6 Global Stability for Multi-Species Models

6 Global Stability for Multi-Species Models

6 Global Stability for Multi-Species Models 6.1. Introduction In the real world, species interact and coexist in somewhat closed environments. It is ...

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Global Stability for Multi-Species Models 6.1. Introduction In the real world, species interact and coexist in somewhat closed environments. It is thus more realistic to model population growths in system settings; that is, modeling the growths of interacting species by systems of equations. Suppose we are to model n interacting species in a closed environment, and the per capita growth rate for the ith species at , ( t )is the density of ith time t is F;(t,q),where z ( t ) = ( z i ( t ) , . ,z n ( t ) ) q species at time t ; then, we have a system of n delay differential equations

..

i = 1 , . . . ,n.

$,(t) = z ; ( t ) F , ( t , q ) ,

(1.1)

The type of interactions (competition, predator-prey, or cooperation) .). among species determines the sign of partial (FrCchet) derivatives of F;(t, For example, systems (1.1.7)-(1.1.9) are delayed predator-prey systems. As we can see from Chapter 4, the study of system (1.1) with the most general form of F;(.) can be very complicated and may not even be well defined. In applications, one often adopts the following general Lotka-Volterra type system as models of multi-species population growths in closed environments:

& j ( t )= bj(ui(t))Gi(t,~ t ( * ) ) , where

1

Gi(t,w(-)) r ; ( t )- a ; ( t )

i = 1 , 2 , . .. , n ,

0

-r

Ui(t

(1.2)

+ 0) 444 0)

+ 2 J_”,U j ( t + 0) + i j ( t , O ) ;

(1.3)

3=1

~ ( t =) ( q ( t ) ,. . . , ~ n ( t ) ) ,ri(t), u ; ( t ) are positive continuous functions; p , ( t , * ) ,p ; j ( t , . ) are of bounded variation; and r, are positive constants, . . 2 , j = 1 , . . . ,n. Note that (1.2) is a special case of (1.1). Our main concern

in this chapter is to establish sufficient conditions for the global asymptotic 205

206

Delay Differeniial Equaiions

stability of the positive steady state (assumed to be unique) in (1.2) with respect to its positive solutions. This chapter is organized as follows: In the next two sections, we describe two types of Lyapunov functionals that are quite effective in analyzing the global stability of system (1.2)-(1.3) when it is autonomous and has strong nondelayed negative feedbacks. In Section 4, we use Razumikhin-type theorems to study the global stability problem of (1.2)(1.3), and we apply the results obtained in that section to autonomous systems in Section 5 to obtain some sharp and easy to use theorems. In Section 6, we deal with the situations that occur when nondelayed diagonal terms do not exist, where both the Razumikhin-type theorem and Liapunov function methods are applied to establish regions of attraction of the positive steady states (assumed to be unique) in initial function spaces. We end this chapter with some remarks and open problems. 6.2. Stability via Liapunov Functionals, I The material of this section is adapted from Leung (1979) and Leung

and Zhou (1988). Consider first the following predator-prey model:

where a , b , c , d , p , q , and r are positive constants and ki(O), 8 E [O,r], i = 1 , . . . , 4 , satisfy the following assumptions: ( H l ) ki(8) 2 0 and continuous for fl E [O,r],ki(r) = 0; (H2) $ ( 8 ) 5 0 and continuous for 8 E (O,r), lime+o+ ki(8) and limg+,- ki(8) both exist; (H3) kY(8) 2 0 and continuous for 8 E ( 0 , r ) . These assumptions amount to saying that delay effects diminish gradually in ever moderating pace as one moves backward in time, and the effects become negligible after a length of time r. Let Jlk;(8)d8 be 6, c, p, ij, respectively, for i = 1,2,3,4. We assume further that (H4) .(P p) > ( b 6)d;

+

+

6. Global Siabiliiy for Mulii-Species Models

207

so that (2.1) has a positive steady state ( z * , y * ) , where

As usual, we assume that initial data for (2.1) are taken from C([-r, O],R$). Theorem 2.1. Suppose ( z ( t ) , y ( t ) ) is a solution of system (2.1) with initial data satisfying z(0) > 0, y(0) > 0. T h e n

provided that f o r all 0 E ( 0 ,r ) , k;(8) (i = 1 , . . . ,4) satisfy (H5) p.-1(ki(8))2 < 2br-lk:(O) and cp-1(ki(fl))2 < 2qr-'k:(O). Proof. Since b and q are positive constants, it is easy to see that the solution ( z ( t ) , y ( t ) )of system (2.1) stays positive and is bounded. Let u ( t ) = z ( t ) - z*, v ( t ) = y ( t ) - y*; system (2.1) becomes

Let ($(s),$(s)) E C([-r,0],R2) besuch that 4 ( s ) + z * s E [-r, 01, 4(0) z* > 0, $ ( O ) y* > 0. Define

+

+

2 0, $(s)+Y* 2 0,

208

Delay Differeniial Equations

We have

Separating the last two integrals preceding into four terms and integrating them by parts leads to

Clearly, (H5) implies that V ( $ , $ ) = 0 if and only if 4(s),$(s) = 0 for s E [-r,O], and V(q5,$) < 0 otherwise. By applying Theorem 2.5.5, we have limt,,(u(t),v(t)) = ( O , O ) , and the conclusion of Theorem 2.1 follows. 0

all

209

6. Global Stability for Multi-Species Models

Note that Theorem 2.1 is valid even if (H2)-(H3) for k2 and k3 are replaced by somewhat weaker assumptions, such as k2(8), k3(8) are only continuously differentiable for 8 E (0, r ) . When k3(8), k4(8) do not satisfy (H2) and (H3), we have the following. Theorem 2.2. Assume (2.1) satisfies ( H l ) and (H2), k1(8), k z ( 0 ) ~ a t i s f y (H2)-(H3) with strict inequality in (H3), and k3(8),k 4 8 ) are continuously differentiable on (0, r ) . Assume further that, for 8 E (0, r ) , (H6) (kh(d))’/k!(@) < 2br-’,

Then limt+m(z(t), y(t)) = (z*,y*). Proof. The proof is almost the same as that of Theorem 2.1, except we define now

One can show that

[Joe

The quadratic expression inside the first is clearly nonpositive. The expression inside the second {.} is also a quadratic in $(O), +(O), 4(-s) ds], and [J,”+(-s) ds]. By completing squares three times with the expression in the two {-}, we see that (H7) and k;’(O) > 0, k t ( 8 ) > 0 imply that V ( + , + ) = 0 if and only if 4(s),+(s) are identically zero on [-r,O]. The 0 rest of the proof follows that of Theorem 2.1. {a}

210

Delay Differential Equations

As an example of applying the preceding approach to higher dimensional systems, we consider the following n dimensional system:

where, for i , j = 1,. . . ,n, e ; , p ; j , s;j are constants and (H3)and

ir

k ; j ( e ) d~

= 1.

kij(0)

satisfy

(2.5)

(H1)(2.6)

Furthermore, we assume that (2.5) has a unique positive steady state (xi,.. . ,x:). Let u i ( t ) = x;(t) - t;,i = 1,. . . , n . System (2.5) becomes

We have

6. Global Stabiliiy for Multi-Species Models

211

Following the proof of Theorem 2.1, we obtain the next theorem. Theorem 2.3. Suppose that (i) there are positive condankr a, > 0, i = 1,. . . ,n, such that n

C aipijwiwj 5 o i,j=l

for all

..

( ~ 1 , . ,w")

E R";

(ii) sii < 0, k::(O) > 0 almost everywhere o n [0,r ] , i = 1 , . . . , n ; (iii) sij = 0 when i # j. Then ( x i , . . . ,xi) is globally asymptotically stable with respect to positive solutions of system (2.5). We conclude this section by giving an interesting application of T h e orem 2.3 (or Theorem 2.1). Consider the following simple predator-prey svstem:

where 6 is a positive constant, k;; (i = 1,2) satisfy (Hl)-(H3), and J{ lcii(8)de = 1. If k::(8) > 0 almost everywhere on [O,r], i = 1,2, then Theorem 2.3 implies that, for 0 < 6 < 1, all positive solutions of (2.8) tend to the unique positive steady state. This is in contrast to the case that when 6 = 0, all positive solutions, except the steady state ( l . l ) , are nonconstant periodic solutions. In this case, we may say that delay stabilizes the system. For more results of this kind, see Leung and Zhou (1988). 6.3. Stability via Liapunov F'unctionals, I1 The results in the previous section depend on some very specific properties of lc;(O), which in the real world may not be easy to verify. In this section, we use a slightly different Liapunov functional to obtain a set of rather different conditions for the global stability of the positive steady state (assumed to be unique) of general Lotka-Volterra type systems. We consider i i i ( t ) = bi(ui(t))Gi(ut(.)), i = 1 , 2 , . .. , n , (3.1) where G i ( u t ( * )= ) Ti

Here, T i ,

aij

and

bij

+ j2 a i j u j ( t ) + 2 bij J u j ( t + e) d p i j ( 0 ) . =l j=1 -' 0

are all real constants,

J-, Idpij(e)I = 1, 0

7

(3.2)

is a positive constant, and

i,j=

1 , e . e

,n;

(3-3)

212

Delay Differential Equations

..

hi(-), i = 1 , . ,n, are continuous and bi(0) = 0. Moreover, we assume that ,ti:). steady state u* = (ti!,

...

strictly increasing functions with (3.1)-(3.2) has a unique positive

Theorem 3.1. If there i s co?st$nt positive diagonal matriz C = diag (c1,. ,G ) such that C A ATC is negative definite, then u* i n (3.1)-(3.2) is globally asymptotically stable. Here, = (ajj), where

+

..

Remark 3.1. When b;j = 0, the theorem reduces to the well-known result of Goh (1977). When n = 1, bl(ul(t)) = ul(t), (3.1) reduces to (2.5.11).

Thus, Theorem 3.1 partially generalizes Theorem 2.5.6. It also generalizes Lemma 2.1 in Kuang and Smith (1991). Proof of Theorem 3.1. Assume that the constant positive diagonal matrix = diag (c1, cp, . . . ,cn) satisfies the assumptions specified in the theorem. Define V : C([--T,01, R;) + R as

c

+

Clearly, V ( $ ) > 0 if $;(O) uf > 0 for i = 1 , 2 , . . . ,n, V ( 0 )= 0. Assume that ut(.) = u* is a solution of (3.1)-(3.2). Let V ( $ t ) denote the derivative of V ( $ t )along the solution of (3.1)-(3.2). We have (see Lemma 2.5.1) $ t ( e )

+

6 . Global Stability for Multi-Species Models

213

+ aTc

The assumption of being negative definite implies that if $ ( t ) # 0, then V ( q 5 t ) c 0. By Theorem 2.5.5, we have 4(t) + 0 as t + 00 and hence the theorem. 0 Note that negative definiteness of ATC requires that aii < 0; that is, roughly speaking, the nondelayed intraspecific competition should be strong enough to dominate the effects of delayed interspecific and intraspecific interactions. In the following, we consider system (3.1) with

CA +

n

n

.t

where yij are real constants, and Fij : functions normalized by

1

00

F&)ds=

1,

[e,m)

+

R+ are continuous

. . = 1 )... , n .

2,]

We assume that the initial data are taken from BC((-w,O],R;) with Ui(0) > 0, i = 1,. . . ,n . We further assume that Fij in (3.4) are convex combinations of functions of the form

214

Delay Differeniial Equations

Using the linear chain trick described in Section 3.6, we introduce new variables u n + l ( t ) ,... ,up of the form

1, F,(t t

S)Uj(S)dS,

p = 1,. . . ,m,

where F,, runs through all functions of the form (3.5) appearing in Flj,. . . , F,j for j = 1,. . . ,n. The new variables un+l,. .. ,up satisfy a system of linear constant coefficient differential equations in u1,. .. ,up. Together with the equations for u1, . . . ,un, we have

P

t q t )= C Ulj"j(t),

1=n

j=1

+ 1,. . . , p ,

(3.7)

for some well-defined coefficients a;,, i, j = 1,.. . , p , with initial conditions UI(0)

= uio

> 0,

2

= 1,2,. . . , n

(3.8)

To simplify the presentation, in the following we do not specify the coefficients aij. The following theorem slightly generalizes Theorem 2 in Wiirz-Busekros (1978). Theorem 3.2. Suppose s y s t e m (3.6)-(3.7) has a unique positive steady state u* = (ui, uf,.. . ,u;). A s u f i c i e n t condition for u* to be globally asymptotically stable with respect to initial values of the f o r m (3.8)-(3.9) i s the existence of positive real numbers d l , .. . ,dn, and a positive definite ( p - n ) x ( p - n ) - m a t r i x 0 4 such that D A + A * D is negative definite, where A = (aij)pxp, D1 = diag ( d l , . . . , d n ) , and D = diag ( D l , D 4 ) . Proof. The proof is almost identical to that of Theorem 2 in WorzBusekros (1978). We define

+

where D4 = ( d i j ) , i,j = n 1,. . . , p . It is easy to see that V 2 0, and V = 0 if and only if u ( t ) = u*. The derivative of V along a solution of

215

6. Global Stability for Multi-Species Models

(3.6)-(3.9) is

+

22

[(uk - u;)aikdij(uj - u;) i,j=n+l k = l (Ui - u:)dijajk(uk - ti;)]. Here, we have used the fact that u* satisfies (3.6)-(3.7). In matrix notation, we have

+

= ( U - u*)(DA

where

A A = (A:

A A:)

+ ATD)(u - u * ) ~ , A1 = (aij)nxn, i,j = n + l , ... ,p.

7

A4 = (aij),

Since DA+ATD is negative definite, V 0 along every solution u of (3.6)(3.7) and equals zero if and only if u = u*. This leads to the conclusion of the theorem. 0 For more results that relate to Theorem 3.2, see W&z-Busekros (1978). 6.4. Stability via Razumikhin-Type Theorems-Theory

In this section, we consider the following general nonautonomous Lotka-Volterra-type system with infinite delays:

i = 1, ... ,n,

(44

216

Delay Differential Equations

where the integral variable on the right hand side is s. In system (4.1), we always assume that, for i = 1,. ,n, the following hold (Hl) bi(0) = 0, b:(ui) > 0; (H2) ri(t) and a i ( t ) are continuous for t 2 0. a i ( t ) > 0 , Ti 2 0; (H3) p i ( t , S ) , pj;(t,s),and vji(t,s) are nondecreasing with respect t o s. Let

..

A

pi(t)=pi(t,O)- pi(t,-Ti),

A p j i ( t ) = p j i ( t , 0) - pji(t7 --oo), A vji(t)= vji(t,0) - vji(t, -w).

We assume that p i ( t ) , p j i ( t ) and vji(t) are continuous for all t 2 0. As usual, we choose the UC, space, where g : (-m,O] + [1,00) satisfies (g1)-(g3) in Section 2.7. We assume our initial conditions satisfy E BC, +(O) > 0. (4.2) (H4) u,,(s) = +(s) 2 0 for s 5 0; Here, +(O) > 0 means +,(O) > 0 for i = 1,. . . ,n. We call such a +(s) an admissible initial function. It is easy to see that if +i(O) = 0, then ui(t) = 0 for t 2 u. The following fundamental lemma is a simple combination of some well-known results. L e m m a 4.1. Assume (Hl)-(H4) hold. Assume also that (H5) there is a g(s), satisfying (g1)-(g3), such that

+

lri

A '

pig(t)=

A

i = 1, ... , n ,

g(s)+i(t,s) < 00,

'

vjig(t)= L r n g ( s )dvji(t,s) < 00,

. .

2,3 = 1,. . . ,n,

and pi,, p j i g , vj;, are continuous for t 2 u. Then (4.1) has a unique solution u(u,+)(t)such that u(u,+)(t)> 0, for t 2 u in its maximal interval of existence. If u(u,+)(t) i s a noncontinuable solution, then, for any M > 0 , there is a t* > u such that Cy=lui(o,+)(t*)> Proof. It is straightforward to verify that the function on the right-hand side of (4.1) is locally Lipschitz in R x UC,. A simple application of Theorem 2.2 in Hale and Kato (1978) yields the existence and uniqueness of u(u,+)(t).If, for some 1 E {1,2,. . . , n } , t > u is such that u,(u, +)(t) = 0, then there is an N > 0 such that Iui(u,+)(s)l< N for i = 1,. . . , n , s I?. Thus, from (Hl)-(H5), we see there is a Q > 0 such that u;(t)2

-Qw(~),

t It,

217

6. Global Stabiliiy for Multi-Species Models

which leads to

3 q+(0)e-Q('-a) > 0,

a contradiction to our assumption. The last statement is an immediate result of Theorem 2.3 in Hale and Kato (1978). 0 For the sake of simplicity, in the rest of this paper, we denote, for i = 1,... , n ,

Gi(t,ut) = r i ( t )- ai(t)ui(t)-

ui(t

+

S) d p i ( t , s )

The results in this and the next sections are taken from Kuang and Smith (1992a). Our object in this section is to establish sufficient conditions for system (4.1) t o have a globally asymptotically stable solution with respect to admissible initial functions. The next lemma presents sufficient conditions for the solution u(o, $ ) ( t ) of (4.1) to exist for all t 2 6. Lemma 4.2. In addition t o (Hl)-(H5), a s s u m e f u r t h e r that (i) lim~~pl,,[r;(t)/~;(t)] < +OO, i = 1,2,. . . ,n; (ii) there i s a 7 E (O,I), sach that rai(t) >

n

C pji(t), j=1

tL

6-

T h e n u(o,$ ) ( t )esists a n d is bounded for t 2 6. Proof. From (i), we see there is an M > 0 such that

i = 1,2,. . . ,n,

r;(t)[(l - r ) a ; ( t ) ] - l < M , Let

t 2 6.

u = max{M, Il$illm : i = 1,. . . ,n}.

(4.4) We claim that U i ( Y , t ) ( t ) 5 ii for t 2 6. Otherwise, there is an 1 E {1,2,. . . ,n } and a t, t 2 o,such that ul(o,$)(t) = U , hi(o,$)(t) 2 0, and q ( a ,$)(s) 5 ii for s 5 2. From (4.1), we have

+ j=1 c J_

h i ( t > 5 b / ( u l ) [ r / ( t-) a i ( t ) u / ( t )

Thus.

"

0 00

uj(t

+

dpjl(t,s)~-

218

Delay Differeniial Equations

0

a contradiction. This proves the lemma. We consider first the following simpler system: ri(t) - ai(t)ui(t)

Assume there exists u* = ( u i ,u;, u; 2 0 and, for t 2 u,

For this u * , we define, for ur

r t ( t ) = ai(t)uf -

cJ "

j=1

+ j=1 cJ

0

--M

uj(t

+

3)

dpji(t, s)

... , u i ) such that, for i = 1,2,. .. ,n,

> 0,

0

-m

"

+c "

u; d p j i ( t , s )

j=1

1 0

00

u; dvji(t, s),

t 2 0; (4.7)

and if ur = 0, we denote rT(t) = ri(t). Clearly, r;(t) 5 r f ( t ) , t 2 u. Proposition 4.1. I n addition to (Hl)-(H5), assume further that (H6) there i s a 0 < 70 c 1 such that, for i = 1,2, ... ,n and t 2 u,

and where

a ( t ) = min{a;(t) : i = 1 , 2 , . . . ,n}.

(4.10)

6. Global Stability for Multi-Species Models

219

corresponding to an admisJible function 4. T h e n lim u i ( t ) = u r ,

t++m

i = 1,2,. . . ,n.

Proof. By substituting (4.8) into (4.10) and letting iii(t) = ui(t) - u r , i = 1,. . . ,n, we obtain

(4.11)

If

ur

> 0, then

the equality holds in (4.11).

Let

V ( i i ) ( t= ) max{ii,2 ( t ) : i = 1,2,. . . ,n, t 2 a } , 114 - u*llm = max(ll4i - urIlm : i = 1,2,. .. ,n}. We prove first that, for t 2 a ,

V ( WI 114 - u*llk.

(4.12)

Otherwise, there is a 2 2 a such that

V(G)(S)= 114 - u*Iloo, and V(ii)(t)2 0, where V ( i i ) always denotes the upper right hand side derivative. We let N ( t ) = {i E ( 1 , ... , n ) : iiP(2) = V ( i i ) ( t ) } . Then, for any i E N ( t ) , we have (regardless of whether ur = 0 or ur

> 0)

220

Delay Diffemntial Equations

< 0.

(4.13)

Since (4.13) is true for all i E N ( j ) , we arrive at a contradiction. This proves (4.12). Denote w 2 = limsupV(ii)(t). t++w

> 0, then there is an 6 > 0 such that, for y = yo + c, 2, - € > y1'2(v + €). By the definition of v, we see there is a tl > such that, for t 2 t l , If v

Clearly, there is a

t2

V(ii)(t) I(v > 0 such that, for s g-'(s)

*

+

(T

€)2.

5 -t2, [Id- u*IIm < v

+

6.

+ t2 such that, for t 2 T , V ( i i ) ( t ) < 0. + t2, such that

We claim that there exists T 2 t l Otherwise, we see there is a t, 1> t l

V ( u ) ( t )2 0. V(ii)(t) 2 (v - q, Again, we define N ( 1 ) as before. Then, for any i E N @ ) , we have

6. Global Stability for Multi-Species Models

221

This is a contradiction. Thus, we have, for t V ( i i ) ( t )< 0,

and

Hence, there is TI > T such that, for t

(v

> T,

lim V (i i )(t )= v2.

t-+m

2 TI,

+ € ) 2 2 V(ii)(t)2 v2.

Define

N* = {i : for any t > 0, there is a 2 > t , such that i E N ( t ) } . Then N* is not empty. Let 1 E N * . If (i) u; = 0, then for any > TI such that 1 E N(Z), we have

+);.

bi(W

= bi(@))

2

bi(v);

(ii) u; > 0, we claim that there is a S ( 1 ) > 0 such that, for any t such that 1 E N(Z), bi(ui(1) u ; ) 2 b(1).

> TI

+

Otherwise, there is a t i , ti > T I , 1 E N ( t i ) , iii(t1) < 0. Since u(t1) = u; > 0, we see iir(t1) > -u;. Thus, for Z > t i such that 1 E N ( t ) , we have (since V ( i i ) ( t )< 0, for t > T ) lii,(t)l< liii(t,)l, which implies that iil(t1)

+

./(I) hence, bl(ui(t)

+ u; > i i i ( t i ) +

21;

A + u ; ) > b i ( i i i ( t i ) + u;)=6(1) > 0.

The preceding argument indicates that there is a 6 > 0 and a T2 such that, for all t > T2 and all i E N ( t ) ,

> TI

+

b;(ii;(t) ur) > 6. Now, for any t 2 T2, i E N ( t ) , by repeating the previous argument, we have zd( G j2( t ) ) l , = i

< 2 b i ( ~ i ( t+) ur)(u + ~ ) ~ [ - + ( rE o) . i ( t )

<-24v

+ €)2.j(Z).

222

Delay Differential Equations

+

Therefore, we have proved that, for t 2 T2, V ( i i ) ( t ) < - 2 4 v ~)~g(t). By (H6), we see that V ( i i ) ( t )+ as t 4 +cq a contradiction. This 0 implies v must be zero, proving the proposition. In the following, we assume that there exist ii = ( i i l , . ,i i n ) such that, for i = 1,. .. , n , iii 2 0, and, for t 2 n, -KJ

..

(4.15)

Proposition 4.2. Assume the same assumptions a3 in Proposition 4.1 for (4.1). Let u ( t ) = u(n,#)(t)be the solution of (4.1). Then limsupt,, u ; ( t ) 5 iii. Proof. Let ii(t) be the solution of

with initial condition (4.2); then, from Proposition 4.1, we see lim i i i ( t ) = 4;.

(4.16)

t-.+oo

Assume i i i ( t ) = u i ( t ) and i i i ( s ) 2 obtain

U,(S)

for s I t. From (4.1), we

As the right hand side of the equation satisfied by ii(t) is a quasimonotone function, a standard comparison argument similar to Proposition 1.1 in Smith (1987) gives u ( n , $ ) ( t )I ii(n,+)(t) = ii(t)

for t 2 n.

This, together with (4.16), implies the proposition. 0 Now, we are ready to state and prove the main theorem in this section. The following assumptions are required: (A) There is a nonnegative vector ii = ( i i l , i i 2 , ... ,Gin) satisfying (4.15).

6. Global Siabiliiy for Multi-Species Models

223

(B) There is a 4 = (41, ... ,fin) such that, for i = l , . . . ,n, 4i

G;(t,C)5 0,

2 0,

and

QiGi(t,ir)= 0.

Theorem 4.1. In system (4.1), assume (Hl)-(H6), (A), and (B) hold, and limsup max t++oo

n

Tibi(ili)

j=l

llcln

I"

+ p i ( t ) + C ( p j i ( t ) + ~ j i ( t ) )= p

< 1. (4.17)

Then the solution u ( t ) = u(o,q5)(t) of (4.1) tends to ir as t + +w. Proof. From Proposition 4.2, we see that lim sup u , ( t ) I4,. t+m

Define ~ ( u ) ( t=> max{(u;

- f i ; 1 2 : i = 1 , 2 , . . . ,n},

v2 = limsup ~ ( u ) ( t ) . t-oo

From (4.17), we see that there is an

i=1, Thus, for

t 2 Tl,

~ ( € 1 )<

Assuming v

€1

> 0 such that, for 0 5 7 5 € 1 ,

...,n}) (

< 1, there exists

a TI = T I ( ( )> u such that, for

> 0, then there is an 6 > 0 such that, for 7 = 70 + C, v - c > max{y'/2(v

+ c>,((v + c)}.

(4.18)

Clearly, assumption (B) implies the assumption (i) in Lemma 4.2, and (H6) implies the assumption (ii) in the same lemma. Therefore, we can define

224

D e l a y Differeniial E q u a t i o n s

U as in (4.4), which satisfies ii 2 Q i , i = 1 , 2,... ,n. Thus, i = 1,2, ...n and t 2 u. Let u1 > 0 be so large that

+

I

u/g(-a1)

and, for t 2 u1,

ui(t)

5 ii for

6

+

V(u)(tI ) (v €)2. Since U 2 i i i 2 0 , we have V ( u ) ( t )I U 2 for all t 2 u. Let u2 = m a x { 2 u ~ , u+ ~ T,T~ + T } , where T = maxl
- C i ) 2 = V(u)('i)2 (v - € ) 2 ,

then

8 E [-Ti,O]. - C i ) ( U i ( T + 8 ) - C i ) 2 0, Otherwise, there is a 8 E [ - q , O ] such that u i ( t + 8) = i i i . Thus, (Ui(2)

L I(. + E ) , where t* E (1,t+ 8). Thus, -6 I

I(v

+

€),

which contradicts (4.18). This proves our claim. Again, arguing as in the proof of Proposition 4.1, we claim that there exists T 2 2 u 2 such that, for t 2 T 2 , V ( u ) ( t ) < 0 . Otherwise, there must be a t 2 u2 such that

V(u)(T)2 (v - €) 2 ,

V ( u ) ( t ) 2 0.

225

6. Global Stability for Multi-Species Models

Then, by combining the conclusion we have just proved (to show that the new term J!ri ui(t s) d p ; ( t ,s) appearing in (4.14) is nonpositive) and a similar argument as the proof of the same claim in Proposition 4.1, we can easily obtain a contradiction, which proves the claim. The rest of the proof is very similar to the proof of Proposition 4.1. We omit the details here to avoid repetition. Our last theorem in this section is essentially the same as Theorem 4.1, although it appears to be more general. Theorem 4.2. I n system (4.1), i n addition to (Hl)-(H5), assume further that (i) there i s a 0 < 70 < 1 and 6; > 0, i = 1,2, ... ,n, such that

+

and

(ii) there is a nonnegative vector 4 = (41,.

a i = 1,2, ... , n } ) =

p

. . ,an) such that

< 1;

(iv) there is a nonnegative vector f = (fl,... , f n ) such that, for

i = 1,2,... , n ,

C; 1 0,

G;(t,6)

5 0,

and

f ; G ; ( t6) , = 0.

Then, the solution u ( t ) = u(0, $ ) ( t ) of (4.1) tends to f as t Proof. By letting w:(t) = S;u:(t),

4 00.

one can transform (4.1) and (4.2) to a system in terms of w(t) = (wl(t), . . . ,w n ( t ) ) . It is easy to see the resulting system satisfies all conditions of Theorem 4.1. Thus, the global stability of w(t) follows, which, in turn, implies the conclusion of our theorem. 0

Delay Diffemntial Equations

226

via R a z u m i k h i n - T y p e T h e o r e m s - A p p l i c a t i o n s In this section, we focus our attention on the autonomous system

6.5. S t a b i l i t y

iii(t) = bi(ui)[ri - a t' u t' -

J_, ui(t +

+

dpji(3) -

0

+urn uj(t

J=1

S)

S ) +i(S)

cJ "

j=1

0

uj(t

--M

+

8)

-

dvji(S)]

(54

For this system, we always assume that, for i = 1 , 2 , . .. ,n, ( A l ) bi(0) = 0, bi(ui) > 0 , and Ti 2 0; (A2) pi(S), P j i ( S ) and v j i ( S ) are nondecreasing for s 5 0, and A

< 00, vji = vji(0) - ~ j i ( - m < ) 00;

pji=pji(O) - p j i ( - m ) A

+

(A3) ai > Cin,l(pji vji). For convenience, we will denote p i = p i ( 0 ) - pi(-Ti). The following lemma is essentially well known. A similar result is given in Haddock e t al. (1989). The argument there can easily be adjusted to prove the present one. Therefore, we omit the proof. Lemma 5.1. I n system (5.1), there exists a 0 < 7 < 1 and a g ( s ) satisfying (g1)-(g3) in Section 2.7 such that, f o r i = 1 , 2 , . . . ,n,

Thus, the assumptions (H5) and (H6) for system (4.1) automatically hold for system (5.1). The next lemma can be found in Berman and Plemmons (1979), p. 274. Lemma 5.2. Suppose the matrix C = ( C i j ) n x n has positive diagonal element8 and is strictly diagonally dominant. Then, for any r E Rn, the linear complementarity problem of finding p E Rn such that

pLO,

C p - r 2 0,

and

[Cp-r]p=O

(5.2)

has a unique solution. We note that Hofbauer and Sigmund (1988) call p a saturated equilibrium for the Lotka-Volterra system n

6. Global Siabiliiy for Multi-Species Models

227

Therefore, Lemma 5.2 indicates that (5.1) always has a unique saturated equilibrium. We will adopt this name for the unique equilibrium of (5.1) that satisfies (5.2). The following theorem gives conditions for such an equilibrium to be globally asymptotically stable. T h e o r e m 5.1. Let U * = (ur,. . . ,u:) be the saturated equilibrium of (5.1) and p = ( P I , . .. ,pn) the saturated equilibrium of

In addition t o (A1)-(A3), a s ~ u m efurther that n

+ + C(/iji + ~ j i ):]i = 1,2,. ..

max{Tibi(pi)[ai pi

,TI}

< 1.

j=1

T h e n every solution of (5.1) corresponding t o a n admissible initial funct i o n tends t o u* a3 t + +m. Proof. This is an immediate consequence of Theorem 4.1 and Lemmas 5.1 and 5.2. 0 The autonomous version of Theorem 4.2 takes the following form. T h e o r e m 5.2. In system (5.1), assume (Al)-(A2), and (i) there esists 6; > 0, i = 1,2,. .. ,n, such that n

ai

> SF' C Gj(pji + Vji). j=1

(ii) For u* and p be defined as in Theorem 5.1,

T h e n , every solution u ( t ) of (5.1) corresponding t o admissible initial function tends to u* as t + +co. In the following, we restrict our attention to scalar equations. We consider

228

Delay Differenfial Equafions

where T O , a, ao, a l , a2, and TO are positive constants, 0 5 71, 72 5 +w. b(0) = 0, b'(u) > 0,p i ( s ) , i = 0,1,2, are nondecreasing and satisfy

Li 0

i = 0,1,2.

dp;(s) = 1,

For simplicity, we denote u* = r o / ( a

+ + a0

-a),

a2

uo = ro/(Q- Ql),

and

T

= max{TO,q,T2}.

The initial condition for (5.3) is admissible, if u(8) = 4(8)

2 0,

4(0)

> 0,

8 E [-T,O],

and

4 is continuous.

The following result is a simple consequence of Theorem 5.1. Theorem 5.3. In (5.3) assume a > a1 a2, and

+

Then every solution u ( t ) of (5.3) corresponding to admissible initial function tends to u* as t + +w. Theorem 5.3 generalizes Theorem 1 in Miller (1966) in two ways: (i) we allow discrete delays; (ii) we don't require the undelayed term to dominate the other terms. Miller (1966) also.studied the scalar equation

k(t)= N ( t ) [ a- b N ( t ) -

/ f(t t

0

- s ) N ( s )d s ] ,

with respect to initial condition N ( 0 ) = NO > 0, where a > 0, 6 > 0, f(t) E C[O,w) n L1[0, w). Assuming

Miller (1966), Theorem 2, proved that lim N ( t ) = N * = a[b

t+m

+/

m

0

f(s) ds]-'.

229

6. Global Siabiliiy for Mulii-Species Models

In the following, we consider

(5-4) where the parameters and functions appearing in (5.4) satisfy (A1)-(A3). We assume

The initial condition of (5.4) takes the form u;(O) = u;o

> 0,

i = 1 , 2 , . . .n.

One can view (5.1) as the limiting equation, as t --t 00, of (5.4). The following theorem improves Theorem 2 in Miller (1966). Theorem 5.4. Let U* = (ti; ,... , u i ) and p = ( P I , ... ,pn) be the saturated equilibria of (5.1) and (5.4), respectively. Assume system (5.4) satisfies all the hypotheses of Theorem 5.1. Then, every solution u ( t ) of (5.4) corresponding to positive initial data tends to u* as t --t +00. Sketch of the Proof. We can show first that u ( t ) is bounded. In fact, we can choose 0 < y < 1 , g ( s ) satisfying (g1)-(g3) of Section 2.7 such that, for i = 1 , 2,... ,n,

then, we can show that

By a standard comparison argument, one can show that the solution of

Delay Differential Equations

230

dominates the solution of (5.4), where both have the same initial function. By essentially the same argument as given in the proof of Theorem 4.1, one can show the solution of (5.5) tends to p as t + +oo. The only tricky step needed here is to note that

= -ai(ui - pi)

Thus, we can show that

By applying the same trick as (5.6) to system (5.4) and making a similar argument as the proof of Theorem 5.3, one can easily arrive at the 0 conclusion of the theorem. The following theorem is obvious: Theorem 5.5. Assume all the hypotheses of Theorem 5.2 are satisfied. Then every solution u ( t ) of (5.4) corresponding to positive initial data tends to u* as t + +oo. It should be mentioned here that we can allow system (5.4) to have variable coefficients. In this case, a result similar to Theorem 4.2 can be established. By a lengthy argument, Gopalsamy (1980a) was able to obtain the following global stability result for a well-known Lotka-Volterra type two species competition model: Theorem A. Assume a l , ag, 71, 72, b l , bz are positive constants satisfying ala,' > 717;' > bib,'. (5-7) Let K 1 ( . ) and 1(2(-) be nonnegative continuous functions defined o n [-T,O] (T i s any positive constant) such that

6. Global Stability for Mulli-Species Models

I

231

T h e n the soIution [ u ( t ) , v ( t ) of ]

h ( t ) = u[rl - a l u - bl

; ( t ) = 4 7 2 - u2

1

44 = h ( t ) L 0,

0

-T

1, 0

I(1(3)V(t

I(2(S)U(t

+ S) dS],

+ s)ds - b w ] ,

t 2 0,

+ l ( O ) > 0 , v(t) = & ( t ) 1 0 , 42(0) > 0 , t E [-T, 01,

satisfies [ u ( t ) , v ( t ) ]+ (u*,v*)

as t + +w,

where

In our next theorem, we will see that this result can be easily extended to the following more general two species competition system:

.

.

where all the parameters are positive constants. It is assumed that p i , k i , i = 1,2, are nondecreasing and satisfy

Li 0

0

dp:(.) = 1 = L r n d k i ( S ) ,

2

= 1,2.

Moreover, we assume, for i = 1,2,

bl(0) = 0,

b’,(*)> 0.

Theorem 5.6. In (5.8), assume (5.7) and (9 max{Tlbl ( ~ ) ( Q . + 2 a 1 ) , 7 2 b z ( ~ ) ( P + b z + ~ b
as

t -+ +w,

(5.9)

232

Delay Differential Equations

Proof. Clearly, (Al)-(A2) hold in (5.8). Since ala,' > blb;', we see b2 > a z b l / a l . Thus, there is an € 1 > 0 such that, for 0 < 6 5 €1,

Let

61 = ( b i

+€)/ail

62

(5.10)

= 1.

Then,

and

(5.11) That is, the assumption (i) in Theorem 5.2 is satisfied. Clearly, we have limsupu(t) 5 y l / a l , t++w

From (5.9), we see there is a 0

and

limsupv(t) t++w

5 72/b2.

< €2 5 € 1 such that, for 0 < 6 5 €2,

< 1. However, (5.11) is precisely the assumption (ii) of Theorem 5.2 if we choose 6 2 as in (5.10) with 0 < c 5 € 2 . It is easy to verify that (u*,v*) is the unique positive equilibrium of (5.8). Now, the conclusion follows Theorem 5.2. 0

61,

Theorem 5.6 improves Theorem A in three ways: (i) We do not require the delays to be bounded; (ii) We allow both discrete and distributed delays; (iii) Most importantly, we can tolerate large coefficients a and p, so long as 71 and 72 are small enough. We would like to mention that results similar to the preceding theorem can be established for general infinite delay Lotka-Volterra type population models involving mixed interactions, such as predator-prey, competition, and cooperation.

6. Global Stability for Multi-Species Models

233

Generally speaking, the application of our main results (Theorems 4.1, 4.2, 5.1, and 5.2) may involve some complicated computations. In the case of Theorems 4.2 and 5.2, some additional guesswork may be required. Nevertheless, the application of Theorem 5.1 is quite routine. Consider the following autonomous system:

where (Al)-(A2) are assumed to hold. In order to apply Theorem 5.1, the following algorithm should be used: Step 1: Check whether or not n

n

j=1

j=1 j#i

If the answer is no, stop here. Otherwise, continue to Step 2. Step 2: First find the saturated equilibrium p = ( P I , . . . , p n ) of

then, find a set of positive numbers

i = 1,2,... , n ,

T,,

i = 1 , 2,... , n , such that, for

Step 3: Check whether or not

If the answer is no, stop here. Otherwise, the conclusion of Theorem 5.1 is valid for system (5.12). A “no” answer to Step 1 or Step 3 implies that Theorem 5.1 does not apply to system (5.12). In these cases, one may want to try Theorem 5.2. In applying Theorem 5.2, one needs to do the additional guesswork of finding positive constants S;, i = 1 , 2 , . . . , n . If one can find such S;,

234

Delay Differential Equations

such that (i) and (ii) of Theorem 5.2 are satisfied for system (5.12), then the conclusion of Theorem 5.2 applies. Otherwise, the task of applying Theorem 5.2 becomes very much involved. It should be mentioned here that the method can be extended to cover more general systems. For instance, we may consider

tii(t) = bi(Ui)Fi(t,ut),

.

u = ( ~ 1 , .. ,u"),

i = 1,. . . ,TL,

where Fi(t,ut) may not be linear but is globally Lipschitz. The bounded delay ri in Eq. (4.3) can be replaced by a positive continuous function q ( t ) . 6.6. When Nondelayed Diagonal Terms Do Not Exist Most of the convergence results appearing so far for delayed LotkaVolterra-type systems require that undelayed negative feedback dominate both delayed feedback and interspecific interactions. Such a requirement is rarely met in real systems. In this section, we present convergence criteria for systems without instantaneous feedback. Roughly, our results suggest that, in a Lotka-Volterra-type system, if some of the delays are small, and initial functions are small and smooth, then the convergence of its positive steady state follows that of the undelayed system or the corresponding system whose instantaneous negative feedback dominates. In particular, we establish explicit expressions for allowable delay lengths for such convergence to sustain. We study first a class of Lotka-Volterra-type infinite delay systems, where delays for the negative feedback are expected to be small. If the system is diagonally dominant (to be defined in what follows), and initial functions are relatively small and smooth in the short past, then we can show that the unique saturated equilibrium or steady state attracts such kind of neighboring solutions. Similar results are found to be true for delayed Volterra-Liapunov stable Lotka-Volterra-type systems. Our approach involves constructing suitable Liapunov-Razumikhin functions, carefully selecting initial function sets, and estimating the length of relevant delays. We consider first the following general autonomous Lotka-Volterratype infinite delay system:

where

6. Global Stability for Multi-Species Models

235

( u i ( t ) , u 2 ( t )., . . , u n ( t ) ) . Throughout the rest of this section, we u(t) assume that, for i = 1,. . . ,n, ( H l ) bi(0) = 0, b i ( . ) is continuously differentiable, and bi(.) > 0; (H2) r i ,a i , and Ti are constants; in particular, a, and T , are positive; (H3) p i ( @ )are nondecreasing, pi(0) - pi(m-7,) = 1; (H4) p i j ( 8 ) are bounded real valued Bore1 measures on (-oo,O] with total variation Ipijl. Note that (6.1)-(6.2) is a special case of (4.1). As usual, we assume the initial conditions satisfy (H5) uo(s) = +(s) 2 0 for s 5 0; E BC, +(O) > 0. In this section, we choose the norm I I in R" as

+

I+(s)I

= max{l4i(s)l : i = 1 , . . . ,n},

where +(s) = ( +1(s),. . . ,&(s)) in Section 2.7, we have

ll+ll,

E

R". Thus, for g ( s ) satisfying (g1)-(g3)

= supmax{ -: 2 = l , s
+

1

..., n ,

where E UC,. We say system (6.1) is diagonally dominant if

It is well known that (see Berman and Plemmons (1979)) system (6.1) has a unique saturated equilibrium if it is diagonally dominant. It is also easy to see that if (6.1) is diagonally dominant, then there is a g ( s ) satisfying (gl)-(g3) such that ai > C;=, l p i j g l , where l p i j g l .Pmg(s)Idpij(@)l, i = 1 , . . . , n . For simplicity, we denote T

= max{.ri : i = I , . . . ,n } .

(6.4)

Throughout this section, we assume that system (6.1) is diagonally dominant, and denote g = g ( s ) as a function that satisfies (g1)-(g3), a, > C;=1 I p i j g l , i = 1,..., n , and g(s) = 1 for s E [-T,O]. Clearly, such a g ( s ) always exists as long as (6.1) is diagonally dominant. Also, we always denote p as the unique saturated equilibrium of (6.1). The material of this section is adapted from Kuang and Smith (1992b). Lemma 6.1. In addition to (Hl)-(H5), assume that system (6.1) i s diagonally dominant. Assume further that

Delay Diffemntial Equations

236

=

+

+

u,(+)(t), u;(t 0) = ui($)(t 8), i = 1, ...,n. Clearly, Zi'implies that ui 5 K + p ; and Iui(t+O)-pi1/g(8) 5 K for i = 1,. . . , T I , 0 I 0. Thus, where u;

I l ~ t ( $ ) ( . > - p 1 15 ~

Note that we have used the assumption that g(8) = 1 for 8 E [-T,O]. This shows that Ihi(4)(t)l di(K)Zi' follows from Ilut(q5)(.)-pllg 5 Ii' fort 1 0. Assume that there is a t > 0 such that llul(+)(.) - p1Ig = Ii' and l l u t ( 4 ) ( - )- p1Ig < ( I for t E [O,i). Then, the definition of 11 [Ig together with the fact that 114 - pllg c Ii' implies that there is an i E (1,. . . ,n} such that I l u j ( + )(-)- pllg = Iui(t) - pi[. The previous arguments ensure that Ih;(t)l 5 d;(Ii')Ii' for t E [-.,I]. Denote

-

~ ( t=)(ui(t) - pi)2.

6. Global Stability for Multi-Species Models

237

(6.5)

It + e,?] is determined by mean value theorem.

where ( = ((0) E then for 4 E [-t,8],

Iu,(t

+ e ) - uj(t)l 5 d,(Zt')Zie

If

t < T,,

< di(Zt')KTi,

and, for 8 E [-q, -t], Iui(t

+ 0) - u;(t)l I Iui(t + 4)- ui(O)l+ Iui(O)- ui(f)l 5 di(Zi)Zt'lt

Hence, in all these cases,

Therefore,

+ 81 + di(Zi)Zt't

= d i ( K ) Z i e 5 di(Zt')KTi.

238

Delay Diffemniial Equaiions

which contradicts (6.5). This implies that no such t exists and, hence, for 0 t L 0, Il.t(4)(.) - Pllg I K. Now we are ready to state the main result of this section. Theorem 6.1. Assume that all the assumptions of Lemma 6.1 are satisfied. Then lim u ( + ) ( t ) = p. t++m

Proof. Let a E ( 0 , l ) be a constant such that ai(a - di(K)Ti) >

n

C lpijg1-

j=1

We claim that there is a TI > T such that, fort 2 2'1, ~ ~ ~ ~ ( q 5 ) ( ~aK. ) - p ~ ~ ~ ~ < There are two possibilities: (i) For any large T , there is a t > T , i E { 1 , . . . ,n}, such that

K2 2

K ( i ) > (Y21i2,

k(i)2 0.

(6.7)

However, a similar argument as the proof of Lemma 6.1 yields

which contradicts (6.7). (ii) There is an i E ( 1 , . . . , n } such that G ( t ) < 0, and Iim ~ ( t=) a 2 ~ { 2 . t++m

In this case, a similar argument as in case (i) implies that

e(t)5 2 a b i ( O l ~ ) [ - u i ( a- a i ( z i ) T i ) +

c n

Ipijsl]K2 < 0,

j=l

which leads t o limt++mK(t) = -00, a contradiction. This proves the claim. By Lemma 6.1, we know that, for t 0, i = 1 , . . . ,n, J u ; ( t )- pi1 5 11'. Since 4 E BC and lims+-mg(s) = +m, we see that there is a S1 > 0 such that (114 - pllm + 1 0 < al(. g(--S1)

6. Global Stability for Multi-Species Models

Thus, for t 21'7

239

+ S1 = ul,

Observe that d , ( K ) is strictly increasing with respect to K. If we replace K by a K in Lemma 6.1, we see that all its assumptions are satisfied. Clearly, n

ai(a

- di(aK)Ti) > a i ( a - d i ( K ) T i ) > C

Ipijgl.

j=l

Hence, we can repeat the previous argument and conclude that there is a u2 > u1 such that, for t 2 u2,

By repeating such an argument again and again, we obtain a sequence < u2 < . * . < u i < u;+l** * , limi++, a, = +oo, such that, for t 2 a;,

u1

This clearly implies that

limt++cc .(W)= P. 0 Equivalently, we can state the preceding theorem as follows: Theorem 6 . 2 . In addition to (Hl)-(H5), assume that system (6.1) is diagonally dominant. Denote, for i = 1 , . . . ,n,ZC > 0,

Assume further that, for some K > 0, 7, 5 T i ( K ) , i = 1,...,n, and satisfies the assumptions (ii) and (iii) of Lemma 6.1. Then

4

lim u(+)(t) = p.

t++m

Proof. Note that ~i 5 T , ( K )is equivalent to (i) in Lemma 6.1. The rest 0 is trivial. An immediate consequence of the preceding theorem is the following corollary. Corollary 6.1. In addition to the assumptions of Theorem 6.2, assume further that limsupt,+, \u(q5)(t)l < K. Then Ti I Ti(K) implies that limt-+, .($)(t) = p.

Delay Differeniiol Equaiions

240

The more general version of Theorem 6.2 takes the following form: Theorem 6.3. In addition to (Hl)-(H5), assume that in (6.1) there ezist 6 ; > O , i = 1 , 2 ,... ( 1 2 , s u c h t h a t a;

> 6;

c n

6;yp;jl.

j=l

Assume further that there is a I(

> 0 such that r; 5 r;(K),where

i = 1 , 2 , . . . ,n, and 4 satisfies the assumptions (ii) and (iii) of Lemma 6.1. Then limt++oo u ( $ ) ( t ) = p. Proof. If we denote U;(t) = 6;u;(t), then U ( t ) = (U,(t), , . . , U n ( t ) ) satisfies i = 1,. . . ,TI, (6.10) ~ ; ‘ (= t ) a;b;(6;’Ui(t))~i(U~i(.)), where

then (6.10) and (6.11) are identical to (6.1) and (6.2) (except the bars). The condition (6.9) thus implies that (6.10) is diagonally dominant. The rest follows Theorem 6.2. 0 Remark 6.1. If we replace a;,!J u,(t 0) d p i ( 0 ) by aju;(t) in the system (6.1), then the assumption of diagonal dominance implies that all solutions of (6.1) tend to the unique saturated equilibrium p. Equivalently, this means that p is globally asymptotically stable with respect to admissible initial functions. Our results indicate that this global asymptotical stability of p is maintained provided that (i) initial functions are bounded (in BC norm), and the derivatives of their ith component are also bounded properly in the interval [-q, 01; and (ii) r; are small enough. In view of the fact that many real systems are studied with bounded initial functions with bounded initial derivatives, and the time delays ~i are usually regarded as small, our results suggest that when the system is diagonally dominant, instantaneous negative feedbacks ( a ; u ; ( t ) )are suitable approximations of delayed negative feedbacks (ai J!ri ui(t 0) d p i ( 0 ) ) . It should be pointed out that the infinite delays appearing in (6.2) do not create any real

+

+

6. Global Stability for Multi-Species Models

24 1

difficulties in our analysis; therefore, restricting them to a finite delay case will not provide any new results from our method. If a uniform bound can be found for solutions of (6.1), then Corollary 6.1 asserts that p is globally asymptotically stable as long as the Ti are small enough. If the initial functions are not differentiable, then one can replace q5 by u~(q5)for some T > max{r, : i = 1,. .. ,n}. This way the initial function becomes differentiable for s 2 -r,. The resulting estimate may change, though. Our estimate of the size of delay is not optimum even in scalar cases. We consider now a finite delay version of system (6.1):

[

hi(t>= bi(ui(t)) ri - ai

Li 0

ui(t

+ 6) dpi(e) + f:Jo u j ( t + 4) dpij(e)], j=1 -rij

(6.12) where rij > 0, i , j = 1,. .. ,n. (H4) thus reduces to (H4)' p ; j ( S ) are bounded real valued Bore1 measures on [-q,01 with total variation IpijI. Denote T = max{q,qj : i,j = 1,. . . , n } . Clearly, (H5) should be replaced by (H5)' ~ o ( s = ) 4(s), s E [-.,O], 4 E BC([-T, 01, R"),4(0) > 0. Our objective in the following is to establish convergence criteria for system (6.12) when it is not diagonally dominant. For notational convenience, we define dpij(O),

i , j = 1,. . . ,n,

D=diag(dl,dz ,...,d"),

d;>0,

i = l , ...,n.

For such a positive diagonal matrix D and a nonnegative steady state p = (PI,. . . ,pn) of (6.12), we define

where R; = ( ( ~ 1 , .. . ,un) : ui 2 0, i = 1,. . . , n } and Int R; denotes the interior of R;. It is easy to show that VD,,(U) > Vo,,(p)= 0 if u # p and u E Int R;. In the rest of this section, we assume that p is the unique positive steady state of system (6.12). For any number VO > 0, we denote

A(V0)= { u : u E R;,VD,,(U) = Vo}.

242

Delay Diffemniial Equaiions

In this section, we adopt the standard Rn norm; if u = ( u i ,. .. ,un) E R", then 1161 = (ELru?)''~.Since limlUI+mVD,~(U) = +oo, we see that A ( b ) is compact. We can thus define

It is easy to see that p(V0) is strictly increasing with respect to VOand 4 0. It should be pointed out here that p ( b ) also p ( b ) + 1 as depends on D and p. Denote A = ( a i j ) n x n , where (1.. II

- -a. I + pi,; -

and a,, = p i ,

when i # j, i, j = 1,. . .,n.

We call system (6.12) VL-stable (Volterra-Liapunov stable [see Hofbauer and Sigmund (1988)]) if there is a positive diagonal matrix D = diag ( d l , . . . ,d,) such that D A + ATD is negative definite. Finally, we define, for any positive numbers M , N ,

F ( M , N ) = { d : d E c'([--7,ol,R;), IId(*)- PI1 c M , lldf(.)ll < N ) , where II-II is the uniform norm (with respect to 1.1 in R")in C([--7,O],R;). We are now ready to state and prove our main result of this section. Theorem 6.4. In addition to (Hl)-(H3) and (H4)', assume that (6.12) is VL-stable. Then for any positive number M , there i s an N = N ( M ) and T ( M ) such that -7 < T ( M ) and E F ( M , N ) implies that limt4+m u(r$)(t)= p, where p as the unique positive steady state of (6.12). Proof. Since (6.12) is VL-stable, there is a D = diag(d1,. . . , d n ) such that E = D A ATD is negative definite. Thus, there is a A > 0 such that, for any u = ( u , .. . ,u n ) ,

+

uEuT

< -Xu. uT = - A

n

C uz. i=l

Denote, for u ( t ) E Int R;,

W(U(t)= ) maxe€[-r,o] V ( 4 t + 6 ) ) . The derivative of V ( u ) ( t )along a solution u ( t )of (6.12) takes the form n

V ( u ) ( t )=

C di(ui(t)- pi)Qi(ut(*)),

i=l

6. Global Siabiliiy f or Mulli-Species Models

243

where

Note that

Denote Vo = % ( M ) = max{V(u) : Ju- pJ 5 M } , u = u ( M ) = max{lu - pI : V ( u )= Vo}, q = q ( M ) = max{bi(p;

+ ii)(ai + 2 Ipjjl) j=l

: i = 1,2,.

.. , n } ,

then Iu($)(t) - pJ < ii, t 2 0. Otherwise, there is a t* > 0 such that v0 = V ( u ( t * )= ) W ( u ( t * )and ) I/i/(u(t*))2 0. We note from the definition of p(V0) that p(&)-’. 5 [up’)- p l 5 u.

244

Delay Diffemniial Equalions

i.e., k ( u ( t * ) )< 0, a desired contradiction. This proves the claim. Denote

Q

= Q($) = limsup Iu($)(t) - PI, t++w

$ E F(M,N).

If the theorem is false, then, for some 4 E F ( M ,N ) , C following that this is the case. Denote

> 0.

Assume in the

V* = limsup V ( u ( $ ) ( t ) ) , t++w

u* = max{lu

- PI : V ( u )= V * } ,

u* = min{lu

- PI : V ( u )= V * } .

The proof of the previous claim clearly implies that V* 5 V,. And it is trivial to see that u* 2 C 2 U, > 0. Since - i X p ( h ) - ' T q cE1 d i ( a i Cj"=,p i j ) < 0, there is an €0 = q(Q)> 0 such that 1

-'XP(VO 2

+ C O ) - ' ( Q - + [ T q C di(ai + C n

n

i=l

j=1

€0)

pij)](C

+

+

+

< 0. (6.13)

€0)

By the definition of Q, we see that there is a t l > 0 such that, for I min{ii,Q €0). Since u+ = p ( v * ) - l u * , we can choose a small positive € 1 such that

+

t 2 t i , I p ( $ ) ( t ) - pi u* - € 1

> p ( v * ) - ' ( u * - €0) 2 p ( &

Clearly, there is a t2 = t 2 ( ~ 1 > ) tl l u ( N t 2 )- PI

+

T

> u* - € 1 ,

+ € O ) - l ( C - €0).

such that

V ( u ( 4 ) ( t 2 )L ) 0.

(6.14)

245

6. Global Siability for Multi-Species Models

Note that, for t 2

tl

+ 7,

Now, the fact that Iu(q5)(t2)- p ( > u, - €1 > p ( h with (6.13) imply that W q 5 ) ( t 2 ) ) < 0,

+ to)-'(& +

€0)

together

which contradicts (6.14). This indicates that & must be zero, proving the theorem. 0 Similar comments to those included in Remark 6.1 can be made for the preceding theorem. It is well known that if a nondelayed Lotka-Volterratype system is VL-stable, then its unique positive steady state (if any) is globally asymptotically stable with respect to positive initial data. Roughly, Theorem 6.4 asserts that if (i) all involved delays are small and (ii) the initial (positive) functions are small and smooth, then this positive steady state continues to attract neighboring solutions. This partially justifies that in some real life systems if delays are expected to be small, one can approximate these systems by models consisting of only ordinary differential equations. 6.7, Remarks a n d O p e n Problems We have restricted our attention on systems of the form (1.2)-(1.3) in this chapter. Some of the results may be extended to more general systems such as (1.1); others may not. The linearity of G ; ( t , u t ( - )with ) respect to

246

Delay Differential Equations

u t ( . ) played an important role in almost all of our discussions. It is thus important and interesting to discuss cases when G,(t,t i t ( . ) ) is not linear with respect to u t ( - ) . As we can see, all the global stability results of Sections 2-5 require some kind of dominance of the nondelayed intraspecific competition. This restriction is very severe and may not be necessary at all. A reason for this is that, for retarded single species models, the global stability of the positive steady state usually persists, so long as the time delays are short. The results in Section 6 gave only estimates of regions of attraction of the positive steady states. We thus propose the following important research problem. Open Problem 6.1. Obtain sufficient conditions for the global asymptotical stability of saturated steady state in system (6.1)-(6.2). The following open problem related to Theorem 4.1. It may be less complicated than Open Problem 6.1. Open Problem 6.2. In system (4.1), assume that (Hl)-(H6), (A), and (4.17) hold. Let u ( ' ) ( t ) and ~ ( ~ )be( tany ) two solutions of (4.1)-(4.2). Is it true that

One can also find some global stability results of delayed multi-species models in Gopalsamy (1992). For local stability results, see Cushing (1977) and Gopalsamy (1992).