINIVAS 146 Usually such reactions from which the differential equation (1) is derived are enzyme reactions. As with all such reactions this reaction follows the law of mass action which states that the rate of a reaction is roortional to the roduct of the concentrations of the reactants. This is a nonlinear henomenon and several eole have exlained the mechanism [12]. This is not surrising since enzymes are biological catalysts. The Brusselator equation as is well-known reresents an autocatalytic reaction (for examle Belousov-Zhabotinsky reaction, see [13]). Enzymes alter the rates of reactions in cells without being changed themselves during the course of a reaction. Our interest in this aer is with (aart from other enzyme reactions) non-allosteric enzymes [1]. Non-allosteric enzymes are a art of enzymes that are involved in the control and regulation of biological rocesses. The Michaelis Menten [12] enzyme reaction has been studied by several authors. We are interested in this model as well and we roose a new model were in we introduce a functional resonse term of the Michaelis Menten tye outut into the generalised Brusselator equations. This is in Section 4. We stu the following equations in this aer (2) dτ = a bx x y q, dτ = x y q cy y + 1. In this aer we discuss, amongst several cases the case where in equations (2), q = 1, b = 0. Equations (2) would then become (3) dτ = a xn y, dτ = xn y cy y The reaction diffusion system of the generalised Brusslator reaction If the entities x and y reresent chemicals then the equations (1) to (3) would be the rate of reactions corresonding to some reaction kinetics. Taking a hint from [13], the reaction diffusion equations can be written. Such equations would stu the instability induced in a reaction (chemical) by diffusion [16]. Thus the corresonding equations with the inclusion of the diffusion term would be x t = D 1 2 x + a b x + x ỹ q, ỹ t = D 2 2 ỹ + b( x ỹ q ỹ), where 2 x and 2 ỹ are the diffusion terms with D 1 and D 2 the diffusion coefficients, a b x + x ỹ q and b( x ỹ q ỹ) would be the reaction terms. We stu the reaction diffusion reaction related to a Brusselator model as the Brusselator is a erfectly accetable model for the stu of cooerative rocesses in chemical kinetics (to quote Prigogine [13]). According to Prigogine the Brusselator model lays a somewhat similar role as the harmonic oscillator or the Heisenberg model in ferromanetism which are studied to illustrate the basic laws of classical

3 QUALITATIVE STUDY OF A GENERALISED BRUSSELATOR TYPE EQUATION 147 and quantum mechanics. Prigogine stresses on the imortance of trimolecular stes ( shown by the x 2 y and y 2 x terms in the equations) because such cubic nonlinearities give rise to cooerative behaviour. 3. Analysis of the generalised Brusselator reaction Let us consider the system of equations (2). cy Choose b = 0 and use cy in lace of (for the sake of some insight into y + 1 a simler system). (4) dt = a x y q, dt = x y q cy. This is a lanar olynomial system. Solving these two equations for the equilibrium oints, (by equating the derivatives on the left hand sides to zero), one of the equilibrium oints is x = a 1/ ( c a )q/, y = a c. Linearising the system about this equilibrium oint would give the Jacobian matrix J as Its eigenvalues are λ 41 = 1 2 and λ 42 = q q (a c ) 1+ ( a J = c )q 1 q q (a c ) 1+ ( a c )q 1+q [ c a 4a 1+2+q 1+q [ c a 4a 1+2+q ( a c )q c q qa 1 q c q ( a c ) 1+q c + qa 1 q c q ( a. c ) 1+q a 1 q c q + ( a c ) 1+q a 1 q c q q] 1 2 a 1 c q ( a c )q c +q a 1 q c q + [a q c ( a c )q c q + a +q a ( c )q a 1 q c q a 1 a ( c )q c +q a 1 q c q q] 2 a 1 q c q + ( a c ) 1+q a 1 q c q q] a 1 c q ( a c )q c +q a 1 q c q + [a q c. + a +q a ( c )q a 1 q c q a 1 a ( c )q c +q a 1 q c q q] 2 To stu the secial trimolecular case referred to earlier, let us choose +q = 3 with = 2 and q = 1. The system is (5) dt = a x2 y, Its equilibrium oints are ( c, a c ) and ( c, a c ). The eigenvalues at ( c, a c ) are dt = x2 y cy. λ 51 = a c a 2 + 2ac 3/2 c, λ 52 = a c + a 2 + 2ac 3/2 c.

4 B.S. LAKSHMI, S.S. PHULSAGAR, M.A.S. SRINIVAS 148 The eigenvalues at ( c, a c ) are λ 53 = a a 2 2ac 3/2, λ 54 = a a + 2 2ac 3/2. c c c c Observing λ 51 and λ 52 it is aarent that they will not take on comlex values for c > 0, a > 0 and hence the equilibrium oint ( c, a c ) is either a node or a saddle oint (deending on the sign of the eigenvalues). λ 53 and λ 54 can take on comlex values for a 2 2ac 3/2 < 0 or a < 2c 3/2. In this case, since the real art of λ 53 and λ 54 is negative one would exect a siral sink for the equilibrium oint ( c, a c ). This case is illustrated in Figure 1. Trajectory and coordinate functions of the solutions y t 8 6 Out[11]= x t Figure 1: A siral sink for the system (5) for some articular values a = 1.8, c = , = 2, q = 1 In Figure 1 a siral sink for the system (5) is shown for some articular values of a, b, c.

5 QUALITATIVE STUDY OF A GENERALISED BRUSSELATOR TYPE EQUATION 149 A simulation of equation (2) We now consider a simulation of equation (2) by relacing the term cy by It can be seen that a limit cycle aears for some values of arameters. cy y + 1. y t 5 Out[14]= 5 5 x t 5 Figure 2: A Limit Cycle for the system (2) for some articular values a = 1.8, b = 0, c = 4.8, = 2, q = 1 4. A generalised Brusselator tye equation with a Michaelis-Menten functional resonse term C.S.Holling studied the factors involved in the utilization of resources by redators. He described the changes in the feeding rate of organisms as the functional resonse term. He showed that there were three categories of functional resonse [15]. Tye 1. Refers to animals which consume food roortional to the rate of their encounter with food items. Tye 2. Where the organisms take some time to eat and to cature their rey.

6 B.S. LAKSHMI, S.S. PHULSAGAR, M.A.S. SRINIVAS 150 Tye 3. In this category the organism will not consume the rey if it is below a certain threshold density. There is a remarkable arallel between enzyme reactions and the redator-rey Holling functional resonse [2]. The Michaelis-Menten enzyme reaction follows a tye 2 functional resonse. Keeing this tye of functional resonse in mind we roose the following model and analyse it. (6) dt = a bx x y q, dt = x y q cxy y + 1. For the urose of understanding and analysis, we take b = 0, = 2, q = 1. Equations (6) reduce to (7) dt = a x2 y, dt = x2 y cxy y + 1. The equilibrium oints of equation (7) are and ( 1 2 [c 4a + c 2 ], 2a + c2 + c 4a + c 2 ) 2a ( 1 2 [c + 4a + c 2 ], 2a + c2 c 4a + c 2 ). 2a Let (x 0, y 0 ) be an equilibrium oint chosen from amongst the two equilibrium oints of equation (7) so that x 0 > 0, y 0 > 0. This is ossible if c 2 > 4a. Linearising the system (7) about its equilibrium oint, the Jacobean matrix is [ 2x 0 y 0 x 2 ] 0 M = 2x 0 y 0 cy 0 x y + cx 0y 0 0 (1 + y 0 ) 2 cx 0, 1 + y 0 where the elements in the linearised matrix are to be treated as the functions of the arameter c. The characteristic equation of this matrix has the form λ 2 sλ + D = 0 where s(c) = the trace of M = 2x 0 y 0 + x cx 0y 0 (1 + y 0 ) 2 cx y 0 and D = determinant of M = 2cx2 0 y2 0 (1+y 0 + cx2 ) 2 0 y 0 1+y 0. Let the two eigenvalues of this matrix be λ 1 and λ 2. These can be reresented as a function of c. λ 1, λ 2 = 1 2 [s(c) ± s 2 (c) 4D(c)]. We will now show that a Hof bifurcation can occur in this equation for some values of the arameter c. A Hof bifurcation condition would require that ([9], see age 91) the real art (Re) of the eigenvalues is equal to zero and

7 QUALITATIVE STUDY OF A GENERALISED BRUSSELATOR TYPE EQUATION 151 the imaginary art (Im) is nonzero Re = θ(c) = 1 2s(c) and Im = ν(c) = 1 2 s 2 (c) 4D(c) (see [8]). Solving for the arameter c after setting the trace to zero (1) (2) x 2 0 2x 0 y 0 + cx 0y 0 (1 + y 0 ) 2 cx 0 = 0 c = (x 0 2y 0 )(1 + y 0 ) y 0 d dc (Trace) = x 0y 0 (1 + y 0 ) 2 x y 0 (1) is the non-hyerbolicity condition and (2) is the transversality condition. Thus showing the existence of a Hof bifurcation for the arameter c. A simulation of the limit cycle is shown in Figure 3. y t x t Figure 3: A limit cycle for the values of the arameter a = 1.8, c = 1, = 2, q = 1

8 B.S. LAKSHMI, S.S. PHULSAGAR, M.A.S. SRINIVAS 152 The generalised Brusselator equation (1) is a lanar olynomial system. Usually in such systems one needs to stu not just its finite equilibria but also the equilibria at infinity. When related to a chemical reaction this would mean that one studies the tendency of the concentrations of the chemicals over a large range. So for this it is aroriate to use a Poincaré Comactification. 5. A Brief Outline of Poincaré Comactification In order to stu the behaviour of the trajectories of a lanar differential system near infinity we use a comactification. A good aroach for stuing the behaviour of trajectories near infinity is to use the Poincaré shere, introduced by Poincaré [14]. It has the advantage that the singular oints at infinity are sread out along the equator of the shere. In order to draw the hase ortrait of a vector field, we would have to work over the comlete real lane R 2, which is not very ractical. If the functions defining the vector field are olynomials, we can aly Poincaré comactification, which will tell us how to draw it in a finite region. It controls the orbits which tend to or come from infinity. Here we use (x 1, x 2 ) as coordinates instead of (x, y) (in order to differentiate). Let X = P / x 1 + Q / x 2 be a olynomial vector field (the functions P and Q are olynomials of arbitrary degree in the variables x 1 and x 2 ), or in other words: x 1 = P (x 1, x 2 ) x 2 = Q(x 1, x 2 ). The degree of X is reresented as d where d is the maximum of the degrees of P and Q. Poincaré comactification works as follows: First we consider R 2 as the lane in R 3 defined by (y 1, y 2, y 3 ) = (x 1, x 2, 1). We consider the shere S 2 = {y R 3 : y1 2 + y2 2 + y2 3 = 1} which we will call here Poincaré shere; it is tangent to R 2 at the oint (0, 0, 1). We may divide this shere into H + = {y S 2 : y 3 > 0} (the northern hemishere), H = {y S 2 : y 3 < 0} (the southern hemishere) and S 1 = {y S 2 : y 3 = 0} (the equator). Now we consider the rojection of the vector field X from R 2 to S 2 given by the central rojections f + : R 2 S 2 and f : R 2 S 2. More recisely, f + (x) (resectively f (x)) is the intersection of the straight line assing through the oint y and the origin with the northern (resectively, southern) hemishere of S 2. f + (x) = ( x 1 (x), x 2 (x), 1 (x) ), where f (x) = ( x 1 (x), x 2 (x), 1 (x) ), (x) = (x 1 ) 2 + (x 2 )

9 QUALITATIVE STUDY OF A GENERALISED BRUSSELATOR TYPE EQUATION 153 Figure 4: Poincaré Shere In this way we obtain induced vector fields in each hemishere. (For more details see age 150 of [5]). We now consider a Poincaré comactification of equation (2). Let ˆP (x, y) = a bx x y q, ˆQ(x, y) = x y q cy y + 1. For the local chart U 1 (see [5]), x is transformed to 1 v and y is transformed to u v. 1 ˆP ( v, u v ) = a b v uq v +q, ˆQ( 1 v, u v ) = uq v +q cu u + v. The extended vector on S 2 which is called the Poincaré comactification of the vector field X on R 2 is denoted by (X). The exression for (X) in the local chart (U 1, ϕ 1 ) is given by u = v d [ u ˆP ( 1 v, u v ) + ˆQ( 1 v, u v )], v = v d+1 ˆP ( 1 v, u v ).

10 B.S. LAKSHMI, S.S. PHULSAGAR, M.A.S. SRINIVAS 154 where d=(maximum of the degrees of ˆP and ˆQ)= + q. Therefore, u = v +q [ au + bu v + uq+1 + u q v +q cu u + v ] v = v +q+1 [a b v uq ]. v+q For the local chart U 2, x is transformed to u v and y is transformed to 1 v. The exression for chart (U 2, ϕ 2 ) is given by Therefore u = v d [ ˆP ( u v, 1 v ) u ˆQ( u v, 1 v )], v = v d+1 ˆQ( u v, 1 v ). u = v +q [a bu v u+1 + u v +q c v + 1 ], u v = v +q+1 [ v +q c v + 1 ]. In order to illustrate this comactification rocess, using these equations we evaluate the local charts U 1 and U 2. We choose some secific values for the various arameters in equation (2) as a = 1, b = 0, c = 1, = 2, q = 1. The system is dt dt The exression for the local chart U 1 is For the local chart U 2 = (1 x 2 y)(y + 1), = x 2 y(y + 1) y. u = u(u + v)(1 + u v 3 ) uv 3, v = v(u + v)(u v 3 ). u = (1 + v)(v 3 u 2 u 3 ) + uv 3, v = v 4 vu 2 (1 + v). In figure 5 there are many equilibrium oints visible on the Poincaré shere. One of them is an unstable node as can be seen from the trajectories moving away from the equilibrium oint and a stable node where the trajectories move towards the equilibrium oint. For the system dt dt = (0.35 x 2 y)(y + 1), = x 2 y(y + 1) 5y.

11 QUALITATIVE STUDY OF A GENERALISED BRUSSELATOR TYPE EQUATION 155 Figure 5: Phase ortrait on the Poincaré shere The exression for the local chart U 1 is u = u(u + v)(1 + u 0.35v 3 ) 5uv 3, v = v(u + v)(u 0.35v 3 ). For the local chart U 2, is u = (1 + v)(0.35v 3 u 2 u 3 ) + 5uv 3, v = 5v 4 vu 2 (1 + v). Figure 6: Phase ortrait on the Poincaré shere

12 B.S. LAKSHMI, S.S. PHULSAGAR, M.A.S. SRINIVAS 156 With the change of the values of the arameters, the hase ortrait on the Poincaré shere has in addition to the stable and unstable nodes, a saddle oint and a focus as can be seen from the figure 6. References [1] J.M. Berg, J.L. Tymoczko, L. Stryer, Biochemistry 5th Edition, W.H. Freeman and Comany, [2] A.A. Berryman, The Origins and Evolution of Predator-rey Theory, The Ecological Society of America, 73(5) (1992), [3] D. Erle, Nonuniqueness of stable limit cycles in a class of enzyme catalyzed reactions, Journal of Mathematical Analysis and Alications, 82 (1981), [4] D. Erle, K.H. Mayer and T. Plesser, The existance of stable limit cycles for enzyme catalyzed reactions with ositive feedback, Mathematical Biosciences, 44 (1979), [5] F. Dumortier, J. Llibre and J.C. Artés, Qualitative Theory of Planar Differential Systems, Sringer-Verlag Berlin Heidelberg, [6] C. Escher, Models of chemical reaction systems with exactly evaluable limit cycle oscillations, Zeitschrift für Physik B Condensed Matter, 35 (4) (1979), [7] P. Glansdorffand, I. Prigogine, Thermonamic Theory of Structure, Stability and Fluctuations, Wiley-Interscience, New York, [8] M.W. Hirsch, S. Smale, R.L. Devaney, Differential Equations, Dynamical Systems & An Introduction to Chaos Second Edition, Elsevier Academic Press (USA), [9] Y.A. Kuznetsov, Elements of Alied Bifurcation Theory, Second Edition, Sringer-Verlag New York, [10] Z. Leng, B. Gao, Z. Wang, Qualitative analysis of a generalized system of saturated enzyme reactions, Mathematical and Comuter Modelling, 49 (2009) [11] J.X. Li, H.Y. Fan, T.L. Jian, X.D. Chen, Qualitative analysis for a differential equation model of multi-molecular reactions, Journal of Biomathematics, 2 (1990), [12] L.M. Michaelis, M.L. Menten, Die kinetik der invertinwirkung, Biochem.Z. 49 (1913),

13 QUALITATIVE STUDY OF A GENERALISED BRUSSELATOR TYPE EQUATION 157 [13] G. Nicolis, I. Prigogine, Self-Organisation in Nonequilibrium Systems, John Wiley and Sons, Inc., [14] H. Poincaré, Sur I intégration des équations différentielles du remier ordre et du remier degré I. Rendiconti del circolo motematico di alermo, 5 (1891), [15] L.A. Real, The Kinetics of Functional Resonse, The American Naturalist, Vol. III No. 978 (1977), [16] A. M. Turing, The chemical basis of morhogeneses. Philosohical Transactions of the Royal Society of London B:Biological Sciences. B 237, 3772 (1952). Acceted: