Persistent sets via Lyapunov functions

Nonlinear Anol~as, Theory, Methods & Applrcatwm. Vol. 3. No. 1. pp 73-80 0 Pergamon Press Ltd. 1979. Prmted in Great Britain

PERSISTENT

03~2-54~x/79/0101-0073

SETS VIA LYAPUNOV

802.W/O

FUNCTIONS

G. W. HARRISON Department

of Mathematics,

University

of Georgia,

Athens,

Georgia

30602, U.S.A.

(Received 9 February 1978) words: Persistent sets, Lyapunov Kq Lotka-Volterra model.

functions,

stability,

continuing

perturbations,

random

environments,

INTRODUCTION THE STATE variables describing a biological or a sociological system are subject to many diverse forces which are only partially understood. In addition, the environment which determines these forces is subject to random fluctuations. Suppose that the known or understood portion of the dynamics under a fixed “average” environment is modelled by

i = f(x). In view of the uncertainty in the system, equation true dynamics are given by 2 = f(x)

(1 .l) is only an approximate

+ g(t, x)

(1.1) equation,

and the

(1.2)

where g(t, .x) is an unknown function representing the part ofthe dynamics which is unknown and/ or subject to the random environment. It will be assumed, however, that enough is known to establish that g is a member of some set of functions G. The functions g(t, x) can be viewed as continuing perturbations of the system (l.l), and the question arises whether the known dynamics of (1.1) can be used to make any predictions about the behavior of the perturbed system (1.2). This is a stability question, but requires more than the usual analysis of Lyapunov stability with respect to perturbations of initial conditions. The problem arises so frequently in ecology that a number of authors [l-6] have indicated that ordinary Lyapunov stability is not sufficient for their needs. Perturbations of the function describing the dynamics must be considered as well as perturbations of initial conditions, and these perturbations may be fairly large. On the other hand, for many applications it is not necessary (or possible) to show that all solutions of (1.2) converge to, or stay close to, some equilibrium point or known trajectory of (1 .l). Indeed many natural systems show fairly large, unpredictable fluctuations but persist within certain acceptable ranges. Techniques are needed to determine these persistent sets, that is, sets which are invariant under the dynamics of system (1.2) for every g E G [4]. The relation of persistent sets to other stability concepts in ecology is discussed in [2]. To be more specific, let R" denote n dimensional Euclidean space and assume that f~ C[R",R"] and G c C[R+ x R",R"]. 73

74

G W. HARRISON

Definition 1.1. A set M c R” is persistent under G if, for any function of equation (2) with x(t,) E M for some t,, remains in M for all t > t,. would be to let the set M be a function

A useful generalization sidered in this paper.

2. USE

OF

A LYAPUNOV

FUNCTION

TO SHOW

x(t)

g E G, every solution

of time, but it will not be con-

PERSISTENT

SETS

It is assumed that the reader is familiar with the use of Lyapunov functions to show stability solutions of differential equations with respect to perturbations of initial conditions [7-93. Notation. The

following

of

will be used throughout: I/E C,[R”, R+]. M, = (x E R”: c(x) = (aV)/(ax) .g(x). Then the time derivative of V(x) of (1.2), denoted V’/+,(x), satisfies

notation

V(x) < L}. v;.(x) = (iYV)/(iTx).f(.x) and

with respect to trajectories

V-+,(x) = V;(x) + 5(x,. LEMMA 2.1.

If V(x) = L implies

(2.1)

I’/ +,(x) < 0 for all g E G, then M, is persistent

under G.

Proof: If any trajectory in M, at time t, leaves M,, then at same time t,, V(x(t,)) = L and for some E > 0, V@(t)) > L for all t, < t < t, + E. But this implies that lif+e(x(t,)) = lim(V(x(t)) - L)/(t - tl) 2 0, which contradicts the hypothesis that v’+e(x(t,)) < 0. t-t, If each set M, in a family of sets is persistent

LEMMA 2.2.

sistent

under

under G, then n M, and u M, are per= a

G.

LEMMA 2.3. If for some 6 > 0, L < V(x) < L + b implies that r’/+&x) < 0 for all g E G, then M,> is persistent under G.

Proof: For M, =

n

0 < E < b, ML+F is persistent is persistent under G. ML.+, every

under

G by Lemma

2.1. By Lemma

2.2

O<&-=b

TO find that V(x) functions it is only satisfying

a function V satisfying the hypothesis of Lemma 2.1 or 2.3, one seeks a 2 0 and as V(x) increases vf(x) becomes more negative, which is typical used to study domains of attraction of asymptotically stable equilibrium known that g E G, it is necessary to assume that there is a function c(x) < U(x)

for all g E G.

function such of Lyapunov points. Since U E C[R”, R] (2.2)

It follows from (2.1) that pf+,(x)

< vf(x) + U(x)

for all g E G.

(2.3)

Now as V(x) increases, U(x) may increase, but if U(x) increases more slowly than vf(.x) decreases, there may be a number L such that V(x) > L implies v,(x) + U(x) < 0, hence v’+,(x) is negative for all g E G. In other words, the restoring forces pushing x toward regions where V(x) is smaller, measured by v$), become stronger than the random diffusive forces, measured by vg(x).

75

Persistent sets via Lyapunov functions

As a simple example, D satisfying

suppose that there is a positive definite function

V(X) defined on a domain

V-(X, < -nV(.X)

(2.4)

and that

(2.5) for all x E D. Then if G = {g : 11g(x) )I < b for all x E D>, vf+,(.x) < -iV(x)

+ Kb

(2.6)

and vf++,(x) is negative if V(x) > Kb/l. Hence, if the set M,d with L = Kb/A is contained in D, it is persistent under G. Furthermore, if x E 0 is a uniformly asymptotically stable equilibrium point of (l.l), then there exists a Lyapunov function V satisfying (2.4) and (2.5). (See [7], p. 276 and references therein.) Thus the total stability theorem (e.g., [7], p. 276) is a special case of persistent sets. The hypotheses for the persistence of M, are in general weaker than those for the total stability theorem, however, since it is not necessary that the functions g E G be bounded, only that (W/(&x)). g grows more slowly than (aV/(a.x)).f d ecreases as x approaches the boundary of M, It is also not necessary that (1.1) have an equilibrium point in M,, although it usually will. On the other hand the results are weaker in that M, heed not be persistent under some G for every E > 0. Thus persistence is closer to the concept of practical stability suggested in [7] and [9]. The following results give practical criteria for finding persistent sets. THEOREM 2.4. Assume

then M, is persistent

that (2.2) is true and let P = {.x E RN:pf(.x) + U(x)3 under G.

O}. If P c M,,

Pro05 If V(x) > L, then .X4 P and from (2.3) it follows that PI+s(.~) < 0 for all g E G. Therefore M, is persistent under G by Lemma 2.3. To find the smallest

set of the form M, which is persistent

take L = max V(x). To assure that ?“P

this maximum exists, assume that P is bounded. Since the continuity of V,(x) + U(x) implies that P is closed, it follows that P is compact and hence V(x) has a maximum on P. For many typical Lyapunov functions, V has no local maxima on the interior of P. (This must be the case, for example, if (av/(ax)) # 0 except at x = 0 and V has a minimum at 0.) In this case max V(x) must occur on the boundary of P where v;.(.x) + U(X) = 0. Thus the following result is true. If P is bounded and V has no local max V(x). Then M,> is persistent under G. r;,(X)+ U(X)= 0

COROLLARY 2.5.

L=

THEOREM 2.6. Let /.I) be a vector norm

maxima

on the

interior

of P, let

and assume that

V&.x) + U(x) < 0 < if rl < (Ix 1)< r2. If for some L and some b > 0, {.x: IIxl) < rI} c M, persistent under G.

and ML+b c {x: I/.x )I < r,} then

(2.7)

M,> is

76

G.

W. HARRISON

Proof. If L < V(x) < L + 6, then the hypotheses imply that r1 < )(.x I/ < r2, and hence from (2.7) vJ(.x) + V(x) < 0. In view ofequation (2.3) the result follows from Lemma 2.3. COROLLARY 2.7. Assume that for any positive a < j? If condition

(2.7) holds and

Proof. Condition c M,_. In addition,

implies

scalars a, /I and any x E R”, V(rx) < V(fl.x).

max V(x) Q L < ,,xyFr, V(x), then M,_ is persistent IIx II = I1

(2.8) implies

c > O.Thus,foranyO < b < c, ML+b c {x: Ii x II < r2}. Therefore are satisfied and M,> is persistent under G.

Suppose

under

G.

max V(x) = ,,II$C, V(x) < L, Hence {x: 11 x 11< rl} IIx IIGr1 x ) 3 ,,x,, min=12 V(x) = L + c, for some V(x) 2 V((r,/II x 11).

that

if 11 x 1)2 r2, then

Remarks. In the event that Theorem 2.4 and Corollary U(Y) < ~(11x II), where p is exist for example (with r2 or in = n and a > 6.

(2.8)

the hypotheses

ofTheorem

2.6

r2 = co, Theorem 2.6 and Corollary 2.7 reduce to special cases of 2.5. Inequality (2.7) can often be established by showing that yf(.x) + a polynomial with p(r) < 0 for rl < r < rz. Such a polynomial will = co) if p+) < --a IIx Ilm,a > 0, and V(x) < 6 11 .XIIn where m > n,

3. EXAMPLE:

LINEAR

SYSTEM WITH NONLINEAR

PERTURBATIONS

that the known

part of the system is linear and asymptotically

stable, so that

_? = Ax + g(t, x)

(3.1)

with A a stable matrix and g E G. It is well known [7] that A is stable if and only if for any symmetric, positive definite mati!ixQ, there is a symmetric, positive definite matrix S satisfying ATS +SA It is convenient

= -Q.

to use V(x) = .xTS.x as a Lyapunov

(3.2)

function

and the norm

I(x 1)= (x~S.X)‘~~.

THEOREM 3.1. Assume that S and Q are symmetric, positive definite, that A, S, and Q satisfy (3.2), and that -2 is the largest (closest to zero) eigenvalue of -- S- ‘Q. If for every g E G, IIsk 4 II < K II .Yl[Ia, on N = {x: II x IJael < 2/(2K)}, of the three cases below: (1) (2) (3) Thus

then

K > 0,

M = (.x: 1)x II < c} = 1.x: V(x) < c”} is persistent

rr > 1, K > 0, c < (il/2K)l’@-l). a=l,K<(A/2),c>O. II!< 1, K > 0, c > (2K/#‘(‘-‘). for every a there isa K such that a persistent

(3.3) in each

set exists.

Proof: Let V(.x) = xTS.x = 1)x l12.Then

r’f+,(x) =

-.xTQ.x

+ 2.xTSg(.x).

(3.4)

Persistent

Applying Schwartz

sets via Lyapunov

77

functions

the definition of 2 and the theory of quadratic forms to the first term and the Cauchyinequality to the second term on the right of (3.4) yields I$+ B(x) d - AXTSX + 2(XTSX)“2(gTSg)“2.

From the definition

of the norm and condition

(3.5)

(3.3) it follows that

V-+,(x) < -+$

+2KIIxI)‘+“.

(3.6)

Hence in each of the three cases, riJ+9(.x) is negative on the set N. M is a set of the form M,> with L =c2; thus the persistence of M follows from Theorem 2.6 with rr = 0, r2 = (1/(2K))“‘“-‘) in case 1, with rl = 0, r2 = co in case 2, and with rl = (2K/ll)‘“‘-“), r2 = 00 in case 3. Remarks. In cases 1 and 2, not only is the set M persistent under G, but for any g E G, x z 0 is an asymptotically stable equilibrium point of (3.1) whose domain of attraction contains M. Case 1 corresponds to the well known fact that if 11g(t, x) )( = o I(x I( as x approaches 0, then asymptotic stability of i = Ax implies asymptotic stability of (3.1), because the linear part dominates close to the origin. Case 3 is the companion result, showing that persistence results when the linear part dominates away from the origin. If more is known about g(t, x), an improved estimate of the persistent set can be obtained by working directly with (3.4). This approach is used in [lo] to give a more detailed discussion of persistence for a generalized version of case 2.

4. EXAMPLE:

Consider

GENERALIZED

a generalized

LOTKA-VOLTERRA PERTURBATTONS

Lotka-Volterra fii =

(

MODEL

WITH

PERSISTENT

system

a,, + i

aijNj Ni, >

j=l

i = 1,2 ,...,

n,

(4.1)

which is often used to model competition and/or predator-prey interactions among n species. As usual, assume that there is an equilibrium point m with each mi > 0 and, for ease of notation, scale the variables so that each q = 1. Then, letting xi = Ni - 1, the system (4.1) becomes j$l aipj

ii = ( Lotka-Volterra systems are often criticized having the form

(xi + 1).

(4.2)

> for being only local approximations,

the true model

ii = Fi(X)(Xi + 1) where F(x) is a nonlinear function of X. It is therefore of interest to consider (4.2) under continuing nonlinear perturbations, that is the persistence of ii = [ i

(4.3) the persistence

of

aij-xj + gi(t3X)](.xi + l).

(4.4)

C di(xi - In(l + xi))

(4.5)

j=l

The Lyapunov

function V(x) =

78

G. W. HARRISON

is positive

definite, and its time derivative

along trajectories

of (4.4) is given by

V(x) = c 1 .XidiUii.Xj + c x&,(t, i

j

x).

(4.6)

I

Goh [ 1 l] assumes that 1gi(t, X) 1 d c for every i and finds a persistent set. The following generalizes Goh’s result by allowing gi(t,x) to satisfy bounds that grow linearly with .x, and gives a more precise estimate of the persistent set. THEOREM 4.1. Assume that for some positive matrix D = diag[d,, (1) There is a positive matrix Q = diag[q,, . . . , q,] such that

. . . , d,] the following

$.xT(ATD + DA)x < -.xTQx,

are true:

(4.7)

where A = [aij] is the matrix of coefficients in (4.2). (2) There are numbers ri, hi with hi > 0 for each i = 1, . . ., n such that T Xidigi(t, X, < 1 (r& I (3) The differences

ci = qi - ri are positive ci >hi

+$

1

+ ‘i 1xi 1).

(4.8)

and satisfy hj2/cj

foreachi=

l,...,n.

(4.9)

j#l

Then the set M, = {x: V(x) < I,}, with L = max V(X) over the set where F(-c& is persistent

+ hil.Xil) = 0,

(4.10)

for (4.4).

Remarks. Condition (1) is true whenever ATD + DA is negative definite, which is also a sufficient condition for global asymptotic stability of the unperturbed system (4.2) [12]. In a predator-prey or simple food chain model with each species having intraspecific competition it is possible to choose D and Q so that equality holds in (4.7). Condition (2) says that on the average the perturbations I gi(t, x) I grow no faster than ri (xi I + hi as Ixi I increases. Corollary 4.2 gives an alternate condition for (2). Actually, condition (2) only needs to hold outside the surface defined by (4.10).

Proof. From (4.6), (4.7), and (4.8) it follows that V(x) < C(-c$;

+ hiI.Xil).

(4.11)

The result now follows directly from Corollary 2.5 with P the set where the right side of (4.11) is non-negative, as long as P is contained in the feasible region, xi > - 1 for each i (i.e. iVi > 0). Otherwise L = max V(x) over P does not exist since V(X) approaches infinity as any xi approaches - 1. But completing the squares in (4.11) shows that P is the set where (4.12) (P is a part of an ellipsoid

in the positive

quadrant,

plus its reflections

in the other quadrants.)

Persistent

sets via Lyapunov

79

functions

The maximum value of 1xi ( for any point in P is (4.13)

which is less than 1 because of condition (4.9). Thus every point in P satisfies xi > - 1, and the result is proved. Now let g,(t, x) be written in the form

gittY‘) = ki(t9X) + 1 COROLLARY

(4.14)

bijt, X)Xj

4.2. Theorem 4.2 is true if inequality (4.8) is replaced

by (4.15)

dibii + + C 1djbji + dibijl < ri

foreachi

= l,...,rz.

(4.16)

if1

Proof: Let B = [bij], R = diag[r,, r2,. . . , 1.1, and k = (k,, k,, . . . , kJT. By Gershgorin’s theorem [ 131, (4.16) implies that 1/2(BTD + DB) - R is negative definite, which indicates that xTDBx < xTRx. Together with (4.15), this implies that xTDbx + xTDk < xTRx + i

hi/xi/,

(4.17)

i=l

which, in view of (4.14), is equivalent to (4.8). Remark. Condition (4.16) shows that the terms b& x) and bji(t, x), reflecting the unknown effects that species i and j have on each other, can be large as long as they have opposite sign (predatorprey relationship) and remain in about the same proportion, dj/di, to each other.

5. CONCLUSION

The trajectories of a dynamic system are determined by various forces, some of which are known by the modeler and represented by f(x) in equation (1.2). Other forces may not be completely predictable by the modeler and are represented by g(t, x) in equation (1.3). To show that some region M of the state space is persistent, that is, that the system’s trajectories stay within M in spite of the random forces g(t, x), it is not necessary to know the behavior of the system in the interior of M, only that on the boundary of M the known forces f(x) dominate the random g(t, X) and are directed toward the interior of M. Lyapunov functions can be a useful tool for comparing the size and direction of these forces. Acknowledgements-The in part by NSF. grants

author would like to thank Dr. T. Hallam no. DEB76-09830 and no. DEB77-02942.

for his encouragement.

The research

was supported

80

G. W. HARRISON REFERENCES

1. BOTKIN D. & SOBEL M., The complexity of ecosystem stability, in Ecosystem Analysis and Prediction, S. Levin (ed.). SIAM, Philadelphia (1974). 2. HARRISON G. W., Stability under environmental stress: resistance, resilience, persistence, and variability, Am. Nutur. (in press). 3. HOLLING C. S., Resilience and stability of ecological systems, Ann. Rev. ecol. Syst. 4, l-24 (1973). 4. INNIS G., Stability, sensitivity, resilience, persistence. What is of interest?, in Ecosystem Analysis and Prediction, S. Levin (ed). SIAM, Philadelphia (1974). 5. LEWONTIN R. C., The meaning of stability, in Diversity and Stability in Ecological Systems, G. M. Woodwell and H. H. Smith (eds.). Brookhaven Nat. Lab. Publ. no. 22. Upton, New York. pp. 13-24 (1969). 6. WV L., On the stability of ecosystems, in Ecosystem Analysisand Prediction, S. Levin (ed.) SIAM. Philadelphia (1974). 7. HAHN W., Stability of Motion. Springer, New York (1967). 8. LAKSHMIKANTHAMV. & LEELA S., Differential and Integral Inequalities. Vol. 1. Academic Press, New York (1969). 9. LASALLE J. P. & LEFSCHETZ S., Stability by Liapunov’s Direct Method with Applications. Academic Press, New York (1961). 10. HARRISON G. W., Stability of linear systems with uncertain parameters, Int. J. syst. Sci. (in press). 11. GOH B. S., Nonvulnerability of ecosystems in unpredictable environments, Theor. Pop. Biol. 10, 83-95 (1976). 12. GOH B. S., Global stability in many species systems, Am. Nutur. 111, 135-143 (1977). 13. WILKINSON J. H., The Algebraic Eigenvulue Problem. Clarendon Press, Oxford (1965).