We present the analysis of the genetic model of autocatalytic chemical reactions proposed by Arnold et al. This model, when subjected to multiplicative white noise modelling environmental fluctuations, can undergo a sudden change from a unimodal state to a bimodal one, while no such transition occurs if the noise is absent. Here, this noise-induced transition is studied analytically by investigating the so-called critical surface in the three-dimensional parameter spac

Pham, T. M.

Virchenko, Y. P.

Journal of Physics Conference Series

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18th International Conference PhysicA.SPbJournal of Physics: Conference Series 769 (2016) 012026IOP Publishingdoi:10.1088/1742-6596/769/1/012026Phase diagram analysis of noise-induced transition in an autocatalytic reactionTuan M inh  P ham , Yu. P. V irchenkoBelgorod National Research University, 14 Studencheskaya, 308015 Belgorod, Russia E-mail: physicsidea@gmail.comAbstract. We present the analysis of the genetic model of autocatalytic chemical reactions proposed by Arnold et al. This model, when subjected to multiplicative white noise modelling environmental fluctuations, can undergo a sudden change from a unimodal state to a bimodal one, while no such transition occurs if the noise is absent. Here, this noise-induced transition is studied analytically by investigating the so-called critical surface in the three-dimensional parameter space.1. In trodu ctionNoise-induced phenomena have become a subject of widespread interest in recent decades, because of their observations in various nonlinear systems in physics, chemistry and biology [1,2]. In particular, autocatalytic reaction occurring in a random environmentis one of the most studied models in chemistry. The reaction dynamics is governed by the following stochastic differential equation (SDE) for the concentration x(t),where a  e  [0,1], A e R, а >  0. This SDE is interpreted in the Stratonovirch sense according to the well-known results of Van Kampen [3], Wong-Zakai [4] and Blankenship [5]. Such chemical interpretation was proposed by Arnold, Horsthemke and Lefever in [6].It was shown in [1] that this equation also describes the time dependence of gene frequency in a haploid population where there are only two possible alleles with corresponding frequencies x and 1 — x. For this reason, the model was named ’’genetic model” .The solutions of Eq.(1) form the Markovian diffusion process x(t) e  [0,1] at any initial condition x(0) e  [0,1]. In thermodynamic equilibrium states, this process has the stationary probability densityA +  X  +  Y  ^  2Y +  A * , B  +  X  +  Y  ^  2X  +  B *,dx(t) =  [a — x(t) +  Ax(t)(1 — x(t))] dt +  ax(t)(1 — x ( t ) )dw(t ) , x(t) e  [0,1] (1)a  — 1 a1 — x x(2)l @  ®  1 Content from this work may be used under the terms o f the Creative Commons Attribution 3.0 licence. Any further distributionK a ^ H ! Z ^ ^ l o f  this work must maintain attribution to the author(s) and the title o f the work, journal citation and DOI.Published under licence by IOP Publishing Ltd 118th International Conference PhysicA.SPb IOP PublishingJournal of Physics: Conference Series 769 (2016) 012026 doi:10.1088/1742-6596/769/1/012026C  =e2/a22K - e  (4а - 2 л/ а ( Г — а ) )1 — аав/2в =2(2а  +  Л -  1)а2here К - в (.) is the modified Bessel function of the second kind with index (—в ) defined asК - в  (z) =  1 /  e-ZCh U+l3'Udu, Re z >  0.2. T h e analysis o f  critical surfaceIn our system coupled to a fluctuating environment, a phase transition can occur whenever the density function p(x) qualitatively changes. In fact, this function may have different mode numbers, depending on the value of parameters Л, а2, a. Therefore, the three-dimensional parameter space is divided into domains in such a way that there is a certain mode number in each of them. This partition, on the analogy of thermodynamics, is called the phase diagram, while the surface £  dividing these domains is called the critical surface. Our paper is devoted to the surface investigation.Since any changes in the mode number of p(x) corresponds to changes in the solution number of the equation dp(x)/dx =  0 which can be written asаS (x) =  a — x  +  Лx(1 — x) — — x(1 — x)(1 — 2x) =  0, x e (0 ,1), (3)we find the degeneracy condition of solutions. This is equivalent to the existence of such a multiple root xo that S (x0) =  S' (x0) =  0. Then apply the Euclid algorithm for polynomials S(x) and S '(x). As a result, we obtain the equation of the critical surface:P (Л, a2 , e) =  Л4 +  Л2  ^1 — 5а2 — а4/2^ — Ле(9а4 +  18а2 — 4Л2) — 4а2  ^1 — а2/4 )  — 27а4е2 =  0 , (4)where е =  а  — 1/2. At this time, the root x0 is given by1 2Л(1 +  2а2) +  18еа2xo =  2 — AA  =  4Л2 +  3а4 — 12а2 .In order that x0 be the bifurcation point of p(x), it is necessary and sufficient that 0 <  x0 <  1 which is equivalent to the following inequalityС +(Л ,а2,е )С _ (Л ,а 2,е) >  0,by introducing the hyperbolasС±(Л, а 2, е) =  A  ±  4[Л(2а2 +  1) +  9еа2] =  0.(5)(6)The inequality (5) defines the allowable domain for the location of the points of the critical surface, whose boundaries are the hyperbolas С ±(Л ,а2,е) =  0. It is crucial as the surface (4) is not connected.Let £ e be the intersection of the surface defined by Eq.(4) with the plane e =  const and (Л*, а^) be the coordinates of the contacting points between the curve £ e and the ellipse A  =  0. Then_  9еа;Л* — —1 +  2а2 • (7)218th International Conference PhysicA.SPb IOP PublishingJournal of Physics: Conference Series 769 (2016) 012026 doi:10.1088/1742-6596/769/1/012026If exclude A* from this relation by using A  =  0, the single-valued function al(e)  is derived from the equation4(а* -  1) 3 =  27а2(1 -  4e2) ,  а ,2 >  1 (8 )due to the convexity of the left-hand side function.Now, we represent the polynomial P(A, a 2, e) as Taylor’s power series of (A -A *) and (а 2 - а 2), which, after introducing the variable substitutionA =  A* +  p cos f ,  a =  a* +  p sin f ,  (9)has the formP(A , a 2 ,e) =  - 3 ( a 2 -  1 ) 2 (p s in f ) 2 Q 2 (z) +  ^ (p s in f ) 3 Q 3 (z) +  -^ (p s in f ) 4 Q 4 ( z ) ,6  16where z =  c tg f, z* =  ctgf* =  A*/3a2, Q 2 (z) =  (z -  z*)2, Q 4 (z) =  (4z2 -  1) 2 and3Q3(z) =  8 z * ( 7 a 2  -  1 ) z 3  -  6 ( a 2  +  5 ) z 2  +  1 8 z * ( 1  +  a2)z +  2 ( a 2  -  3) •The Eq.(4) now becomes- 1  p2 Q 4 (z) sin2 f  +  1  pQ3 (z) sin f  -  3(a 2 -  1 ) 2 Q 2 (z) =  0  , ( 1 0 )16 6and its solutions readp±( f  =  3 q ( z/) sin f  (  -  Q3(z) ±  \JQ3(z) +  2 7 ( a 2  -  1)2Q4(z)Q 2(z) )  • (11)These two functions describe the curve at such f  that p ± ( f )  >  0. This means that is defined by p + (f)  for f  e  [0, n] and by p _ ( f )  for f  e  [ - n ,0] . From Eq.(11), it is not difficult to show that, , v/5|Q3(±1/2)| ^ ^  u , ,1p+(^ ) =  1 2 ( z 2 -  1 / 4 ) 2  ( 1 +  ° (1)) when z 2  •L em m a 1. is a biconnected curve consisting of two components X+ and £ _  where £+  =  p + ( f ) at f  e  [ф, n -  ф], ф =  arcctg (1 / 2 ), while X_ is the glue of simple curvesp+(f ^  f  £ [0 ,ф ); p _ (f ^  f  £ [ - n 0]; p+(f ^  f  £ (п -  ф ,п].□  This statement follows from the fact that connected curve component must be defined as a continuous function and both p ± ( f )  tend to the same finite limit at f  ^  0 ,n. ■T h eorem  1. X+ is composed of two branches stitched at the contacting point (A*,a^).□  In the neighborhood of the point (A*, a2), the asymptotic formula of £+ , in terms of local Cartesian coordinates u and v, has the formu =  const | v |2 / 3  when v ^  0 . ( 1 2 )In fact, from Eq.(11), we find\ 2 I „ Л 3Л ___________ _  „ _ 81а4(1+  z 2 ) 1 / 2p + ( f ) =  k (z  -  z*)2 +  O ((z -  z*)3) when z ^  z* , к  =4(а2 -  1)318th International Conference PhysicA.SPb IOP PublishingJournal of Physics: Conference Series 769 (2016) 012026 doi:10.1088/1742-6596/769/1/012026Hence p+(^>) is presented by u2 +  v2 =  к 2 arctan4(v/u ) where ф — ф* =  arctan(v/u) and consequently it follows Eq.(12). ■In order to establish which component of £ e corresponds to the critical curve, we find the following expressions of G±(A, a2, e) =  0 in the polar coordinates (p, ф)p(±) [4(1 ±  sin2^>) — sin2 ф] + 2  [2 cos ^>(2(A* ±  a2) ±  1) +  sin ф(3а* ±  4A* — 6 ±  18e)] = 0 .Then such angles that p(±)(^ (±)) =  0 are given byz(±) =  6z* T 3(a2 — 2) d 3 )2(2a2 +  1) ±  12a2z* ( )where z (± ) =  ctg ф(± ) . Note that the denominator is greater than zero due to a* >  1.T h eorem  2. £+  is the intersection of the critical surface £  and the plane e =  const.□  It can be seen from Eq.(4) and Eq.(6) that (0,0) and (A*, a2) are two common points of £  and the hyperbolas G±  =  0. Furthermore, the former is the intersection of £ -  and G±  =  0 (as p -(^ o) =  A* +  a4 where ctg^o =  A*/al), while the latter is the intersection of £+  and G± =  0 (Р+(Ф*) =  0 ).We now prove that the component £ -  is contained inside the domain G+G_ <  0. In fact, the derivative (da2/dA) of the implicit function a2(A) defined by Eq.(4) vanishes at (0,0) while those derivatives of functions a 2 (A) described by G± =  0 areda2~d~A± '  = [3 (1  T 3e)]-1 =  0.(0,0)This means that for the points of £ - , being close enough to (0,0), G +G -  <  0 since G±(A, 0,e) =  ±4A +  O (A2) when A ^  0. Moreover, because (0,0) is the unique common point of £ -  and G± =  0, if the inequality G +G_ <  0 is satisfied in the neighborhood of (0,0), it definitely holds for all others points (A, a2) of £ _ . Consequently, £ _  is confined in this domain.In contrast, the half-plane (A, a2 >  0) is divided by the component £+  into two separate parts. Whenever one of the hyperbolas G± =  0 crosses this component, it must go from one part to another part when ф continuously changes, passing through the contacting point (p(+)(^ (+)) =  0 and p( - ) (^ ( - ) ) =  0 respectively). However, this is impossible because of the existence of the cusp point (0, ф*). Since tangent of £+  at this point is a ray radiating from it, for the proof of this fact it is sufficient that ф(-) >  ф* >  ф(+) i.e. z (+) <  z* <  z (-) which follows from Eq. (13). Therefore, only £+  fulfills the condition (5) and represents the critical curve. ■As an illustration for the obtained result, we demonstrate here the critical curve in the case of a  =  1/2, which is defined asA2= 2Ly  +  5a2 — 1 — (2a2 +  1)3/2The inverse function a 2 (A) is depicted in Pic. 1.418th International Conference PhysicA.SPbJournal of Physics: Conference Series 769 (2016) 012026IOP Publishingdoi:10.1088/1742-6596/769/1/012026Pic. 1. Phase diagram in the symmetrical case а  =  1/2. The components £ +  and £ -  are presented bythe solid lines, while the hyperbolas G± =  0 and the ellipse A  =  0 are drawn by dotted lines.3. C onclu sionIn contrast to the previous work of Horsthhemke et al., the complete analysis of the critical surface £  of the model is obtained. This surface divides the parameter space into two domains where qualitatively different stationary dynamical regimes of the system are observed. From the physical point of view, for sufficiently slow variations of the parameters, the switch between these regimes represents a transition between two ” phases” : the unimodal one and the bimodal one. The dynamical regime in the bimodal phase consists of temporal intervals consequentially replacing each other and having random lengths. In these intervals reactant relative concentrations fluctuate nearby the two maxima of the density function p(x).4. R eferences[1] Horsthemke W  and Lefever R 2006 Noise-Induced Transitions: Theory and Applications in Physics, Chemistry,and Biology (Berlin: Springer)[2] Landa P S and McClintock P V E 2000 Physics Reports 323 1[3] Van Kampen N G I 1981 J. Stat. Phys. 24 175[4] Wong E and Zakai M 1965 Ann. Math. Stat. 36 1560[5] Blankenship G and Papanicolaou G C 1978 SIAM J.Appl.Math. 34 437[6] Arnold L, Horsthemke W  and Lefever R 1978 Zs. Phys. B 29 3675

Phase diagram analysis of noise-induced transition in an autocatalytic reaction

Tuan Minh Pham

Yu. P. Virchenko

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Phase diagram analysis of noise-induced transition in an autocatalytic reaction

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