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Positive periodic solutions for an integrodifferential model of mutualism
Applied Mathematics Letters
Applied Mathematms Letters 14 (2001) 525-530
PERGAMON
www elsewer nl/locate/aml
P o s i t i v e P e r i o d i c S o l u t i o n s for a n I n t e g r o d i f f e r e n t i a l M o d e l of M u t u a l i s m Y O N G K U N L I AND G U I T O N G X U Department of Mathematics, Yunnan Umvermty Kunmmg, Yunnan 650091, P R China (Recewed and accepted August 2000)
Commumcated by M Slemrod A b s t r a c t - - S u f f i c i e n t conditions are obtained for the existence of positive penodm solutions of the followmg mtegrodtfferentml model of mutuahsm "gl(t) + a l ( t ) f o ~ g 2 ( s ) N 2 ( t - s ) d s dNl(t) = r l ( t ) N l ( t )
al(t))
,
N 2 ( t - a2(t))
,
- Nl(t-
dt
1+
J2(s)N2(t - s)ds
K2(t) + c~2(t) d l ( s ) N l ( t - s) d s dN2(t) _ r 2 ( t ) N 2 ( t ) . --- ~ . . . . dt [ 1+/0 Jl(S)Nl(t-s)ds
where r ~ , K ~ , a ~ , a ~ , ~ = 1,2 are positive continuous w-permdm functmns, a~ > K~, z = 1,2, J~ E C([0, c~], [0, c~)), and / o J~(s) ds = 1, z = 1, 2 @ 2001 Elsewer Scmnce Ltd All rights reserved K e y w o r d s - - M u t u a h s m model, Pomtlve penodm solution, Integrodlfferentlal equation, Fredholm mapping
1. I N T R O D U C T I O N Consider t h e following lntegrodlfferentlal model of mutuallsm K1 + al - - - ~
dNl(t) _rlNl(t) dt
[
l+fo
J2(s)N2(t - s) as . . . . .
Nl(t
- al)
,
J2(s)N2(t-s)ds
(1 1) dN2(t) dt
K2 + c~2 = r2N2(t)
Jl(s)Nl(t
- s) ds
" J0// foo
1+
- N 2 ( t - 62)
dl (s)N1 (t - s) ds
]
,
where r,, K,, at are positive constants, a , > K , , , = 1, 2, J~ E C([0, c~], [0, co)) and f o J ~ ( s ) d s = 1, z = 1, 2 D e p e n d m g on the n a t u r e of K, (z = 1,2), system (1 1) can be classffied as facultatlve, Thin work is partially supported by the ABF of Yunnan Province of China 0893-9659/01/$ - see front matter (~) 2001 Elsevmr Science Ltd All rights reserved PII S0893-9659(00)00188-9
Typeset by .A,,~IS-TEX
Ll AND
Y
526
obligate, or a combmatlon System (1 1) means that
of both For more dettils of mutuahstlc mteractlons, we refer to [l-5] the mutuahstlc or cooperative effects are not realized Instantaneously
but take place with time delays and the references environment
G XU
For the further
cited therem
This motivates
Reahstlc
ecological
models
us to consider
meaning
require
of system
the mcluslon
the followmg
(1 l), we refer to [6]
of the effect of changing
nonautonomous
model
co
d%(t) -
dt
= ~1 (t)Nl (t)
=
Jz(s)N2(t - s) ds
0
J
- N(t
owJ2(s)Nz(t
1+
K2(t)
dNz (t) dt
-
J
Kl(q + al(t)
J
1+
7 I (12)
+,,(t$=Jl(spv~(t0
r2(tW2(t)
- aI
- s) ds
s)ds
-
Nz(t
mJl(s)N,(t
- s) ds
- g2(t)) !
0
In addltlon, the effects of a perlodlcally varying environment are important for evolutionary theory as the selective forces on systems m a fluctuatmg environment differ from those m a stable environment Therefore, the assumptions of perlodlclty of the parameters are a way of mcorporatmg the perlodlclty of the environment (e g , seasonal effects of weather, food supplies, mating habits, etc ), which leads us w-perlodlc functions, cy, > K,, z = Our purpose m this article 1s based positive periodic solutions of system
2. EXISTENCE
to assume that rZ, K,, Q,, (T%,z = 1,2 are posltlve contmuous 1,2, J, E C([O, 001, [O,oo)), and SOWJ,(s),ds = 1, z = 1,2 on the comcldence degree theory, to study the existence of (1 2)
OF A POSITIVE
PERIODIC
SOLUTION
In this section, by using Mawhm’s contmuatlon theorem, we shall show the existence of at least one posltlve periodic solution of (1 2) To do so, we need to make some preparations Let X, Y be real Banach space, L Dom L c X + Y a Fredholm mappmg of mdex zero, and Y --) Y contmuous proJectors such that Im P = Ker L, KerQ = Im L, and P X+X,& X = Ker L CBKer P, Y = Im L $ Im Q Denote by Lp the restrlctlon of L to Dom L n Ker P, Kp Im L -+ Ker P n Dom L the Inverse (to Lp), and J Im Q + Ker L an lsomorphlsm of Im Q onto Ker L For convenience, LEMMA
2 1
we introduce
Mawhm’s
Let il c X be an open
which IS L-compact
contmuatlon
bounded
theorem
set and N
(a) for each X E (0, l), z E dR n DomL, Lx # XNz, (b) for each x E dCl n Ker L, QNx # 0, and deg{JQN, Then Lz = N~c has at least one solution LEMMA
In fi
R
fi --+ Y are compact)
operator Assume
n Ker L, 0) # 0
n Dom L
Let
22
f (2, y) = and fl = {(~,y)~ E R2 and A > max{I ln(az/c,)(,
(~1- e
-
qe",
a2
-e -
c2eg)
1x1+ IyI < A}, wh ere A, a$, b,, c, E R+ are constants, a, > b,, a = 1,2, I ln(b,/c%)(, z = 1,2} Then de@,
PROOF
X -+ Y be a contmuous
a -+ Y and Kp (I - Q)N
on fi [I e , QN
[7, p 401 as follows
fi,
(0,0)) #
0
Set
H(x,y,~)=
(
al---
~1
-bl
l+peg
clez,a2
a2 -
h
- 1+/.X"
-
OI/.JSl,
PosItwe Perlodlc Solutions
527
then it 1s easy to see that, for (x, Y, PL)~E R2 x [0,11, a1 -b1 al----
l+peg
cle’
a2 - b2 a2 - -1 + pex al---
al -bl 1 + peg
-
5
al
cle’
<
A 0,
asx>
cleY 5 a2 - c2e” < 0,
asYL
clez 2 bl - cle’
asx<---,
> 0,
-,2 A 2, A 2
and a2 - b2 -a’ - 1 + pex
c2ey 2 b2 - c2ey > 0,
asy<--
A 2
Hence, for (5, Y, 14 E 80 x [o, 11
H(x, Y, P) # 0,
It follows from the property of invariance under a homotopy that
deg{f(x, Y>, Q, (O,O))= dedH(x, ~~0)~fl, (OTO)) By a straightforward computation, we find
dedH(x, y, O),fl, (O,O))= -1 # 0 The proof 1s complete Now we state our fundamental theorem about the existence of a posltlve w-periodic solution of (13),(14) THEOREM 2 1 PROOF
System (1 2) has at least one poshve
w-perlock
solution
,
Consider the system of mtegrodlfferentlal equations
dx(t) dt
KlN) =
+ m(t)
-Mt) = ?-p(t) dt
mJ2(s)ey(t-")
00
q(t)
1+
K2(t)
+
s0
s
a2(t)
s0
_
,G-mW)
Jz(s)e V(t--s)&
J 0
1+
ds
0
(2 1)
mJl(s)e dt-s) &
00 J1 (s)ez(t-s) ds
_
e”(t-“z(t))
whele rz, I(z, cr,, Jz, a,, z = 1,2 are the same as those in system (1 2) It 1s clear that if system (2 1) has an w-periodic solution (z*(t),~*(t))~, then (Nr(t),NG(t))T IS a positive w-periodic solution of system (1 2) So, to complete the proof, it suffices to show that system (2 1) has an w-periodic solution Take
x = y = {(x(t),YWT and
x:(t),y(t) E C(R,RI, x(t + w) = x(t), y(t + w) = y(t)}
528 Wdh
Y
this norm,
X 1s a Banach
LI AND G XU
space
Let
Kl(t)
+ ul(t)~mJ2(s)ey(t-s'ds
(t)
0
1+
! I
- ez(t-“l(t))
co Y(t--5)
J2(s)e
&
J0
Nx= [I Y
Kz(t)
+
J JO3
02(t)
0
1-t
wJI(sk dt-$1 &
- eYCteo2(t))
J1 (s)ezctms) ds
0
Since Ker L = R2 and Im L 1s closed m X, L IS a Fredholm
mappmg
of mdex zero
we have that N 1s L-compact on fi (see [7]), h ere s2 1s any open bounded to equation Lx = XNx, X E (0, l), we have
(t) + Ql (t)
Kl T
=
_I"
Xrl(t)
J
If
0
1(2(t)
-dy(t) =
+
a2(t)
h-2(t)
dt
1+
J
Assume
that
(x(t),
y(t))T
E
Furthermore, Correspondmg
52 ( s)eyctms) ds 0
_
Jds)e
Y(t-s)
00 coJ
e”(t-a
(t))
&
(2 2)
Jl (Sk d--S) ds
0
_
,Yhn(t))
Jl (Sk dt-s) ds
0
integrating
set m X
X 1s a solution
of system
I
(2 2) for a certam
X E (0,l)
By
(2 2) over [0, w], we obtain
_ e”(t-ul(t))
&
=
0
dt
=
0
(2 3)
I
and Kz(t)
w
’ [ 0
rdt)
1+
It 1s easy to see that we can rewrite
s
w rl(t)(al(t)
O 1+
Jrn 0
and
J0
Y(t-s)
ds
J0
mJ1(sk z(t-s)&
_ eY(t-42))
ds
dt
+
Jw
rl(t)e”(t-Ul(t))
as
&
=
0
Jw
r2(t)eY(t-U2(t))
0
(2 4)
I
(2 3) and (2 4), respectively,
- 1i’l(t))
Jz(s)e
MJl(s)e dt--S)
+ a2(t)
Jw r1
0
dt
(tbl (t) dt
(2 5)
-J r2(tbdt) dt _
w
0
(2 6)
Posltlve Perlodlc Solutions
529
,
Thus, from (2 2) and (2 5), lt follows that 03
Kl w [z’(t)1 dt < X
w
I I
r1
0
=
+ &~-~Iw)
co
&(s)e+)
1+
<
J2 ( s)&‘(~-~) ds
(t) + a1 (t)
2
(t)m (t) dt +
ds
w n(t)(w(t)
IO I
- Kl(4)
dt
co
J2(s)e”(t-s)
1+
+
I
w
I dt
rl(t)e”(t-dt))
&
0
ds
0
w
rI(t)al(t)
dt dgfMl,
0
that is, w Ix’(t)1 dt < Ml s0
(27)
In the slmllar way, by (2 2) and (2 6), we have
s Moreover,
oy Iy’(t)l dt < 2
I0
w r2(t)a(t)
dt Ef A42
(28)
from (2 5), It follows that
I
w
rl(t)w(t)
I
dt 2
w
rl(t)ex(t-b’(t))
dt > -
I
r1
0
0
0
w
(t)Kl (t) dt,
which lmphes that there exists a point t\ E [0, w] and a constant Cl > 0 such that - &))I
I+; Denote ti - al(t\)
< Cl
= tl + nw, tl E [0, w] and n 1s an Integer, then Iz(t1)l < Cl
(2 9)
Slmllarly, by (2 6)) we can obtain that there exists a point t2 E [0, w] and a constant C2 > 0 such that (2 10)
Iv(tn)l < G
Therefore, it follows from (2 7)-(2 10) that
max Ix(t)/ I I4tl)l +
E[OWl
SW Ix’(t)1dt < Cl + MI, 0
ma Iy(t)l I tE [GJI
Iv(tl)l + IU
Iv’(t)1 dt < G + M2
0
Clearly, M, and C, (z = 1,2) are independent of X Denote M = Ml + M2 + Cl + C2 + D, where D > 0 1s taken sufficiently large such that M > max{ I ln(a/F$) I, ( ln(rzKz/Ft) 1, z = 1,2} Now we take R = {(z(t),y(t))T E X Il(z,y)T)I < M} This satisfies Condltlon (a) m Lemma 2 1 When (z, Y)~ E dR n Ker L = dCl n R2, (z, Y)~ 1s a constant vector m R2 with 1x( + IyI = M Then
_ r1a1
-
-
r2a2 -
rlal-
r1K1
1 + ey
cm
- nK2 1 + ex
- Flez - F2eY I
where F%= (l/w) Jt r,(t) dt, T,Q2 = (l/w) Jr rz(t)az(t) dt, r,K, = (l/w) sr r,(t).&(t) dt, 2 = I,2 Furthermore, take J = I Im Q + Ker L, (cc,Y)~ H (z, Y)~ By Lemma 2 2, we have deg { JQN@,
Y)~, at, (O,(9)
= deg {
QNk:,
By now we know that C? verifies all the requirements
one w-periodic solution
The proof 1s complete
Y)~, f12,(0, O)} # 0
m Lemma 2 1 and then (2 1) has at least
530
Y
h
AND
G XV
REFERENCES 1 2 3 4 5 6 7
J H Vandermeer and D H Boucher, Varieties of mutuahstic mteractlon models, J The-or Bzol 74, 549-558 (1978) D H Boucher, S James and K H Keeler, The ecology of mutuahsm, Ann Rev Syst 13, 315-347 (1982) A M Dean, A simple model of mutuahsm, Amer Natural 121, 409-417 (1983) C L Wolm and L R Lawlor, Models of facultatlve mutuahsm Density effects, Amer Natural 144, 843-862 (1984) D H Boucher, The Bzology of Mutualzsm Ecology and Evolutaon, Croom Helm, London, (1985) K Gopalsamy, Stab&y and Oscallatzons an Delay Dafferentaal Equataons of Populataon Dynamacs, Kluwer Academq Boston, (1992) R E Games and J L Mawhm, Comcldence degree and nonlinear differential equations, In Lecture Notes an Math , Vol 568, Springer-Verlag, Berhn, (1977)
Applied Mathematms Letters 14 (2001) 525-530
PERGAMON
www elsewer nl/locate/aml
P o s i t i v e P e r i o d i c S o l u t i o n s for a n I n t e g r o d i f f e r e n t i a l M o d e l of M u t u a l i s m Y O N G K U N L I AND G U I T O N G X U Department of Mathematics, Yunnan Umvermty Kunmmg, Yunnan 650091, P R China (Recewed and accepted August 2000)
Commumcated by M Slemrod A b s t r a c t - - S u f f i c i e n t conditions are obtained for the existence of positive penodm solutions of the followmg mtegrodtfferentml model of mutuahsm "gl(t) + a l ( t ) f o ~ g 2 ( s ) N 2 ( t - s ) d s dNl(t) = r l ( t ) N l ( t )
al(t))
,
N 2 ( t - a2(t))
,
- Nl(t-
dt
1+
J2(s)N2(t - s)ds
K2(t) + c~2(t) d l ( s ) N l ( t - s) d s dN2(t) _ r 2 ( t ) N 2 ( t ) . --- ~ . . . . dt [ 1+/0 Jl(S)Nl(t-s)ds
where r ~ , K ~ , a ~ , a ~ , ~ = 1,2 are positive continuous w-permdm functmns, a~ > K~, z = 1,2, J~ E C([0, c~], [0, c~)), and / o J~(s) ds = 1, z = 1, 2 @ 2001 Elsewer Scmnce Ltd All rights reserved K e y w o r d s - - M u t u a h s m model, Pomtlve penodm solution, Integrodlfferentlal equation, Fredholm mapping
1. I N T R O D U C T I O N Consider t h e following lntegrodlfferentlal model of mutuallsm K1 + al - - - ~
dNl(t) _rlNl(t) dt
[
l+fo
J2(s)N2(t - s) as . . . . .
Nl(t
- al)
,
J2(s)N2(t-s)ds
(1 1) dN2(t) dt
K2 + c~2 = r2N2(t)
Jl(s)Nl(t
- s) ds
" J0// foo
1+
- N 2 ( t - 62)
dl (s)N1 (t - s) ds
]
,
where r,, K,, at are positive constants, a , > K , , , = 1, 2, J~ E C([0, c~], [0, co)) and f o J ~ ( s ) d s = 1, z = 1, 2 D e p e n d m g on the n a t u r e of K, (z = 1,2), system (1 1) can be classffied as facultatlve, Thin work is partially supported by the ABF of Yunnan Province of China 0893-9659/01/$ - see front matter (~) 2001 Elsevmr Science Ltd All rights reserved PII S0893-9659(00)00188-9
Typeset by .A,,~IS-TEX
Ll AND
Y
526
obligate, or a combmatlon System (1 1) means that
of both For more dettils of mutuahstlc mteractlons, we refer to [l-5] the mutuahstlc or cooperative effects are not realized Instantaneously
but take place with time delays and the references environment
G XU
For the further
cited therem
This motivates
Reahstlc
ecological
models
us to consider
meaning
require
of system
the mcluslon
the followmg
(1 l), we refer to [6]
of the effect of changing
nonautonomous
model
co
d%(t) -
dt
= ~1 (t)Nl (t)
=
Jz(s)N2(t - s) ds
0
J
- N(t
owJ2(s)Nz(t
1+
K2(t)
dNz (t) dt
-
J
Kl(q + al(t)
J
1+
7 I (12)
+,,(t$=Jl(spv~(t0
r2(tW2(t)
- aI
- s) ds
s)ds
-
Nz(t
mJl(s)N,(t
- s) ds
- g2(t)) !
0
In addltlon, the effects of a perlodlcally varying environment are important for evolutionary theory as the selective forces on systems m a fluctuatmg environment differ from those m a stable environment Therefore, the assumptions of perlodlclty of the parameters are a way of mcorporatmg the perlodlclty of the environment (e g , seasonal effects of weather, food supplies, mating habits, etc ), which leads us w-perlodlc functions, cy, > K,, z = Our purpose m this article 1s based positive periodic solutions of system
2. EXISTENCE
to assume that rZ, K,, Q,, (T%,z = 1,2 are posltlve contmuous 1,2, J, E C([O, 001, [O,oo)), and SOWJ,(s),ds = 1, z = 1,2 on the comcldence degree theory, to study the existence of (1 2)
OF A POSITIVE
PERIODIC
SOLUTION
In this section, by using Mawhm’s contmuatlon theorem, we shall show the existence of at least one posltlve periodic solution of (1 2) To do so, we need to make some preparations Let X, Y be real Banach space, L Dom L c X + Y a Fredholm mappmg of mdex zero, and Y --) Y contmuous proJectors such that Im P = Ker L, KerQ = Im L, and P X+X,& X = Ker L CBKer P, Y = Im L $ Im Q Denote by Lp the restrlctlon of L to Dom L n Ker P, Kp Im L -+ Ker P n Dom L the Inverse (to Lp), and J Im Q + Ker L an lsomorphlsm of Im Q onto Ker L For convenience, LEMMA
2 1
we introduce
Mawhm’s
Let il c X be an open
which IS L-compact
contmuatlon
bounded
theorem
set and N
(a) for each X E (0, l), z E dR n DomL, Lx # XNz, (b) for each x E dCl n Ker L, QNx # 0, and deg{JQN, Then Lz = N~c has at least one solution LEMMA
In fi
R
fi --+ Y are compact)
operator Assume
n Ker L, 0) # 0
n Dom L
Let
22
f (2, y) = and fl = {(~,y)~ E R2 and A > max{I ln(az/c,)(,
(~1- e
-
qe",
a2
-e -
c2eg)
1x1+ IyI < A}, wh ere A, a$, b,, c, E R+ are constants, a, > b,, a = 1,2, I ln(b,/c%)(, z = 1,2} Then de@,
PROOF
X -+ Y be a contmuous
a -+ Y and Kp (I - Q)N
on fi [I e , QN
[7, p 401 as follows
fi,
(0,0)) #
0
Set
H(x,y,~)=
(
al---
~1
-bl
l+peg
clez,a2
a2 -
h
- 1+/.X"
-
OI/.JSl,
PosItwe Perlodlc Solutions
527
then it 1s easy to see that, for (x, Y, PL)~E R2 x [0,11, a1 -b1 al----
l+peg
cle’
a2 - b2 a2 - -1 + pex al---
al -bl 1 + peg
-
5
al
cle’
<
A 0,
asx>
cleY 5 a2 - c2e” < 0,
asYL
clez 2 bl - cle’
asx<---,
> 0,
-,2 A 2, A 2
and a2 - b2 -a’ - 1 + pex
c2ey 2 b2 - c2ey > 0,
asy<--
A 2
Hence, for (5, Y, 14 E 80 x [o, 11
H(x, Y, P) # 0,
It follows from the property of invariance under a homotopy that
deg{f(x, Y>, Q, (O,O))= dedH(x, ~~0)~fl, (OTO)) By a straightforward computation, we find
dedH(x, y, O),fl, (O,O))= -1 # 0 The proof 1s complete Now we state our fundamental theorem about the existence of a posltlve w-periodic solution of (13),(14) THEOREM 2 1 PROOF
System (1 2) has at least one poshve
w-perlock
solution
,
Consider the system of mtegrodlfferentlal equations
dx(t) dt
KlN) =
+ m(t)
-Mt) = ?-p(t) dt
mJ2(s)ey(t-")
00
q(t)
1+
K2(t)
+
s0
s
a2(t)
s0
_
,G-mW)
Jz(s)e V(t--s)&
J 0
1+
ds
0
(2 1)
mJl(s)e dt-s) &
00 J1 (s)ez(t-s) ds
_
e”(t-“z(t))
whele rz, I(z, cr,, Jz, a,, z = 1,2 are the same as those in system (1 2) It 1s clear that if system (2 1) has an w-periodic solution (z*(t),~*(t))~, then (Nr(t),NG(t))T IS a positive w-periodic solution of system (1 2) So, to complete the proof, it suffices to show that system (2 1) has an w-periodic solution Take
x = y = {(x(t),YWT and
x:(t),y(t) E C(R,RI, x(t + w) = x(t), y(t + w) = y(t)}
528 Wdh
Y
this norm,
X 1s a Banach
LI AND G XU
space
Let
Kl(t)
+ ul(t)~mJ2(s)ey(t-s'ds
(t)
0
1+
! I
- ez(t-“l(t))
co Y(t--5)
J2(s)e
&
J0
Nx= [I Y
Kz(t)
+
J JO3
02(t)
0
1-t
wJI(sk dt-$1 &
- eYCteo2(t))
J1 (s)ezctms) ds
0
Since Ker L = R2 and Im L 1s closed m X, L IS a Fredholm
mappmg
of mdex zero
we have that N 1s L-compact on fi (see [7]), h ere s2 1s any open bounded to equation Lx = XNx, X E (0, l), we have
(t) + Ql (t)
Kl T
=
_I"
Xrl(t)
J
If
0
1(2(t)
-dy(t) =
+
a2(t)
h-2(t)
dt
1+
J
Assume
that
(x(t),
y(t))T
E
Furthermore, Correspondmg
52 ( s)eyctms) ds 0
_
Jds)e
Y(t-s)
00 coJ
e”(t-a
(t))
&
(2 2)
Jl (Sk d--S) ds
0
_
,Yhn(t))
Jl (Sk dt-s) ds
0
integrating
set m X
X 1s a solution
of system
I
(2 2) for a certam
X E (0,l)
By
(2 2) over [0, w], we obtain
_ e”(t-ul(t))
&
=
0
dt
=
0
(2 3)
I
and Kz(t)
w
’ [ 0
rdt)
1+
It 1s easy to see that we can rewrite
s
w rl(t)(al(t)
O 1+
Jrn 0
and
J0
Y(t-s)
ds
J0
mJ1(sk z(t-s)&
_ eY(t-42))
ds
dt
+
Jw
rl(t)e”(t-Ul(t))
as
&
=
0
Jw
r2(t)eY(t-U2(t))
0
(2 4)
I
(2 3) and (2 4), respectively,
- 1i’l(t))
Jz(s)e
MJl(s)e dt--S)
+ a2(t)
Jw r1
0
dt
(tbl (t) dt
(2 5)
-J r2(tbdt) dt _
w
0
(2 6)
Posltlve Perlodlc Solutions
529
,
Thus, from (2 2) and (2 5), lt follows that 03
Kl w [z’(t)1 dt < X
w
I I
r1
0
=
+ &~-~Iw)
co
&(s)e+)
1+
<
J2 ( s)&‘(~-~) ds
(t) + a1 (t)
2
(t)m (t) dt +
ds
w n(t)(w(t)
IO I
- Kl(4)
dt
co
J2(s)e”(t-s)
1+
+
I
w
I dt
rl(t)e”(t-dt))
&
0
ds
0
w
rI(t)al(t)
dt dgfMl,
0
that is, w Ix’(t)1 dt < Ml s0
(27)
In the slmllar way, by (2 2) and (2 6), we have
s Moreover,
oy Iy’(t)l dt < 2
I0
w r2(t)a(t)
dt Ef A42
(28)
from (2 5), It follows that
I
w
rl(t)w(t)
I
dt 2
w
rl(t)ex(t-b’(t))
dt > -
I
r1
0
0
0
w
(t)Kl (t) dt,
which lmphes that there exists a point t\ E [0, w] and a constant Cl > 0 such that - &))I
I+; Denote ti - al(t\)
< Cl
= tl + nw, tl E [0, w] and n 1s an Integer, then Iz(t1)l < Cl
(2 9)
Slmllarly, by (2 6)) we can obtain that there exists a point t2 E [0, w] and a constant C2 > 0 such that (2 10)
Iv(tn)l < G
Therefore, it follows from (2 7)-(2 10) that
max Ix(t)/ I I4tl)l +
E[OWl
SW Ix’(t)1dt < Cl + MI, 0
ma Iy(t)l I tE [GJI
Iv(tl)l + IU
Iv’(t)1 dt < G + M2
0
Clearly, M, and C, (z = 1,2) are independent of X Denote M = Ml + M2 + Cl + C2 + D, where D > 0 1s taken sufficiently large such that M > max{ I ln(a/F$) I, ( ln(rzKz/Ft) 1, z = 1,2} Now we take R = {(z(t),y(t))T E X Il(z,y)T)I < M} This satisfies Condltlon (a) m Lemma 2 1 When (z, Y)~ E dR n Ker L = dCl n R2, (z, Y)~ 1s a constant vector m R2 with 1x( + IyI = M Then
_ r1a1
-
-
r2a2 -
rlal-
r1K1
1 + ey
cm
- nK2 1 + ex
- Flez - F2eY I
where F%= (l/w) Jt r,(t) dt, T,Q2 = (l/w) Jr rz(t)az(t) dt, r,K, = (l/w) sr r,(t).&(t) dt, 2 = I,2 Furthermore, take J = I Im Q + Ker L, (cc,Y)~ H (z, Y)~ By Lemma 2 2, we have deg { JQN@,
Y)~, at, (O,(9)
= deg {
QNk:,
By now we know that C? verifies all the requirements
one w-periodic solution
The proof 1s complete
Y)~, f12,(0, O)} # 0
m Lemma 2 1 and then (2 1) has at least
530
Y
h
AND
G XV
REFERENCES 1 2 3 4 5 6 7
J H Vandermeer and D H Boucher, Varieties of mutuahstic mteractlon models, J The-or Bzol 74, 549-558 (1978) D H Boucher, S James and K H Keeler, The ecology of mutuahsm, Ann Rev Syst 13, 315-347 (1982) A M Dean, A simple model of mutuahsm, Amer Natural 121, 409-417 (1983) C L Wolm and L R Lawlor, Models of facultatlve mutuahsm Density effects, Amer Natural 144, 843-862 (1984) D H Boucher, The Bzology of Mutualzsm Ecology and Evolutaon, Croom Helm, London, (1985) K Gopalsamy, Stab&y and Oscallatzons an Delay Dafferentaal Equataons of Populataon Dynamacs, Kluwer Academq Boston, (1992) R E Games and J L Mawhm, Comcldence degree and nonlinear differential equations, In Lecture Notes an Math , Vol 568, Springer-Verlag, Berhn, (1977)