Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol166.Issue2.Aug2000:

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 non-autonomous 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 as-ymptotically 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 refer-ences 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.}

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 refer-ences 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 environ-ment. 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 dy-namics 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 ef-fectively 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 Cr _{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 time-heterogeneous model for the dynamics of one species with general feed-back, formulated by a scalar retarded functional di€erential equation

_

We derive conditions for permanence, i.e., for the existence of constants 0< <c<1such that

6lim inf

t!1 x…t†6lim sup_t!1 x…t†6c

with;c not depending on the initial conditionsxr ˆ/ as long asx…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

LetX be a set,r0 2R, and

Dˆ f…t;s†;r06s6t<1g:

A mappingW:DX !X is called a (non-autonomous) semi¯owon X(anchored atr0) if

W…t;s;W…s;r;x†† ˆW…t;r;x†; W…r;r;x† ˆx 8tPsPrPr0; x2X:

IfXis topological space and the mappingWis continuous and a semi¯ow,Wis called acontinuous

semi¯ow.The semi¯ows we will consider are not necessarily continuous.

Wis called an autonomous semi¯ow ifW…t‡r;r;x†does not depend onrPr0fortP0; x2X.

Further let

q:X ! ‰0;1†

be a non-negative functional onXand

XqˆX \ 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† ˆq…W…t‡r;r;x††; tP0; x2X; rPr0: …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 x2X; rPr0.

We notice the following relation between rand W:

r…t;W…s;r;x†;s† ˆq…W…t‡s;s;W…s;r;x†† ˆq…W…t‡s;r;x††;

r…0;x;r† ˆq…x†: …2:2†

We introduce the following notation:

r1…x;r† ˆlim sup

t!1

r…t;x;r†; r1…x;r† ˆlim inf

De®nition 2.1. Wis called

· _{weaklyq-persistentif}

r1…x;r†>0 8x2Xq; rPr0;

· _{strongly q-persistentif}

r1…x;r†>0 8x2Xq; rPr0;

· _{uniformly weakly q-persistentif there exists some >0 such that}

r1…x;r†> 8x2Xq; r r0;

· _{uniformly (strongly) q-persistentif there exists some >0 such that}

r1…x;r†> 8x2Xq; r r0;

· _{weaklyq-dissipative if there exists somec>0 such that}

r1…x;r†<c 8x2X; rPr0;

· _{strongly q-dissipativeif there exists some c>0 such that}

r1…x;r†<c 8x2X; rPr0;

· _{q-permanent if Wis both uniformly stronglyq-persistent and stronglyq-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. Freeingqfrom 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.Wis uniformly weaklyq-persistent if and only if, for suciently small >0, no x2Xq

and no rPr0 can be found such that r…t;x;r†6 for all tP0.

also [12,42]). Actually one can replace compactness completely by equi-continuity requirements

forr. This makes it possible to use persistence techniques to derive conditions for

point-dissip-ativity 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.

LetBbe a subset ofX;J ˆNor 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 inX. 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 thatB absorbs …xj†, if

…/† there exists somej0 2J such that xj2Bfor all j2J; jPj0.

For semi¯ows on topological spaces that come from problems with in®nite memory (e.g., age-dependent equations without ®nite maximum age), the following interpretation is more

appro-priate: We say thatB attracts …xj†, if

….† for any open setU B there exists some jU 2J such that xj2U for all j2J; jPjU.

IfX is a metric space,B attracts…xj† if the distance ofxj fromB tends to 0 asj! 1.

Wis said to have property (CA) if the following holds:

(CA) There exists a subsetB inXwith the following two properties:

· _{For allx2Xq; s}Pr₀, we haveW…t;s;x† !B; t! 1.

· _{If…rj† is a sequence of real numbers and…yj† a sequence inXsuch that rj! 1 andyj!B as}

j! 1 and, for some >0; q…yj† ˆfor allj2N, then the continuity ofr…;yj;rj†is uniform

inj2N, possibly after choosing a sub-sequence.

In both conditions, of course, `!B' has to be consistently interpreted as B being either

ab-sorbing or attracting. IfW…t;s;x† !B is interpreted in the ®rst way…/†, we say that Bis an

ab-sorbing setforW, while in the second interpretation ….† we say thatBis anattracting set.IfBis

absorbing, the second property of (CA) is more succinctly stated as the continuity ofr…;y;r†to be

uniform inrPr0 andy 2B; q…y† ˆ for ®xed, but arbitrary >0.

Recall that we assume throughout this section thatr…;x;r†is continuous for allx2X; rPr0.

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]. IfXis

topological space, we call a subsetBofX relativelyq-compactif all its intersections with q-shells

have compact closure inX. Aq-shell is a set fx2X;16q…x†62gwith 0< 1 < 2<1.

Remark.(CA) holds if the following holds:

· _{W…t;s;x† ˆW…tÿs;0;x† ˆU…tÿs;x†is a continuous autonomous semi¯ow on a metric spaceX.}

· _{There exists a relatively q-compact setBwithU…t;x† !B ast! 1 for all x2X.}

· _{qis continuous onX ifBis absorbing, and uniformly continuous if Bis attracting.}

Proof.Assume that r…t;yj;sj† ˆq…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 ®ndtP0 andtj!tasj! 1 such

d…yj;zj† !0 as j! 1. Sinceqis (uniformly) continuous on the metric spaceXandq…yj† ˆ for

large enough j, the zj are in the intersection of B with a q-shell and so have a converging

sub-sequence. 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.LetX ˆ …0;1†andq…x† ˆx. Construct a pair of functionsx},aon‰0;1†with the following properties:

_

x}ˆ ÿx}‡a…t†; tP0; x}…0† ˆ2;

0<x}…t†62; 06a…t†63 8tP0;

lim sup

t!1

x}…t† ˆ2; lim inf

t!1 x}…t† ˆ0:

Now let Wbe the semi¯ow induced by the solutions of

_

xˆ ÿx‡a…t†; tPr; x…r† ˆx0;

i.e., W…t;r;x0† ˆx…t†. Since x…t† ÿx}…t† !0 as t! 1, we have that Wis uniformly weakly

per-sistent, but not strongly perper-sistent, though Wis 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 setR…†andR…t; †as follows (Bis the absorbing or attracting

set in (CA)):

R…t; † consists of continuous functions r~:‰0;tŠ ! ‰0; Š;

~

r…t† ˆ0<r~…0† ˆ;

~

r…s† ˆ lim

j!1r…s;yj;sj† uniformly in s2 ‰0;tŠ

for sequences …sj† in‰r0;1†; …yj† inX withsj! 1; yj !B;asj! 1:

R…† consists of continuous functions r~:‰0;1† ! …0; Š;

0<r~…0† ˆ;

~

r…s† ˆ lim

j!1r…s;yj;sj† locally uniformly in sP0

for sequences …sj† in‰r0;1†; …yj† inX withsj! 1; yj !B;asj! 1:

…2:3†

The semi¯ow Wis said to have property (PS) if the following holds:

Remark.(PS) automatically holds for a continuous autonomous semi¯ow Uwhich is uniformly

weaklyq-persistent, provided that there is a relativelyq-compact attracting setBforUsuch that

Xq\B is forward invariant underU and that q is uniformly continuous on the metric space X.

This statement holdsMutandis mutatisfor asymptotically autonomous semi¯ows [40].

Proof.Following the proof of the Remark subsequent (CA) we notice that the functionsr~inR…† or R…t; †, respectively have the form r~…s† ˆq…U…s;y†† for some y2B; q…y† ˆ. R…t; † is empty

becauseXq\Bis forward invariant.R…† ˆ ;follows from the uniform weak persistence ofUfor

suciently small >0.

Theorem 2.3. If a semiflowWis uniformly weakly q-persistent and has properties(CA) and(PS), then it is uniformly strongly persistent.

Proof.Assume thatWis uniformly weakly persistent. Then there exist >0 such that

r1…x;r†> 8x2Xq; rPr0:

By (PS) we can arrange that the setsR…†and R…t; † are empty for allt>0 for this .

Step 1: Letx2Xq and rPr0 such that lim inft!1r…t;x;r†< 1 < .

Sinceq…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 8k2N;

q…W…s‡sk;r;x††6 whenever 06s6tk:

Step 2:Suppose thatWis not uniformly strongly persistent.

Then there exist sequences …rj† in‰r0;1†and …xj† 2Xq such that

lim inf

t!1 q…W…t;rj;xj†† !0; j! 1:

Using step 1 and a diagonalization procedure we obtain sequences…rj†and…sj†in‰r0;1†, and…tj†

in‰0;1†such that sjPrj‡jfor all j2N, and

q…W…sj;rj;xj†† ˆ;

q…W…tj‡sj;rj;xj†† !0; j! 1;

q…W…s‡sj;rj;xj††6; 06s6tj:

Since, for every j2N;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:

SetyjˆW…sj;rj;xj†. Using the de®nition of rin (2.1), we have the following situation:

q…yj† ˆ;

r…tj;yj;sj† !0; j! 1;

r…s;yj;sj†6; 06s6tj;

We claim: tj ! 1 asj! 1.

If not, after choosing sub-sequences,tj!t andr…;yj;sj† is continuous uniformly inj. By the

second assumption of (CA), there exist somed>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† ˆlim

j!1r…s;yj;sj†6; 06s6t;

with convergence holding uniformly for s2 ‰0;t‡dŠ. Obviously r~…0† ˆ >0. Again using that

r…;yj;sj† is continuous uniformly inj,

0ˆlim sup

j!1

r…t;yj;sj† ˆr~…t†:

Hence we have found an element in R…t; † which is a contradiction to (PS).

So,tj ! 1asj! 1. LettP0 be arbitrary. Then tj>t for suciently largej and

Pr…s;yj;sj† 8s2 ‰0;tŠ;

ifjis large enough. This implies thatr…;yj;sj†is bounded on ®nite intervals in‰0;1†uniformly in

j2N. The second assumption in (CA) implies thatr…;yj;sj†is continuous on‰0;1†uniformly in

j, possibly after choosing a sub-sequence. By the Arzela±Ascoli theorem, there exists a functionr~

on ‰0;1†such that, after choosing another sub-sequence,

r…t;yj;sj† !r~…t†; j! 1; locally uniformly in tP0

and

~

r…0† ˆPr~…t† 8tP0:

Since R…t; † ˆ ;for all t>0,r~is strictly positive. So we have an element r~2R…† which again

contradicts (PS).

The ¯exibility ofq-persistence (compared with the topological concept of persistence) allows to

prove boundedness results.

For every j>0; t>0 we de®ne sets R~…j† and R~…t;j† as follows (B is the absorbing or

at-tracting set in (CA)):

~

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

~

r…0† ˆj; r~…s† ˆlim

j!1r…s;yj;sj† locally uniformly in sP0

for sequences …sj† in‰r0;1; …yj† inX with sj! 1; yj !B asj! 1:

~

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

r…0† ˆj; lim

s!tÿr~…s† ˆ 1; ~

r…s† ˆlim

j!1r…s;yj;sj† uniformly in s2 ‰0;t†

Theorem 2.4. Let Wbe a semiflow that has the property (CA).Assume that, if jis chosen

suffi-ciently large, the setsR~…j† andR~…t;j†are empty for all tP0.

ThenWis stronglyq-dissipative whenever it is weakly q-dissipative.

Proof.Assume the Wis weaklyq-dissipative. Set

~

q…x† ˆ 1 1‡q…x†:

Then q~…x†>0 for allx2X and Wis uniformly weakly q~-persistent. Moreover W satis®es (CA)

and (PS) withq~replacingq. It follows from Theorem 2.3 thatWis uniformly stronglyq~-persistent,

i.e., stronglyq-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]).

LetN be an index set and assume that we have a family of (non-autonomous) semi¯ows

Wn :DX !X; n2N:

…Wn† is said to have property…CA†N if the following holds:

…CA†_N There exists a subsetBinX with the following two properties:

· _{For all x2X; s}Pr0; n2N, we haveWn…t;s;x† !B; t! 1.

· _{If…rj†is a sequence of real numbers and …yj† a sequence inX; …nj†a sequence inN}

such thatrj! 1andyj!Bast! 1and, for some >0; q…yj† ˆfor allj2N,

then the continuity of

t7!q…Wnj…t‡sj;sj;yj††

on ‰0;1†is uniform in j2N, possibly after choosing a sub-sequence.

Setr…t;y;r† ˆq…Un…t‡r;r;x††foryˆ …x;n†. For everytP0; >0, setsR…†andR…t; †are now

de®ned in obvious analogy to (2.3) using sequences yiˆ …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 setsR…† and R…t; †are empty for alltP0.

Theorem 2.5. Let N be a set and Wn; n2N, be a family of semiflows on X that has properties

…CA†_N and…PS†_N with absorbing or attracting set B. Further assume that

t7!q…Wn…t‡s;s;x††

is continuous on ‰0;1†for any x2X; n2N; sPr0. Finally assume there exists some >0 such

that

lim sup

t!1

q…Wn…t;s;x††P 8x2Xq; n2N; sPr0:

Then there exists some >0 such that

lim inf

Proof.LetZ ˆX N and

~

q…x;n† ˆq…x†:

De®ne a semi¯owW~ :_DZ !Z by

~

W…t;s;…x;n†† ˆ …Wn…t;s;x†;n†:

Then W~ is uniformly weaklyq~-persistent. Further W~ satis®es (CA) with absorbing or attracting

set BN (in the second case endow N with the trivial topology). One easily checks that the

other assumptions of Theorem 2.3 are satis®ed as well. It follows that W~ is uniformly strongly

~

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¯owsU, whereXqis not

forward invariant under all, but only under total orbits ofU. We recall that/:R!X is a total

orbit of U, ifU…t†/…s† ˆ/…t‡s† for all tP0; s2R.

Theorem 2.6.LetUbe a continuous autonomous semiflow on a metric space X which has a compact

attracting set K, i.e., a compact set K such thatdist…U…t;x†;K† !0; t! 1. We further assume for

any total orbit /:R!X of Uwith relatively compact range:

· _{Ifs2Randq…/…s††>0, thenq…/…t††>0 for allt>s.}

ThenUis uniformly strongly q-persistent whenever it is uniformly weaklyq-persistent.

Proof. We can assume that the compact attracting setK is invariant under U. Otherwise we

re-place it by the closure of the union of allx-limit sets ofU. The assumption (CA) is satis®ed as we

have shown in the remark following (CA).

Using the compactness ofKin the metric spaceX, for every elementr~inR…†or inR…t; † we

haver~…s† ˆq…U…s;y††for somey 2K. So uniform weak persistence implies thatR…†is empty for

suciently small >0. Sincey 2KandKis invariant,U…s;y†can be extended to a total orbit ofU

with relatively compact range. This implies R…t; † ˆ ;, because any r~ in this set would satisfy

~

r…0†>0ˆr~…t†, while, by; r~…0†>0 would imply r~…t†>0.

Situations whereXq 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 ofUwith relatively compact range,s2R, andq…/…r†† ˆ0 for allr6s,

thenq…/…t†† ˆ0 for all tPs.

(ii) There exist somes>0; d>0 such that q…U…t;x††>0 for all t2 …s;s‡d†; x2Xq.

Proof.Ifx2Xq, it follows from the semi¯ow property and repeated use of (ii) thatq…U…t;x††>0 for allt2 …ms;m…s‡d††; m2N, and soq…U…t;x††>0 for alltPms, wheremis the ®rst natural

number withmd>s. Let now/be a total orbit ofU; s2Randq…/…s††>0. By (i), there exists a

sequence…sj†in…ÿ1;s†withsj! ÿ1asj! 1 and q…/…sj††>0. Choose somesj<sÿms. By

To interpret condition (ii) of the above Remark, let us assume thatqUis 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-hetero-geneity 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 in-fection 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

re-covered, R, parts, and the contraction of the disease is modeled by the law of mass action

in-volving susceptibles and infectives

N ˆS‡I‡R;

dI

dt ˆ ÿl…t†I ‡a…t†SIÿc…t†I; …3:1†

dR

dt ˆ ÿl…t†R‡c…t†I ÿn…t†R:

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 thatais a periodic function and thatN;l;c and nare 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

so-lutions exist whose periods are integer multiples of the period ofaand 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 bi-furcations 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,

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

R_{0ˆaN=…l‡c†61; thenI…t† !0; ast! 1 for every solution of (3.1). If}R_{0 >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

f ˆlim

t!1

1

t

Z t‡r

r

f…s†ds

exists, is independent ofrand the convergence is uniform inr. Almost periodic functions are

equi-mean-convergent, and so are asymptotically almost periodic functions and weakly asymptotically

almost periodic solutions (see Appendix A). If aN; c; lare equi-mean-convergent, we de®ne

R_{0 ˆ} …aN†

l‡c:

Theorem 3.1. Let N be uniformly continuous on‰0;1†andaN; landcbe equi-mean-convergent.

(a) If R_{0 <1, the disease dies out, i.e., for every solution of (3.1) we have I…t†;}

R…t† !0 as t! 1.

(b)IfR_{0>1,the disease persists uniformly strongly in the population, in the sense that there}

ex-ists some >0such thatI1ˆlim inf t!1I…t†> for all solutions of(3.1)withI…r†>0for some

rP0.

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†andaN; l andcbe equi-mean-convergent

andR_0>1.Then

1

t

Z t 0

a…s†S…s†ds!l‡cˆ…aN†

R₀ ; t! 1

and

1

t

Z t 0

a…s†…I…s† ‡R…s††ds! …aN† ÿ …l‡c† ˆ …aN† 1

ÿ 1

R₀

Proof.From (3.4) we obtain

The last equality follows fromI 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 alltPrP0 where it is de®ned wheneverI…r† 0. The same property holds for

R. Adding the di€erential equations forI and R it is easy to see that every solution satis®es

I…t† ‡R…t†6 maxfI…r† ‡R…r†;supNg; t>rP0:

A standard continuation argument now tells us that, given non-negative initial data at timerP0,

the solutions are de®ned for alltr.

Though this is not important for our mathematical considerations, epidemiologically one

would likeS ˆNÿIÿRto be non-negative as well. This can be achieved by assuming thatNis

continuously di€erentiable and that the population birth rate,B, satis®es

B…t†:ˆ d

dtN…t† ‡l…t†N…t†P0: …3:2†

Then SˆN ÿIÿRsatis®es the di€erential equation

dS

dt ˆB…t† ÿl…t†Sÿa…t†SI‡n…t†R

from which we learn that, wheneverS…r†P0; rP0; thenS…t†P0 for alltPr. This implies

06S…t†;I…t†;R…t†6N…t† 8tPr:

Integrating theI-equation we obtain

1

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 ast! 1.

Proof. For simplicity we restrict the proof to a solution with initial data given at time 0. Since

S6N and R}_{<1, there exists some >0; r>0 such that}

1

t

Z t

0

a…s†S…s†dsÿ1

t

Z t

0

‰c…s† ‡l…s†Šds6 ÿ 8tPr:

By (3.3),

1

t ln I…t†

I…r†6 ÿ 8tPr:

Hence

I…t†6I…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

…aN†_} ˆ lim inf

t!1

1

t

Z t 0

a…s†N…s†ds;

c} ˆ lim sup

t!1

1

t

Z t 0

c…s†ds; l} ˆ ;

R_{} ˆ} …aN†}

l}_‡c}

and let I1 be the limit superior ofI…t† ast! 1.

Theorem 3.4. LetR_{}>1.Then the disease uniformly weakly persists in the population, in the sense}

that there exists some >0such thatI1> for all solutions of(3.1)withI…r†>0for somerP0.

does not depend on n.

Proof. Let us suppose that for every >0, there is some solution with I1_{< : Since} R16 sup…c=l†I1;

I1‡R1<c

withc>0 being independent of the solution, of, and ofn. ByN ˆS‡I‡R and (3.3),

d

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

For suciently large timet, we have

1

In order to formulate a strong persistence results, letI1 be the limit inferior ofI…t†: Further

R_:ˆ …aN†

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 >0such that I1> for all

solutions of(3.1) withI…r†>0 for somerP0.LetI₁ be the limit inferior of I…t† ast! 1.

endowed with the standard metric. Further we set

q…I;R† ˆI:

In particular

W…t;s;x† ˆ …I…t†;R…t††; tPsP0;

whereI;R solve (3.4) and …I…s†;R…s†† ˆx. Further

r…t;x;s† ˆI…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

I1<supN; R16supc

inflI

1

and the set

Bˆ fxˆ …I;R†;I;R>0;I ‡R6supNg

is absorbing and forward invariant. A standard Gronwall argument implies that r…;x;s† is

continuous on ‰0;1†uniformly inx2B; sP0:

Wis uniformly weaklyq-persistent as a consequence of Theorem 3.4. In order to check property

(PS), let us describe the elements ofR…t; †andR…†in (2.3) in terms of system (3.4). To this end we

consider sequencessj! 1; yj !B inX, asj! 1: LetIj; Rj be the solutions of

dIj

dt ˆ ÿl…t‡sj†I‡a…t‡sj†…N…t‡sj†† ÿIjÿRj†Ijÿc…t‡sj†Ij;

dRj

dt ˆ ÿl…t‡sj†Rj‡c…t‡sj†Ijÿn…t‡sj†Rj; …Ij…0†;Rj…0†† ˆyj:

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 intP0;whereN~ is a bounded continuous function. Moreover, sinceL1‰0;1†is

separable, the Alaoglu±Bourbaki theorem implies that, after choosing a sub-sequence,

a…sj‡t† !a~…t†; j! 1;

c…sj‡t† !~c…t†; j! 1;

l…sj‡t† !l~…t†; j! 1;

wherea~;c~;l~are elements ofL1‰0;1†and the convergence holds in the weak topology carried by

L1‰0;1† as dual space ofL1‰0;1†.

The derivatives of Rj and Ij are bounded, uniformly in j2N. Again by the Arzela±Ascoli

theorem we have that, after choosing a sub-sequence,

locally uniformly in tP0; where I~ is absolutely continuous, R~ is continuous, and both are

The same consideration as in the proof of Theorem 3.4 now implies that such an~I cannot exist,

if >0 is chosen small enough, provided thatR~_}>1;whereR~_{}is the analog of}R_{} in Theorem}

Returning to Theorem 3.4 we notice that the weak disease persistence does not depend on the

removal raten. Similarly, checking the proof of Theorem 3.5, we notice that ndid not enter the

proof. This means that the property (PS) is satis®ed uniformly for any continuous non-negativen.

Theorem 3.6. Let N be uniformly continuous and bounded on‰0;1†andR _{>1:LetNbe an}

equi-bounded family of non-negative continuous functions n on ‰0;1†: Then the disease persistence is

uniform inn, in the sense that there exists some >0that is independent ofn2Nsuch thatI1>

for all solutions of (3.1)with I…r†>0 for somerP0.

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 sequencessj ! 1; nj 2N; xj !B, whereBis chosen

as in proof of Theorem 3.5. After choosing a sub-sequence we can assume that xj!x2B and

nj !n2N as j! 1; with the latter holding in the weak topology of L1‰0;1†. Let Ij; Rj be

solutions of

dIj

dt ˆ ÿl…t‡sj†I‡a…t‡sj†…N…sj‡t† ÿIjÿRj†Ijÿc…t‡sj†Ij;

dRj

dt ˆ ÿl…t‡sj†Rj‡c…t‡sj†Ijÿnj…t‡sj†Rj; …Ij…0†;Rj…0†† ˆxj:

Arguing as in the proof of Theorem 3.5, elements inR…† or R…t; † are given by ~I;

~

I0…t†P ÿl~…t†_I~_{t† ‡a~…t†…N}~_ÿ~_IÿR~_{† ÿ~c…t†}~_{I…t†; for almost allt}P0;

withR~satisfying (3.6) and~a;~c;l~being the weak_{limits ofa… ‡s}

j†;c… ‡sj†;l… ‡sj†andN~ 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 n2N.

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 ˆS‡I‡R;

dI

dt ˆ ÿl…t†I ‡a…t†SIÿc…t†I:

…4:1†

Again we assume that the disease causes no fatalities and that the population sizeN…t† is a given

function of timet.l…t†is the instantaneous per capita mortality rate.c…t†is the instantaneous per

capita rate of leaving the infective stage.

R…t† ˆ Z a0

0

u…t;a†da; …4:2†

whereu…t;†is the class age density of removed individuals. Class age is the time that has elapsed

since entering the class.a02 ‰0;1Šis 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€er-ential equation

capita rate of leaving the removed stage at class ageaand timet. 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±3.5hold 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

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 inI 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 thatR is absolutely continuous and

we de®ne

n…t† ˆ 1

R…t† Z a0

0

g…a;t†u…t;a†da if R…t†>0;

n…t† ˆ0 if R…t† ˆ0:

Then

dR

dt ˆc…t†I…t† ÿl…t†R…t† ÿn…t†R…t† for almost all t>0

and

n…t†6 supg 8tP0:

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

CˆC‰ÿs;0Š; C‡ˆC‡‰ÿs;0Š

denote the space of real-valued continuous functions on the interval‰ÿs;0Šand the cone of

non-negative functions, respectively. C is endowed with the supremum-norm and C‡ becomes a

complete metric space under the induced metric.

ForrP0 and/2C‡‰ÿs;0Š consider

_

x…t† ˆx…t†f…t;xt†; t>r;

x…r‡s† ˆ/…s†; s2 ‰ÿs;0Š; …5:1†

where

f :‰0;1† C‡!R

is continuous. Here xt 2C‡‰ÿs;0Š is given by

xt…s† ˆx…t‡s†; s2 ‰ÿs;0Š:

An example is Hutchinson's [22] equation

x0…t† ˆx…t†‰a…t† ÿm…t†x…tÿs†Š; …5:2†

wherea is a continuous function andm 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.ais 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

(see [24, Chapters 4 and 9], and the references therein). In this paper, we try to let the time

de-pendence of a and m as general as possible formulating our assumptions in terms of their time

averages.

ar…t† ˆ

1

t

Z t 0

a…s‡r†ds:

Permanence of the population follows if the time averages ofaand mare bounded and bounded

away from zero for larget and r and ifaand m are uniformly locally integrable.

A locally integrable functiona:‰0;1† !Ris called locally uniformly integrableif

Z t

r

a…s†ds!0; tÿr!0; tPr:

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. Letaandmbe uniformly locally integrable and let there existsc0> 0 >0andt0;r0 >

0such that the time averages of aand msatisfy

06ar…t†;mr…t†6c0 8tPt0; rPr0:

Then the population is permanent in the sense that there exists somej>1(independent on the initial

data) such that

1

j<lim inft!1 x…t†6lim sup_t!1 x…t†<j

for all non-negative solutions x of (5.2)which are not identically equal to 0.

In particular we have permanence if bothaandmare bounded and bounded away from 0. But

in populations with seasonal reproduction,arepeatedly 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 thatfis continuous and satis®es the following Lipschitz condition: There exists a

Borel measurable functiong :‰0:1†2 ! ‰0;1†such that

jf…t;/† ÿf…t;w†j6g…t;k/k ‡ kwk†k/ÿwk 8tP0; /;w2C‡:

g…t;†is non-decreasing for everytP0 andg…;c† is uniformly locally integrable for everyc>0.

(b) We assume thatf…t;0† is uniformly locally integrable.

(c) Finally, we assume that there exists a uniformly locally integrable functionf~such that

f…t;/†6f~…t† 8tP0; /2C‡:

In order to show global existence and uniqueness of solutions for initial data inC‡ we extendf

f…t;/† ˆf…t;/_‡†

with /_‡ denoting the positive part of /;/_‡…s† ˆmaxf0;/…s†g. Standard FDE theory (for

ex-ample [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†P0 for allt>r or x…t†>0 for all t>r whenever the

respective property holds fortˆr. Uniqueness implies that

W…t;r;/† ˆxt

with x solving (5.1), de®nes a non-autonomous semi¯ow anchored at rˆ0. 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 conditions for the semi¯owWto be point-dissipative

(cf. [16]), i.e., that there exists some c>0 such that

lim sup

t!1 k

W…t;r;/†k6c 8/2C‡; rP0:

Proposition 5.3. Let Assumption 5.2 hold. Further assume that there exists a locally integrable

function f1 such that

lim sup

inf/!1

f…t;/† f1…t†; uniformly in tP0 …5:3†

and

lim sup

t!1

1

t

Z t 0

f1…s†ds<0:

Then there exists somej>0 such that

lim inf

t!1 x…t†6j

for all solution x of (5.1).

By (5.3) we mean that, for everyd>0 there exists some c>0 such that

f…t;/†6f1…t† ‡d 8tP0;

whenever/2C‡; /…s†Pc 8s2 ‰ÿs;0Š:

Proof.Choose >0 and t0>0 such that

1

t

Z t 0

f1…s†ds6 ÿ; tPt0: …5:4†

By (5.3), there exists somej>0 such that

f…s;/†6f1…s† ‡=4 8sP0; whenever inf /Pj:

Suppose that the statement of the theorem does not hold. Then there exists a solution xof

_

for some rPs such that

lim inf

t!1 x…t†>j:

Choosingr large enough we can assume that

x…t†Pj 8tPrÿs:

Hence, for large enought,

x…t†6x…r†eÿ…tÿr†=2_:

This implies that x…t† !0 as t! 1 contradicting

lim inf

t!1 x…t†Pj>0:

In order to apply Theorem 2.4 we choose as metric space X ˆC‡ endowed with the metric

induced by the supremum norm. We set

q…/† ˆ/…ÿs†:

This choice of q will work because, in Assumption 5.2, we have required a global estimate of f

from above, while, via the Lipschitz condition, only a local estimate offfrom below. If it were the

other way round, q…/† ˆ/…0† would be a more appropriate choice.

We turn to the study of the function r in Section 2. In this context, r…t;/;s† ˆx…t‡sÿs†,

wherexis the solution of

_

x…t† ˆx…t†f…t;xt†; tPs; xs ˆ/:

Equivalentlyr…t;/;s† ˆy…tÿs†, whereyis the solution of

_

y…t† ˆy…t†f…t‡s;yt†; tP0; y0 ˆ/: …5:5†

Lemma 5.4.Hypothesis (CA) is satisfied under Assumption 5.2.

_

c₀ <1 by Assumption 5.2(c). By (5.1) and Assumption 5.2(a),

_ x…t†

x…t† Pf…t;0† ÿg…t;q…xt†c0†q…xt†c0:

This means that the following set Bis an absorbing set for the semi¯ow W:

Bˆ f/2C‡; /…s†6/…r†c…s;r†; /…s†P/…r†~c…s;r;q…/††; ÿs6r; s60g

solution yof (5.5) that

y…t†6/…0†exp

The ®rst estimate together with Assumption 5.2(c) shows that y…t†6n…t† with a non-decreasing

It follows from Assumption 5.2 that the continuity of y does not depend on r and /, so the

Due to the Lipschitz condition in Assumption 5.2(a),

lny…t†

Theorem 5.5. Let f satisfy Assumption5.2.Assume that there exists a locally integrable functionf1

on‰0;1†with

lim sup

inf/!1

and

lim sup

t!1

lim sup

r!1

1

t

Z t

0

f1…s‡r†ds<0: …5:7†

Then there exist some j>0such that

lim sup

t!1

x…t†6j

for all solutions x of

_

x…t† ˆx…t†f…t;xt†; tPr; xr ˆ/2C‡;

where rP0; /2C‡ are arbitrary initial data.

Proof. Choosing >0 small enough, it follows from the considerations preceding this theorem

thatyis exponentially decreasing. So the setsR~…j†in Theorem 2.4 are empty for all/2X ifjis

chosen large enough. All other assumptions of Theorem 2.4 have been already checked. By

Proposition 5.3, the semi¯ow generated by (5.1) is uniformly weaklyq-persistent and so uniformly

strongly q-persistent. The statement now follows fromq…xt† ˆx…tÿr†.

Using Theorem 2.3 rather than Theorem 2.4 we can obtain that the solutions are bounded away from 0. The assumptions of Theorem 2.3 are veri®ed in complete analogy.

Theorem 5.6. Let f satisfy Assumption 5.2andf0 a locally integrable function such that

lim inf

/!0 f…t;/†

Pf0…t†; uniformly in tP0;

lim inf

t!1 lim infr!1

1

t

Z t 0

f0…s‡r†ds>0:

Then there exist some >0 such that

lim inf

t!1 x…t†P

for all solutions of x

dx

dt ˆx…t†f…t;xt†; tPr; xrˆ/2C‡;

where rP0; /2C‡ are arbitrary initial data with /…0†>0.

Theorem 5.1 follows from Theorems 5.5 and 5.6 withf~…t† ˆa…t†; g…t;x† ˆm…t†x, and takingf1

and f0 in the forma…t† ÿgm…t† with suciently large or small g>0.

Acknowledgements

Appendix A. Equi-mean-convergent functions

A Borel measurable functionf :‰0;1† !Ris calledequi-mean-convergentor to haveuniformly

convergent means if the (asymptotic) mean value

M‰fŠ ˆlim

t!1

1

t

Z t‡r

r

f…s†ds

exists, is independent of r and the convergence is uniform in rP0. Obviously the

equi-mean-convergent functions form a vector space. Important examples of equi-equi-mean-convergent functions are asymptotically almost periodic functions and weakly asymptotically almost periodic functions.

A bounded function f :‰0;1† !R is called asymptotically almost periodic if f is uniformly

continuous and if every sequence tj! 1 …j! 1† has a sub-sequence tjk such that f…tjk‡ †

converges uniformly on ‰0;1†ask! 1.

A continuous bounded function f :‰0;1† !Ris calledweakly asymptotically almost periodic

if the set of translatesfft;tP0g; ft…s† ˆf…t‡s†; is relatively compact in the weak topology of

the space of bounded continuous functions on‰0;1†(with sup-norm).

The following characterization holds (see [27], or [28], for references, also concerning

gener-alizations to Banach space-valued functions). A bounded continuous function f :‰0;1† !Ris

weakly asymptotically almost periodic if, for any two sequences …sn† and …tn† in‰0;1†,

lim

n!1 mlim!1f…sm‡tn† ˆmlim!1 nlim!1f…sm‡tn†;

provided that both iterated limits exists.

Obviously equi-mean-convergence makes sense for locally integrable function while weak as-ymptotic almost periodicity requires the function to be bounded and continuous. In the following we extend the class of equi-mean-convergent functions beyond the class of weakly asymptotic almost periodic functions.

Lemma A.1.A locally integrable functionf :‰0;1† !Ris equi-mean-convergent if there exists a weakly almost periodic function g such that f ± g is equi-mean-convergent.

The following results are easily checked.

Lemma A.2.A locally functionh:‰0;1† !Ris equi-mean-convergent to 0 if one of the following holds:

(i) h2Lp_{‰0;1†for somep2 ‰1;1†.}

(ii)For every >0 there exist somec>0 such that

Z t

r

h…s†ds

6c‡…tÿr† 8tPrP0:

Proposition A.3.A locally integrable functionf :‰0;1† !Ris equi-mean-convergent if there exists a weakly almost periodic function g such that one of the following holds:

(ii)For every >0 there exist some c> 0 such that

Z t

r

‰f…s†

ÿg…s†Šds

6c‡…tÿr† 8tPrP0:

Obviously the asymptotically almost periodic functions (with values in R) form an algebra

under point-wise multiplication. We conclude with the observation that asymptotic almost

peri-odicity is preserved under convolution with an L1_-function.

Lemma A.4. Let f :‰0;1† !R be asymptotically almost periodic and k2L1_{‰0;1†. Then the}

convolution kf,

…kf†…t† ˆ Z t

0

k…s†f…tÿs†ds; tP0

is also asymptotically almost periodic.

Proof.This follows from a classical characterization (see Refs. [27,28]). A function is

asymptot-ically almost periodic if and only if there exists a uniquely determined almost periodic functiong

such that f…t† ÿg…t† !0 as t! 1.

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