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Mathematical Biosciences 166 (2000) 173±201
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Uniform persistence and permanence for non-autonomous
semi¯ows in population biology q
Horst R. Thieme *
Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804, USA
Received 21 February 2000; accepted 19 April 2000

Abstract
Conditions are presented for uniform strong persistence of non-autonomous semi¯ows, taking uniform
weak persistence for granted. Turning the idea of persistence upside down, conditions are derived for nonautonomous semi¯ows to be point-dissipative. These results are applied to time-heterogeneous models of
S±I±R±S type for the spread of infectious childhood diseases. If some of the parameter functions are asymptotically almost periodic, an almost sharp threshold result is obtained for uniform strong endemicity
versus extinction in terms of asymptotic time averages. Applications are also presented to scalar retarded
functional di€erential equations modeling one species population growth. Ó 2000 Elsevier Science Inc.
All rights reserved.
MSC: 34C35; 34D05; 34D40; 34K25; 92D25; 92D30
Keywords: Persistence; Permanence; Dissipativity; Dynamical systems; Epidemic models; Functional di€erential
equations; (Asymptotically) almost periodic functions; Time averages

1. Introduction

Persistence (or permanence) is an important property of dynamical systems and of the
systems in ecology, epidemics etc., they are modeling. Persistence addresses the long-term
survival of some or all components of a system, while permanence also deals with the limits of
growth for some (or all) components of the system, For background information and references we refer to Thieme [40]. We show that uniform weak persistence implies uniform (strong)

q
*

Research partially supported by NSF grants DMS-9403884 and DMS-9706787.
Tel.: +1-480 965 4772; fax: +1-480 965 8119.
E-mail address: h.thieme@asu.edu (H.R. Thieme).

0025-5564/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 1 8 - 3

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H.R. Thieme / Mathematical Biosciences 166 (2000) 173±201

persistence. Loosely speaking, a population is uniformly weakly persistent if its size, while it

may come arbitrarily close to 0 every now and then, always climbs back to a level that
eventually is independent of the initial data. The population is uniformly (strongly) persistent,
if its size is bounded away from 0 and the bound does not depend on the initial data after
suciently long time. The population is permanent, if it is uniformly (strongly) persistent and if
the population size is bounded with the bound not depending on the initial data after suciently long time.
While a general persistence/permanence theory is available for autonomous semi¯ows [17,23]
and special non-autonomous systems have been considered in the past [3,41] (for more references see [40]), a general non-autonomous theory is still under development. The importance of
a non-autonomous theory is obvious because all ®eld populations live in a seasonal environment. The approach in this paper is based on uniform weak persistence [11,39]; alternatively one
can try to reduce the non-autonomous case to the autonomous one by using skew product ¯ows
[43]. Both approaches show the existence of positive lower bounds which, for suciently large
times, do not depend on the initial conditions, but they do not provide estimates of these
bounds.
It is worth mentioning that permanence of the biological system in particular involves the
point-dissipativity of the semi¯ow (existence of a bounded absorbing set) that models the dynamics of the system. While point-dissipativity has often been assumed to prove persistence [17],
we will in turn use persistence techniques to derive conditions for point-dissipativity.
The results for non-autonomous semi¯ows cannot be so elegantly stated as in the autonomous
case [11,39], as we need additional conditions which appear quite technical though they can effectively be checked in many applications. As a trade-o€, compactness requirements for the state
space or at least for an attracting set can be replaced by appropriate equi-continuity conditions for
the semi¯ow.
We apply our results to establish threshold criteria for disease extinction and disease persistence

in time-hetergeneous S±I±R±S epidemic models and to establish permanence for a one species
model consisting of a scalar retarded functional di€erential equation.
This paper is organized as follows. In Section 2, we generalize the result in [40] that uniform
weak persistence implies uniform strong persistence under appropriate extra conditions. Among
other things, it now covers situations with relaxed invariance (cf. [10, Section 6]). We show how
persistence theory can be turned around to show ultimate boundedness (or point dissipativity) for
non-autonomous semi¯ows. We also illustrate the versatility of the framework by establishing
persistence that holds uniformly with respect to parameters. This is related to robust permanence
[21,32], where permanence of semi¯ows induced by ordinary di€erential equations is preserved
under small C r perturbations of the vector ®eld.
In Section 3, we derive threshold results for disease extinction and disease persistence for an
unstructured epidemic model of S±I±R±S type. In Section 4, we show that these results also hold if
the sojourn time in the removed class has a general distribution. In Section 5, we study a timeheterogeneous model for the dynamics of one species with general feed-back, formulated by a
scalar retarded functional di€erential equation
x_ …t† ˆ x…t†f …t; xt †;

t P r:

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175

We derive conditions for permanence, i.e., for the existence of constants 0 <  < c < 1 such that
 6 lim inf x…t† 6 lim sup x…t† 6 c
t!1

t!1

with ; c not depending on the initial conditions xr ˆ / as long as x…r† > 0.
The conditions we obtain involve asymptotic time averages and are reminiscent of conditions
obtained by Burton and Hutson [3] for prey±predator models and by Wu et al. [41] for almost
periodic Kolmogorov equations. We have collected some material on asymptotic time averages
and their connection to (asymptotic) almost periodicity in Appendix A. Each application section
(Sections 3±5) presents a typical persistence or permanence result shortly after the explanation of
the model equations. These results are not the most general possible, but their meaning can be
grasped independently of Section 2 and the rest of the respective sections.
2. Uniform weak is uniform strong
Let X be a set, r0 2 R, and
D ˆ f…t; s†; r0 6 s 6 t < 1g:
A mapping W : D  X ! X is called a (non-autonomous) semi¯ow on X (anchored at r0 ) if

W…t; s; W…s; r; x†† ˆ W…t; r; x†;

W…r; r; x† ˆ x

8t P s P r P r0 ; x 2 X :

If X is topological space and the mapping W is continuous and a semi¯ow, W is called a continuous
semi¯ow. The semi¯ows we will consider are not necessarily continuous.
W is called an autonomous semi¯ow if W…t ‡ r; r; x† does not depend on r P r0 for t P 0; x 2 X .
Further let
q : X ! ‰0; 1†
be a non-negative functional on X and
Xq ˆ X \ fq > 0g:
Xq is not necessarily forward invariant under W. We consider the function
r : ‰0; 1†  X  ‰r0 ; 1† ! ‰0; 1†
de®ned by
r…t; x; r† ˆ q…W…t ‡ r; r; x††;

t P 0; x 2 X ; r P r0 :

…2:1†

We make the following assumption throughout this section, namely that the real-valued function
r…
; x; r† is continuous on ‰0; 1† for all x 2 X ; r P r0 .
We notice the following relation between r and W:
r…t; W…s; r; x†; s† ˆ q…W…t ‡ s; s; W…s; r; x†† ˆ q…W…t ‡ s; r; x††;
r…0; x; r† ˆ q…x†:
We introduce the following notation:
r1 …x; r† ˆ lim sup r…t; x; r†;
t!1

r1 …x; r† ˆ lim inf r…t; x; r†:
t!1

…2:2†

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H.R. Thieme / Mathematical Biosciences 166 (2000) 173±201

De®nition 2.1. W is called
· weakly q-persistent if
r1 …x; r† > 0 8x 2 Xq ; r P r0 ;
· strongly q-persistent if
r1 …x; r† > 0 8x 2 Xq ; r P r0 ;
· uniformly weakly q-persistent if there exists some  > 0 such that
r1 …x; r† >  8x 2 Xq ; r  r0 ;
· uniformly (strongly) q-persistent if there exists some  > 0 such that
r1 …x; r† >  8x 2 Xq ; r  r0 ;
· weakly q-dissipative if there exists some c > 0 such that
r1 …x; r† < c 8x 2 X ; r P r0 ;
· strongly q-dissipative if there exists some c > 0 such that
r1 …x; r† < c 8x 2 X ; r P r0 ;
· q-permanent if W is both uniformly strongly q-persistent and strongly q-dissipative.
If no misunderstanding about the functional q is possible, we use persistent rather than
q-persistent etc.
In the topologically oriented persistence theory, q…x† is the distance of the point x from a
certain set, the boundary of extinction. Freeing q from this interpretation makes it possible to use
persistence techniques to derive strong from weak dissipativity. It will also come in handy for
studying persistence in physiologically structured population models, where the appropriate state

spaces are formed by regular Borel measures on a locally compact Hausdor€ space and the
semi¯ows are continuous not in the strong, but in the weak topology (see [5] for the linear
foundation of the construction of such semi¯ows and [6] for a ®rst non-linear extension).
Obviously uniform strong persistence implies uniform weak persistence. Deriving conditions
for the converse to hold is not only of theoretical but also of practical interest, because uniform
weak persistence can often be checked in concrete situations with relative ease. Here the following
remark is helpful.
Remark. W is uniformly weakly q-persistent if and only if, for suciently small  > 0, no x 2 Xq
and no r P r0 can be found such that r…t; x; r† 6  for all t P 0.
One direction in this statement is obvious, while the other follows from the semi¯ow property.
Even for autonomous semi¯ows, uniform weak persistence does not imply uniform strong
persistence without further assumptions as shown by the simplest Lotka±Volterra predator±prey
system [11]. Typically some compactness has been assumed in deriving uniform strong from
uniform weak persistence. The metric space is supposed to be locally compact [11] and the
semi¯ow point-dissipative, or the semi¯ow is supposed to be asymptotically smooth and pointdissipative [17]. In [39] we have relaxed these compactness assumptions and presented several
applications in epidemics. We have followed this route for non-autonomous semi¯ows in [40] (see

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177

also [12,42]). Actually one can replace compactness completely by equi-continuity requirements
for r. This makes it possible to use persistence techniques to derive conditions for point-dissipativity of semi¯ows.
We will still use the idea of an absorbing or attracting set, but we will not assume that this set is
bounded or even compact.
Let B be a subset of X ; J ˆ N or an interval of the form ‰a; 1†. We will associate two di€erent
interpretations with the notation
xj ! B as j ! 1;
where …xj †j2J is a sequence or family of elements in X. The ®rst interpretation is often appropriate
for semi¯ows that operate on sets rather than topological spaces, or on locally compact spaces, or
come from problems with ®nite memory. We say that B absorbs …xj †, if
…/† there exists some j0 2 J such that xj 2 B for all j 2 J ; j P j0 .
For semi¯ows on topological spaces that come from problems with in®nite memory (e.g., agedependent equations without ®nite maximum age), the following interpretation is more appropriate: We say that B attracts …xj †, if
….† for any open set U  B there exists some jU 2 J such that xj 2 U for all j 2 J ; j P jU .
If X is a metric space, B attracts …xj † if the distance of xj from B tends to 0 as j ! 1.
W is said to have property (CA) if the following holds:
(CA) There exists a subset B in X with the following two properties:
· For all x 2 Xq ; s P r0 , we have W…t; s; x† ! B; t ! 1.
· If …rj † is a sequence of real numbers and …yj † a sequence in X such that rj ! 1 and yj ! B as
j ! 1 and, for some  > 0; q…yj † ˆ  for all j 2 N, then the continuity of r…
; yj ; rj † is uniform

in j 2 N, possibly after choosing a sub-sequence.
In both conditions, of course, `! B' has to be consistently interpreted as B being either absorbing or attracting. If W…t; s; x† ! B is interpreted in the ®rst way …/†, we say that B is an absorbing set for W, while in the second interpretation ….† we say that B is an attracting set. If B is
absorbing, the second property of (CA) is more succinctly stated as the continuity of r…
; y; r† to be
uniform in r P r0 and y 2 B; q…y† ˆ  for ®xed, but arbitrary  > 0.
Recall that we assume throughout this section that r…
; x; r† is continuous for all x 2 X ; r P r0 .
The notation (CA) has been chosen to recall the historical connection to the existence of a
compact attracting set. The following generalization has been proved useful in studying the
persistence of host and/or parasite populations in time-autonomous epidemic models [39]. If X is
topological space, we call a subset B of X relatively q-compact if all its intersections with q-shells
have compact closure in X. A q-shell is a set fx 2 X ; 1 6 q…x† 6 2 g with 0 < 1 < 2 < 1.
Remark. (CA) holds if the following holds:
· W…t; s; x† ˆ W…t ÿ s; 0; x† ˆ U…t ÿ s; x† is a continuous autonomous semi¯ow on a metric space X.
· There exists a relatively q-compact set B with U…t; x† ! B as t ! 1 for all x 2 X .
· q is continuous on X if B is absorbing, and uniformly continuous if B is attracting.
Proof. Assume that r…t; yj ; sj † ˆ q…U…t; yj †† is not continuous in t uniformly in yj with yj ! B as
j ! 1; q…yj † ˆ . After choosing a sub-sequence of …yj † we ®nd t P 0 and tj ! t as j ! 1 such
that jq…U…tj ; yj †† ÿ q…/…t; yj ††j P d > 0 for all j 2 N. Since yj ! B, we ®nd zj 2 B such that

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H.R. Thieme / Mathematical Biosciences 166 (2000) 173±201

d…yj ; zj † ! 0 as j ! 1. Since q is (uniformly) continuous on the metric space X and q…yj † ˆ  for
large enough j, the zj are in the intersection of B with a q-shell and so have a converging subsequence. Letting y be the limit and using the continuity of q and U, we have
q…U…tj ; yj †† ! q…U…t; y††; q…U…t; yj †† ! q…U…t; y†† contradicting our previous statement.
Even in a locally compact metric space, uniform weak persistence will not imply uniform strong
persistence if the semi¯ow is non-autonomous.
Example 2.2. Let X ˆ …0; 1† and q…x† ˆ x. Construct a pair of functions x} , a on ‰0; 1† with the
following properties:
x_ } ˆ ÿx} ‡ a…t†;
0 < x} …t† 6 2;

t P 0; x} …0† ˆ 2;

0 6 a…t† 6 3 8t P 0;

lim sup x} …t† ˆ 2;

lim inf x} …t† ˆ 0:

t!1

t!1

Now let W be the semi¯ow induced by the solutions of
x_ ˆ ÿx ‡ a…t†; t P r;

x…r† ˆ x0 ;

i.e., W…t; r; x0 † ˆ x…t†. Since x…t† ÿ x} …t† ! 0 as t ! 1, we have that W is uniformly weakly persistent, but not strongly persistent, though W is point-dissipative and asymptotically smooth if
considered on ‰0; 1†.
This example teaches us that we need some condition that takes care of the non-autonomous
nature of the semi¯ow.
For every  > 0; t > 0 we de®ne set R…† and R…t; † as follows (B is the absorbing or attracting
set in (CA)):
R…t; † consists of continuous functions r~ : ‰0; tŠ ! ‰0; Š;
r~…t† ˆ 0 < r~…0† ˆ ;
r~…s† ˆ lim r…s; yj ; sj † uniformly in s 2 ‰0; tŠ
j!1

for sequences …sj † in ‰r0 ; 1†; …yj † in X with sj ! 1; yj ! B; as j ! 1:
R…† consists of continuous functions r~ : ‰0; 1† ! …0; Š;
0 < r~…0† ˆ ;
r~…s† ˆ lim r…s; yj ; sj † locally uniformly in s P 0

…2:3†

j!1

for sequences …sj † in ‰r0 ; 1†; …yj † in X with sj ! 1; yj ! B; as j ! 1:
The semi¯ow W is said to have property (PS) if the following holds:
(PS) If  > 0 is chosen suciently small, the sets R…† and R…t; † are empty for all t P 0.

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179

Remark. (PS) automatically holds for a continuous autonomous semi¯ow U which is uniformly
weakly q-persistent, provided that there is a relatively q-compact attracting set B for U such that
Xq \ B is forward invariant under U and that q is uniformly continuous on the metric space X.
This statement holds Mutandis mutatis for asymptotically autonomous semi¯ows [40].
Proof. Following the proof of the Remark subsequent (CA) we notice that the functions r~ in R…†
or R…t; †, respectively have the form r~…s† ˆ q…U…s; y†† for some y 2 B; q…y† ˆ . R…t; † is empty
because Xq \ B is forward invariant. R…† ˆ ; follows from the uniform weak persistence of U for
suciently small  > 0.
Theorem 2.3. If a semiflow W is uniformly weakly q-persistent and has properties (CA) and (PS),
then it is uniformly strongly persistent.
Proof. Assume that W is uniformly weakly persistent. Then there exist  > 0 such that
r1 …x; r† >  8x 2 Xq ; r P r0 :
By (PS) we can arrange that the sets R…† and R…t; † are empty for all t > 0 for this .
Step 1: Let x 2 Xq and r P r0 such that lim inf t!1 r…t; x; r† < 1 < .
Since q…W…
; r; x†† is continuous on ‰r; 1†, there exist sequences …sk † in ‰r0 ; 1† and …tk † in …0; 1†
such that sk ! 1…k ! 1† and
q…W…sk ; r; x†† ˆ ;

q…W…tk ‡ sk ; r; x†† < 1

q…W…s ‡ sk ; r; x†† 6 

8k 2 N;

whenever 0 6 s 6 tk :

Step 2: Suppose that W is not uniformly strongly persistent.
Then there exist sequences …rj † in ‰r0 ; 1† and …xj † 2 Xq such that
lim inf q…W…t; rj ; xj †† ! 0;

j ! 1:

t!1

Using step 1 and a diagonalization procedure we obtain sequences …rj † and …sj † in ‰r0 ; 1†, and …tj †
in ‰0; 1† such that sj P rj ‡ j for all j 2 N, and
q…W…sj ; rj ; xj †† ˆ ;
q…W…tj ‡ sj ; rj ; xj †† ! 0;
q…W…s ‡ sj ; rj ; xj †† 6 ;

j ! 1;
0 6 s 6 tj :

Since, for every j 2 N; W…t; rj ; xj † ! B as t ! 1 by the ®rst property in (CA), we can further
achieve that
W…sj ; rj ; xj † ! B;

j ! 1:

Set yj ˆ W…sj ; rj ; xj †. Using the de®nition of r in (2.1), we have the following situation:
q…yj † ˆ ;
r…tj ; yj ; sj † ! 0;
r…s; yj ; sj † 6 ;
yj ! y 2 B;

j ! 1;
0 6 s 6 tj ;

j ! 1:

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H.R. Thieme / Mathematical Biosciences 166 (2000) 173±201

We claim: tj ! 1 as j ! 1.
If not, after choosing sub-sequences, tj ! t and r…
; yj ; sj † is continuous uniformly in j. By the
second assumption of (CA), there exist some d > 0 such that
r…s; yj ; sj † 6 ;

06s6t ‡ d

for suciently large j. By the Arzela±Ascoli theorem, after choosing another sub-sequence, we
have
r~…s† ˆ lim r…s; yj ; sj † 6 ; 0 6 s 6 t;
j!1

with convergence holding uniformly for s 2 ‰0; t ‡ dŠ. Obviously r~…0† ˆ  > 0. Again using that
r…
; yj ; sj † is continuous uniformly in j,
0 ˆ lim sup r…t; yj ; sj † ˆ r~…t†:
j!1

Hence we have found an element in R…t; † which is a contradiction to (PS).
So, tj ! 1 as j ! 1. Let t P 0 be arbitrary. Then tj > t for suciently large j and
 P r…s; yj ; sj †

8s 2 ‰0; tŠ;

if j is large enough. This implies that r…
; yj ; sj † is bounded on ®nite intervals in ‰0; 1† uniformly in
j 2 N. The second assumption in (CA) implies that r…
; yj ; sj † is continuous on ‰0; 1† uniformly in
j, possibly after choosing a sub-sequence. By the Arzela±Ascoli theorem, there exists a function r~
on ‰0; 1† such that, after choosing another sub-sequence,
r…t; yj ; sj † ! r~…t†;

j ! 1; locally uniformly in t P 0

and
r~…0† ˆ  P r~…t† 8t P 0:
Since R…t; † ˆ ; for all t > 0, r~ is strictly positive. So we have an element r~ 2 R…† which again
contradicts (PS).
The ¯exibility of q-persistence (compared with the topological concept of persistence) allows to
prove boundedness results.
~
~ j† as follows (B is the absorbing or atFor every j > 0; t > 0 we de®ne sets R…j†
and R…t;
tracting set in (CA)):
~
R…j†

consists of continuous functions r~ : ‰0; 1† ! ‰j; 1†;
r~…0† ˆ j;

r~…s† ˆ lim r…s; yj ; sj † locally uniformly in s P 0
j!1

for sequences …sj † in ‰r0 ; 1; …yj † in X with sj ! 1; yj ! B as j ! 1:
~ j†
R…t;

consists of continuous functions r~ : ‰0; t† ! ‰j; 1†;
r~…0† ˆ j; lim r~…s† ˆ 1;
s!tÿ

r~…s† ˆ lim r…s; yj ; sj † uniformly in s 2 ‰0; t†
j!1

for sequences …sj † in ‰r0 ; 1†; …yj † in X with sj ! 1; yj ! B as j ! 1:

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181

Theorem 2.4. Let W be a semiflow that has the property (CA). Assume that, if j is chosen suffi~
~ j† are empty for all t P 0.
ciently large, the sets R…j†
and R…t;
Then W is strongly q-dissipative whenever it is weakly q-dissipative.
Proof. Assume the W is weakly q-dissipative. Set
q~…x† ˆ

1
:
1 ‡ q…x†

Then q~…x† > 0 for all x 2 X and W is uniformly weakly q~-persistent. Moreover W satis®es (CA)
and (PS) with q~ replacing q. It follows from Theorem 2.3 that W is uniformly strongly q~-persistent,
i.e., strongly q-dissipative.
In some applications one would like persistence to hold uniformly with respect to a parameter
(cf. [9, end of Section 4], [36, Section 3] and [37, Section 4]).
Let N be an index set and assume that we have a family of (non-autonomous) semi¯ows
Wn : D  X ! X ;

n 2 N:

…Wn † is said to have property …CA†N if the following holds:
…CA†N There exists a subset B in X with the following two properties:
· For all x 2 X ; s P r0 ; n 2 N, we have Wn …t; s; x† ! B; t ! 1.
· If …rj † is a sequence of real numbers and …yj † a sequence in X ; …nj † a sequence in N
such that rj ! 1 and yj ! B as t ! 1 and, for some  > 0; q…yj † ˆ  for all j 2 N,
then the continuity of
t 7! q…Wnj …t ‡ sj ; sj ; yj ††
on ‰0; 1† is uniform in j 2 N, possibly after choosing a sub-sequence.
Set r…t; y; r† ˆ q…Un …t ‡ r; r; x†† for y ˆ …x; n†. For every t P 0;  > 0, sets R…† and R…t; † are now
de®ned in obvious analogy to (2.3) using sequences yi ˆ …xj ; nj † with xj ! B. We say that the
family …Wn † satis®es property …PS†N if the following holds:
…PS†N

If  > 0 is chosen suciently small, the sets R…† and R…t; † are empty for all t P 0.

Theorem 2.5. Let N be a set and Wn ; n 2 N, be a family of semiflows on X that has properties
…CA†N and …PS†N with absorbing or attracting set B. Further assume that
t 7! q…Wn …t ‡ s; s; x††
is continuous on ‰0; 1† for any x 2 X ; n 2 N; s P r0 . Finally assume there exists some  > 0 such
that
lim sup q…Wn …t; s; x†† P  8x 2 Xq ; n 2 N; s P r0 :
t!1

Then there exists some  > 0 such that
lim inf q…Wn …t; s; x†† P 
t!1

8x 2 Xq ; n 2 N; s P r0 :

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H.R. Thieme / Mathematical Biosciences 166 (2000) 173±201

Proof. Let Z ˆ X  N and
q~…x; n† ˆ q…x†:
~ : D  Z ! Z by
De®ne a semi¯ow W
~ s; …x; n†† ˆ …Wn …t; s; x†; n†:
W…t;
~ is uniformly weakly q~-persistent. Further W
~ satis®es (CA) with absorbing or attracting
Then W
set B  N (in the second case endow N with the trivial topology). One easily checks that the
~ is uniformly strongly
other assumptions of Theorem 2.3 are satis®ed as well. It follows that W
q~-persistent. This implies the assertion of the theorem.
Similarly one can prove boundedness results for families of semi¯ows.
Theorem 2.3 has an interesting twist for continuous autonomous semi¯ows U, where Xq is not
forward invariant under all, but only under total orbits of U. We recall that / : R ! X is a total
orbit of U, if U…t†/…s† ˆ /…t ‡ s† for all t P 0; s 2 R.
Theorem 2.6. Let U be a continuous autonomous semiflow on a metric space X which has a compact
attracting set K, i.e., a compact set K such that dist…U…t; x†; K† ! 0; t ! 1. We further assume for
any total orbit / : R ! X of U with relatively compact range:
· If s 2 R and q…/…s†† > 0, then q…/…t†† > 0 for all t > s.
Then U is uniformly strongly q-persistent whenever it is uniformly weakly q-persistent.
Proof. We can assume that the compact attracting set K is invariant under U. Otherwise we replace it by the closure of the union of all x-limit sets of U. The assumption (CA) is satis®ed as we
have shown in the remark following (CA).
Using the compactness of K in the metric space X, for every element r~ in R…† or in R…t; † we
have r~…s† ˆ q…U…s; y†† for some y 2 K. So uniform weak persistence implies that R…† is empty for
suciently small  > 0. Since y 2 K and K is invariant, U…s; y† can be extended to a total orbit of U
with relatively compact range. This implies R…t; † ˆ ;, because any r~ in this set would satisfy
r~…0† > 0 ˆ r~…t†, while, by ; r~…0† > 0 would imply r~…t† > 0.
Situations where Xq is not forward invariant, but q-positivity is preserved by total orbits, are
met in stage-structured population or epidemic models, see, [10, Section 6].
Remark. Property  holds, e.g., if the following two conditions are satis®ed:
(i) If / is a total orbit of U with relatively compact range, s 2 R, and q…/…r†† ˆ 0 for all r 6 s,
then q…/…t†† ˆ 0 for all t P s.
(ii) There exist some s > 0; d > 0 such that q…U…t; x†† > 0 for all t 2 …s; s ‡ d†; x 2 Xq .
Proof. If x 2 Xq , it follows from the semi¯ow property and repeated use of (ii) that q…U…t; x†† > 0
for all t 2 …ms; m…s ‡ d††; m 2 N, and so q…U…t; x†† > 0 for all t P ms, where m is the ®rst natural
number with md > s. Let now / be a total orbit of U; s 2 R and q…/…s†† > 0. By (i), there exists a
sequence …sj † in …ÿ1; s† with sj ! ÿ1 as j ! 1 and q…/…sj †† > 0. Choose some sj < s ÿ ms. By
condition (ii) and the orbit property, q…/…t†† > 0 for all t P s.

H.R. Thieme / Mathematical Biosciences 166 (2000) 173±201

183

To interpret condition (ii) of the above Remark, let us assume that q  U is the infection rate in
an epidemic model. If the infection rate is positive at the beginning, it can be 0 somewhat later
because all infectious individuals have recovered in the meanwhile and the infected ones are still in
the latency period. Eventually however, after an elapse of time which only depends on the length
of the various periods of infection, the infection rate will be positive again. Condition (i) is related
to the fact that infectious diseases cannot come out of nothing. If infectives have not been around
in the past, then they will not be in the future (excluding, of course, that they are introduced
from outside, or that viruses mutate and so become able to infect the host species under
consideration).

3. The time-heterogeneous S±I±R±S model
As an application of the theory developed in Section 2, we consider a model for the spread of
infectious childhood diseases. It has been argued that the school system induces a time-heterogeneity in the per capita/capita infection rate, a, because the chain of infections is interrupted or at
least weakened by the vacations and new individuals are recruited into a scene with higher infection risk at the beginning of each school year [8,31]. Here we consider a model without exposed
period; the total population, with size N, is divided into its susceptible, S, infective, I, and recovered, R, parts, and the contraction of the disease is modeled by the law of mass action involving susceptibles and infectives
N ˆ S ‡ I ‡ R;
dI
ˆ ÿl…t†I ‡ a…t†SI ÿ c…t†I;
dt
dR
ˆ ÿl…t†R ‡ c…t†I ÿ n…t†R:
dt

…3:1†

We assume that the disease causes no fatalities and that the population size N …t† is a given
function of time t, l…t† is the instantaneous per capita mortality rate and c…t† and n…t† are the
instantaneous per capita rates of leaving the infective stage or removed stage, respectively.
In the ®rst step, one can assume that a is a periodic function and that N ; l; c and n are constant.
This case (sometimes for the S±I±R, sometimes for the S±E±I±R model with exposed stage) was
studied numerically by London and Yorke [25] and Dietz [7], formally by Grossman et al. [13,14],
and analytically by Smith [34,35] and Schwartz and Smith [33]. It was shown that periodic solutions exist whose periods are integer multiples of the period of a and that co-existence of stable
periodic solutions with di€erent periods is possible [33±35]. In the S±E±I±R model, periodic
forcing of the infection rate can even lead to a sequence of period doubling sub-harmonic bifurcations and ®nally chaos [1,2,26,29,30]. Whether the same holds for the S±I±R model, is not
known to me; Theorem 3.2 below suggests that the trajectories cannot be completely wild (see
[8,19] for more detailed reviews).
Throughout this section we will assume that N ; l; a; c; and n are arbitrary non-negative,
continuous, bounded functions on [0, 1) and that l is bounded away from 0. Under these

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assumptions we derive conditions for extinction and persistence of the disease. In case that
N; l; a and c (but not necessarily n) are almost periodic and we will obtain an almost sharp
threshold result.
Unfortunately our approach does not seem to extend to the S±E±I±R±S model.
To set the stage, we recall the following result, if N ; l; a; c; and n are constant [18]. If
R0 ˆ aN =…l ‡ c† 6 1; then I…t† ! 0; as t ! 1 for every solution of (3.1). If R0 > 1, there exists
a unique endemic equilibrium which attracts all solutions with I…0† > 0, in particular
I…t† ! I  > 0 as t ! 1 with a uniquely determined I  > 0. While we cannot retain the global
asymptotic stability, we can preserve the threshold result concerning disease extinction versus
endemicity using asymptotic time averages.
A function f : ‰0; 1† ! R is called equi-mean-convergent (or to have uniformly convergent
means) if the (asymptotic) mean value
1
f ˆ lim
t!1 t

Z

t‡r

f …s† ds
r

exists, is independent of r and the convergence is uniform in r. Almost periodic functions are equimean-convergent, and so are asymptotically almost periodic functions and weakly asymptotically
almost periodic solutions (see Appendix A). If aN ; c; l are equi-mean-convergent, we de®ne
R0 ˆ

…aN†
:
l ‡ c

Theorem 3.1. Let N be uniformly continuous on ‰0; 1† and aN; l and c be equi-mean-convergent.
(a) If R0 < 1, the disease dies out, i.e., for every solution of (3.1) we have I…t†;
R…t† ! 0 as t ! 1.
(b) If R0 > 1, the disease persists uniformly strongly in the population, in the sense that there exists some  > 0 such that I1 ˆ lim inf t!1 I…t† >  for all solutions of (3.1) with I…r† > 0 for some
r P 0.
The next result shows that strong persistence implies that the solutions cannot be arbitrarily
wild.
Theorem 3.2. Let N be uniformly continuous on ‰0; 1† and aN ; l and c be equi-mean-convergent
and R0 > 1. Then
1
t

Z

1
t

Z

t

a…s†S…s† ds ! l ‡ c ˆ
0

…aN †
;
R0

t!1

and
t
0



1
> 0:
l ‡ c† ˆ …aN † 1 ÿ
a…s†…I…s† ‡ R…s†† ds ! …aN† ÿ …
R0

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Proof. From (3.4) we obtain

 Z t

1

l ‡ c† 6 lim sup
a…s†S…s† ds ÿ …
lim sup 
t 0
t!1
t!1



1  I…t† 
ˆ 0:
ln
t  I…0† 

The last equality follows from I being bounded away from 0 and bounded. This implies the ®rst
statement, the second is now obvious.
We will obtain Theorem 3.1 as a special case of extinction and persistence results which do not
assume equi-mean-convergence. To get started we notice that, for every solution of (3.1), I…t† is
non-negative for all t P r P 0 where it is de®ned whenever I…r†  0. The same property holds for
R. Adding the di€erential equations for I and R it is easy to see that every solution satis®es
I…t† ‡ R…t† 6 maxfI…r† ‡ R…r†; sup N g;

t > r P 0:

A standard continuation argument now tells us that, given non-negative initial data at time r P 0,
the solutions are de®ned for all t  r.
Though this is not important for our mathematical considerations, epidemiologically one
would like S ˆ N ÿ I ÿ R to be non-negative as well. This can be achieved by assuming that N is
continuously di€erentiable and that the population birth rate, B, satis®es
B…t† :ˆ

d
N …t† ‡ l…t†N …t† P 0:
dt

…3:2†

Then S ˆ N ÿ I ÿ R satis®es the di€erential equation
dS
ˆ B…t† ÿ l…t†S ÿ a…t†SI ‡ n…t†R
dt
from which we learn that, whenever S…r† P 0; r P 0; then S…t† P 0 for all t P r. This implies
0 6 S…t†; I…t†; R…t† 6 N …t†

8t P r:

Integrating the I-equation we obtain
1
I…t†
ln
ˆ
t ÿ r I…r†



1
tÿr

Z

t
r

1
a…s†S…s† ds ÿ
tÿr

Z

t
r

1
c…s† ds ÿ
tÿr

Z

t

r

Set
R} ˆ

…aN †}
;
l} ‡ c}

Z
1 t
a…s†N …s† ds;
t 0
t!1
Z
1 t
c…s† ds
c} ˆ lim inf
t!1
t 0

…aN†} ˆ lim sup

and l} analogously to c} . We have the extinction result as follows.


l…s† ds
8t > r P 0:

…3:3†

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Theorem 3.3. Let R} < 1. Then the disease dies out, i.e., for every solution of (3.3) we have
I…t†; R…t† ! 0 as t ! 1.
Proof. For simplicity we restrict the proof to a solution with initial data given at time 0. Since
S 6 N and R} < 1, there exists some  > 0; r > 0 such that
Z
Z
1 t
1 t
a…s†S…s† ds ÿ
‰c…s† ‡ l…s†Š ds 6 ÿ  8t P r:
t 0
t 0
By (3.3),
1 I…t†
ln
6 ÿ
t I…r†

8t P r:

Hence
I…t† 6 I…r†eÿt ;

tPr

and I…t† ! 0; t ! 1. The R-equation in (3.1) implies R…t† ! 0 as t ! 1.
In order to obtain a weak persistence result we set
Z
1 t
…aN †} ˆ lim inf
a…s†N …s† ds;
t!1
t 0
Z
1 t
c…s†ds; l} ˆ


;
c} ˆ lim sup
t
t!1
0
R} ˆ

…aN †}
l} ‡ c }

and let I 1 be the limit superior of I…t† as t ! 1.
Theorem 3.4. Let R} > 1. Then the disease uniformly weakly persists in the population, in the sense
that there exists some  > 0 such that I 1 >  for all solutions of (3.1) with I…r† > 0 for some r P 0. 
does not depend on n.
Proof. Let us suppose that for every  > 0, there is some solution with I 1 < : Since
R1 6 sup…c=l†I 1 ;
I 1 ‡ R1 < c
with c > 0 being independent of the solution, of , and of n. By N ˆ S ‡ I ‡ R and (3.3),
d
ln I P a…t†…N…t† ÿ c† ÿ c…t† ÿ l…t†
dt

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for suciently large t: Hence there exists some t > 0 such that

Z t
Z t
1
I…t†
1
1
a…s†N …s† ds ÿ c
a…s† ds
ln
P
t ÿ t I…t †
t ÿ t t
t ÿ t t

Z t
Z t
1
1
c…s† ds ÿ
l…s† ds ; t > t :
ÿ
t ÿ t t
t ÿ t t
For suciently large time t, we have
1
I…t†
ln
P …aN †} ÿ ca} ÿ c} ÿ l} ÿ :
t ÿ t I…t †
Since R} > 1; …aN †} ÿ c} ÿ l} > 0. Hence
1
I…t†
ln
P d
t ÿ t I…t †
for large times t, with d > 0 provided  > 0 is chosen small enough. Thus


I…t† P I…t †ed…tÿt †
for suciently large t and I…t† ! 1 as t ! 1; a contradiction to the fact that I is bounded.
In order to formulate a strong persistence results, let I1 be the limit inferior of I…t†: Further
…aN†
;
l ‡ c
Z
1 t
…aN† :ˆ lim inf
a…r ‡ s†N …r ‡ s† dr;
t;s!1 t
0
Z t
1
c :ˆ lim sup
c…r ‡ s† dr
t 0
t;s!1
R :ˆ

and l is de®ned analogously to c :
Theorem 3.5. Let N be uniformly continuous on ‰0; 1† and R > 1: Then the disease uniformly
strongly persists in the population, in the sense that there exists some  > 0 such that I1 >  for all
solutions of (3.1) with I…r† > 0 for some r P 0. Let I1 be the limit inferior of I…t† as t ! 1.
Proof. In order to get into the framework of Section 2, consider the semi¯ow W that is induced by
the system
dI
ˆ ÿl…t†I ‡ a…t†…N ÿ I ÿ R†I ÿ c…t†I;
dt
dR
ˆ ÿl…t†R ‡ c…t†I ÿ n…t†R
dt
on
X ˆ fx ˆ …I; R†; R P 0; I > 0g;

…3:4†

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endowed with the standard metric. Further we set
q…I; R† ˆ I:
In particular
W…t; s; x† ˆ …I…t†; R…t††;

t P s P 0;

where I; R solve (3.4) and …I…s†; R…s†† ˆ x. Further
r…t; x; s† ˆ I…t ‡ s†:
It may happen that, for certain initial data, we get solutions that are not epidemiologically
meaningful because I…t† ‡ R…t† > N …t† may occur, but this is irrelevant mathematically.
Condition (CA) is satis®ed because
sup c 1
I
I 1 < sup N ; R1 6
inf l
and the set
B ˆ fx ˆ …I; R†; I; R > 0; I ‡ R 6 sup N g
is absorbing and forward invariant. A standard Gronwall argument implies that r…
; x; s† is
continuous on ‰0; 1† uniformly in x 2 B; s P 0:
W is uniformly weakly q-persistent as a consequence of Theorem 3.4. In order to check property
(PS), let us describe the elements of R…t; † and R…† in (2.3) in terms of system (3.4). To this end we
consider sequences sj ! 1; yj ! B in X, as j ! 1: Let Ij ; Rj be the solutions of
dIj
ˆ ÿl…t ‡ sj †I ‡ a…t ‡ sj †…N …t ‡ sj †† ÿ Ij ÿ Rj †Ij ÿ c…t ‡ sj †Ij ;
dt
dRj
ˆ ÿl…t ‡ sj †Rj ‡ c…t ‡ sj †Ij ÿ n…t ‡ sj †Rj ; …Ij …0†; Rj …0†† ˆ yj :
dt
After choosing a sub-sequence we may assume that yj converges to some element x. Since N is
uniformly continuous and bounded on ‰0; 1†; the Arzela±Ascoli theorem implies that, after
choosing a sub-sequence,
~
N …sj ‡ t† ! N…t†;

j!1

locally uniformly in t P 0; where N~ is a bounded continuous function. Moreover, since L1 ‰0; 1† is
separable, the Alaoglu±Bourbaki theorem implies that, after choosing a sub-sequence,
a…sj ‡ t† ! a~…t†;
c…sj ‡ t† ! c~…t†;

j ! 1;
j ! 1;

l…sj ‡ t† ! l~…t†;

j ! 1;

where a~; c~; l~ are elements of L1 ‰0; 1† and the convergence holds in the weak topology carried by
L1 ‰0; 1† as dual space of L1 ‰0; 1†.
The derivatives of Rj and Ij are bounded, uniformly in j 2 N. Again by the Arzela±Ascoli
theorem we have that, after choosing a sub-sequence,
~
Ij …t† ! I…t†;

~
Rj …t† ! R…t†;
j!1

H.R. Thieme / Mathematical Biosciences 166 (2000) 173±201

189

locally uniformly in t P 0; where I~ is absolutely continuous, R~ is continuous, and both are
bounded and satisfy
~ ‡ a~…t†…N~ ÿ I~ ÿ R†
~ I…t†
~ ÿ c~…t†I…t†;
~
I~0 …t† P ÿ l~…t†I…t†

for almost all t P 0:

…3:5†

Since
Rj …t† 6 Rj …0†e

ÿl t

Z

‡ sup c

t

Ij …t ÿ s†eÿl s ds;

l ˆ inf l

0

we have
ÿl t
~ 6 R…0†e
~
R…t†
‡ sup c

Z

t

~ ÿ s†eÿl s ds
I…t

0

and, by Fatou's lemma,
sup c ~
I1:
R~1 6
inf l

…3:6†

~ > 0 implies I…t†
~ > 0 for all t P 0; so R…t; † is empty whatever  > 0.
We ®rst realize that I…0†
The elements of R…e† in (2.3) can be identi®ed as
~
r~…t† ˆ I…t†;
~ > 0; I…t†
~ 6  for all t P 0:
where I~ satis®es (3.5) with R~ satisfying (3.6) and I…0†
The same consideration as in the proof of Theorem 3.4 now implies that such an I~ cannot exist,
~ } > 1; where R
~ } is the analog of R} in Theorem
if  > 0 is chosen small enough, provided that R
3.4 with a~ and c~ replacing a and c: But
Z
Z
1 t
1 t
a~…r† dr ˆ lim inf
a~} ˆ lim inf
lim a…sj ‡ r† dr
t!1
t!1
t 0
t 0 j!1
Z t
1
ˆ lim inf lim
a…sj ‡ r† dr
t!1
t j!1 0
Z t
1
a…s ‡ r† dr
P lim inf lim inf
t!1
t s!1 0
Z
1 t
P lim inf
a…s ‡ r† dr ˆ a :
t;s!1 t
0
~ } : Hence R
~ } P R > 1: This ®nishes the
Similarly considerations hold for the other terms in R
proof.
Theorem 3.1 now follows from Theorems 3.3 and 3.5 because R0 ˆ R ˆ R} :
Returning to Theorem 3.4 we notice that the weak disease persistence does not depend on the
removal rate n. Similarly, checking the proof of Theorem 3.5, we notice that n did not enter the
proof. This means that the property (PS) is satis®ed uniformly for any continuous non-negative n.
We obtain the following endemicity result from Theorem 2.5.

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Theorem 3.6. Let N be uniformly continuous and bounded on ‰0; 1† and R > 1: Let N be an equibounded family of non-negative continuous functions n on ‰0; 1†: Then the disease persistence is
uniform in n, in the sense that there exists some  > 0 that is independent of n 2 N such that I1 > 
for all solutions of (3.1) with I…r† > 0 for some r P 0.
Proof. Theorem 3.6 is derived from Theorem 2.5 similarly as Theorem 3.5 from Theorem 2.3. In
order to check condition …PS†N , we consider sequences sj ! 1; nj 2 N; xj ! B, where B is chosen
as in proof of Theorem 3.5. After choosing a sub-sequence we can assume that xj ! x 2 B and
nj ! n 2 N as j ! 1; with the latter holding in the weak topology of L1 ‰0; 1†. Let Ij ; Rj be
solutions of
dIj
ˆ ÿl…t ‡ sj †I ‡ a…t ‡ sj †…N …sj ‡ t† ÿ Ij ÿ Rj †Ij ÿ c…t ‡ sj †Ij ;
dt
dRj
ˆ ÿl…t ‡ sj †Rj ‡ c…t ‡ sj †Ij ÿ nj …t ‡ sj †Rj ; …Ij …0†; Rj …0†† ˆ xj :
dt
~
Arguing as in the proof of Theorem 3.5, elements in R…† or R…t; † are given by I;
~ ‡ a~…t†…N~ ÿ I~ ÿ R†
~ ÿ c~…t†I…t†;
~
I~0 …t† P ÿ l~…t†I…t†
for almost all t P 0;
with R~ satisfying (3.6) and a~; c~; l~ being the weak limits of a…
‡ sj †; c…
‡ sj †; l…
‡ sj † and N~ being
the locally uniform limit of N …
‡ sj †; after choosing appropriate sub-sequences.
It follows that (PS)N holds, in the same way as in the proof of Theorem 3.5.
Checking the proof of Theorem 3.2 one notices that, under the assumptions of Theorem 3.6, the
convergence in Theorem 3.2 holds uniformly in n 2 N.

4. The time-heterogeneous S±I±R±S model with distributed removed class
In this section, we model the spread of infection in the same way as in Section 3
N ˆ S ‡ I ‡ R;
dI
ˆ ÿl…t†I ‡ a…t†SI ÿ c…t†I:
dt

…4:1†

Again we assume that the disease causes no fatalities and that the population size N…t† is a given
function of time t. l…t† is the instantaneous per capita mortality rate. c…t† is the instantaneous per
capita rate of leaving the infective stage.
Di€erently from Section 3, we assume that passage through the removed stage is heterogeneous
not only in time. One possible reason may be that the removed stage splits into substages, e.g., the
®rst removed stage may be a quarantine stage, where the infected individuals are still potentially
infectious, but are kept from spreading the disease, while in the second removed stage the individuals have recovered from the disease. Another possible scenario has a recovery period of ®xed
length. A convenient general formulation uses the concept of stage (or class age),

H.R. Thieme / Mathematical Biosciences 166 (2000) 173±201

R…t† ˆ

Z

191

a0

…4:2†

u…t; a† da;
0

where u…t;
† is the class age density of removed individuals. Class age is the time that has elapsed
since entering the class. a0 2 ‰0; 1Š is the maximum possible class age, i.e., the maximum duration
of the removed stage.
One way to model the dynamics of the removed individuals consists in using a partial di€erential equation
ou ou
‡
ˆ ÿ…l…t†† ‡ g…t; a††u;
ot oa
u…t; 0† ˆ c…t†I…t†;

…4:3†

u…r; a† ˆ u0 …a†:
In these equations we have t > r; a > 0: r P 0 is the time at which we start observing the course
of the epidemic and u0 is the stage age density at time r, called the initial density. g…t; a† is the per
capita rate of leaving the removed stage at class age a and time t. Autonomous or asymptotically
autonomous versions of this model have been considered by Hethcote et al. [20], Stech and
Williams [38] and Castillo-Chavez and Thieme [4].
We make the same assumptions concerning N ; l; a and c as in the previous sections. g is
assumed to be a non-negative, bounded continuous function on ‰0; 1†  ‰0; a0 †:
The results of the previous section still hold.
Theorem 4.1. Theorems 3.1±3.5 hold verbatim.
We brie¯y indicate how the proofs need to be modi®ed and supplemented.
Integrating along characteristics one can transform the boundary value problem (4.3) into an
integral equation

 Z t
‰l…s† ‡ g…s; s ‡ a ÿ t†Š ds ; t ÿ r > a;
u…t; a† ˆ c…t ÿ a†I…t ÿ a† exp ÿ
tÿa

u…t; a† ˆ u0 …a ‡ r ÿ t† exp



ÿ

Z

r

t


‰l…s† ‡ g…s; s ‡ a ÿ t†Š ds ;

t ÿ r < a:

This formula can be used to establish existence and uniqueness of non-negative solutions of (4.1)±
(4.3): substituting it into (4.2) reduces (4.1) to an integro-di€erential equation in I which can be
solved by Banach's ®xed point theorem. In general, the partial di€erential equation in (4.3) will
not be satis®ed in a classical sense, but in a generalized sense, which is strong enough to imply
uniqueness. Anyway one can show that R is absolutely continuous and
Z a0
dR
g…a; t†u…t; a† da a:e:
ˆ c…t†I…t† ÿ l…t†R…t† ÿ
dt
0
Noticing that
Z a0
g…a; t†u…t; a† da 6 sup g R…t†
0

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we de®ne
1
n…t† ˆ
R…t†
n…t† ˆ 0
Then

Z

a0

g…a; t†u…t; a† da if R…t† > 0;

0

if R…t† ˆ 0:

dR
ˆ c…t†I…t† ÿ l…t†R…t† ÿ n…t†R…t† for almost all t > 0
dt
and
n…t† 6 sup g

8t P 0:

The assertion now follows from Theorem 3.6 and the remark at the end of Section 3.

5. A scalar functional di€erential equation
We consider a scalar retarded functional di€erential equation that models the growth of a one
species population. We assume that state and size of the population couple back to its growth rate
with some delay which does not exceed a number s > 0. Let
C ˆ C‰ÿs; 0Š;

C‡ ˆ C‡ ‰ÿs; 0Š

denote the space of real-valued continuous functions on the interval ‰ÿs; 0Š and the cone of nonnegative functions, respectively. C is endowed with the supremum-norm and C‡ becomes a
complete metric space under the induced metric.
For r P 0 and / 2 C‡ ‰ÿs; 0Š consider
x_ …t† ˆ x…t†f …t; xt †;
x…r ‡ s† ˆ /…s†;

t > r;

s 2 ‰ÿs; 0Š;

…5:1†

where
f : ‰0; 1†  C‡ ! R
is continuous. Here xt 2 C‡ ‰ÿs; 0Š is given by
xt …s† ˆ x…t ‡ s†;

s 2 ‰ÿs; 0Š:

An example is Hutchinson's [22] equation
x0 …t† ˆ x…t†‰a…t† ÿ m…t†x…t ÿ s†Š;

…5:2†

where a is a continuous function and m a continuous non-negative function on R. This equation
models intra-speci®c competition increasing the per capita mortality rate. If competition is not
direct but mediated by exhaustion of a vital resource, the density-dependent feed-back is typically
delayed. a is the di€erence of the per capita birth and death rates under optimal conditions, while
m gives the increase of the per capita death rate per unit population density. The autonomous
version (constant a and m) has been widely studied and generalized in various directions

H.R. Thieme / Mathematical Biosciences 166 (2000) 173±201

193

(see [24, Chapters 4 and 9], and the references therein). In this paper, we try to let the time dependence of a and m as general as possible formulating our assumptions in terms of their time
averages.
Z
1 t
a…s ‡ r† ds:
ar …t† ˆ
t 0
Permanence of the population follows if the time averages of a and m are bounded and bounded
away from zero for large t and r and if a and m are uniformly locally integrable.
A locally integrable function a : ‰0; 1† ! R is called locally uniformly integrable if
Z t
a…s†ds ! 0; t ÿ r ! 0; t P r:
r

The following result for Hutchinson's equation, (5.2), gives a ¯avor of the general results for Eq.
(5.1) (Theorems 5.5 and 5.6).
Theorem 5.1. Let a and m be uniformly locally integrable and let there exists c0 > 0 > 0 and t0 ; r0 >
0 such that the time averages of a and m satisfy
0 6 ar …t†; mr …t† 6 c0

8t P t0 ; r P r0 :

Then the population is permanent in the sense that there exists some j > 1 (independent on the initial
data) such that
1
< lim inf x…t† 6 lim sup x…t† < j
t!1
j
t!1
for all non-negative solutions x of (5.2) which are not identically equal to 0.
In particular we have permanence if both a and m are bounded and bounded away from 0. But
in populations with seasonal reproduction, a repeatedly becomes negative and this is why we use
time averages to come up with realistic conditions for permanence. The proof of Theorem 5.1
(end of this section) checks the assumptions of the results we will now derive for the general scalar
equation (5.1).
Assumption 5.2.
(a) We assume that f is continuous and satis®es the following Lipschitz condition: There exists a
Borel measurable function g : ‰0:1†2 ! ‰0; 1† such that
jf …t; /† ÿ f …t; w†j 6 g…t; k/k ‡ kwk†k/ ÿ wk

8t P 0; /; w 2 C‡ :

g…t;
† is non-decreasing for every t P 0 and g…
; c† is uniformly locally integrable for every c > 0.
(b) We assume that f …t; 0† is uniformly locally integrable.
(c) Finally, we assume that there exists a uniformly locally integrable function f~ such that
f …t; /† 6 f~…t†

8t P 0; / 2 C‡ :

In order to show global existence and uniqueness of solutions for initial data in C‡ we extend f
to ‰0; 1†  C by setting

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H.R. Thieme / Mathematical Biosciences 166 (2000) 173±201

f …t; /† ˆ f …t; /‡ †
with /‡ denoting the positive part of /; /‡ …s† ˆ maxf0; /…s†g. Standard FDE theory (for example [15,24]) implies that we have unique local solutions of the modi®ed equation. Obviously,
from the form of (5.1), we have that x…t† P 0 for all t > r or x…t† > 0 for all t > r whenever the
respective property holds for t ˆ r. Uniqueness implies that
W…t; r; /† ˆ xt
with x solving (5.1), de®nes a non-autonomous semi¯ow anchored at r ˆ 0. Assumption 5.2(b)
and non-negativity of solutions now imply that a solution is bounded on every ®nite interval of
existence. It follows from Assumption 5.2(a) that the derivative of a solution is integrable on every
®nite interval of existence. So a solution de®ned on a ®nite interval can be extended to the closure
of the interval and further extended. This means that solutions exist for all forward times.
We ®rst want to use Theorem 2.4 to derive condi