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Hierarchical behavior in fit dynamical systems
HIERARCHICAL SYSTEMS n
BEHAVIOR IN FIT DYNAMICAL
CHARLESJ.LUMSDEN Department of Medicine, University of Toronto, Toronto, Canada
We consider a class of differentiable dynamical systems that fulfill a criterion of adaptation, or being fit. The adaptation criterion is found to be sufficient for the existence of hierarchical behavior in these systems, once they become complex. Commutative diagram methods are used to show that under reasonable conditions the hierarchical behavior takes the form of a closed dynamics of aggregate variables. This dynamics characterizes the system functions as a whole. Existence conditions for the aggregate dynamics are obtained, together with their differential form and their connections to the original system description. The results indicate that evolutionary processes are likely to produce systems that are hierarchically organized in terms of their function as well as their structure.
Introduction. Darwinian theory (Darwin, 1857; Mayr, 1970) observes that an adaptive fit often exists between organisms and the environments they occupy. Natural selection is viewed as the prime mover shaping organic diversity and phenotypic design. The fit is created as natural selection acts on individuals through their differential survival and reproduction. The winners of the ensuing evolutionary competitions are postulated to be those whose fitness is greatest. In mathematical versions of the theory, fitness is typically a function that integrates reproductive success with survival likelihood over an individual’s lifetime. The concept can be extended to cover genes that related individuals share by common descent (Hamilton, 1964; Maynard Smith. 1964: Kurland, 1980), and to selection processes that act on groups, societies, and even entire populations (Levins, 1970: Eshel, 1972: Boorman and Levitt, 1973; Wilson, 1980). Substitute measures of quantitative fitness (such as physiological efficiency) more amenable to observation than net reproductive success are often used in modeling (Rosen, 1967; Oster and Wilson, 1978). Fit organisms correspond to the phenotypes or gene variants at which the fitness functional takes on its maximal values. In this report we are interested in relating adaptive design to hierarchical organization. Rather than structure, the focus will be on the dynamical order characterizing the phenotype. A class of differentiable dynamical systems will be introduced for this purpose. Its members are naturally described as 1.
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C. J. LUMSDEN
fit in the sense that they obey a principle of adaptive design. An approach to hierarchical organization is then presented that scales among levels of dynamical order. Commutative diagram methods are used to show that such levels of order exist in fit systems once they attain a certain degree of complexity. Complexity is measured merely by the number of independent state variables (equivalently, the number of differential equations) required to describe the phenotype. Thus the fact that a complex system is the product of adaptive design suffices to establish a degree of hierarchical order within its dynamical behavior. The notion that biological systems have a hierarchical design, with many interconnected levels of structure and function, has a long and productive history (Novikoff, 1945; Wilson, 1969; Pattee, 1973). There is an extensive literature deaiing with biological complexity and hierarchical order in a qualitative way (e.g. Wilson, 1969; Whyte et al. 1969; Dresden, 1974; Cottrell, 1977; Allen and Starr, 1982). In addition, there has recently been interest in exact mathematical models of biological processes that have multiple levels of organization. Some investigations have concentrated on making rigorous the meaning of multiple alternative descriptions (Rosen, 1968; 1976; 1977a, b; Primas, 1977). Others have analyzed the types of hierarchical order present in specific models (Simon and Ando, 1961; Simon, 1962; Wilson, 1968; Mesarovic, Madco and Takabarz, 1970; Weidlich, 1972; Demetrius, 1977; Haken, 1977; Vemuri, 1978; Lumsden and Trainor, 1979; Lumsden and Wilson, 1981; Lumsden, 1982; Auger, 1983). The increasing number of studies of the second kind raises an interesting question: what can be said in formal terms about the hierarchical organization of dynamics as an adaptive trait in itself? The purpose of this study is to establish the existence of a connection between fit design and this form of organization. The models to be used in the investigation are defined and their dynamics specified in Section 2. Section 3 introduces a scaling procedure and uses it to relate the system dynamics to the organization of its aggregate variables. Theorem 2 establishes the connection linking adaptation to multilevel ordering of the fit dynamics. The mathematical findings are then discussed in the paper’s concluding section. We consider a system 9 with state variables xk E w, k = 2. Fit Systems. I N characterizing its morphology, behavior, and consequences of this * * 7 behavior (e.g., proportion of body fat, number of matings per unit time, number of offspring produced per unit time). Depending on the traits of relevance, the same state variables might be used to compare members of different biological species, or their application might be restricted to deal
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593
with intraspecific variation. Activity of Ywithin its environment is characterized by temporal changes in xk. The dynamics of the state vector x=(X1,. . . , xN) follow the first-order differential equations dx, -ZX dt The the For tion
.
k=Xk(x,
t),
k= I,...
.N.
form of the Xk for systems of biological interest ultimately expresses interaction of genotype and environment in shaping the phenotype. the systems relevant to the present investigation the principal restricon the .& is smoothness of a minimal sort, modeled by @“-continuity on
RN+‘. x:lRN+‘+lR~:(x,
t) p
x
where t E W denotes the time variable and we adopt the mapping conventions of Arbib and Manes (1975). The equations (1) induce the flow 4 for Ythrough the configuration phase spaceRN: $:RN+r+RN:
(x,, to) I+ x(t 1x0, to).
For every # on R N+’ induced by equations (1) we define the reward function or fitness as a scalar functional J that integrates over the life history [to, tf ] of9 J=
tfG(x, x, t)dt, I to
(2)
where G(O) is the reward rate on [to, tf I. The reward rate of particular interest in population biology is the net reproductive success of SPper unit time, or (more commonly) a phenotype variable believed to determine a significant proportion of the variance in this fitness measure. The choice of G(*> is influenced by the type of phenotypic designs to be compared and the envisaged nature of their variation under genetic change (Rosen, 1967; Oster and Wilson, 1978). For example, biological organisms have a finite reproductive success, and generally G( ) is constructed such that J < UJfor each3 Similarly, among the possible G(O) are only those with support in the subsets of pertinent initial states. Conversely, not all formally conceivable initial states x(t,) will enter into the determination of a G( l ) [e.g., ant societies based on claustral founding always start from x(to) = 1 individual, the fertilized queen, irrespective of other possible modes of variation (Oster and Wilson, 1978; Lumsden, 1982)]. In view of these considerations we introduce the following definition. Definition. The differentiable system 9 is a fit system F iff for each x(t 1x0, to) solving (1) the fitness J is maximal on the Life history [to, tf ] : l
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C.J.LUMSDEN
= max.
J[x(*)]
(3)
The set of all such fit systems will be denoted byP. Remark. This definition is clearly over-inclusive in the sense that all systems (1) for which (3) is true are admissible fit systems, and this set may include elements that are not biological in any reasonable sense. However, because processes of competition coupled with differential (re)production influence a broad range of natural and man-made systems, in a manner often resembling classical natural selection, it is useful to relax the notion somewhat and for the time being include systems of the latter types within 9. Equally intriguing is the question of whether 9 includes any biological systems at all, or whether near-optimally designed phenotypes rarely appear during biological evolution. Our definition would then be too restrictive for biological purposes. Although much work remains to be done on this issue, the available evidence from studies of physiological (Rosen, 1967; Alexander, 1982) ethological (McFarland and Houston, 1981; Krebs and McCleery, 1984), and societal (Wilson, 1980a, b) design indicates that Scontains some (but by no means necessarily all) real biological systems, and that nearoptimal phenotypic traits may be rather more common than previously supposed. The flows $ in which equation (3) is true follow from the variational principle rfG(x, i, t)dt = 0. s to
6J=S
(4)
In differential form this condition is expressible as the control equations -aG ---=d aG dt a.& aXk
0,
k=
l,...,N
(Bryson and Ho, 1969; Vagners, 1974). It is useful to transform symmetric notation in which costate variables h&6x/8X&,
k=
i.,...,
N
(5) a more (6)
replace the x&s. Using the generating function Q=G-5
h&x& k=l
(7)
in the fitness principle (3), one obtains from the control equations (5) the costate dynamics dXk/dt
=-aQl&Yk,
k = 1,. . . ,N
(84
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595
as well as the state variable equations dx-,/dt = aQ/ah,,
li = I,. . . >N.
(8b)
The natural canonical or Hamiltonian structure of equations (8) follows from the variational structure of the fitness problem (3) and the use of costate variables to express the control equations as a first-order differential dynamics. Unlike physical systems, whose microscopic dynamics are also essentially canonical, fit systems are not restricted to reward functions G( l) that have the form of an energy function. This flexibility opens the possibility of new and unexpected behaviors from a physical point of view. Moreover, treatments of physical systems far from thermodynamic equilibrium have demonstrated that even rather conventional energy-Hamiltonian systems can exhibit entirely novel models of spatio-temporal organization relative to the characteristics of near-equilibrium (Prigogine, 1980; Prigogine and Stengers. 1984). The present development will not restrict the fitsystem dynamics to orders analogous to ‘at’ or ‘near’ some condition of aggregate equilibrium. For some applications it is advantageous to incorporate constraints among the state variables into the explicit statement (3) of the fitness problem. The resulting control equations preserve the basic canonical structure of equations (6)-(g), with modifications introduced by the presence of the constraining conditions (Bryson and Ho, 1968; Oster and Wilson, 1978). Methods treating the collective dynamics of constrained systems have recently become a subject of analysis and application (Lumsden and Trainor, 1979? 1985). In this study we will assume that any constraining conditions among the state variables have already been incorporated by eliminating variables (1) and appropriately rewriting equations (l)-(3). When N %=1 in equations (l)-(8) it becomes useful to introduce variables that characterize the system, or large parts of it, as a whole. The original description in terms of (x, x) or (x, h) contains an abundance of detail. Dejhitiorz. The %’ map
A,:lRZN+ lR:(x, A) PA,(x,
h) ER,
(9)
is an aggregate variable of F iff dim [dom (A,)] /2N % 1. In other words, A, depends on most or all of the state variables xk or state and costate variables Sk> hk: k = l> . . . ) N. Remark. A system may possess a number of aggregate variables A,, 111= 1, . . , > M relevant to a particular set of properties, so we introduce the aggregate descriptiolz d s {A, I nz = 1, . . . , M}, along with the following definition. Dejkitiorl. An aggregate description dis interesting iff M Q N.
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C. J. LUMSDEN
Interesting aggregate descriptions characterise F by means of relatively few relevant variables A,. They are economical relative to state-variable descriptions in terms of the xk and h k. In biological applications the A, represent such important concepts as the local concentration of a biochemical species (sum over many molecules), the number of individuals in a population (sum over many individuals), and net matter-energy fluxes though trophic networks (sum over activities of many producers and consumers). The best-understood aggregate variables are in fact in the form of simple sums over the state and costate variables. The set J$ is a subset of 9, the set of all observables f:lRvv + lR defined on (1). A weak criterion for %?, sufficient for the present study, is that it comprises the set of all %‘I complex functions on lRzN which vanish at infinity. The behavior at infmity models the finite capacities of biological systems; formally, it allows meaningful expectation values to be constructed. Since lR2N is locally compact, relative to the sup norm llfll ~,~;2JfoI
(10)
and the involution f t+ f*, * G complex conjugation,@ is an Abelian B*algebra over the system phase space lRZN (Ruelle, 1969). Through d the flow 4 in lRN (equivalently, IR2N) connects the aggregate variables at t,, with their values at any t E [to, tfl . 9
I XI), to>
a$----
x0
d
54 I A@ I At,, to>
I AO,)
(Dl)
If there exists a t-autonomous where A(t) G (A r(t), . . . , AM(t)). differentiable flow GAon RM such that # x(to)------+
x(t I x0, to>
Nto)
A@IAo, to)
-
V1-
@A
with d* $A = 9 at we say that &f?constitutes a closed aggregate description of F. The corresponding differential equations can be written as l
k,=Y,(A,
)...)
A,),
m=
I)...)
M.
(11)
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591
For fit systems that admit closed aggregate descriptions the system
+ aA?3a9 --=
ahk axk
Ym(A1,.
.
. ,A,),
m = l,..
. ,M(12)
is solvable. On the basis of present knowledge it appears that pairs (Y,4qd) for which (D2) is true are either rare or quite difficult to find (e.g., Coxson, 1984, p. 435). Commonly there is some leakage from the x-level to the A-level that prevents closure in the sense of(D2) (see below). All fit systems are, however, A-closed in the following sense. THEOREM 1.Let p(x, h) be an element of Xw, the set of integrable, unitnormaIizable (‘probability ‘) functions in R 2N.Xw is a convex subset of 4V’. the dual of the algebra of system observabIes S (Ruelle, 1969). Let P(a) EX”, where ,XM is defined similarly to X2N. Then there exist Gp and q$ such that the diagram (03) commutes.
(D3)
P(a, toI
d?P
b
P(a, t).
Remark. The operation sdenotes construction of the probability distribution P [system ensemble (Lumsden and Trainor, 1979)] on lRM for a given specification of aggregate variables. Notation. As context warrants we shall write equivalently p(x, X, t) G p(x, X) = p(t) for the value of p at time t. We prove Theorem 1 in several steps.
LEMMA
1.In the (x, h) variables the differential form of & is apiat = -i.C$p
(13)
where
(14)
598
C. J. LUMSDEN
Proof. Since
s
dx dX p(x, X, t) = 1
RZN
vt
1,151
p(t) has a local balance equation
(16) (Lumsden and Trainor, 1979). Using equation (8),
+I:
N --------_oo. ap39 ap w kzl
axk ahk
ahk axk
(17)
The generating function Q(x, X) is X-linear and the formal difference a2QjaXk a& - a2Q]&aXk
(18)
n vanishes. When .Y is explicitly independent of time, the propagator T(tl to) that takes p(x, h, to) into p(x, A, t) for t > to is by operator integration
T(t
1 to) = e-t(r-rO)~
(1%
where @p: p(to) 5 T(t I to) : p(to) /-+p(t) = e-‘(r-ro)9p(to).
(20)
Explicit time dependence may occur in Ywhen the fit system F is coupled to an external environment or in the presence of certain forms of dissipation (Lumsden and Trainor, 1979). This inclusion requires the use of timeordering operators (Fetter and Walecka, 1971) and makes the development more cumbersome, but entails essentially no new formal results. To keep the presentation streamlined we treat the case of explicit time-independence, and without loss of generality set co = 0. LEMMA 2. The constructive procedure 5 exists for diagram (03). Proof. She A,,, = A&x, X), m = 1, . . . , M, an A, contour with value a, is a hypersurface in lR2N. Intersection of the hypersurfaces A,(x,X)=a,(t),
m = 1,. . . ,M
(21)
FITDYNAMICALSYSTEMS
599
generates at any time t a 2N -M-dimensional subset IYa,. . . . , uM) - I’(a) C R2” consistent with A having value a(t) - @r(t), . . . I a,~&t>). Define P(a,t)da~Prob(a,
that a,,
(a) t = I& left side of(D3).
the value of A,? lies in [a,! a,
(22) + da,]
for
The initial aggregate state is aoE a(t = to).
(23)
Then
JYa,to> =
fiS(u, - ~0~)
(24)
m=l
while in lR 2N. (x0, ho) E l?(ao). Since no additional information tion of F in lR2” at t = to is given, p(to) is uniform on r(ao): dto) = WA(x, A) - ao)WA’(ao)
on the loca(15)
where Wa0>
=
s
2Ndx dX 6(A(x. A) - ao),
(‘6)
a hypersurface measure of F(a) in RZN (Khinchin, 1949). This completes the left side of (D3). fb) t > to riglz t side of (03). Consider p = p(t > to). Then p(t)
= T(t I to)p(to)
= e-w(r-ro)p(to).
(27)
By definition P(a, t) dt gives the likelihood of finding the system with a E [a, a + da] at t. P(t) must therefore equal the total probability density p(x, h. t) with support r(a), where a = a(t). Thus
Ra, t) = =
jR’”dx dh &(A -
IIR 2N
a)p(x. h, t)
dx dX 6(A - a)e-i(f-fo)y 6(A - ao).
(28)
These considerations show that the entire set tiN of admissible p(x, h! ty is not pertinent to A-level descriptions of the system. It is sufficient to consider the subset X:” C x 2N that contains only those p with functional dependence of the fonn
600
C. J. LUMSDEN
(29)
P(X, N = PL-UX,Nl .
Xi” can be characterized by the action of a map 9 that extracts from a state function f(x, X) that part, f’ =Yf, dependent
(30)
on (x, X) via the A(x, h). By definition ofX
y,
pf = W” [AllRzN dx’ dh’ 6 [ A(x’, A’) - A(x, h)] f (x’, X’)
(31)
where SPf
= Pf, g2 =s?
(32)
LEMMA 3 (Zwanzig). Let tip be a dynamics of form (13) for p(t). Th!n @, has the equivalent differential form iap’ja t =92zp’(t)
(33) where
P’W =W(t) p”(t) = (I-9)p(t)
(34)
with to = 0 and I the identity. Proof. Since 9is linear and explicity time independent 9$=&9?
(35)
Thus iap’/& =@sp’
+ p”)
iap”/at = (I -g)Y(p’
(36a) + p”).
(36b)
The operator (I - 9) projects onto the subset X2N - Xy : 9 (I - 9 ) = g2 - 9 = B - 9 = 0, and equation (36b) then formally integrates to p”(t)
= e+t(Z+W
p”(O) - i
’
s0
ds e-Is(z-@y(I -g)_@‘( t - s)
(37) n
by Laplace transforms. Using equation (37) in (36a) gives the desired result.
FIT DYNAMICAL
SYSTEMS
601
Remark. The form (31) 9 and its use to develop the expression (33) for the projected density p’ were first documented by Zwanzig (1960, I96 1a, b).
In this study his method wiIl be used to help establish the commutativity of the diagrams [Di] linking the descriptive levels of F. Remark. The dynamical content of equation (33) is partly evident from its structure. The first RHS term gives the Markovian part of the p’(t) motion caused by contributions that remain in e” for 0 < s < t. In the second RHS term, (I - 9 )UP,p(t - s), is a flow of probability from Xp into XzN - Xim at prior times (t - s). The factor e-ir(lFb) y maps this probability forward in time while keeping it confined to X 2N - %fN. 96p then determines the effects of this leaked probability density on the p’-motion. The time integration sums leak these effects over the time interval that the probability contribution is in Xuv - XT”? The third RHS term represents the influence of that part of p(t) which fell in 3yzN - Xy at t = 0.
COROLLARY
1. p”(0) = (1 --B)p(O) = 0.
(38)
Proof. Since F is describable by the initial aggregate state (23), condition a (24) implies that ~(t = 0) EXfN. Although the projected density p’(t) relates the dynamics (1) of F to the aggregate variables Am, the connection is highly implicit. An explicit representation is guaranteed through the following
4.Let p(t) EpN have a p’(t) EXy that obeys equation (33). Then for P(a, t) EX”, #p in (03) exists and is given in differential form by
LEMMA
!$a,f) = -m$l $ [v, WYa, t>1 m
(3%
such that v,(a) =
K,,(a,
a’, s) = (kme-‘s(z--S?p(I --B)i$(A
(40a) - a') 1a>,
(40b)
where (* I a> is a linear functional on@ (* Ia):¶V+lR:
f(x,h>/-+(f(x,h)Ja)EW.
(41)
Remark. We note that equation (39) is a closed-form relation for P(a, t), expressed in terms of the aggregate variables.
602
C. J. LLJMSDEN
PPOOJ:From (28) and the definition of9 W-l(a)P(a, t) = p’(x, h, t)
(43)
A(x,A)=a’
Differentiate (42) with respect to time and, use the differential 9, from (33); then W-‘(a)$P(a,
t) = -ij+
form of
da’ W-‘(a’) tJfG(A - a) I a )&a’, t) RY
-
s2ss o
da'
W-‘(a’) IUe -ir(zmB)y(J-
9)lps(A
-a’)1 a>P(a’, t -3)
lRM
where forf(x,
(43)
A) E4
h) I a> si W-1(a)JR2H dx dX 6[A(x, h) - al f(x, h).
(44)
The action of (* la> is to average an observable over the cell r(a) C R*? Now M a U’6(A - a) I a> = -i W-l(a) x -[WW(a)v,(a)G(a m=l
- a’)1
(45)
aam
and W’e-*(z-*9(I
s
-P)LbS(A - a) I a) =
W%‘)m$l ngl &$
da’
RM
[ W(a)K,,(a,
a’, s) 1 W-l(a’)Pta’, t -s).
n
(46) Equation (39) follows from an integration tions of the result and (45) into (43). Remark. The identity les[A(x,h)-al
of (46) by parts and substituW
M a
=-im~l~6(A-a)--$ m
dA
(471
(Sewell, 1965) is useful in constructing identities such as (45) and (46). Lemmas l-4 establish that p(t) and P(t) are linked as indicated along each side of diagram (D3). Theorem 1 is therefore established. The arguments leading to Theorem 1 are an example of system aggregation, whereby certain properties characterizing the components of a system are demonstrated to support one or more interesting aggregate descriptions.
FITDYNAMICALSYSTEMS
603
This descriptive process will be taken further in the next section and related explicitly to notions of hierarchical organization. In concluding the present section it is useful to note that in other, somewhat simpler, systems algorithms for aggregation are known (cf. discussion in Section 1) and can be compared to the present schema without resorting to a complete formal derivation. For the best-understood cases a natural linearity in the dynamical problem facilitates the argument. If in system (1) the Xk are all linear combinations of the state variables, Xk = ZjCkiXj, then for aggregation in the sense of diagram (D2) it suffices to find a linear map that takes the matrix of coefficients Ilcki 11 into a matrix of much-reduced dimensionality. Conditions for the existence of such maps have been established (Soong, 1977). Sometimes the system possesses intrinsically stochastic properties that can be better modelled with master equations of the Pauli type, rather than with differential equations. The Pauli equations link the system states via a matrix of transition probabilities per time step (discrete time scale) or of transition rates (continuous time), and again comprise a linear description (Haken, 1977). Aggregation then seeks to combine the states into state-sets, between which transitions again occur in a stochastic fashion, governed by relatively simple master equations. Conditions allowing aggregation in these stochastic systems are readily seen in physical terms: the set of all states must divide naturally into subsets containing states linked by high transition probability, such that the likelihood of transition between the subsets is much smaller. The Simon-Ando theorems and subsequent developments (Simon and Ando, 196 1; Courtois, 1977) provide formal criteria which transition matrices must fulfil in discrete time or continuous time in order for this aggregation to be possible. Theorem 1, by comparison, applies to systems best described in terms of a continuous time variable, and which are deterministic in their microscopic specification (1). The systems can, and in the cases of greatest biological interest will, involve nonlinear dynamics. They are not restricted to linear differential structure. Out of this class of nonlinear systems, Theorem 1 deals specifically with those designed according to a fitness principle of the form (3), rather than with a subset constructed on the basis of analytically convenient characteristics such as linearity or Pauli form. 3. o-Hierarchies. As functions of many variables xk, hk, the time-rates Of change Of the A,,, hVOhe allthe r&S ik, ik : (48) In general the ,k and ik will be of VatiOUSmagnitudes and signs for any t > tW If this is the case the sums (48) will smooth over changes on the
604
C.J. LUMSDEN
timescale of microvariable dynamics. The aggregate variables A, change relatively Slowly on the timescales of xk and hk. Definition. Let A, f IR and all xk and hk change by a what are agreed to be ‘significant amounts’ A,, v = 1, . . . , 2N. Let T, be the time t - to required for the change A,,. Then r=maxr, V
is said to be a characteristic time scale for change in the (x, X)-variables. Definition. Let &,, E W and all aggregate variables A, change by what is agreed to be ‘significant amounts’ .& m = 1, ...,M. Let 7; be the time t - to required for the change t,. Then
is said to be a characteristic time scale for change in the A-variables. Observa?ion.When N > 1, T Q 7’. If F E Scpossesses two such well-separated time scales one suspects that it might be possible to remove the residual references to (x, A>variables entirely from the A-level description, at least in approximate terms. This motivates the following definition. Definition. Let Q be a scaling procedure based on r and T’, where r 4 7’. If u maps (pP into a closed dynamics $A of aggregate variables on lRM we say that F is a u-hierarchy or u-hierarchical system. If closure is true at most to order K of a perturbation parameter in u, then F is a UK-hierarchyor #-hierarchical system. The properties of F as a system with hierarchically organized dynamical properties are characterized by the following theorem.
THEOREM 2.A fft dynamical system with T 4 T’ is a &hierarchy. In order to establish Theorem 2 we will require some results on the properties of W(a, t)/at [equation (39)] under the scaling operation u.
LEMMAS.~
03
M a $Ya, 0 = -mTl-[urn b)P(a, 01 aa
m
Proof. In a timescale relationship like T Q T’ the rates A, are small on the timescale for change in the & and &. Let us consider a scaling method that
FITDYNAMICALSYSTEMS
605
uses perturbation terms to approximate v,,, and K,,,,,. The approximation becomes exact in the limit A,,, + 0 V m. The kinetic coefficients v, are explicitly f”lrst order in the small quantities A,, while the K,, contain the A, explicitly to second order. Define u2: Expand all terms in (49) in A, and retain these rates to second order. Operating on an observable F(x, A) with the projector 9retains only the of F, while using 2’ on a 9F brings out a factor of A, Applying c2:
A(x,h)-dependence
e-rs(z-@2(1 -9)
= (I -9)
- is (I -9)
_9 ( I-9)
+ 4 (is)2(l -9)14(1-P) S (I -9)
-
js(l -S)Y+
9(1 -9) f
+...
(is)2(f -9)Ie2
+ ,..
= (I -9)e+"Y
(50)
Similarly,
upon expanding the exponential
and using equation (47). Then
Km, (a, a’, s) S 6(a - a’)[ti,e-‘SyAn
la> - (A, la)(e+4in
la>1 + 0(A3> (52)
Writing (em%&
I a) =
(53)
we see that Kmn(a, a’, s) 2 6(a - a’)K,,(a,
s)
(54)
where K,, (a, s) E
la> -
bi,(O)A',(-s) Ia> -
(A, (-s) la> I aHA,
C-s) Id.
Wa) (55b)
s) evidently has the form of a correlation function between the aggregate variables A, and A,. Equation (54) can now be used to carry out l the integral over a’ in (39). The result is equation (49). Remark. In the case that $J induces sufficiently ‘statistical’ wandering in the state-space path of F, correlations between the A, decay quickly on the time scale r’ and the functional &,(a, S) can be modeled by a rapidly (typically exponentially) decreasing function of time (Prigogine,
Kmn(a,
C. J. LUMSDEN
606
1962). The relevance of such properties for the hierarchical organization of F will be discussed below. It now remains to construct a relationship between P(a, t) and the rate laws for the aggregate variables. LEMMA 6. Let ii,(t)
be the expectation value for A,,, under P(a, t). Suppose that the aggregate description {Al, , . , , AM} is at least weak@ useful in that F responds in a similar way each time it occurs in the same initial aggregate state A(t,). Then in a2 the Z,(t) obey the&l -4 N integro-differential equations
g&l = -v,(ii,> dt
+s2sxlKm,E(t - s>,sl M
0
A,
[a0 - s)l
+
$-Kmn [a(t -
s), S]
n
n=l
1 (56)
where A, (a) = Proof.
$
(57)
In W(a). n
By definition c?,(t) =
s
daa,P(a, t) =
lRM
s R2N
dx dhA,(x, h)p(x, A, t>,
(58)
so that
dam_ dt-
s
dala,;:P(a, lRM
t>l.
(5%
Using Lemma 5, integrating the vm terms by parts once and the Km, parts twice gives after rearrangement darn -=-
dt
s
da v,(a>P(a, t>
RM
+ Jdds/_Mda$
W-‘(a)& n=l
[K,,(a,
s)W(a)l P(a, t - s).
(60)
n
Since the description {AIT . . . , AM} is at least weakly useful it must be that P(a, t) develops from its initial state P(a. to) in such a way that it remains sharply concentrated about the means G,(t). If this were not the case, the variances of P(a, t) would increase significantly with time and the observed
FITDYNAMICALSYSTEMS
607
a-values would not closely repeat. Instead, they would drift between repetitions in which F is in A at t = to and the description {A r, . . . , A,} would not be at least weakly useful. For such descriptions
I
RM
@la)
da rj,(a)P(a, t) s v,[ii(t>l
~~‘(a)$ilU,,(a, sW(a)lP(a, f- s> n
2 5
IV-r[ti(t -s)]$&Jr(t
-s),
s]Wl$t--)I}.
n
(61b)
?I
?I=1
The combined action of the scaling procedure u2 and the averaging operation over P(a, t), which we denote in general by (0 >,
(9
:gfaWgT;;)$RM dag(a)P(a,0,
(62)
is to induce a map x relating the P-dynamics (39) through its o-image @f: [equation (49)] to @$ the motion of the expectation values ii(t) [equation (56)l: fP @p,
*
4;:
(D4) a commutative diagram. We may therefore formalize the relationships of the closed dynamical descriptions of F in terms of its aggregate variables: LEMMA
7.Diagram (D5) is commutative:
(J35)
608
C. J. LUMSDEN
By (49) $5 exists. For any t E [to, tf 1, (* > recovers g from P. By n (56) and (D4) $I exists and takes S(t,J into 3(t). Remark. The full set of dynamical relationships including the probability densities is given by the hourglass diagram Proof.
@P PO01
. P(f)
d
d 1
1
* P(t)
POO) f#JP
(W
P(to)
P(t) (0) t
WI
Since the dynamics of g are closed in terms of involving only measures of the a,-variables, one would begin again with the a-level of dynamical description (56) and seek superaggregate variables B, = B&i 1, . . . , AM), I= 1, . . . , L Q M Q N, and so on through the available levels of dynamical order in F. Lemma 5 establishes the existence of the scaling procedure a2. Lemmas 6 and 7 show that o2 induces a closed dynamics for the aggregate variable values a,,, given the very reasonable assumption that {A,, . . . , AM} is at least a weakly useful description of F. These conclusions suffice to prove Theorem 2. In some circumstances the integro-differential form of 4: will be considerably simplified. If the Km, decay rapidly with time, the (7, will change little
FITDYNAMICALSYSTEMS
609
during the period that the 1Km, ) differ appreciably from zero. Equations (49) can then be well approximated by ordinary differential equations da,,, dt
= -J&(g) + $
n=l
&?&)A,@)
+ C,(t),
m= 1,....
M
(63)
where &&)
(64a)
= SddS K,,(s)
C,(t)s/‘dsg
0
K,,ti3t--s,s)l,
m, n = 1,. . . ,M.
(64b)
n=l
For sufficiently rapid decay of correlations, B,, and C,,, are the order of dt; when they can be neglected in this way $$ is given by the explicitly time-independent system dZ,/drZ-i&a),
m = 1,. . . ,M.
(65)
The size of the subset ss of Famenable to description by systems of the form (63) is not known at present. Earlier results indicate that whenN% 1 and the X*(x) link variables Xj and xk, i # k, in (1) only weakly, approximations of the type (63) may be justified (e.g., Prigogine, 1962). Further characterization ofFs would clearly be useful. It is certain that Fis nonempty (Lumsden, 1982). Fit dynamical systems, which obey a specific rule of adap4. Discussion. tive design, have a hierarchy of dynamical organization when they are complex. This hierarchy (diagram (D6)J contains at least two levels, (4) and (56). The dynamics on each level is specified only by the variables pertinent to that level and, possibly, the time t. In terms of the nomenclature introduced in Section 3, a complex fit system has the dynamical properties of a o’hierarchy . The aggregate dynamics will be detected by all observers interacting with F E 9 by means of Am-measuring instruments. It therefore comprises part of the empirically verifiable repertory of the system’s behavior. When F is complex the A, are generally small on the r timescale and the density &a, t) remains concentrated on the means Z, over some time T such that r Q T’< T. If the P(a, t) variances increase rapidly T - 7’; otherwise T > 7’ and the interval of small variances lasts a finite and possible large time [r’, T S 7’1 (Gortz, 1978). Over [ 0, T] (D6) is valid and an observer at the A-level of system aggregation detects a dynamical ordering of the form (56). It will be of great interest to determine whether analogous principles apply to
610
C. J. LUMSDEN
biological systems governed by fitness rules quite different in their formal structure from equation (2) [e.g., criteria involving explicit constraints among the state variables (Oster and Wilson, 1978; Wagner, 1984), or incorporating hierarchical levels of selection (Gould, 1982 ; Ho and Saunders, 1984)). In a classic paper Simon (1962) argued persuasively that natural selection would favor phenotypes made of hierarchical building blocks. Hierarchical aggregation of subunits (molecules, macromolecules, cells, . . .> is an evolutionarily advantageous strategy allowing incorrectly formed components in a complex system to be rapidly excluded and replaced. All processes of molecular and cellular development are susceptible to some degree of error, and the existence of a simple means of error correction is, arguably, quite sensible. Simon’s principle of hierarchical organization is concerned with levels oi‘ structure rather than function. Function in such systems is realized by the dynamics of the organized structure. It is interesting to find that adapted systems are predicted to be hierarchically organized in their function as well, once they reach a certain degree of complexity. Given the evident advantages attained through a division of labor, in which task-specialized subsystems work together to achieve common goals, the emergence of such organized activity is evidently a natural step in biological evolution. The author is grateful to Bessie Moshopoulos for her careful preparation of the typescript. Financial support was provided in part by the Natural Sciences and Engineering Research Council of Canada and by the Medical Research Council of Canada.
LITERATURE Alexander, R. M. 1982. Optima for Animals. London: Edward Arnold. Allen, T. F. H. and T. B. Starr. 1982. Hierarchy: Perspectives for Ecological Complexity. Chicago: University of ChicagoPress. Arbib, M. A. and E. G. Manes. 1975. Arrows, Structures, and Functors: The Categorical Imperative. New York: Academic Press. Auger, P. 1983. “HierarchicallyOrganized Populations: Interactions between Individual, Population, and Ecosystem Levels.”Math1 Biosci. 65, 269-289. Boorman, S. A. and P. R. Levitt. 1973. “Group Selection on the Boundary of a Stable Population.” Theoret. PopuL Biol. 4, 85-128. Bryson, A. and Y. C. Ho. 1969. Applied Optimal Control. Waltham, Massachusetts: Blaisdell. Cottrell, A. 1977. “Emergent Properties of Complex Systems.” In The Encyclopedia of Ignorance, R. Duncan and M. Weston-Smith(Eds), pp. 129-135. New York: Pergamon Press. Courtois, P. J. 1977. Decomposability: Queueing and Computer System Applications.
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New York: Academic Press. Coxson, P. G. 1984. “Lumpability and Observability of Linear Systems.” J. Math1 Anal),sis Applic. 99, 435-446. Darwin, C. 1964. (1857). On the Origin of Species: A Facsimile of the First Edition. Cambridge, Massachusetts: Harvard University Press. Demetrius, L. 1977. “Macroscopic Parameters and Ecological Systems.” Math1 Biosci. 36, 15-23. Dresden, M. 1974. “Reflections on Fundamentahty and Complexity.” In Physical Realit), and Mathematical Description, C. P. Enz and J. Mehra (Eds), pp. 132-166. Boston: Reidel. Eshel, I. 1972. “On the Neighborhood Effect and the Evolution of Altruistic Traits.” Theoret. Popul. Biol. 3, 258-277. Fetter, A. L. and J. D. WaIecka. 1971. Quantum Theory of Many-particle Systems. New York: McGraw-Hill. Gortz, R. 1978. “A Connection Between Deterministic Motion and the Steady-state Probability Distribution.” Physica 90A, 360-363. Gould, S. J. 1982. “Darwinism and the Expansion of Evolutionary Theory.” Science 216, 380-387. Haake, F. 1973. “Statistical Treatment of Open Systems by Generalized Master Equations.” Springer Tracts in Modern Physics 66, 98-168. Haken, H. 1977. Synergetics. New York: Springer. Hamilton, W. D. 1964. “The Genetical Evolution of Social Behavior.” I, II. J. theor. BioL 7, l-52.. Ho, M.-W. and P. T. Saunders (Eds). 1984. Beyond Neo-Darwinism: An introduction to the New Evolutionary Paradigm. New York: Academic Press. Khinchin, A. I. 1949. Mathematical Foundations of Statistical Mechanics. New York: Dover. Krebs, J. R. and R. H. McCleery. 1984. “Optimization in Behavioral Ecology.” In Behavioural Ecofogy: An Evolutionary Approach, J. R. Krebs and N. B. Davies (Eds), pp. 9 l-l 2 1. Sunderland, Massachusetts: Sinauer. Kurland, J. A. 1980. “Km Selection Theory: A Review and Selective Critique.” EthoZ. Sociobiol. 1, 255-274. Levins, R. 1970. “Extinction.” Ann. N.Y. Acad. Sci. 231, 123-138. Lumsden, C. J. 1982. “The Social Regulation of Physical Caste: the Superorganism Revived.” J. theor. Biol. 95, 749-781. and L. E. H. Trainor. 1979. “On the Statistical Mechanics of Constrained Biophysical Systems.” J. statist. Phys. 20, 657-669. 1985. “Hamiltonian Statistical Mechanics and Biological Order: Problems and -. and Progress.” In Mathematical Essays on Growth and the Emergence of Form, P. L. Antonelh (Ed.). Calgary: University of Alberta Press (in press). __ and E. 0. Wilson. 1981. Genes, Mind, and Culture: The Coevolutionary Process. Cambridge, Massachusetts: Harvard University Press. Maynard Smith, J. 1964. “Group Se&ection and Km Selection: A Rejoinder,” Nature 201, 1145-l 147. Mayr, E. 1970. Popularions, Species and Evolution. Cambridge, Massachusetts: Harvard Univeristy Press. McFarland, D. and A. Houston. 1981. Quantitative Ethology: The State Space Approach. Marshfield, Massachusetts: Pitman. Mesarovic, M. D., D. Macko, and Y. Takahara. 1970. Theory of Hierarchical MuZtiZeveZ Systems. New York: Academic Press, Novikoff, A. B. 1945. “The Concept of Integrative Levels and Biology.” Science 101, 209-215. Oster, G. F. and E. 0. Wilson. 1978. Caste and Ecology in the Social Insects. Princeton, New Jersey: Princeton University Press.
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H. 1973. Hierarchy
Pattee,
Theory:
The Challenge
of Complex
Systems.
New York:
George Braziller.
Prigogine, I. 1962. Nonequilibrium Statistical Mechanics. New York: John Wiley. -. 1980. From Being to Becoming: Time and Complexity in the Physical Sciences. San Francisco: W. H. Freeman. and I. Stengers. 1984. Order Out of Chaos: Man’s New Dialogue with Nature. New York: Bantam Books. Primas, H. 1977. “Theory Reduction and NonBoolean Theories.” J. math. Biol. 4, 281-301.
Rosen, R. 1967. Optimality Principles in Biology. London: Butterworths. -. 1968. “On Analogous Systems.” Bull. math. Biophys. 30,481-492. -. 1976. “Structural Stability, Alternate Descriptions and Information.” J. theor. Biol. 63, 19-31. -. 1977a. “Observation and Biological Systems.” Bull. math. Biol. 39, 663-677. -. 1977b. The Measurement and Representation of Natural Systems, New York: Elsevier. Ruelle, D. 1969. Statistical Mechanics: Rigorous Results. Reading, Massachusetts: Benjamin. Sewell, G. L. 1965. “Quantum-statistical Theory of Irreversible Processes: Dynamics of Gross Variables.” Physica al, 1520-l 536. Simon, H. A. 1962. “The Architecture of Complexity.” Proc. Am. Phil. Sot. 106, 467482. -and A. Ando. 1961. metrica 29, 111-138.
Soong,
“Aggregation
T. T. 1977. “On Model Reduction
of Variables
in Dynamic
of Large-scale Systems.”
Systems.”
Econo-
J. math. Analysis
Applic. 60,477-482.
Vagners, J. 1974. “Optimization Techniques.” In Handbook of Applied Mathematics, C. E. Pearson (Ed.), pp. 1104-l 180. New York: Van Nostrand Reinhold. Vemuri, V. 1978. Modelling of Complex Systems. New York: Academic Press. Wagner, G. P. 1984. “Coevolution of Functionally Constrained Characters: Prerequisites for Adaptive Versatility.” Biosystems 17, 51-55. Weidlich, W. 1972. “The Use of Statistical Models in Sociology.” Collect. Phenomena 1, 5 1-59. Whyte, L. L., A. G. Wilson and D. Wilson (Eds) 1969. Hierarchical Structures. New York: Elsevier. Wilson, D. 1969. “Forms of Hierarchy: A Selected Bibliography.” Gen. Syst. XIV, 3-l 5. Wilson, D. S. 1980. The Natural Selection of Populations and Communities. Menlo Park, California: Benjamin/Cummings. Wilson, E. 0. 1968. “The Ergonomics of Caste in the Social Insects.” Am. Nat. 102,4 l-66. -. 1980a. “Caste and Division of Labor in Leaf-cutter Ants (Hymenoptera: Formicidae: Atta). I. The Overall Pattern in A. sexdens.” Behavl Ecol. Sociobiol. 7, 143156. -.
1980b. “Caste and Division of Labor in Leaf-cutter Ants (Hymenoptera: Formicidae: Optimization of Leaf Cutting.” Behavl Ecol. Sociobiol. 7,
Attu). II. The Ergonomic 1.57-165.
Zwanzig,
R. 1960. “Ensemble
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33, 1338-1341. -.
1961a.
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124,
983-992. -. 196 1b. “Statistical
In Lectures in Theoretical Physics, Methods of Irreversibility.” W. E. B&tin (Ed.). Volume VIII, pp. 106-141. New York: Interscience. RECEIVED REVISED
1 l-S-84 4-30-84
BEHAVIOR IN FIT DYNAMICAL
CHARLESJ.LUMSDEN Department of Medicine, University of Toronto, Toronto, Canada
We consider a class of differentiable dynamical systems that fulfill a criterion of adaptation, or being fit. The adaptation criterion is found to be sufficient for the existence of hierarchical behavior in these systems, once they become complex. Commutative diagram methods are used to show that under reasonable conditions the hierarchical behavior takes the form of a closed dynamics of aggregate variables. This dynamics characterizes the system functions as a whole. Existence conditions for the aggregate dynamics are obtained, together with their differential form and their connections to the original system description. The results indicate that evolutionary processes are likely to produce systems that are hierarchically organized in terms of their function as well as their structure.
Introduction. Darwinian theory (Darwin, 1857; Mayr, 1970) observes that an adaptive fit often exists between organisms and the environments they occupy. Natural selection is viewed as the prime mover shaping organic diversity and phenotypic design. The fit is created as natural selection acts on individuals through their differential survival and reproduction. The winners of the ensuing evolutionary competitions are postulated to be those whose fitness is greatest. In mathematical versions of the theory, fitness is typically a function that integrates reproductive success with survival likelihood over an individual’s lifetime. The concept can be extended to cover genes that related individuals share by common descent (Hamilton, 1964; Maynard Smith. 1964: Kurland, 1980), and to selection processes that act on groups, societies, and even entire populations (Levins, 1970: Eshel, 1972: Boorman and Levitt, 1973; Wilson, 1980). Substitute measures of quantitative fitness (such as physiological efficiency) more amenable to observation than net reproductive success are often used in modeling (Rosen, 1967; Oster and Wilson, 1978). Fit organisms correspond to the phenotypes or gene variants at which the fitness functional takes on its maximal values. In this report we are interested in relating adaptive design to hierarchical organization. Rather than structure, the focus will be on the dynamical order characterizing the phenotype. A class of differentiable dynamical systems will be introduced for this purpose. Its members are naturally described as 1.
591
592
C. J. LUMSDEN
fit in the sense that they obey a principle of adaptive design. An approach to hierarchical organization is then presented that scales among levels of dynamical order. Commutative diagram methods are used to show that such levels of order exist in fit systems once they attain a certain degree of complexity. Complexity is measured merely by the number of independent state variables (equivalently, the number of differential equations) required to describe the phenotype. Thus the fact that a complex system is the product of adaptive design suffices to establish a degree of hierarchical order within its dynamical behavior. The notion that biological systems have a hierarchical design, with many interconnected levels of structure and function, has a long and productive history (Novikoff, 1945; Wilson, 1969; Pattee, 1973). There is an extensive literature deaiing with biological complexity and hierarchical order in a qualitative way (e.g. Wilson, 1969; Whyte et al. 1969; Dresden, 1974; Cottrell, 1977; Allen and Starr, 1982). In addition, there has recently been interest in exact mathematical models of biological processes that have multiple levels of organization. Some investigations have concentrated on making rigorous the meaning of multiple alternative descriptions (Rosen, 1968; 1976; 1977a, b; Primas, 1977). Others have analyzed the types of hierarchical order present in specific models (Simon and Ando, 1961; Simon, 1962; Wilson, 1968; Mesarovic, Madco and Takabarz, 1970; Weidlich, 1972; Demetrius, 1977; Haken, 1977; Vemuri, 1978; Lumsden and Trainor, 1979; Lumsden and Wilson, 1981; Lumsden, 1982; Auger, 1983). The increasing number of studies of the second kind raises an interesting question: what can be said in formal terms about the hierarchical organization of dynamics as an adaptive trait in itself? The purpose of this study is to establish the existence of a connection between fit design and this form of organization. The models to be used in the investigation are defined and their dynamics specified in Section 2. Section 3 introduces a scaling procedure and uses it to relate the system dynamics to the organization of its aggregate variables. Theorem 2 establishes the connection linking adaptation to multilevel ordering of the fit dynamics. The mathematical findings are then discussed in the paper’s concluding section. We consider a system 9 with state variables xk E w, k = 2. Fit Systems. I N characterizing its morphology, behavior, and consequences of this * * 7 behavior (e.g., proportion of body fat, number of matings per unit time, number of offspring produced per unit time). Depending on the traits of relevance, the same state variables might be used to compare members of different biological species, or their application might be restricted to deal
FITDYNAMICALSYSTEMS
593
with intraspecific variation. Activity of Ywithin its environment is characterized by temporal changes in xk. The dynamics of the state vector x=(X1,. . . , xN) follow the first-order differential equations dx, -ZX dt The the For tion
.
k=Xk(x,
t),
k= I,...
.N.
form of the Xk for systems of biological interest ultimately expresses interaction of genotype and environment in shaping the phenotype. the systems relevant to the present investigation the principal restricon the .& is smoothness of a minimal sort, modeled by @“-continuity on
RN+‘. x:lRN+‘+lR~:(x,
t) p
x
where t E W denotes the time variable and we adopt the mapping conventions of Arbib and Manes (1975). The equations (1) induce the flow 4 for Ythrough the configuration phase spaceRN: $:RN+r+RN:
(x,, to) I+ x(t 1x0, to).
For every # on R N+’ induced by equations (1) we define the reward function or fitness as a scalar functional J that integrates over the life history [to, tf ] of9 J=
tfG(x, x, t)dt, I to
(2)
where G(O) is the reward rate on [to, tf I. The reward rate of particular interest in population biology is the net reproductive success of SPper unit time, or (more commonly) a phenotype variable believed to determine a significant proportion of the variance in this fitness measure. The choice of G(*> is influenced by the type of phenotypic designs to be compared and the envisaged nature of their variation under genetic change (Rosen, 1967; Oster and Wilson, 1978). For example, biological organisms have a finite reproductive success, and generally G( ) is constructed such that J < UJfor each3 Similarly, among the possible G(O) are only those with support in the subsets of pertinent initial states. Conversely, not all formally conceivable initial states x(t,) will enter into the determination of a G( l ) [e.g., ant societies based on claustral founding always start from x(to) = 1 individual, the fertilized queen, irrespective of other possible modes of variation (Oster and Wilson, 1978; Lumsden, 1982)]. In view of these considerations we introduce the following definition. Definition. The differentiable system 9 is a fit system F iff for each x(t 1x0, to) solving (1) the fitness J is maximal on the Life history [to, tf ] : l
594
C.J.LUMSDEN
= max.
J[x(*)]
(3)
The set of all such fit systems will be denoted byP. Remark. This definition is clearly over-inclusive in the sense that all systems (1) for which (3) is true are admissible fit systems, and this set may include elements that are not biological in any reasonable sense. However, because processes of competition coupled with differential (re)production influence a broad range of natural and man-made systems, in a manner often resembling classical natural selection, it is useful to relax the notion somewhat and for the time being include systems of the latter types within 9. Equally intriguing is the question of whether 9 includes any biological systems at all, or whether near-optimally designed phenotypes rarely appear during biological evolution. Our definition would then be too restrictive for biological purposes. Although much work remains to be done on this issue, the available evidence from studies of physiological (Rosen, 1967; Alexander, 1982) ethological (McFarland and Houston, 1981; Krebs and McCleery, 1984), and societal (Wilson, 1980a, b) design indicates that Scontains some (but by no means necessarily all) real biological systems, and that nearoptimal phenotypic traits may be rather more common than previously supposed. The flows $ in which equation (3) is true follow from the variational principle rfG(x, i, t)dt = 0. s to
6J=S
(4)
In differential form this condition is expressible as the control equations -aG ---=d aG dt a.& aXk
0,
k=
l,...,N
(Bryson and Ho, 1969; Vagners, 1974). It is useful to transform symmetric notation in which costate variables h&6x/8X&,
k=
i.,...,
N
(5) a more (6)
replace the x&s. Using the generating function Q=G-5
h&x& k=l
(7)
in the fitness principle (3), one obtains from the control equations (5) the costate dynamics dXk/dt
=-aQl&Yk,
k = 1,. . . ,N
(84
FITDYNAMICALSYSTEMS
595
as well as the state variable equations dx-,/dt = aQ/ah,,
li = I,. . . >N.
(8b)
The natural canonical or Hamiltonian structure of equations (8) follows from the variational structure of the fitness problem (3) and the use of costate variables to express the control equations as a first-order differential dynamics. Unlike physical systems, whose microscopic dynamics are also essentially canonical, fit systems are not restricted to reward functions G( l) that have the form of an energy function. This flexibility opens the possibility of new and unexpected behaviors from a physical point of view. Moreover, treatments of physical systems far from thermodynamic equilibrium have demonstrated that even rather conventional energy-Hamiltonian systems can exhibit entirely novel models of spatio-temporal organization relative to the characteristics of near-equilibrium (Prigogine, 1980; Prigogine and Stengers. 1984). The present development will not restrict the fitsystem dynamics to orders analogous to ‘at’ or ‘near’ some condition of aggregate equilibrium. For some applications it is advantageous to incorporate constraints among the state variables into the explicit statement (3) of the fitness problem. The resulting control equations preserve the basic canonical structure of equations (6)-(g), with modifications introduced by the presence of the constraining conditions (Bryson and Ho, 1968; Oster and Wilson, 1978). Methods treating the collective dynamics of constrained systems have recently become a subject of analysis and application (Lumsden and Trainor, 1979? 1985). In this study we will assume that any constraining conditions among the state variables have already been incorporated by eliminating variables (1) and appropriately rewriting equations (l)-(3). When N %=1 in equations (l)-(8) it becomes useful to introduce variables that characterize the system, or large parts of it, as a whole. The original description in terms of (x, x) or (x, h) contains an abundance of detail. Dejhitiorz. The %’ map
A,:lRZN+ lR:(x, A) PA,(x,
h) ER,
(9)
is an aggregate variable of F iff dim [dom (A,)] /2N % 1. In other words, A, depends on most or all of the state variables xk or state and costate variables Sk> hk: k = l> . . . ) N. Remark. A system may possess a number of aggregate variables A,, 111= 1, . . , > M relevant to a particular set of properties, so we introduce the aggregate descriptiolz d s {A, I nz = 1, . . . , M}, along with the following definition. Dejkitiorl. An aggregate description dis interesting iff M Q N.
596
C. J. LUMSDEN
Interesting aggregate descriptions characterise F by means of relatively few relevant variables A,. They are economical relative to state-variable descriptions in terms of the xk and h k. In biological applications the A, represent such important concepts as the local concentration of a biochemical species (sum over many molecules), the number of individuals in a population (sum over many individuals), and net matter-energy fluxes though trophic networks (sum over activities of many producers and consumers). The best-understood aggregate variables are in fact in the form of simple sums over the state and costate variables. The set J$ is a subset of 9, the set of all observables f:lRvv + lR defined on (1). A weak criterion for %?, sufficient for the present study, is that it comprises the set of all %‘I complex functions on lRzN which vanish at infinity. The behavior at infmity models the finite capacities of biological systems; formally, it allows meaningful expectation values to be constructed. Since lR2N is locally compact, relative to the sup norm llfll ~,~;2JfoI
(10)
and the involution f t+ f*, * G complex conjugation,@ is an Abelian B*algebra over the system phase space lRZN (Ruelle, 1969). Through d the flow 4 in lRN (equivalently, IR2N) connects the aggregate variables at t,, with their values at any t E [to, tfl . 9
I XI), to>
a$----
x0
d
54 I A@ I At,, to>
I AO,)
(Dl)
If there exists a t-autonomous where A(t) G (A r(t), . . . , AM(t)). differentiable flow GAon RM such that # x(to)------+
x(t I x0, to>
Nto)
A@IAo, to)
-
V1-
@A
with d* $A = 9 at we say that &f?constitutes a closed aggregate description of F. The corresponding differential equations can be written as l
k,=Y,(A,
)...)
A,),
m=
I)...)
M.
(11)
FITDYNAMICALSYSTEMS
591
For fit systems that admit closed aggregate descriptions the system
+ aA?3a9 --=
ahk axk
Ym(A1,.
.
. ,A,),
m = l,..
. ,M(12)
is solvable. On the basis of present knowledge it appears that pairs (Y,4qd) for which (D2) is true are either rare or quite difficult to find (e.g., Coxson, 1984, p. 435). Commonly there is some leakage from the x-level to the A-level that prevents closure in the sense of(D2) (see below). All fit systems are, however, A-closed in the following sense. THEOREM 1.Let p(x, h) be an element of Xw, the set of integrable, unitnormaIizable (‘probability ‘) functions in R 2N.Xw is a convex subset of 4V’. the dual of the algebra of system observabIes S (Ruelle, 1969). Let P(a) EX”, where ,XM is defined similarly to X2N. Then there exist Gp and q$ such that the diagram (03) commutes.
(D3)
P(a, toI
d?P
b
P(a, t).
Remark. The operation sdenotes construction of the probability distribution P [system ensemble (Lumsden and Trainor, 1979)] on lRM for a given specification of aggregate variables. Notation. As context warrants we shall write equivalently p(x, X, t) G p(x, X) = p(t) for the value of p at time t. We prove Theorem 1 in several steps.
LEMMA
1.In the (x, h) variables the differential form of & is apiat = -i.C$p
(13)
where
(14)
598
C. J. LUMSDEN
Proof. Since
s
dx dX p(x, X, t) = 1
RZN
vt
1,151
p(t) has a local balance equation
(16) (Lumsden and Trainor, 1979). Using equation (8),
+I:
N --------_oo. ap39 ap w kzl
axk ahk
ahk axk
(17)
The generating function Q(x, X) is X-linear and the formal difference a2QjaXk a& - a2Q]&aXk
(18)
n vanishes. When .Y is explicitly independent of time, the propagator T(tl to) that takes p(x, h, to) into p(x, A, t) for t > to is by operator integration
T(t
1 to) = e-t(r-rO)~
(1%
where @p: p(to) 5 T(t I to) : p(to) /-+p(t) = e-‘(r-ro)9p(to).
(20)
Explicit time dependence may occur in Ywhen the fit system F is coupled to an external environment or in the presence of certain forms of dissipation (Lumsden and Trainor, 1979). This inclusion requires the use of timeordering operators (Fetter and Walecka, 1971) and makes the development more cumbersome, but entails essentially no new formal results. To keep the presentation streamlined we treat the case of explicit time-independence, and without loss of generality set co = 0. LEMMA 2. The constructive procedure 5 exists for diagram (03). Proof. She A,,, = A&x, X), m = 1, . . . , M, an A, contour with value a, is a hypersurface in lR2N. Intersection of the hypersurfaces A,(x,X)=a,(t),
m = 1,. . . ,M
(21)
FITDYNAMICALSYSTEMS
599
generates at any time t a 2N -M-dimensional subset IYa,. . . . , uM) - I’(a) C R2” consistent with A having value a(t) - @r(t), . . . I a,~&t>). Define P(a,t)da~Prob(a,
that a,,
(a) t = I& left side of(D3).
the value of A,? lies in [a,! a,
(22) + da,]
for
The initial aggregate state is aoE a(t = to).
(23)
Then
JYa,to> =
fiS(u, - ~0~)
(24)
m=l
while in lR 2N. (x0, ho) E l?(ao). Since no additional information tion of F in lR2” at t = to is given, p(to) is uniform on r(ao): dto) = WA(x, A) - ao)WA’(ao)
on the loca(15)
where Wa0>
=
s
2Ndx dX 6(A(x. A) - ao),
(‘6)
a hypersurface measure of F(a) in RZN (Khinchin, 1949). This completes the left side of (D3). fb) t > to riglz t side of (03). Consider p = p(t > to). Then p(t)
= T(t I to)p(to)
= e-w(r-ro)p(to).
(27)
By definition P(a, t) dt gives the likelihood of finding the system with a E [a, a + da] at t. P(t) must therefore equal the total probability density p(x, h. t) with support r(a), where a = a(t). Thus
Ra, t) = =
jR’”dx dh &(A -
IIR 2N
a)p(x. h, t)
dx dX 6(A - a)e-i(f-fo)y 6(A - ao).
(28)
These considerations show that the entire set tiN of admissible p(x, h! ty is not pertinent to A-level descriptions of the system. It is sufficient to consider the subset X:” C x 2N that contains only those p with functional dependence of the fonn
600
C. J. LUMSDEN
(29)
P(X, N = PL-UX,Nl .
Xi” can be characterized by the action of a map 9 that extracts from a state function f(x, X) that part, f’ =Yf, dependent
(30)
on (x, X) via the A(x, h). By definition ofX
y,
pf = W” [AllRzN dx’ dh’ 6 [ A(x’, A’) - A(x, h)] f (x’, X’)
(31)
where SPf
= Pf, g2 =s?
(32)
LEMMA 3 (Zwanzig). Let tip be a dynamics of form (13) for p(t). Th!n @, has the equivalent differential form iap’ja t =92zp’(t)
(33) where
P’W =W(t) p”(t) = (I-9)p(t)
(34)
with to = 0 and I the identity. Proof. Since 9is linear and explicity time independent 9$=&9?
(35)
Thus iap’/& =@sp’
+ p”)
iap”/at = (I -g)Y(p’
(36a) + p”).
(36b)
The operator (I - 9) projects onto the subset X2N - Xy : 9 (I - 9 ) = g2 - 9 = B - 9 = 0, and equation (36b) then formally integrates to p”(t)
= e+t(Z+W
p”(O) - i
’
s0
ds e-Is(z-@y(I -g)_@‘( t - s)
(37) n
by Laplace transforms. Using equation (37) in (36a) gives the desired result.
FIT DYNAMICAL
SYSTEMS
601
Remark. The form (31) 9 and its use to develop the expression (33) for the projected density p’ were first documented by Zwanzig (1960, I96 1a, b).
In this study his method wiIl be used to help establish the commutativity of the diagrams [Di] linking the descriptive levels of F. Remark. The dynamical content of equation (33) is partly evident from its structure. The first RHS term gives the Markovian part of the p’(t) motion caused by contributions that remain in e” for 0 < s < t. In the second RHS term, (I - 9 )UP,p(t - s), is a flow of probability from Xp into XzN - Xim at prior times (t - s). The factor e-ir(lFb) y maps this probability forward in time while keeping it confined to X 2N - %fN. 96p then determines the effects of this leaked probability density on the p’-motion. The time integration sums leak these effects over the time interval that the probability contribution is in Xuv - XT”? The third RHS term represents the influence of that part of p(t) which fell in 3yzN - Xy at t = 0.
COROLLARY
1. p”(0) = (1 --B)p(O) = 0.
(38)
Proof. Since F is describable by the initial aggregate state (23), condition a (24) implies that ~(t = 0) EXfN. Although the projected density p’(t) relates the dynamics (1) of F to the aggregate variables Am, the connection is highly implicit. An explicit representation is guaranteed through the following
4.Let p(t) EpN have a p’(t) EXy that obeys equation (33). Then for P(a, t) EX”, #p in (03) exists and is given in differential form by
LEMMA
!$a,f) = -m$l $ [v, WYa, t>1 m
(3%
such that v,(a) =
K,,(a,
a’, s) = (kme-‘s(z--S?p(I --B)i$(A
(40a) - a') 1a>,
(40b)
where (* I a> is a linear functional on@ (* Ia):¶V+lR:
f(x,h>/-+(f(x,h)Ja)EW.
(41)
Remark. We note that equation (39) is a closed-form relation for P(a, t), expressed in terms of the aggregate variables.
602
C. J. LLJMSDEN
PPOOJ:From (28) and the definition of9 W-l(a)P(a, t) = p’(x, h, t)
(43)
A(x,A)=a’
Differentiate (42) with respect to time and, use the differential 9, from (33); then W-‘(a)$P(a,
t) = -ij+
form of
da’ W-‘(a’) tJfG(A - a) I a )&a’, t) RY
-
s2ss o
da'
W-‘(a’) IUe -ir(zmB)y(J-
9)lps(A
-a’)1 a>P(a’, t -3)
lRM
where forf(x,
(43)
A) E4
h) I a> si W-1(a)JR2H dx dX 6[A(x, h) - al f(x, h).
(44)
The action of (* la> is to average an observable over the cell r(a) C R*? Now M a U’6(A - a) I a> = -i W-l(a) x -[WW(a)v,(a)G(a m=l
- a’)1
(45)
aam
and W’e-*(z-*9(I
s
-P)LbS(A - a) I a) =
W%‘)m$l ngl &$
da’
RM
[ W(a)K,,(a,
a’, s) 1 W-l(a’)Pta’, t -s).
n
(46) Equation (39) follows from an integration tions of the result and (45) into (43). Remark. The identity les[A(x,h)-al
of (46) by parts and substituW
M a
=-im~l~6(A-a)--$ m
dA
(471
(Sewell, 1965) is useful in constructing identities such as (45) and (46). Lemmas l-4 establish that p(t) and P(t) are linked as indicated along each side of diagram (D3). Theorem 1 is therefore established. The arguments leading to Theorem 1 are an example of system aggregation, whereby certain properties characterizing the components of a system are demonstrated to support one or more interesting aggregate descriptions.
FITDYNAMICALSYSTEMS
603
This descriptive process will be taken further in the next section and related explicitly to notions of hierarchical organization. In concluding the present section it is useful to note that in other, somewhat simpler, systems algorithms for aggregation are known (cf. discussion in Section 1) and can be compared to the present schema without resorting to a complete formal derivation. For the best-understood cases a natural linearity in the dynamical problem facilitates the argument. If in system (1) the Xk are all linear combinations of the state variables, Xk = ZjCkiXj, then for aggregation in the sense of diagram (D2) it suffices to find a linear map that takes the matrix of coefficients Ilcki 11 into a matrix of much-reduced dimensionality. Conditions for the existence of such maps have been established (Soong, 1977). Sometimes the system possesses intrinsically stochastic properties that can be better modelled with master equations of the Pauli type, rather than with differential equations. The Pauli equations link the system states via a matrix of transition probabilities per time step (discrete time scale) or of transition rates (continuous time), and again comprise a linear description (Haken, 1977). Aggregation then seeks to combine the states into state-sets, between which transitions again occur in a stochastic fashion, governed by relatively simple master equations. Conditions allowing aggregation in these stochastic systems are readily seen in physical terms: the set of all states must divide naturally into subsets containing states linked by high transition probability, such that the likelihood of transition between the subsets is much smaller. The Simon-Ando theorems and subsequent developments (Simon and Ando, 196 1; Courtois, 1977) provide formal criteria which transition matrices must fulfil in discrete time or continuous time in order for this aggregation to be possible. Theorem 1, by comparison, applies to systems best described in terms of a continuous time variable, and which are deterministic in their microscopic specification (1). The systems can, and in the cases of greatest biological interest will, involve nonlinear dynamics. They are not restricted to linear differential structure. Out of this class of nonlinear systems, Theorem 1 deals specifically with those designed according to a fitness principle of the form (3), rather than with a subset constructed on the basis of analytically convenient characteristics such as linearity or Pauli form. 3. o-Hierarchies. As functions of many variables xk, hk, the time-rates Of change Of the A,,, hVOhe allthe r&S ik, ik : (48) In general the ,k and ik will be of VatiOUSmagnitudes and signs for any t > tW If this is the case the sums (48) will smooth over changes on the
604
C.J. LUMSDEN
timescale of microvariable dynamics. The aggregate variables A, change relatively Slowly on the timescales of xk and hk. Definition. Let A, f IR and all xk and hk change by a what are agreed to be ‘significant amounts’ A,, v = 1, . . . , 2N. Let T, be the time t - to required for the change A,,. Then r=maxr, V
is said to be a characteristic time scale for change in the (x, X)-variables. Definition. Let &,, E W and all aggregate variables A, change by what is agreed to be ‘significant amounts’ .& m = 1, ...,M. Let 7; be the time t - to required for the change t,. Then
is said to be a characteristic time scale for change in the A-variables. Observa?ion.When N > 1, T Q 7’. If F E Scpossesses two such well-separated time scales one suspects that it might be possible to remove the residual references to (x, A>variables entirely from the A-level description, at least in approximate terms. This motivates the following definition. Definition. Let Q be a scaling procedure based on r and T’, where r 4 7’. If u maps (pP into a closed dynamics $A of aggregate variables on lRM we say that F is a u-hierarchy or u-hierarchical system. If closure is true at most to order K of a perturbation parameter in u, then F is a UK-hierarchyor #-hierarchical system. The properties of F as a system with hierarchically organized dynamical properties are characterized by the following theorem.
THEOREM 2.A fft dynamical system with T 4 T’ is a &hierarchy. In order to establish Theorem 2 we will require some results on the properties of W(a, t)/at [equation (39)] under the scaling operation u.
LEMMAS.~
03
M a $Ya, 0 = -mTl-[urn b)P(a, 01 aa
m
Proof. In a timescale relationship like T Q T’ the rates A, are small on the timescale for change in the & and &. Let us consider a scaling method that
FITDYNAMICALSYSTEMS
605
uses perturbation terms to approximate v,,, and K,,,,,. The approximation becomes exact in the limit A,,, + 0 V m. The kinetic coefficients v, are explicitly f”lrst order in the small quantities A,, while the K,, contain the A, explicitly to second order. Define u2: Expand all terms in (49) in A, and retain these rates to second order. Operating on an observable F(x, A) with the projector 9retains only the of F, while using 2’ on a 9F brings out a factor of A, Applying c2:
A(x,h)-dependence
e-rs(z-@2(1 -9)
= (I -9)
- is (I -9)
_9 ( I-9)
+ 4 (is)2(l -9)14(1-P) S (I -9)
-
js(l -S)Y+
9(1 -9) f
+...
(is)2(f -9)Ie2
+ ,..
= (I -9)e+"Y
(50)
Similarly,
upon expanding the exponential
and using equation (47). Then
Km, (a, a’, s) S 6(a - a’)[ti,e-‘SyAn
la> - (A, la)(e+4in
la>1 + 0(A3> (52)
Writing (em%&
I a) =
(53)
we see that Kmn(a, a’, s) 2 6(a - a’)K,,(a,
s)
(54)
where K,, (a, s) E
la> -
bi,(O)A',(-s) Ia> -
(A, (-s) la> I aHA,
C-s) Id.
Wa) (55b)
s) evidently has the form of a correlation function between the aggregate variables A, and A,. Equation (54) can now be used to carry out l the integral over a’ in (39). The result is equation (49). Remark. In the case that $J induces sufficiently ‘statistical’ wandering in the state-space path of F, correlations between the A, decay quickly on the time scale r’ and the functional &,(a, S) can be modeled by a rapidly (typically exponentially) decreasing function of time (Prigogine,
Kmn(a,
C. J. LUMSDEN
606
1962). The relevance of such properties for the hierarchical organization of F will be discussed below. It now remains to construct a relationship between P(a, t) and the rate laws for the aggregate variables. LEMMA 6. Let ii,(t)
be the expectation value for A,,, under P(a, t). Suppose that the aggregate description {Al, , . , , AM} is at least weak@ useful in that F responds in a similar way each time it occurs in the same initial aggregate state A(t,). Then in a2 the Z,(t) obey the&l -4 N integro-differential equations
g&l = -v,(ii,> dt
+s2sxlKm,E(t - s>,sl M
0
A,
[a0 - s)l
+
$-Kmn [a(t -
s), S]
n
n=l
1 (56)
where A, (a) = Proof.
$
(57)
In W(a). n
By definition c?,(t) =
s
daa,P(a, t) =
lRM
s R2N
dx dhA,(x, h)p(x, A, t>,
(58)
so that
dam_ dt-
s
dala,;:P(a, lRM
t>l.
(5%
Using Lemma 5, integrating the vm terms by parts once and the Km, parts twice gives after rearrangement darn -=-
dt
s
da v,(a>P(a, t>
RM
+ Jdds/_Mda$
W-‘(a)& n=l
[K,,(a,
s)W(a)l P(a, t - s).
(60)
n
Since the description {AIT . . . , AM} is at least weakly useful it must be that P(a, t) develops from its initial state P(a. to) in such a way that it remains sharply concentrated about the means G,(t). If this were not the case, the variances of P(a, t) would increase significantly with time and the observed
FITDYNAMICALSYSTEMS
607
a-values would not closely repeat. Instead, they would drift between repetitions in which F is in A at t = to and the description {A r, . . . , A,} would not be at least weakly useful. For such descriptions
I
RM
@la)
da rj,(a)P(a, t) s v,[ii(t>l
~~‘(a)$ilU,,(a, sW(a)lP(a, f- s> n
2 5
IV-r[ti(t -s)]$&Jr(t
-s),
s]Wl$t--)I}.
n
(61b)
?I
?I=1
The combined action of the scaling procedure u2 and the averaging operation over P(a, t), which we denote in general by (0 >,
(9
:gfaWgT;;)$RM dag(a)P(a,0,
(62)
is to induce a map x relating the P-dynamics (39) through its o-image @f: [equation (49)] to @$ the motion of the expectation values ii(t) [equation (56)l: fP @p,
*
4;:
(D4) a commutative diagram. We may therefore formalize the relationships of the closed dynamical descriptions of F in terms of its aggregate variables: LEMMA
7.Diagram (D5) is commutative:
(J35)
608
C. J. LUMSDEN
By (49) $5 exists. For any t E [to, tf 1, (* > recovers g from P. By n (56) and (D4) $I exists and takes S(t,J into 3(t). Remark. The full set of dynamical relationships including the probability densities is given by the hourglass diagram Proof.
@P PO01
. P(f)
d
d 1
1
* P(t)
POO) f#JP
(W
P(to)
P(t) (0) t
WI
Since the dynamics of g are closed in terms of involving only measures of the a,-variables, one would begin again with the a-level of dynamical description (56) and seek superaggregate variables B, = B&i 1, . . . , AM), I= 1, . . . , L Q M Q N, and so on through the available levels of dynamical order in F. Lemma 5 establishes the existence of the scaling procedure a2. Lemmas 6 and 7 show that o2 induces a closed dynamics for the aggregate variable values a,,, given the very reasonable assumption that {A,, . . . , AM} is at least a weakly useful description of F. These conclusions suffice to prove Theorem 2. In some circumstances the integro-differential form of 4: will be considerably simplified. If the Km, decay rapidly with time, the (7, will change little
FITDYNAMICALSYSTEMS
609
during the period that the 1Km, ) differ appreciably from zero. Equations (49) can then be well approximated by ordinary differential equations da,,, dt
= -J&(g) + $
n=l
&?&)A,@)
+ C,(t),
m= 1,....
M
(63)
where &&)
(64a)
= SddS K,,(s)
C,(t)s/‘dsg
0
K,,ti3t--s,s)l,
m, n = 1,. . . ,M.
(64b)
n=l
For sufficiently rapid decay of correlations, B,, and C,,, are the order of dt; when they can be neglected in this way $$ is given by the explicitly time-independent system dZ,/drZ-i&a),
m = 1,. . . ,M.
(65)
The size of the subset ss of Famenable to description by systems of the form (63) is not known at present. Earlier results indicate that whenN% 1 and the X*(x) link variables Xj and xk, i # k, in (1) only weakly, approximations of the type (63) may be justified (e.g., Prigogine, 1962). Further characterization ofFs would clearly be useful. It is certain that Fis nonempty (Lumsden, 1982). Fit dynamical systems, which obey a specific rule of adap4. Discussion. tive design, have a hierarchy of dynamical organization when they are complex. This hierarchy (diagram (D6)J contains at least two levels, (4) and (56). The dynamics on each level is specified only by the variables pertinent to that level and, possibly, the time t. In terms of the nomenclature introduced in Section 3, a complex fit system has the dynamical properties of a o’hierarchy . The aggregate dynamics will be detected by all observers interacting with F E 9 by means of Am-measuring instruments. It therefore comprises part of the empirically verifiable repertory of the system’s behavior. When F is complex the A, are generally small on the r timescale and the density &a, t) remains concentrated on the means Z, over some time T such that r Q T’< T. If the P(a, t) variances increase rapidly T - 7’; otherwise T > 7’ and the interval of small variances lasts a finite and possible large time [r’, T S 7’1 (Gortz, 1978). Over [ 0, T] (D6) is valid and an observer at the A-level of system aggregation detects a dynamical ordering of the form (56). It will be of great interest to determine whether analogous principles apply to
610
C. J. LUMSDEN
biological systems governed by fitness rules quite different in their formal structure from equation (2) [e.g., criteria involving explicit constraints among the state variables (Oster and Wilson, 1978; Wagner, 1984), or incorporating hierarchical levels of selection (Gould, 1982 ; Ho and Saunders, 1984)). In a classic paper Simon (1962) argued persuasively that natural selection would favor phenotypes made of hierarchical building blocks. Hierarchical aggregation of subunits (molecules, macromolecules, cells, . . .> is an evolutionarily advantageous strategy allowing incorrectly formed components in a complex system to be rapidly excluded and replaced. All processes of molecular and cellular development are susceptible to some degree of error, and the existence of a simple means of error correction is, arguably, quite sensible. Simon’s principle of hierarchical organization is concerned with levels oi‘ structure rather than function. Function in such systems is realized by the dynamics of the organized structure. It is interesting to find that adapted systems are predicted to be hierarchically organized in their function as well, once they reach a certain degree of complexity. Given the evident advantages attained through a division of labor, in which task-specialized subsystems work together to achieve common goals, the emergence of such organized activity is evidently a natural step in biological evolution. The author is grateful to Bessie Moshopoulos for her careful preparation of the typescript. Financial support was provided in part by the Natural Sciences and Engineering Research Council of Canada and by the Medical Research Council of Canada.
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