J. rheor. Biol. (1977) 68, 331-354

Competitive Processes II.

Stability

of Random Systems

G. S. LADDE Department of Mathematics, The State University of New York at Potsdam, New York 13676, U.S.A. (Received 29 April 1976, and irz revisedform 21 January 1977) In this work, by employing the concept of vector Lyapunov functions and the theory of random differential inequalities, the stability analysis of random competitive systemsis initiated in a systematicand unified way. Furthermore, an attempt has beenmadeto formulate and partially resolvethe “deterministicvs. stochastic”,and the “complexity vs. stability” problemsin stochasticcompetitive processes, Finally, the usefulness of the stability analysisof the competitive systemshas been demonstratedby exhibiting severalwell-known examplesof competitive processesin biological. medical, physical and socialsciencesin a coherent way.

1. Introduction The concept of competitive processesprovides a suitable meansof description for several wide range dynamical processesin biological, physical and social sciences. Very recently, the stability analysis of deterministic competitive systems called compartmental systems and hereditary competitive systems has been made by Ladde (1977b), in a systematic and unified way. It is natural to expect an extension of this stability analysis to random competitive processes.It is well recognized that the probabilistic models in multispecies communities (Goel & Richter-Dyn, 1974; May, 1973; Smith, 1974; Capocelli WCRicciardi, 1974a, b, 1975), chemical kinetics (Batholomay, 1972; Goel & Richter-Dyn, 1974; McQuarrie, 1968; Montroll, 1967; Teramoto, Shiegesada. Nakajima & Sato, 1971), population genetics (Crow & Kimura, 1971; Goel & Richter-Dyn, 1974), firing of neuron (Goel & Richter-Dyn, 1974: Capocelli & Ricciardi, 1973), compartmental systems (Jacquez. 1972: Soong, 1971), statistical mechanics (Rosen, 1970) and economic systems (Turnovsky & Weintraub, 1971) are more realistic than the deterministic models. However, they remained almost invariably restricted to differential equations with Markov and white noise coefficients, since the methods of the theory of Markov processescould be utilized. 331

332

G.

S.

LADDE

Very recently (Ladde & Siljak, 1975), by employing the It6 differential equations as community models, sufficient conditions for stability in the mean of the equilibrium population are given in terms of the dominant diagonal property of community matrices. In this work, we use the differential equations with non-white noise coefficients (Morozan, 1969) as models for competitive processes. An attempt to use non-white noise excitation was made by Goel & Richter-Dyn (1974), Chuang & Lloyd (1974), Soong ( I97 1) and Turnovsky & Weintraub (1971). It is important to note that our approach does not demand linearization, approximation, computational or numerical techniques for analyzing the stability properties of random competitive systems. Moreover, it provides a suitable mechanism to formulate and partially resolve several important issues that exist in the stochastic competitive processes, for example, the “deterministic vs. stochastic”, the “stochastic vs. stability” and the “complexity vs. stability”. By introducing the concepts of connective stability in probability and with probability one as natural analogs to the concept of connective stability in the mean (Ladde & Siljak, 1975) and employing the concept of logarithmic norm for a random matrix (Ladde, 1976c, 1977a), which is a natural analog of the logarithmic norm for a deterministic matrix (Lakshmikantham & Leela, 1969), I provide a considerable insight into the structural properties of multispecies competitive process under both deterministic and random perturbations. From the stability analysis, one can draw several important conclusions about the measurability of the complexity of the system, the measurability of the random effects, the invariability of the stability of the system under both deterministic and stochastic structural perturbations, the tolerance of the complexity by the stable system, the tolerance of the random disturbances by the stable system and the reliability of the stability of the system. Furthermore, the results can be used to estimate the stabilizing and destabilizing effects of random environmental fluctuations on the competitive system. For linear random competitive systems, the present stability conditions are algebraic and computationally more attractive. For complex random competitive systems, the stability analysis is based on the stability of a comparison or auxiliary random competitive system associated with the complex system and in the context of vector Lyapunov function and random differential inequalities (Ladde, 1976~). Furthermore, the stability analysis of hierarchic random competitive systems was considered by means of decomposition-aggregation method coupled with Lyapunov’s second method. In fact, by decomposing a hierarchic random competitive system into simpler and suitable units, the stability of each unit is aggregated by a scalar random Lyapunov function, and an auxiliary or comparison random

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333

competitive system is formed on the basis of the scalar random Lyapunov functions associated with each subsystem and the nature of interactions among the subsystems of a system. Then, the stability of the overall system is obtained by testing the stability of the auxiliary competitive system. The paper is organized as follows. The stability analysis of the linear, non-linear and hierarchic random competitive processes is developed in sections 2, 3, and 4, respectively. The stability analysis is directly applicable to deterministic competitive systems under structural perturbations, whenever the system ignores the random disturbances. Finally, in section 5, the scope of the stability analysis of random competitive systems is demonstrated by exhibiting several well-known examples of random competitive processes from the biological, medical, physical, and social sciences, namely, chemical systems, ecosystems, genetic systems, pharmacological systems, physiological systems, economic systems, armament systems, psychological systems, and social systems.

2. Linear Random Competitive Systems Consider a competitive process of n interacting species described by a linear random differential system k(r, co) = A(w)x(t, 0,) + k(o), (1) where x is an n-vector x = (xi, -Ye, . . ., qJT, and T stands for transpose of a vector; for iEl= (1, 2, , . ., n>, xi represents the state of the ith species in the process; A(w) is the n x II random process rate matrix whose elements ~,,~(a) are random variables defined on a complete probability space ((2, F, P); k(w) E R” is the random input rate vector from external sources. Because of the practical aspects of the competitive process, one assumes that all species in the process are self-inhibitory, that is, the diagonal elements of the process rate matrix A(o) satisfy the relations: ai, < 0, with probability 1 (w.p.1) every (V)~E I. (2) Such an assumption reflects an increase in the instantaneous state of ith species si will cause the rate of change of xi to be decreased w.p.1, or alternatively, a decrease in the instantaneous state of ,Ui will cause the rate of change of si to be increased w.p.1. In the case off-diagonal elements of A(W), one does not specify signs, thus allowing for “mixed” competitivrinhibitive-activative-cooperative-neutral interactions among the species in the competitive process with probability one. Furthermore, in order to assure the existence of a solution process of (I), it is essential that A(tu) is bounded w.p. 1.

334

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In order to avoid monotonicity, hereafter, it will be understood, unicns otherwise specified, that all equalities, inequalities, relations and statement,, that involve random processes, will hold with probability one. The equilibrium x+(o) of (1) is given by the equation: A(o)x+k(o)

Now, one uses the transformation

= 0.

y = x-x”,

j(t, w) = A(o)y(r,

(3)

and rewrites (I) as:

w) + A(o)x*(o)

+ k(o),

which reduces to: k(t, u) = A(o)x(t,

co)

(4) with y replaced by x. Thus the stability of the equilibrium state X*(Q) of (I) equivalent to the stability of the equilibrium state x*(w) E 0 of (4). Depending on the mode of convergence in the theory of random calculus, one has different kinds of stability concepts, namely, the stability in probability, the stability in the moment and the stability with probability one or the stability in the almost sure sense. 1 shall formulate the definition of the stability in the mean. However, the remaining types of stability definitions can be formulated, similarly. For details, see Morozan (1969) and Ladde (1976~). The equilibrium x*(o) = 0 of (4) is said to be stable in the mean, if for < every& > 0, toER+ = [0, cc], there exists a 6 > 0 such that M[l[x,(o)jl] 6 implies M[l/x(t, w)II] < E t B t, where M(.) stands for expectation or moment of a random variable, and x(t, o) = x[t, t,, X,(W), w] is a solution process of (4) through [to, -Q(W)]. If, in addition, there exists a S > 0 such that M[~lx,(w)/IJ < 6e impiies M[/x(r, w)//] + 0 as t --f cry, then the equilibrium state x*(w) z 0 of (4) is said to be asymptotically stable in the mean. The main objective of the present work is to study the effects of random disturbances in the competitive processes and the effects of the strength of the interactions among the species on the stability of the equilibrium state of the competitive process. In this section, I discuss the linear systems, and the non-linear systems will be discussed in succeeding sections. To achieve the above objectives, one assumes that the elements aij(w) 01 the matrix A(m) have the form : -dW)+

Uij(U)

where ai {p E I:

c(pi(W)

C

e,P,i(w)+

i #j. 1 eijaij(u>, > 0, crij(o) are random
= I .

i.(ua)ea[A(w)l

where cr[A(w)J is the spectrum or the set of all eigenvalues of A(w), and u is a positive real number. From the practical point of view, the problem with this stability condition is that it does not connect in any way with the structural properties of process rate matrix A(U). In addition, the problem of computation of eigenvaluesof a random matrix is also complex; moreover, if the dimension of the matrix is large, then the problem becomes worse. In the present work, I propose an alternative method which is an algebraically simple, as well as powerful tool to study the structural properties of the system. For this purpose, one needsthe concept of logarithmic norm

336 for the random

G.

matrix A(o) p[A(w)]

S.

LADDE

(Ladde, 1976c, 1977n), = ,,-rg+ lim :, CIII+Wdl/

- 11.

(6)

Note that the value of p[A(o)] depends on the particular norm used foi vectors and matrices. For example, if lixllE re p resents the euclidean norm, p[A(o)] is the largest eigenvalue of j[A(o)+AT(~)], AT(w) being the transpose of A(o), whereas the corresponding matrix norm A(w) is the square root of the Iargest eigenvalue of AT(w if:

then :

PCNW>I=

sup i

ajj(m)+

itI IQij(w)l

i+j

if:

1 ;

(7)

/Ix/Id= jI dilxil for some di > 0, i E Z, then :

P[A(o)I = SUP an(w) +dj ’ $, ~il~~j(~)ll* i i#j 1

(8)

For more details see Ladde (1977~). Denote p[A(o)] in (7) by p,[A(w)] and p[A(o)] in (8) by p,,[A(w)]. It is easy to see that ~JA(o)] = p,[D-‘A (w)D], where D is a diagonal matrix diag (d,, d2, . . ., (i,) and D-l is the inverse matrix of D. Now I shall present a very general sufficient condition for the connective stability of the system (1). For some real a > 0, we assume that p[A(o)] in (6) satisfies the relation P(o: p[A(to)]

< -a;. = I.

(9)

From (9), one can conclude that the equilibrium x”(w) of (1) is asymptotically stable in the mean as well as with probability one (Morozan, 1969). However, further detailed justification of (9) can be seen in the following sections. Depending on norm, one further notes that the stability condition (9) provides different kinds of stability conditions. Further details are given in Ladde (1976c, 1977~). As indicated above, the objective is not only to study stability but also to study the effects of the strengths of the interactions among the species on the stability of the equilibrium state of the system under random perturbations.

STABILITY

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337

For this purpose, hereafter, I am going to use the logarithmic norm p[A(o)] defined in (8), becauseit provides the simple algebraic relationship between the elements of the process rate matrix A(Q). Note that the relation (8) includes (7), whenever di = 1 for all i E I. Furthermore, one observes that the relation (8) satisfying (9) implies : P(W:

< 0, Vi E f1 = 1

rrjj(o)

(10)

aud P

W: /Ujj(")l-d,~'

~ dilU,j(w)l

2 ff, V’i E i‘

i=l i#j

i

= 1,

(11)

i

and conversely. the relations (8) (10) and (11) imply the relation (9). The relation (11) means that the random matrix A(w) is the quasidominant diagonal matrix with probability one. From (7), the condition (11) includes the diagonal dominance property P CO:IUjj(W)I-

f

Irlij(W)I 2 U, Vj E I

= 1

(1’)

1 /’

i=l i#j

{ as,a special case whenever di = 1. In the context of (I), the condition (12) states that the self-inhibitory affects are larger than the activatory-inhibitory effects. Note that the stability condition (9) leads toward the extension of the quasidominance and diagonal properties of deterministic matrices (Newman. 1959) to random matrices. To conclude the connective stability of (I), it is enough to assume that the relations (10) and (11) hold for a given fundamental interconnection matrix E. This is becauseof the facts that: eijlaij(4(

6 eijlaij(W)I

(13)

and P(A) < P(B),

(14)

whenever njj < bjj,

laij[ < ]bijl for i # ,i. i,,i E 1.

The preceding stability analysis can be extended to the linear time-varying random competitive system

K(t, co) = A(r, co)x(t,w)+k(co). (IS) where x E R”, the II x11 random matrix function A([, w) = [aij(t, co)] has elements aij(t, CL))which are product measurable and M[laii(f, w)l] are locally integrable on R,. In case of A(r. o) the, relations similar to (2). (5). (6). (7) and (8) can be formulated by replacing aij(w). c(Jo). ccij(tu) and p[A(cu)] by aijt, w), sr,(t, w), rij(t, w) and p[A(t, w)], respectively. However. 23 T.B.

33s

G.

to insure the asymptotic replaced by :

S. LADDE

stability with probability

one, the relation

(9) 1s

for some TV> 0. ( 16) From (II), the relation (16) is equivalent to the following relation

(17) -d,~ I ~ dilu,j(S, w)J ds 2 u =l. i=l i#j 11 1 In order to study the effects of the strengths of interactions among the species as well as the strengths of the random disturbances on the process, assume that the random matrix function A(r, w) in (15) has the following form A(t, w) = B(r) + C(r, w), (18) where the II x n deterministic time-varying matrix B(t) = [bij(t)] whose elements bij(t) have the form: -PiCt>

+

,Ez,i,

IpiPpiCt)

+

,gi,

IpiPpiCf)*

i=j,

bij(t) =

(1’))

i # j, 1 /ijPijCt>* Zij are the elements of deterministic interconnection matrix L, and the nature of lij and their functions are similar to the eij; furthermore,

the II x N random matrix function C(r, w) = [C,,(r, oj] whose elements cij(t, o) have the form similar to (5) with ai(o) and ccij(to) are replaced by qi(t, o) and qij(t, o), respectively. In this set up, the interconnection matrix E will be referred as random interconnection matrix. In order to study the deterministic and stochastic interactions, the stability condition (17) can bc remodified as

-[Cjj(S, CO)]- dj ’ i i=l i#j

dif~ij(s, CO)/

STABILITY

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Note that one has made use of the fact that /t[A(t, o)] = ,u[B(t)+C(t, w)] < MW) +dW 41. In particular, if the II x IZ matrix B(t) is constant matrix and the elements c,~I:~, IO) of the random matrix C(r, CO)in (18) are strictly stationary metrically transitive random process (Doob, 1953), then the stability condition (21) reduces to : P QJ; -[bjj+cjj(f,, c

~)l-d~li$l

di[lbijl i#j

+)Cij(t,,

W)]] 2 U, Vi E I

= 1. (22)

1. Note that the special form of A(r, W) in (18) is not very restrictive assumption. In fact, any A(t, o) in (15) that satisfies the relation M[A(t, Q)] < co, can be rewritten as (18). For example, set w)]. Bit) = M[A(r, N-J, and C(t, o) = Aft, w)-M[A(r, (23)

Remark

It is also appropriate to remark that all our stability conditions yield the stability in probability, stability in the mean and the stability with probability one. From the above stability analysis, one can draw a few conclusions about thle measurability of the complexity of the system, the measurability of the random effects, the invariability of the stability of the system under both deterministic and stochastic structural perturbations, the tolerance of the complexity by the stable system, the tolerance of the random disturbances by the stable system and the reliability of the stability of the system. These issues are very important in the stability analysis of the stochastic competitive system. In the following I shall analyze the above-mentioned conclusions relative to (1); however, the conclusions relative to (I 5) can be formulated, analogously. The satbility condition (11) of (l} gives an estimate of the magnitude of hoth deterministic and stochastic interactions. P CO:cljaj(o)

> i$ d(]Uij(CO)l +dj [ C pel-(i) it i

as a measure of complexity state of (1).

iipjcCpj(W)

that assures the stability

of the equilibrium

340

G.

S.

LADDE

If one assumes that the random matrix A(o) in (1) has the special form A(o) = B+ C(o) as in (IS), then the corresponding stability condition (21 J -bjj-d,”

i$l dilbijl + iij I[

-c,CCO)-L~J”~$,

1

diIcij(w)( i+j

2 a, YjE I I = 1, (25)

which is explicit in terms of deterministic and stochastic interactions. Therefore, for the given deterministic process rate matrix B, (25) gives an estimate of a random parameter matrix C(W), -bjj

-dJ

’ i$l di[b,jl

cjj(w> +di ’ ,,tl diICij(W)[ i#j

9 I

VjjEI =1, I

as a measure of random effects that assures the stability of the equilibrium state of the system (1). Similarly, for the given stochastic rate matrix C(w), (25) gives an estimate of a deterministic rate matrix B. Thus, in the stochastic competitive process of self-inhibitory species, the estimate of the magnitude of both deterministic and stochastic interactions as a measure of complexity and the estimate of a random disturbance as a measure of random effects are determinted by the quasidominant diagonal property of the matrix A(w) associated with the stochastic competitive system (1). The stability condition (25) is valid for both deterministic and stochastic fundamental matrices L and E. However, (25) is also valid for all deterministic and stochastic interconnection matrices L and E. In fact, as indicated before, if one replaces the unit elements of L and E in (25) by any elements lij nd eij that are between zero and one, then the relation (25) still remains true. In addition, this relation will also be valid for all both deterministic and stochastic structural perturbations. Thus, the stability of the equilibrium state x*(w) of (1) is indeed connective stability, and thus showing the invariability of the stability of the stochastic competitive process under both deterministic and stochastic structural perturbation. Furthermore, note that, if L = 0, that is, the species in the competitive system (1) ignore the deterministic links bewteen them, then the stability condition (25) reduces to: P 01 (/lj-~jj(~)-d,r’~$, i

Similarly,

dilcij(o)l

2 U, Vj E I

i#j

if E = 0, that is, the species in the competitive

= 1. 1

system (1) ignore

STABILITY

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SYSTEMS

the random links between them, then the stability condition -bjj-d,”

i

341

(25) becomes

dilbijl 2 ~1,Vj e I.

i= 1

This yields the stability of the equilibrium state of the ordinary competitive system corresponding to (1). The problem of complexity vs. stability in model ecosystems (Ladde & Siljak, 1975) can be extended to the competitive systems in a unified and natural way. As shown (Ladde & Siljak, 1975), the diagonal dominance c’ondition provides a partial solution to resolve this problem. Analogously it can be stated that the condition (11 j or (25) provides the partial solution to the complexity vs. stability problem. In the case of white noise coefficients (.Ladde & Siljak, 1975) the random disturbances are destabilizing agents, however, in the case of non-white noise coefficients such as strictly stationary and metrically transitive random coefficients, the random disturbances may be stabilizing agents. This statement can be justified from (11). Note that this fact confirms the remark of Khas’minskii (1966). Hence, it leads us to the problem of “deterministic vs. stochastic” in the competitive process. A partial answer to this problem is given by the diagonal dominance condlition (I 1) or (25). Thus, in the competitive process of self-inhibitory species under random disturbances, the tolerance of a random shock and the tolerance of a complexity of both deterministic and stochastic by the stable isolated system, are given by the quasidominant diagonal property of the matrix A(o) associated with the system (I). It should be noted that the stability conditions are only sufficient, therefore, it is not possible to use them to establish the critical limit of both deterministic and stochastic complexity. This drawback of the conditions is balanced by the reliability of the stability that follows from the quasidominant diagonal property of the random matrix A(o). In fact (11) implies ihat any decrease of the magnitude of the strength of interactions among the species can only increase the margin of stability. Further we remark that so long as the self-inhibitory rates are strong enough to overcome the sum of the magnitude of both deterministic and stochastic interactions, the competitive system (1) is stable regardless of the nature ot interactions (activational, inhibitory, neutral, etc.) among the species. 3. Non-linear Random Competitive Systems For competitive processes, the linear systems of differential equations with random coefficients represent the simplest mathematical approximations for actual competitive processes in randomly varying environment. In this

342

G. s. LADlIE

section, 1 develop the stability analysis of the preceding section closer tu the reality with the expense of a more refined mathematical analysis. Consider the non-linear time-varying stochastic competitive systems described by the system of random differential equations: k(t, co) = A[t, x(t, w), w]x(t, Q) (26) where x E S[O(O, p)] is the state vector, D(0, p) is defined by: D(O,p) = {XERY [Xi/ < piviEz) 07) andW(O, PII denote the set of random vectors that belong to D(0, p) w.p. 1, and some pi > 0, V i E I; the n x n process rate random matrix function A:R+ x D(0, p) x Sz + R”‘. I shall assume that A is smooth enough to assure the existence and uniqueness of sample solutions x(t, w) = x[t, t,, x,,(o), w] of (26) for all t > to, (r,,, xJR+ x S[D(O, p)]. Because of the physical significance, I shall also assume that for [to, x,(o)] E R, x S [D+(O, p)], x(t, w) E S[D+(O, p)] for all t 3 t,, where D+(O, p) = (x E D(0, p): 0 < Xi < pL V i E I}. That is, all solution processes of (26) are nonnegative for all t > to, whenever their initial data are non-negative. For detail sufficient conditions for the non-negativity of solution will be reported elsewhere. The non-linear systems, in general, admit a multiplicity of equilibrium states. This fact forces one to investigate the stability behavior in the vicinity of a given equilibrium state in which one is interested. In the following stability analysis, it is enough to assume that x*(o) = 0 is the only equilibrium state of (26) in S[D(O, p)] w.p. 1. Otherwise, for a given isolated equilibrium state x*(o) # 0 of a given system, one can always find pi > 0 for V i E I so that the system that is defined on : s[D(x*, p)] = {x E S(R”): I.+Xil < pi, Vi), possesses only one equilibrium state x*(w) in S[D(x*, p)], where S(R”) is a set of all n-dimensional random vectors that belong to R”. Now, by using the linear transformation y = s-x*, the given system with the equilibrium state x* # 0 and the region S[D(x*, p)] reduces to the system with the equilibrium state y* = 0 and the region S[D(O, p) 1. In order to derive the conditions for connective stability of the system (26), assume that the elements Uij(t, x, w) of the process rate random matrix function A(t, x, w) have the following form: - 4i(r7 X3 W>+ C epi(t)4,i(t7 X3 W> pEr-(i) Uij(l,

X,

0)

=

+

C

e,i(r>$,i(t,

X,

CO), i -

.i.

(28)

PEI(i)

i

eij(t)$ij(t,

X9

0)~

i Z

.i3

for all (I, x) E R+ x D(0, p), where @,, q5ij:R+ x D(0, p) x R -+ R are random

STABILITY

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343

rate functions of (t, x), Z-(i) = {pi /:~Fi(t,X, W) < 0 for all (t, x) E R+D(O, p) w.p.l), I(i) = Z-Z-(i). Note that di is the rate of excretion from the ith species, rpij is the rate of the jth species that influences the ith species in the sornpetitive systems under random perturbations; the functions eij.. R, + [0, 11 which are elements of the n x n interconnection continuous matrix function E(t) = [eij(t)]. To show the usefulness of the special nature of the process rate random matrix A(t, x, w), E(r) can be described in the framework of itheearlier work (Ladde, 19766). Note that the interconnection function E(t) whose elements are deterministic, however, these deterministic coefficient\ Gus. can be replaced by the random coefficients eii(t, w):R+ x Q -+ [O. I]. ,4ssume that the random functions c$~,+ij in (28) satisfy the constraints: ‘$,tt. x* cl)) 2 Xi(t* W), Zpj@pitt>0) < -Epi(t)$pi(f.

X-9U4 d L.piCcpi(t,(0) for p E I I

and ICpi(f)(/Jp,(f+X, OJ)~< PpiZpi(t, W) for /j E I(i), (1. .Y)E R,

X

D(0, 11) ViEI,

(29)

where Ri(t, 0) > 0, j3Jt, w) > 0 for p E I-(i). cc,,(t, ~0) 3 0 for /j, i E I, furthermore cli, ,8,i, ccPi:R+x Q --) R are sufficiently smooth random functions, and satisfy:

Note that the constraints (29) reflect the saturation effects. In fact. it is reasonable to expect that if the state of the process is nearer to the equilibrium then the rate coefficients tend to constants. On the other hand, if the state of the system is far away from the equilibrium, then the evolution will take place. Furthermore, these conditions include any sign pattern, thus allowing mixed competitive-inhibitive-activative-cooperative-neutral interactions among the species in the process. Now, one may state the definition of connective stability for random systems. D+tilion I. The equilibrium state x”(o) = 0 of the system (26) is said to be: (i) connectively stable in the mean if and onlv if it is stable in the meanfor all irrterconnection matrices E(t); (ii) connectively stable in the probability if and ouly if it is stable in the probability for all interconnection matrices E(t); (iii) connectively stable with probability one if amf o& {f it is stable with probability onefor all irtterconnected matrices E(t ).

I shall state below the stronger definition significant quantitative aspect of stability.

of stability

that provides a

344

G.

S.

LADDE

Dejinition

2. The equilibrium state x*(o) = 0 of the system (26) is said to be. connectively stable in the mean if and only lf there exist positive numbers K, a, and a positive vector 6 independent of initial data (to, x0) E R, x S[D(O, p)] such that (31) M[IIx(f, u~)jl] < KM[ll.~,(w)ij] exp [-aft-f,)], t 3 to for all (to, x0) E R, x S[D(O, S)], 0 < di < pi, V i E I and fbv all interconnection matrices E(t); (ii) exponentiallJ1 connectively stable in theprobabilit~* if and only iffor any E > 0, q > 0 there exist positive numbers a and b snck tltat (32) P{o:I\x(t, co)/\ > cj d tj exp [-a(t-to)], t >, to for ah interconnection matrices E(t), u~henever P(o: //.Y~(uJ)/~ 3 fl) 6 '1; (iii) exponentially connectively) stable with probability one if and onlyl (f there exist positive numbers K, a, and a positive vector 6 independent of initial data (to, x0) E R, x S[D(O, /?)I such that

(i) exponentially

IIx(t, w)l/ 6 IIM~O(co)IIKexp [-act-t,)], t > to w.p.1 (33) for all (t o, so) E R, x D[S(O, 6)], 0 < 6, < pi, V i E I and all interconnection matrices E(t). Remark 2. Note that our stability definitions are local in nature. If, on the other hand, pi = co, V i E I, then D(0, p) = R”, S[D(;O, p)] = S(R”), and

the corresponding concepts of stability would

be of global character.

A(t, ~1 = [Eij(t, w>], re Presents the II x IZ time-varying random matrix defined by - ai(t, W>+ 1 ep$pi(t> W) + 2 epi”pi(t, 019 i = j9 [ PEI-(l) 0) = (34) zijcrij(t, w), i # j, { where ~ij are entries of the fundamental interconnection matrix E that is defined in section 2; CQ,clij, fiij are as defined in (29) and (30). Consider the linear auxiliary or comparison random competitive process described by the system of random differential equations e(t, 0) = A(& to)u(t, IJJ), (35) where u E S(R;), and A(r, W) is as defined in (34). For the system (35), one first proves the following result. The validity of the following result will also justify the algebraic stability conditions that are stated in section 2. In the following, I will use the genera1 stability condition (I 6), however, other special types of stability conditions (21)and (25)can beused similarly. BY

Sij(t,

1 PEI(i)

Lemma 1. Assume that the II x n random n1atri.x function &t, OJ) in (34) satisjies the relation (16). Then, the equilibrium state u*(w) = 0 of‘ the comparison competitive system (35) is exponentially stable with probability one.

STABILITY

Proof. As mentioned

OF

RANDOM

COMPETITIVE

345

SYSTEMS

in section 2, I will be using the norm of u [/u/ld

=

i; i=l

dilUil

for some cl, > 0 and norm of A (l,w) as defined in section 2. Choose v(u) = /(u/Id as a Luapunov function for (35). Note that: ~(llull) G w G c(lIuJl) whxe

(36)

: Mllull)

=

4&/(,

c(ljul/)

=

lld(l

. ljujj,

4

= $;

di.

Using (6) and (7), one computes D&,,v(u) as: DA,,v(u) = lim sup i [v(u+It;l(t, h-+0+

w)--\.(u)].

(37) where bij = 0 for i # j and hij = 1 for i = j, j E 1. From (37), one has

i#j

+di

’ i$,

i=l

5 p[X((t, o)]v(u) w.p.1.

di)sjj(t,

IIV(U) it-j

346

By the application

G.

of comparison

S.

LADDE

theorem 4.1 in Ladde (1976c), we have

v[u(t, co)] I v[u,,(w)] exp j /&X(s, o)] ds , t 2 f,,. I L(0 From (16) and (39), one obtains

(39)

v[u(t, w,] < v[u,(Cu)]Rexp where c1> 0 as in (16) and

(40)

[-+-lJJ

t > f,,,

iT 2 exp i p[A(s, co)] ds I Lto for I, < t < T, and the existence of T follows from (16). From (36). (40) reduces to the inequality (33) with K = d;’ ildl]E. Thus establishing the global exponential stability property of (35). This completes the proof of the lemma. Remark 3. From (40) it follows that the equilibrium state u*(w) = 0 of (35) is exponentially state in the mean. This together with the Markov inequality (Wong, 1971) implies that the equilibrium state u*(w) = 0 of (35) is also exponentially stable in probability.

We have the following result which establishes the stability ibrium state x*(o) = 0 of (26) in the finite domain D(0, p).

of the cqull-

Theorem 1. Assume that the system (26) satisfies the constraints (29 )and (30). Further assume that the random matrix function A(t, o) in (34) satisfies ( 16). Then the equilibrium state x + = 0 of (26) is exponentiallv connectively stable with probability one.

Under the hypothesis (16), the equilibrium state II::: = 0 of (35) is exponentially stable with probability one. Furthermore, note that the random function &, o)(u is quasimonotone nondecreasing in u for fixed t w.p.1 (Ladde, 1976~). Let r(t, w) be maximum solution process of the random auxiliary equation (35). To conclude the exponential stability with probability one of the cquilibrium state s* = 0 of (26), choose the vector Lyapunov function V:D(O, I’) --f RF defined by

Proof

Vx) = [Vl(Xl>, V*(x,>, . * . 9 CICG, where Vi(xi) = Ixil. It is obvious that V(x) is Lipschitzian note that the function, V(X)

= @ d,l/,(Xi) i=l

(41)

in x. Further

STABILITY

OF RANDOM

satisfies the relation (36). definite and decrescent in as follows : 1, if fFi = 0, if ( - 1, if

COMPETITIVE

347

SYSTEMS

This implies that the function v(x) is positive x. For V i E Z, we also define pi relative to (26) Xi > 0, or if Xi = 0 and fi > 0, xi = 0, and pi = 0, xi < 0, or if xi = 0 and pi < 0.

(42)

Now, by using (28, )(29), (30), (34), (35), (41) and (42), one computes D&,&(.q)

= aiii = (Ti ~ Uii(t, X, 0)Xj j=l

I jil

Sij(t> W>Vj(Xj>qVj E J, (1, X) e R+ X D(0.t p)q

which is equivalent to: D&)!‘(x)

I z(r, w)V(x) w.p.1 for (t, x) E R, x D(0, p).

By applying the comparison

(43)

theorem 4.1 in Ladde (1976c), one obtains

Jqxk o)J < r(t, w), 2 2 f, (44) Provided V(x,) < uO, where r(t, w) = r(t, to, ue, w) is the maximal solution process of (35), and x(t, w) = x(t, to, x0, o) is a solution process of (26) through (to, so), From (36), (40), (44) and the definition of V, one has:

d+(C w)jll Q v[xO,41 G v[rO, ~11 < v(u,)R exp [ - a(? - f,)]

(45)

as long as x(r, w) E S[D(O, p)] w.p. 1. Let us consider: R,(O, p) = (x E R”: v(x) < yj,

(46)

where : y = min b(pi). isI

Note that R,(O, p) c D(0, p). Choose V(x,) = ue for x0 E S[D(O, p)]. Since v(x) is continuous, it is possible to find a vector 6 > 0 such that x0 E S[D(O, S)] implies v(xo) c y/K w.p.1. Note that S[D(O, a)] c S[R,CO, p)] c S[D (0, p)] w.p.1. Now one claims that: (47) whenever .u,S[D(O, S)]. If (47) is not true, then there exists t, and 52, c R with P(Q,) > 0, v(x, t, w) < /Aw.p.1 for t E (to, tr) and v[x(to, to, x0, CD)] = y for co E 12,. This implies that x(t, to, x0) E S[D(O, p)] w.p.1 for l,, ], t 2 to, w.p.1,

348

G.

S.

LADDE

t < 1,. From (45) and choices of x0 and u0 leads us to a contradiction Y = v[x(t~, t0, x0, o>] G y[uo(w)] exp [--act, -to)] < y for w E 52,. This proves the validity of (47), provided x,, E s[D(O, S)]. From (47), (4.5) holds for all t > t,,, whenever .x0 E S[D(O, S)]. Therefore, from (4.5) and the nature of b, one has: (48) Ilx(t, 411 < ~llxo(w>II w [-W-to)], t 3 to, for all (to, x0) E R+ x S[D(O, S)], where K = d,’ /djlK, Further we note that the inequality (43) is valid for all interconnection matrices E(r). Therefore, the exponential stability w.p.1 of the equilibrium state x* = 0 of (26) is indeed connective property. This completes the proof of Theorem 1.

In the following, I formulate the result that establishes the global exponential connective stability of the equilibrium state x” = 0 of (26). Theorem 2. Let the hypotheses of Theorem 1 be satisjed with pi = CYJfor ever!* i E I so that D(0, p) = R”. Then, the equilibrium state x* = 0 of (26) i.r globally exponentially connectively stable with probability 0~. Proof. By imitating

the proof of Theorem 1 and noting the fact that the relation (43) holds for all (t, x) E R, x R”, the proof of the Theorem follows immediately. 4. All remarks, conclusions and comments made in section 1 relative to linear random competitive system (1) can be reformulated in the non-linear random competitive system (26).

Remark

4. Hierarchic

Random Competitive Systems

Several competitive systems in biological, physical and social sciences are hierarchic in nature. In this section, I would like to preserve partially the originality of hierarchic random competitive systems with respect to its stability hahavior. This is achieved by extending the decomposition aggregation scheme (Ladde & siljak, 1975) that was developed for It6 type stochastic differential equations, to the system of differential equations with random coefficients in context of vector Lyapunov functions (Ladde, 1976c). Further details about the important features of this method, see Ladde, 1976a; Ladde & siljak, 1975. Consider a random competitive system described by the system of random differential equations k(t, w) = f Cc x0, WI, 01

(49)

STABILITY

OF RANDOM

COMPETITIVE

SYSTEMS

349

wlherc x = x(t) E S[D(O, p)] is the state variable, the random rate function ,f: R+ X D(O, p) X R --) S(Rm), D(O, /-)) = :X E R”: llX.ill

(.u2)T, . . ., (x,)‘]‘, and T stands for transpose of a vector or matrix; for V i E f, pi is a positive real number. 1 shall suppose that f is: smooth enough to assure the existence and uniqueness of sample solution xfr, t,, x0, m) of (49) for all t 2 to, (to, x0) E R, x S[D(O, p)]. Briefly, one formulates the decomposition-aggregation technique relative to (49), however, for more details see Ladde & Siljak (1975). Decompose the system (49) into n interconnected subsystems described by

x = [(x,)‘,

ai(t,

W)

=

C!i[t,

Xi(tp

O),

W]

+bj[t,

X(t,

CO).

011,

i E 1,

(SO)

where .I+;E S[D(O, pi)], D(0, pi) = (X E R”fi)ll~iII < fli), and .yi is the state of the ith subsystem, and represents the ith component of the state vector x. lm (50), the random function a,:R+ x D(0, pi) x Sz --f S(R”‘lj represents the interactions among species within the ith subsystem: the random function bi:R+ x D(0, p) x Q --f S(Rm’) represents the interactions between the subs:ystems. Assume that: hj(tt X, W) = 6,(t, ei,X’, eizX*, . . .f Ein.Yn,lIr3J,V i E I, (51) where eij:R+ --f [0, I] are coupling functions which are elements of the n x II interconnection continuous matrix function E(r). Without loss in generality, assume that a,(t, 0, (!I) z 0, b,(t, 0, III) s 0. This implies that x*(o) = 0 is the equilibrium of the system (49). When E(r) E 0, from (50), one gets the isolated subsystems described by ~i(t, W) = Ui(t, .~i(t, 0). (O),V i E I

(52)

which has the equilibrium state x*(w) = 0. For each subsystem in (52), assume that there exists ;I scaler random function Vi:R+ x D(0, pi) x !S + R, such that vi(f, xi, o) is sample continuous on R, x D(0, pi). Further assume that the scalar random functions I’i(t, Si, u), the interaction functions satisfy the following conditions: (53) IViCr9xi, w)- T/,tt9 Yi, O)l G ki(t9 t”)Ilsi-!‘ijj7 for (t, x,), (t, Jpi) E R, x D(0, pi), where ki(t, OJ = ki(t, w, /-,i) :> 0 is a product measurable random function defined on R, x R -+ R, : bil(llxill> d li,(r7 .yi7co) < ai2(ll*~ill)

(54)

ibr (t, xi) E R+ x D(0, pi), where LZil. ai, E K;

D;52)q(t, Xi, (0) 5 -ai(t, O-()Wi[l’i(t, Xi, o)]

(55)

350

Ci.

S.

LADDE

for (t, xi) E R, x D(0, pi), where a,(t, 0)) is a product measurable, random function defined on R, x CJ-+ R,, W, E K, where DGzjVi(t, xi, o) is defined by D&,I/,(t,

X,

O) = lim

~p~{~[t+h,

xi+hUi(t,

x;,

CD)]-&(t,

xi,

CO)}

h-+0+

which implies by the Rademacher theorem (1918) DGz)I/i(t,

Xi*

O) = i q(t,

Xi,

o)+[grad

almost everywhere on its domain; and for (t, x) E R, x D(0, p) [grad K(t, Xi, 0)‘. bi(t, X, 0)] 5 -t C pd(i)

ZpiEpi(t,

1

0) l+$[K(t,

Xi,

q(t,

Xi,

UJ)lTUi(t+

(56)

Xi,W),

C Zprppi(t. W) 1Per- (0

O)] + i

Sij@ij(t,

O,>Wj[Vj(t,

Xi9

O)],

(57)

j=l j#l

where clij(t, o), ppi(t, o) are product measurable random functions defined on R,xQ-+ R+, Pi, are elements of the fundamental interconnection matrix E. Further assume that the first moments of cli, clij, /jij and ki are locally integrable on R,, and also assume that they satisfy the relation (30). In the following, I shall make use of the definition of&t, o) in (34) with respect to the cli, gij, /Iii, Zrj that are as defined in (55), (56) and (57). Consider the comparison random competitive system described by the system of random differential equations li(t, 0) = A(t, w) W[u(t, co)] (58) where u E R; ; X(t, w) is as in (34); and W(u) = [W,(u,), . . ., W,(u,)]‘. Now one may state the following result whose proof can be formulated by following the arguments used in Theorem 1 (Theorem 5.3 in Ladde, 1976c, and Theorem 3 in Ladde & Siljak, 1975). Theorem 3. The equilibrium x * = 0 of the system (49) is asymptotically connectively &table with probability one if the random matrix function in (34) satisfies the relation (16). To appreciate constraints (57) imposed on the interconnection functions, under specialized conditions, a corollary similar to Corollary 2 in Ladde & Siljak (1975) can be formulated analogously. To avoid monotonicity, I do not want to discuss further details. Note that the theorem similar to theorem 2 for (49) can be formulated in the same manner. Furthermore, the remark similar to remark 4 can be made relative to the system (49). I omit the details.

STABILITY

OF

RANDOM

COMPETITIVE

SYSTEMS

351

5. Examples In this section, I present several well-known examples of competitive cooperative processes in biological, physical and social sciences. To avoid monotonicity, 1 do not want to discuss them in detail: however, 1 will identify the terminology used in the preceding sections with the specific terminology used in the specific competitive processes in biological, physical and social sciences. (A)

CHEMICAL

SYSTEMS

A chemical process can be considered as one of the members of the family fi)f competitive processes. By identifying the concepts of state vector, rate matrices both stochastic and deterministic and rate functions both stochastic and deterministic of the competitive process by the concentration vector, reaction rate matrices both stochastic and deterministic and rate functions both stochastic and deterministic, respectively, our stability analysis of random competitive systems reduces to the stability analysis of random chemical reaction systems. In chemical systems, the randomness arises mostly because of the fluctuations in the environment such as in the ionic strength, temperature, acidity or basicity, concentration of certain substance. These facts have been surveyed in a recent monograph by Goel & Richter-Dyn (1974). (B)

ECOSYSTEMS

,411ecosystem is another example of the competitive systems. A great deal of experimental as well as analytical evidence has been surveyed by Goel & Richter-Dyn (1974) May (1973), Smith (1974) and Pielou (1969). If one replaces the concepts of state vector, rate matrices, rate functions, inhibitionact.ivation-cooperation and subsystems by the population vector, community matrices, community rates, competitive-symbiotic-saprophytic and sulbcommunities, respectively, then the preceding stability analysis reduces to the stability analysis of ecosystems under randomly varying environment. Furthermore, we note that the present stability analysis differs from the earlier stability analysis (Ladde & Siljak, 1975). (C)

GENETIC

SYSTEMS

If one replaces the concepts of state vectors, inhibition, activation, rate matrices, rate functions and subsystems in the preceding stability analysis of competitive system by the concepts of concentration vectors, induction, repression, reaction rate matrices, reaction rate functions and operon units in the genetic systems, respectively, then the stability analysis for stochastic

352

G.

S.

LADDE

genetic systems follows, analogously. The randomness in the genetic system arises due to selection intensities, change in environmental conditions and sampling of gametes. For further details, see Gole & Richter-Dyn (1974). (D)

PHARMACOLOGICAL

SYSTEMS

As stated in the earlier examples, if one substitutes the concepts state vector, rate matrices, rate functions and subsystems by the drug concentration vector, transport rate matrices, transport rate functions and units, then the stability analysis of the pharmacoiogical systems follows directly from the previous discussion. Very recently, by considering the random rate constants, the joint probability distribution of the drug concentration is determined by Soong (1971), and further, the analysis and identification of stochastic compartmental models in pharmacokinetics has been studied by Chuang & Lloyd (1974). The randomness in rate functions are due to cxperimental data, environmental effects, variations of patient parameters, etc. Those factors have been well documented by Jacquez (1972) in the context 01 compartmental systems. (E)

PHYSIOLOGICAL

SYSTEMS

Several physiological systems can be considered as the competitive systems. It is natural to assume that the rate functions are subject to fluctuations in living things, for example, the renal excretion of most materials depends on renal blood flow (Jacquez, 1972), transmission of nerve impulses (Goel & Richter-Dyn, 1974). The stability analysis of the preceding sections can be applied directly to the physiological systems such as circulatory system and nervous system in a systematic and unified way. (F)

ECONOMIC

SYSTEMS

The stability analysis of the random competitive systems described in the previous sections also represents the stability analysis of the competitive market systems. In this case, the slate vector, rate functions and subsystems are replaced by the price vector, excess demand function and subeconomics, respectively. In the framework of this, our stability analysis for stochastic economic systems contains not only gross substitutes (Arrow & Hahn, 1972) but also gross complements, and further extends to the non-linear timcvarying random excess demand adjustment process in the competitive market systems. The randomness in the excess demand function arises due to unforeseen exogenous factors such as changes in taste or technological uncertainties, etc. Furthermore, note that our stability analysis not only extends and generalizes the stability analysis of Turnovsky & Weintraub (1971) but also provides a structural insight to the stability analysis.

STABILITY

OF

RANDOM (G)

COMPETITIVE

ARMAMENT

353

SYSTEMS

SYSTEMS

Hy replacing the state vector, inhibition, activation, rate matrices, rate functions and species by the armament vector, fatigue, defense. armament rate matrices, armament rate functions and countries, respectively, the stability analysis exhibits the stability analysis of random armament systems. The random disturbances in armament system arises due to several unforeseen exogeneous factors such as changes in economics, changes in governmental policies, etc. We note that the formulation and stability of random armament systems extends the formulation and stability of the deterministic armament systems (Richardson, 1960: Saaty. 1968; !%ljak, 1975a.b.c) in ii natural

way. (H)

PSYCHOLOGICAL

SYSTEMS

The preceding stability analysis is directly applicable to psychological systemssuch as learning patterns (Grossberg, 1972). Here the statle vectors of the competitive processare replaced by the short-term-long-term memoq traces. The randomness in learning patterns can arise due to past inputs, change in size or spiking thresholds and arousal level, and etc. (1)

SOCIAL

SYSTEMS

Finally, 1 present another competitive process, namely. the competitive social process. The study of deterministic competitive social processhas been initiated by Homans (1950) and Simon (1957). Very recently. Sandbcrg (1974) has attempted to develop a mathematical theory of interaction in social groups in a systematic way. The present stability analysis of random competitive systemsprovides a natural framework for both formulating and testing the stability properties of stochastic competitive social systems. Note that the exogeneous functions or parameters that are incorporated in Simon (1957) and Sandberg (1974) can be considered as random functions 01 parameters which take into account the number of externally imposed uncertain activities. The researchreported herein was supported by SUNY ResearchFoundatloll Fi.tculty FellowshipNO. 0407-01-030-76-O. REFERENCES ARROLV,

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