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Hyperbolic law of reliability and its logarithmic effects in ecology
Ecological Modelling, 55 (1991) 75 88
75
Elsevier Science Publishers B.V., Amsterdam
Hyperbolic law of reliability and its logarithmic effects in ecology B.S. Fleishman Institute of Oceanology of the USSR Academy of Sciences, Moscow, USSR (Accepted 30 May 1990)
ABSTRACT Fleishman, B.S., 1991. Hyperbolic law of reliability and its logarithmic effects in ecology. Ecol. Modelling, 55: 75-88. The paper deals with the exponent of hyperbolic decrease in a regenerative system's reliability as the function of the similar exponent of its nonvitality. Ecological bases for the hyperbolic law of the decreasing nonvitality of the system are provided. Stochastic models of the system's death at metabolic and structural levels, as well as a general model of its death are considered. The minimum order of the increase in the number of the system's elements as function of its age is found to result in its reliability arbitrary close to unity. This order is of logarithmic and logarithmic-power nature. Biological effects of the regularities so discovered are dicussed.
INTRODUCTION
In his previous papers (Fleishman, 1980, 1982, 1984) the author developed the theory of adaptation of regenerative systems Aab in a stationary regime and also estimated the time of the transient conditions. This time can be of the same order as the life span of the system TA which is related to its reliability. In view of this, the present paper considers the 'juvenile' period in the life of the regenerative system related to the transient period, as well as the conditions under which it can survive till its 'mature' age while passing into the stationary regime. These conditions require at least a logarithmic increase in the number of its elements. One may think that this result contradicts the familiar logistic law of the increase in biological systems. However, if we discriminate in the logistic curve striving to make constant its asymptotic part following the exponential one, the former will prove to be quite comparable with the logarithmic dependence at a limited interval of the system's life. 0304-3800/91/$03.50
© 1991 - Elsevier Science Publishers B.V.
76
B.S. FLEISHMAN
R E L I A B I L I T Y AS R E L A T E D T O V I T A L I T Y A N D T H E G E N E R A L MODEL OF VITALITY
ECOLOGICAL
In ecology there is a concept of instantaneous probability /~(s) of the biological system's death at m o m e n t s as well as that of the probability P ( t ) of its survival till age t (Pianka, 1978). In the theory of potential efficiency of complex systems, the probability 1 - / ~ ( s ) [Ms)] is called vitality [nonvitality] and the probability P ( t ) is called reliability. The following dependence has been found between them (see Appendix 1): If:
ii(s)<<.es -2
fors=l,2
..... t
(1)
then
P ( t ) > 1 - 2e
for t = 1, 2 . . . .
(2)
where ~ is an arbitrary positive quantity. If /_t(s) > es -1 and the deaths at different moments are independent, then (Fleishman, 1971):
P(t) The hyperbolic decrease of ~(s) and P ( t ) are known from a number of ecological papers. However, the relation between the exponents of the respective hyperbolas were not known up to now. The relations (A.1.2) and (A.1.3) reveal the hyperbolic decrease in the system's reliability resulting from the hyperbolic decrease of its nonvitality as a function of its age. The latter decrease is due to the system's adaptive improvement domination over the environmental pressure with the system's age. The interaction between these factors can be found in its most general form by assessing the death probability of the system's nonvitality due to numerous external causes. Let us assume that at the m o m e n t s the system is exposed to the N, of potential causes of the death, the danger of each cause being described with the separate probability xsj ( j = 1, N,). We'll say as well that the system "is under pressure of N, of possible death causes". Then, taking in mind the ecological fact that the system's death cannot be repeated, in the asymptotic case of N, ~ ~v the following relationships will take place (see Appendix 2). If the average probability Ys of the system's death due to N, reasons is sufficiently low: 2s zx ]
N~ x,j < CN, -<~+q') Y'~ j=l
(3)
HYPERBOLIC
LAW OF RELIABILITY
77
then the system's nonvitality will have its upper estimation: i x ( s ) ~ N s Y s = C N ] -~"
(4)
where C and q, are some positive constants. It is natural to suggest that, in the course of the system's life, the number of causes of its possible death increases. The linear dependence N s = C ' s is substantiated in Appendix 2. In general, one can admit the power dependence: (5)
U s = C ' s 4"
where C' and ~p are some positive constants. Substituting expression (5) into estimation (4) and relating it with condition (1), we obtain the expression: Ys < ( e / C ' )
s-{2+'~)
(6)
METABOLIC A N D STRUCTURAL LEVELS OF SYSTEM A a b
In order to construct stochastic models of the dependence between the reliability of the system mal, and the number of its elements, it is necessary to provide a deeper ecological analysis of the internal causes of its death. The generative system Aab = {A~}, as a supersystem, is composed of systems A~ which are vitally important for it. Each of the systems A~ = { eT} is structured, that is it contains the quantity m '~ of elements e 7 (i = 1, m ") and a certain number of links between the elements (the structural level). Each of the elements e 7 = {e,~} is non-structured, containing the number n~ of homogeneous replaceable primary elements e;'~ ( k = 1, n~) (metabolic level). Let us now interpret the supersystem Aab firstly in terms of biology. It can be interpreted first of all as an organism (individual) composed of the n u m b e r a of vitally important organs A , ( a = 1, a). Then the tissues of this organism have to be assumed to be the structure-forming elements e 7 (structural level), and the cells take the part of the primary elements e;~ (metabolic level). Another way of interpretation (Fleishman, 1982, 1984) is the ecology. In this case, the supersystem Aa~ is a biocenose, including vitally important populations A~ of lower trophic level. Age groups of populations were regarded as the structure-forming elements e? (structural level), individuals being considered as primary elements e~ of the metabolic level. There are no objections against interpreting the supersystem Aab with its contained community as the metabolical level. However, up to now there is no c o m m o n opinion of which biological systems are disposed here at the structural level (Beklemishev, 1969).
78
B.s. FLEISHMAN
We identify the two above levels because at each of them the models of system's A s death are principally different. At the metabolic level, the death of system A s is due to that of at least one of its elements e 7. The death of the last one is caused by the loss of an unacceptable number of its primary elements. At the structural level the death of system A s is due to the loss of its integrity. The death of the supersystem Aab is due to that of at least one system A s. Biologically, these determinations can be justified in the following way. The metabolic death of tissues, which preserve their structure, may occur when the amount of their cells diminishes to below the tolerable number. The same can be said about the metabolic death of populations due to the sharply decreasing number of their individuals. Treating structural death as the loss of the system's A s, integrity can be justified in the following way. When a biological system is separated into two parts, there can be three possible cases: (1) both parts remain alive; (2) one part remains alive, while the other dies; (3) both parts die. The first case is found in asexual reproduction. The second case takes place when an organism is separated from its nonvital or unimportant parts, or when gregarious organisms are separated from their flocks, or higher trophic levels - from biocenoses. We shall not dwell on these cases in our further discussion. We shall consider only the third case, which corresponds to the case of the structural death of the system A s. The supersystem's Aab death can be associated with the death of an organism when at least one of the latter's vital organs dies, or with the death of populations at the lower trophic level.
S T O C H A S T I C M O D E L S O F VITALITY IN B I O L O G I C A L SYSTEMS N A M I C S O F T H E I R M E T A B O L I C A N D S T R U C T U R A L LEVELS
AND
DY-
On the basis of the above multilevel concepts, one can build stochastic models of nonvitality in biological systems, i.e. probabilities of their death at the moment s. For simplicity, we shall omit, further on, the index a in all the parameters of the system As, replacing them by the index s of the time. In case of the metabolic death of the system A, we define the probability 1 - P~i of the death of its primary element esik, and assume that these elements die independently of one another. We also assign the critical number rn~s~= 8s, ns~ (0 < Osi < 1) of primary elements e~,k such that if their number m si is smaller than m~, so the element esi is thought to be dead. In case of the structural death of the system A we assign the probability 1 - ds~ of the loss of links between the elements, and assume that they are lost independently of one another. We define such a critical quantity l~ r = Osl s ( 0 < O~ < 1 < l~ <_ m s - 1) such that, if the number of links between
79
HYPERBOLIC LAW OF RELIABILITY
elements of the two parts of the system A is smaller than /s~r, these parts are thought to be isolated. Then, in the asymptotic case, l s --+ o¢ and 0s -- constant, the vitality I ~ ( s ) of the system A, as the probability of its vitality coincidence at the metabolic and structural levels, will look as follows [see Appendix 1, (A.1.16) and (A.I.10)]: 11 - C m s ~
1 -/x(s) =
e x p [ - (1 +
8)m~s]
1+ 6
if
fin ms
ns=~mslm~
(7a) (7b)
where m~
FIs ~
E
Flsi
i=1
and F, c, ~" and 6 > 0 are constants defined in Appendix 1. When 8 < 0, then 1 - / , ( s ) --+ 0. Finally, the vitality 1 - # a ( s ) of the supersystem Aab asymptotically coincides with the minimum vitality of the system A determined to be the 'bottleneck' of the supersystem. In the singular structureless metabolic case, when m s = 1, the system's vitality will look as follows: 1 -/2(s) = 1 - exp(-nkl)
(8)
where k 1 is a positive constant defined in Appendix 1. Let us now consider the dynamics of the metabolic and structural levels of system A (depending on the age). So far, we have discussed the dynamics of the metabolic level of tissue and of individuals in populations. In the course of analysis of these dynamics the differential and finite-difference equations of the material-energy balances were used. The corresponding solutions resulted in the logistic curves growth, as mentioned above. Below we describe another possible approach to the same dynamics, based on the vitality requirement (1) expressed through the relations (7) and (8). This requirement leads the system's reliability to the value getting arbitrary close to one. Let us substitute relations (7) and (8) into (1). We shall obtain, then, the requirement for the number of the elements in the general case: (c/e)l/Ss"
ms =
c2(ln s ) 1/;
( c3s C' In s
(9a)
ns -~ ~~c4(ln s) ('+¢)/~
(9b)
In the singular case these numbers are: m s=l
n s.-~c 5 In s
(10)
80
B.S. FLEISHMAN
where the positive constants c, c 1, c2, c3, c4 and c 5 are defined in Appendix 1. Relation (10) was originally obtained from Fleishman (1964). Only in the case of this relation, the logarithmic law of increase in the number of primary elements takes place in its pure form. However, it may also take place in the variant (b) of the general case, if the value of the parameter ~ is sufficiently great. EMPIRICAL LOGARITHMIC SYSTEMS ON THEIR AGE
DEPENDENCE
OF THE GROWTH
OF BIOLOGICAL
M a n y investigators found, empirically, the logarithmic dependence of the growth of the volume B inhabited by the biological system or its 'biological' age (or else, of the age indicator) on the physical time t; let us note this dependence as B = B ( t ) . This dependence (temporal law of 'logarithmic') looks as follows: B = B ( t ) = B 1 In t + B(1)
(11)
where B 1 and B(1) are some positive constants reflecting the specific features of the biological system. In different cases the value B ( t ) is of the following biological nature: wound healing area (Le Compte du Nouy, 1936), the crystalline lens dimension of the rat N e s o k i a indica (Shaher, 1982) as many as 23 different indicators of biological age (Hofecker et al., 1981) and, last, the biomass growth B ( t ) - B ( 1 ) , beginning with some fixed m o m e n t t = 1. The last regularity is most universal and is called the law of 'organic time' (Backman, 1943). These examples prove that the logarithmic law is a sufficiently general law of biological growth. Qualitatively, this law demonstrates that, once emerging, a biological system continues its growth all its life, but the rate of this growth is steadily slowing down. However, besides the logarithmic function B ( t ) this property is possessed by all the functions with the negative second derivative: B " ( t ) < 0. Therefore it is important to substantiate theoretically just the logarithmic dependence (11). This theoretical substantiation is provided by the theoretical relations (9b) with large values of the parameter and the relation (10). In both of these cases the metabolic effect dominates over the structural one. This observation agrees well with 'good' Backman's law (purely logarithmic), for instance, in case of the growing wood biomass. Assuming that B ( t ) = n t b , where b is the mean value of biomass in the primary element of the biological system, and taking into account the asymptotic nature of the considered relations, as well as assuming that B 1 = c4b and B1 = csb, we can provide for formal coincidence of the discussed empiric (11) and theoretical (9b and 10) equations.
HYPERBOLIC
81
LAW OF RELIABILITY
This coincidence is not accidental, because the biological concepts of the so-called metabolic time (Meien, 1983) are in full agreement with the concepts used to derive relations (9) and (10). The concept of metabolic time in its broad sense just suggests that any biological system of a given level has exchangeable element structure of lower metabolic level. The very definition of the metabolic level postulates the variability (interchangeability) of its elements (Hofman-Kadoshnikov, 1984). This regularity is true for organelles at the molecular level, for organs at the cellular level, for an age-group at the level of its individuals, and for a community at the level of populations. However, it is not valid for a cell at the level of organelles, for an individual at the level of organs, for a population at the level of age-groups, or the biosphere at the level of communities. So, the metabolic process is based on the elements exchange of a biological system. But this is just the very process studied, in its general form, by the systematology which builds models of reliable systems made of unreliable elements. APPENDIX 1 Reliability is the probability P(t) that the system will stay alive till the fixed moment of time t. We assume that P(1) = 1 - I*(1), and the time s is a discrete value s -- 1, 2 . . . . . t, with an interval taken for unity. The vitality [nonvitality] of the system is the instantaneous probability 1 - / , ( s ) [~(s)] of its life (death) at the moment s. By their very definition these probabilities are connected by the relation: l
P(t)=l-[1-P(q)]
I]
t
[1-#(s)]>P(q)-
s=t+l
Y'. t~(s)
(A.I.1)
s=t+l
where 1 ~< t I ~< t, P(1) = 1 - / , ( 1 ) and the system's death is supposed to be independent at various moments of time. In a general case we find the lower estimate (A.1), according to Bull's formula (Feller, 1957). If for all values of s (1 ~< t ~< s ~< t) we have:
~ ( s ) <~I~(tl) (tl/S) 1+"
(A.1.2)
where 71 = t 1 ~ ( t l ) / e and e > 0, then, by using the estimation (A.I.1) and the integral estimation of the sum, we obtain:
P ( t ) >i P ( t l ) - e ( l - ( t , / t ) ' )
>~P ( q ) - e
(A.1.3)
and the asymptotic (with t --) 0o) estimation: P ( o o ) >~ P ( t l ) - e which is nontrivial for e < P ( t l ) and rl > tl ~ ( q ) / P ( q ) .
(A.1.4)
82
B.SFLEISHMAN
In the practical case of t 1 = 1, assumed P(1) = 1 - ~(1) = 1 - e, we have 7/= 1, the upper condition (A.1.2) leads to the condition: /l(s) ~ e(1/s) 2
(A.1.5)
with which: P(t)
>/1 - e ( 2 - ( l / t ) )
and
P ( ~ ) > 1 - 2e
(1.1.6)
where e < 0.5 is an arbitrary small positive quantity. Let us assume that the system A at the moment s includes n s of primary elements grouped into m s of enlarged elements ei having each ns~ of primary elements (metabolic level), my
y" ~lsi = n s
(A.1.7)
i=1
At the level of elements e, the system possesses the structure determined by the graph Gs = (U,, /,1,). Here, U s = {ei} ( i = 1 , ms) and V,= {(ei, ei,)} c U 2 are the sets of vertexes and edges corresponding to the sets of elements and their links (the structural level). In the partial case of m s = 1, the structure degenerates and the system turns into a metabolic one composed by ns, = n s of primary elements. We suppose the element e i to become dead, if within the time interval from s till s + 1 it loses as many of its primary elements that their remaining quantity is v, ~< n,~ = Os,ns,, where 0,~ we assume to be equal to some fixed quantity (0 < Os~ < 1). The death of system A at the metabolic level is due to that of at least one of its elements ep The death of system A at the structural level is due to its loss of integrity. A system will be formally regarded as integral if the respective graph G s of its structure is b~r-edge-connected. It means that the vertexes of any of its subgraphs have at least Icr= Osls of common edges with the vertexes of another part of the graph, where 0s and ls ( 0 < 0 s < l < I s < r n s - l ) are arbitrary constants. It should be noted that the metabolic death of the element e s may also lead to integrity loss in system A. Let us consider the probability model of death for the elements and the system. We shall assume that at the moment s the primary elements of the element G survive [die] independently with the probabilities Ps~ [1-Ps~l, and that the links between the elements e~ and e~, (the edges between the vertices of the graph) are preserved (disappear) independently of each other with the probabilities ds(1 - d s ) . Then, asymptotically, with ns~ ~ ov and Os~ < P s i , probability P ( e , ) of the survival of the element e i will look as follows (Fleishman, 1971): P ( e ~ ) = 1 - e x p ( - ns,ks, )
(1.1.8)
83
HYPERBOLIC LAW OF RELIABILITY
where L i = k(Osi, Psi) and k ( O , p ) = 0 l n ( O / p ) + (1 - 0) ln[(1 - 0 ) / ( 1 - p ) ]
Asymptotically, with Is ~ ~ and 0s < d , , the probability D, of nonseparation of element e i from the system A will look in the following way: (A.1.9)
D, = 1 - e x p ( - l s k , )
where k , = k ( Os, ds)
F r o m the relation (A.1.8), with: gt s ---) ~ n si =
Fn,/ksi
and F=
lim
(m
)
-1
Y'~k~l/rns
m~---~ ~
i=1
we have the maximum metabolic vitality (Fleishman, 1971): 1 - ~ l ( s ) = [1 - e x p ( - F n s / m s ) ]
m`
e x p ( - c m ] '1) ~ 0
Inm s lnc
=
1-cm;
8
1 - e x p [ - (1
if
n s-
F
i
1+8 m , + ~ m
8 <0
In rn, ) s' m~
J~
6>0
+ ~)m~] (A.I.10)
where c, f > 0 and 6 are arbitrary constants. Hence we derive the nonvitality of the system A when it is too finely divided into structural elements: 8<0,
n , i < [(1
-
[ ~ [ ) / k s i ] In m s
In another degenerated metabolic case of the system A with its ultimately enlarged elements, when ms = 1, k 1 = lims~o~k,1 and n,i = n s, we have the following formula for the vitality of the system A: 1 -/21(s ) = 1 - e x p ( - n , k l )
(A.I.ll)
To derive the asymptotic estimate of the structural vitality, let us generalize the estimates of probability of the one-edge-connectivity of the graph
84
B.S.FLEISHMAN
for the same probabilistic model (Gilbert, 1959). Let us note as E the event of the /~r-edge-connectivity of the graph Gs, and as X, the event of the presence of less than l cr of edges at the vertex i. Strictly repeating the deriving of Gilbert (1959) we will have for our general case: m s
1-
Y'~P(X,)
=P(E)
<1-
~_,P(X,)+ i=l
Y~
P ( X , NX~,)
(A.1.12)
l
We have P( X~) = 1 - D s, P( X, n X~,) = P( X~)P( X~,/X~) = ( 1 Ds) P( Xi,/X~); 1 - D s = e x p ( - l s K s ) , where k s = k(Os, ds). If the vertexes i and i" have not common edge, the P ( X , , / X / ) = 1 - D s ; if they do have a common edge, then P ( X ~ , / X , ) = e x p [ - ( / s - 1 ) k ' ] , where k" = k(O', d s ) > k s = k(Os, ds) because of: l cr- 1 l cr
o/-
Therefore:
P ( X i , / X , ) < e x p [ - (l s - 1)ks] And from the relation (A.l.12) we have:
1--m~ exp(-lsks) < l-/~2(s) < l-m +
s exp(-l~k~)
m ~ ( m s - 1) e x p [ _ 2 ( / s _ 1)ks] 2 (A.1.13)
Inasmuch as Is < rn, - 1 and Is ~ ~ , with m s e x p ( - l s k s ) - ~ 0 we have precise asymptotic estimates: (A.l.14)
1 - / , 2 ( s ) = 1 - rn s e x p ( - l s k s )
Let us denote as I the event of the system's metabolic survival at the m o m e n t s. Then, the total vitality 1 - ~(s) of the system A as the probability of P ( I , E) of the coincidence of the dependent events I and E may have, by using the formula of complete probability, the following identical presentation: 1 --#(s)
=
P(I, E)= P(I) P(E) + P(I)
[P(E)-
=[1-th(s)][1-#2(s)]+lZ2(s)[P(E)-e(E/i)]
P(E/I)] (A.l.15)
Counting the relations (A.I.10), (A.1.14) and (A.1.15), the m a x i m u m value of 1 - # ( s ) in the asymptotic case of rn, --+ oc is obtained when 1 - / , l ( s ) -
HYPERBOLIC LAW OF RELIABILITY
85
1 - ~2(s), which is equivalent to 1-/.t (s)= 1-/z,(s)-/x2(s
ls ~ Fn,/m,k
s
and it will look as follows:
) = 1- 2/q(s)= 1-/Xl(S )
(1.1.16)
Now let us use the conditions of the hyperbolic law of reliability (A.1.2), expressed as /_t(s)< e ( 1 / s ) 2, and with its help establish the dependence between the number n, of primary elements and the number m, of elements on the age of the system A. From relations (A.1.2) and (A.1.16), accounting relations (A.1.10) and ( A . I . l l ) , we get for the general case:
l
m,--
{
C3Sq In s
ns
(ln s ) l / ~ '
t c2
c 4 (In s)~l
(A.1.17)
+~)/~
where c1 = 2 / 3
c 2 = [2/(1 + 3)1 '/~
C3 ~
2_~33(
C/E
)l/n
and C4
=
c~+~(1 + 3)/F
For the degenerated case we have: rn, = 1
n s = c5
Ins
(A.1.18)
where c s = 2/K
1
In this way we have derived a set of logarithmic relations between the n u m b e r of elements and the lifetime s of the system A, making its reliability arbitrarily close to one. APPENDIX
2
Let us now consider the vitality model of system A experiencing environmental pressure at the m o m e n t s. The environment is identified with the source N, of potential causes for the system's death with probabilities X s j ( j = 1, N,). Then, regarding the ecological fact that the repeated death of the system is impossible (which results in the so-called formula of 'four minus ones', we shall obtain the following expression of the system's vitality (Fleishman, 1982a, b): 1-/*(s)=l-
1+
~(x~.'-l)-' j=l
- t 1 - Ns(x/(1 - x ) ) , with Ns(X/(1 1/N,(x/(1
- x)),
x))[
---+0 I, ----, oo
(1.2.1)
86
B.S. FLEISHMAN
N 4000 woo
~OB ~o
~00
~ -
tit:,,st~.a,sbu~La~
200 ::,>
0 ,~E CYE~R$) Fig. 1. The number N 1000 P ( t ) of t-year-old male and female lizards Uta stansburiana (Tinkle, 1967) surviving till the age t. P(t) is the probability of survival till age t, reached by an individual (reliability).
where (x/t1
- X))s=
Y'.
-
j~l
From the relation (A.2.1), it follows that for striving the vitality to one with the increase of Ns, it is sufficient that in the case of s --, oo the relation ( x / ( 1 - x ) ) s ~ < c6Nj
(A.2.2)
j=l
where c6 > 0 is an arbitrary and ~ > 0 an arbitrary small constant. Hence, also using relation (A.2.1) we get: 1 - ~(s) > 1 - c6NS ¢
(A.2.3)
Considering the dynamics, let us assume that system A emerges in a stationary environment and occasionally encounters new and new again factors which suppress it. Or, on the contrary, the stationary system, due to some environmental calamity, is experiencing new and new again environmental phenomena which suppress it. In terms of mathematics, both of these situations are equivalent for consideration of a flow of suppressing factors; the latter appear at the moments o = 1, 2, 3 . . . . , s independently of one another. Each m o m e n t there appears a random number x ( c ) of them with the mathematical expectation (ME) E x ( o ) . Each of them has a r a n d o m action time % with the same distribution function P(% < t ) = F o ( t ) . Then the ME E ~,(s) of the r a n d o m number ~,(s) of factors acting at the m o m e n t s, will look as below (Fleishman, 1971, 1984): S
E u(s)=
Y'~ E x ( o ) [ 1 - F o ( s - o ) ] o=1
(A.2.4)
HYPERBOLIC
87
LAW OF RELIABILITY
Let us assume that, upon their emergence, the factors act for a time which is sufficiently long, as compared to the lifetime t A of system A. It is convenient to formulate this assumption in the following probabilistic form for all o = 1, 2, 3 . . . . . / A :
P('ro
< tA) =
Fo( tA) <~O
(A.2.5)
where 0 is an arbitrary small value. In other words, none of the factors terminates its action throughout the entire life of system A. Then, inasmuch as Fo(t) is a function which does not decrease with the growth of t, and assuming that N, = E 1,(s), obtain from relations (A.2.4) and (A.2.5) the following expression: s
N, = Y'~ E × ( o ) + O ( 0 ) --~ sEx
(A.2.6)
o=1
where 1
E×=-
s
s
Y'~ E x(O)s__,
)constant
o=1
If the density of the probability Fo'(t) is symmetrical, the medians Mo (Fo(Mo) = 0 . 5 ) coincide with the ME E%=Mo; and t A is estimated by o-quantiles of distributions Fo(t), i.e. by their upper estimates t A < t~ o (Fo(tQ) = 0)- In this way, we have here t A << E% and the number of factors, according to the relation (A.2.6), is linearly increasing with the lifetime of the system A. It should be noted that tA ~ m leads to E u(s) = constant. In the general case, one can suggest any power law for the increase in the number of factors with the lifetime of the system:
Ns = c7s~
~ > 0
(A.2.7)
We should like to note that the hyperbolic decrease with the time s of the average probability of the system's death Ys, due to the separate factors, reflects its enhancing resistance to environmental pressure. This may result from the system's adaptation to the environment. ACKNOWLEDGEMENTS
Thanks are due to Dr. M. Stra~kraba and to Dr. A. Kurakin for useful remarks. REFERENCES Backman, G., 1943. Wachstum und organische Zeit. Barth, Leipzig, 195 pp. Beklemishev, K.V., 1969. Regulation at the biocoenotic level of life organization. Bull. M O I P Ser. Biol., 74 (3): 144-157 (in Russian).
88
B.S. FLEISHMAN
Feller, W., 1957. Introduction to probability theory and its Applications, Vol. 1 (2nd Edition). Wiley, New York/Chapman & Hall, London, 320 pp. Fleishman, B.S., 1964. Statistical sequential analysis and automatic self-control. In: Scientific Notes on Statistics, Vol. 8. Nauka, Moscow, pp. 57-77 (in Russian). Fleishman, B.S., 1971. Elements of potential efficiency theory for complex systems. Sovetskoe Radio, Moscow, 224 pp. Fleishman, B.S., 1980. Information theory of collective behaviour as applied for fish schools. ISEM J., 2: 33-41. Fleishman, B.S., 1982a. Principles of Systematology. Radio and Sviaz, Moscow, 368 pp. (in Russian). Fleishman, B.S., 1982b. Stochastic theory of ecological interactions. Ecol. Modelling, 17: 65-73. Fleishman, B.S., 1984. Contribution to the theory of adaptation with application to ecology. Ecol. Modelling, 26: 21-31. Gilbert, E.N., 1959. Random graphs. Ann. Math. Stat., 30: 1141-1144. Hofecker, G., Skalicky, M., Kment, A. and Niddermuller, M., 1981. Mathematical models of the biological age of the rat. In: Proc. 28th. Int. Congr. Physiological Sciences, Budapest. Pergamon, Oxford, Vol. 34, pp. 47-52. Hofman-Kadoshnikov, P.B., 1984. Systematical analysis of the hierarchy of life levels. In: System Investigations. Yearbook 1984. Nauka, Moscow, pp. 322-328 (in Russian). Lecompte du Nouy, P., 1936. Biological Time. Methuen, London, 180 pp. Meien, S.V., 1983. Concept of time and typology of objects (with examples from geology and biology). In: Dialectics in Sciences on Nature and on Man. Nauka, Moscow, pp. 311-317 (in Russian). Pianka, E.R., 1978. Evolutionary Ecology (2nd Edition). Harper and Row, New York, 255 PP. Shaher, B.U, 1982. Ageing of Nesokia indica by weighing the eye lens. Pakistan J. Zool., 14 (1): 103-105. Tinkle, D.W., 1967. The life and demography of the sideblotched lizard Uta stansburiana. Misc. Publ. Mus. Zool. Univ. Mich. 132, 182 pp.
75
Elsevier Science Publishers B.V., Amsterdam
Hyperbolic law of reliability and its logarithmic effects in ecology B.S. Fleishman Institute of Oceanology of the USSR Academy of Sciences, Moscow, USSR (Accepted 30 May 1990)
ABSTRACT Fleishman, B.S., 1991. Hyperbolic law of reliability and its logarithmic effects in ecology. Ecol. Modelling, 55: 75-88. The paper deals with the exponent of hyperbolic decrease in a regenerative system's reliability as the function of the similar exponent of its nonvitality. Ecological bases for the hyperbolic law of the decreasing nonvitality of the system are provided. Stochastic models of the system's death at metabolic and structural levels, as well as a general model of its death are considered. The minimum order of the increase in the number of the system's elements as function of its age is found to result in its reliability arbitrary close to unity. This order is of logarithmic and logarithmic-power nature. Biological effects of the regularities so discovered are dicussed.
INTRODUCTION
In his previous papers (Fleishman, 1980, 1982, 1984) the author developed the theory of adaptation of regenerative systems Aab in a stationary regime and also estimated the time of the transient conditions. This time can be of the same order as the life span of the system TA which is related to its reliability. In view of this, the present paper considers the 'juvenile' period in the life of the regenerative system related to the transient period, as well as the conditions under which it can survive till its 'mature' age while passing into the stationary regime. These conditions require at least a logarithmic increase in the number of its elements. One may think that this result contradicts the familiar logistic law of the increase in biological systems. However, if we discriminate in the logistic curve striving to make constant its asymptotic part following the exponential one, the former will prove to be quite comparable with the logarithmic dependence at a limited interval of the system's life. 0304-3800/91/$03.50
© 1991 - Elsevier Science Publishers B.V.
76
B.S. FLEISHMAN
R E L I A B I L I T Y AS R E L A T E D T O V I T A L I T Y A N D T H E G E N E R A L MODEL OF VITALITY
ECOLOGICAL
In ecology there is a concept of instantaneous probability /~(s) of the biological system's death at m o m e n t s as well as that of the probability P ( t ) of its survival till age t (Pianka, 1978). In the theory of potential efficiency of complex systems, the probability 1 - / ~ ( s ) [Ms)] is called vitality [nonvitality] and the probability P ( t ) is called reliability. The following dependence has been found between them (see Appendix 1): If:
ii(s)<<.es -2
fors=l,2
..... t
(1)
then
P ( t ) > 1 - 2e
for t = 1, 2 . . . .
(2)
where ~ is an arbitrary positive quantity. If /_t(s) > es -1 and the deaths at different moments are independent, then (Fleishman, 1971):
P(t) The hyperbolic decrease of ~(s) and P ( t ) are known from a number of ecological papers. However, the relation between the exponents of the respective hyperbolas were not known up to now. The relations (A.1.2) and (A.1.3) reveal the hyperbolic decrease in the system's reliability resulting from the hyperbolic decrease of its nonvitality as a function of its age. The latter decrease is due to the system's adaptive improvement domination over the environmental pressure with the system's age. The interaction between these factors can be found in its most general form by assessing the death probability of the system's nonvitality due to numerous external causes. Let us assume that at the m o m e n t s the system is exposed to the N, of potential causes of the death, the danger of each cause being described with the separate probability xsj ( j = 1, N,). We'll say as well that the system "is under pressure of N, of possible death causes". Then, taking in mind the ecological fact that the system's death cannot be repeated, in the asymptotic case of N, ~ ~v the following relationships will take place (see Appendix 2). If the average probability Ys of the system's death due to N, reasons is sufficiently low: 2s zx ]
N~ x,j < CN, -<~+q') Y'~ j=l
(3)
HYPERBOLIC
LAW OF RELIABILITY
77
then the system's nonvitality will have its upper estimation: i x ( s ) ~ N s Y s = C N ] -~"
(4)
where C and q, are some positive constants. It is natural to suggest that, in the course of the system's life, the number of causes of its possible death increases. The linear dependence N s = C ' s is substantiated in Appendix 2. In general, one can admit the power dependence: (5)
U s = C ' s 4"
where C' and ~p are some positive constants. Substituting expression (5) into estimation (4) and relating it with condition (1), we obtain the expression: Ys < ( e / C ' )
s-{2+'~)
(6)
METABOLIC A N D STRUCTURAL LEVELS OF SYSTEM A a b
In order to construct stochastic models of the dependence between the reliability of the system mal, and the number of its elements, it is necessary to provide a deeper ecological analysis of the internal causes of its death. The generative system Aab = {A~}, as a supersystem, is composed of systems A~ which are vitally important for it. Each of the systems A~ = { eT} is structured, that is it contains the quantity m '~ of elements e 7 (i = 1, m ") and a certain number of links between the elements (the structural level). Each of the elements e 7 = {e,~} is non-structured, containing the number n~ of homogeneous replaceable primary elements e;'~ ( k = 1, n~) (metabolic level). Let us now interpret the supersystem Aab firstly in terms of biology. It can be interpreted first of all as an organism (individual) composed of the n u m b e r a of vitally important organs A , ( a = 1, a). Then the tissues of this organism have to be assumed to be the structure-forming elements e 7 (structural level), and the cells take the part of the primary elements e;~ (metabolic level). Another way of interpretation (Fleishman, 1982, 1984) is the ecology. In this case, the supersystem Aa~ is a biocenose, including vitally important populations A~ of lower trophic level. Age groups of populations were regarded as the structure-forming elements e? (structural level), individuals being considered as primary elements e~ of the metabolic level. There are no objections against interpreting the supersystem Aab with its contained community as the metabolical level. However, up to now there is no c o m m o n opinion of which biological systems are disposed here at the structural level (Beklemishev, 1969).
78
B.s. FLEISHMAN
We identify the two above levels because at each of them the models of system's A s death are principally different. At the metabolic level, the death of system A s is due to that of at least one of its elements e 7. The death of the last one is caused by the loss of an unacceptable number of its primary elements. At the structural level the death of system A s is due to the loss of its integrity. The death of the supersystem Aab is due to that of at least one system A s. Biologically, these determinations can be justified in the following way. The metabolic death of tissues, which preserve their structure, may occur when the amount of their cells diminishes to below the tolerable number. The same can be said about the metabolic death of populations due to the sharply decreasing number of their individuals. Treating structural death as the loss of the system's A s, integrity can be justified in the following way. When a biological system is separated into two parts, there can be three possible cases: (1) both parts remain alive; (2) one part remains alive, while the other dies; (3) both parts die. The first case is found in asexual reproduction. The second case takes place when an organism is separated from its nonvital or unimportant parts, or when gregarious organisms are separated from their flocks, or higher trophic levels - from biocenoses. We shall not dwell on these cases in our further discussion. We shall consider only the third case, which corresponds to the case of the structural death of the system A s. The supersystem's Aab death can be associated with the death of an organism when at least one of the latter's vital organs dies, or with the death of populations at the lower trophic level.
S T O C H A S T I C M O D E L S O F VITALITY IN B I O L O G I C A L SYSTEMS N A M I C S O F T H E I R M E T A B O L I C A N D S T R U C T U R A L LEVELS
AND
DY-
On the basis of the above multilevel concepts, one can build stochastic models of nonvitality in biological systems, i.e. probabilities of their death at the moment s. For simplicity, we shall omit, further on, the index a in all the parameters of the system As, replacing them by the index s of the time. In case of the metabolic death of the system A, we define the probability 1 - P~i of the death of its primary element esik, and assume that these elements die independently of one another. We also assign the critical number rn~s~= 8s, ns~ (0 < Osi < 1) of primary elements e~,k such that if their number m si is smaller than m~, so the element esi is thought to be dead. In case of the structural death of the system A we assign the probability 1 - ds~ of the loss of links between the elements, and assume that they are lost independently of one another. We define such a critical quantity l~ r = Osl s ( 0 < O~ < 1 < l~ <_ m s - 1) such that, if the number of links between
79
HYPERBOLIC LAW OF RELIABILITY
elements of the two parts of the system A is smaller than /s~r, these parts are thought to be isolated. Then, in the asymptotic case, l s --+ o¢ and 0s -- constant, the vitality I ~ ( s ) of the system A, as the probability of its vitality coincidence at the metabolic and structural levels, will look as follows [see Appendix 1, (A.1.16) and (A.I.10)]: 11 - C m s ~
1 -/x(s) =
e x p [ - (1 +
8)m~s]
1+ 6
if
fin ms
ns=~mslm~
(7a) (7b)
where m~
FIs ~
E
Flsi
i=1
and F, c, ~" and 6 > 0 are constants defined in Appendix 1. When 8 < 0, then 1 - / , ( s ) --+ 0. Finally, the vitality 1 - # a ( s ) of the supersystem Aab asymptotically coincides with the minimum vitality of the system A determined to be the 'bottleneck' of the supersystem. In the singular structureless metabolic case, when m s = 1, the system's vitality will look as follows: 1 -/2(s) = 1 - exp(-nkl)
(8)
where k 1 is a positive constant defined in Appendix 1. Let us now consider the dynamics of the metabolic and structural levels of system A (depending on the age). So far, we have discussed the dynamics of the metabolic level of tissue and of individuals in populations. In the course of analysis of these dynamics the differential and finite-difference equations of the material-energy balances were used. The corresponding solutions resulted in the logistic curves growth, as mentioned above. Below we describe another possible approach to the same dynamics, based on the vitality requirement (1) expressed through the relations (7) and (8). This requirement leads the system's reliability to the value getting arbitrary close to one. Let us substitute relations (7) and (8) into (1). We shall obtain, then, the requirement for the number of the elements in the general case: (c/e)l/Ss"
ms =
c2(ln s ) 1/;
( c3s C' In s
(9a)
ns -~ ~~c4(ln s) ('+¢)/~
(9b)
In the singular case these numbers are: m s=l
n s.-~c 5 In s
(10)
80
B.S. FLEISHMAN
where the positive constants c, c 1, c2, c3, c4 and c 5 are defined in Appendix 1. Relation (10) was originally obtained from Fleishman (1964). Only in the case of this relation, the logarithmic law of increase in the number of primary elements takes place in its pure form. However, it may also take place in the variant (b) of the general case, if the value of the parameter ~ is sufficiently great. EMPIRICAL LOGARITHMIC SYSTEMS ON THEIR AGE
DEPENDENCE
OF THE GROWTH
OF BIOLOGICAL
M a n y investigators found, empirically, the logarithmic dependence of the growth of the volume B inhabited by the biological system or its 'biological' age (or else, of the age indicator) on the physical time t; let us note this dependence as B = B ( t ) . This dependence (temporal law of 'logarithmic') looks as follows: B = B ( t ) = B 1 In t + B(1)
(11)
where B 1 and B(1) are some positive constants reflecting the specific features of the biological system. In different cases the value B ( t ) is of the following biological nature: wound healing area (Le Compte du Nouy, 1936), the crystalline lens dimension of the rat N e s o k i a indica (Shaher, 1982) as many as 23 different indicators of biological age (Hofecker et al., 1981) and, last, the biomass growth B ( t ) - B ( 1 ) , beginning with some fixed m o m e n t t = 1. The last regularity is most universal and is called the law of 'organic time' (Backman, 1943). These examples prove that the logarithmic law is a sufficiently general law of biological growth. Qualitatively, this law demonstrates that, once emerging, a biological system continues its growth all its life, but the rate of this growth is steadily slowing down. However, besides the logarithmic function B ( t ) this property is possessed by all the functions with the negative second derivative: B " ( t ) < 0. Therefore it is important to substantiate theoretically just the logarithmic dependence (11). This theoretical substantiation is provided by the theoretical relations (9b) with large values of the parameter and the relation (10). In both of these cases the metabolic effect dominates over the structural one. This observation agrees well with 'good' Backman's law (purely logarithmic), for instance, in case of the growing wood biomass. Assuming that B ( t ) = n t b , where b is the mean value of biomass in the primary element of the biological system, and taking into account the asymptotic nature of the considered relations, as well as assuming that B 1 = c4b and B1 = csb, we can provide for formal coincidence of the discussed empiric (11) and theoretical (9b and 10) equations.
HYPERBOLIC
81
LAW OF RELIABILITY
This coincidence is not accidental, because the biological concepts of the so-called metabolic time (Meien, 1983) are in full agreement with the concepts used to derive relations (9) and (10). The concept of metabolic time in its broad sense just suggests that any biological system of a given level has exchangeable element structure of lower metabolic level. The very definition of the metabolic level postulates the variability (interchangeability) of its elements (Hofman-Kadoshnikov, 1984). This regularity is true for organelles at the molecular level, for organs at the cellular level, for an age-group at the level of its individuals, and for a community at the level of populations. However, it is not valid for a cell at the level of organelles, for an individual at the level of organs, for a population at the level of age-groups, or the biosphere at the level of communities. So, the metabolic process is based on the elements exchange of a biological system. But this is just the very process studied, in its general form, by the systematology which builds models of reliable systems made of unreliable elements. APPENDIX 1 Reliability is the probability P(t) that the system will stay alive till the fixed moment of time t. We assume that P(1) = 1 - I*(1), and the time s is a discrete value s -- 1, 2 . . . . . t, with an interval taken for unity. The vitality [nonvitality] of the system is the instantaneous probability 1 - / , ( s ) [~(s)] of its life (death) at the moment s. By their very definition these probabilities are connected by the relation: l
P(t)=l-[1-P(q)]
I]
t
[1-#(s)]>P(q)-
s=t+l
Y'. t~(s)
(A.I.1)
s=t+l
where 1 ~< t I ~< t, P(1) = 1 - / , ( 1 ) and the system's death is supposed to be independent at various moments of time. In a general case we find the lower estimate (A.1), according to Bull's formula (Feller, 1957). If for all values of s (1 ~< t ~< s ~< t) we have:
~ ( s ) <~I~(tl) (tl/S) 1+"
(A.1.2)
where 71 = t 1 ~ ( t l ) / e and e > 0, then, by using the estimation (A.I.1) and the integral estimation of the sum, we obtain:
P ( t ) >i P ( t l ) - e ( l - ( t , / t ) ' )
>~P ( q ) - e
(A.1.3)
and the asymptotic (with t --) 0o) estimation: P ( o o ) >~ P ( t l ) - e which is nontrivial for e < P ( t l ) and rl > tl ~ ( q ) / P ( q ) .
(A.1.4)
82
B.SFLEISHMAN
In the practical case of t 1 = 1, assumed P(1) = 1 - ~(1) = 1 - e, we have 7/= 1, the upper condition (A.1.2) leads to the condition: /l(s) ~ e(1/s) 2
(A.1.5)
with which: P(t)
>/1 - e ( 2 - ( l / t ) )
and
P ( ~ ) > 1 - 2e
(1.1.6)
where e < 0.5 is an arbitrary small positive quantity. Let us assume that the system A at the moment s includes n s of primary elements grouped into m s of enlarged elements ei having each ns~ of primary elements (metabolic level), my
y" ~lsi = n s
(A.1.7)
i=1
At the level of elements e, the system possesses the structure determined by the graph Gs = (U,, /,1,). Here, U s = {ei} ( i = 1 , ms) and V,= {(ei, ei,)} c U 2 are the sets of vertexes and edges corresponding to the sets of elements and their links (the structural level). In the partial case of m s = 1, the structure degenerates and the system turns into a metabolic one composed by ns, = n s of primary elements. We suppose the element e i to become dead, if within the time interval from s till s + 1 it loses as many of its primary elements that their remaining quantity is v, ~< n,~ = Os,ns,, where 0,~ we assume to be equal to some fixed quantity (0 < Os~ < 1). The death of system A at the metabolic level is due to that of at least one of its elements ep The death of system A at the structural level is due to its loss of integrity. A system will be formally regarded as integral if the respective graph G s of its structure is b~r-edge-connected. It means that the vertexes of any of its subgraphs have at least Icr= Osls of common edges with the vertexes of another part of the graph, where 0s and ls ( 0 < 0 s < l < I s < r n s - l ) are arbitrary constants. It should be noted that the metabolic death of the element e s may also lead to integrity loss in system A. Let us consider the probability model of death for the elements and the system. We shall assume that at the moment s the primary elements of the element G survive [die] independently with the probabilities Ps~ [1-Ps~l, and that the links between the elements e~ and e~, (the edges between the vertices of the graph) are preserved (disappear) independently of each other with the probabilities ds(1 - d s ) . Then, asymptotically, with ns~ ~ ov and Os~ < P s i , probability P ( e , ) of the survival of the element e i will look as follows (Fleishman, 1971): P ( e ~ ) = 1 - e x p ( - ns,ks, )
(1.1.8)
83
HYPERBOLIC LAW OF RELIABILITY
where L i = k(Osi, Psi) and k ( O , p ) = 0 l n ( O / p ) + (1 - 0) ln[(1 - 0 ) / ( 1 - p ) ]
Asymptotically, with Is ~ ~ and 0s < d , , the probability D, of nonseparation of element e i from the system A will look in the following way: (A.1.9)
D, = 1 - e x p ( - l s k , )
where k , = k ( Os, ds)
F r o m the relation (A.1.8), with: gt s ---) ~ n si =
Fn,/ksi
and F=
lim
(m
)
-1
Y'~k~l/rns
m~---~ ~
i=1
we have the maximum metabolic vitality (Fleishman, 1971): 1 - ~ l ( s ) = [1 - e x p ( - F n s / m s ) ]
m`
e x p ( - c m ] '1) ~ 0
Inm s lnc
=
1-cm;
8
1 - e x p [ - (1
if
n s-
F
i
1+8 m , + ~ m
8 <0
In rn, ) s' m~
J~
6>0
+ ~)m~] (A.I.10)
where c, f > 0 and 6 are arbitrary constants. Hence we derive the nonvitality of the system A when it is too finely divided into structural elements: 8<0,
n , i < [(1
-
[ ~ [ ) / k s i ] In m s
In another degenerated metabolic case of the system A with its ultimately enlarged elements, when ms = 1, k 1 = lims~o~k,1 and n,i = n s, we have the following formula for the vitality of the system A: 1 -/21(s ) = 1 - e x p ( - n , k l )
(A.I.ll)
To derive the asymptotic estimate of the structural vitality, let us generalize the estimates of probability of the one-edge-connectivity of the graph
84
B.S.FLEISHMAN
for the same probabilistic model (Gilbert, 1959). Let us note as E the event of the /~r-edge-connectivity of the graph Gs, and as X, the event of the presence of less than l cr of edges at the vertex i. Strictly repeating the deriving of Gilbert (1959) we will have for our general case: m s
1-
Y'~P(X,)
=P(E)
<1-
~_,P(X,)+ i=l
Y~
P ( X , NX~,)
(A.1.12)
l
We have P( X~) = 1 - D s, P( X, n X~,) = P( X~)P( X~,/X~) = ( 1 Ds) P( Xi,/X~); 1 - D s = e x p ( - l s K s ) , where k s = k(Os, ds). If the vertexes i and i" have not common edge, the P ( X , , / X / ) = 1 - D s ; if they do have a common edge, then P ( X ~ , / X , ) = e x p [ - ( / s - 1 ) k ' ] , where k" = k(O', d s ) > k s = k(Os, ds) because of: l cr- 1 l cr
o/-
Therefore:
P ( X i , / X , ) < e x p [ - (l s - 1)ks] And from the relation (A.l.12) we have:
1--m~ exp(-lsks) < l-/~2(s) < l-m +
s exp(-l~k~)
m ~ ( m s - 1) e x p [ _ 2 ( / s _ 1)ks] 2 (A.1.13)
Inasmuch as Is < rn, - 1 and Is ~ ~ , with m s e x p ( - l s k s ) - ~ 0 we have precise asymptotic estimates: (A.l.14)
1 - / , 2 ( s ) = 1 - rn s e x p ( - l s k s )
Let us denote as I the event of the system's metabolic survival at the m o m e n t s. Then, the total vitality 1 - ~(s) of the system A as the probability of P ( I , E) of the coincidence of the dependent events I and E may have, by using the formula of complete probability, the following identical presentation: 1 --#(s)
=
P(I, E)= P(I) P(E) + P(I)
[P(E)-
=[1-th(s)][1-#2(s)]+lZ2(s)[P(E)-e(E/i)]
P(E/I)] (A.l.15)
Counting the relations (A.I.10), (A.1.14) and (A.1.15), the m a x i m u m value of 1 - # ( s ) in the asymptotic case of rn, --+ oc is obtained when 1 - / , l ( s ) -
HYPERBOLIC LAW OF RELIABILITY
85
1 - ~2(s), which is equivalent to 1-/.t (s)= 1-/z,(s)-/x2(s
ls ~ Fn,/m,k
s
and it will look as follows:
) = 1- 2/q(s)= 1-/Xl(S )
(1.1.16)
Now let us use the conditions of the hyperbolic law of reliability (A.1.2), expressed as /_t(s)< e ( 1 / s ) 2, and with its help establish the dependence between the number n, of primary elements and the number m, of elements on the age of the system A. From relations (A.1.2) and (A.1.16), accounting relations (A.1.10) and ( A . I . l l ) , we get for the general case:
l
m,--
{
C3Sq In s
ns
(ln s ) l / ~ '
t c2
c 4 (In s)~l
(A.1.17)
+~)/~
where c1 = 2 / 3
c 2 = [2/(1 + 3)1 '/~
C3 ~
2_~33(
C/E
)l/n
and C4
=
c~+~(1 + 3)/F
For the degenerated case we have: rn, = 1
n s = c5
Ins
(A.1.18)
where c s = 2/K
1
In this way we have derived a set of logarithmic relations between the n u m b e r of elements and the lifetime s of the system A, making its reliability arbitrarily close to one. APPENDIX
2
Let us now consider the vitality model of system A experiencing environmental pressure at the m o m e n t s. The environment is identified with the source N, of potential causes for the system's death with probabilities X s j ( j = 1, N,). Then, regarding the ecological fact that the repeated death of the system is impossible (which results in the so-called formula of 'four minus ones', we shall obtain the following expression of the system's vitality (Fleishman, 1982a, b): 1-/*(s)=l-
1+
~(x~.'-l)-' j=l
- t 1 - Ns(x/(1 - x ) ) , with Ns(X/(1 1/N,(x/(1
- x)),
x))[
---+0 I, ----, oo
(1.2.1)
86
B.S. FLEISHMAN
N 4000 woo
~OB ~o
~00
~ -
tit:,,st~.a,sbu~La~
200 ::,>
0 ,~E CYE~R$) Fig. 1. The number N 1000 P ( t ) of t-year-old male and female lizards Uta stansburiana (Tinkle, 1967) surviving till the age t. P(t) is the probability of survival till age t, reached by an individual (reliability).
where (x/t1
- X))s=
Y'.
-
j~l
From the relation (A.2.1), it follows that for striving the vitality to one with the increase of Ns, it is sufficient that in the case of s --, oo the relation ( x / ( 1 - x ) ) s ~ < c6Nj
(A.2.2)
j=l
where c6 > 0 is an arbitrary and ~ > 0 an arbitrary small constant. Hence, also using relation (A.2.1) we get: 1 - ~(s) > 1 - c6NS ¢
(A.2.3)
Considering the dynamics, let us assume that system A emerges in a stationary environment and occasionally encounters new and new again factors which suppress it. Or, on the contrary, the stationary system, due to some environmental calamity, is experiencing new and new again environmental phenomena which suppress it. In terms of mathematics, both of these situations are equivalent for consideration of a flow of suppressing factors; the latter appear at the moments o = 1, 2, 3 . . . . , s independently of one another. Each m o m e n t there appears a random number x ( c ) of them with the mathematical expectation (ME) E x ( o ) . Each of them has a r a n d o m action time % with the same distribution function P(% < t ) = F o ( t ) . Then the ME E ~,(s) of the r a n d o m number ~,(s) of factors acting at the m o m e n t s, will look as below (Fleishman, 1971, 1984): S
E u(s)=
Y'~ E x ( o ) [ 1 - F o ( s - o ) ] o=1
(A.2.4)
HYPERBOLIC
87
LAW OF RELIABILITY
Let us assume that, upon their emergence, the factors act for a time which is sufficiently long, as compared to the lifetime t A of system A. It is convenient to formulate this assumption in the following probabilistic form for all o = 1, 2, 3 . . . . . / A :
P('ro
< tA) =
Fo( tA) <~O
(A.2.5)
where 0 is an arbitrary small value. In other words, none of the factors terminates its action throughout the entire life of system A. Then, inasmuch as Fo(t) is a function which does not decrease with the growth of t, and assuming that N, = E 1,(s), obtain from relations (A.2.4) and (A.2.5) the following expression: s
N, = Y'~ E × ( o ) + O ( 0 ) --~ sEx
(A.2.6)
o=1
where 1
E×=-
s
s
Y'~ E x(O)s__,
)constant
o=1
If the density of the probability Fo'(t) is symmetrical, the medians Mo (Fo(Mo) = 0 . 5 ) coincide with the ME E%=Mo; and t A is estimated by o-quantiles of distributions Fo(t), i.e. by their upper estimates t A < t~ o (Fo(tQ) = 0)- In this way, we have here t A << E% and the number of factors, according to the relation (A.2.6), is linearly increasing with the lifetime of the system A. It should be noted that tA ~ m leads to E u(s) = constant. In the general case, one can suggest any power law for the increase in the number of factors with the lifetime of the system:
Ns = c7s~
~ > 0
(A.2.7)
We should like to note that the hyperbolic decrease with the time s of the average probability of the system's death Ys, due to the separate factors, reflects its enhancing resistance to environmental pressure. This may result from the system's adaptation to the environment. ACKNOWLEDGEMENTS
Thanks are due to Dr. M. Stra~kraba and to Dr. A. Kurakin for useful remarks. REFERENCES Backman, G., 1943. Wachstum und organische Zeit. Barth, Leipzig, 195 pp. Beklemishev, K.V., 1969. Regulation at the biocoenotic level of life organization. Bull. M O I P Ser. Biol., 74 (3): 144-157 (in Russian).
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B.S. FLEISHMAN
Feller, W., 1957. Introduction to probability theory and its Applications, Vol. 1 (2nd Edition). Wiley, New York/Chapman & Hall, London, 320 pp. Fleishman, B.S., 1964. Statistical sequential analysis and automatic self-control. In: Scientific Notes on Statistics, Vol. 8. Nauka, Moscow, pp. 57-77 (in Russian). Fleishman, B.S., 1971. Elements of potential efficiency theory for complex systems. Sovetskoe Radio, Moscow, 224 pp. Fleishman, B.S., 1980. Information theory of collective behaviour as applied for fish schools. ISEM J., 2: 33-41. Fleishman, B.S., 1982a. Principles of Systematology. Radio and Sviaz, Moscow, 368 pp. (in Russian). Fleishman, B.S., 1982b. Stochastic theory of ecological interactions. Ecol. Modelling, 17: 65-73. Fleishman, B.S., 1984. Contribution to the theory of adaptation with application to ecology. Ecol. Modelling, 26: 21-31. Gilbert, E.N., 1959. Random graphs. Ann. Math. Stat., 30: 1141-1144. Hofecker, G., Skalicky, M., Kment, A. and Niddermuller, M., 1981. Mathematical models of the biological age of the rat. In: Proc. 28th. Int. Congr. Physiological Sciences, Budapest. Pergamon, Oxford, Vol. 34, pp. 47-52. Hofman-Kadoshnikov, P.B., 1984. Systematical analysis of the hierarchy of life levels. In: System Investigations. Yearbook 1984. Nauka, Moscow, pp. 322-328 (in Russian). Lecompte du Nouy, P., 1936. Biological Time. Methuen, London, 180 pp. Meien, S.V., 1983. Concept of time and typology of objects (with examples from geology and biology). In: Dialectics in Sciences on Nature and on Man. Nauka, Moscow, pp. 311-317 (in Russian). Pianka, E.R., 1978. Evolutionary Ecology (2nd Edition). Harper and Row, New York, 255 PP. Shaher, B.U, 1982. Ageing of Nesokia indica by weighing the eye lens. Pakistan J. Zool., 14 (1): 103-105. Tinkle, D.W., 1967. The life and demography of the sideblotched lizard Uta stansburiana. Misc. Publ. Mus. Zool. Univ. Mich. 132, 182 pp.