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- INTRODUCTION In the following, we consider the nonautonomous Lotka-Volterra type delay differential equation fi(,) = W(t)) r(t) - aowqt) [ @(W(Pi(4) + g i=l ~j(w(*j(t)) j=l I , (1.1) where h, T, ai, bj : R+ + R+ (0 i < n, < j m) and pi : R+ -+ R, pi(t) < t (t 0), i < t (t > 0), j 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) = (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 = (t 0, < i 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 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 (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 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 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) = and h(zl) > 0, IL > zL-++oo, 0, h(u) ‘fco, (ii) r : R+ + R is locally Lebesgue integrable and locally bounded, (iii) : R+ + R+ (0 < i n) and 63 : R+ + R+ (1 < j m) are locally Lebesgue and locally bounded on R+, (iv) pi : R+ -+ R (1 i n) and qj : R+ R (1 j m) are locally and locally bounded on R+, moreover pi(t) < t (t 0), lim,,+,pi(t) andqj(t)O), Let tll(s) min{t!,(s), lim t++co&) Lebesgue integrable = +co (1 i n) = +cQ (1 j m) = minl -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 = (2.1) if it continuous In this paper, we 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 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 > 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 > Let et(t) = a(t)h(N(t)), t > Then (2.3) yields m r^.a(t) +K a(t) &yy - uw&(t))l 8.e t Integrating both sides of the above inequality, we find N(t) c +li I t4s) where c = N(0) + F SOWa(t) dt < co Let u(t) = m=~_,(o)+~t{N(s)), u(t) g.y& t s t 0, and ~1 = m={c, Cl + 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 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 This yields and hence, N(oo) = limt-,+ooN(t) exists and is finite It remains to show that N(oo) = Otherwise, N(co) > 0, and there is a constant T > such that N(t) (1/2)N(co) > for t > T 13 Global Asymptotic Stability Problem Thus, @Jr ,8 = inftkT{h(M(t))} a(t) dt = foe : u (1/2)N(co)} 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 (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) = Therefore, (2.5) yields that there exists a T and c > such that h(N(t)) r(t) - ao( - ai(t)N + for t T Therefore, equation c, I t N(t) 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 > 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 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 ? < -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 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 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 Thus, jqt) T such that N(t) < M, But, in that case (2.8)-(2.10) t-l(O) 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 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^ < in the above theorem is important THEOREM 2.3 Assume that Conditions (i)-( v) are satisfied, moreover al(t) = = u,(t) r(t) 0, t If either = 0, 15 Global Asymptotic Stability Problem and &=l r(t) > 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)), = 0, i TZ,t 0, from equation (2.1) we have t > Since N(t) = e- cds) fdds) ds d”N(0) + e- solution of equation (2.1) t s wlh4) 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) foe, as t -+ fco Otherwise, liminft++m we obtain 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) & > or 60 = and ye > Therefore, is complete 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 > 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)) = 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 = 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) Vt) j=1 lim ~ t*+c= uo(t) moreover qo=cuo+po 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 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 > then for any positive solution N : [t-l(O), oo) 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 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) a(t)- ‘@)- a(t)N(t) e- R+ of equa- r,” d”) d”N(()) + a(u)d” e- t > Then + a(t) 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) = ts s’ 8.e t > 0, + e- J;4%) du &” 4~) du 4s) bj(s) cmp-N(e(s))ds, so(s) t> 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 co/(1 - ,f30)5 On the other hand, from the nonnegativity of N, we have 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 > and let N : [t-l(O), co) + equation (2.1) By Theorem 2.2, we have R+ be any positive solution of mo = l;lmpinfN(t) 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 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 + e- I; 4s) j=l ds t e.C’ 4s) mN(qj(u)) ds+) s0 so(u) for t By using Lemma 2.1, we find MO I co - a3mo + POMO du, 17 Global Asymptotic Stability Problem and mo 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 ~~ww~(t)) + $Jb(w(qd~)) j=l I a.e t > , (3.2) For a locally Lebesgue integrable and locally bounded function g : R+ + R, the equation k(t)=K(t) [ + ~(t)-~O(t)K(t)-~ui(t)Ko) i=l I ~(?,(t)K.(p,O) +S(t) j=l a.e t > 0, (3.3) is called the g-perturbed equation of equation (3.2) It is clear that when h(u) = u, u E R and (vi) is satisfied then the function K,,(t) = c/(1 + a - P), t t-1(0), is a solution of the go perturbed equation (3.3), where go : R+ -+ R is a suitable function such that go(t) -+ 0, t + +m In the next result, we generalize Theorem 3.1 in the following sense: we prove that under certain conditions, for any positive solution N : [t-l (0), m) + R+ of equation (3.2) one has where KS is a solution of the related g-perturbed equation (3.3) In fact, if Kg is a bounded function on R+, then (3.5) yields t$&(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 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 Kg(t), i 72, Kg(qj(t)) K,(t), L j m, (3.5) b > 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) ai(t)(N(pi(t)) - K,(M))) N(t) - ao(t)(N(t)- Kg(t))- Kg(t) i=l +$j(W%j(t)) -Kg(&))) -g(t) , j=l for a.e t Let a(t) = ao( and go(t) = (g(t)/ao(t)K,(t)), k(t) = -CX(t)X(t) - Q(t) t Then $ ui~~~c~j)).(pi(t)) + Because of c< a(t) fJb~(t)Kg(q4-C(qj(t)) ao(t 1, there exist T > 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) t I T, are satisfied, where cr = [z(T)1 + sup,2Tlga(u)l a.e t and M > (3.8) Global Asymptotic Stability Now, we show ]z(t)] A4 for any t 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 + M < M, - which is a contradiction So ]z(t)] < M, t L t-1(0), and hence, = 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 qf, and hence, E = This means z(t) = - -+ 0, Kg(t) theorem < 00 on R+ then limt_++W(N(t) dt z= that t ++cm, - K,(t)) = 1s a 1so satisfied The proof of the is complete 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 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 > 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 0, WL,~ = maxo such that the fundamental solution v, of equation (4.2) is positive on {(t, s) : T, s 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 > and equation (2.1) has property P(Ks,n,) N : [t-1(0), 0;) + R+ of equation (2.1), the fundamental solution VN : {(t, s) : s t} -+ R of the delay differential equation GN(t) a.e t 0, = -h(N(t))~ui(t)~N(~~(t)), (4.3) i=o is positive on TN s t < 00, where TN 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 > such that the fundamental solution v, : {(t, s) : s t} -+ R of the equation u(t) = -wKo+ a.e., t 0, ~h(t)%(p,(t)), (4.4) i=o is positive on T, s 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 T, and N(t) Ko + e, t TN t TN, and for the fundamental solution In that case, h(N(t)) WK,,, = mmO s T,, and hence, ai(t)t $f(t,s) =-wKo& ai(t)vE(fh(t),s) L -h(N(t)) i=o s), i=o a.e t > s TN By using a comparison theorem for the positive solutions of linear delay differential equations from [lo], we find < %(r,a) '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 + R+ of equation (2.1) and for any T > TN, $rnpmf g(t) lim inf : vlv(t, u)e(U)h(N(U))S(u) du t_+oo s t limsup vN(t,U)a('lL)h(N(U))g('IL)d'U.I lims: t-++m s T assuming that sow a(s)h(N(s)) ds = tco, where a(t) = ~~.!o ai( g(t), t to PROOF Let T TN be a fixed number and y : [t-l(T), co ) + R+ be a positive solution of equation (4.3), where N : [t-l(T), m ) -+ R+ of equation (2.1) such that s;;” u(s)h(N(s)) ds = +co We show that y(t) -+ 0, as t + +cm In fact, y satisfies Q(t) = -h(N(t)) ei(t)y(pi(t)), i=o t T, Global Asymptotic 21 Stability Problem exists Assume that y(m) and hence, g(t) 0, t T Thus, y(oo) = 1im++,y(t) there exists a constant ti > T such that y@,(t)) c > 0, t ti But in that case, B(t) I -ch (N(t)) F c%(t), t > Then t1, %=a and hence, t ch s t1 y(t) I yl(t1) This is a contradiction, (N(s)) i=o y(t) -+ 0, t -+ +co and hence, a.Y++co ds + co, Now, let v(t) = 1, t > tl(T)(T > T,y) Then e(t) = -h(N(t)) F ei(+?(P%(t)) + h(N(t)) F i=O Thus, by using the variation t T G(t), i=o of constants formula this yields t v(t) where a(t) = C~~o~i(t)7 with initial condition = ~1 (G T) + ~(4 IT t 0, and yl(.;T) ~l(t;T) uMu)h(N(u)) : [t-l(T), = 1, t-l(T) 03) + R is the solution of equation (4.3) t T, and UN is the fundamental solution of equation (4.3) Under our conditions yl(t;T) > 0, t T, and uN(t,s) limt,+ooyr(t; T) = On the other t L T, du, > 0, t s > TN, and hence, hand, t IT ~(4ub(u)h(~(u)) as t + +ce, = - w(t; T) -+ 1, for any fixed T > TN Now, we show that for any fixed Tl such that Tl > T (2 TN) Tl t”$% 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 = 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) 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 + Tl VN(t, u)a(zL)h(N(u))g(u) I and gkI = fixed E > 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 = + E But E > 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 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, and the limits c= 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+P 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 s 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)) ai(t)N(pi(t)) + h(N(t)) bj(W(qj(t)) + h(N(t))r(t), t> 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) 0, t -+ +co t 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, 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 = 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 -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 < 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 > 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 > is a fixed constant t > t-1(0), I GY~RI 24 Then from equation (2.1)) we have i(t) = r(t) - ai(t)(c a.e t + 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) (c+ E) c&(t) i=o a.e t>T, For any u E R, - e” < -u, and thus, k(t) I -(c + &) F a.e t T, ui(t)z(pi(.t)), (4.9) i=O Under the conditions of Theorem 4.1, if E > is small enough and T, is large enough, then the fundamental solution 21,: {(t, s) : s 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 s > T, On the other hand, so” CrJ?o 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) 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) y(t), limsup,_+,x(t) 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 > can be arbitrarily small, so limsup,_+, I: c The proof of the theorem is N(t) complete COROLLARY 4.1 Let T > 0, a0 0, 0, pi > (1 i n) and bj 0, aj > (1 j < m) be given constants and assume (i) C,“=, bj < a0 and there exists an index no E (1, , n} such that ai+eb, 2=no+1 j=l < Fai, z=o where Cycno+l = whenever no = n by definition; (ii) the equation X = - (Ko + E) aiemxrT, (70 = Oh (4.11) i=O has a real root for any E > small enough, where Ko= rm, a0 - C bj j=l m if < c bj < a0 and if bl = = 6, = (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) - aJv(t - 7J + i=l [ 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) 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 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 - aiN(t - - Tii) + bjN(t - ~j) j=l i=l (5.1) , where r > 0, a0 0, 0, are given constants From Theorem 2.3 and Corollary is globally asymptotically stable ri 0, (1 < i < n), 4.1 it follows that bj20, (l~jlm) if = (1 i 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 i n) and uj > (1 j m) if and only if eai+Fbj i=l It is clear that Lao and j=l if = (1 i n), then Fbj j=1 condition to the positive and sufficient) stability states that solutions 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 or bl = = b, = 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 > or bl = = b, = According to our best knowledge, no result is know for the case when a0 > 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 < bl < al, r > 0, asymptotically equation > (5.7) for equation (5.7) is not sufficient (5.6) to be globally for global asymptotic T > 0, al > 0, ~1 > 01 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 < 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 > Let us consider N(t) delay ~1 is small enough, (5.6) is a global attractor (5.8) conditions: < bl < ai, + al, 5.6 of Kuang (see [2, Section if (5.9) holds and bl a0 -al - ol)] , 71 > 0‘1 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 > If 5.1 < 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 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 > small enough < e are strongly related to one of the seems to be extremely difficult if ~1 > 0, ~1 > 0, al > are given +a1 e -T(T1-“I) > for a given constant a0 > 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 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 > and ~1 is small enough 28 I GY~RI REFERENCES K Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht , ( 1992) Y Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, MA, (1993) Y Kuang, Global stability in delayed nonautonomous Lotka-Volterra type systems without saturated equilibria, Diflerential Integral Equations (3), 557-567, (1996) H Bereketoglu and I GyBri, Global asymptotic stability in a nonautonomous Lotka-Volterra type system with infinite delay, J Math Anal Appl 210, 279-291, (1997) X.Z He, Global stability in nonautonomous Lotka-Volterra systems of “pure-delay type”, Di# Int Eqns 11, 293-310,(1998) H.L Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, (1995) I GyGri, Asymptotic stability and oscillations of linear non-autonomous delay differential equations, In Proceedings of International Conference, Columbus, OH, March 21-25, University Press, Athens, pp 389397, (1989) I GyGri, Interaction between oscillation and global asymptotic stability in delay differential equations, Differential and Integral Equations 3, 181-200, (1990) I Gyiiri, Global attractivity in nonoscillating perturbed linear delay differential equations, In Proceedings of the International Symposium on finctional Differential Equations and Related Topics, August 30September 2, 1990, Kyoto, Japan, pp 95-101, World Scientific, Singapore, (1991) 10 I GyGri and M Pituk, Comparison theorems and asymptotic equilibrium for delay differential and difference equations, Dynamics Systems and Applications 5, 277-302, (1996) 11 R Bellman and K.L Cooke, Diflerential Diflerence Equations, Academic Press, New York, (1963) 12 J.K Hale, Theory of Functional Differential Equations, Springer-Verlag, (1977) 13 I Gyiiri and G Ladas, Oscillation Theory of Delay Differential Equations: With Applications, Oxford University Press, Oxford, (1991) 14 S.M Lenhart and C.C Travis, Global stability of a biological model with time delay, Proc Amer Math Sot 96, 75-78, (1986) 15 S Jianhua and W Zhicheng, Global attractivity in a nonautonomous delay-logistic equation, Tamkang Jownal of Mathematics 26, 159-164, (1995) [...]... 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... > 0 and ~1 is small enough 28 I GY~RI REFERENCES 1 K Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht , ( 1992) 2 Y Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, MA, (1993) 3 Y Kuang, Global stability in delayed nonautonomous Lotka- Volterra type systems without saturated... stability in delay differential equations, Differential and Integral Equations 3, 181-200, (1990) 9 I Gyiiri, Global attractivity in nonoscillating perturbed linear delay differential equations, In Proceedings of the International Symposium on finctional Differential Equations and Related Topics, August 30September 2, 1990, Kyoto, Japan, pp 95-101, World Scientific, Singapore, (1991) 10 I GyGri and M Pituk,... equilibria, Diflerential Integral Equations 9 (3), 557-567, (1996) 4 H Bereketoglu and I GyBri, Global asymptotic stability in a nonautonomous Lotka- Volterra type system with infinite delay, J Math Anal Appl 210, 279-291, (1997) 5 X.Z He, Global stability in nonautonomous Lotka- Volterra systems of “pure -delay type”, Di# Int Eqns 11, 293-310,(1998) 6 H.L Smith, Monotone Dynamical Systems, An Introduction to. .. to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, (1995) 7 I GyGri, Asymptotic stability and oscillations of linear non-autonomous delay differential equations, In Proceedings of International Conference, Columbus, OH, March 21-25, University Press, Athens, pp 389397, (1989) 8 I GyGri, Interaction between oscillation and global asymptotic stability in. .. 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... 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... 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... not surprising, because equation (5.1) has a positive steady state if and only if gbj j=l 0 or bl = 9 = b, = 0 additional restriction... Comparison theorems and asymptotic equilibrium for delay differential and difference equations, Dynamics Systems and Applications 5, 277-302, (1996) 11 R Bellman and K.L Cooke, Diflerential Diflerence Equations, Academic Press, New York, (1963) 12 J.K Hale, Theory of Functional Differential Equations, Springer-Verlag, (1977) 13 I Gyiiri and G Ladas, Oscillation Theory of Delay Differential Equations: ... 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 > t++m a(t) and (2.5) then for any positive... of equation (2.1) Let t 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... 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 then for any solution N : [~_~(O),CQ)