oscillations and multiple steady states in a cyclic gene model with repression

J. Math. Biol. (1987) 25:169-190 ,Journal ol'

Mathematical Wology

(~) Springer-Verlag 1987

Oscillations and multiple steady states in a cyclic gene model with repression*

Hal Smith**

Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, RI 02912, USA

Abstract. In this paper we study the cyclic gene model with repression considered by H. T. Banks and J. M. Mahaffy. Roughly, the model describes a biochemical feedback loop consisting of an integer number G of single gene reaction sequences in series. The model leads to a system of functional differential equations. We show that there is a qualitative difference in the dynamics between even and odd G if the feedback repression is sufficiently large. For even G, multiple stable steady states can coexist while for odd G, periodic orbits exist.

Key words: Cyclic gene m o d e l - - B i o c h e m i c a l f e e d b a c k - - R e p r e s s i o n - - Functional differential equations

Introduction

Beginning with the early investigations of Goodwin [8, 9], there has been considerable interest in mathematical models of protein synthesis involving end-product repression. In this paper, we consider a model cyclic gene system which derives from the suggestions of Goodwin. To the best of our knowledge, explicit equations first appeared in work of Banks and Mahaffy [4] who studied the model using analytical techniques, although Fraser and Tiwari [7] had earlier made numerical simulations of a discrete analogue. We follow the former reference in our brief description of the model. In addition to [4], the reader interested in more details may also wish to consult the survey articles of Tyson and Othmer [29] and Banks and Mahaffy [3]. An interesting and more recent survey of Rapp [36] is recom- mended.

It is assumed that a positive integer number, G, of genes control the synthesis of certain end products (proteins). The first gene is transcribed producing mRNA(yl ) which in turn is translated to produce an enzyme (Y2) which in turn produces another enzyme (Y3) and so on until an end product repressor molecule (yp) is produced. This end product molecule of gene one acts to repress the

* This research was supported in part by the Air Force Office of Scientific Research under Contract # AFOSR-84-0376 and by the US Army Research Office under Contract # DAAG29-84-K-0082

**Present address: Department of Mathematics, Arizona State University, Tempe, AZ 85287, USA

170 H. Smith

transcription of the second gene. The transcription of the second gene produces mRNA(zl) which, as before, is the first in a series of enzymes (Zl, z 2 , . . . , zt-1) produced leading up to the production of a repressor molecule (z~) which inhibits the transcription of the third gene. And so on until the end product of the Gth gene is produced. This molecule acts to inhibit transcription of the first gene. Each of the species is assumed to degrade and each reaction in the chain is assumed to be first order except for the inhibition of an end product molecule of one gene on the transcription of the following gene which is assumed to be of Michaelis-Menten type. Time delays are introduced to account for time involved in transcription, translation and transport.

In order to avoid writing an equation with multi-indices on each variable, we give below the model equations only in case of two participating genes ( G = 2). The reader should have no trouble imagining the general case which we will also refer to (with some abuse of language) as (0.1) although we will usually specify G.

y~(t) = fl(Nlz~) - - c~ty I (t)

y~(t)= Li_lY~_l-cc~y~(t), 2<~i<~p

z',( t) = f2( Lpy;) - f i ,zl( t) (0.1)

z~(t) = Nj ,z~_,- f l jz j( t) , 2<~j<~l

where

with

a ; > 0 , / 3 : > 0

fa,f2: [0, oo)~ (0, oc)

f ~ , f ~ < O

fo L~y~ = y~(t+ O) dvi(O), - - r i

~z'~ = zj(t+ o) dnj(o), sj

l<~i<~p

l<~j<~l

ri/> 0, vi nondecreasing on [ - 5 , 0]

vi(O)>vi(-r~), O>-r~, l<~i<~p

sj t> 0, ~5 nondecreasing on [ - s j , 0]

~b(O)>~b(-s:) , O > - s j , l<~j<~l.

In addition we require that v~ and ~b be left continuous at each point of the interior of their domain. The reader will note that we allow quite general distributed delay terms in (0.1) which include as special cases, ordinary differential equations, if

ri, sj = O

v~(O) = a~g(o), 71j(O) = bill(O) (0.2)

Liy~ = a~yi( t)

N z j = b:j( t)

Oscillations and multiple steady states in a cyclic model with repression 171

where H is the Heaviside function, and discrete delay equations, if

r i>0 , s j > 0

~i( o) = aiH( o + r~) •j( O ) = b~H ( O + sj ) (0.3)

Liy~ = aiYi( t - ri)

N : j = b:j( t - sj). In the general case, initial conditions of the form

yi(O)=cb~(O), -r,<~O<~O, l<~i<~p

z:(O)=Oj(O), -sj<~O<~O, l<~j<~l

must be given where ~b~>0 and 0:/>0. The Eqs. (0.1) are then solved for t~>0. The Eqs. (0.1) are linear except for the equations for y~ and z~ which contain the nonlinear terms f l and f2 representing the inhibitory effect of the end product of one gene on the transcription of the other. The nonlinearities, which can be derived from Michaelis-Menten enzyme kinetics, (see [3, 29]) and which have received the most attention in the literature are of the form

a f ( u ) l + k u q , a > 0 , k > 0 , q~>l. (0.4)

The parameter q, usually a positive integer, called the Hill coefficient, will play an important role in our considerations. It appears that the case q = 1 is of primary importance for most applications. Biologically, q is the number of end-product molecules (having concentration represented by z~ or yp) which must combine with a repressor molecule in order to inhibit transcription of mRNA (with concentration Yl or zl). The case of q > 1 represents cooperative inhibition of the transcription of yl or zl. Mathematically, q is a measure of the strength of the nonlinearity.

It should be noted that no delays appear in each species first order decay term (e.g. -a~y~(t) in the equation for Yl)- It has been observed [19] and is easy to see that the introduction of delays in these terms allow solutions of (0.1) to become negative for some nonnegative initial conditions.

The effects of diffusion of biochemical species within the cell have been modelled by inclusion of time delays in (0.1) to account for transport time. Recently, more realistic models have explicitly allowed for spacial variation of species concentration. We mention particularly the papers of C. V. Pao and J. Mahaffy [34, 35] and S. Busenberg and J. Mahaffy [37]. The latter article is particularly interesting, showing how to represent the effects of diffusion by a distributed delay.

There are two important questions to address in a study of (0.1). The most important question is whether or not the equations (0.1) allow stable undamped oscillations (periodic solutions). Intuitively, we expect negative feedback control loops to give rise to oscillations under suitable conditions. The biological implica- tions of sustained oscillations in systems similar to (0.1) are many. We refer the interested reader to the article of Tyson and Othmer [29] for a discussion. Oscillatory solutions of mathematical model biochemical control loops involving end-product repression have been suggested as models for circadian rhythms and

172 H. Smith

as a developmental clock during morphogenesis. A second fundamental question concerning the system (0.1) is whether or not it possesses multiple stable steady states. Associated with each of these questions is the problem of determining the parameter region in which sustained oscillations or multiple steady state are possible. If these domains lie outside of biologically relevant ones, then the phenomena are only of mathematical interest.

It is the goal of this paper to make a qualitative study of the system (0.1) for general G with the primary focus on exposing the range of possible dynamical behavior implicit in the equations. We will be particularly interested in the special class of nonlinearities given by (0.4) but our primary results will be given for general nonlinearities.

The question posed above has motivated most of the previous studies of mathematical models of biochemical control loops. Much of this work concerns the single gene system ( G = 1, note that z does not appear in (0.1) and the argument in f l is Lj'p). We briefly describe some results which have been obtained for the single gene system. Allwright [1] and Banks and Mahaffy [5] have showed that there is a single globally asymptotically stable steady state in case the negative feedback term (fl) is "weak" (e.g. f i given by (0.4) with q = 1). Stronger feedback

(q sufficiently large depending on 1~ ai) can destabilize this steady state [2, 13, 14, i = 1

18, 21]. In this case, a pair of complex conjugate eigenvalues with positive real part exist for the characteristic equation. MacDonald [18] has shown that a supercritical (stable) Hopf bifurcation of a small amplitude periodic solution takes place for the ordinary differential equation (ODE) system. Mahaffy [32, 33] has extended this Hopf bifurcation analysis to include the case of discrete delays. The existence of not necessarily small amplitude periodic solutions has been established for the ODE single gene system by several authors. In [28], Tyson used the method of Poincare sections to obtain periodic solutions for the three dimensional system. This work was refined and extended to arbitrary dimension by Hasting, Tyson and Webster in [14]. Not surprisingly, one has no information on stability or multiplicity of the periodic solutions obtained. More is known for the three dimensional ODE system. In this case, Hastings [13] gives sufficient conditions for almost all solutions to approach a periodic orbit when f l is a piecewise linear function. This author establishes this result for general f l in [26] obtaining considerable additional information (see also a recent article of Levine [ 17]). The above-mentioned results assume the instability of the steady state.

Much less is known concerning the existence of not necessarily small amplitude solutions of the single gene system with delays. What is known concerns systems with discrete delays (0.3) following techniques due to an der Heiden [2] who uses fixed point arguments to obtain periodic solutions for one and two dimensional systems. Mahaffy [21] extends this analysis to arbitrary dimension but unfortunately for large dimension (p/> 4), his result requires the sum of all the delays to be large. Hence, Mahaffy's result does not contain the result of [14] for the ODE case.

In a broad sense then, our understanding of the single gene system, so far as the questions posed above are concerned, is fairly satisfactory. Our understanding of the multi-gene system (G/> 2), on the other hand, is more fragmentary. Early,


numerical simulations by Fraser and Tiwari [7] of a discrete time analog of (0.1) led these authors to predict that if G is odd and the nonlinearities f~ are sufficiently strong, then undamped oscillations can occur. However, analytical results obtained by Banks and Mahaffy [4] and by Allwright [1] showed, at least for the case that the f~ are given by (0.4) with q = 1, that there exists only one steady state and this steady state is globally asymptotically stable. Both works allowed for delay terms (Allwright allows discrete delays (0.3) and Banks and Mahaffy allow general finite delays). These results appear t o c a s t some doubt on the findings of Fraser and Tiwari. This apparent discrepancy between computer simulations and analytical results is only the latest in a series of controversies which have marked earlier investigations of oscillatory behavior in biochemical control loops (see [3] and [29] for an interesting discussion of the phenomena). Such controversy focuses attention on the need for rigorous mathematical arguments and the possibilities of being mislead by computer simulations. In this particular case, it turns out that the numerical simulations by Fraser and Tiwari and the analytical work of Banks and Mahaffy and of Allwright are not in conflict.

We will show that there is a fundamental difference between even G and odd G systems which may result in entirely different qualitative behavior. Roughly speaking, in a control loop containing an even number of negative feedback terms, there is the potential fo r cancellation of the negative feedback. We will show that the even G systems act essentially like inducible (positive feedback) systems (see e.g. [23, 24, 29]). More precisely, we will show that stable oscillations are impossible for even G but that multiple stable steady states are possible if the nonlinearities are sufficiently strong. For even G, almost all solutions tend to equilibrium monotonically and if there is only one equilibrium, then it is a global attractor. The magnitude of delays in the even G case have essentially no qualitative effect on the dynamics.

In the case of odd G, the cancellation effect mentioned above does not occur. While our results in the case of odd G are far less complete than for even G, we can say that most of the results mentioned above for G = 1 carry over to the case of general odd G. For odd G there is exactly one steady state which is a global attractor if the feedback nonlinearities are sufficiently weak. In the special case of odd G and no time delays, we find that the result of Hastings et al. [14] applies: if the steady state is unstable (which can be caused by strong nonlinearities) then there exists a non-constant periodic orbit. This result continues to hold for small discrete delays. It appears that the strength of the nonlinearities required for this result decrease with increasing G.

We should point out, however, that if in (0.1) we take (0.4) with q = 1, then the dynamics are independent of G. In this case there is a single globally asymptotically stable steady state. This result is due to Banks and Mahaffy [4] and follows also from a slight modification of the work of Allwright [1] who considered only discrete delays but whose arguments carry over easily to distributed delays. There is a subtle difference between even and odd G even in this case which is worth noting. For even G the approach to steady state is most likely to be monotone while for odd G we find an oscillatory approach.

Several authors [3, 29] have questioned whether the parameter regions for which one finds stable oscillations or multiple stable steady state might not have

174 H. Smith

empty intersection with the biologically relevant parameter domain. In particular, large values of q (strong nonlinearity) seem to be unlikely in many biological systems [3]. Throughout this work, however, our aim has been to thoroughly explore the possible dynamics implicit in the equations (0.1), leaving for more qtialified workers the question of relevance to particular applications. Systems of the form (0.1) have been used to model a fairly broad range of biochemical control circuits (see [ 14, 29]). Consequently, gaining a full knowlege of the range of dynamical behavior possible in such models appears to be a worthwhile objective.

There is a rather striking observation which we can make concerning the dynamics of (0.1) for arbitrary G but which we cannot adequately explain. This is the observation that the dynamics of (0.1) is mimicked by the dynamics of a scalar discrete dynamical system U,+l = g(un), n =0, 1, 2 , . . . which arises quite naturally by searching for steady states of (0.1). For example, in the case of G = 2, the steady states of an appropriately scaled version of (0.1) are in one to one correspondence with the fixed points of the map g given by

g ( u ) = ( ( ' y21f2 ) o ( , y l l f l ) ) ( U )

p I

"y, = 11 ~,, ~2 = II/3;- i--1 j = l

Given our hypotheses on f l and f2, g is a strictly increasing function taking positive values on 0 ~< u ~< oo. Hence every orbit of the discrete dynamical system generated by g tends to a fixed point of g in a monotone fashion. This is precisely the behavior of solutions of (0.1). Moreover, the stability of a steady state of (0.1) is identical to the stability of the corresponding fixed point of g. In the case of odd G, for example G = 1, the corresponding map g is given by g = yllf~. In this case, g is a monotone decreasing function taking positive values on 0 ~< u ~< oo. Hence, there can be only one positive fixed point of g and so (0.1) can have only one positive steady state. This steady state of (0.1) is globally attracting so long as the fixed point of g is globally attracting (Allwright [1], see also section three). Moreover, the approach to steady state (fixed point) is oscillatory in both cases. However, the steady state of (0.1) loses stability and a nontrivial periodic orbit appears (at least for the nondelay case) when the fixed point of g undergoes a period two bifurcation from its fixed point. We will have more to say concerning this close relation between the dynamics of (0.1) and that generated by an appropriate map g throughout this work. We point out that this phenomena should be very useful both for the modeller of biochemical control circuits and for the person faced with the task of predicting the behavior of a particular system similar to (0.1).

The organization of this paper follows. In Sect. 1 we examine some aspects of the dynamics of (0.1) that are independent of G. In this section we show that there is a compact attractor to which all solutions of (0.1) are attracted. In Sect. 2, we consider the dynamics of (0.1) for even G. In Sect. 3, the dynamics of (0.1) for odd G are considered. Throughout this work, we have made an attempt to keep the length of the proofs of our results to a bare minimum. We have succeeded in this to a.targe degree because many of the ideas used in the present work have


their inception in other works of the author (particularly [25] and [27]) and in work of Allwright [ 1 ]. Finally, in Sect. 4 we summarize our conclusions and make further remarks regarding the intriguing parallel between the dynamics exhibited by (0.1) and the dynamics of an associated scalar mapping.

Some notation follows. We will denote by q~ an initial condition of (0.1), for example if G = 2 , CI):(4)I,4)2,...,4)p,I/II,~lt2,...,lltl)ECrlXCr2x'''XCrp>( Csx x . �9 �9 • Cs, --- C where Cr = C ( [ - r , 0], R), the space of continuous functions mapping I - r , 0] into the reals. Our assumptions are such that (0.1), together with the initial condition, @, at t = 0 (see following (0.3)) generate a unique solution which we write as x(t, cP)(=(y(t), z(t)) if G = 2 ) . We regard C as the real state space of (0.1) and we write x , (~ ) for the state of the system (0.1) at time t, xt(CIa)=(y,,z,), yt=(Ytl,y~,...,ytp), Zt: (Z t l , . . . ,Z~) where y[cCr, is given by y ~(0) = y~ (t + 0), - rl ~< 0 ~< 0, and similarity for zj. Since only nonnegative initial data make biological sense we use the notation C + for the subset (cone) of p + 1 tuples of nonnegative functions. I f @ e C is such that x,(@) is defined for t 1> 0 and Ut>~oX,(r is precompact , we write w ( r for the omega limit set. That is, w ( ~ ) = { gt ~ C: there exists t~ --> co as n -+ oo such that x,, (45) --> gt } (the topology of C is the product topology with the uniform topology on each C~).

The author would like to acknowledge a useful conversation with J. M. Mahaffy who brought to our attention the articles [32-37].

1. Preliminary results

In this section we obtain some preliminary results for (0.1) which are independent of the parity of G. For simplicity, we set G = 2, although it will be apparent that our arguments are quite general. We begin by appropriately scaling our equations in order to reduce the number of independent parameters. Then the solutions of the resulting equations with nonnegative initial conditions are shown to be continuable, to remain nonnegative, and to approach a compact attractor in C §

Consider the equations (0.1) with G = 2 which we rewrite for convenience.

y~( t) = f~( Nlz~) - c~lYl(t )

yl(t)=L~_ly~_m-cqyi(t), 2 ~ i < ~ 0 (1.1)

z~( t) = f2( Lpyp) -- fllZl(t)

z j ( t )= Nj_~z'~_l-C~jzj(t), 2<~j<l

where

a i > 0 , & > 0

f l , f2; [0, o0) --> (0, cO)

f~ , f '2>O

f_ Li4) = &(O) dv,(O), 4) ~ C([-r , , 0], R) rl

vi is nondecreasing on [ - r i , 0], r~ t> 0

v~(0) > v~(-ri), O> -r ,

(1.2)

176 H. Smith

N:I, = ,I,(0) gnj(O), 0 ~ C( [ - s j , 0] , n ) sj

~Tj is nondec reas ing on [ - s j , 0] , sj/> 0

nj(o) > nj(-sj), O>-sj. Make the fol lowing change of variables

Y, = f,(0)371

y, =f, (o)( , , ,_ , (o)- ,,,_,(- r,_,))y,, 2<~ i<~p

Z1 = f 2 ( 0 ) z ,

zj =f2(0)( rb_l (0 ) - ~Tj-l(-Sj-1))~j, 2 < ~ j ~ < l

and let

f l (uf2( O)( ~Tt-,(O) - ~Tt-,(--st-,))/(~71(0) - ~Tt(-st)) =--- f , ( O ) f ,( u)

f2( v f , (O)(vp_,(O)- vv_ , ( -rv_l ) ) / (vp(O ) -- vv( -rp) ) )=--- f2(O)f2( v)

L, = (v,(O) - v,(-r,))[,, N = (hA 0) - nj(-sj))~.

The new set o f equat ions is, after d ropp ing the bars, the set (1.1) where now, in addi t ion to (1.2) we have

f l (0) =f2(0) = 1

v~(O) - v~(-r~) = 1 (1.3)

~7:(0) - ~b(-s j ) = 1.

In our first result we will show that if 7/1> 1, ~:~> 1, then the set p l

B(~q, ~) -= H [ 0 , r l ( a , a 2 ' ' ' ol i) - 1 ] x I I [0 , ~ ( f } l �9 " �9 ~ j ) - l ] c_. Rp+ +l i=1 j= l

is posit ively invar iant for (1.1). This observa t ion goes back to Griffith [10] for the single gene system.

Lemma 1.1. Let �9 = (~bl, ( o2 , . . . , &p, 01, . . . , Ol) be given such that q9 takes values in B(T1, ~) for some ~7 >~ 1, ~>~ 1, i.e. 0 <~ 4~i <~ ~7(a1" �9 " ai) -~, 0 ~ Oj ~ ~ ( ~ 1 " " " /~ j ) - - l .

Then the solution x ( t, ~ ) exists for t ~ O and x ( t, ~ ) ~ B(~, ~:) for t ~ O.

Proof One only needs to check [27] that i f q) takes values in B(~7, ~) in the sense descr ibed and if 4h(0) = 0(~bi(0) = ~ (aa �9 �9 �9 a~) -~) then xl(0, q~)/> 0(x~(0, q~) ~< 0), 1 ~< i ~< p, and if Cj(0) = 0(0j(0) = ~(/31 �9 . . f l j )- l) then x~+j(0, q~) i> 0(x~+i(0, q~) <~ 0), 1 ~<j ~< I. This is easily seen with the aid of (1.2) and (1.3). Since the m a p p i n g of C + into R p+! given by the right hand side of (1.1), maps b o u n d e d sets to b o u n d e d sets, the extendabi l i ty of solutions to [0, oo) follows f rom [12, thin. 3.2, ch. 2]. []

The ma in result of this section is the following:

Theorem 1.2. There exists a compact attractor A ~_ B (1, 1) for (1.1). More precisely, A is compact, invariant and a connected subset o f C + such that for all qJ c C +, w(@) is a subset o f A. I f q~=(4~l , - - - ,4~p, 0 ~ , . . . , 0 ~ ) ~ A then O~&~<~ ( a l a z " " ai) -~ and O ~ t p 3 < ~ ( ~ 2 . . " 13:) -~.


Proof If we can show that B(1, 1) attracts all solutions x(t , ~) , q~ c C +, then the result follows from a known result [12, Lemma 2.2, Chap. 13]. This argument is a straightforward application of differential inequality arguments (e.g. y~(t) <~ 1 - aly~(t) implies y ~ ( t ) < a T ~ + e for large enough t depending on e > 0 ) and is g~ven in [25] for the case of ordinary differential equations. []

2. Even G: Convergence to equilibria

We examine the case of an even number G of genes. For simplicity, we examine in detail the case G = 2 only. However our conclusions hold for arbitrary even G. Consider (t.1) where (1.2) and (1.3) hold. The first task is to find steady states of (1.1). Steady states x = (y, z) c *-+~ of (1.1) must satisfy

fl (Zt)-a,yl = 0

Yi_l--O~iYi=O 2<~i<~p

f2( Yp) -- [3,Zl = O

Zj_l --/3jzj = 0 2<~j<~l

and hence

Yp-l=tXpYp, Yp-2=OlpOlp-lYp," " ",Yl=tXpap 1""" a2Yp

L(Z, ) = YlYp, Yl = ~1~2" " ' %

/ 2 ( y p ) = y 2 z , , ')/2 = / ~ , �9 �9 � 9

It follows that steady states are in one to one correspondence with nonnegative solutions of

3'i-'fl(Z,) = yp

3"2~f2(Yp)= Zt

or, equivalently, to solutions of

g(z,) --= (3 '2% o 3 ' l l f l ) ( Z , ) = Z 1.

In view of our assumptions on f , g: [0, ao) --> (0, o0) satisfies g(O) = 3"21f2(3"~ 1) > O, g ' ( u ) > O for u~>O and g(oo)<y21. We assume

( H ) g ( u ) = u implies g ' ( u ) r 1

This nondegeneracy assumption implies that the number of steady states of (1.1) is odd. In Fig. 1 we sketch a possible graph o fg . Let Z I < z 2 < " " "<Z m, m odd, be the set of positive fixed points of g. The corresponding steady states of (1.1) are given by

X~=(y~,z~r) , O'= 1 , 2 , . . . , m

y ~ [ = ( a ~ ' . , oe~)-~f~(z ~) i = l , . . . , p (2.1)

Z ; = 3 ' 2 ( f l l " " " l~j) -1Zty j = 1 , . . . , I.

The main result of this section is the following:

Theorem 2.1. Assume ( H) . Then (1.1) has an odd number m o f steady states.given by (2.1). The odd indexed steady states x 1, x 3 , . . . , x ~ are asymptotically stable,

178 H. Smith

g(z)

i

t i

r I

z I z 2 z 3

> z

Fig. 1. Atypical g

the even indexed steady state are unstable. I f B(xl ) , B ( x 3 ) , . . . , B(x m) denote the basins of attraction of x 1, x 3 , . . . , x m in C +, then U o ' o d d B(X~ is an open dense subset of C +. I f m = 1, then x 1 is a global attractor in C +.

For each odd o-, B(x ~) ={45c C+;x(t , 45 )~x '~ as t~oo}. Thus, Theorem 2.1 says that "almost every" solution of (1.1) is asymptotic, as t tends to infinity, to one of the stable steady states. The set of initial conditions 45 e C + with the property that x(t, 45) approaches one of the stable steady states as t tends to infinity, namely mJ~odd B(x~), has two nice properties. First, if 450 belongs to this set, then all nearby 45 e C + belong to this set. Secondly, if 450 does not belong to this set, then 45o can be approximated in C § to any described accuracy by a point 45 belonging t o U c r o d d B(x~) �9

Theorem 2.1 does not imply that (1.1) cannot have 'a periodic solution or a strange invariant set unless m = 1. However, it does imply that any such behavior, together with it's domain of attraction occupie~ a negligible subset of C + and thus would not be detected by numerical simulations.

It is important to consider the question of when (1.1) can have multiple steady states and hence multiple stable steady states. I f the f are given by (0.4) with q = 1, then g is necessarily a linear fractional transformation and, as such, can have either one or two fixed points. Since g must have an odd number of positive fixed points, it follows that g has precisely one positive fixed point. Hence, in the case that t h e f are given by (0.4) with q -- 1, (1.1) has precisely one positive steady state which is globally asymptotically stable in C +. This constitutes one half of a result obtained by Banks and Mahaffy in [4]. In a crude way, if the nonlinearities f l and f2 are weak in the sense that

f~ ( z ) f~ (y )<y lY2 , y,z>~O

then g ' (u )< 1 and there can only be one fixed point of g and thus only one steady state of (1.1).


We might expect, then, that if the nonlinearities f l and f2 are sufficiently strong then multiple steady states may result. We explore this possibility in the very simple situation for which both genes have the same parameters. Namely

OLi ~ j ~ j ~ Ol

l = p (2.2)

1 f l =- f z ( u ) ~ f ( u ) = 1 + ku ~ - - - - ~ ' q > 1.

Observe that a - t f ( u ) = u has precisely one positive root ti which is a fixed point of g. I f we can give conditions for a - ~ f ' ( t i ) < - 1 then g ' ( t i )> 1 and hence g ( u )

must have (recall g ( 0 ) > 0 and g(co)< 0o) at least three fixed points. This is the idea behind the following proposition.

Proposition 2.2. I f in (1.1), (2.2) holds, and

q 10l Iq k > (1 - q - 1 ) q + l (2.3)

then (1.1) has at least three s teady states.

Proo f Set c = a-lk 1/q and v = kl/qu. I f o~-~c(zi)= ff and a - ~ f ' ( t i ) < - 1 then c = v + v q+~ and vq+l /c > 1/q. I f v = ~5(c) is the unique positive solution of c = v + v q+~

for c > 0, then a little calculus shows that ~q+~(c)/c is a strictly increasing function of c approaching one as c ~ co. It follows that the above equation and inequality hold simultaneously if and only if c > 6 where v q+1(6)/6 = q-~. An easy calculation gives ~= [q-~/(1-q-1)q+l] l /q . The result now follows. []

For example, if q = 2, (2.3) becomes k > 4~ 2t. Also note that if a ~ 1 then the right-hand side of (2.3) tends to zero monotonically as q tends to infinity. Hence, if a <~ 1 and k are fixed, (1.1) with (2.2) will have at least two asymptotically stable steady states if q is sufficiently large. We point out that the same idea and the same inequality (2.3) hold for arbitrary even G.

It is worth emphasizing that, in a qualitative sense, Theorem 2.1 implies that the dynamics of (1.1) are mimicked by the dynamics of the difference equation zn = g(zn ~), n = 1, 2 , . . . , Zo~ 0: every orbit approaches (monotonically) one of the fixed points z ~, z 2, . . . , z m where z ~, z 3 , . . . , z m are asymptotically stable and z 2, z 4, . . . , z m-1 are unstable.

The proof of theorem 2.1, sketched below, uses the ideas developed in [27] and follows very closely the application of these ideas in [27] to the single gene control loop in Sect. 5 of that paper. The ordinary differential equation version of Theorem 2.1 is proved in [25]. We might add that the underlying reason we are able to give an essentially complete analysis of the even gene case is that in this case, the semiflow on C + generated by (1.1) (and the corresponding set of equations for general even G) is strongly monotone in the sense of Hirsch [ 15, 16]. Hirsch has obtained very powerful results for such semiflows (see also [25] and [27] where these results and others have been applied). Roughly speaking, a monotone semiflow is generated by a system in which every loop of interactions contains an even number of negative feedback interactions. This is the case for our G-gene control loop precisely when G is even, the negative feedback terms given by the f .

180 H. Smith

An indirect consequence of the monotonicity o f the dynamics of even G systems is that for each steady state, the "most unstable" eigenvalue is well defined and real. As a consequence, solutions asymptotic to a stable steady state are most likely to approach the steady state in a monotone (nonoscillatory) fashion.

Before proceeding to the proof of Theorem 2.1, we indicate briefly the modifications required in the case of general even G. Really, there is only one: the mapping g defined above, the fixed points of which are in one to one correspondence with the steady state of (1.1), will be given by the composition of an even number G of strictly decreasing functions. Hence g ,will be a positive increasing function with finite limit at infinity, exactly as in the case G = 2. The statement of Theorem 2.1 is, then, unchanged and its proof unaffected by the complication of general even G.

Proof of Theorem 2.1. As indicated above, the proof of Theorem 2.1 has been given in [25] for the case of ordinary differential equations and the ideas needed to extend the analysis given there to the functional differential equations (1.1) are developed in [27]. In addition, in [27, Sect. 5], the ideas are applied to a closely related equation. We will not repeat this entire analysis, but we will sketch briefly how to apply this earlier work to the present situation. First, we should see that (1.1) can be considered to be, in an appropriate sense, a cooperative and irreducible system as described in [27]. In order to see this, make the change of variables (y, 2) = (y, - z ) in (1.1) and let f l ( u ) = f l ( - u ) , f2(v) = -f2(/)). The resulting system has the form (1.1) with bars on Yi, z j , f l , f 2 , and of course, the relevant domain has changed from Re+ +t to RP+ • (-Rt+). In the new variables, all of the off diagonal interaction terms represent positive feedback. Hence, the new set of equations is cooperative and irreducible in the sense of [27] and the analysis of these equations does not differ in any essential way from that given in [27, Sect. 5].

We conclude this sketch of the proof of Theorem 2.1 by applying the results of [27, Sect. 3] to the determination of stability of the x ~. We use the notation developed in [27] and we consider the transformed set of equations as described above. According to Corollary 3.2 of [27], the stability of g~= ( y ~ , - z ~) is determined by the signs of the principal minors of the matrix

--0/1 0 0

1 --a2 0

0 1 --aS

0 1 --%

- ! -o" f 2(Ym)

0

0 - - t -o - f l ( z , ) 0 0

0 0

--[~1 0 0 1 -~2 0 0

0 0 1 -~z

0 0


The reader may easily check that the signs of the principal minors alternate according to that required in Corollary 3.2 of [27], except possibly for the requirement that (--1) p+! times the determinant of the above matrix be positive. This latter requirement for stability holds only if

(--1)P+l[(--1)P+lala2 " �9 �9 Odp]~l]~2 " " " ]~l q- (--1)P+'+lf~(~Df~(YT,)] > 0

that is, only if

3,71f~(2~) 3,2~f~(y,~) < 1

or, in terms of the original variables, only if

g'(zT) = y 2 ' f ' 2 ( y ~ ) y T l f ~ ( z ~ -) < 1 ~:

In view of our hypotheses on f l and f2 and in particular (H) , this last condition holds only for odd tr and is violated for even ~r. This concludes our sketch of the proof. []

3. Odd G: Convergence or undamped oscillations

In this section we consider systems consisting of an odd number G of genes. The case G = 1 has been considered by numerous authors [1, 2, 3, 5, 13, 14, 17, 18, 19, 21, 28, 29, 32, 33] with and without delays. Basically, we will show that most of these results hold for odd G. The analysis given here will be far less complete than was possible for even G. Even gene systems are fundamentally simpler since in a sense, an even number of negative feedbacks in a single loop system effectively cancel each other, the system essentially acts as if it were a positive feedback system (see [23, 27, 29] for a treatment of single gene positive feedback systems). For an odd number of negative feedback terms in a single loop system, this cancellation does not occur, and the potential for stable oscillations is present.

We will consider in detail the case G = 3 but our conclusions hold generally for odd G. The equations to be considered are the following extension of (0.1).

where

y't( t) =fl(M~w'~) - crly,(t)

y l ( t ) = L i _ , y l _ a - c q y , ( t ) 2 ~ i < ~ p

z~( t) = f2( Lpytp) - f l , z l ( t)

z j ( t ) = N j _ l Z ; _ , - f i j z j ( t ) 2<~j<-I

w~( t) = fa( Ntz;) - t3, Wl( t)

W 'k ( t )=Mk_lWtk_ l - -3kWk( t ) 2<~k<~s

(3.1)

a~, fij, G > 0

j~,f2,A:[O, co)-+ (0, 1]

f ~ , f ~ , f ' 3 < O , fl(0) =f2(0)=f3(0) = 1

fo Mk)( = X(O) dpk(O), X ~ C([- - tk , 0], R) tk

(3.2)

182 H. S m i t h

P k i s nondecreasing on [ - t k , 0 ] , t k >1 0

p k ( O ) -- p k ( - - t k ) = 1

p k ( 0 ) > p k ( - t 0 , 0 > - t , , .

N, L as in (1.2), (1.3).

We recall that by Theorems 1.1 and 1.2, all orbits o f (3.1) with nonnegat ive initial condi t ions app roach asymptot ica l ly the posit ively invariant set

p l

B(1, 1, 1) = [I [0, ( a , a 2 - �9 �9 a i ) - 1 ] x I~ [0, (~1/~2 " " " ~ j ) - - I 1 i=1 j - -1

X ~ I [ 0 , ( a l a 2 - �9 �9 a k ) - l ] . k = l

Hence, all the interesting dynamics of (3.1) occurs inside this ( p + l + s ) -

dimensional rectangle (in fact, inside a compac t a t t ractor in C+). x ~ p + l + s Steady states of (3.1), x = (y, z, w) ~ ~:+ , must satisfy

Y p - 1 = a p y p , Y p - 2 = % % lYp , . . � 9 Y l = % % - 1 �9 " " a 2 Y p

f , ( w s ) = y~yp,

Z l _ 1 = f l l Z l , �9 . . ,

f 2 ( Y p ) = y2zt,

W s _ ~ = ~ s W s , . . . ,

f 3 ( z , ) = 3"3w~,

3'1 ~ 0/1 " " " Ofp

zl = / 3 . e , _ 1 �9 �9 �9 g 2 z ,

3'2 = f l l " " �9 ,81

W 1 = ~ s ~ s _ l ~ �9 �9 ~ 2 W s

3"3 = ~ 1 " " " 6 s "

It follows that s teady states are in one to one cor respondence with nonnegat ive solutions o f

3 " 7 1 f a ( w s ) = yp

3'flf2(Yp) = z,

3'31f3(z,) = we

or equivalently, to nonnegat ive fixed points of

g(~d) = (3'31f3 o 3'21f2 o 3'11fl)(u). (3.3)

In view of our assumpt ions on f , g : [0, oo) -+ (0, oo) satisfies g(0) > 0, g(oo) < 00 and g'(u)<0. It is immediate that there is precisely one positive fixed point of g and hence exactly one s teady state ff = (y, ~, ~ ) o f (3.1) given by

y, = (oe,o~2 �9 �9 �9 o~i)-lf,(a)

= ( J ~ l f l 2 " " " ~ j ) l f 2 ( 3 ' l l f l ( a ) ) (3.4)

Wk = 3 ' 3 ( 6 1 6 2 " " " 6 k ) - 1 ~

The fol lowing result, due to Allwright [1], relates the dynamics o f (3.1) to that o f the difference equat ion u,+l = g (u , ) .


Theorem 3.1. Assume that the map g of (3.3) has no 2-periodic points other than ~. Then ,2 is globally asymptotically stable in C +.

Proof In [1, Theorem 1], Allwright proves a general result which implies the above result in the special case o f discrete delays. His p roo f goes th rough for our distributed delays. One only need observe that

I ~ lim inf x( t ) ~< lim inf O) drl(O) ~< lim sup x( t+O) d~7(O) t ~ o o t ~ c o _ t ~ c o r

~< lira sup x( t )

in case ~/is an increasing funct ion on [ - r , 0] with ~/(0) - 7 / ( - r ) = 1. This completes our sketch o f the proof. []

Since g o g is m o n o t o n e increasing on [0, co), it follows that every orbit {u,}n~0 of u, +1 = g (u , ) , u0 ~> 0 approaches a per iod two solution of the difference equation. The hypothesis o f Theorem 3.1 requires that the only period two solution o f the difference equat ion be actually the fixed point ft. It is apparen t that the hypothesis of Theorem 3.1 is entirely equivalent to assuming that ~ be globally attracting (for u ~> 0) for g. Hence Theorem 3.1 asserts that if ~i is a global at tractor for g then ~ is a global at tractor for (3.1). Notice that if ti is a global attractor for g then all orbits approach ~ in an oscillatory manner (e.g. ul < u3 <" �9 �9 < ti < . �9 �9 < u4< u2). We are led to expect an oscillatory approach to steady state or (3.1). This expectat ion will be borne out later in this section.

The sufficient condi t ion for global stability o f )7 given in Theorem 3.1 can be viewed, in a rough way, as requiring weak negative feedback terms f l , f2, f3. For example, if the f are o f the form (0.4) with q = 1, then t h e f are linear fractional t ransformat ions (LFT) and hence so is g and g o g. But a LFT can have at most two fixed points while failure of the hypotheses o f Theorem 3.1 to hold requires g o g to have at least three fixed points. Hence, the hypotheses of Theorem 3.1 hold whenever f have the form (0.4) with q = 1. In this way we have the other half of the result o f Banks and Mahaffy [4].

A sufficient condi t ion for g to have no two-per iodic points other than ti is for

g'(u) > - 1

for all two-per iodic points o f g. A rather crude sufficient condi t ion for the last inequali ty to hold is

If~(w)f'2(y)f;(z)l <~ y~7273

for all w ~> 0, y ~> 0, z/> 0. This inequality, in rough terms, requires the negative feedback terms in (3.1) to be weak relative to the decay rates o f the species.

I f the nonlinearit ies fa, f2, f3 are sufficiently strong, we expect the hypotheses o f theorem 3.1 to fail. Indeed, if

g ' ( a ) < - I

then g will have two-per iodic points different than ~i and Theorem 3.1 no longer applies. Al though the conclus ion of Theorem 3.1 may still hold in this case, we might again look to the dynamics of the difference equat ion U,+l = g(u,) as a

184 H . S m i t h

guide to the possible dynamics of (3.1) (it has been a faithful comparison so far). If the inequality above holds, then ti is an unstable fixed point of g and, due to the monotonicity of g o g, all orbits of the difference equation tend to a nontrivial period two solution (there may be several). We are thus led to expect nontrivial periodic solutions of (3.1) when the nonlinearities are sufficiently strong.

We investigate the possibility of undamped oscillations in (3.1) with strong nonlinearities by considering the simpler ordinary differential equation version of (3.1). Thus, make the following simplifications in (3.1) (see (0.2))

Liy~ ~ yi( t)

N~zj ~ zg(t) (3.5)

MkWtk ~ Wk(t).

By making a simple change of variables in the resulting system of ordinary differential equations, we are able to apply the result of Hastings et al. [14] to obtain the following theorem (assume f l , f2,f3 c C~).

Theorem 3.2. Let E be the Jacobian matrix of the right-hand side of the ordinary differential systems (3.1) and (3.5), evaluated at 2. Suppose that E has no repeated eigenvalues. I f E has any eigenvalues with positive real parts, then (3.1) with (3.5) has a noneonstant periodic solution.

Proof By Theorem 1.2, we know that the omega limit set of every orbit belongs to B(1, 1, 1). Hence we need only consider (3.1) with (3.5) in this domain. If, in (3.1) with (3.5), we make the change of variables )7=y, ~ = w , and 2i= (~l~2"''~j)-l--zj, l<-j<~l, and define f l=- f l , f2(fip)=-l-f2(fip), f3(~z)------ f a (y i l - f f l ) , then we obtain again (3.1) with (3.5) except with bars on y, z, w, f l , f 2 , f3 . It is now a straightforward calculation to check that all the hypotheses of [14, Theorem 1] hold on B(1, 1, 1). []

We make two remarks before considering the spectral properties of E. First, Theorem 3.2 asserts nothing about stability properties of the nontrivial periodic orbit. Secondly, if the hypotheses of Theorem 3.2 hold so that (3.1) with (3.5) has a nontrivial periodic orbit, then by a result of Perello [30] (see also the supplementary remarks in [12, Chap. 11]), (3.1) with small discrete delays a la (0.3) (r~, st, tk<< 1) also has a nontrivial periodic orbit.

The characteristic polynomial associated with E at ~ is given by

p 1

Q(A)= [I (A+a, ) l] (A+flj) ~I ( A + 6 k ) + P i=1 j = l k = l

P = -f~(ws)f'z(fip)f~(~,) = -'Y,'Y2T3g'(~) > 0. (3.6)

Observe that if

g '(~) ~>-1 (3.7)

then all roots of (3.6) have negative real part. Indeed, if )t is a root of (3.6) with Re A/> 0, then, by taking modulus in (3.6),

j = l k = l


with equal i ty holding in the left inequali ty if and only if A = 0. The last inequali ty follows f rom (3.7). Thus, if (3.7) holds, the only possibil i ty for an eigenvalue with nonnegat ive real pa r t is for zero to be a root o f Q ( A ) = 0 . But this is not the case since the constant te rm (in fact , all the coefficients) o f the characterist ic po lynomia l is positive.

We remark that the above a rgument showing that all characterist ic roots A of the characteris t ic po lynomia l associa ted to (3.1) with (3.5), satisfy Re A < 0 if (3.7) holds, can be easily carr ied over to the characterist ic equat ion associa ted with the l inear izat ion of the funct ional differential equat ion (3.1) abou t 9~ to obtain the same result.

Thus we see that

g'(a) < - 1 (3.8)

is a necessary condi t ion for there to be eigenvalues of E with posit ive real part. Observe that any such eigenvalues must have nonzero imaginary par t since Q has strictly posi t ive values on R § Recall that (3.8) is also a sufficient condi t ion for the hypotheses of T h e o r e m 3.1 to fail.

We examine the possibi l i ty of characterist ic roots o f (3.6) with posit ive real par t in a very simplified si tuation, namely , in the si tuat ion where all three genes have the same pa ramete r s

Oli=[~j=~k=Ol

p = l = s (3.9)

1 fa(u)=f2(u)=f3(u) l+kuq , q > l

It fol lows f rom symmet ry that #s = 37p = ~1 ~ u where

1 t a u (3.10)

l + k u q

We then have the fol lowing result.

Proposit ion 3.3. Suppose in (3.1) that both (3.5) and (3.9) hold. Then (3.6) has a root A with Re A > 0 if and only if both

1(;,) e --=q cos > 1 (3.11a)

and

Og qlE k > ( l _ e ) q + l (3.11b)

Moreover, all roots are distinct if (3.11) holds.

Proof The character is t ic equat ion becomes

(A + ol)3t+ P = 0

186 H. Smith

where P can be written, making use of (3.9) and (3.10), as P = (qkol21uq+l)3. A computation shows that there will be characteristic roots with positive real part if and only if

1 q + l "/T qka u cos > 1.

Note also that, in this case, all roots are distinct. We need to determine the dependence of u on the parameters.

From (3.10), it follows that

U = k-1/qo(ol-lk 1/q)

where v = v(c) is the unique positive root of

C=V~-~) q+l, c~O.

Putting this information in the above inequality yields

C q-1 sec~ll ' c = a-lkUq

Recall from Proposition 2.2 that vq+l(c)/c < 1 and monotonely increases to one as c tends to infinity. Thus the above inequality holds if and only if

q s e c ~ < 1

and c > ~ where g is the positive solution of

vq+'( 6) - q - l ( sec ~l) I.

This ~ can be computed explicitly, from which (3.11) follows. []

Thus, for the ordinary differential system (3.1) with (3.5) and (3.9), conditions (3.11) give sufficient conditions for the existence of a nontrivial periodic orbit. Observe that both (3.11) (a) and (b) will hold for fixed l, k and a ~< 1 if q is sufficiently large. Since (cos z r / G l ) ~ 1 as l~oo, increasing l has the effect of weakening the largeness requirement for q in (3.11) (a) (clearly q must exceed one). Hence, if q > 1 and a <1 then (3.11) will hold for sufficiently large l. However, if a > 1 then (3.11) (b) can be a problem. As an example, if l = 2 and q =2 then (3.11) will hold if k > 18a 4.

We remark that for general odd G, Proposition 3.3 holds with the modification that G replaces "3" in (3.11 (a). Hence, increasing G makes (3.11) more likely to hold.

Another approach to finding undamped oscillations in (3.1), with or without assuming (3.5), is to apply the Hopf bifurcation theorem to find small oscillations about s This approach has the virtue of yielding, with considerable effort, stability properties of the bifurcating periodic orbit. N. MacDonald [18] and later, Tyson and Othmer [29] have shown that for a restricted set of parameter values for the single gene ordinary differential equations, the bifurcating periodic orbit is locally stable. Mahaffy has extended this to the single gene model with discrete delays in [32, 22]. We do not attempt to carry out such an analysis here. However, such


a calculation, for a suitably restricted set of parameters as above, should not vary significantly from those carried out in [18]. The reason for this expectation is that, in our indication of the proof of Theorem 3.2, we subjected the equations (3.1) with (3.5) to a change of variable which in essence changed a three gene feedback loop into a single gene feedback loop- -on ly the nonlinearity f l remained as a negative feedback term while the others became positive "feed forward" terms. This is the setting of N. MacDonald 's calculations in [18] except for the complication of the nonlinear terms f2 and f3-

J. M. Mahaffy has shown the existence of (not necessarily small amplitude) periodic solutions when G = 1, f l has the form (0.4) and the delays are discrete in [21]. Basically, he requires the sum of the discrete delays to be sufficiently large and the nonlinearity to be sufficiently strong (see Corollary 5.2 in [21]). Discrete delays are very convenient in [21] for they allow a simplifying transformation of variables, pointed out by an der Heiden [2], to be utilized which allows one to work in a simpler state space. This trick of an der Heiden works in the multigene models which we study in case that all delays are discrete. Thus, it seems likely that the results of Mahaffy for G = 1 carry over to general odd G if the delays are discrete.

4. Conclusion

In this section we summarize our results and make some remarks. Our main point in this article has been to point out a qualitative difference between a control loop with an odd number and one with an even number of negative feedback terms. This qualitative difference can manifest itself in a qualitative difference in dynamics. We emphasize the word can, since for (0.1) with weak nonlinearities (e.g. f given by (0.4) with q = 1) the dynamics are essentially independent of G, the number of genes. Banks and Mahaffy [4] and Allwright [1] show that all biologically relevant solutions approach the unique positive steady state. The only manifestation of a qualitative difference between even and odd G in this case is subtle one, the approach to steady state is monotone for even G and oscillatory for odd G.

The contribution of this paper has been to point out that for strong nonlinearities ( f l large relative to the decay terms ai,/~j, etc.) there is a real qualitative difference in dynamics between odd and even G for (0.1). For even G, strong nonlinearities lead to multiple stable steady s ta tes--almost all biologically relevant solutions approach a stable steady state (Theorem 2.1). For odd G, strong nonlinearities destabilize the unique positive steady state and result in the appear- ance of a nontrivial periodic solution, at least in the ordinary differential equation case (Theorem 3.2) and, most likely, in general.

There remains the important question of whether or not strong nonlinearities are biologically relevant in (0.1). In any case, we feel that it is important to expose their effect.

Throughout our work, a curious parallel has been noted between the dynamics of (0.1), for arbitrary G, and the dynamics of a certain discrete dynamical system u~+l = g ( u n ) , n = 1, 2 , . . . , generated by the mapping

g = (TS~fo) o (y~l_,f~_,) . . . . . (T71f1)

188 H. Smith

where the constant y~ is the product of the decay rates for the ith gene and f is the nonlinear term representing the inhibiting effect of the presence of the end-product of the i-lst gene on transcription of the ith genes mRNA. Since the f are strictly decreasing, g is either strictly decreasing or strictly increasing on [(~, oo) as G is odd or even. In either case g is a positive valued function with a finite limit at infinity. Allwright [1] observed that if g has a single fixed point which is globally (on [0, oo)) attracting for g then (0.1) has a single globally attracting steady state. On the other hand, when G is even, g may have multiple fixed points and hence so will (0.1). I f G is odd, the single fixed point of g can lose stability to a period two solution and when this happens the single steady state of (0.1) losses stability and a periodic orbit appears (at least for the ordinary differential equation version of (0.1)).

The remarkable fact is that the dynamics of the two systems appear to correspond both at the level of local stability analysis and at the level of global dynamics. This is potentially a very useful fact, both for model construction and for analysis of particular models. Although the dynamics generated by one dimensional maps need not be simple, one can learn a lot about the dynamics associated with the map g by some simple hand calaculator and back-of-the- envelope calculations. Finally, if such a correspondence holds even in the case of nonmonotone f (and hence nonmonotone g), then one is led to expect complicated dynamics in (0.1) for nonmonotone feedback (see e.g. [31]). We remark that several authors [6, 20, 22] have noted a similar cor respondence- - particularly between singularity perturbed delay equations with nonmonotone nonlinearity and an associated difference equation. Chow and Mallet-Paret [6] and Mallet-Paret and Nussbaum [22] have made progress in relating the dynamics of the delay and difference equation.

In order to make a rough connection between (0.1) and an associated discrete map, we take G = 1 and p = 2 for ease of exposition. The ideas we develop in this case apply equally to the general case. Thus, consider

y~l( t) = f l ( L 2 y ; ) - aay l ( t)

y'~( t) = L~y'I - ~ y ~ ( t) (4.1)

Recall that there is a compact attractor for (4.1) in C + and that solutions of (4.1) with initial conditions on this attractor exist for all t c •, are bounded, and remain on the attractor [12, Chap. 13]. Since all solutions of (4.1) approach the attractor as t becomes large, it is precisely these solutions on the attractor which are important. Hereafter, we will consider only these solutions which are defined for all t c R and are bounded.

Now if g(t) is a bounded function on R there is a unique bounded solution of

y'=-ay+g(t) , a > 0

given by

y ( t ) = a - l K ( a ) g = a - ~ f ~ ae~Sg(t+s) ds = c~ -1 f]oo g(t+s) d(e ~s)


Notice that the weighting factor, ae ~ has total weight equal to one. Hence, for the solutions of (4.1) which exist and are bounded on R, we have

y l ( t ) = 0/Taglfa(L2y~)

y2(t) = 0/21K2Lly~

where w e have written K1 for K(0/1) and K2 for K(0/2). Not ice , that L 1 and L2 are operators very s imilar to K1 and / s they are also given in terms o f probabil i ty measures . Hence , if w e v iew these operators as acting on BC(R), the Banach space o f b ou n d ed cont inuous funct ions on ~ with the uni form t o p o l o g y then w e may write (4.1)

Y2 = T?~L2LIKlf(L2Y2), Y2 c BC(R)

~/1 = O/10/2

The Eq. (4.2) establishes a connection between (4.1) and the map g = y71fl obtained from (4.2) formally by setting K2 = L1 = K~ = L2 = 1. For every solution (yl(t), ye(t)), t~R, of ( 4 . 1 ) o n the compact attractor associated with (4.1), it fol lows that Y2 is a solution of (4.2), y2~BC(R) +. Conversely, if y2c BC(E) + satisfies (4.2) and Yl -= 0/11Klf(L2Y2) theft (y~(t), y2(t)) is a solution of (4.1) which is bounded on R and hence is a solution o f (4.1) lying on the compact attractor [12, Chap. 13].

Our remarks above, o f course, explain very little. Clearly, there remains an interesting mathematical problem. The first steps toward its solutions, we feel, are being made in [6, 22].

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Received March 28, 1986/Revised September 15, 1986

oscillations and multiple steady states in a cyclic gene model with repression

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