Stability Analysis for a Class of Non-Weakly Reversible Chemical Reaction Networks

Transcription

1 Specia Issue on SICE Annua Conference 217 SICE Journa of Contro, Measurement, and System Integration, Vo. 11, No. 3, pp , May 218 Stabiity Anaysis for a Cass of Non-eay Reversibe Chemica Reaction Networs Hiroazu KOMATSU and Hiroyui NAKAJIMA Abstract : e consider ordinary differentia equations (ODEs) that describe the time evoution of the concentrations of species in chemica reaction networs (CRNs). In order to anayze the convergence of soutions to the ODEs, the chemica reaction networ theory has estabished an important theorem caed Deficiency Zero Theorem (DZT). This theorem provides a sufficient condition for any soution to the ODEs to converge to an equiibrium point, based ony on the graph structures of the CRNs and the agebraic properties of ODEs. In the present paper, we consider a cass of non-weay reversibe chemica reaction networs, to which the DZT cannot be appied since one of the conditions, wea reversibiity, is not satisfied. In order to mae up for the faiure of this important condition, by decomposing the networ into weay reversibe sub-networs and appying the DZT to them, we show any soution to the ODEs for our cass of networs with positive initia vaues converges to an equiibrium point on the boundary of the positive orthant. Key ords : chemica reaction networs, wea reversibiity, ordinary differentia equations, stabiity, equiibrium point. 1. Introduction It is widey nown that soutions to ordinary differentia equations (ODEs) that describe the time evoution of the concentrations of species in chemica reaction networs (CRNs) can be osciatory or behave chaoticay. For most of the equations, however, the soution converges to an equiibrium point that corresponds to a dynamica equiibrium for the chemica reaction networ [1]. Athough it is very important in theoretica chemistry to determine whether the dynamics of a CRN converges to a state of dynamica equiibrium or not, no genera method to determine that has been estabished. In order to formuate and anayze the differentia equations describing CRNs, M. Feinberg and his coeagues [1] [4] have deveoped the chemica reaction networ theory. This theory estabished the Deficiency Zero Theorem (DZT), which provides a sufficient condition for the ODEs describing the dynamics of a chemica reaction networ to have an asymptoticay stabe equiibrium point in a positive stoichiometric compatibiity cass, based ony on their graph structures of the CRNs and the agebraic properties of the ODEs. The conditions of the theorem do not depend on the size of the system and the vaues of system parameters. However, there are many CRNs that do not satisfy the two conditions of the DZT, wea reversibiity and zero deficiency, and hence the positive soution to the ODE that describes the dynamics of such a CRN cannot be proved to converge to an equiibrium point based on the DZT. Athough some methods have been proposed for anayzing the convergence to a positive equiibrium point of a soution to ODEs for a cass of CRNs that are weay reversibe but with Graduate Schoo of Systems Engineering, Kindai University, Hiroshima , Japan E-mai: @hiro.indai.ac.p, naaima@hiro.indai.ac.p (Received October 31, 217) (Revised December 21, 217) non-zero deficiencies [5],[6], no methods have been estabished to anayze the dynamics of non-weay reversibe CRNs with zero deficiency, as far as we now. In the recent wor [7], we showed that any soution to the ODE that describes the dynamics of CRNs in a cass of nonweay reversibe networs with positive initia vaues converges to an equiibrium point. The proof given in the paper was, however, in an outine form for a specific exampe networ in our cass. In the present paper, we sha give a genera and rigorous proof for the convergence of a soution to an equiibrium point for our cass. The outine of this paper is as foows. In Section 2, we briefy introduce the chemica reaction networ theory, which pays an important roe throughout this paper. In Section 3, first we provide a cass of chemica reaction networs that do not satisfy the wea reversibiity, one of the conditions of the Deficiency Zero Theorem. Next, we state two theorems with respect to convergence of a soution to the ODE describing the non-weay reversibe networs in our cass. In Section 4, we give proofs of these theorems. In Section 5, we appy our resuts to an exampe networ. Finay, we give a concusion of this paper in Section Chemica Reaction Networ Theory In this section, we briefy introduce the chemica reaction networ theory [1] [3]. A chemica reaction networ (CRN) in the sense of Feinberg is mathematicay defined by a tripet (S, C, R), where the eements S, C and R are defined by S: thesetofn species in the networ, which are denoted by X 1, X 2,...,X n, C: the set of a compexes y in the networ, R: the set of a chemica reactions y y in the networ. Here a compex is a inear combination of species, y = y 1 X y n X n, with coefficients of non-negative integers, y 1, y 2,...,y n, and a chemica reaction y y denotes that a JCMSI 3/18/ c 217 SICE

2 SICE JCMSI, Vo. 11, No. 3, May compex y is produced from y. e may associate y with the vector of these cofficients, (y 1, y 2,...,y n ) T,andyand y are caed a reactant and a product, respectivey. Next, we denote by x i the moar concentration of the species X i Sand define the non-negative vector of concentrations by x = (x 1, x 2,...,x n ) R n. Then, the time-evoution of x is given by the foowing ordinary differentia equation: d dt x(t) = y y R K y y (x(t))(y y), t, (1) whereafunctionk y y : R n R, which is caed a inetics of the chemica reaction y y, is a function of cass C 1. In particuar, the inetics given by K y y (x) = y y x y := y y x y 1 1 xy 2 2 xy n n, x R n, (2) is caed the mass action inetics. In [3],[4], it has been proved that the ODE (1) is non-negative (resp. positive), that is, any soution x(t) to (1) with an initia vaue x R n (resp. x R n >, which is the positive orthant of R n ) remains in R n (resp. Rn > )forat. Next, we define a subspace H of R n, which is caed the stoichiometric subspace of a chemica reaction networ (S, C, R), by where m = C, the number of compexes, and is the number of the inage casses [1],[3]. For a chemica reaction networ with the zero deficiency, the foowing theorem, which is caed the Deficiency Zero Theorem, has been proved by M. Feinberg in [1],[3]. Theorem 2.1 (The Deficiency Zero Theorem) Suppose that a chemica reaction networ (S, C, R) has zero deficiency. Then, the foowing statements hod: (i) If the networ (S, C, R) is not weay reversibe, then the ODE (1) for (S, C, R) admits neither a positive equiibrium nor a periodic orbit containing a point in R n >. (ii) If the networ (S, C, R) is weay reversibe, then within each PSCC, the ODE (1) for (S, C, R) with the mass action inetics (2) has a unique equiibrium point, which is ocay asymptoticay stabe reative to the PSCC. 3. A Cass of Non-eay Reversibe Networs e sha dea with a cass of chemica reaction networs as shown in Fig. 1, which do not satisfy the wea reversibiity, one of the conditions of the Deficiency Zero Theorem. H := span{y y y y R}, (3) and a stoichiometric compatibity cass of (S, C, R) bytheset c + H := {c + h h H} for c R n, i.e. the subset of R n obtained by transating H by c. Moreover, we refer to the set (c+h) R n >, c Rn >,asapositive stoichiometric compatibiity cass of the networ (S, C, R). By integrating (1), for a t wehave ( t ) x(t) x = K y y (x(s))ds (y y), y y R and hence the soution x to the ODE (1) with the initia vaue x remains in x + H for a t. Together with a positivity of the soution, we see that the soution x remains in the positive stoichiometric compatibiity cass (x + H) R n > for a t. Hereafter we abbreviate a stoichiometric subspace, a stoichiometric compatibiity cass and a positive stoichiometric compatibiity cass to an SS, an SCC and a PSCC, respectivey. A chemica reaction networ (S, C, R) can be regarded as a directed graph. Actuay, the nodes of the graph are the compexes and a directed edge of the graph is given by a reaction from a compex y to a compex y if and ony if y y R. Each connected component of the directed graph is caed a inage cass of the networ. hen each inage cass is a strongy connected subgraph, the chemica reaction networ (S, C, R) is caed weay reversibe. In other words, a networ (S, C, R) is weay reversibe if and ony if for any reaction y y Rthere exist compexes y 1, y 2,...,y such that a of the reactions y y 1, y 1 y 2,..., y y are in R. At the end of this section, we define an index, which is caed the deficiency of (S, C, R), as foows. Definition 2.1 The deficiency, a non-negative integer denoted by δ, of a chemica reaction networ (S, C, R) is defined by: δ := m dim(h), (4) Fig. 1 A cass of non-weay reversibe networs (N = 4). A networ in this cass can be decomposed into N + 1 subnetwors (S i, C i, R i ) and for each sub-networ (S i, C i, R i )(i = 2,...,N+1), there are N reactions with reactants in (S 1, C 1, R 1 ) and products in (S i, C i, R i ) as denoted by thic arrows in Fig. 1. Obviousy, this networ is not weay reversibe since these reactions are in one direction. A chemica reaction networ in Fig. 1 can be mathematicay described by the foowing networ (S, C, R ): N+1 N+1 S := S i, C := C i, N+1 N+1 R := R i { y (1) (p i ) y (i) (q i ) }, (5) i=2 where 1 p 2 < p 3 <...<p N+1 m 1 (:= C 1 ), 1 q i m i (:= C i ), (i = 2,...,N + 1) and S i, C i and R i (i = 1, 2,...,N + 1) satisfy the foowing three conditions: 1. S i = {X (i) 1, X(i) 2,...,X(i) n i } and S i S =, (i ). 2. C i = {y (i) (1), y (i) (2),...,y (i) (m i )}, wherey (i) ( ) Z n i ( = 1,...,m i ), and C i does not contain a zero compex, a zero vector in Z n i. Moreover for a y(i) (), y (i) () C i,, supp(y (i) ()) supp(y (i) ()) =. Here, supp(x)forx R n is a subset of species S such that X i supp(x) if and ony if x i. 3. Each (S i, C i, R i ) consists of a singe inage cass and is weay reversibe. The condition 1 means that each of species in one subnetwor is not in any other. The condition 2 means that species

3 14 SICE JCMSI, Vo. 11, No. 3, May 218 composing one compex are not contained in any other compex. For simpicity, we denote by (P )and(p i ) the initia vaue probem of the ODE (1) with the mass action inetics (2) for the networ (S, C, R ) and the sub-networ (S i, C i, R i )(i = 1,...,N +1), respectivey. e denote by x (i), ( = 1,...,n i, i = 1,...,N + 1) the moar concentration of species X (i), ( = 1,...,n i, i = 1,...,N + 1), and define the vector of concentrations of species in the subnetwor (S i, C i, R i ), i = 1, 2,...,N+1 by x (i) := (x (i) 1,...,x(i) n i ) T R n i, (i = 1,...,N + 1) and put x := (x (1)T,...,x (N+1)T ) T R n,wheren := n n N+1. The main resuts of this paper are in the foowing two theorems, the proofs of which wi be given in Section 4. Theorem 3.1 The probem (P ) admits neither a positive equiibrium nor a periodic orbit containing a point in R n >. Theorem 3.2 Any positive soution x(t) to(p ) with an initia vaue x() R n > converges to an equiibrium point on the boundary of the PSCC containing x(), that is, some concentrations of species in chemica reaction networ (S, C, R ) converge to zero and a others converge to some positive vaues. By the definition of wea reversibiity and the fact widey nown in the graph theory, we see that any non-weay reversibe CRN consists of strongy connected inage casses at east two of which are ined in one direction. In other words, any non-weay reversibe CRN (any directed graph) can be decomposed into weay reversibe sub-networs (strongy connected components). However, it is hard to tace such genera networs, hence in the present paper we choose one of the simpest singe inage non-weay reversibe networ in Fig. 1 as a target of study, athough our stabiity anaysis is potentiay appicabe to more genera non-weay reversibe CRNs, which wi be deat with in future wor. 4. The Proofs of Theorems 3.1 and 3.2 First, we consider the initia vaue probem (P i ) of the subnetwor (S i, C i, R i )(i = 1, 2,...,N + 1). From the conditions 2 and 3 of of the networ (S i, C i, R i )and Theorem 2.1 (ii), we have the foowing theorem immediatey. Theorem 4.1 The probem (P i )(i = 1, 2,...,N + 1) has a unique equiibrium point within each PSCC, and this equiibrium point is ocay asymptoticay stabe reative to the PSCC. Proof: e see from the conditions 2 and 3 of the networ (S i, C i, R i ) that the SS, H i, is spanned by m i 1 ineary independent vectors as H i = span { y (i) (2) y (i) (1),...,y (i) (m i ) y (i) (1) }, (6) which impies that dim(h i ) = m i 1. Since we see from the condition 3 that the number of inage casses i of the networ is unity, the deficiency δ i of the networ is δ i = C i i dim(h i ) = m i 1 (m i 1) =. Besides, from the condition 3 the networ (S i, C i, R i )is weay reversibe, and hence Theorem 2.1 (ii) can be appied to each networ (S i, C i, R i ). The proof is compete. e sha show that any positive soution to the probem (P i )(i = 1,...,N + 1) is bounded gobay in time. In order to do this, we start by defining conservativity of a chemica reaction networ, giving a emma [8] and proving some properties are equivaent to the conservativity. Definition 4.1 Consider a chemica reaction networ (S, C, R) with n species S = {X 1,...,X n }.LetHbe the SS of (S, C, R). The networ (S, C, R) issaidtobeconservative if there exists a vector a R n > such that a h = forah H. Lemma 4.1 (Stieme [8]) For u 1,...,u m R n, exacty one of the foowing two conditons hods: 1. There exists c R n such that m c i u i, = 1,...,n, and at east one of the inequaities is strict. 2. There is a w R n > such that w u i = for each i = 1,...,m. Theorem 4.2 For any CRN (S, C, R), the foowing three conditions are equivaent to each other: 1. The CRN (S, C, R) is conservative. 2. Any non-zero vector in H consists of both positive and negative components. 3. There are n s ineary independent vectors of positive integers orthogona to H. Here s = dim(h). (Proof for 1 2) Suppose that the condition 1 hods. Then by the definition of conservativity there exits a vector a R n > such that a h = forah H. Now we assume that 2 does not hod. Then there exits a non-zero vector h H R n. Thus we see a h >, which is in contradiction with the conservativity. (Proof for 2 1) Suppose that the condition 2 hods. Letting {h 1,..., h s } beabasisforh we see that the inear combination s c ih i has both positive and negative components for a nonzero c R s. Therefore by virtue of Lemma 4.1, we see that there exists a positive vector a R n > such that a h i = (1 i s). This impies a h = forah H because {h 1,..., h s } is a basis of H. This competes the proof. (Proof for 1 3) It is trivia that 3 impies 1, hence we sha prove that 1 impies 3. Suppose that 1 hods. Then there exists a vector b 1 R n > that is orthogona to H. e can aso find a set of ineary independent positive vectors {b 1,..., b n s } each vector in which is orthogona to the s-dimensiona subspace H. By the definition of an SS, we have ineary independent integervaued vectors h 1,..., h s Z n such that H = span(h 1,..., h s ). Let A := [h 1,..., h s ] Z s n. As shown above, there exist ineary independent positive-vaued vectors {b 1,..., b n s } such that Ab i = forai = 1,..., n s. The soution space for the simutaneous inear equations Ax = with integer coefficient matrix A is spanned by a set of ineary independent vectors of rationa numbers, {q 1,..., q n s }. Since rationa numbers are dense in the set of rea numbers, we can find a set of ineary independent positive-vaued rationa vectors orthogona to H by sighty changing the above positive-vaued vectors {b 1,..., b n s }. Besides, by mutipying this vector by a sufficienty arge integer we obtain n s ineary independent vectors of positive integers orthogona to H. This competes the proof. e see that each chemica reaction networ (S i, C i, R i ), i = 1, 2,..., N + 1 is conservative.

4 SICE JCMSI, Vo. 11, No. 3, May Lemma 4.2 Each chemica reaction networ (S i, C i, R i ), i = 1, 2,...,N + 1 is conservative. Furthermore, for the SS of (S i, C i, R i ), H i, there exist n i dim(h i ) positive integer vectors () = ( 1 (),...,a(i) n i ()) T Z n i >, = 1,...,n i dim(h i ) satisfying () h = foranyh H i. Proof e show that the chemica reaction networ (S i, C i, R i ) satisfies the condition 2 of Theorem 4.2, that is, any non-zero vector in H i consists of both positive and negative components. For a non-zero vectors h H i, there exist α 2,...,α mi R such that h = α (y (i) () y (i) (1)) + α (y (i) () y (i) (1)), I + I where index sets I + and I are defined by I + := { {2,...,m i } α > }, I := { {2,...,m i } α < }. In the case where I + and I, we see from the condition 3 of the networ (S i, C i, R i )thath s > foras such that X s (i) supp(y (i) ( )) for I + and h s < foras such that X s (i) supp(y (i) ()) for I. In the case where I + and I =, we aso see from the condition 3 that h s > forassuch that X s (i) supp(y (i) ( )) for I + and h s < foras such that X s (i) supp(y (i) (1)). In the case where I + = and I, we can show in the same way as above that h s < foras such that X s (i) supp(y (i) ( )) for I and h s > foras such that X s (i) supp(y (i) (1)). Hence, the networ (S i, C i, R i ) satisfies the condition 2 of Theorem 4.2. Therefore, the emma foows from the equivaence of the conditions 1, 2 and 3. Using this emma, we can show the foowing theorem. Theorem 4.3 Any positive soution x (i) (t) to the probem (P i ), i = 1, 2,...,N + 1 is bounded gobay in time, that is, sup x (i) (t) < +. (7) t Proof Since the networ (S i, C i, R i ) is conservative, we see from Theorem 4.2 that there exist n i dim(h i ) ineary independent positive integer vectors () = ( 1 (),...,a(i) n i ()) T Z n i >, = 1,...,n i dim(h i ) satisfying () h = fora h H i. Now, we define inear functions : R n i R, = 1,...,n i dim(h i )by (x(i) ):= n i =1 ()x (i), = 1,...,n i dim(h i ).(8) By considering the time derivative Ṫ (i) of these functions aong the positive soution x (i) (t) to(p i ), we easiy see that (x(i) (t)) = (x(i) ()), = 1,...,n i dim(h i )forat. Hence by putting M(x (i) ()) := max 1 n i dim(h i ) (x(i) ()), we see that x (i) (t) M(x (i) ()), = 1,...,n i for a t, which impies that im sup t x (i) (t) <. Moreover, it is easy to see that the inear functions characterize the PSCC (c + H i ) R n i > for any c Rn i > as { P (i) (c) := x (i) R n i } > (x(i) ) = () c,, = 1,...,n i dim(h i ) i = 1,...,N + 1. (9) e sha show that the unique equiibrium point within each PSCC of the probem (P i )(i = 1, 2,...,N + 1), which has been shown to exist in Theorem 4.1, is gobay asymptoticay stabe reative to the PSCC. First we give the definitions of a semiocing set and semi-conservativity [2]. Definition 4.2 For a networ (S, C, R), a non-empty subset of S is caed a semi-ocing set if supp(y) for any reaction y y Rsuch that supp(y ). hen does not contain any other semi-ocing set, it is caed a minima semi-ocing set. Definition 4.3 Foranetwor(S, C, R), et be a non-empty subset of S and H be the SS of (S, C, R). The networ (S, C, R) is said to be semi-conservative with respect to if there exists a vector a R n satisfying supp(a) = and a h = fora h H. The foowing emma pays an important roe in showing gobay asymptotic stabiity of the unique equiibrium point within each PSCC of the probem (P i )(i = 1, 2,...,N + 1). Lemma 4.3 The networ (S i, C i, R i )(i = 1,...,N+1) is semiconservative with respect to each minima semi-ocing set. Proof Let be a minima semi-ocing set. Then we see from the conditions 2 and 3 of the networ (S i, C i, R i )thatevery minima semi-ocing set consists of m i species each of which appears in exacty one compex. mi } S i, That is, there exist X (i) supp(y (i) ()), = 1,...,m i such that = {X (i) 1,...,X (i) and X (i) X (i),. Nowwedenotebyc (i) the -th component of y (i) (), = 1,...,m i and define () = ( 1,...,a(i) n i ) T R n i mi =1, := c(i) >, { 1,..., mi },, (otherwise). Then, from the condition 2 we have () (y (i) () y (i) (1) ) m i m i = =1 c (i) =1 c (i) by (1) =, = 2,...,m i. (11) Henceweseefrom(6)that () h = forah H i. Here we introduce the foowing two emmas [2] to show gobay asymptotic stabiity of the unique equiibrium point within each PSCC of the probem (P i )(i = 1, 2,...,N + 1). Lemma 4.4 Consider the ODE (1) for (S, C, R). Let S be a non-empty subset. If there exists an initia vaue x() R n > such that ω(x()) L, then is a semiocing set. Here L is a subset of R n defined by L := { x R n x i = X i }.

5 142 Lemma 4.5 Suppose the ODE ϕ = f (ϕ) with a C 1 function f defined on an open set Ω R n has a unique equiibrium point ϕ Ω and there exists a gobay defined Lyapunov function V satisfying the foowing conditions: (1) V(ϕ), and the equaity hods if and ony if ϕ = ϕ. (2) d dtv(ϕ(t)), and the equaity hods if and ony if ϕ(t) = ϕ for a t. (3) V(ϕ) as ϕ R n. Then it hods that for any soution ϕ(t), either ϕ(t) ϕ or ϕ(t) Ω in R n as t,where Ωis the boundary of Ω. By using Lemmas 4.3, 4.4 and 4.5, we have the foowing theorem. Theorem 4.4 The probem (P i )(i = 1, 2,...,N + 1) has a unique equiibrium point within each PSCC, and this equiibrium point is gobay asymptoticay stabe reative to the PSCC. Proof: From Theorem 4.1, (P i ) has a unique equiibrium point within each PSCC, and this equiibrium point is ocay asymptoticay stabe reative to the PSCC. Moreover, we see from Theorem 4.3 that any positive soution to (P i ) with an initia vaue x (i) () R n i > is bounded, hence its ω-imit set ω(x(i) ()) is non-empty. The function V : R n i > R defined by n i V(x (i) ):= (x (i) (n(x (i) ) n(x (i) ) 1) + x (i) ), =1 x (i) P (i) (x (i) ()), where x (i) = (x (i) 1,...,x(i) n i ) T is the unique equiibrium point on the PSCC P (i) (x (i) ()), satisfies (1) (3) of Lemma 4.5 [1],[2], and hence we see from Lemma 4.5 that a positive soution x (i) (t) to (P i ) satisfies either x (i) (t) x (i) or x (i) (t) P (i) (x (i) ()) as t. e assume that x (i) (t) P (i) (x (i) ()), t, that is, ω(x (i) ()) P (i) (x (i) ()). Since the boundary P (i) (x (i) ()) is a union of the sets L for some S, ω(x (i) ()) L for some hods. Thus, there exists w (i) ω(x (i) ()) L,and we see from Lemma 4.4 that is a semi-ocing set. For a minima semi-ocing set, since from Lemma 4.3 the networ (S i, C i, R i ) is semi-conservative with respect to, there exists an ( ) R n i satisfying supp( ( )) = and ( ) h = forah H i.now, we define a inear function : R n i R by (x (i) ):= n i =1 ( )x (i). (12) By considering the time derivative Ṫ (i) of these functions aong the positive soution x (i) (t) to(p i ), we easiy see that (x (i) (t)) = (x (i) ()) > forat. Hence, by the continuity of (x (i) (t)) with respect to t, wesee (w (i) ) = (x (i) ()) >, which is in contradiction with w (i) L. Therefore it hods that x (i) (t) x (i). The proof of this theorem is compete. Now, we sha give the proof of Theorem 3.1. Proof of Theorem 3.1: e first prove SICE JCMSI, Vo. 11, No. 3, May 218 N+1 y (i) (q i ) y (1) (p i ) span y y y y R (13) for a i = 2, 3,...,N + 1. Since supp(y (i) ()) supp(y ( ) ()) =, = 1,...,m i, = 1,...,m, i, = 1,...,N + 1, i, if (13) does not hod for some i,wefind y (i) (q i ) y (1) (p i ) span { y y y y R 1 R i }. (14) Besides, for supp(y (i) (q i )) supp(y (1) (p i )) =,weseey (i) (q i ) is in span {y y y y R i }, i.e. in H i. However, from the proof of Lemma 4.2, any non-zero vector in H i consists of both positive and negative components, whie y (i) (q i ) is non-zero and non-negative. Hence, (13) hods for a i = 2, 3,..., N + 1. From the proof of Theorem 4.1, the deficiency δ i of each networ (S i, C i, R i ), i = 1, 2,...,N + 1isgivenbyδ i = m i 1 s i =, which impies s i = m i 1wheres i = dim(h i ). Defining the number of compexes, the numbers of inage casses and the SS of (S, C, R )bym, and H respectivey, we easiy see from the conditions 2 and 3, and (14) that m = N+1 m i, = 1ands := dim(h ) = N+1 s i + N, and hence we have δ = m s = N+1 m i 1 ( N+1 s i + N) = N+1 (m i 1 s i ) + N N =. Therefore, the deficiency of the networ (S, C, R ) is zero. Besides, it is cear that the networ is not weay reversibe, hence this theorem foows from Theorem 2.1 (i). Next, we sha give the proof of Theorem 3.2. First, by using (8) in the proof of Theorem 4.3 we show in the foowing two theorems that any positive soution to the probem (P ) of the networ (S, C, R ) is bounded gobay in time. Theorem 4.5 For any positive soution x(t) to(p ), x (1) (t) is bounded gobay in time, that is, sup x (1) (t) < +. (15) t Proof: By taing the time derivative of the functions T (1), = 1,...,n 1 dim(h 1 ) given by (8) aong the positive soution x(t) to (P ), we have d dt T (1) (x (1) (t)) = N+1 = n 1 =1 a (1) n 1 y (1) (p i ) y (i) (q i ) i=2 =1 () d dt x(1) (t) a (1) i ()y (1) (p i ) =1 (t), t. (16) Hence from the positivity of the soution to (P )wehave (x (1) (t)) T (1) (x (1) ()) for a t and putting T (1) M (1) (x (1) ()) := max T (1) 1 n 1 dim(h 1 ) (x (1) ()), we see that for a t, x (1) (t) M (1) (x (1) ()), = 1,...,n 1 which impies sup t x (1) (t) < +. Lemma 4.6 t im t The foowing imit exists: (s)ds, i = 2, 3,...,N + 1. (17)

6 SICE JCMSI, Vo. 11, No. 3, May Proof: Integrating (16) from to t,wehave T (1) (x (1) ()) T (1) (x (1) (t)) N+1 n 1 = y (1) (p i ) y (i) (q i ) a (1) ()y (1) (p i ) i=2 =1 t x (1) y (1) (s)ds. Hence from the positivity of the soution to (P )wehave T (1) (x (1) ()) y (1) (p i ) y (i) (q i ) n1 t =1 a (1) ()y (1) (p i ) (s)ds, t, i = 2,...,N + 1. Since the above integra is an increasing function of t, (17) hods. Theorem 4.6 For any positive soution x(t) to(p ), x (i) (t), i = 2,...,N + 1 is bounded gobay in time, that is, sup x (i) (t) < +. (18) t Proof: In the same way as the proof of Theorem 4.5, we first tae the time derivative of the functions, = 1,...,n i dim(h i ) given by (8) aong the positive soution x(t) to(p ). Then for a i = 2,...,N + 1wehave d dt (x(i) (t)) = = y (1) (p i ) y (i) (q i ) n i =1 n i =1 () d dt x(i) (t) ()y (i) Hence, from Lemma 4.6 we have (x(i) (t)) + y (1) (p i ) y (i) (q i ) (x(i) ()) n i =1 (q i ) ()y (i) (q i ) (t). (19) x (1) y (1) (s)ds =: (x(i) ()), t, (2) and putting M (i) (x (i) ()) := max 1 n i dim(h i ) (x(i) ()), we see that for a t, x (i) (t) M (i) (x (i) ()), = 1,...,n i which impies sup t x (i) (t) < +. By integrating (19) from to t and appying Lemma 4.6, we have the foowing theorem. Theorem 4.7 For any positive soution x(t) to(p ), we have im t (x(i) (t)) = (x(i) ()) n i + y (1) (p i ) y (i) (q i ) =1 ()y (i) (q i ) x (1) y (1) (s)ds =: (x(i) ()), = 1,...,n i dim(h i ), i = 2,...,N + 1. On the basis of the boundedness, Lemma 4.6 and Theorems 4.5 and 4.6, we show the convergence of the soution to (P ). First, in order to show the convergence of x (1),wegivethe foowing emma. Lemma 4.7 e consider the positive soution x(t) C 1 ([, ); R n > ) to the foowing ODE: d dt x i(t) = b i f (x(t)), i = 1,...,n, (21) x() = x R n >, (22) where f C(R n ; R) andb i >, i = 1,...,n. If the soution satisfies x b i i (t) =, (23) im t then there exists {1,...,n} such that im t x (t) =, and the imit, im t x i (t), exists for any other i {1,...,n}. Proof: Putting a i := n =1, i b, i = 1,...n, wehave a i ẋ i (t) a ẋ (t) = b i b b b fx(t) =, =1, i =1, 1 i, n, i, t, (24) which impies that a i x i (t) a x (t) = a i x i () a x () for a t. Now we choose an index satisfying a x () = min 1 i n a i x i (). Then since for a t x i (t) = a x (t) + a i x i () a x (), i = 1,...,n, i, a i (25) we have x b i i (t) = x b (t),i [ ] bi a x (t) + a i x i () a x (), t. e assume that im sup t x (t). Then we see from a i x i () a x () that im sup t x i (t) >, hence im sup t a i x b i i (t) >, (26) which is in contradiction with (23). Therefore we have im t x (t) =. Furthermore, we see from (25) that any other variabe x i (t) converges to a nonnegative vaue. From Lemma 4.7, we obtain the foowing emma. Lemma 4.8 For any positive soution x(t) to(p ), there exists X (1) supp(y (1) (p )) satisfying im t x (1) (t) = forany = 2,...,N + 1. Moreover, any other concentration of the species contained in supp(y (1) (p )) converges to a non-negative vaue. Proof: e easiy see from the ODE (1) with mass action inetics (2) and Theorem 4.5 that for a t, d dt (p ) (t) < +, = 2,...,N + 1. (27) Hence, by combining (27) with Lemma 4.6 and appying Barbaat s emma [9], we have im t (p ) (t) =, = 2,...,N + 1. (28)

7 144 SICE JCMSI, Vo. 11, No. 3, May 218 Now, for any index satisfying X (1) supp(y (1) (p )), the ODE of x (1) can be given by ẋ (1) (t) = y (1) (p ) f (1) (x(t)) where f (1) (x) := xy(1)(p) y (1) (p ) y (t) y (1) (p ) y R + y y (1) (p )x y (t). (29) y y (1) (p ) R Hence, from Lemma 4.7 this emma hods. From the above resut, the convergence of x (1) can be shown immediatey. Theorem 4.8 For any positive soution x(t)to(p ), there exists a vector x (1) R n 1 satisfying (x(1) ) y(1) = foray (1) C 1 such that x (1) (t) x (1), t. Proof: From Lemma 4.8, there exists X (1) supp(y (1) (p )) for any = 2,...,N + 1 satisfying im t x (1) (t) =. e see from Theorem 4.5 that the derivative ẋ (1) (t) is uniformy continuous with respect to t, hence, by appying Barbaat s emma we have ẋ (1) (t), t. From the proof of Lemma 4.8, we have ẋ (1) (t) + y (1) (p ) xy(1)(p) y (1) (p ) y (t) y (1) (p ) y R = y (1) (p ) y (1) y (1) (p )x (1)y(1) (t), (3) y (1) y (1) (p ) R 1 and im t y(1) (p ) y (1) y (1) (p ) R 1 y (1) y (1) (p )x (1)y(1) (t) =, (31) which impies that im t x (1)y(1) (t) = foray (1) C 1 satisfying y (1) y (1) (p ) R 1. Hence, we see from Lemma 4.8 that for a y (1) C 1 satisfying y (1) y (1) (p ) R 1 a variabes contained in supp(y (1) ) converge. By repeating the above argument and using wea reversibiity of (S 1, C 1, R 1 ), we can show that this theorem hods. Finay, we show the convergence of x (i), i = 2,...,N + 1, of the soution x(t) to(p ) with the initia vaue x() = x R n >, preparing the foowing two emmas. Lemma 4.9 The ω-imit set ω(x()) of (P ) is a non-empty, compact, and invariant subset of X of R n defined by x (1) = x (1), X := x R n (x(i) ) = (x(i) ()),, (32) = 1,...,n i dim(h i ), i = 2,...,N + 1 where x (1) is the vector introduced in Theorem 4.8. Proof: This foows from the genera theory of ODEs and the direct consequence of Theorems 4.6, 4.7 and 4.8, so we omit the proof of this theorem here. From the above resuts, we obtain the foowing emma immediatey. Lemma 4.1 For any semi-ocing set of the networ (S, C, R ) incuding 1 S 1 such that (S 2... S N+1 ), it hods that ω(x ) L =. Here, the set 1 is given by { } 1 := X (1) S 1 x (1) =. (33) Proof: e easiy see that for any semi-ocing set of the networ (S, C, R ) incuding 1 such that (S 2... S N+1 ), there exists i {2,...,N + 1} such that 1 i,where i is a minima semi-ocing set of the networ (S i, C i, R i ). e assume that ω(x ) L. Since i is a minima semi-ocing set of the networ (S i, C i, R i ), by taing the time derivative of i given by (12) aong a positive soution x(t) to (P ), we have im t i (x (i) (t)) = i (x (i) ())+ n i y (1) (p i ) y (i) (q i ) ( i )y (i) (q i ) =1 (s)ds >, (34) However, for a w ω(x ) L we see from the continuity of i (x (i) (t)) with respect to t that i (w (i) ) >, which is in contradiction with w L and i. Therefore, it hods that ω(x ) L =. Now, we define a subset X of X by X := { x X x (i) R n i >, 2 i N + 1 }. (35) e easiy see that the set X is a forward invariant set, that is, any positive soution x(t) to(p ) with an initia vaue x() X remains x(t) X for a t. Aso, together with Lemmas 4.4 and 4.1 we see that ω(x ) X. By using the resuts above and the foowing theorem [1], we can prove Theorem 3.2. Theorem 4.9 Suppose that C is a nonempty, compact and invariant set of a dynamica system generated by a system ẋ = f (x) defined on some set Ω in R n and that x is a gobay asymptoticay stabe equiibrium point. Then C = {x}. The proof of Theorem 3.2: ehavex (1) = x (1) for a x X, which impies that =, i = 2,...,N + 1. Hence, we see that soutions to the probem (P ) restricted on the set X are equivaent to those to independent probems of (P i ), i = 2,...,N + 1. Here we see that the conditions (x(i) ) = (x(i) ()), = 1,...,n i dim(h i ) in (35) determine PSCCs of (P i ), i = 2,...,N + 1. Hence we see from Theorem 4.4 that the probem (P )onthesetx has a unique gobay asymptoticay stabe equiibrium point x := (x (1)T, x (2)T,...,x (N+1)T ) T X since X is forward invariant. Here, the vector x (i)t represents a unique gobay asymptoticay stabe equiibrium point on the PSCC of each (P i ), i = 2,...,N + 1. Besides, since from Lemmas 4.9 and 4.1 ω(x()) of (P ) is a non-empty, compact, and invariant subset of X, by appying Theorem 4.9 we have ω(x()) = {x}, which competes the proof.

8 SICE JCMSI, Vo. 11, No. 3, May Exampe As an appication of Theorem 3.2, we consider the CRN [1] given in Fig. 2. This networ consists of three sub-networs (S i, C i, R i )(i = 1, 2, 3), where S i, C i and R i are given by: S 1 = {X 1, X 2, X 3 }, S 2 = {X 4, X 5, X 6 }, S 3 = {X 7, X 8, X 9 }, C 1 = {X 1 + X 2, X 3 }, C 2 = {X 4 + X 5, X 6 }, C 3 = {X 7, X 8, 2X 9 }, R 1 = {X 1 + X 2 X 3, X 3 X 1 + X 2 }, R 2 = {X 4 + X 5 X 6, X 6 X 4 + X 5 }, R 3 = {X 7 X 8, X 8 2X 9, 2X 9 X 8, 2X 9 X 7 }. Since this networ is not weay reversibe, we can not appy Theorem 2.1 (DZT) to it. However, it is easiy verified that the three sub-networs (S i, C i, R i ) satisfy the three conditions given in Section 3. Hence, by appying Theorem 3.2 we can prove that any positive soution to the system (1) for the CRN converges to an equiibrium point on the boundary of the PSCC containing the initia vaue. A numerica exampe is iustrated in Fig. 3 and Tabe 1. The numerica cacuation is executed for t 1 on the basis of Eq. (1) with the mass action inetics (2), using the initia vaues shown in Tabe 1 and the rate constants in Fig. 2. Actuay, we ceary see from Fig. 3 that x 3 (t) vanishes and x 6 (t) andx 8 (t) converge to some positive vaues, besides other variabes of concentrations converge to some non-negative vaues as shown in Tabe 1. Consequenty, we can confirm that the soution of Eq. (1) converges to an equiibrium point on the boundary of the PSCC. Fig. 2 Exampe of a non-weay reversibe networ in our cass. networs. By decomposing the networ into weay reversibe sub-networs and appying the DZT to them, we have proven any soution with positive initia vaues converges to an equiibrium point on the boundary of the positive orthant. In future wor, we wi dea with a more genera cass of nonweay reversibe chemica reaction networs than that deat with in this wor, and construct an extended Deficiency Zero Theorem. Acnowedgments Funding from Furuawa Giutsu Shino Zaidan is gratefuy acnowedged. References [1] M. Feinberg: Lectures on chemica reaction networs, Math. Res. Cent., U. isc.-mad, LecturesOnReactionNetwors, [2] D.F. Anderson: Goba asymptotic stabiity for a cass of noninear chemica equations, SIAM J. App. Math., Vo. 68, No. 5, pp , 28. [3] V. Cheaboina, S.P. Bhat,.M. Haddad, and D.S. Bernstein: Modeing and anaysis of mass-action inetics: Nonnegativity, reaizabiity, reducibiity, and semistabiity, IEEE Contro Systems Magazine, Vo. 29, August, pp. 6 78, 29. [4] E.D. Sontag: Structure and stabiity of certain chemica networs and appications to the inetic proofreading mode of T-ce receptor signa transduction, IEEE Transactions on Automatic Contro, Vo. 46, No. 7, pp , 21. [5] M.A. A-Radhawi and D. Angei: New approach to the stabiity of chemica reaction networs: Piecewise inear in rates Lyapunov functions, IEEE Transactions on Automatic Contro, Vo. 61, No.1, pp , 216. [6] P. Donne and M. Banai: Loca and goba stabiity of equiibria for a cass of chemica reaction networs, SIAM J. App. Dyn. Syst., Vo. 12, No. 2, pp , 213. [7] H. Komatsu, M. Ugawa, and H. Naaima: Stabiity anaysis of chemica reaction networs by decomposing them into weay reversibe sub-networs, Proc. SICE Annua Conference 217, pp , 217. [8] D.F. Anderson: Boundedness of traectories for weay reversibe, singe inage cass reaction system, J. Math. Chem., Vo. 49, No.1, pp , 211. [9] H.K. Khai: Noninear Systems, 3rd edition, Prentice Ha, 22. [1] D. Angei, P.D. Leenheer, and E.D. Sontag: Persistence resuts for chemica reaction networs with time-dependent inetics and no goba conservation aws, SIAM J. App. Math., Vo. 71, No. 1, pp , 211. Fig. 3 The time evoution of concentrations of species x 3, x 6 and x 8. Tabe 1 The initia vaues and the convergence vaues. x i initia vaue convergence vaue x x x x x x x x x Concusion In the present paper, we have discussed a stabiity anaysis for a certain cass of non-weay reversibe chemica reaction Hiroazu KOMATSU He received his B.E. and M.E. degrees from Kindai University, Japan, in 214, 216, respectivey. He is currenty a doctora student at the Graduate Schoo of System Engineering, Kindai University. His research interests incude noninear dynamica systems. He is a memberofjsiamandmsj. Hiroyui NAKAJIMA (Member) He received his B.E., M.E., and Ph.D. degrees from Kyoto University, Japan, in 1984, 1986, and 1996, respectivey. In 1996, he oined Kindai University, where he is currenty a Professor of the Facuty of Engineering. His research interests incude noninear dynamica systems. He is a member of JSIAM.