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Persistence in stochastic food web models
Bulletin of Mathematical Biology Vol. 46, No. 3, pp. 357-370, 1984. Printed in Great Britain
PERSISTENCE MODELSt •
0092-8240/8453.00 + 0.00 Pergamon Press Ltd. © 1984 Society for Mathematical Biology
IN S T O C H A S T I C F O O D W E B
THOMAS C. GARD Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A.
A sufficient condition is given for stochastic boundedness persistence of a top predator in generalized Lotka-Volterra-type stochastic food web models in arbitrary bounded regions of state space. The main result indicates that persistence in the corresponding deterministic system is preserved in the stochastic system if the intensities of the random fluctuations are not too large.
1. Introduction.
A principle problem of population ecology is how to explain the stability amidst variability phenomenon observed in real ecosystems. To deal explicitly with variability, stochastic models have been introduced; such models can be classified according to the source of random effects considered. [See the review articles of Chesson (1978) and Turelli (1984), and the book by Nisbet and Gurney 1982.] When the random effects taken into account are primarily environmental, Markov diffusion processes, represented by solutions of stochastic differential equations, arise as models of population density dynamics. Here the stochastic differential equations can be thought of as interpreting the mathematical situation when an appropriate parameter, for example the intrinsic growth rate, in a deterministic differential equation model is replaced by an average value plus a random noise fluctuation term. This is one of a number of ways in which diffusion processes provide useful approximations to mathematically less tractable models. Persistence, as the opposite of extinction, has emerged as the most important stability-type notion for population models. Chesson (1982) has suggested stochastic boundedness persistence (s.b. persistence) as the appropriate stability concept for stochastic population models in particular. Suppose X(t), t > 0, is a stochastic process representing a population density at time t; s.b. persistence for x means, given e > 0, that there is a positive lower bound Le such that
P(X(t) >~L~, all t > 0)/> 1 --e.
(1)
T h i s w o r k w a s s u p p o r t e d , in p a r t , b y t h e U. S. E n v i r o n m e n t a l P r o t e c t i o n A g e n c y u n d e r G r a n t No. CR 807830.
357
358
T . C . GARD
Chesson (1982) points out that s.b. persistence interprets biologically as: trends to ever lower population densities are impossible; and the average frequency of fluctuations to low density does not increase with time. The main result of this paper, presented in Section 3, gives a sufficient condition for s.b. persistence for top predators in Lotka-Volterra type multispecies food web diffusion models restricted to an arbitrary bounded region. Basically it says that top predator persistence in a deterministic food web model is preserved in the analogous stochastic model if the noise intensity is not too large. As such this result extends to the stochastic case persistence criteria previously established by the author for corresponding deterministic food web models. In the next section, as a prelude to the presentation of the main theorem, the mathematical relationship between s.b. persistence and other stability-type concepts is discussed in the simplified setting o f one-dimensional diffusion-type population models. Following the presentation of the main result in Section 3 for Lotka-Volterra food chain-type models, extension to more general food web models for multispecies stochastic models given in the concluding section. 2. Persistence in Single-species Diffusion Process Models. Single-species populations in random environments have been discussed extensively in recent literature. [See Turelli (1977) for a review.] In this section it will be assumed that the population density X ( t ) satisfies the stochastic differential equation d X = X(a -- b X ) dt + X x / ( a ( X ) ) dW,
(2)
where a and b are positive constants and o(x) is a positive continuous function of x. The solution X ( t ) of (2) with X(0) = x E (0, oo) realizes the onedimensional diffusion process with infinitesimal mean (or drift coefficient) x(a -- bx) and variance (or diffusion coefficient) x2a(x) on an interval [0, r). Equation (2) serves as a general stochastic analogue of the deterministic logistic equation dx --
dt
=
x(a
--
bx).
(3)
From a heuristic point of view, if the intrinsic growth rate in (3) is replaced by the sum of an average rate a and a fluctuation termx/(o(X))~ representing white noise of intensity a(x), then (2) corresponds to one possible interpretation of the resulting model. A well-known stability concept for (2) is the existence of a stable invariant distribution in (0, oo). If such a distribution has compact support, then solutions of (2) are stochastically bounded. Stochastic boundedness for a process X requires, for each e > 0, the existence of a lower bound Le
PERSISTENCE IN STOCHASTIC FOOD WEB MODELS
359
and an upper b o u n d Ue such t h a t
P(Le ~ X ( t ) <~ Ue, all t) ~> 1 -- e.
(4)
In particular, for the stochastic differential e q u a t i o n (2), if o(a/b) = O, stochastic stability o f the equilibrium X = a/b implies stochastic boundedhess, and hence s.b. persistence. But s.b. persistence does n o t require the existence o f an invariant distribution. A weaker c o n c e p t is (0, oo)-invariance. Let r ° a n d r~* be the extinction and explosion times, respectively, for the process X ( t ) w i t h X(0) = x, i.e. r ° = inf{t: X ( t ) = 0} r x = inf{T: X ( t ) -+ oo as t ~ T}; (0, oo)-invariance is equivalent to r ° = oo and r~* = 0% w.p.1. In this case, for each t > 0 and x > 0 and for all Borel sets B _C(0, oo),
Pt, x(B) = P(X(t) E. B) defines a probability measure on (0, oo) which as such m u s t be tight: for every e > 0 there is a c o m p a c t set K _C(0, co) such t h a t
Pt, x (K) ~> 1 -- e.
(5)
Billingsley (1968) discusses tightness in c o n n e c t i o n w i t h weak convergence o f probability measures. The relationship b e t w e e n stochastic boundedness and tightness is clear; stochastic b o u n d e d n e s s (4) requires t h a t for each x the family o f probability measures {Pt,~ : t > 0) is tight, which means t h a t for each x (5) holds u n i f o r m l y in t. C o n s e q u e n t l y , s.b. persistence is a stronger c o n d i t i o n t h a n r ° = oo. Example. The solution X ( t ) o f the linear m o d e l
dx = aX dt + x / ( a ) X dW with X(O) = x is
X ( t ) = x exp{(a -- ½a)t + W(t)}. oo
It is clear t h a t rx° = oo and rx = oo for a n y x > 0. Since W(t)/t -+ 0 as T -+ oo w. p. 1. t h e n i f a < 0/2
X ( t ) -+ 0 as t -+0% w. p. 1. The example illustrates t h a t (0, oo)-invariance m a y h o l d w i t h o u t s.b. persistence. The m o s t closely related m a t h e m a t i c a l concept to s.b. persistence for diffusion processes appears to be the n o t i o n o f natural boundary at 0 (from Feller's b o u n d a r y classification scheme for singular diffusion): for a < t3, let
360
T . C . GARD
rx [a, /3] denote the first exit time o f X(t) from the interval (a,/3), where X(0) -- x E (a,/3), i.e. Tx [a,/3] = inf(t: X(t) q~ (a,/3)); 0 is a natural boundary if
P(X(-rx [a,/3] ) = a)
(6)
is arbitrarily small for a sufficiently close to zero. Let x0 and xl be arbitrary points in (0, oo), and let ~b(x) =fxX' ~(y) dy, where
y 2z(a -- bz) z~a(z)
~(y) = e x p { - fx °
dz}.
The criterion o f Feller asserts that 0 is a natural boundary for any solution o f (2) if and only if ~ ( 0 ) - lim ~b(x) = + ~ (7) x..+o +
[or equivalently that ~(x) is not integrable on any interval (0, /3)]. This follows from the fact that
P(X(rx [~,/3] ) = a)
-
~(x) - ~(/3) ~(~) ~(~)
-
(Gihman and Skorohod, 1972). The next result gives the relationship between s.b. persistence and natural boundary at 0. THEOREM 1. I f (2) is s.b. persistent, then 0 is a natural boundary for (2). Let 7 be an arbitrary positive number and consider the stopped process
X.r(t)=Y(t),
O<.t<.r~,
where T~ is the first hitting time o f X at 7, Le. ~-~ = inf{t: X(t) = 3'}.
Then, i f O is a natural boundary for (2), the process X~(t) is s.b. persistent. Proof. S.b. persistence means that, for any e > 0, P(X(t) >1 L~, all t) 1> 1 -- e for some positive n u m b e r L~. So for any/3 > x
PERSISTENCE IN STOCHASTIC F O O D WEB MODELS
361
occurs with probability less t h a n e if a < L~, i.e. 0 is a n a t u r a l b o u n d a r y . F o r the converse, n o t e t h a t
P ( X ~ ( t ) < a, some t) < . P ( X ( r x [ a , 3'1 ) = a). So
P(X.~(t) >~ a, all t) >~ 1 - - P ( X ( T x [ a , 3'1 ) = a) -
¢(a)-~(x)
(8)
Since ~(a) ~ ~(0) = +oo as a ~ 0, the last expression in (8) can be made arbitrarily close to 1 by choosing ~ sufficiently small. So s.b. persistence holds for the s t o p p e d process X. r (t). The conclusion o f this t h e o r e m can be written for X(0) = x as: the
P ( X ( t ) >~ o~, t <~ T~ )
(9)
can be m a d e arbitrarily close to 1 by choosing a sufficiently small. In addition to a natural b o u n d a r y at 0 for (2), ~(3') < +oo was required in the proof. Also, it is n o t e d t h a t s.b. persistence follows if 0 is a natural b o u n d a r y and either oo is n o t a natural b o u n d a r y or stochastic b o u n d e d n e s s f r o m above holds. In the first case 3" can be replaced by oo in the p r o o f o f the theorem, while in the second case
P ( X ~ ( t ) = X ( t ) , all t) --, 1 a s 3' ---~ oo.
Examples. Suppose a(x) = a0 x2(p-1), where a 0 is a positive constant. A necessary c o n d i t i o n for s.b. persistence in (2) is t h a t p ~> 1. Suppose p
I /
~(x) = exp
---(70
az I-2p -- bz 2-2p dz o
J
Since 1 -- 2p > --1,
~(x)
Z3--2p
exp
--vo
2 -- 2p
3 -- 2p
Xo
and so ~ is integrable on a n y interval (0,/3) as 2 -- 2p and 3 -- 2p are positive. Thus 0 is n o t a natural b o u n d a r y , and hence s.b. persistence does n o t hold if p < 1. Consider, once again, the linear model
d X = a X dt + x / ( a ) X dW.
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T.C. GARD
Here $(x) = exp
alnzl~ 0
=
Zero is a natural boundary if a i> e/2, and oo is not natural if a > o/2, so s.b. persistence holds if a > e/2.
3. F o o d Web Diffusion Process Models. For notational simplicity, consider first the stochastic analogue o f the simple Lotka-Volterra food chain: X ( t ) , Y(t) and Z ( t ) represent the densitites at time t of prey, intermediate predator and top predator, respectively, whose dynamics are given by the stochastic system dX = X
-- b X --
dt + ~"
~/(oii)
dWi
]=1
dY= Y
dZ= Z
C(--d + e X - - f Z )
E(--g + h X )
3
(10)
dt + ~ X,/(o2]) d ]=1
3
dt + ~ X/(a3i) d__ j=l
,
where the W1. are independent scalar Wiener or Brownian motion processes, a-h are positive constants and the oi! are non-negative constants. The sums o f Brownian differentials in each equation o f (10) allow for the possibility o f correlated noise fluctuations among species. For an arbitrary triple 3' = (3"1, 3'2, 3"3) of positive numbers 3"i, denote by R v the rectangular region: 0 ~
0 ~< Y ~<"/2 ,
0 ~Z~3`3,
with interior R ° . Of interest here are solutions (X(t), Y(t), Z ( t ) ) of (10) with X(0) = x, Y(0) = y, Z(0) = z and (x, y, z) C R ° ; i f r v denotes the first exit time of (X(t), Y(t), Z ( t ) ) from R ° ,
r.~ =---r(x,y,z) [Rv], the basic theory of stochastic differential equations yields, for any (x, y, z) E R °, the existence o f a unique solution (X(t), Y(t), Z ( t ) ) satisfying X(0) = z, Y(0) = y and Z(0) = z, at least for all t in the interval [0, rv). The main result can now be stated. THEOREM 2. L e t t2 = a -- (b/e)d -- (c/h)g and let
PERSISTENCE IN STOCHASTIC FOOD WEB MODELS
v
=
ta -½ ~.
ai] +
a2i +
363
a3i
i=1
3 = a -½
E
3 ",,-
)
3
d + ½ )5_ o~,
-
i=l
i=1
)
g + ½ Y
,,~,
•
i=1
If v>O
then s.b. persistence holds for the top predator Z in (10) up to the first exit f r o m R~, i.e. given e > O, there is a lower bound L~ > 0 such that P(Z(t) >~Le, 0 <~ t <, r~) >~ 1 -- e. Proof. For a solution (X(t), Y(t), Z(t)) with initial value in R °, consider the process V(t) = X(t) [ Y(t)] r [Z(t)] s, where r = b/e and s = c/h. Using Ito's formula, the stochastic differential of V can be calculated:
dV =
[OV bV OV ~ (X(a -- b X -- cY)) + - ~ ( g ( - - d + eX - - f Z ) ) + ~ (Z(--g + hY))
o2V (x~ 3
)
]=1
+~
z2 ]=x X a3j
+ OYOZ
,
~2v ( ~-~
3
)
]=x
+ -O-X O Y xY )
YZ E 4(02j03] ) nt]=1 OZ~X
~/(oljo=j) ,
Z X E 4(o3jolj) j=l
)]
dt
OV 3 OV 3 OV 3 + - ~ X ~. x / ( a l j ) dW] + - - Y ~ ~/(o2j ) dW/+ - - Z ~_ ~/(a3]) dW]; ]=1 OY ]=1 ~Z ]=1 = V[a--rd--sg + (re--b)X + (sh--c)Y--rfZ + ½{r(r - -
1)722 +
s(s - -
1)7133}
+ r r h 2 + rsr~23 +
+ V[(V'(on) + rx/(o21) + sx/'(o31)dW 1
st/31 ] dt
364
T.C. GARD
+ (~/(o12) + rx/(022) + sx/(o32)dW2 + (X/(o13) + rx/(o23) + sx/(o33)dW31, where 3
n~/= X x / ( a . ~ j k ) • k=l
Noting the choice r = b/e and s = c/h and letting "0i = ~/(ax/) + (b/e)X/(a2l) + (c/h)~/(%]),
dV= V [ P - - ( @ f ) z + ½ / ( b )
(eb---1) r/22+
+ (b)r/12+ ( b ) (h)r123+(h)r/311 dt
+
3
d,,l.
Since 3
~'v Y ~jdWj • 0
j=l
is a Brownian motion with clock 3
T(t) =
fo
V 2 ~_. B~" d~ 1=I
(McKean, 1969, p. 45), V can be viewed as a diffusion process with unit diffusion coefficient and drift 1
y(k~ - k:)
where /~+
--
-~-+rh2
kl-
+ h 3 ]=1
and k2-
bf 3
eY~ /=1
2
T}31 -~'e-h--f/23
PERSISTENCE IN STOCHASTIC FOOD WEB MODELS
365
More precisely, one has
V(t) = V(O) + /o t ~ l
{k 1 --k2Z(s))ds + W(t),
where V(s) = V(T-I(s)) and Z,(s) = Z(T-I(s)), with T-l(s) = min{t: T(t) = s}. Now let T3 = re~bf, "y = (')'1, ")'2, 73) and define R~, R ° and ~-~ analogously to R~, R ° and r~. First the result o f the theorem is established for solutions (X(t), Y(t), Z(t)) with (X(0), Y(0), Z(0)) E R ° and for t up to re. Indeed, one has for t < T¢,
v--(b~fe)Z(t)>~O.
(11)
Adding 3
1=1 to both sides o f (11) and dividing by twice this quantity leads to, for t =
T-l(s), kl -- k2Z(s) = kl -- k2Z(t) >t ½.
(12)
By a basic comparison theorem for stochastic differential equations (e.g. Ikeda and Watanabe, 1981, p. 352)
(/(t) >~ U(t), t <~r~ w.p.l.,
(13)
where U(t) satisfies the stochastic initial value problem dU = (1/2U) dt + dW u(o) = v(o).
Since U has a natural boundary at 0 ~(u)=exp
--
2--dr uo
=
,
2v
then for a O,
P(U(r[a,/3] ) = oO < el
(14)
for sufficiently small a. But (13) implies that if l~ hits c~ before/3, then U must hit ~ before/3, i.e. P(I?(~[c~,/3] ) = a) <~P(U(r[~,/3] ) = a). So (14) and (15) imply that, given e 1 > 0, there is an o~such that
P(X(t) [ Y(t)] b/e [Z(t)] o/h hits a before/3, t ~< rq) < el, or equivalently, taking/3 > 7172~73 and
(15)
366
T.C. GARD
Lel ~
I
~b2/e ] h/c ----oq,
P(Z(t) >~Lel, t ~ 1 -- ea.
(16)
N o w the restriction Z(t) <~~/3 = ue/bf is removed b y the following sequential random stopping time argument. To begin this part of the proof, let 6 = P (there exist T, t with T < t < r~ such that Z(T) > "Y3and Z(t) <~~a); 6 is the probability o f return to [0, ~3] for Z before (X, Y, Z) exits R ° . Since Z is the non-degenerate diffusion process on [~3, ~/3 ] there is a positive probability that Z exits [73, "/3] at 73 before returning to q3. Hence 6 < 1. Now let r n be the nth exit time o f Z from [oq, ~3] prior to r~, given that
Z(t) > ~1 for t < rn, i.e. define T1 = TZ [OL1, ~t3]
rl -- inf{t > fl : Z(t) = ~3 and there exists T with fl < T < t such that Z(T) > ~3}, and continuing, for n t> 2, fn = inf(t > T n - - l : Z(t) ql (OL1, ' ~ 3 ) and Z(T) > aa, T < t}, where fn = oo if the defining conditions are n o t met. Finally, let
rn = min{fn, r~}. Now, it follows that
P(Z(t) < c~1, some t < r~) ~ ~. P(Z(rn) = al).
(17)
n=l
If one sets 1 - - e 2 = P(Z&z [ch, ~a] ) = ~a), then one can estimate the probability that the nth exit o f Z from [oq, % ] occurs through a l , given that the previous n -- 1 exits were through ~a b y
P(Z(%)
= oL1) ~ (1 - - ~ 2 ) n - - l ~ j n - - l e l .
From (17), setting e = e J ( 1 -- 6), it follows
PERSISTENCE IN STOCHASTIC FOOD WEB MODELS
367
61
P(Z(t) < 12/1,some t ~< %) ~< 1 -- (1 -- e2)8 < e.
(18)
So one can take L~ to be any positive number not exceeding ~1, and obtain via (18) P(Z(t) >~Le, all t ~< %) ~> 1 -- e,
which completes the proof. 4. Discussion and Extension to F o o d Web Models. The condition ~ > 0 guarantees top predator (in fact, all species) persistence in the corresponding Lotka-Volterra food chain model, in the sense that, given positive initial densities for all species, ~the top predator density cannot tend to zero in finite or infinite time (Gard and Hallam, 1979). Theorem 2 extends this as well as previous work on prey-predator stochastic models (Gard and Kannan, 1976). Theorem 2 has the flavor of many known stability results for stochastic differential equations in general (e.g. Khas'minskii, 1980) and for multispecies models specifically (e.g. May and MacArthur, 1972) in that stability is preserved in the corresponding stochastic model provided the noise intensity is sufficiently small. In particular, the criterion ~ > 0 is similar to the condition required by Barra et aL (1979) in obtaining the weaker (at least for top predator) positive cone invariance property; furthermore, the result is slightly less restrictive than Polansky's (1979) condition for the existence of a stable invariant distribution, as the following generalizations emphasize. Indeed, Theorem 2 can be extended to cover the situations of: (i) arbitrary number of Brownian motion processes; (ii) the a-h and the oii, non-negative continuous functions of species densities with a, e, h and, for each i, at least some oij bounded below by positive constants; (iii) arbitrary number of trophic levels; (iv) more than one species on any trophic level; (v) arbitrary omnivory; and (vi) competition for space among species on the same trophic level. Extensions to these cases follow analogously to the deterministic counterparts (Gard, 1980, 1982, !984). For example, the next theorem states the result for non-constant parameters, arbitrary numbers of species per trophic level and full omnivory. Consider the food web model
= x,
[
a,- E %xj c,j /=1 1=i
+ X i [ j =~l x/(oi~)dWJ] '
"
y
j=l
l<.i<~k
] dt
368
T.C. GARD
d Y i = Y, --d, + ~. eqX/-- ~. fqZ/ 1=1
dZi = Zi --gi +
/=1
dt
(19)
1--1
hqY/ + ~" giiX/ dt /=1
/=1
The parameters a,, aq, bq, cq, d,, eq, fq, g,, gq, hq and aq are non-negative continuous functions, and as such are b o u n d e d on any region R,y, 3' = (~'1, • • •, %0, n = k + m + p, defined as in Section 2:
O<~X,<~.yi, l ~ i ~ k ;
O~Y~[,,k
O~Zi~'yi, k + m~i~n, i.e. there exist constants ~ and di such that with X = {Xi}, Y = {Yi} and Z = {Z,),
O<.ai <-ai =ai(X, Y, Z)<~d,, and all (X, Y, Z) E R~, and similarly for the other parameters. Observe that, in general, these bounds and hence the persistence criterion obtained below depend on % as opposed to the criterion given in Theorem 2. Also note that at least some o f the lower bounds for the ai, eq, gq, hq and the o~ must be positive for the inequalities involved to be feasible and for top predator densities to be non-degenerate diffusions. THEOREM 3. Let ri, 1 <~ i <~ m, ands,, 1 ~ i <~p, be non-negative numbers which minimize the expression
r, ~, + ½ y /=1
subject to the constraints
1=1
0
+ y s, ~, + ½ 5i=l
]=1
~,~
(20)
PERSISTENCE IN STOCHASTIC F O O D WEB MODELS
m
p
j=l
i=a
369
k
Z e~irj+ Z ~,si--E bY,>~0,
l <~i <~k
j=l k
l~i~m. j=l
j=l
k
m
p
~ = y a/-Y i=1
r,~,-y
i=1
s,e,
i=1 --Z r,0~ + y s,o,
Y ~ +
-½Z ]=1
/=1
i=1
>0,
(21)
i=1
then, for each solution (X(t), Y(t), Z(t)) o f (19) with initial value (x, y, z) E R °, there is some index i (possibly dependent on (x, y, z)) such that Zi(t) is s.b. persistent until first exit o f (X(t), Y(t), Z(t)) from R~. Outline o f the proof. The p r o o f follows similarly to that for Theorem 2. Given a solution (X, Y, Z) o f (19) in R~, one computes the stochastic differential of the product k
m
p
v = I I x, I-I ~r, I-I zg,, i=1
i=l
t=1
where the constants r i and si solve the linear programming problem (20). Consider, first of all, such solutions that for some time t (with no loss o f generality, t = 0) are in a corresponding b o u n d e d region R s for = (71, • • •, 7k+m,~k+m+l, • • • , % ) . For qi, k + m + 1 <~ i ~< n, sufficiently small, (21) leads to an estimate like (12) for the deterministic part o f the stochastic differential o f V until (X, Y, Z) exits R s . So the comparison theorem can be applied, and consequently (16) is obtained for each Zi. Finally, the sequential stopping time argument, given in Theorem 2, can be repeated for any Z~ represented by a non-degenerate diffusion, which removes the restriction that (X, Y, Z) remains in Rq. (Note that the conclusion o f the theorem is immediate for any solution o f (19) that never hits R s before exiting R~ .) The author wishes to thank D. Kannan for helpful discussions in preparing this paper.
370
T.C. GARD LITERATURE
Barra, M., G. Del Grosso, A. Gerardi, G. Koch and F. Marchetti. 1979. "Some Basic Properties of Stochastic Population Models." In Systems Theory in Immunology, Proceedings of the Working Conference, Rome, 1978, Ed. C. Bruni, Lecture Notes in Biomath 32. New York: Springer-Verlag. Billingsley, P. 1968. Convergence of Probability Measures. New York: Wiley. Chesson, P. 1978. "Predator-Prey Theory and Variability." Ann. Rev. ecoL Syst. 9, 323-347. 1982. "The Stabilizing Effect of a Random Environment." J. math Biol. 15, 1-36. Gard, T. C. 1980. "Persistence in F o o d Chains with General Interactions." Math. BioscL 51, 165-174. --. 1982. "Top Predator Persistence in Differential Equation Models of F o o d Chains: the Effects of Omnivory and External Forcing of Lower Trophic Levels." J. math. Biol. 1 4 , 2 8 5 - 2 9 9 . --. 1984. "Persistence in F o o d Webs." In Autumn Course in Ecology, Proceedings of the International Conference, Trieste, 1982, Ed. S. A. Levin, Lecture Notes in Biomathematics. New York: Springer-Verlag (in press). and T. G. Hallam. 1979. "Persistence in F o o d W e b s - - I . Lotka-Volterra F o o d Chains." Bull. math. Biol. 4 1 , 8 7 7 - 8 9 1 . and D. Kannan. 1976. "On a Stochastic Differential Equation Modeling of PreyPredator Evolution." J. appl. Prob. 1 3 , 4 2 9 - 4 4 3 . Gihman, I. I. and A. V. Skorohod. 1972. Stochastic Differential Equations. New York: Spfinger-Verlag. Ikeda, N. and S. Watanabe. 1981. Stochastic Differential Equations and Diffusion Processes. Amsterdam: North-Holland. Khas'minskii, R. Z. 1980. Stochastic Stability of Differential Equations. The Hague: Sijthoff and Noordhoff. May, R. M. and R. H. MacArthur. 1972. "Niche Overlap as a Function of Environmental Variability." Proc. Natl. Acad. Scl. USA 69, 1109-1113. McKean, H. P. 1969. Stochastic Integrals. New York: Academic Press. Nisbet, R. M. and W. S. C. Gurney. 1982. Modelling Fluctuating Populations. New York: Wiley. Polansky, P. 1979. "Invariant Distributions for Multi-population Models in Random Environments." Theor. Pop. Biol. 16, 25-34. Turelli, M. 1977. " R a n d o m Environments and Stochastic Calculus." Theor. Pop. BioL 12, 140-178. --. 1984. "Stochastic Community Theory: a Partially-guided Tour." In A Course in Mathematical Eeology, Ed. T. G. Hallam and S. A. Levin. New York: SpringerVerlag (in press). -
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RECEIVED
9-13-83
PERSISTENCE MODELSt •
0092-8240/8453.00 + 0.00 Pergamon Press Ltd. © 1984 Society for Mathematical Biology
IN S T O C H A S T I C F O O D W E B
THOMAS C. GARD Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A.
A sufficient condition is given for stochastic boundedness persistence of a top predator in generalized Lotka-Volterra-type stochastic food web models in arbitrary bounded regions of state space. The main result indicates that persistence in the corresponding deterministic system is preserved in the stochastic system if the intensities of the random fluctuations are not too large.
1. Introduction.
A principle problem of population ecology is how to explain the stability amidst variability phenomenon observed in real ecosystems. To deal explicitly with variability, stochastic models have been introduced; such models can be classified according to the source of random effects considered. [See the review articles of Chesson (1978) and Turelli (1984), and the book by Nisbet and Gurney 1982.] When the random effects taken into account are primarily environmental, Markov diffusion processes, represented by solutions of stochastic differential equations, arise as models of population density dynamics. Here the stochastic differential equations can be thought of as interpreting the mathematical situation when an appropriate parameter, for example the intrinsic growth rate, in a deterministic differential equation model is replaced by an average value plus a random noise fluctuation term. This is one of a number of ways in which diffusion processes provide useful approximations to mathematically less tractable models. Persistence, as the opposite of extinction, has emerged as the most important stability-type notion for population models. Chesson (1982) has suggested stochastic boundedness persistence (s.b. persistence) as the appropriate stability concept for stochastic population models in particular. Suppose X(t), t > 0, is a stochastic process representing a population density at time t; s.b. persistence for x means, given e > 0, that there is a positive lower bound Le such that
P(X(t) >~L~, all t > 0)/> 1 --e.
(1)
T h i s w o r k w a s s u p p o r t e d , in p a r t , b y t h e U. S. E n v i r o n m e n t a l P r o t e c t i o n A g e n c y u n d e r G r a n t No. CR 807830.
357
358
T . C . GARD
Chesson (1982) points out that s.b. persistence interprets biologically as: trends to ever lower population densities are impossible; and the average frequency of fluctuations to low density does not increase with time. The main result of this paper, presented in Section 3, gives a sufficient condition for s.b. persistence for top predators in Lotka-Volterra type multispecies food web diffusion models restricted to an arbitrary bounded region. Basically it says that top predator persistence in a deterministic food web model is preserved in the analogous stochastic model if the noise intensity is not too large. As such this result extends to the stochastic case persistence criteria previously established by the author for corresponding deterministic food web models. In the next section, as a prelude to the presentation of the main theorem, the mathematical relationship between s.b. persistence and other stability-type concepts is discussed in the simplified setting o f one-dimensional diffusion-type population models. Following the presentation of the main result in Section 3 for Lotka-Volterra food chain-type models, extension to more general food web models for multispecies stochastic models given in the concluding section. 2. Persistence in Single-species Diffusion Process Models. Single-species populations in random environments have been discussed extensively in recent literature. [See Turelli (1977) for a review.] In this section it will be assumed that the population density X ( t ) satisfies the stochastic differential equation d X = X(a -- b X ) dt + X x / ( a ( X ) ) dW,
(2)
where a and b are positive constants and o(x) is a positive continuous function of x. The solution X ( t ) of (2) with X(0) = x E (0, oo) realizes the onedimensional diffusion process with infinitesimal mean (or drift coefficient) x(a -- bx) and variance (or diffusion coefficient) x2a(x) on an interval [0, r). Equation (2) serves as a general stochastic analogue of the deterministic logistic equation dx --
dt
=
x(a
--
bx).
(3)
From a heuristic point of view, if the intrinsic growth rate in (3) is replaced by the sum of an average rate a and a fluctuation termx/(o(X))~ representing white noise of intensity a(x), then (2) corresponds to one possible interpretation of the resulting model. A well-known stability concept for (2) is the existence of a stable invariant distribution in (0, oo). If such a distribution has compact support, then solutions of (2) are stochastically bounded. Stochastic boundedness for a process X requires, for each e > 0, the existence of a lower bound Le
PERSISTENCE IN STOCHASTIC FOOD WEB MODELS
359
and an upper b o u n d Ue such t h a t
P(Le ~ X ( t ) <~ Ue, all t) ~> 1 -- e.
(4)
In particular, for the stochastic differential e q u a t i o n (2), if o(a/b) = O, stochastic stability o f the equilibrium X = a/b implies stochastic boundedhess, and hence s.b. persistence. But s.b. persistence does n o t require the existence o f an invariant distribution. A weaker c o n c e p t is (0, oo)-invariance. Let r ° a n d r~* be the extinction and explosion times, respectively, for the process X ( t ) w i t h X(0) = x, i.e. r ° = inf{t: X ( t ) = 0} r x = inf{T: X ( t ) -+ oo as t ~ T}; (0, oo)-invariance is equivalent to r ° = oo and r~* = 0% w.p.1. In this case, for each t > 0 and x > 0 and for all Borel sets B _C(0, oo),
Pt, x(B) = P(X(t) E. B) defines a probability measure on (0, oo) which as such m u s t be tight: for every e > 0 there is a c o m p a c t set K _C(0, co) such t h a t
Pt, x (K) ~> 1 -- e.
(5)
Billingsley (1968) discusses tightness in c o n n e c t i o n w i t h weak convergence o f probability measures. The relationship b e t w e e n stochastic boundedness and tightness is clear; stochastic b o u n d e d n e s s (4) requires t h a t for each x the family o f probability measures {Pt,~ : t > 0) is tight, which means t h a t for each x (5) holds u n i f o r m l y in t. C o n s e q u e n t l y , s.b. persistence is a stronger c o n d i t i o n t h a n r ° = oo. Example. The solution X ( t ) o f the linear m o d e l
dx = aX dt + x / ( a ) X dW with X(O) = x is
X ( t ) = x exp{(a -- ½a)t + W(t)}. oo
It is clear t h a t rx° = oo and rx = oo for a n y x > 0. Since W(t)/t -+ 0 as T -+ oo w. p. 1. t h e n i f a < 0/2
X ( t ) -+ 0 as t -+0% w. p. 1. The example illustrates t h a t (0, oo)-invariance m a y h o l d w i t h o u t s.b. persistence. The m o s t closely related m a t h e m a t i c a l concept to s.b. persistence for diffusion processes appears to be the n o t i o n o f natural boundary at 0 (from Feller's b o u n d a r y classification scheme for singular diffusion): for a < t3, let
360
T . C . GARD
rx [a, /3] denote the first exit time o f X(t) from the interval (a,/3), where X(0) -- x E (a,/3), i.e. Tx [a,/3] = inf(t: X(t) q~ (a,/3)); 0 is a natural boundary if
P(X(-rx [a,/3] ) = a)
(6)
is arbitrarily small for a sufficiently close to zero. Let x0 and xl be arbitrary points in (0, oo), and let ~b(x) =fxX' ~(y) dy, where
y 2z(a -- bz) z~a(z)
~(y) = e x p { - fx °
dz}.
The criterion o f Feller asserts that 0 is a natural boundary for any solution o f (2) if and only if ~ ( 0 ) - lim ~b(x) = + ~ (7) x..+o +
[or equivalently that ~(x) is not integrable on any interval (0, /3)]. This follows from the fact that
P(X(rx [~,/3] ) = a)
-
~(x) - ~(/3) ~(~) ~(~)
-
(Gihman and Skorohod, 1972). The next result gives the relationship between s.b. persistence and natural boundary at 0. THEOREM 1. I f (2) is s.b. persistent, then 0 is a natural boundary for (2). Let 7 be an arbitrary positive number and consider the stopped process
X.r(t)=Y(t),
O<.t<.r~,
where T~ is the first hitting time o f X at 7, Le. ~-~ = inf{t: X(t) = 3'}.
Then, i f O is a natural boundary for (2), the process X~(t) is s.b. persistent. Proof. S.b. persistence means that, for any e > 0, P(X(t) >1 L~, all t) 1> 1 -- e for some positive n u m b e r L~. So for any/3 > x
PERSISTENCE IN STOCHASTIC F O O D WEB MODELS
361
occurs with probability less t h a n e if a < L~, i.e. 0 is a n a t u r a l b o u n d a r y . F o r the converse, n o t e t h a t
P ( X ~ ( t ) < a, some t) < . P ( X ( r x [ a , 3'1 ) = a). So
P(X.~(t) >~ a, all t) >~ 1 - - P ( X ( T x [ a , 3'1 ) = a) -
¢(a)-~(x)
(8)
Since ~(a) ~ ~(0) = +oo as a ~ 0, the last expression in (8) can be made arbitrarily close to 1 by choosing ~ sufficiently small. So s.b. persistence holds for the s t o p p e d process X. r (t). The conclusion o f this t h e o r e m can be written for X(0) = x as: the
P ( X ( t ) >~ o~, t <~ T~ )
(9)
can be m a d e arbitrarily close to 1 by choosing a sufficiently small. In addition to a natural b o u n d a r y at 0 for (2), ~(3') < +oo was required in the proof. Also, it is n o t e d t h a t s.b. persistence follows if 0 is a natural b o u n d a r y and either oo is n o t a natural b o u n d a r y or stochastic b o u n d e d n e s s f r o m above holds. In the first case 3" can be replaced by oo in the p r o o f o f the theorem, while in the second case
P ( X ~ ( t ) = X ( t ) , all t) --, 1 a s 3' ---~ oo.
Examples. Suppose a(x) = a0 x2(p-1), where a 0 is a positive constant. A necessary c o n d i t i o n for s.b. persistence in (2) is t h a t p ~> 1. Suppose p
I /
~(x) = exp
---(70
az I-2p -- bz 2-2p dz o
J
Since 1 -- 2p > --1,
~(x)
Z3--2p
exp
--vo
2 -- 2p
3 -- 2p
Xo
and so ~ is integrable on a n y interval (0,/3) as 2 -- 2p and 3 -- 2p are positive. Thus 0 is n o t a natural b o u n d a r y , and hence s.b. persistence does n o t hold if p < 1. Consider, once again, the linear model
d X = a X dt + x / ( a ) X dW.
362
T.C. GARD
Here $(x) = exp
alnzl~ 0
=
Zero is a natural boundary if a i> e/2, and oo is not natural if a > o/2, so s.b. persistence holds if a > e/2.
3. F o o d Web Diffusion Process Models. For notational simplicity, consider first the stochastic analogue o f the simple Lotka-Volterra food chain: X ( t ) , Y(t) and Z ( t ) represent the densitites at time t of prey, intermediate predator and top predator, respectively, whose dynamics are given by the stochastic system dX = X
-- b X --
dt + ~"
~/(oii)
dWi
]=1
dY= Y
dZ= Z
C(--d + e X - - f Z )
E(--g + h X )
3
(10)
dt + ~ X,/(o2]) d ]=1
3
dt + ~ X/(a3i) d__ j=l
,
where the W1. are independent scalar Wiener or Brownian motion processes, a-h are positive constants and the oi! are non-negative constants. The sums o f Brownian differentials in each equation o f (10) allow for the possibility o f correlated noise fluctuations among species. For an arbitrary triple 3' = (3"1, 3'2, 3"3) of positive numbers 3"i, denote by R v the rectangular region: 0 ~
0 ~< Y ~<"/2 ,
0 ~Z~3`3,
with interior R ° . Of interest here are solutions (X(t), Y(t), Z ( t ) ) of (10) with X(0) = x, Y(0) = y, Z(0) = z and (x, y, z) C R ° ; i f r v denotes the first exit time of (X(t), Y(t), Z ( t ) ) from R ° ,
r.~ =---r(x,y,z) [Rv], the basic theory of stochastic differential equations yields, for any (x, y, z) E R °, the existence o f a unique solution (X(t), Y(t), Z ( t ) ) satisfying X(0) = z, Y(0) = y and Z(0) = z, at least for all t in the interval [0, rv). The main result can now be stated. THEOREM 2. L e t t2 = a -- (b/e)d -- (c/h)g and let
PERSISTENCE IN STOCHASTIC FOOD WEB MODELS
v
=
ta -½ ~.
ai] +
a2i +
363
a3i
i=1
3 = a -½
E
3 ",,-
)
3
d + ½ )5_ o~,
-
i=l
i=1
)
g + ½ Y
,,~,
•
i=1
If v>O
then s.b. persistence holds for the top predator Z in (10) up to the first exit f r o m R~, i.e. given e > O, there is a lower bound L~ > 0 such that P(Z(t) >~Le, 0 <~ t <, r~) >~ 1 -- e. Proof. For a solution (X(t), Y(t), Z(t)) with initial value in R °, consider the process V(t) = X(t) [ Y(t)] r [Z(t)] s, where r = b/e and s = c/h. Using Ito's formula, the stochastic differential of V can be calculated:
dV =
[OV bV OV ~ (X(a -- b X -- cY)) + - ~ ( g ( - - d + eX - - f Z ) ) + ~ (Z(--g + hY))
o2V (x~ 3
)
]=1
+~
z2 ]=x X a3j
+ OYOZ
,
~2v ( ~-~
3
)
]=x
+ -O-X O Y xY )
YZ E 4(02j03] ) nt]=1 OZ~X
~/(oljo=j) ,
Z X E 4(o3jolj) j=l
)]
dt
OV 3 OV 3 OV 3 + - ~ X ~. x / ( a l j ) dW] + - - Y ~ ~/(o2j ) dW/+ - - Z ~_ ~/(a3]) dW]; ]=1 OY ]=1 ~Z ]=1 = V[a--rd--sg + (re--b)X + (sh--c)Y--rfZ + ½{r(r - -
1)722 +
s(s - -
1)7133}
+ r r h 2 + rsr~23 +
+ V[(V'(on) + rx/(o21) + sx/'(o31)dW 1
st/31 ] dt
364
T.C. GARD
+ (~/(o12) + rx/(022) + sx/(o32)dW2 + (X/(o13) + rx/(o23) + sx/(o33)dW31, where 3
n~/= X x / ( a . ~ j k ) • k=l
Noting the choice r = b/e and s = c/h and letting "0i = ~/(ax/) + (b/e)X/(a2l) + (c/h)~/(%]),
dV= V [ P - - ( @ f ) z + ½ / ( b )
(eb---1) r/22+
+ (b)r/12+ ( b ) (h)r123+(h)r/311 dt
+
3
d,,l.
Since 3
~'v Y ~jdWj • 0
j=l
is a Brownian motion with clock 3
T(t) =
fo
V 2 ~_. B~" d~ 1=I
(McKean, 1969, p. 45), V can be viewed as a diffusion process with unit diffusion coefficient and drift 1
y(k~ - k:)
where /~+
--
-~-+rh2
kl-
+ h 3 ]=1
and k2-
bf 3
eY~ /=1
2
T}31 -~'e-h--f/23
PERSISTENCE IN STOCHASTIC FOOD WEB MODELS
365
More precisely, one has
V(t) = V(O) + /o t ~ l
{k 1 --k2Z(s))ds + W(t),
where V(s) = V(T-I(s)) and Z,(s) = Z(T-I(s)), with T-l(s) = min{t: T(t) = s}. Now let T3 = re~bf, "y = (')'1, ")'2, 73) and define R~, R ° and ~-~ analogously to R~, R ° and r~. First the result o f the theorem is established for solutions (X(t), Y(t), Z(t)) with (X(0), Y(0), Z(0)) E R ° and for t up to re. Indeed, one has for t < T¢,
v--(b~fe)Z(t)>~O.
(11)
Adding 3
1=1 to both sides o f (11) and dividing by twice this quantity leads to, for t =
T-l(s), kl -- k2Z(s) = kl -- k2Z(t) >t ½.
(12)
By a basic comparison theorem for stochastic differential equations (e.g. Ikeda and Watanabe, 1981, p. 352)
(/(t) >~ U(t), t <~r~ w.p.l.,
(13)
where U(t) satisfies the stochastic initial value problem dU = (1/2U) dt + dW u(o) = v(o).
Since U has a natural boundary at 0 ~(u)=exp
--
2--dr uo
=
,
2v
then for a O,
P(U(r[a,/3] ) = oO < el
(14)
for sufficiently small a. But (13) implies that if l~ hits c~ before/3, then U must hit ~ before/3, i.e. P(I?(~[c~,/3] ) = a) <~P(U(r[~,/3] ) = a). So (14) and (15) imply that, given e 1 > 0, there is an o~such that
P(X(t) [ Y(t)] b/e [Z(t)] o/h hits a before/3, t ~< rq) < el, or equivalently, taking/3 > 7172~73 and
(15)
366
T.C. GARD
Lel ~
I
~b2/e ] h/c ----oq,
P(Z(t) >~Lel, t ~
(16)
N o w the restriction Z(t) <~~/3 = ue/bf is removed b y the following sequential random stopping time argument. To begin this part of the proof, let 6 = P (there exist T, t with T < t < r~ such that Z(T) > "Y3and Z(t) <~~a); 6 is the probability o f return to [0, ~3] for Z before (X, Y, Z) exits R ° . Since Z is the non-degenerate diffusion process on [~3, ~/3 ] there is a positive probability that Z exits [73, "/3] at 73 before returning to q3. Hence 6 < 1. Now let r n be the nth exit time o f Z from [oq, ~3] prior to r~, given that
Z(t) > ~1 for t < rn, i.e. define T1 = TZ [OL1, ~t3]
rl -- inf{t > fl : Z(t) = ~3 and there exists T with fl < T < t such that Z(T) > ~3}, and continuing, for n t> 2, fn = inf(t > T n - - l : Z(t) ql (OL1, ' ~ 3 ) and Z(T) > aa, T < t}, where fn = oo if the defining conditions are n o t met. Finally, let
rn = min{fn, r~}. Now, it follows that
P(Z(t) < c~1, some t < r~) ~ ~. P(Z(rn) = al).
(17)
n=l
If one sets 1 - - e 2 = P(Z&z [ch, ~a] ) = ~a), then one can estimate the probability that the nth exit o f Z from [oq, % ] occurs through a l , given that the previous n -- 1 exits were through ~a b y
P(Z(%)
= oL1) ~ (1 - - ~ 2 ) n - - l ~ j n - - l e l .
From (17), setting e = e J ( 1 -- 6), it follows
PERSISTENCE IN STOCHASTIC FOOD WEB MODELS
367
61
P(Z(t) < 12/1,some t ~< %) ~< 1 -- (1 -- e2)8 < e.
(18)
So one can take L~ to be any positive number not exceeding ~1, and obtain via (18) P(Z(t) >~Le, all t ~< %) ~> 1 -- e,
which completes the proof. 4. Discussion and Extension to F o o d Web Models. The condition ~ > 0 guarantees top predator (in fact, all species) persistence in the corresponding Lotka-Volterra food chain model, in the sense that, given positive initial densities for all species, ~the top predator density cannot tend to zero in finite or infinite time (Gard and Hallam, 1979). Theorem 2 extends this as well as previous work on prey-predator stochastic models (Gard and Kannan, 1976). Theorem 2 has the flavor of many known stability results for stochastic differential equations in general (e.g. Khas'minskii, 1980) and for multispecies models specifically (e.g. May and MacArthur, 1972) in that stability is preserved in the corresponding stochastic model provided the noise intensity is sufficiently small. In particular, the criterion ~ > 0 is similar to the condition required by Barra et aL (1979) in obtaining the weaker (at least for top predator) positive cone invariance property; furthermore, the result is slightly less restrictive than Polansky's (1979) condition for the existence of a stable invariant distribution, as the following generalizations emphasize. Indeed, Theorem 2 can be extended to cover the situations of: (i) arbitrary number of Brownian motion processes; (ii) the a-h and the oii, non-negative continuous functions of species densities with a, e, h and, for each i, at least some oij bounded below by positive constants; (iii) arbitrary number of trophic levels; (iv) more than one species on any trophic level; (v) arbitrary omnivory; and (vi) competition for space among species on the same trophic level. Extensions to these cases follow analogously to the deterministic counterparts (Gard, 1980, 1982, !984). For example, the next theorem states the result for non-constant parameters, arbitrary numbers of species per trophic level and full omnivory. Consider the food web model
= x,
[
a,- E %xj c,j /=1 1=i
+ X i [ j =~l x/(oi~)dWJ] '
"
y
j=l
l<.i<~k
] dt
368
T.C. GARD
d Y i = Y, --d, + ~. eqX/-- ~. fqZ/ 1=1
dZi = Zi --gi +
/=1
dt
(19)
1--1
hqY/ + ~" giiX/ dt /=1
/=1
The parameters a,, aq, bq, cq, d,, eq, fq, g,, gq, hq and aq are non-negative continuous functions, and as such are b o u n d e d on any region R,y, 3' = (~'1, • • •, %0, n = k + m + p, defined as in Section 2:
O<~X,<~.yi, l ~ i ~ k ;
O~Y~[,,k
O~Zi~'yi, k + m~i~n, i.e. there exist constants ~ and di such that with X = {Xi}, Y = {Yi} and Z = {Z,),
O<.ai <-ai =ai(X, Y, Z)<~d,, and all (X, Y, Z) E R~, and similarly for the other parameters. Observe that, in general, these bounds and hence the persistence criterion obtained below depend on % as opposed to the criterion given in Theorem 2. Also note that at least some o f the lower bounds for the ai, eq, gq, hq and the o~ must be positive for the inequalities involved to be feasible and for top predator densities to be non-degenerate diffusions. THEOREM 3. Let ri, 1 <~ i <~ m, ands,, 1 ~ i <~p, be non-negative numbers which minimize the expression
r, ~, + ½ y /=1
subject to the constraints
1=1
0
+ y s, ~, + ½ 5i=l
]=1
~,~
(20)
PERSISTENCE IN STOCHASTIC F O O D WEB MODELS
m
p
j=l
i=a
369
k
Z e~irj+ Z ~,si--E bY,>~0,
l <~i <~k
j=l k
l~i~m. j=l
j=l
k
m
p
~ = y a/-Y i=1
r,~,-y
i=1
s,e,
i=1 --Z r,0~ + y s,o,
Y ~ +
-½Z ]=1
/=1
i=1
>0,
(21)
i=1
then, for each solution (X(t), Y(t), Z(t)) o f (19) with initial value (x, y, z) E R °, there is some index i (possibly dependent on (x, y, z)) such that Zi(t) is s.b. persistent until first exit o f (X(t), Y(t), Z(t)) from R~. Outline o f the proof. The p r o o f follows similarly to that for Theorem 2. Given a solution (X, Y, Z) o f (19) in R~, one computes the stochastic differential of the product k
m
p
v = I I x, I-I ~r, I-I zg,, i=1
i=l
t=1
where the constants r i and si solve the linear programming problem (20). Consider, first of all, such solutions that for some time t (with no loss o f generality, t = 0) are in a corresponding b o u n d e d region R s for = (71, • • •, 7k+m,~k+m+l, • • • , % ) . For qi, k + m + 1 <~ i ~< n, sufficiently small, (21) leads to an estimate like (12) for the deterministic part o f the stochastic differential o f V until (X, Y, Z) exits R s . So the comparison theorem can be applied, and consequently (16) is obtained for each Zi. Finally, the sequential stopping time argument, given in Theorem 2, can be repeated for any Z~ represented by a non-degenerate diffusion, which removes the restriction that (X, Y, Z) remains in Rq. (Note that the conclusion o f the theorem is immediate for any solution o f (19) that never hits R s before exiting R~ .) The author wishes to thank D. Kannan for helpful discussions in preparing this paper.
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