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

GY~RI asymptotic stability of delay differential equations without steady state solutions.. Our focus in this paper, is the global asymptotic stability analysis of the delay differential

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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 GY~RI Department of Mathematics and Computing

University of Veszprem

8201 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

Keywords-Lotka-VoIterra differential equation, Global attractivity, The effect of negative feed-

back terms with delays, Open problems

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 ~j(w(*j(t)) 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

In, qj: R+ -+ R, e(t) < t (t > 0), 1 5 j 5 m, are some given functions

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 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

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10 I GY~RI

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) is a dominating term, which essentially means that Cy=“=, ai + C,“=, bj(t) I quo(t), 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)) in equation (l.l), are not taking into account On the other hand, the Liapunov technique is much less effective if equation (1.1) is so called “pure-delay type”, that is when au(t) = 0 (t > O)-even if the equation has a saturated equilibria Recently, He [5] proved some global asymptotic 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 methods to be used in this paper are different from those used in the above mentioned papers To some extent, our approaches are related to some “simple” techniques, basically applying variation of constants 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, but the nonautonomous equation does not have a steady state solution Our sufficient conditions for global asymptotic stability of equation (1.1) are delay independent when the asymptotic equilibrium is constant 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 co- efficients ui (1 I i I no) are small enough and Cz =no+l as(t) f c3n,i &(Q < Q c;: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

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Global Asymptotic Stability Problem 11

Kuang in [2, p 351 We also construct examples which show the sharpness of our results and also

that the delay in the negative feedback term can destroy the global asymptotic stability property

if it is greater than the delay in a positive feedback term Interesting to note that this fact is

true even for small delays We also raise up some open questions based on our investigations and

examples

2 A PRIORI BOUNDS

Consider the following Lotka-Volterra equation:

M) = h(N(t)) [ r(t) - ao( - -$i(t)N(Pi(t)) i=l + -&(t)N(qj(t)) j=l 1 , (2.1)

where

(i) h : R+ -+ R+(R+ = [O,w)) is a continuous function, h(0) = 0 and h(zl) > 0, IL >

(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 integrable

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 Lebesgue integrable

and locally bounded on R+, moreover pi(t) < t (t 2 0), lim,,+,pi(t) = +co (1 5 i 5 n)

andqj(t)<t(t>O), lim t++co&) = +cQ (1 5 j 2 m)

Let tll(s) = minl<i<,{inft~s{pi(t)}}, ttl(s) = minllj~,{inft~s{q~(t)}}, and t-l(s) =

min{t!,(s), t2_l(s)} f or any s 2 0 We assume that til(s) > 00 and t?l(s) > -co

DEFINITION 2.1 A function N : [L1(0),oo) -f R is called positive solution of equation (2.1) if it

is positive on [t-l(O), co), Lebesgue integrable and bounded on [t-l(O), 01, absolutely continuous

on [0, co) and N satisfies equation (2.1) for almost every t 2 0

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 (2.1) exist on [t-l(O), co) It is worth noting that the uniqueness of the

solutions are not needed in our results

For any function c : R+ -+ R the functions c+ : R+ + R+ and c- : R+ -+ R+ are defined by

the relations 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 (2.1) with respect

to the existence of the integral sooo a(t)h(N(t)) dt, where a : R+ + R+ is a given function

THEOREM 2.1 Assume (i)-(iv) and suppose that there exists a locally Lebesgue integrable,

locally bounded function a : R+ -+ (0, co) such that

moreover,

Then,

(A) for any solution N : [t-l(O), 00 + R+ of equation (2.1) the relation )

u(t)h(N(t)) dt < 03,

(2.3)

(2.4)

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12 GY~RI

yields limt-++ooN(t) = 0,

(B) if

liminffi > 0

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

PROOF A Assume that the solution N : [L1(0), cc + R+ satisfies (2.4) 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

I t

N(t) 5 c +li 4s) g.y& v%j(SN) ds, t L 0,

where c = N(0) + F SOW a(t) dt < co

u(t) 5 Cl + 4 s t 4sMs) ds, t 2 0,

0 and by using the Gronwall inequality, we obtain *

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 =

0

I

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

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

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Thus, ,8 = inftkT{h(M(t))} 2 inf{h(u) : u 2 (1/2)N(co)} > 0, and hence, J,“a(t)h(N(t))dt 2

@Jr a(t) dt = foe contradicting (2.4) The proof of Statement (A) is complete

PROOF B For the seeking of contradiction, 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 bj(W(&)) > a(t)h(N(t)) c,

I

for t 2 T Therefore, equation (2.1) implies

and hence,

s

t

T

This yields that lim++,N(t) > 0 which is a contradiction The proof of Statement (B) is complete

In the next theorem, we give an upper bound for the solutions of equation (2.1)

THEOREM 2.2 Assume that (i)-(iv) are satisfied, and

(v) uo(t) > 0, t > 0, J,“uo(t)dt = +oo, moreover

y = limsup- t-++w so(t)

Then for any solution N : [t-l(O), 00 -+ R+ of equation ) (2.11, one has

? lim supN(t) < -

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

of a similar result proved in [ll] in that case when u(t) = a~(> 0), t > 0, is a constant The proof of the next lemma is omitted

LEMMA 2.1 Let a : R, f R+ and g : R, t R be locally Lebesgue integrable functions such that

s

Co u(t) dt = +cc and sup Is(t)1 < co

0 t>0

Then, for any T 2 0,

vmpmfg(t) < $mpWfe- Jk a(s)ds & “‘“‘%(u)g(u) du

T

< limsup e-lGa(s)ds

THE PROOF OF THEOREM 2.2 Let N : [t_1(0), cc + R+ be any solution ) of equation (2.1) Let

a(t) = uo(t)h(N(t)), t 2 0 Then either so” o(t) dt < 00 or sooo a(t) dt = +CQ If sooo a(t) dt < co

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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

m b(t) ti(t) I -a(t)N(t) + o(t)% + a(t) c -wfm>

j=r oo(t) a.e t 1 0

Thus,

jqt)<e-J’w)ds [ N 0 ( )+~e~:‘+)dsc+) [$$+$$V(qj(u))] -1 , t 2 0 (2.8)

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

lim sup e- .I,” a(s) ds s t d&p<1

t r+03 0

and

-ddu<co

So, for arbitrary /?I E (3, 1) we may (and do) fix a constant T > 0 such that

e- .r: 43) ds t &’ a(s) da+) 2 -du bj(u) <PI < 1,

j=l so(u)

t 2 T

Let M be a constant such that

M > max N(O) + MT

l-01 ’ m=t_l(0)+~T~(s)

( 2.9)

(2.10)

Now, we show that N(t) < A4 for all t > 0 Otherwise, there exists a Tl > T such that

0 5 N(t) < M, t-l(O) 5 t < TI, and N(Tl) = M

But, in that case (2.8)-(2.10) yield

N(Tl) L N(O) + MT + PlM < M,

which is a contradiction 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

t++cx, so(t) + EsumpF 3=1 @3(t) - dv=y+pyfi, 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) = 0, )

r(t) 2 0, t 2 0

If either

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&=l and qo = lim inf ~ r(t) > 0

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 solution ) of equation (2.1) 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)), t > 0 Since

ai = 0, 0 5 i 5 TZ, t 2 0, from equation (2.1) we have

s t

N(t) = e- .c ds) d”N(0) + e- .fd ds) ds wlh4)

and hence, by using Lemma 2.1, we find

r(t) ljrn&f N(t) > lim inf - =~rJ>O (2112)

On the other hand, sow so(t) dt = +oo, and hence, (2.12) and Theorem 2.1 yield sooo uo(t)h(N(t))

dt = +cm This is a contradiction and so sooo uo(t)h(N(t)) dt = +CQ It remains to show that

N(t) 4 foe, as t -+ fco Otherwise, liminft++m N(t) < co and by Lemma 2.1 and (2.11)!

we obtain co > liminft-,+ooN(t) 2 70 + &liminf t++WN(t), which cannot be satisfied if either

& > 1 or 60 = 1 and ye > 0 Therefore, N(t) + +cc as t + too, and the proof of the theorem

is complete

WHEN so(t) IS DOMINATING

In this section, we give global asymptotic stability results for equation (2.1) when uo(t) > 0,

t > 0

DEFINITION 3.1 We say that equation (2.1) is globally asymptotically stable if for any two positive solutions NI, NZ : [t_l(O),co) t R+ of equation (2.1),

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

t++cc

It is clear that equation (2.1) is globally asymptotically stable if and only if there exists a function K:[t_i(O), CQ) + R+ such that for any positive solution N : [t-l(O), co) -+ R+ of

equation (2.1) one has

lim (N(t) - K(t)) = 0,

t-i+co

that is K is a global attractor with respect to the positive solutions of equation (2.1)

Now, assume that the following hypothesis is satisfied

(vi) a0(t) > 0, t L 0, Jr uo(t) dt = +m, and the limits

r(t)

CO = lim -

t++muo(t) ’

t-++oo q)(t) DO = t*+c= lim ~ uo(t)

exist and are finite, 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

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16 GYBRI

THEOREM 3.1 Assume (i)-(iv) and (vi) Then

(A) if CO 5 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

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) + R+ of equa-

tion (2.1) If Jooo ao(t)h(N(t)) dt < co then by Theorem 2.1 we have limt_++ooN(t) = 0

Now, assume that sooo cr(t) dt = +CXI,, a(t) = ao(t)h(N(t)), t > 0 Then

‘@)

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

so(t) a(t)N(t) + a(t) 2 wN(g,(t)), j=l so(t) 8.e t > 0,

and hence,

s t

N(t) 5 e- .r,” d”) d”N(()) + e- .6 a(u) d”

0

+ e- .J; 4%) du s ’ &” 4~) du m p-N(e(s))ds, bj(s) t > 0

j=l so(s)

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 sooo a(t) dt < co The proof of Statement (A) is complete

PROOF B Assume that CO > 0 and let N : [t-l(O), co) + R+ be any positive solution of

equation (2.1) By Theorem 2.2, we have

0 1 mo = l;lmpinf N(t) 2 MO = limsup N(t) < co,

t-+m

and by Statement (B) of Theorem 2.1 we know that soM a(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) ds

s

t e.C’ 4s) ds+) mN(qj(u)) du,

for t 2 0

By using Lemma 2.1, we find

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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) - 2 ~~ww~(t)) + $J b(w(qd~)) , a.e t > 0 (3.2)

DEFINITION 3.2 For a locally Lebesgue integrable and locally bounded function g : R+ + R,

the equation

k(t)=K(t)

[

I 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

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

s

M so(t) > 0, t 2 0, a0(t) dt = +m

0

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)

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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

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

Then

k(t) = - - - -

N(t)

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

i=l

for a.e t 2 0

Let a(t) = ao( and go(t) = (g(t)/ao(t)K,(t)), t 2 0 Then

k(t) = -CX(t)X(t) - Q(t) $ ui~~~c~j)).(pi(t))

9

+ a(t) fJ b~(t)Kg(q4-C(qj(t)) - a(t)

Because of c< 1, there exist T > 0 and q E (0,l) such that

n ai(t)Kg(pi(t))

In that case,

Let M be a constant for which

Iz(t)J I M, t_,(O) 5 t I T, and M > 5

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

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