1. theor. Biol. (1977) 66, 345-359
The Stability of Ecosystems-A
Finite-Time Approach
LILIAN SHIAO-YEN WV
IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598, U.S.A. (Received 24 December 1975, and in revised form 27 July 1976) The objective of this paper is to present some concepts of stability which originate from ecological considerations, discuss their relationship with existing concepts of stability, and furthermore to establish sufficient conditions for stability which are both robust in the sense of Levins and Slobodkin, and constructive. The essential notion of stability for unper-
turbed ecosystems, applied in most cases,is that although fluctuations in particular variables exist, each remains within certain bounds (which may be time-varying). The use of this notion within a finite-time horizon is the framework in which total, essential, and terminal stability for both perturbed
and unperturbed ecosystems are defined. These concepts are studied and sufficient conditions for stability are given. In particular, theorems 2 and 3 provide conditions to ensure that each variable stays within bounds and theorem 1 a way to identify the occurrence of large fluctuations outside the bounds, by examining the system at “boundary”points only. The strength of these results lies in the criteria for stability and instability: in addition to being robust and constructive, they can be applied directly to the differential (or difference) equations, without any knowledge of the solutions. Relationships between our concepts and the existing concepts of asymptotic and neighborhood stability, and Qersistance
are also stated.
1. Introduction and Motivation Previous mathematical approaches to studying stability of ecosystems have frequently involved the application of established mathematical concepts and theorems to suitable ecosystem models. Here we try a different approach. The objective of this paper is to present some concepts of stability which originate fromecologicalconsiderations,discusstheirrelationshipswithexistingconcepts of stability and furthermore to establish sufficient conditions for stability which are both robust (Levins, 1966; Slobodkin, 1974) and constructive.? 7 Constructive sticient conditions for stability are conditions which can be directly applied to determine the stability of the system, as opposed to sufficient conditions which depend on the existence of functions with certain properties. For example most results of the Liapunov theory of stability (La Salle & Lefschetz, 1961; Brauer % Nohel, 1969, Chapter 5) are of this nature. A difficulty with such results is that their applicability is always limited by methods for constructing the necessary functions. 345
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wu
After reading much of the literaturet on the stability of ecosystems, we observe that there are basically two groups of deterministic concepts. One group originating from physical systems, and containing concepts such as neighborhood stability and asymptotic stability, is attractive because the mathematics is well-developed. The other group includes the conceptually more appealing but mathematically more difficult concepts such as structural stability and global stability. All the proposed concepts are based on the notion that some unperturbed “behavior” defines a standard and stability is studied by comparing the perturbed system with the standard. However these concepts frequently diverge from the notion of stability in an ecological context. Furthermore, they often seem ill-suited for the purpose of managing ecosystems. Others (Slobodkin, 1965; Preston, 1969; Cohen, 1971; Innis, 1974; Holling, 1974; Botkin & Sobel, 1975, etc.) have criticized such approaches. In particular, Botkin & Sobel (1975) also propose other concepts of stability, one of which will be discussed in section 5. To motivate our concepts, we first discuss some of the shortcomings of these proposed definitions. The difficulty of specifying a dynamical system and its inputs for large times often makes the use of criteria of stability which depend on an infinite-time horizon unsuitable. In this paper we propose concepts of stability which lie within a finite-time framework.8 The concept of stability over a finite-time interval is not new. Weiss and Infante first introduced and developed a theory of finite-time stability parallel to the classical Liapunov theory of stability (Weiss & Infante, 1965). Their work has since then been extended by them and others (Weiss & Infante, 1967; Weiss, 1968; Lam &Weiss, 1974; Kayande & Wong, 1968; Garrard, 1969; Kaplan, 1969; Kayande, 1969; Michel & Wu, 1969; Ben Lashiher & Storey, 1972). However, their concepts were largely motivated by problems in control theory and engineering where stability means: given that the initial state of a system belongs to a fixed set, trajectories of the system then stay within fixed bounds over a prespecificed interval of time. Hence their definitions do not always coincide with concepts of stability originating from ecological considerations. Another difficulty with using this theory is being a theory parallel to the Liapunov theory of stability, sufficient conditions for stability depend on the existence of “Liapunov-like” functions. The usefulness of the theory is therefore limited by there being no general method of constructing these “Liapunov-like” functions. A shortcoming of those definitions of stability which involve steady states is that special emphasis is placed on equilibrium points which are a con1 For an overview on the subject, see Woodwell & Smith (1969), and May (1973). $ Note, however, that kite-time does not rule out considering large finite-times approaching infinity when an infinite-time scale is desired.
STABILITY
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347
sequenceof the chosen model. Furthermore, in the definitions of neighborhood and asymptotic stability, the size of the relevant neighborhood is not specified. Thus it may be unlimitedly small and therefore unmeasurable, and as indicated by Lewontin (1969) and Boorman (1972), provides no information about what is likely to occur under larger perturbations. The essential notion of stability in many instances is not that particular state variables of interest? (e.g. density of species,rate of oxygen consumption) remain constant, but that although fluctuations in these variables exist, each remains within certain bounds which may vary with time. It is obvious that bounds which vary with time are more general than bounds which do not. But more important for natural populations there are frequently cyclic variations in (external) inputs to a system which cause well marked population cycles, such as the seasonalvariations in population density of phytoplankton in northern temperate lakes (Odum, 1971) thus from the standpoint of data collection in the modelling of ecosystems there is empirical value in a definition of stability whose bounds vary with time. The notion of stability meaning particular variables remain within certain bounds arises in several ecological concerns. The minimum criterion for ecological stability is the existence of an ecosystem, as stated by Preston (196% “An ecological system may be said to be stable, from my point of view, during that period of time when no species become extinct (thereby creating a vacant “niche”) and none reaches plague proportions, except momentarily, thereby destroying the niches of other speciesand causing them to become extinct.” We use bounds (which may vary with time) to denote the “healthy” range of each variable in the ecosystem, or for the ecosystem as a whole. For the population$ variables, this range is defined as follows. The lower bound of the range is a number such that the population either faces the danger of becoming extinct or seriously disrupts the ecosystem (Paine, 1966) when it falls below this value.5 This concept is also referred to in Allee’s principle (Alle, Emerson, Park 8z Schmidt, 1949; Odum, 1971). The upper bound is harder to define. Certainly it can be no greater than the carrying capacity of t The term “state variablesof interest” is usedto distinguishbetweenthe variables includedto specifythe systembut are not of interestto us,and thosethat are. We make thisdistinctionbecause oftenin practicethisdistinctionismade,We makethe assumption that all the variablesof interestare(in the language of systems theory) observables, or that theyarevariableswhichcan bemeasured. $ The word populationis usedvery looselyhere, it can Lx one specific species or an aggregate of species with similarfunctionalroles(i.e. a functionalgroup). $ Thisdefinitionfor lowerboundis like a thresholdconceptandthereforeis subjectto someof the samedisputations. T.R. 23
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the system; and intuitively, it is a number such that the population either faces hunger from overcrowding or endangers the survival of another population of interest, when it rises above this value. However, since the definition of existence (i.e. not becoming extinct) does not require the specification of an upper bound, it can in principle be left unspecified (or equivalently set equal 00). But an important problem would then be to determine such an upper limit Ml(t) for each variable i using the following definition. If variable Xi is greater than M,(t), at any t, it reaches “plague proportions” and will cause some other variable to fall below its lower bound. Alternatively, these bounds need not have any intrinsic meaning, but at time t be a measure OfUi(t) above and b,(t) below a chosen value Z,(t). See Fig. 1.
FIG. 1. The bounds for the ith co-ordinate generated by a reference function .7,(t), and deviations a,(r) and b,(t), r[O, Tl.
In many cases stability means that “perturbations” of short duration do not lead to large changes in the solution relative to the “unperturbed” system and that “perturbation” effects are damped with time (Margalef, 1968 ; Odum, 1971; May, 1973 ; Woodmansee, 1974). If a system is defined by an initial state, input, and a system of differential or difference equations i = f or Ax = f(see section 2) a perturbation in the system is then a change in the system which can be a change in input, initial state, the functionf, or any combination of these three elements.? Given a set of perturbations, this notion of stability may be expressed by using for each variable of interest i, a lower bound of xi(t)--hi(t) and an upper bound of Ei(t)+ai(t) where the state of the unperturbed system at t is Z,(t), and ai( b,(r) are decreasing functions
in time.
See Fig. 2. In the case of ecosystem
management,
these
bounds can simply be an a priori management decision. t Note that the common definition of perturbation, as in the cases of asymptotic, neighborhood, and global stabilities, is a change in initial state only. It is clear that ecosystem perturbations are not restricted to changes in initial state but can occur in inputs to the system and the system itself. For discussions of these more general notations of perturbation and stability see Halanay (1966).
STABILITY
Fro.
2.
OF
349
ECOSYSTEMS
&convergence to n,(t) in
[0,
T].
The many notions of stability presently used suggest that there is more than one appropriate notion-the choice of which depends on the questions asked. In sections 2 and 4 we define concepts of stability over finite-time for unperturbed and perturbed ecosystems. Measures of instability are also given in section 4. These concepts avoid the objections stated above. In addition, all definitions are unit independent. By this we mean: the stability (and instability) of an ecosystem is independent of the expressed units for the state variables. In section 3 these concepts are studied and sufficient conditions and necessary conditions for stability are given. In particular, theorems 2 and 3 provide ways to ensure each variable stays within the bounds and theorem 1 a way to identify the occurrance of large fluctuations outside the bounds, by examining the system at “boundary” points only. The approach of characterizing the behavior of a system in terms of its behavior at “boundary” points is of course used often in the fields of optimization (Zangwill, 1969; Luenberger, 1973; Hestenes, 1975) and optimal control (Leitmann, 1966; Lee & Markus, 1967) when there are constraints placed on the states of a system. The strength of our results lies in the criteria for stability and instability: in addition to being robust in the sense of Levins (1966) and Slobodkin (1974), and constructive, they can be applied directly to the differential (or difference) equations without any knowledge of the solutions. Section 5 relates the existing concepts of neighborhood and asymptotic stability, and e-persistence to our definitions. In conclusion we discuss directions for extention of our stability concepts in section 6. 2. Stability of Unperturbed Systems For an ecosystem model E, the state x is a point in R”. An input is a function U: z --t R where T is the time domain. z is either a finite set of points (0 = t,, tz, . . . ) t, = T}, or a time interval [0, T]. The dynamics of E, described by the state change between “neighboring” points in time, is given
350
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as either
ds & =.f(x,U(C)at
?E[O, T]
or x(fi+
ll-X(fiJ
=
f(x(ti)3
u(tJ,
ti,)
i=O,...,N-1.
Thus an ecosystem is described by a model which includes: the initial state, input, and the change in state between neighboring points in time. Since our definitions of stability for continuous time models can be straightforwardly extended to discrete time models and results analogous to those given in section 3 can be obtained for difference equations, we shall restrict our discussion to continuous time models. To distinguish between the state variables which are not of interest and those that are (see footnote? on page 347), we denote the entire set of state variables {x,, . . . , x,,,} as X and the subset of state variables which are of interest {xii, . . . , Xi”} or {y,, . . . , y.} as Y.1 Without loss of generality we assume y, = xi, i = 1, . . . , n. The state of the variables of interest is then a point y in R”. With each yi and m, there is an upper bound function M,(t): r + R and a lower bound function m,(t): z + R, i = 1, . . . , n. We assume that Mi(t) and mitt) are continuously differentiable. For notational simplicity, we denote the interval [m,(t), Mi(r)] by I&). We also denote the direct product i=l
as I(t), where
is the collection of all points (z,, . . . , z,) having the property that zidi(t). Given an ecosystem model E which consists of a dynamical system and set of “healthy” states, we are concerned with the behavior of the system over a finite-time interval. Does a system which starts out in a “healthy” state always remain “healthy”? Or does it become a “healthy” state by time T even though the system did not initially start out in such a state? The t We assume for the existence and uniqueness of solutions to dx/dt = J that f and 11 are continuously differentiable. We remark that an input in the form of a pulse can be approximated, as close as we wish, by a continuous function. i The distinction between the state variables in X-Y and the input variables is the followina. The variables in X-Y effect those in Y and the variables in Y effects those in X-Y. 0; the other hand, the input variables effect those in Y but the variables in Y do not effect the input variables. For example, if the density of phytoplankton is the variable of interest, then the density of zooplankton would be a state variable in the set X-Y, whereas temperature could be considered an input variable.
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following definitions of total stability and terminal stability are natural concepts for studying these questions. Definition 1 For an initial state x0 and input u, we say the system is totally stable for (x0, u), if for all t&z y(t)EZ(t); and terminally stable for (x0, u), if y(T)eZ(T) where T = sup {t : t&r). Intuitively, we say a system is totally stable if each variable remains within a “healthy” range at all times,? and terminally stableT if eventually at time T, each variable is within a “healthy range”. See Figs 3 and 4. These
FIG. 3. Total stability in the ith co-ordinate. The solid line denotes y dM,(f)/dt tinuity, there exists E > 0 such that
dYi(t) > dMj(t) -dt
dt
t&u, f +&I.
holds. Then by con-
STABILITY
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353
ECOSYSTEMS
This implies for all ts[t, I+&), t dy,M ds I ds
YjW = yjw+ > y~@)+~ =
Mjtf)
+
’ dM,@) ds ds t dMj(s) [ ds
ds
= M,(t). But yj(t) ) M,(t) contradicts our hypothesis of total stability. Hence dyj(t)/dt I dMj(t)/dt and the theorem is proved. To discuss sufficiency conditions for stability, we need the following definitions. For each j = 1, . . . , n, we define the sets M,(t) and m,(t) is follows, M,(t) = {YlYj = M,(t) and yiCZi(t), i # j} and mj(t) = {y(yj = mj(t)
and
yisZi(t), i #,.i}.
Note that
is the boundary of Z(t). Theorem 2 Let E be an ecosystem model with y,sZ(O). Then denoting x by Cy, x,+ ,, . . . ) x,) a sufficient condition for total stability for (x0, U) is for all ts[O, T), (I)
fj(X,
U(t),
t)
~~”
JXIllj(t),
and XieSi i = n + I, . . . , m.
j = 1, . . . , a; XiESj, i = n+l,
. . .) m.
Proof. Let A be the following set of real numbers: t&A if and only if te[O, T] and for all t’s[O, t], y(t’)&Z(t’). Since y,sZ(O), A # 4. Also A is bounded above by T, hence sup A exists.
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First we show that sup A&A. Suppose the contrary is true. Let t = sup A. t # 0 and there exists j = 1, . . . , n such that vi(I) > Mj(Z) or yj(f) < mj(2). Let us assume yj(i) > i%fj(Z). Then by continuity, there exists E > 0 such that vj(t) > M,(t) for ts[t--E, t]. This leads to a contradiction to t = sup A. A similar contradiction arises if we assume rj(r) < mj(t). Hence sup A&A. Next we show i = T. Suppose not or f < T. Let Q denote the set of indices of y(f) with the property that y,(f) = M,(t) and S denote the set of indices of y(f) with the property that y,(Z) = m,(Z). Also let R = (1, . . . , n>-(Q u 5’). Then if R # 4, for r&R, m,(Z) < u,(i) < M,(Z). By continuity, there exists E, > 0 such that m,(t) -zzy,(t) < M,(t) for ts[f, t+.s,]. Let sR = min {.s,lreR}. sR > 0 since R is finite. We next show that if Q # 4 then for q&Q, there exists sq > 0 such that v,(t) 5 M&f) for &[I, f+sJ. By our hypothesis and continuity, there exists sq > 0 such thatf,([y(t), x,+i(t), . . . , x,,,(l)J, u(t), t) < dk&(t)dt for te[t, f+.s,J. But
As in the proof of theorem 1, this implies y,(t) 5 MJt), ts[t, t+s,J. Let EQ = min {c41qEQ} > 0. We can similarly show for SES # I$ there exists E, > 0 such that y,(t) 2 m,(t), te[t, Z+E,]. Also let aS = min {E,IsES) > 0. Now let E = min {~a, .sQ,es}, Clearly E > 0. But this implies for te[Z, t+s] andj = 1,. . . , it we have mj(t) I uj(t) I M,(t). Again we have a contradiction to t = sup A. Hence sup A = T and this proves our theorem. In cases where there are points on the boundary of Z(t) with direction of change pointing along the boundary, the following is a sufficient condition for total stability. Theorem 3 Let E be an ecosystem model with y,sZ(O). Then denoting x by (JJ, x,+ 1, , x,,,) a sufficient condition for total stability for (x,, U) is : for all ts[O, T), (l)for eachj = 1, . . . , n, y&M,(t) and XiSSi where Xi(t)ESi i = n + 1, . . . , m, either (A) fj(x, u(t), t) < ‘y or (B) fj(x, u(t), t) = dMj(t) -- ~dt
STABILITY
355
OF ECOSYSTEMS
and there exists 8, > 0 such that dM,(O fj(X,
U(t’),
t’)
5
__--
dt
for
t’E[t.
t
+
Ej]
and (2) a set of similar conditions for the lower bound mj(t). Proof. The proof proceeds as in that of theorem 2, with the only change in showing that if Q # $I then for q&Q, there exists .sq> 0 such that v,(t) I M,(t) for &[t, t+&,J. This has been shown if condition (A) holds and the statement follows trivially if (B) is true. The remaining portion of the proof is the same as that of theorem 2. Theorem 2 says, if the initial state is “healthy”, and for all points on the boundary of Z(t) the direction of change points inward, then without any knowledge of the solutions of the differential equation, we can conclude that all variables will always be within bounds. Theorem 3 also gives a sufficient condition for the total stability of a system with points on the boundary which points either inward or along the boundary. These conditions also show that the definition of total stability is robust. Theorem 4 Let E be an ecosystem model. If there exists t&CO, T] such that y(t)sl(f) and conditions (1) and (2) of theorem 3 holds for all I@, T], then E is terminally stable. Proof. If f = T, there is nothing to prove, so let t < T. As in the proof of theorem 2, we can show that forj = 1, . . . , IZ and ts[f, T], y,(t)&Zj(t). Hence E is terminally stable. 4. Stability of Perturbed Systems and Instability
An initial state, input, and a system of differential or difference equations define a system. A perturbation in the system is a change in the system denoted by (R,, 0,3),t and the stability of a perturbed system is treated by comparing its behavior with that of the unperturbed system. If stability for the unperturbed system is defined to be a fixed trajectory which we can assume to be the equilibrium state, then the use of concepts such as neighborhood stability and asymptotic stability is appropriate [a great deal of work has been done in this area, see for example La Salle & Lefschetz (1961), Cesari (1963)]. But for most ecosystem considerations this is not the case. Therefore we study the stability of perturbed systems by first establishing what defines stability in the unperturbed ecosystem model, and then discuss the stability of pert Successive discrete perturbations in a system at to, . . . , t, can be treated by considering p separate systems where the ith one is from ti-l to t, with r, = T.
356
L. S.-Y.
wu
turbed systems with reference to the definitions. Hence referring to the concepts of stability developed in the previous section, we can, for example, speak of the essential stability of a system subject to the set of perturbations {CR,, a) : IlRo -x0 II < a,? Q(t,)+/?}. Theorems similar to theorems 2 and 3 which assure total stability under perturbations can be constructed without much difficulty, and are therefore omitted. If a system subject to a set of perturbations is unstable, instability can be quantified to reflect the different notions emphasized in each definition of stability. The measure of instability for a non-totally stable system is a function N(t) : [0, T] + R” where
Ni(r) = CYi(r)-Mil+- Cmdt)-Ydt>3 +P i= I,...,n. This value gives the amount of deviation from Ii(t) for each ts[O, T]. For a non-terminally stable system, the measure of instability is the n-vector
CYitT)-M*(T)I+ - [mj(T)-YdT)l+,
i = 1,. . . , n;
and for a non-essentially stable system, the n-vector ~{[yi(t)-M,(r)l’fCmi(f)--yi(f)]‘~
dl-Ci,
5. Relations to Asymptotic and Neighborhood Stability,
i = 1, . . . , n.
and O-Persistance
A consideration which arises when new stability concepts are defined is how do the existing stability concepts reIate to these new definitions? This section will focus on answering this question. We fist study asymptotic and neighborhood stability. For this discussion we are constrained by the definitions of asymptotic and a neighborhood stability. Hence the only meaningful dynamics are described by a system of ordinary differential equations such that the set of state variables of interest equals the entire set of state variables, and also z = [0, co). The concepts of neighborhood and asymptotic stability are concerned with the stability of a fixed solution which we can assume to be the equilibrium solution x z 0. We say a solution is stable in a neighborhood of x E 0, or neighborhood stable, if for every E > 0 and every to 2 0 there exists 6(e, to) > 0 such that for t 2 to the solution Ilx(t)II c E whenever Ilx(t,)il < 6. We emphasize that 6 may depend on both E and to. It is easy to show that neighborhood stability is equivalent to total stability in POY co) under the set of perturbations in initial state {R. : x(to) = to, llRo II < S>, if for E > 0 we choose I;(t) = [ -6, E], i = 1, . . . , n, t 2 0. In terms of essential stability with respect to the equilibrium state or t In this paper 11. /I always denotes the euclidean norm.
STABILITY
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ECOSYSTEMS
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Zi(t) = [0, 0] for each i and t 2 0, neighborhood stability implies: for every T > t, 2 0 and E > 0, there exists a(&, to) > 0 such that subject to the set of initial state perturbations {& .* x(t,) = Jlo, ljllo(I < S> the system is essentially stable in [to, T] for Ci 2 E . (T- to), i = 1, . . . , n.t We remark the stringency of the neighborhood concept with regard to behavior close to the equilibrium state is clear from these relationships, a characteristic which could also make it an inappropriate concept to use for analyzing the stability or instability of an ecosystem model. To discuss asymptotic stability, we need another concept. If A and B are two sets in R”, we define the difference between A and B to be the maximal distance between points in A and points in B. We denote this difference by d(A, B), thus d(A, B) = sup { 11x--y\l : x&A, y&B}. Let f be a set function f: [0, co) + 2R”. We say f(t) -+f * as t + 03, if the maximal distance between points in f(t) and points in f* approach 0 or dCf(t),f*] + 0 as 1--* co. For perturbations in the initial state of a system, the relationship between asymptotic stability and total stability is given in the following theorem. Theorem 5 Let Z(z) + (0) as t + co and for each to 2 0 let there exist a(?,) > 0 for which the perturbations {R,, : x(t,) = jO, II&ii < a(to)) are totally stable in [to, co). Then neighborhood stability implies asymptotic stability. Proof. The theorem follows immediately if each solution x(t; t,, 2,) + 0 as t + co. We now show this is implied by the hypothesis. Suppose the contrary. Then there exists some E > 0 and solution x(t; t,, 2,) such that for every T > to, there exists some t 2 T for which d[O, x(t; to, R,)] r E. Total stability implies for each i t 2 to. xi(f; fO, aO)szi(f) But then we have a contradiction with the hypothesis that Z(t) + (0) as t-+ co. We close this section by discussing the relationship between the concept of e-persistence given by Botkin & Sobel (1975) and total stability. A posterity (trajectory) is e-persistent about the point x’ if for all t 2 0. ll~‘--~(0II s 8, f The converse is not true. Given any C, > 0 and Z,(t), by concentrating the deviations from Z,(t) in sufficiently short time intervals, the amount of deviation can be made as large as we wish and the system will be essentially stable. Therefore essential stability does not imply neighborhood stability. The same holds true for terminal stability.
358
From the definition of equivalent to total stability and i. Thus the concept of total stability, where for [XI-0, x;+q.
L.
S.-Y.
wu
&persistence, it is clear that this concept is if we choose C(t) = [xi-8, xi+/31 for each t e-persistence can be viewed as a special case of each t and i, ii(t) is given by the interval
6. Some Extensions Several directions for extension of these concepts present themselves. One is to consider analogous notions of stability for systems with stochastic inputs. Then rather than requiring “trajectories” to stay within bounds, stability would be with certain probability “trajectories” stay within bounds.? Another is to consider a more general notion of stability which not only requires individual variables xi to remain within bounds but also restricts the variation of 2-triples, 3-triples, . . . n-triples of variables. This results in stable sets for each value of t which are more general subsets in R” than the n-dimensional rectangles I(t) discussed in this paper.$ We conclude this paper with a remark. A difficulty with applying our concepts can be in defining the upper and lower bounds, Mi(t) and m
The Stability of Ecosystems-A
Finite-Time Approach
LILIAN SHIAO-YEN WV
IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598, U.S.A. (Received 24 December 1975, and in revised form 27 July 1976) The objective of this paper is to present some concepts of stability which originate from ecological considerations, discuss their relationship with existing concepts of stability, and furthermore to establish sufficient conditions for stability which are both robust in the sense of Levins and Slobodkin, and constructive. The essential notion of stability for unper-
turbed ecosystems, applied in most cases,is that although fluctuations in particular variables exist, each remains within certain bounds (which may be time-varying). The use of this notion within a finite-time horizon is the framework in which total, essential, and terminal stability for both perturbed
and unperturbed ecosystems are defined. These concepts are studied and sufficient conditions for stability are given. In particular, theorems 2 and 3 provide conditions to ensure that each variable stays within bounds and theorem 1 a way to identify the occurrence of large fluctuations outside the bounds, by examining the system at “boundary”points only. The strength of these results lies in the criteria for stability and instability: in addition to being robust and constructive, they can be applied directly to the differential (or difference) equations, without any knowledge of the solutions. Relationships between our concepts and the existing concepts of asymptotic and neighborhood stability, and Qersistance
are also stated.
1. Introduction and Motivation Previous mathematical approaches to studying stability of ecosystems have frequently involved the application of established mathematical concepts and theorems to suitable ecosystem models. Here we try a different approach. The objective of this paper is to present some concepts of stability which originate fromecologicalconsiderations,discusstheirrelationshipswithexistingconcepts of stability and furthermore to establish sufficient conditions for stability which are both robust (Levins, 1966; Slobodkin, 1974) and constructive.? 7 Constructive sticient conditions for stability are conditions which can be directly applied to determine the stability of the system, as opposed to sufficient conditions which depend on the existence of functions with certain properties. For example most results of the Liapunov theory of stability (La Salle & Lefschetz, 1961; Brauer % Nohel, 1969, Chapter 5) are of this nature. A difficulty with such results is that their applicability is always limited by methods for constructing the necessary functions. 345
346
L. S.-Y.
wu
After reading much of the literaturet on the stability of ecosystems, we observe that there are basically two groups of deterministic concepts. One group originating from physical systems, and containing concepts such as neighborhood stability and asymptotic stability, is attractive because the mathematics is well-developed. The other group includes the conceptually more appealing but mathematically more difficult concepts such as structural stability and global stability. All the proposed concepts are based on the notion that some unperturbed “behavior” defines a standard and stability is studied by comparing the perturbed system with the standard. However these concepts frequently diverge from the notion of stability in an ecological context. Furthermore, they often seem ill-suited for the purpose of managing ecosystems. Others (Slobodkin, 1965; Preston, 1969; Cohen, 1971; Innis, 1974; Holling, 1974; Botkin & Sobel, 1975, etc.) have criticized such approaches. In particular, Botkin & Sobel (1975) also propose other concepts of stability, one of which will be discussed in section 5. To motivate our concepts, we first discuss some of the shortcomings of these proposed definitions. The difficulty of specifying a dynamical system and its inputs for large times often makes the use of criteria of stability which depend on an infinite-time horizon unsuitable. In this paper we propose concepts of stability which lie within a finite-time framework.8 The concept of stability over a finite-time interval is not new. Weiss and Infante first introduced and developed a theory of finite-time stability parallel to the classical Liapunov theory of stability (Weiss & Infante, 1965). Their work has since then been extended by them and others (Weiss & Infante, 1967; Weiss, 1968; Lam &Weiss, 1974; Kayande & Wong, 1968; Garrard, 1969; Kaplan, 1969; Kayande, 1969; Michel & Wu, 1969; Ben Lashiher & Storey, 1972). However, their concepts were largely motivated by problems in control theory and engineering where stability means: given that the initial state of a system belongs to a fixed set, trajectories of the system then stay within fixed bounds over a prespecificed interval of time. Hence their definitions do not always coincide with concepts of stability originating from ecological considerations. Another difficulty with using this theory is being a theory parallel to the Liapunov theory of stability, sufficient conditions for stability depend on the existence of “Liapunov-like” functions. The usefulness of the theory is therefore limited by there being no general method of constructing these “Liapunov-like” functions. A shortcoming of those definitions of stability which involve steady states is that special emphasis is placed on equilibrium points which are a con1 For an overview on the subject, see Woodwell & Smith (1969), and May (1973). $ Note, however, that kite-time does not rule out considering large finite-times approaching infinity when an infinite-time scale is desired.
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sequenceof the chosen model. Furthermore, in the definitions of neighborhood and asymptotic stability, the size of the relevant neighborhood is not specified. Thus it may be unlimitedly small and therefore unmeasurable, and as indicated by Lewontin (1969) and Boorman (1972), provides no information about what is likely to occur under larger perturbations. The essential notion of stability in many instances is not that particular state variables of interest? (e.g. density of species,rate of oxygen consumption) remain constant, but that although fluctuations in these variables exist, each remains within certain bounds which may vary with time. It is obvious that bounds which vary with time are more general than bounds which do not. But more important for natural populations there are frequently cyclic variations in (external) inputs to a system which cause well marked population cycles, such as the seasonalvariations in population density of phytoplankton in northern temperate lakes (Odum, 1971) thus from the standpoint of data collection in the modelling of ecosystems there is empirical value in a definition of stability whose bounds vary with time. The notion of stability meaning particular variables remain within certain bounds arises in several ecological concerns. The minimum criterion for ecological stability is the existence of an ecosystem, as stated by Preston (196% “An ecological system may be said to be stable, from my point of view, during that period of time when no species become extinct (thereby creating a vacant “niche”) and none reaches plague proportions, except momentarily, thereby destroying the niches of other speciesand causing them to become extinct.” We use bounds (which may vary with time) to denote the “healthy” range of each variable in the ecosystem, or for the ecosystem as a whole. For the population$ variables, this range is defined as follows. The lower bound of the range is a number such that the population either faces the danger of becoming extinct or seriously disrupts the ecosystem (Paine, 1966) when it falls below this value.5 This concept is also referred to in Allee’s principle (Alle, Emerson, Park 8z Schmidt, 1949; Odum, 1971). The upper bound is harder to define. Certainly it can be no greater than the carrying capacity of t The term “state variablesof interest” is usedto distinguishbetweenthe variables includedto specifythe systembut are not of interestto us,and thosethat are. We make thisdistinctionbecause oftenin practicethisdistinctionismade,We makethe assumption that all the variablesof interestare(in the language of systems theory) observables, or that theyarevariableswhichcan bemeasured. $ The word populationis usedvery looselyhere, it can Lx one specific species or an aggregate of species with similarfunctionalroles(i.e. a functionalgroup). $ Thisdefinitionfor lowerboundis like a thresholdconceptandthereforeis subjectto someof the samedisputations. T.R. 23
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the system; and intuitively, it is a number such that the population either faces hunger from overcrowding or endangers the survival of another population of interest, when it rises above this value. However, since the definition of existence (i.e. not becoming extinct) does not require the specification of an upper bound, it can in principle be left unspecified (or equivalently set equal 00). But an important problem would then be to determine such an upper limit Ml(t) for each variable i using the following definition. If variable Xi is greater than M,(t), at any t, it reaches “plague proportions” and will cause some other variable to fall below its lower bound. Alternatively, these bounds need not have any intrinsic meaning, but at time t be a measure OfUi(t) above and b,(t) below a chosen value Z,(t). See Fig. 1.
FIG. 1. The bounds for the ith co-ordinate generated by a reference function .7,(t), and deviations a,(r) and b,(t), r[O, Tl.
In many cases stability means that “perturbations” of short duration do not lead to large changes in the solution relative to the “unperturbed” system and that “perturbation” effects are damped with time (Margalef, 1968 ; Odum, 1971; May, 1973 ; Woodmansee, 1974). If a system is defined by an initial state, input, and a system of differential or difference equations i = f or Ax = f(see section 2) a perturbation in the system is then a change in the system which can be a change in input, initial state, the functionf, or any combination of these three elements.? Given a set of perturbations, this notion of stability may be expressed by using for each variable of interest i, a lower bound of xi(t)--hi(t) and an upper bound of Ei(t)+ai(t) where the state of the unperturbed system at t is Z,(t), and ai( b,(r) are decreasing functions
in time.
See Fig. 2. In the case of ecosystem
management,
these
bounds can simply be an a priori management decision. t Note that the common definition of perturbation, as in the cases of asymptotic, neighborhood, and global stabilities, is a change in initial state only. It is clear that ecosystem perturbations are not restricted to changes in initial state but can occur in inputs to the system and the system itself. For discussions of these more general notations of perturbation and stability see Halanay (1966).
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&convergence to n,(t) in
[0,
T].
The many notions of stability presently used suggest that there is more than one appropriate notion-the choice of which depends on the questions asked. In sections 2 and 4 we define concepts of stability over finite-time for unperturbed and perturbed ecosystems. Measures of instability are also given in section 4. These concepts avoid the objections stated above. In addition, all definitions are unit independent. By this we mean: the stability (and instability) of an ecosystem is independent of the expressed units for the state variables. In section 3 these concepts are studied and sufficient conditions and necessary conditions for stability are given. In particular, theorems 2 and 3 provide ways to ensure each variable stays within the bounds and theorem 1 a way to identify the occurrance of large fluctuations outside the bounds, by examining the system at “boundary” points only. The approach of characterizing the behavior of a system in terms of its behavior at “boundary” points is of course used often in the fields of optimization (Zangwill, 1969; Luenberger, 1973; Hestenes, 1975) and optimal control (Leitmann, 1966; Lee & Markus, 1967) when there are constraints placed on the states of a system. The strength of our results lies in the criteria for stability and instability: in addition to being robust in the sense of Levins (1966) and Slobodkin (1974), and constructive, they can be applied directly to the differential (or difference) equations without any knowledge of the solutions. Section 5 relates the existing concepts of neighborhood and asymptotic stability, and e-persistence to our definitions. In conclusion we discuss directions for extention of our stability concepts in section 6. 2. Stability of Unperturbed Systems For an ecosystem model E, the state x is a point in R”. An input is a function U: z --t R where T is the time domain. z is either a finite set of points (0 = t,, tz, . . . ) t, = T}, or a time interval [0, T]. The dynamics of E, described by the state change between “neighboring” points in time, is given
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as either
ds & =.f(x,U(C)at
?E[O, T]
or x(fi+
ll-X(fiJ
=
f(x(ti)3
u(tJ,
ti,)
i=O,...,N-1.
Thus an ecosystem is described by a model which includes: the initial state, input, and the change in state between neighboring points in time. Since our definitions of stability for continuous time models can be straightforwardly extended to discrete time models and results analogous to those given in section 3 can be obtained for difference equations, we shall restrict our discussion to continuous time models. To distinguish between the state variables which are not of interest and those that are (see footnote? on page 347), we denote the entire set of state variables {x,, . . . , x,,,} as X and the subset of state variables which are of interest {xii, . . . , Xi”} or {y,, . . . , y.} as Y.1 Without loss of generality we assume y, = xi, i = 1, . . . , n. The state of the variables of interest is then a point y in R”. With each yi and m, there is an upper bound function M,(t): r + R and a lower bound function m,(t): z + R, i = 1, . . . , n. We assume that Mi(t) and mitt) are continuously differentiable. For notational simplicity, we denote the interval [m,(t), Mi(r)] by I&). We also denote the direct product i=l
as I(t), where
is the collection of all points (z,, . . . , z,) having the property that zidi(t). Given an ecosystem model E which consists of a dynamical system and set of “healthy” states, we are concerned with the behavior of the system over a finite-time interval. Does a system which starts out in a “healthy” state always remain “healthy”? Or does it become a “healthy” state by time T even though the system did not initially start out in such a state? The t We assume for the existence and uniqueness of solutions to dx/dt = J that f and 11 are continuously differentiable. We remark that an input in the form of a pulse can be approximated, as close as we wish, by a continuous function. i The distinction between the state variables in X-Y and the input variables is the followina. The variables in X-Y effect those in Y and the variables in Y effects those in X-Y. 0; the other hand, the input variables effect those in Y but the variables in Y do not effect the input variables. For example, if the density of phytoplankton is the variable of interest, then the density of zooplankton would be a state variable in the set X-Y, whereas temperature could be considered an input variable.
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following definitions of total stability and terminal stability are natural concepts for studying these questions. Definition 1 For an initial state x0 and input u, we say the system is totally stable for (x0, u), if for all t&z y(t)EZ(t); and terminally stable for (x0, u), if y(T)eZ(T) where T = sup {t : t&r). Intuitively, we say a system is totally stable if each variable remains within a “healthy” range at all times,? and terminally stableT if eventually at time T, each variable is within a “healthy range”. See Figs 3 and 4. These
FIG. 3. Total stability in the ith co-ordinate. The solid line denotes y dM,(f)/dt tinuity, there exists E > 0 such that
dYi(t) > dMj(t) -dt
dt
t&u, f +&I.
holds. Then by con-
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This implies for all ts[t, I+&), t dy,M ds I ds
YjW = yjw+ > y~@)+~ =
Mjtf)
+
’ dM,@) ds ds t dMj(s) [ ds
ds
= M,(t). But yj(t) ) M,(t) contradicts our hypothesis of total stability. Hence dyj(t)/dt I dMj(t)/dt and the theorem is proved. To discuss sufficiency conditions for stability, we need the following definitions. For each j = 1, . . . , n, we define the sets M,(t) and m,(t) is follows, M,(t) = {YlYj = M,(t) and yiCZi(t), i # j} and mj(t) = {y(yj = mj(t)
and
yisZi(t), i #,.i}.
Note that
is the boundary of Z(t). Theorem 2 Let E be an ecosystem model with y,sZ(O). Then denoting x by Cy, x,+ ,, . . . ) x,) a sufficient condition for total stability for (x0, U) is for all ts[O, T), (I)
fj(X,
U(t),
t)
~~”
JXIllj(t),
and XieSi i = n + I, . . . , m.
j = 1, . . . , a; XiESj, i = n+l,
. . .) m.
Proof. Let A be the following set of real numbers: t&A if and only if te[O, T] and for all t’s[O, t], y(t’)&Z(t’). Since y,sZ(O), A # 4. Also A is bounded above by T, hence sup A exists.
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First we show that sup A&A. Suppose the contrary is true. Let t = sup A. t # 0 and there exists j = 1, . . . , n such that vi(I) > Mj(Z) or yj(f) < mj(2). Let us assume yj(i) > i%fj(Z). Then by continuity, there exists E > 0 such that vj(t) > M,(t) for ts[t--E, t]. This leads to a contradiction to t = sup A. A similar contradiction arises if we assume rj(r) < mj(t). Hence sup A&A. Next we show i = T. Suppose not or f < T. Let Q denote the set of indices of y(f) with the property that y,(f) = M,(t) and S denote the set of indices of y(f) with the property that y,(Z) = m,(Z). Also let R = (1, . . . , n>-(Q u 5’). Then if R # 4, for r&R, m,(Z) < u,(i) < M,(Z). By continuity, there exists E, > 0 such that m,(t) -zzy,(t) < M,(t) for ts[f, t+.s,]. Let sR = min {.s,lreR}. sR > 0 since R is finite. We next show that if Q # 4 then for q&Q, there exists sq > 0 such that v,(t) 5 M&f) for &[I, f+sJ. By our hypothesis and continuity, there exists sq > 0 such thatf,([y(t), x,+i(t), . . . , x,,,(l)J, u(t), t) < dk&(t)dt for te[t, f+.s,J. But
As in the proof of theorem 1, this implies y,(t) 5 MJt), ts[t, t+s,J. Let EQ = min {c41qEQ} > 0. We can similarly show for SES # I$ there exists E, > 0 such that y,(t) 2 m,(t), te[t, Z+E,]. Also let aS = min {E,IsES) > 0. Now let E = min {~a, .sQ,es}, Clearly E > 0. But this implies for te[Z, t+s] andj = 1,. . . , it we have mj(t) I uj(t) I M,(t). Again we have a contradiction to t = sup A. Hence sup A = T and this proves our theorem. In cases where there are points on the boundary of Z(t) with direction of change pointing along the boundary, the following is a sufficient condition for total stability. Theorem 3 Let E be an ecosystem model with y,sZ(O). Then denoting x by (JJ, x,+ 1, , x,,,) a sufficient condition for total stability for (x,, U) is : for all ts[O, T), (l)for eachj = 1, . . . , n, y&M,(t) and XiSSi where Xi(t)ESi i = n + 1, . . . , m, either (A) fj(x, u(t), t) < ‘y or (B) fj(x, u(t), t) = dMj(t) -- ~dt
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and there exists 8, > 0 such that dM,(O fj(X,
U(t’),
t’)
5
__--
dt
for
t’E[t.
t
+
Ej]
and (2) a set of similar conditions for the lower bound mj(t). Proof. The proof proceeds as in that of theorem 2, with the only change in showing that if Q # $I then for q&Q, there exists .sq> 0 such that v,(t) I M,(t) for &[t, t+&,J. This has been shown if condition (A) holds and the statement follows trivially if (B) is true. The remaining portion of the proof is the same as that of theorem 2. Theorem 2 says, if the initial state is “healthy”, and for all points on the boundary of Z(t) the direction of change points inward, then without any knowledge of the solutions of the differential equation, we can conclude that all variables will always be within bounds. Theorem 3 also gives a sufficient condition for the total stability of a system with points on the boundary which points either inward or along the boundary. These conditions also show that the definition of total stability is robust. Theorem 4 Let E be an ecosystem model. If there exists t&CO, T] such that y(t)sl(f) and conditions (1) and (2) of theorem 3 holds for all I@, T], then E is terminally stable. Proof. If f = T, there is nothing to prove, so let t < T. As in the proof of theorem 2, we can show that forj = 1, . . . , IZ and ts[f, T], y,(t)&Zj(t). Hence E is terminally stable. 4. Stability of Perturbed Systems and Instability
An initial state, input, and a system of differential or difference equations define a system. A perturbation in the system is a change in the system denoted by (R,, 0,3),t and the stability of a perturbed system is treated by comparing its behavior with that of the unperturbed system. If stability for the unperturbed system is defined to be a fixed trajectory which we can assume to be the equilibrium state, then the use of concepts such as neighborhood stability and asymptotic stability is appropriate [a great deal of work has been done in this area, see for example La Salle & Lefschetz (1961), Cesari (1963)]. But for most ecosystem considerations this is not the case. Therefore we study the stability of perturbed systems by first establishing what defines stability in the unperturbed ecosystem model, and then discuss the stability of pert Successive discrete perturbations in a system at to, . . . , t, can be treated by considering p separate systems where the ith one is from ti-l to t, with r, = T.
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turbed systems with reference to the definitions. Hence referring to the concepts of stability developed in the previous section, we can, for example, speak of the essential stability of a system subject to the set of perturbations {CR,, a) : IlRo -x0 II < a,? Q(t,)+/?}. Theorems similar to theorems 2 and 3 which assure total stability under perturbations can be constructed without much difficulty, and are therefore omitted. If a system subject to a set of perturbations is unstable, instability can be quantified to reflect the different notions emphasized in each definition of stability. The measure of instability for a non-totally stable system is a function N(t) : [0, T] + R” where
Ni(r) = CYi(r)-Mil+- Cmdt)-Ydt>3 +P i= I,...,n. This value gives the amount of deviation from Ii(t) for each ts[O, T]. For a non-terminally stable system, the measure of instability is the n-vector
CYitT)-M*(T)I+ - [mj(T)-YdT)l+,
i = 1,. . . , n;
and for a non-essentially stable system, the n-vector ~{[yi(t)-M,(r)l’fCmi(f)--yi(f)]‘~
dl-Ci,
5. Relations to Asymptotic and Neighborhood Stability,
i = 1, . . . , n.
and O-Persistance
A consideration which arises when new stability concepts are defined is how do the existing stability concepts reIate to these new definitions? This section will focus on answering this question. We fist study asymptotic and neighborhood stability. For this discussion we are constrained by the definitions of asymptotic and a neighborhood stability. Hence the only meaningful dynamics are described by a system of ordinary differential equations such that the set of state variables of interest equals the entire set of state variables, and also z = [0, co). The concepts of neighborhood and asymptotic stability are concerned with the stability of a fixed solution which we can assume to be the equilibrium solution x z 0. We say a solution is stable in a neighborhood of x E 0, or neighborhood stable, if for every E > 0 and every to 2 0 there exists 6(e, to) > 0 such that for t 2 to the solution Ilx(t)II c E whenever Ilx(t,)il < 6. We emphasize that 6 may depend on both E and to. It is easy to show that neighborhood stability is equivalent to total stability in POY co) under the set of perturbations in initial state {R. : x(to) = to, llRo II < S>, if for E > 0 we choose I;(t) = [ -6, E], i = 1, . . . , n, t 2 0. In terms of essential stability with respect to the equilibrium state or t In this paper 11. /I always denotes the euclidean norm.
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Zi(t) = [0, 0] for each i and t 2 0, neighborhood stability implies: for every T > t, 2 0 and E > 0, there exists a(&, to) > 0 such that subject to the set of initial state perturbations {& .* x(t,) = Jlo, ljllo(I < S> the system is essentially stable in [to, T] for Ci 2 E . (T- to), i = 1, . . . , n.t We remark the stringency of the neighborhood concept with regard to behavior close to the equilibrium state is clear from these relationships, a characteristic which could also make it an inappropriate concept to use for analyzing the stability or instability of an ecosystem model. To discuss asymptotic stability, we need another concept. If A and B are two sets in R”, we define the difference between A and B to be the maximal distance between points in A and points in B. We denote this difference by d(A, B), thus d(A, B) = sup { 11x--y\l : x&A, y&B}. Let f be a set function f: [0, co) + 2R”. We say f(t) -+f * as t + 03, if the maximal distance between points in f(t) and points in f* approach 0 or dCf(t),f*] + 0 as 1--* co. For perturbations in the initial state of a system, the relationship between asymptotic stability and total stability is given in the following theorem. Theorem 5 Let Z(z) + (0) as t + co and for each to 2 0 let there exist a(?,) > 0 for which the perturbations {R,, : x(t,) = jO, II&ii < a(to)) are totally stable in [to, co). Then neighborhood stability implies asymptotic stability. Proof. The theorem follows immediately if each solution x(t; t,, 2,) + 0 as t + co. We now show this is implied by the hypothesis. Suppose the contrary. Then there exists some E > 0 and solution x(t; t,, 2,) such that for every T > to, there exists some t 2 T for which d[O, x(t; to, R,)] r E. Total stability implies for each i t 2 to. xi(f; fO, aO)szi(f) But then we have a contradiction with the hypothesis that Z(t) + (0) as t-+ co. We close this section by discussing the relationship between the concept of e-persistence given by Botkin & Sobel (1975) and total stability. A posterity (trajectory) is e-persistent about the point x’ if for all t 2 0. ll~‘--~(0II s 8, f The converse is not true. Given any C, > 0 and Z,(t), by concentrating the deviations from Z,(t) in sufficiently short time intervals, the amount of deviation can be made as large as we wish and the system will be essentially stable. Therefore essential stability does not imply neighborhood stability. The same holds true for terminal stability.
358
From the definition of equivalent to total stability and i. Thus the concept of total stability, where for [XI-0, x;+q.
L.
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wu
&persistence, it is clear that this concept is if we choose C(t) = [xi-8, xi+/31 for each t e-persistence can be viewed as a special case of each t and i, ii(t) is given by the interval
6. Some Extensions Several directions for extension of these concepts present themselves. One is to consider analogous notions of stability for systems with stochastic inputs. Then rather than requiring “trajectories” to stay within bounds, stability would be with certain probability “trajectories” stay within bounds.? Another is to consider a more general notion of stability which not only requires individual variables xi to remain within bounds but also restricts the variation of 2-triples, 3-triples, . . . n-triples of variables. This results in stable sets for each value of t which are more general subsets in R” than the n-dimensional rectangles I(t) discussed in this paper.$ We conclude this paper with a remark. A difficulty with applying our concepts can be in defining the upper and lower bounds, Mi(t) and m
