a new approach to the global asymptotic stability problem in a delay lotka voltrra differential equation

MATHEMATICAL
COMPUTER
MODELLING
PERGAMON

Mathematical

and Computer Modelling 31 (2000) 9-28
www.elsevier.nl/locate/mcm

A New Approach to the
Global Asymptotic Stability Problem
in a Delay Lotka-Volterra Differential Equation
I.
Department

GY~RI

of Mathematics
University

8201 Veszprem,

and Computing

of Veszprem

P.O. Box 158, Hungary

(Received and accepted November

1999)

Abstract-In
this paper, some new global attractivity results are given for scalar LotkaVolterra
differential equation with delays. Some examples and related open problems are also raised up at the
end of the paper. @ 2000 Elsevier Science Ltd. All rights reserved.
Keywords-Lotka-VoIterra
differential equation, Global attractivity,
back terms with delays, Open problems.

The effect of negative feed-

1. INTRODUCTION
In the following, we consider the nonautonomous Lotka-Volterra type delay differential equation

fi(,) = W(t))

r(t) - aowqt) [

2 @(W(Pi(4) + g
i=l

~j(w(*j(t))

j=l

I
,

(1.1)

where h, T, ai, bj : R+ + R+ (0 5 i < n, 1 < j 5 m) and pi : R+ -+ R, pi(t) < t (t 2 0), 1 5 i
< t (t > 0), 1 5 j 5 m, are some given functions.
qj: R+ -+ R, e(t)
Equations like (1.1) are important in the single species models and there is a considerable
work done by several authors on the theory of global asymptotic stability of LotkaVolterra type
In,

equations with delay. Especially, the books of Gopalsamy [l] and Kuang [2] are good sources for
global attractivity results for Lotka-Volterra equations.
A large portion of the investigated models assume the so-called negative feedback effect which
often appears as a nondelayed force, such as the term -ao(
in equation (1.1).
When
such a term dominates the others, there are several kinds of conditions which guarantee that
equation (1.1) is globally asymptotically stable. But it seems to us that even in this special case
there are interesting open problems. One of these questions is how to handle equations without
steady state solutions.

In his recent work [3], Kung established sufficient conditions for global

Supported by Hungarian National Foundation for Scientific Research Grant No. TO 129846 and by Hungarian
Ministry of Education Grant No. FKFP 1024/1997.
0895-7177/00/s
- see front matter @ 2000 Elsevier Science Ltd. All rights reserved.
PII: SO895-7177(00)00043-l

Typeset

by &&‘I@

I. GY~RI

10

asymptotic stability of delay differential equations without steady state solutions. These results
and the applied methods were essentially developed further in a recent paper [4] by Bereketoglu
and the recent author. The method applied in these papers are based on Liapunov functionals,
and it is effective if -as(t)N(t)
C,“=, bj(t)

I quo(t),

is a dominating term, which essentially means that Cy=“=,ai
+
t > T, q E (0,l).
In this condition, the possible stabilization effects of

the negative feedback terms with delays, such as the terms -ai(t)N(pi(t))
not taking into account.

in equation (l.l),

equation (1.1) is so called “pure-delay type”, that is when au(t) = 0 (t > O)-even
has a saturated

are

On the other hand, the Liapunov technique is much less effective if

equilibria.

Recently,

He [5] proved some global asymptotic

if the equation
stability

results

for “pure-delay type” equations under the a priori assumption that the equations are uniform
persistent and the solutions are ultimately bounded on [0,00). However, he also remarked that
proving the uniform persistence and finding the ultimate upper bounds of the solutions-which
are necessary in the application

of his theorem-are

still open problems for “pure-delay type”

equations.
Our focus in this paper, is the global asymptotic stability analysis of the delay differential
equation (1.1) when it does not have a steady state solution. We derive new sufficient and also
necessary, in some special case necessary and sufficient conditions for equation (1.1) to be globally
asymptotically

stable.

the above mentioned

techniques,

basically

The methods to be used in this paper are different from those used in
papers.

To some extent,

our approaches

applying variation of constants

are related to some “simple”

formula and some basic comparison

and

oscillatory results from the theory of delay differential equations and inequalities.
This paper is organized as follows.
In Section 2, we obtain a priori upper bounds of the positive solutions of the nonautonomous
logistic equation

(l.l), based on the fact that the solutions can be classified with respect to
the existence of the integral $O” u(t)h(N(t)) dt, where a : R+ -+ R+ is a suitable function (see
Theorem 2.1). In this section, we also show that the conditions which are sufficient for the
existence of an upper bound for the positive solutions are also necessary if the negative feedback
terms with delays are not present, that is ai
= 0 (t 2 0, 1 < i 5 n) in equation (1.1).

In Section 3, we give global asymptotic stability results when the negative feedback term
without delay is dominating,
solution.

but the nonautonomous

equation does not have a steady state

Our sufficient conditions for global asymptotic

independent when the asymptotic equilibrium is constant.

stability

of equation

(1.1) are delay

Otherwise, the delays play role in our

conditions but only trough the time dependent global attractor.
Section 4 contains an entirely new approach of the global asymptotic stability analysis of nonautonomous delay logistic equations. We show that the negative feedback terms with “small delay”
help in the stabilization of equation (1.1). Roughly speaking, we separate the negative feedback
terms into two classes al,. . . , ano and uno+i, . . . , a, such a way that the delays belonging to the coefficients ui (1 I i I no) are small enough and Cz__=no+las(t) f c3n,i &(Q < Qc;:s
ai
(t L 0)
with a constant q E (0,l). Our approach is related to the method based on the monotone semiflow
theory and applied for autonomous differential equations. The reader is refereed to the recent
book of Smith [6] for an extensive discussion of this theory and also for some applications to
autonomous population model equations. Although these results are not directly applicable to

our nonautonomous case, they influenced us how we can apply our method introduced in 1989
in the paper [7] (see also [8,9]).
In [7], the global attractivity results were proved for delay differential equations which are
controlled by a time dependent negative delayed feedback term. In Section 4 of the present
paper, the main idea of [7] is combined with some comparison results from [lo] and also some
basic facts from the oscillation theory of delay differential equations to prove the global asymptotic
stability results.
In Section 5, we discuss the sharpness of our results in autonomous cases and we compare
them to some know results. Especially, we show that one of our result generalize Theorem 5.6 of


Global Asymptotic

Kuang

in [2, p. 351. We also construct

that the delay in the negative
if it is greater
true

than

feedback

examples

even for small delays.

which show the sharpness

term can destroy

the delay in a positive

11

Stability Problem

feedback

of our results

the global asymptotic

term.

Interesting

We also raise up some open questions

and also

stability

to note that

property
this

fact

based on our investigations

is

and

examples.

BOUNDS

2. A PRIORI
Consider

the following

M)

Lotka-Volterra

[

= h(N(t))

1, (2.1)

equation:

r(t) - ao(

- -$i(t)N(Pi(t))

+ -&(t)N(qj(t))

i=l

j=l

where
(i) h : R+ -+ R+(R+

= [O,w))

is a continuous

function,

h(0) = 0 and h(zl) > 0, IL >

zL-++oo,
0, h(u) ‘fco,
(ii) r : R+ + R is locally Lebesgue integrable and locally bounded,
(iii) ai : R+ + R+ (0 < i 5 n) and 63 : R+ + R+ (1 < j 5 m) are locally Lebesgue
and locally bounded on R+,
(iv) pi : R+ -+ R (1 5 i 5 n) and qj : R+ 4 R (1 5 j 5 m) are locally
and locally bounded on R+, moreover pi(t) < t (t 2 0), lim,,+,pi(t)
andqj(t)<t(t>O),
Let tll(s)
min{t!,(s),

lim t++co&)

Lebesgue

integrable

= +co (1 5 i 5 n)

= +cQ (1 5 j 2 m).

= minlttl(s)
= minllj~,{inft~s{q~(t)}},
-> --00 and t?l(s)
t2_l(s)} for any s 2 0. We assume that til(s)

DEFINITION 2.1. A function

integrable

N : [L1(0),oo)

---f R is called positive

solution

and
> -co.

t-l(s)

of equation

is positive on [t-l(O), co), Lebesgue integrable and bounded on [t-l(O), 01, absolutely
on [0, co) and N satisfies equation (2.1) for almost every t 2 0.

=

(2.1) if it
continuous

In this paper, we do not deal with the existence and uniqueness
of the solutions.
We assume
that some additional
conditions
are satisfied for the right-hand
side of equation (2.1) such that
the solutions of equation
solutions are not needed
For any function

(2.1) exist on [t-l(O), co). It is worth noting
in our results.

c : R+ -+ R the functions

that the uniqueness

of the

c+ : R+ + R+ and c- : R+ -+ R+ are defined

c+(t) = max{O, c(t)} and c-(t) = max{O, -c(t)},
t 2. 0.
The following result gives a possible separation of the solutions of equation

by

the relations

to the existence

of the integral

THEOREM 2.1. Assume

locally

bounded

function

soooa(t)h(N(t))

(2.1) with respect

dt, where a : R+ + R+ is a given function.

(i)-(iv) and suppose that
a : R+ -+ (0, co) such that

there

exists

a locally

u(t) dt = +co,

Lebesgue

integrable,

(2.2)

moreover,

(2.3)

Then,
(A) for any solution N : [t-l(O), 00 ) + R+ of equation
u(t)h(N(t))

(2.1) the relation

dt < 03,

(2.4)

I.GY~RI

12
yields limt-++ooN(t) = 0,
(B) if
liminffi
> 0
t++m a(t)

and

(2.5)

then for any positive solution N : [t-l (0), co ) + R+ one has J;;” a(t)h(N(t))
PROOF

A. Assume that the solution N : [L1(0), cc ) +

R+ satisfies (2.4).

dt = +m

Then from equa-

tion (2.1), we obtain

a.e. t > 0.

Let et(t) = a(t)h(N(t)),

t > 0. Then (2.3) yields

m 5 r^.a(t) +K a(t) . &yy---

uw&(t))l

7

8.e. t 2 0.

Integrating both sides of the above inequality, we find

N(t)

5 c

+li

I

t4s) .

0

where c = N(0) + F. SOWa(t) dt < co.
Let u(t) = m=~_,(o)+~t{N(s)),

u(t) 5

g.y&

--

t
s

t 2 0, and ~1 = m={c,

Cl + 4

0

t L 0,

v%j(SN) ds,

4sMs) ds,

and by using the Gronwall inequality, we obtain

mac,(o)~s~o{N(s)}}.

Then

t 2 0,

*

This yields that the solution N of equation (2.1) is bounded on [t-l (0), CXJ),and hence, under
our conditions, we find

A41 =
J0

O”
a(t)h(N(t))

dt < 03.
I

On the other hand, from equation (2.1), we obtain

=

N(0)

+

ds I N(0) + Ml < 00,

t 2

0.

This yields

and hence, N(oo) = limt-,+ooN(t) exists and is finite. It remains to show that N(oo) = 0.
Otherwise, N(co) > 0, and there is a constant T > 0 such that N(t) 2 (1/2)N(co)

> 0 for t > T.


13

Global Asymptotic Stability Problem
Thus,
@Jr

,8 = inftkT{h(M(t))}
a(t) dt = foe

: u 2 (1/2)N(co)}

2 inf{h(u)

contradicting

> 0, and hence,

(2.4). The proof of Statement

PROOF B. For the seeking of contradiction,

J,“a(t)h(N(t))dt

2

(A) is complete.

assume that under condition

(2.5) there is a positive

solution N : [t-l(O), 00 ) -+ R+ of equation (2.1) such that JOWa(t)h(N(t))
dt < co. Then by
Statement
(A) we have limt-++m N(t) = 0. Therefore, (2.5) yields that there exists a T 2 0 and
c > 0 such that

h(N(t))

r(t) - ao(

- 2

ai(t)N

+ 5

for t 2 T. Therefore,

equation

. c,

I

t
N(t) 2 N(T) +

yields

> a(t)h(N(t))

(2.1) implies

and hence,

This

bj(W(&))

j=l

i=l

that

lim++,N(t)

u(s)h(N(s))

ds

c,

t > T.

sT

> 0 which

is a contradiction.

The proof of Statement

(B) is

complete.
In the next theorem,
THEOREM 2.2.

we give an upper

Assume

that

(i)-(iv)

(v) uo(t) > 0, t > 0, J,“uo(t)dt

bound

are satisfied,

of equation

(2.1).

and

= +oo, moreover

5 b(t)

r+(t) < o.

and

y = limsupt-++w so(t)
Then for any solution

for the solutions

(2.6)

N : [t-l(O), 00 ) -+ R+ of equation
lim supN(t)
t-++co

(2.11, one has

?
< 1 -B’

(2.7)

In the proof of the above theorem, we need the following lemma which is a simple generalization

result proved in [ll] in that case when u(t) = a~(> 0), t > 0, is a constant.
The

of a similar

proof of the next lemma
LEMMA 2.1.

is omitted.

Let a : R,

--f R+ and g : R,

that
s0
Then,

--t R be locally Lebesgue

Co
u(t) dt = +cc

and

for any T 2 0,

vmpmfg(t)
-i

< $mpWfe- Jk a(s)ds
+
-< limsup e-lGa(s)ds
t-++cc

whenever limt-++oog(t)

functions

such

sup Is(t)1 < co.
t>0

t

& “‘“‘%(u)g(u)

du

T

IT

t ej; “(“)d”u(u)g(u)

du < limsup
t++cc

g(t),

exists.

THE PROOF OF THEOREM 2.2.
a(t) = uo(t)h(N(t)),

I

integrable

Let N : [t_1(0), cc ) + R+ be any solution

of equation

(2.1). Let

t 2 0. Then either so” o(t) dt < 00 or soooa(t) dt = +CQ. If sooo a(t) dt < co


I.GY~RI

14

then by Statement (A) of Theorem 2.1 we have lim+,+&V(t)
= 0, and hence, (2.7) is satisfied.
Now, assume that &” o(t) dt = +m. From equation (2.1), we have

ti(t)

I -a(t)N(t)

+

b(t)
o(t)%+a(t) j=r
cm-wfm>
oo(t)

a.e. t 1 0.

Thus,

jqt)
[N

(0

)+~e~:‘+)dsc+)
[$$+$$V(qj(u))]
-1, t2 0.

(2.8)

Since Joooa(t) dt = co, by Lemma 2.1 and condition (2.6) we find

lim

sup

e-

.I,”
a(s) ds

t--r+03

st

d&p<1

0

and
-dduSo, for arbitrary /?I E

e-

(3,1) we

.r:43) ds

may (and do) fix a constant T > 0 such that

t

&’ a(s) da+)

2 -du

bj(u)
j=l

< 1,


t 2 T.

so(u)

(2.9)

Let M be a constant such that
M > max

N(O) +

MT

l-01

Now, we show that N(t)

’

< A4 for all t > 0. Otherwise, there exists a Tl > T such that

0 5 N(t) < M,
But, in that case (2.8)-(2.10)

t-l(O)

5 t < TI,

and

N(Tl)

= M.

yield
N(Tl) L N(O) +

which is a contradiction.

(2.10)

m=t_l(0)+~T~(s)

MT

+ PlM

<

M,

So IV is bounded on [t-l(O), co), and hence, N = limsup,_+,l\r(t)

< 00. By Lemma 2.1, from (2.8), it follows
m b,(t)
h
r+(t)
G 5 limsup
dv=y+pyfi,
t++cx, so(t) + EsumpF3=1 -@3(t)
and hence, (2.7) is satisfied. The proof of the theorem is complete.
The next result shows that the conditionp^ < 1 in the above theorem is important.
THEOREM 2.3. Assume that Conditions (i)-( v) are satisfied, moreover al(t) = . . . = u,(t)
r(t) 2 0, t 2 0.
If either

= 0,


15

Global Asymptotic Stability Problem

and

&=l

r(t) > 0
qo = lim inf ~
t-+m so(t)
’

then for any positive solution N : [t-l(O), 03) -+ R+ of equation

(2.1), one has limt_+aN(t)

=

+oO.
PROOF.

Let N : [t_,(O), 00 ) + R+ be a positive

First,

we show that

&O”ac(t)h(N(t))
dt = fco. Otherwise, &” a(t) dt < cm, where a(t) = ao(t)h(N(t)),
ai
= 0, 0 5 i 5 TZ,t 2. 0, from equation (2.1) we have

t > 0. Since

N(t)

= e-

.cds)

.fdds) ds

d”N(0) + e-

solution

of equation

(2.1).

t
s

wlh4)

0

and hence,

by using Lemma

2.1, we find
r(t)
=~rJ>O.
N(t) > lim inf t++oo so(t)

ljrn&f
+

(2112)

sow so(t) dt = +oo, and hence, (2.12) and Theorem

On the other hand,

I

2.1 yield sooo uo(t)h(N(t))

dt = +cm. This is a contradiction
and so sooouo(t)h(N(t)) dt = +CQ. It remains to show that
N(t) < co and by Lemma 2.1 and (2.11)!
N(t) 4 foe, as t -+ fco.
Otherwise,
liminft++m
we obtain

2 70 + &liminf t++WN(t), which cannot be satisfied if either
N(t) + +cc as t + too, and the proof of the theorem

co > liminft-,+ooN(t)

& > 1 or 60 = 1 and ye > 0. Therefore,
is complete.

3. GLOBAL
WHEN
In this section,

ASYMPTOTIC
STABILITY
so(t) IS DOMINATING

we give global asymptotic

stability

results

for equation

(2.1) when uo(t) > 0,

t > 0.
DEFINITION 3.1.

We say that

(2.1) is globally

equation

positive solutions NI, NZ : [t_l(O),co)

--t R+ of equation

asymptotically

stable if for any two

(2.1),

lim (Nl(t) - Nz(t)) = 0.

t++cc
It is clear that
function
equation

equation

K:[t_i(O), CQ) +
(2.1) one has

(2.1) is globally
R+ such that

asymptotically
for any positive

lim (N(t) - K(t))
t-i+co

exist and are finite,

lim -r(t)
t++muo(t)

’

a0 =

5 c%(i)
lim s,

t-++oo q)(t)

if and only if there

exists

a

N : [t-l(O), co) -+ R+ of

= 0,

that is K is a global attractor
with respect to the positive
Now, assume that the following hypothesis is satisfied.
(vi) a0(t) > 0, t L 0, Jr uo(t) dt = +m, and the limits

CO=

stable
solution

solutions

DO=

of equation

(2.1).

2 Vt)
j=1
lim ~
t*+c= uo(t)

moreover
qo=cuo+po<1.

In the following result, we show that co/(1 + CEO- PO) is the limit of all positive solutions of
equation (2.1) whenever cc > 0, otherwise the solutions of equation (2.1) tend to zero at +co.


I.GYBRI

16

3.1. Assume (i)-(iv)

THEOREM

and (vi). Then

(A) if CO5 0 then for any solution N : [~_~(O),CQ) -+ R of equation (2.1),
one has limt++oo
N(t) = 0, and hence, the zero solution of equation (2.1) is a global attractor,
(B) if co > 0 then for any positive solution N : [t-l(O), oo) 4 R+ of equation (2.1) one has
&N(t)

=

Co
l+cro-po’

(3.1)

and hence, equation (2.1) is globally asymptotically stable.
PROOF

A. Let ~0 5 0 and fix an arbitrary positive solution N : [t_l(O),m)

tion (2.1). If Joooao(t)h(N(t))

Now, assume that sooocr(t) dt = +CXI,,a(t) = ao(t)h(N(t)),

l-+(t) 5 a(t)-

‘@)- a(t)N(t)

5

e-

R+ of equa-

.r,”
d”) d”N(()) +

.6 a(u)d”

e-

t > 0. Then

+ a(t) 2 wN(g,(t)),
j=l so(t)

so(t)

and hence,

N(t)

+

dt < co then by Theorem 2.1 we have limt_++ooN(t) = 0.

ts
s’

8.e. t > 0,

0

+

e-

.J;4%) du.

&” 4~) du

4s)

0

bj(s)
cmp-N(e(s))ds,
so(s)

t>

0.

j=l

We know, in virtue of Theorem 2.1, that @ = lim sup ,_+,N(t)

< w, and by using Lemma 2.1

we find

that is fi 5 co/(1 - ,f30)5 0. On the other hand, from the nonnegativity of N, we have 0 5 fi,
so either N(t) + +CKI,t --+ too,

or soooa(t) dt < co. The proof of Statement (A) is complete.

PROOF B. Assume that CO > 0 and let N : [t-l(O), co) +
equation (2.1). By Theorem 2.2, we have

R+ be any positive solution of

0 1. mo = l;lmpinfN(t) 2 MO = limsup N(t) < co,
t-+m
and by Statement (B) of Theorem 2.1 we know that soMa(t) dt = +q a(t) = ao(t)h(N(t)),
t 2 0.
But, by using the variation of constant formula, equation (2.1) can be written in the following
equivalent form:
N(t) = e - .f, 4s) d”N(0) + e- .F a(s) ds

s

t ef,”a(s) dSCy(&?&
,&
so(u)

_~e-,~~~~d~,t~~~~I.~ds~(~)~N(~~(u))d~
i=l
+ 2

e- .I; 4s)

j=l

ds

t e.C’ 4s)

mN(qj(u))

ds+)

s0

so(u)

for t 2 0.
By using Lemma 2.1, we find

MO I co -

a3mo

+

POMO

du,


17

Global Asymptotic Stability Problem
and
mo 2 co - aoh’lo + Porno.
From that, it follows

and since qo < 1, we obtain MO = mo. This means that N(W) = limt,+ooN(t)

exists and N(t)

satisfies the required relation (3.1). The proof of the theorem is complete.
Now, we investigate the case when h(u) = u, IL> 0, that is when equation (2.1) reduces to

ii+) = iv(tj +) - ~~(tuv(t) -

i=l

[
DEFINITION

3.2.

2 ~~ww~(t)) + $Jb(w(qd~))
j=l

I

a.e. t > 0.

,

(3.2)

For a locally Lebesgue integrable and locally bounded function g : R+ --+ R,

the equation

k(t)=K(t)
[

+

~(t)-~O(t)K(t)-~ui(t)Ko)
i=l

I

~(?,(t)K.(p,O)
+S(t)
3

j=l

a.e. t > 0, (3.3)

is called the g-perturbed equation of equation (3.2).
It is clear that when h(u) = u, u E R and (vi) is satisfied then the function K,,(t)

=

c/(1 + a - P), t 2 t-1(0), is a solution of the go perturbed equation (3.3), where go : R+ -+ R is
a suitable function such that go(t) -+ 0, t --+ +m.
In the next result, we generalize Theorem 3.1 in the following sense: we prove that under
certain conditions, for any positive solution N : [t-l (0), m) + R+ of equation (3.2) one has

where KS is a solution of the related g-perturbed equation (3.3).
In fact, if Kg is a bounded function on R+, then (3.5) yields

t$&(N(t) - K,(t)) = 0,

(3.4)

which means that Kg is a global attractor with respect to the positive solutions of equation (3.2),
or equivalently equation (3.2) is globally asymptotically stable with respect to the positive solutions.
THEOREM 3.2.

Assume (ii)-

and
M

so(t) > 0,

t 2 0,

s0

a0(t)

dt

=

+m.

Let g : R+ -+ R be a locally Lebesgue integrable and locally bounded function and KS? :
[t-l(O), 00) -+ R+ is a solution of ( 3.3) such that

Kg(Pi(t)>2 6. Kg(t),

1 5 i 5 72,

Kg(qj(t))

2 6 .K,(t),

1 L j 5 m,

(3.5)

b > 0 is a constant and

g(t)
ao(W&

+

”

t-++co,

(3.6)


18

I. GY~RI

moreover,

Then, for any positive solution N : [t-l(O), cc ) -+ R+ of equation (3.2) relation (3.4) holds, and
if Kg is bounded on R+ then Kg is a global attractor with respect to the positive solutions of
equation (3.21, and hence, (3.2) is globally asymptotically

stable.

PROOF. Let

fixed positive solution N : [t-l(O), co) --) R+ of equation (3.2).

for any arbitrarily
Then

Nt)

k(t)

= -

K,(t)

-

ri,@)

- -N(t) -

Kg@)Kg(t)

2 ai(t)(N(pi(t)) - K,(M)))

N(t) - ao(t)(N(t)- Kg(t))-

Kg(t)

i=l

1

+$j(W%j(t))
-Kg(&)))

-g(t) ,

j=l

for a.e. t 2 0.
Let a(t) = ao(

and go(t) = (g(t)/ao(t)K,(t)),

k(t) = -CX(t)X(t) - Q(t)

t 2 0. Then

$ ui~~~c~j)).(pi(t))
9

+

Because of c<

a(t)

fJb~(t)Kg(q4-C(qj(t))
ao(t

1, there exist T > 0 and q E (0,l)
n ai(t)Kg(pi(t))
c
+l
ao(t

- a(t)

j=l

+c

such that

m W)Kg(gj(t))
j=l

ao(t

< q
-

t > T.

’

In that case,

Let M be a constant for which
Iz(t)J I M,

t_,(O) 5 t I T,

are satisfied, where cr = [z(T)1 + sup,2Tlga(u)l.

a.e. t 2 0.

and

M > 5

(3.8)


Global Asymptotic

Stability

Now, we show ]z(t)] 5 A4 for any t 2 T. Otherwise,
]z(t)] < M,

T < t < TI,

19

Problem

there exists a Tl > T such that
[x(T~)l = M.

and

But, from (3.9), it follows
IG’i)I

IG’)I

I

+ ,““>pTIgo

+ 4. M < M,

-

which is a contradiction.
So ]z(t)] < M, t L t-1(0), and hence, 2 = limsup,_+,]z(t)]
On the other hand, from (3.6) and (3.8) it follows:

and hence,
+co.

by using Statement

So, in virtue

of Lemma

(B) of Theorem

2.1, we have

N(t)

and if Kg is bounded

ao(t)h(N(t))

&? o(t) dt = s.

2.1, (3.9) yields f 5 qf, and hence, E = 0. This means
z(t) = - 1 -+ 0,
Kg(t)

theorem

< 00.

on R+ then limt_++W(N(t)

dt z=

that

t--++cm,

- K,(t))

= 0 1s
. a 1so satisfied.

The proof of the

is complete.

4. GLOBAL ASYMPTOTIC
STABILITY WHEN
so(t) IS NOT NECESSARILY
DOMINATING
of so(t) is important

The positivity
previous

section,

Those theorems

where

equation

in the global

(2.1) is controlled

do not take into account

with delay.

In this section,

terms

with small delay may help to stabilize

stability

by an undelayed

the possible

terms

asymptotic
stabilization

we prove some new results
equation

results

negative

term.

the negative

(2.1). The technique

feedback
feedback

of the proof applied

is original in the literature of delay population
model equations.
borrowed from some earlier papers of the author [7-g], where the stabilization
feedback terms with small delays was initiated and analyzed.
DEFINITION4.1.

in the

effect of the negative

to show that

in this section

We need the following

proved
feedback

This technique is
effect of negative

definitions.

We say that

the solutions of equation

(2.1) are eventually bounded by the

constant KO > 0 if for any solution N : [t-l(O), m) -+ R+ of equation (2.1)
limsupN(t)
t-++m
Let L > 0, 720E (0,. . . , TX} and consider

I Ko.

the delay differential

GE(r) = --W&E F ui(t)%(Pi(t)),
i=o
where m(t)=
t, t 2 0, WL,~ = maxo<,The function 21, : {(t,s) : 0 5 s < t} + R is the fundamental
%(t,

s) =

-WL$

F %(t)Q(pz(t),S),

(4.1)
equation
a.e. t 2 0,

solution

(4.2)

of equation

(4.2) if

a.e. t 2 s 2 Cl

2=0

and

From
solution

the general
21, exists

theory

of the delay

(see, e.g., [11,12]).

differential

equations

it is known

that

the fundamental


I. GY~RI

20

DEFINITION 4.2. We say that equation (2.1) has property P(L,ns) if there exist E > 0 and
T, > 0 such that the fundamental solution v, of equation (4.2) is positive on {(t, s) : T, 5 s 5 t}.
In the oscillation theory of delay differential equations, there are several explicit conditions
which guarantee that the fundamental solution of a linear delay differential equation is positive.
In this direction, we refer to the book [13] and to the papers [7-91.
We need the following lemmas.
LEMMA 4.1. Assume that the solutions of equation (2.1) are eventually bounded by the conThen for any positive solution
stant Ks > 0 and equation (2.1) has property P(Ks,n,).
N : [t-1(0), 0;) 1 + R+ of equation (2.1), the fundamental solution VN : {(t, s) : 0 5 s 5 t} -+ R
of the delay differential equation
GN(t)

a.e. t 2 0,

= -h(N(t))~ui(t)~N(~~(t)),

(4.3)

i=o

is positive on TN 5 s 5 t < 00, where TN 2 0 is a suitable constant depending on the solution
N(t).
PROOF. Let N : [t-l(O), m ) -+ R+ be a positive solution of equation (2.1). Since equation (2.1)
has property P(Ks, no) there exists an E > 0 such that the fundamental solution v, : {(t, s) : 0 5
s 2 t} -+ R of the equation
u(t) = -wKo+

a.e., t 2 0,

~h(t)%(p,(t)),

(4.4)

i=o

is positive on T, 5 s 5 t < 00, where T, is a suitable constant. On the other hand, the solutions
of equation (2.1) are eventually bounded with Ko, and hence, there exists TN such that TN 2 T,
and
N(t) 5 Ko + e,

t 2 TN.

t 2 TN, and for the fundamental solution
In that case, h(N(t)) 5 WK,,, = mmO- v, of equation (4.4) we have that ve(t,s) > 0, t > s 2 T,, and hence,
ai(t)t
$f(t,s)
=-wKo& 2 ai(t)vE(fh(t),s) L -h(N(t)) 2
i=o

s),

i=o

a.e. t > s 2 TN.

By using a comparison theorem for the positive solutions of linear delay differential equations
from [lo], we find
0 < %(r,a) 5 'UN(trS),

t>s>T~,

where ZiN is the fundamental solution of equation (4.3). The proof of the lemma is complete.
LEMMA 4.2. Assume that the conditions of Lemma 4.1 are satisfied and g : R+ + R is a
locally Lebesgue integrable function such that sup tlo [g(t)/ < co. Then for any positive solution
N : [t-l(O), co 1 + R+ of equation (2.1) and for any T > TN,
$rnpmf g(t) 5 lim
inf : vlv(t, u)e(U)h(N(U))S(u) du
t_+oo
s
t
2 limsup

vN(t,U)a('lL)h(N(U))g('IL)d'U.I lims:
t-++m s T
assuming that sow a(s)h(N(s))

ds = tco,

where a(t) = ~~.!o ai(

g(t),

t 2 to.

PROOF. Let T 2 TN be a fixed number and y : [t-l(T), co ) + R+ be a positive solution of
equation (4.3), where N : [t-l(T), m ) -+ R+ of equation (2.1) such that s;;” u(s)h(N(s)) ds =
+co. We show that y(t) -+ 0, as t --+ +cm. In fact, y satisfies
Q(t) = -h(N(t))

2 ei(t)y(pi(t)),
i=o

t 2 T,


Global Asymptotic

21

Stability Problem

exists. Assume that y(m)

and hence, g(t) 5 0, t 2 T. Thus, y(oo) = 1im++,y(t)
there exists a constant ti > T such that y@,(t)) 2 c > 0, t 2 ti. But in that case,

B(t) I -ch (N(t))

F

c%(t),

t 2

> 0. Then

t1,

%=a

and hence,
t ch
s t1

y(t) I yl(t1) This is a contradiction,

(N(s))

2 ai
i=o

y(t) -+ 0, t -+ +co.

and hence,

a.Y++co.

ds + --co,

Now, let v(t) = 1, t > tl(T)(T

> T,y).

Then
e(t) = -h(N(t))

F

ei(+?(P%(t))

+ h(N(t))

F

i=O

Thus,

by using the variation

t 2 T.

G(t),

i=o

of constants

formula

this yields

t
v(t)

where a(t) = C~~o~i(t)7
with initial condition

=

~1 (G T)

+

~(4

.IT

t 2 0, and yl(.;T)
~l(t;T)

uMu)h(N(u))

: [t-l(T),

= 1,

t-l(T)

03) + R is the solution

of equation

(4.3)

5 t 5 T,

and UN is the fundamental
solution of equation (4.3)
Under our conditions
yl(t;T)
> 0, t 2 T, and uN(t,s)
limt,+ooyr(t;
T) = 0.
On the other

t L T,

du,

> 0, t 2 s > TN, and

hence,

hand,
t
IT

~(4ub(u)h(~(u))

as t + +ce,

= 1 - w(t; T) -+ 1,

for any fixed T > TN.
Now, we show that

for any fixed Tl such that

Tl > T (2 TN)

Tl
t”$% s T
In fact, as t --+ +co,

I

Tl

v,,,(t, u)a(u)h(N(U)) du =

T

I

vN(t, ~)a(~)h(N(u))

du = 0.

t
vN(t,

u)a(u)h(N(u))

du. -

T

Now, we turn

I

t

VN(t,u)a(u)h(N(u))

to the proof of the required

inequality.

Let ?j = limsup,_+,g(t)

suptzTNg(t),

Then ?j < 00 and gM < co. Moreover, for an arbitrarily
T, > T (2 TN) such that g(t) 5 Tj + E, t > T,. Thus,

lim sup
t++m

I

t

vp,(t,u)a(u)h(N(u))g(u)

t-++m

du I limsup

T

t-+CC
t

+lim sup
s T,

du + 0.

Tl

VN(t, u)a(zL)h(N(u))g(u)

I

and gkI =

fixed E > 0 there exists

a

T,

T

vN(t,

u)a(~)f~(N(u))g(~)

du

T,

du I gnllimsup
t-‘+m

sT

uN(t,

u)a(u)h(N(u))

du

t
+(g+~).

lim
vN(t, u)u(u)h(l\r(u))
t-++m s TE

du = 9 + E.

But E > 0 was arbitrary,
and hence, the upper estimation
part of the required
proved. The proof of the lower part is similar, and hence, it is omitted.
Now, we state

and prove the main result

of this section

which is analogous

inequality

to Theorem

3.1.

is

I.GY~RI

22
THEOREM
P(&,no),

4.1. A&ume that Conditions (i)-(’IV) are satisfied and equation (2.1) has property
no
t 2 0. Suppose
where KJ > 0, 720E (0,. . . , n}. Let a(t) = C ai(
i=o

s
M

a(t) dt = +ca,

t > 0,

a(t) > 0,

0

and the limits

c=

2

r(t)

a =

lim > 0,
t++oo a(t)

lim
t++cc

i=no+l

ai(t)

fit

p =

a(t)

b(t)

lim 3=1
t-++co a(t)

’

exist and are finite, moreover
q=cu+PIf the solutions of equation (2.1) are eventually KO bounded, then for any solution

N : [t_,(O), cm)

-+ R+ of equation (2.1), one has
l&N(t)

=

’

=

1+a-P

r(t)

lim
t-+mi$oai(t)

(4.5)

- 5

j=l

bj(t)’

and hence, equation (2.1) is globally asymptotically stable.
PROOF. Let N : [tL1(0), co ) --+ R+ be a positive solution of equation (2.1). Then by Theorem 2.1
we have that Joooa(t)h(N(t)) dt = fco, since c > 0. On the other hand, N is eventually bounded

by Ko, moreover, equation (2.1) has property P(Ko,no).
So, by Lemma 4.1, the fundamental
solution UN of equation (4.3) is positive on TN 5 s 5 t < 0;).
Equation (2.1) can be written in the form

&(t) = -h(N(t))

F ai(t)N(Pi(t)) + f(t),

8.e. t > 0,

(4.6)

i=O

where

f(t) = -h(N(t))

2

ai(t)N(pi(t)) + h(N(t))

2 bj(W(qj(t))

+ h(N(t))r(t),

t>

0.

j=l

i=no+l

t
s

By using the variation of constants-formula for equation (4.6), we have

N(t) = y(t) +

vdt, s)f(s) ds,

~>TN,

(4.7)

TN

where y : [t-l(TN),

co) + R is a function satisfying

G(t)= -h(N(t))

c ai(tMpi(t))>

a.e. t > TN:
(4.8)

i=O

y(t) = N(t),
Since equation (4.8) is not oscillatory and so” h(N(t))a(t)
we know that y(t) 4 0, t -+ +co.

t

5 TN.

dt = +co, from the proof of Lemma 4.2,


23

Global Asymptotic Stability Problem

On the other hand, N is bounded on [~_~(O),KI), and hence, 0 5 g
YV = lim sup ,,+,N(t)

= lim inf
t++a

-i=g+l

t
s
TN

Ui(S)N(Pi(S))
+

UN(& s)h(N(s))a(s)

and a similar way,
lim sup
t-++m
Since y(t) + 0, t

4

= liminft_++ooN(t)

:i

< 00. By applying Lemma 4.2, under our conditions, we find

a(s)

a(s)

t
s

wN(tr s)f(s)

J!El
b(S)N(%(S))

ds 5 -aN

+ PX + c.

TN

+co, (4.7) and the above estimations

yield

N>-aN+pE+c
and
NI

-a&+PN+c.

so
05%NI(ck++)(N-N),
which implies that N = N, since a + p < 1. This means that (4.7) is satisfied and the proof of
the theorem is complete.
In the above theorem,

all of the conditions are explicit except the assumption on the even-

tually boundedness of the solutions of equation (2.1). In the following theorems, we give some
applications of Theorem 4.1 for some cases when the eventually boundedness of the solutions of
equation (2.1) is guaranteed by explicit conditions.
THEOREM 4.2.

Assume that the conditions of Theorem 2.2 are satisfied and the conditions of

Theorem 4.1 also hold with the constant Ko = F/(1 - p^), where T and p^are defined in (2.6).
Then the solutions of equ‘ation (2.1) are eventually bounded by Ko and for any positive solution
N

: [t-l(o),

PROOF.

03 ) --+

R+ of equation (2.1) relation (4.7) holds.

The proof is an easy consequence of Theorem 2.2 and Theorem 4.1.

THEOREM 4.3.

Assume that h(u) = u, u > 0, and there is no positive feedback term in equation (2.1), that is bl(t) = ... = b,(t) = 0, t > 0. Assume further that the conditions of
Theorem 4.1 are satisfied whenever KO = c.
Then for any positive solution N : [t-l(O), w) + R+ of equation (2.1) one has
limsup N(t) = &
t++oo

PROOF.

=

In virtue of Theorem 4.1, it is enough to show that the solutions of equation (2.1) are

eventually bounded by the constant c.

To show this, let N : [t-l(O), co) -+ R+ be a positive solution of equation (2.1) and define

N(t)

z(t) = In -,

C+E

where E > 0 is a fixed constant.

t >

t-1(0),


I. GY~RI

24

Then from equation (2.1)) we have
i(t)

= r(t) -

2 ai(t)(c

a.e. t 2 0.

+ e)ezcpict)),

i=O

But, from the definition of c it follows the existence of a T, such that T(t) I (c + E) alto

ai(

t > T,, and hence,
(1 - ez(pi(t))) ,

k(t) 5 (c+ E) c&(t)
i=o

a.e. t>T,.

For any u E R, 1 - e” < -u, and thus,
k(t) I -(c + &) F

a.e. t 2 T,.

ui(t)z(pi(.t)),

(4.9)

i=O

Under the conditions of Theorem 4.1, if E > 0 is small enough and T, is large enough, then the
fundamental solution 21,: {(t, s) : 0 5 s 5 t} + R of the equation
ti(t) = -(c + E)

-$y

ai(t)u(pi(t))

(4.10)

i=O

dt = fm, and hence, (see, e.g., the
is positive for t 2 s > T,. On the other hand, so” CrJ?o ai
proof of Lemma 4.2) any solution of equation (4.12) tends to zero as t + +a.
Inequality (4.9) yields that z is a solution of the equation
k(t) = -(c + E) 5

G(t)z(Pi(t))

-

a.e. t > T,,

f(t),

i=O

where f(t) = -(c + E) ~~~, ai(t)s(pi(t)) - i(t) > 0, a.e., t > T,. So, by using the variation of
constant-formula, x can be written in the form
t
t L T,,
46 s)f(s) ds,
x(t) = y(t) s T,
where y is a solution of the homogeneous equation (4.10). Thus, x(t) 5 y(t),
limsup,_+,x(t)

5 limsup,_+,
y(t) = 0, which yields that
limsup N(t)
t-++m

t > T,, and hence,

limsup eztt) < (c+E).
t++CO

= (c+E)

But E > 0 can be arbitrarily small, so limsup,_+,

I: c. The proof of the theorem is

N(t)

complete.
COROLLARY 4.1. Let T > 0, a0 2 0, ai 2 0, pi > 0 (1 5 i 2 n) and bj 2 0, aj > 0 (1 5 j < m)
be given constants and assume
(i) C,“=, bj < a0 and there exists an index no E (1,. . . , n} such that
2

ai+eb,

2=no+1

j=l

< Fai,
z=o

where Cycno+l ai = 0 whenever no = n by definition;
(ii) the equation
X = - (Ko + E) 2

aiemxrT,

(70 = Oh

(4.11)

i=O

has a real root for any E > 0 small enough, where

Ko=

rm,

a0 - C bj
j=l

m

if 0 < c

bj < a0

and

if bl = . . . = 6, = 0. (4.12)

Ko=&,

j=l
zFo

@


Global Asymptotic

Then for any positive

solution

Stability Problem

N : [-y, 00) --$ R+

25

(Y = max{maxl~i~,{~~},

the equation

tip)

r - aoN

= N(t)

-

2 aJv(t

- 7J +

i=l

[

5 b,N

(t - Uj)

j=l

one has

PROOF.

The proof of the corollary

is an easy consequence

that equation- (4.2) has property
P(&,no),

equation (4.2) is equivalent to the equation
t&(t) = -(Ko

and by assumption
theory

its characteristic

of delay differential

of equation
Theorem

+ E) Faiu,(t
i=o
equation

equations

(4.14) is not oscillatory.

where

- 7i),

(4.13)

2 0,

in (4.2).

But

in our case,

(4.14)

t > 0,
By using

the oscillation

(see, e.g., [8,13]) this implies that the fundamental
So equation

(4.14) has property

4.3 the corollary

is proved.

5. SOME

REMARKS

AND

OPEN

of

4.2 and 4.3 if we show

(4.11) has a real root.

4.2 and Theorem

Let us consider

t
I
)

of Theorems

Ko is defined

maxl~j+rL{a,}})

P(Ko,

solution

no) and by applying

PROBLEMS

the equation

B(t)

= Iv(t)

aoN

T -

2 aiN(t

-

- Tii) +

2 bjN(t - ~j)
j=l

i=l

(5.1)

,
1

where
r > 0,

a0

2 0,

ai

2 0,

are given constants.
From Theorem 2.3 and Corollary
is globally

asymptotically

stable

ri 2 0,

(1 < i < n),

4.1 it follows that

bj20,

(l~jlm)

if ai = 0 (1 5 i 5 n) then equation

(5.1)

if and only if
m
c

bj < ao.

(5.2)

j=l

It is worth
conditions
Lenhart

to note that

there

are only few results

giving

necessary

(necessary

for a population
model equation to guarantee its global asymptotic
and Travis studied equation (5.1) in [14] and their main theorem

tion (5.1) is globally asymptotically
stable with respect
(1 2 i 2 n) and uj > 0 (1 2 j 5 m) if and only if
eai+Fbj

i=l

It is clear that

Lao

and

j=l

if ai = 0 (1 5 i 5 n), then

Fbj
j=1

condition

to the positive

and sufficient)

stability.
states that

solutions


equa-

for all 7i 2 0

(5.3)
2=1

(5.3) reduces

to (5.2),

so our result

is a special case of the statement
Lenhart and Travis. However, the proof of “only if” part of
Lenhart and Travis theorem is not correct, as it was mentioned by Kuang in [2]. The next result
shows that the second condition in (5.3) is, in fact, necessary for global asymptotic
stability of
equation (5.1).


I. GY~RI

26

THEOREM 5.1.

If

(5.4)
j=l

i=l

then for any positive solution N : [to - r, co)
lim sup,,+,N(t)

+

R+ (to 2

0) of equation (5.1) one has

= $00.

PROOF. Let us assume (5.4) and that equation (5.1) has a positive solution N : [to -T, co) + R+
such that s = sup,z,,_,N(t)

< 00.

From equation (5.1), we have
N(t) = N(to)e

r(t--to)--ao J,t,N(S)da-

2 ai ,J,i N(s-pi)
*=I

dsfjgl

bj I;“, N(s-oj)

ds
7

t L to,

that is
N(t) = N(to)e
where
cl

=

N(S) ds

-

+2
j=l

bj

Ito N(s) ds,
to-o,

and

But

and from condition (5.4) it follows
N(t) 2 c2 +er(t-to),

t 2 to,

with a suitable positive constant c2 > 0. This yields that N(t) + fco
contradiction. The proof of the theorem is complete.

as t + +m, which is a

Condition (5.3) is not surprising, because equation (5.1) has a positive steady state if and only
if
gbj

j=l


(5.5)

i=l

In Section 4, our main goal was (see Corollary 4.1) to find additional conditions to (5.4)
to guarantee the global asymptotic stability property of the positive steady state solution of
equation (4.1). But one can see from Corollary 4.1 that either a0 > 0 or bl = 9.. = b, = 0
additional restriction should be satisfied to get the required result. So the following open question
arises.
PROBLEM 5.1. What are the most general delay dependent or independent additional conditions
to (5.4) for equation (4.1) to be globally asymptotically stable?
From Theorem 4.1, it can be seen that to answer Problem 5.2 one should answer first the
following.
PROBLEM 5.2. What are the most general additional conditions to (5.4) which guarantee that

the solutions of equation (4.1) are eventually bounded by a constant independent of the solutions?
Some partial answers can be given for Problem 5.2 (see the results in the earlier sections of
this paper) if either a0 > 0 or bl = . . . = b, = 0. According to our best knowledge, no result is
know for the case when a0 > 0 and some ai, bj are positive, at the same time.


27

Global Asymptotic Stability Problem
Consider

the simple

“pure delay-type”

fi(t)
with both

negative

= N(t) [r - a1N(t

and positive

delayed

F’rom Theorem
stability

- 71) + b,N(t

71

>

5.1, it follows that (5.7) is a necessary
stable.

for any constants

crl

0,

condition

But it turns out that, in general,

if 71 > 01. More precisely,

t>o

- 01)],

(5.6)

terms

0 < bl < al,

r > 0,

asymptotically

equation

>

0.

(5.7)

for equation

(5.7) is not sufficient

(5.6) to be globally
for global asymptotic

T > 0, al > 0, ~1 > 01 2 0 there exists

a

constant bl E (0,al) such that equation (5.6) has an unbounded
solution on [O,W). In fact, let
bl = a1e+‘-T’)
. Then 0 < bl < al and it is easy to verify that the function N(t) = ert is an
unbounded

solution

of equation

Based on our results
CONJECTURE

5.1.

If (5.7) is satisfied

then the positive

steady

positive

of equation

solutions

(5.6) on [0, co).

we state the following

One more remark

state solution

conjecture

as an open problem.

and the positive
of equation

where the coefficients

From the Theorem
asymptotically
stable

= N(t) [r - aoN

- alN(t

to the

a0 > 0,

the equation

- TV) + blN(t

and the delays satisfy the following

r > 0,

say 71 < gl,

with respect

(5.6).

on the case a0 > 0. Let us consider
N(t)

delay ~1 is small enough,

(5.6) is a global attractor

(5.8)

conditions:

0 < bl < ai, + al,

5.6 of Kuang (see [2, Section
if (5.9) holds and bl 5 a0 -al.

- ol)] ,

71

>

0‘1

0,

>

0.

(5.9)

2.51) we have that equation (5.8) is globally
On the other hand, our Corollary 4.1 implies

the following.
PROPOSITION

Let 71 > 0, g1 > 0, T > 0. If

5.1.

0 < bl < a0

then equation

(5.8) is globally

The above proposition

and

r
alTl ,@olaa-01
~
ao - bl

asymptotically

of Corollary

A=-(&+E)
also [13]).
Thus, Corollary

4.1 generalizes

Theorem

and
ffla0

4.1, since the equation

(ao+a&“)

if (5.10) is satisfied
9.6 of Kuang

b

1

as it can be shown

easily

(see

given in [2, p. 351 when the delay 71 is

small enough and the other delay gl is arbitrary.
It also worth to note that the above results and examples
open problems in [3] and the results in [15].
Unfortunately,
the question of global asymptotic
stability
bl > a0 even assuming (5.9). To illustrate this, assume that
constants,
such that 71 > (~1. Let r and bl be defined by

71

(5.10)

stable.

is an easy consequence

has a real root for any E > 0 small enough

< 1
e

are strongly

related

to one of the

seems to be extremely
difficult if
~1 > 0, ~1 > 0, al > 0 are given

+a1

e

-T(T1-“I)

1

>

for a given constant a0 > 0. Then it can be easily shown that bl = aOerol + ale-T(T1-ul),
and
for any small enough constant a0 > 0, condition
(5.9) 1s
. satisfied and bl > a~. At the same
time, the function N(t) = ert, t 2 0, is an unbounded
positive solution of equation
(5.8), and
is
not
globally
asymptotically
stable.

In
this
sense,
our
condition
bl < a0
hence, equation
(5.8)
in Proposition
5.1 seems to be the best possible if al > 0, ~1 > 01 > 0 and ~1 is small enough.


28

I. GY~RI

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