Electronic Journal of Differential Equations, Vol. 2005(2005), No. 31, pp. 1–14.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
PERIODIC SOLUTIONS FOR A DELAYED PREDATOR-PREY
SYSTEM WITH DISPERSAL AND IMPULSES
QIMING LIU, SHIJIE DONG
Abstract. A delayed predator-prey system with prey dispersal in n-patch environments and impulse effects is investigated. By using Gaines and Mawhin’s
continuation theorem of coincidence degree theory, a set of easily verifiable
sufficient conditions are derived for the existence of positive periodic solutions
to the system.
1. Introduction
An important and ubiquitous problem in mathematical ecology concerns the
effect of environment change in the growth and diffusion of a species in a heterogenous habitat. There have been many studies in the literatures that investigate the
population dynamics with diffusion process [5, 6, 10, 11].
In most of the models considered so far, it has been assumed that all biological
and environmental parameters are constants in time. However, any biological or
environmental parameters are naturally subject to fluctuation in time. The effects
of a periodically varying environment are important for evolutionary theory as
the selective forces on systems in a fluctuating environment differ from those in a
stable environment. Thus, the assumptions of periodicity of the parameters are a
way of incorporating the periodicity of the environment (such as seasonal effects
of weather, food supplies, mating habits and so forth); on the other hand, it is
generally recognized that some kinds of time delays are inevitable in population
interactions. Time delay due to gestation is a common example, because generally
the consumption of prey by the predator throughout its past history governs the
present birth rate of the predator. Therefore, more realistic models of population
interactions should take into account the seasonality of the changing environment
and the effect of time delays.
There are still some other possible exterior effects under which the population
densities change very rapidly. For example, impulse reduction of the population
density of a given species is possible after its partial destruction by catching or
by poisoning with chemical used at some transitory slots in fishing or agriculture
[1, 7, 12]. Recently, many authors studies the existence of positive periodic solution
in population models by using power and effective method of coincidence degree [8,
1991 Mathematics Subject Classification. 34K13, 34K45, 92D25.
Key words and phrases. Predator-prey system; impulse; dispersion; periodic solution;
coincidence degree.
c
2005
Texas State University - San Marcos.
Submitted November 1, 2004. Published March 18, 2005.
1
2
Q. LIU, S. DONG
EJDE-2005/31
13], In the present paper, we are concerned with the study on the combined effects of
dispersion, periodicity of environment, time delays and impulses on the dynamics
of predator-prey system. To do so we are devoted to the study of the following
delayed periodic predator-prey system with prey dispersal in n-patch environments
and impulses
ẋi (t) =xi (t)[ri (t) − aii (t)xi (t) − ai,n+1 (t)xn+1 (t)]
+
n
X
Dj (t)[xj (t) − xi (t)],
i = 1, . . . , n,
j=1,j6=i
n
h
X
ẋn+1 (t) =xn+1 (t) − rn+1 (t) +
an+1,j (t)xj (t − σj )
(1.1)
j=1
i
− an+1,n+1 (t)xn+1 (t − σn+1 ) ,
∆(xi (τk )) = −cik xi (τk ),
t = τk , k ∈ Z+ ,
t 6= τk , k ∈ Z+ ,
i = 1, 2, . . . , n + 1.
with initial conditions
xi (s) = φi (s),
s ∈ [−σ, 0],
φi (0) > 0,
i = 1, . . . , n + 1,
(1.2)
where xi (t) (i = 1, . . . , n) denote the densities of prey species in patch i, xn+1
denote the densities of all predator for all patches. ri (t) is the intrinsic growth rate
of the prey, aii (t) is the density-dependent coefficient of the prey species. ai,n+1 (t)
are the capturing rates of the predator and an+1,i /ai,n+1 are the conversion rates
of nutrients into the reproduction of predator, rn+1 (t) are the death rates of the
predator. Di (t) (i = 1, 2, . . . , n) is dispersal rate of prey species, σn+1 ≥ 0 denotes
the delay due to negative feedback of the predator species σi (i = 1, 2, . . . , n) are the
time delays due to gestation, that is, mature adult predators can only contribute
to the production of predator biomass. aii (t), ai,n+1 (t), an+1,i (i, j = 1, . . . , n),
ri (t) (i = 1, . . . , n + 1) and Di (t) (i = 1, 2, . . . , n) are continuously positive periodic
functions with period ω > 0. cik are positive constants and 0 < cik < 1, Z+ is the
set of all positive integers and there exists an integer p > 0 such that ci(k+p) = cik ,
τk+p = τk + ω.
It is well known that by the fundamental theory of functional differential equations [4], system (1.1) has a unique solution x(t) = (x1 (t), . . . , xn+1 (t)) satisfying
initial conditions (1.2). It is easy to verify that solutions of (1.1) corresponding to
initial conditions (1.2) are defined on [0, +∞) and remain positive for all t ≥ 0. In
this paper, the solution of system (1.1) satisfying initial conditions (1.2) is said to
be positive.
We shall use the following notation: Let J ⊂ R. Denote by P C(J, R) the set of
function ψ : J → R which are continuous for t ∈ J, t 6= τk , are continuous from the
left for t ∈ J and have discontinuities of the first kind at the points τk ∈ J. Denote
the Banach space of ω-periodic functions by P Cω = {ψ ∈ P C[0, ω], R}|ψ(0) =
ψ(ω)} and we denote
Z
1 ω
f¯ =
f (t)dt, f L = min f (t), f M = max f (t),
ω 0
t∈[0,ω]
[0,ω]
where f ∈ P Cω .
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PERIODIC SOLUTIONS
3
The organization of this paper is as follows. In the next section, by using Gaines
and Mawhin’s continuation theorem of coincidence degree theory, sufficient conditions are derived for the existence of positive periodic solutions of system (1.1) with
initial conditions (1.2).
2. Main result
In this section, by using Gaines and Mawhin’s continuation theorem of coincidence degree theory, we show the existence of positive ω-periodic solutions of
(1.1)-(1.2). To this end, we first introduce the following notations.
Let X, Z be real Banach spaces, let L: Dom L ⊂ X → Z be a linear mapping, and
N : X → Z be a continuous mapping. The mapping L is called a Fredholm mapping
of index zero if dim ker L = codim Im L < +∞ and Im L is closed in Z. If L is a
Fredholm mapping of index zero and there exist continuous projectors P : X → X,
and Q : Z → Z such that Im P = ker L, ker Q = Im L = Im(I − Q), then the
restriction LP of L → Dom L ∩ ker P : (I − P )X → Im L is invertible. Denote the
inverse of LP by KP . If Ω is an open bounded subset of X, the mapping N will be
called L-compact on Ω̄ if QN (Ω̄) is bounded and KP (I − Q)N : Ω̄ → X is compact.
Since Im Q is isomorphic to ker L, there exists isomorphism J : Im Q → ker L.
For convenience, we introduce the continuation theorem of coincidence degree
theory [3] and compactness criterion for set F ⊂ P Cω [2] as follows.
Lemma 2.1. Let Ω ⊂ X be an open bounded set. Let L be a Fredholm mapping of
index zero and N be L-compact on Ω̄. Assume
(a) For each λ ∈ (0, 1), x ∈ ∂Ω ∩ Dom L, Lx 6= λN x
(b) For each x ∈ ∂Ω ∩ ker L, QN x 6= 0
(c) deg{JQN, Ω ∩ ker L, 0} =
6 0.
Then Lx = N x has at least one solution in Ω̄ ∩ Dom L.
Lemma 2.2 (Compactness criterion). A set F ⊂ P Cω is relative compact if and
only if
(a) F is bounded, that is, kψk = sup{|ψ| : t ∈ J} ≤ M for each x ∈ F and
some M > 0
(b) F is quasiequicontinuous in J.
We are now in a position to state our main result on the existence of a positive
periodic solution to system (1.1).
Theorem 2.3. System (1.1) with initial conditions (1.2) has at least one strictly
positive ω-periodic solution provided that
(H1)
n
X
n
X
aM
n+1,j (rj −
j=1
Dk −
k=1,k6=j
p
p
k=1
k=1
1 Y
1
1 Y
1
ln
)/aM
ln
jj > rn+1 +
ω
1 − cjk
ω
1 − cn+1,k
(H2) For i = 1, 2, . . . , n,
ri −
n
X
p
Dj −
j=1,j6=i
>
aM
i,n+1
A
1 Y
1
ln
ω
1 − cik
k=1
Pn
M
j=1 an+1,j
− rn+1 −
1
ω
aL
n+1,n+1
ln
Qp
1
k=1 1−cn+1,k
,
4
Q. LIU, S. DONG
where A = max1≤i≤n {[(ri −
Pn
j=1,j6=i
EJDE-2005/31
Pn
Dj )M +
j=1,j6=i
DjM ]/aL
ii }.
Proof. Since solutions of (1.1)-(1.2) remain positive for all t ≥ 0, we let
yi (t) = ln[xi (t)], i = 1, . . . , n + 1.
(2.1)
Substituting (2.1) into system (1.1), we derive
ẏi (t) =ri (t) −
+
n
X
Dj (t) − aii (t)eyi (t) − ai,n+1 (t)eyn+1 (t) ]
j=1,j6=i
n
X
Dj (t)eyj (t)−yi (t) ,
i = 1, . . . , n,
j=1,j6=i
ẏn+1 (t) = − rn+1 (t) +
n
X
(2.2)
an+1,j (t)eyj (t−σj ) − an+1,n+1 (t)eyn+1 (t−σn+1 ) ,
j=1
t 6= τk , k ∈ Z+ ,
∆(yi (τk )) = ln(1 − cik ),
t = τk , k ∈ Z+ ,
i = 1, 2, . . . , n + 1.
∗
It is easy to see that if (2.2) has one ω-periodic solution (y1∗ (t), . . . , yn+1
(t))T ,
∗
∗
∗
T
∗
∗
T
then x (t) = (x1 (t), . . . , xn+1 (t)) = (exp[y1 (t)], . . . , exp[yn+1 (t)]) is a positive
ω-periodic solution of system (1.1). Therefore, to complete the proof, it suffices to
show that system (2.2) has one ω-periodic solution. Let
X = {x ∈ P C(R, Rn+1 ) : y(t + ω) = y(t)},
Pn+1
with the norm kxk = supt∈[0,ω] i=1 |yi (t)|, where x = (y1 , y2 , . . . , yn+1 )T . Let
Z = X × Rnp with the norm k(x, a1 , . . . , ap )k = (kxk2 + |a1 |2 + · · · + |ap |2 )1/2 .
Then X and Z are Banach spaces. Let
L : Dom L → Z
L(x)(t) = (ẋ, ∆(y(τ1 ), . . . , ∆(y(τp ))
where Dom L consist of functions x ∈ X such that x is continuous for t 6= τk , x is
continuous from the left for t = τk , and ẋ(τk ) exists. Let N : X → Z, be defined as
(N x)(t) = (f (t, x(t)), C1 , C2 , . . . , Cp ),
where f (t, x) equals


Pn
Pn
r1 (t) − j=2 Dj (t) − a11 (t)ey1 (t) − a1,n+1 (t)eyn+1 (t) ] + j=2 Dj (t)eyj (t)−y1 (t)


..


.


Pn−1
Pn−1

yn (t)
yn+1 (t)
yj (t)−yn (t) 
− an,n+1 (t)e
] + j=1 Dj (t)e
rn (t) − j=1 Dj (t) − ann (t)e

Pn
−rn+1 (t) + j=1 an+1,j (t)eyj (t−σj ) − an+1,n+1 (t)eyn+1 (t−σn+1 ) ,
and
Ck = (ln(1 − c1k ), ln(1 − c2k ), . . . , ln(1 − cn+1,k ))T ,
k = 1, 2, . . . , p.
Define two projectors P and Q as
P : X → ker L,
Q : Z → Z,
Q(y, C1 , . . . , Cp) =
Py =
1 Z
ω
0
1
ω
Z
ω
ydt,
0
ω
ydt +
p
X
k=1
 
0 o
 ..  p
Ck ,  . 
n
0
k=1
,
EJDE-2005/31
PERIODIC SOLUTIONS
5
It is clear that
ker L = {x : x ∈ X, x = h, h ∈ Rn+1 },
Z ω
p
X
Im L = {z = (y, C1 , . . . , Cp ) ∈ Z,
y(t)dt +
Ck = 0} is closed in Z,
0
k=1
dim ker L = codim Im L = n + 1.
Therefore, L is a Fredholm mapping of index zero. It is easy to show that P and
Q are continuous projectors such that
Im P = ker L,
ker Q = Im L = Im(I − Q).
Furthermore, the inverse KP of LP exists and is given by KP : Im Lto Dom L∩ker P ,
t
Z
y(s)ds −
KP (z) =
0
1
ω
Z
ω
Z
t
y(s)ds −
0
0
p
X
Ck .
k=1
Then QN : X → Z and KP (I − Q)N : X → X read
QN x =
1 Z
ω
ω
0
 
0 o
p
n
X
1
  p
f (t, x)dt +
Ck ,  ... 
,
ω
k=1
k=1
0
KP (I − Q)N x
Z
p
1 Z ωZ t
X
1 X
1 t
f (t, x)dt +
Ck −
f (t, x)ds +
Ck
=
ω 0
ω t>τ
ω 0 0
k=1
k
Z ω
p
t
1 1
1X
− ( − )
f (t, x)dt +
Ck .
ω 2 ω 0
ω
k=1
Clearly, QN and KP (I − Q)N are continuous.
To apply Lemma 2.1, we need to search for an appropriate open, bounded subset
Ω. Corresponding to the operator equation Lx = λN x, λ ∈ (0, 1), we obtain
n
h
i
X
ẏi (t) =λ ri (t) −
Dj (t) − aii (t)eyi (t) − ai,n+1 (t)eyn+1 (t)
j=1,j6=i
+
n
X
Dj (t)eyj (t)−yi (t) ],
i = 1, . . . , n,
j=1,j6=i
h
ẏn+1 (t) =λ − rn+1 (t) +
n
X
(2.3)
i
an+1,j (t)eyj (t−σj ) − an+1,n+1 (t)eyn+1 (t−σn+1 ) ,
j=1
t 6= τk , k ∈ Z+ ,
∆(yi (τk )) = λ ln(1 − cik ),
t = τk , k ∈ Z+ , i = 1, 2, . . . , n + 1.
6
Q. LIU, S. DONG
EJDE-2005/31
Suppose that (y1 (t), . . . , yn+1 (t))T ∈ X is a solution of (2.3) for some λ ∈ (0, 1).
Integrating system (2.3) over [0, ω], for i = 1, 2, . . . , n, we have
Z
ω
aii (t)eyi (t) dt +
ω
Z
0
ai,n+1 (t)eyn+1 (t) dt + ln
0
Z
n
X
ω
(ri (t) −
=
0
p
Y
k=1
Z
n
X
Dj (t))dt +
(2.4)
ω
yj (t)−yi (t)
Dj (t)e
dt,
0
j=1,j6=i
j=1,j6=i
1
1 − cik
and
Z
ω
ω
Z
yn+1 (t−σn+1 )
rn+1 (t)dt +
0
=
an+1,n+1 (t)e
dt + ln
0
n Z ω
X
j=1
p
Y
k=1
yj (t−σj )
an+1,j (t)e
1
1 − cn+1,k
(2.5)
dt.
0
Multiplying the ith equation (2.3) by eyi (t) and integrating over [0, ω] gives
Z ω
aii (t)e2yi (t) dt
0
Z
≤
ω
(ri (t) −
0
n
X
Z
n
X
Dj (t))eyi (t) dt +
j=1,j6=i
j=1,j6=i
ω
Dj (t)eyj (t) dt +
0
p
X
k=1
in which ∆(eyi (τk ) ) = [(1 − cik )λ − 1]eyi (τk ) ≤ 0, then we have
Z ω
Z ω
Z
n
n
X
X
2yi (t)
M
yi (t)
M
aL
e
dt
≤
[(r
−
D
)
]
e
dt
+
D
i
j
ii
j
0
0
j=1,j6=i
∆(eyi (τk ) ),
ω
eyj (t) dt . (2.6)
0
j=1,j6=i
Using the inequality
Z
ω
eyi (t) dt
Z
2
ω
e2yi (t) dt,
≤ω
0
0
it follows from (2.6) that
Z
Z
Z ω
n
n
X
X
ω yi (t)
1 L ω yi (t) 2 aii (
e
dt) ≤ (ri −
Dj )M
e
dt +
DjM
eyj (t) dt.
ω
0
0
0
j=1,j6=i
j=1,j6=i
(2.7)
Using the fact that if i = k,
Z ω
Z
eyk (t) dt ≥ max
0
eyi (t) dt, i = 1, . . . , n
0
this , together with (2.7), leads to
Z ω
1 L
akk (
eyk (t) dt)2
ω
0
Z
n
X
≤ (rk −
Dj )M
j=1,j6=k
ω
0
ω
eyk (t) dt + (
n
X
j=1,j6=k
DjM )
Z
0
ω
eyk (t) dt,
EJDE-2005/31
PERIODIC SOLUTIONS
which implies
Z ω
Z
yi (t)
max{
e
dt, i = 1, . . . , n} ≤
ω
0
7
eyk (t) dt
Pn
Pn
[(rk − j=1,j6=k Dj )M + j=1,j6=k DjM ]ω
0
≤
aL
kk
(2.8)
.
Set
n (ri − Pn
j=1,j6=i
A = max
Dj )M +
Pn
j=1,j6=i
aL
ii
1≤i≤n
DjM o
,
we have from (2.8) that
ω
Z
eyi (t) dt ≤ Aω,
i = 1, . . . , n.
(2.9)
0
Since yi ∈ P Cω , there exists ti , Ti ∈ [0, T ] ∪ {τ1+ , τ2+ , . . . , τp+ } such that
yi (ti ) = min yi (t),
yi (Ti ) = max yi (t),
t∈[0,ω]
i = 1, 2, . . . , n.
t∈[0,ω]
It follows from (2.9) that
y(ti ) ≤ ln A,
i = 1, 2, . . . , n.
(2.10)
We derive from (2.5) that
Z ω
an+1,n+1 (t)eyn+1 (t−σn+1 ) dt
0
≤
n
X
aM
n+1,j
Z
=
dt − rn+1 ω − ln
0
aM
n+1,j
Z
n
X
p
Y
k=1
p
Y
ω
eyj (t) dt − rn+1 ω − ln
0
j=1
≤ (A
yj (t−σj )
e
j=1
n
X
ω
k=1
aM
n+1,j − rn+1 )ω − ln
j=1
p
Y
k=1
1
1 − cn+1,k
(2.11)
1
1 − cn+1,k
1
,
1 − cn+1,k
which yields
yn+1 (tn+1 ) ≤ ln
A
Pn
j=1
aM
n+1,j − rn+1 −
1
ω
ln
Qp
1
k=1 1−cn+1,k
an+1,n+1
,
and
Z
ω
eyn+1 (t) dt =
0
≤
Z
ω
eyn+1 (t−σn+1 ) dt
0
Pn
A j=1 aM
n+1,j − rn+1 −
1
ω
aL
n+1,n+1
ln
(2.12)
Qp
1
k=1 1−cn+1,k
ω.
8
Q. LIU, S. DONG
EJDE-2005/31
It follows from (2.4), (2.5), (2.9) and (2.12) that
ω
Z
|ẏn+1 (t)|dt
0
Z
ω
λ| − rn+1 (t) +
=
0
Z
n
X
an+1,j (t)eyj (t−σj ) − an+1,n+1 (t)eyn+1 (t−σn+1 ) |dt
j=1
n
X
ω
≤
[rn+1 (t) +
0
an+1,j (t)eyn+1 (t−σj ) + an+1,n+1 (t)eyn+1 (t−σn+1 ) ]dt
j=1
Z
≤2
ω
0
< 2Aω
n
X
an+1,j (t)eyj (t−σj ) dt − ln
j=1
n
X
p
Y
k=1
(2.13)
1
1 − cn+1,k
aM
n+1,j := dn+1 ,
j=1
and
Z
ω
n
X
ω
Z
|ẏi (t)|dt =
λ|ri (t) −
0
0
Dj (t) − aii (t)eyi (t) − ai,n+1 (t)eyn+1 (t)
j=1,j6=i
n
X
+
Dj (t)eyj (t)−yi (t) |dt
j=1,j6=i
ω
Z
≤
[ri (t) +
0
n
X
Dj (t) + aii (t)eyi (t) + ai,n+1 (t)eyn+1 (t)
j=1,j6=i
n
X
+
Dj (t)eyj (t)−yi (t) ]dt
j=1,j6=i
ω
Z
[aii (t)eyi (t) + ai,n+1 (t)eyn+1 (t) ]dt + ln
≤2
0
p
Y
k=1
≤ 2 aM
ii
ω
Z
eyi (t) dt + aM
i,n+1
0
Z
ω
1
1 − cik
p
Y
eyn+1 (t) dt + ln
0
k=1
n
X
p
Y
L
aM
)/a
ω
+
ln
n+1,j
n+1,n+1
j=1
k=1
M
< 2A aM
ii + ai,n+1 (
1
1 − cik
1
:= di ,
1 − cik
(2.14)
i = 1, . . . , n. From (2.10), (2.14) and (2.12), (2.13), we have
yn+1 (t)
Z
ω
≤ yn+1 (tn+1 ) +
p
Y
|ẏn+1 (t)|dt + ln
0
≤ ln
A
Pn
M
j=1 an+1,j
k=1
− rn+1 −
1
ω
an+1,n+1
ln
Qp
1
1 − cn+1,k
1
k=1 1−cn+1,k
p
Y
+ dn+1 + ln
1
,
1 − cn+1,k
k=1
(2.15)
EJDE-2005/31
PERIODIC SOLUTIONS
9
and
Z
ω
yi (t) ≤yi (ti ) +
p
Y
|ẏi (t)|dt + ln
0
k=1
p
Y
≤ ln A + di + ln
k=1
1 ,
1 − cik
1 1 − cik
(2.16)
i = 1, 2, . . . , n.
On the other hand, it follows from (2.4) and (2.9) that
n
X
yi (Ti )
aM
ω ≥ (ri −
ii e
Dj )ω − ln
j=1,j6=i
n
X
≥ (ri −
Dj )ω − ln
j=1,j6=i
p
Y
k=1
p
Y
k=1
1
− aM
i,n+1
1 − cik
Z
ω
eyn+1 (t) dt
0
1
1 − cik
p
n
X
1 Y
1
M
− aM
A
a
−
r
−
ln
ω/aL
n+1
i,n+1
n+1,j
n+1,n+1 ,
ω
1
−
c
n+1,k
j=1
k=1
which implies
yi (Ti ) ≥ ln
p
n
h 1 X
1 Y
1
r
−
D
−
ln
i
j
ω
1
−
cik
aM
ii
j=1,j6=i
k=1
− aM
i,n+1 (A
n
X
p
aM
n+1,j − rn+1 −
j=1
i
1 Y
1
ln
)/aL
:= ρi .
n+1,n+1
ω
1 − cn+1,k
k=1
(2.17)
From (2.13) and (2.17) it follows that for i = 1, 2 . . . , n,
Z
ω
yi (t) ≥ yi (Ti ) −
|ẏi (t)|dt − | ln
0
k=1
Note that
Z ω
Z
aii (t)eyi (t) dt +
0
p
Y
p
Y
1
1
| ≥ ρi − di − | ln
|, (2.18)
1 − cik
1 − cik
k=1
ω
ai,n+1 (t)eyn+1 (t) dt ≥ (ri −
0
n
X
j=1,j6=i
Dj )ω − ln
p
Y
k=1
1
,
1 − cik
which, together with (2.5), leads to
aM
n+1,n+1 yn+1 (Tn+1 )ω
Z ω
n
X
≥
aL
eyj (t) dt − rn+1 ω
n+1,j
0
j=1
≥
n
X
aL
n+1,j
(rj −
Pn
k=1,k6=j
Dk −
j=1
− rn+1 ω − ln
p
Y
k=1
1
1 − cn+1,k
1
ω
ln
Qp
1
k=1 1−cjk
aM
jj
− aj,n+1 eyn +1(Tn+1 ) )ω
10
Q. LIU, S. DONG
EJDE-2005/31
which yields
yn+1 (Tn+1 ) ≥
aM
n+1,n+1
n
X
1
Pn
M
+ ( j=1 aM
n+1,j aj,n+1 )/ajj
j=1
p
n
X
aM
n+1,j rj −
×
Dk −
k=1,k6=j
1 Y
1 M
ln
/ajj
ω
1 − cjk
(2.19)
k=1
p
− rn+1 −
1 Y
1
ln
:= ρn+1 .
ω
1 − cn+1,k
k=1
From what has been discussed above, we finally derive that for i = 1, . . . , n,
max |yi (t)|
t∈[0,ω]
p
Y
≤ max | ln A| + di + | ln
k=1
p
Y
1
1
|, |ρi | + di + | ln
| := Bi ,
1 − cik
1 − cik
k=1
and
max |yn+1 (t)
t∈[0,ω]
≤ | max{| ln
A
Pn
j=1
1
aM
n+1,j − rn+1 ω ln
Qp
1
k=1 1−cn+1,k
an+1,n+1
|
p
Y
1
1
|, |ρn+1 | + dn+1 + | ln
| := Bn+1 .
1 − cn+1,k
1 − cn+1,k
k=1
k=1
(2.20)
Pn+1
Clearly, Bi (i = 1, 2, . . . , n + 1) are independent of λ. Denote B = i=1 Bi + B0 ,
∗
here B0 is taken sufficiently large such that each solution (v1∗ , v2∗ , ..., vn∗ , vn+1
)T of
the system of algebraic equations
+ dn+1 + | ln
ri −
n
X
Dj −
j=2
p
Y
p
n
X
1 Y
1
ln
− aii evi − ai,n+1 evn+1 +
Dj evj −vi = 0,
ω
1 − cik
k=1
j=1,j6=i
i = 1, 2, . . . , n,
p
−rn+1 −
n
X
1 Y
1
ln
+
an+1,j evj − an+1,n+1 evn+1 = 0,
ω
1 − cn+1,k j=1
k=1
∗
k(v1∗ , v2∗ , . . . , vn+1
)T k
satisfies
where for i = 1, 2, . . . , n,
=
Pn+1
∗
i=1 |vi |
< B (if it exists) and
Pn+1
p
n
1 X
1 Y
1
Ci = max | ln A0 |, ri −
Dj − ln
aii
ω
1 − cik
j=1,j6=i
− ai,n+1 (A0
n
X
j=1
an+1,j
k=1
− rn+1 )/an+1,n+1 i=1
(2.21)
Ci < B,
EJDE-2005/31
PERIODIC SOLUTIONS
11
and
Cn+1
= max | ln
A0
Pn
j=1
an+1,j − rn+1 −
1
ω
ln
Qp
1
k=1 1−cn+1,k
|,
an+1,n+1
Pn
Pn
Qp
1
)/aii − rn+1 −
i=1 an+1,i (ri − j=1,j6=i Dj − ω1 ln k=1 1−c
ik
Pn
an+1,n+1 + ( j=1 an+1,j aj,n+1 )/ajj
1
ω
ln
Qp
1
k=1 1−cn+1,k
(2.22)
in which
A0 = max {ri /aii }.
1≤i≤n
Now, we take Ω = {y ∈ X : kyk < B}. Thus, the condition (a) of Lemma 2.1 is
satisfied. When y ∈ ∂Ω ∩ ker L = ∂Ω ∩ Rn+1 , y = (y1 , y2 , ..., yn+1 )T is a constant
vector in Rn+1 with kyk = B. If system (2.21) has solutions, then
QN (y1 , . . . , yn , yn+1 )T


Qp
Pn
Pn
1
r1 − j=2 Dj − ω1 ln k=1 1−c
− a11 ey1 − a1,n+1 eyn+1 + j=2 Dj eyj −y1
1k

..



.
,

= 
Pn−1
Qp
Pn−1
1
1
yn
yn+1
yj −yn 
e
−
a
e
+
D
e

rn − j=1 Dj − ω lnQ k=1 1−cnk − aP
nn
n,n+1
j
j=1
p
n
1
+ j=1 an+1,j eyj − an+1,n+1 eyn+1
−rn+1 − ω1 ln k=1 1−cn+1,k
 
n 0 op  .. 
6= (0, . . . , 0, 0)T .
.
k=1
0
If system (2.21) does not have a solution, then we can directly derive


 
 
y1
n 0 on+1 n 0 op 

 .. 
 
QN  ...  6=
,  ... 
.
.
yn+1
0
i=1
0
k=1
Thus, the condition (b) in Lemma 2.1 is satisfied.
Finally, we will prove that the condition (c) in Lemma 2.1 is satisfied. To this
end, we define φ : DomL × [0, 1] → X by
φ(y1 , . . . , yn+1 , µ)


Qp
Pn
1
r1 − j=2 Dj − ω1 ln k=1 1−c
− a11 ey1
1k


..


.


=
Qp
Pn−1

1
1
yn
rn − j=1 Dj − ω ln k=1 1−cnk − ann e


Qp
Pn
1
1
−rn+1 − ω ln k=1 1−cn+1,k + j=1 an+1,j eyj − an+1,n+1 eyn+1
Pn


−a1,n+1 eyn+1 + j=2 Dj eyj −y1


..


.
+ µ
,
P
−ann eyn − an,n+1 eyn+1 + n−1 Dj eyj −yn 
j=1
0
where µ is a parameter. When (y1 , y2 , . . . , yn+1 )T ∈ ∂Ω ∩ Rn+1 , (y1 , y2 , . . . , yn+1 )T
is a constant vector in Rn+1 with kyk = B.
12
Q. LIU, S. DONG
EJDE-2005/31
We will show that when (y1 , y2 , . . . , yn+1 )T ∈ ∂Ω ∩ ker L, φ(y1 , y2 , . . . , yn+1 , µ) 6=
0. Otherwise, there is a constant vector (y1 , . . . , yn+1 )T ∈ Rn+1 with kyk = B
satisfying φ(y1 , y2 , . . . , yn+1 , µ) = 0, that is
ri −
n
X
Dj −
p
n
X
1 Y
1
ln
− aii eyi − µai,n+1 eyn+1 + µ
Dj eyj −yi = 0,
ω
1 − cik
k=1
j=1,j6=i
j=1,j6=i
i = 1, 2, . . . , n,
p
−rn+1 −
n
X
1 Y
1
an+1,j eyj − an+1,n+1 eyn+1 = 0.
ln
+
ω
1 − cn+1,k j=1
k=1
By similar argument in (2.10), (2.12), (2.17) and (2.19), we have
|yi | ≤ Ci , i = 1, . . . , n + 1,
Pn+1
Pn+1
where Ci is defined by (2.22). Thus, we have i=1 |yi | ≤ i=1 Ci < B, which is
leads to a contradiction. Using the property of topological degree and taking

  


y1
y1
n 0 op  ..   .. 


J : Im Q → X,
→  ...  ,
 .  , .
k=1
yn+1
0
yn+1
we have
deg(JQN (y1 , . . . , yn+1 )T , Ω ∩ ker L, (0, . . . , 0)T )
= deg(φ(y1 , . . . , yn+1 , 1), Ω ∩ ker L, (0, . . . , 0)T )
= deg(φ(y1 , . . . , yn+1 , 0), Ω ∩ ker L, (0, . . . , 0)T )
p
n
n
X
X
1 Y
1
= deg r1 −
Dj − ln
− a11 ey1 , . . . , rn −
Dj
ω
1 − c1k
j=2
j=2
k=1
−
+
1
ln
ω
n−1
X
p
Y
k=1
p
1
1 Y
1
− ann eyn , −rn+1 − ln
1 − cnk
ω
1 − cn+1,k
k=1
an+1,j eyj − an+1,n+1 eyn+1
T
, Ω ∩ ker L, (0, . . . , 0)T .
j=1
Under assumption (H1), one can easily show that the system of algebraic equations
ri −
n
X
p
Dj −
j=1,j6=i
1 Y
1
ln
− aii ui = 0,
ω
1 − cik
i = 1, 2, . . . , n,
k=1
p
(2.23)
n
X
1 Y
1
−rn+1 − ln
+
an+1,j uj − an+1,n+1 un+1 = 0,
ω
1 − cn+1,k j=1
k=1
has a unique solution (u∗1 , . . . , u∗n+1 )T which satisfies
Qp
Pn
ri − j=1,j6=i Dj − ω1 ln k=1
∗
ui =
aii
1
1−cik
>0
EJDE-2005/31
PERIODIC SOLUTIONS
13
for i = 1, . . . , n, and
u∗n+1 =
≥
n
X
1
an+1,n+1
j=1
1
ln
ω
1
n
X
− rn+1 −
k=1
1
1 − cn+1,k
aM
n+1,j
j=1
1
ln
ω
p
Dj −
n
X
rj −
j=1,j6=i
p
Y
k=1
1
1 − cn+1,k
1 Y
1 ln
/(ajj )
ω
1 − cik
k=1
j=1,j6=i
p
Y
− rn+1 −
an+1,n+1
n
X
an+1,j rj −
p
1 Y
1 Dj − ln
/(ajj )
ω
1 − cik
k=1
> 0.
A direct calculation shows that
deg(JQN (y1 , y2 , . . . , yn+1 )T , Ω ∩ KerL, (0, 0, . . . , 0)T )
= sgn
n+1
Y
(−aii ) = (−1)n+1 6= 0.
i=1
Finally, easily we show that the set {KP (I − Q)N u|u ∈ Ω̄} is equicontinuous and
uniformly bounded. Using Lemma 2.2 and the Arzela-Ascoli Theorem, we see that
KP (I − Q)N : Ω → X is compact. Moreover, QN (Ω̄) is bounded. Consequently,
N is L−compact.
We have proved that Ω satisfies all the requirements in Lemma 2.1. Hence,
system (2.2) has at least one ω-periodic solution. Accordingly, system (1.1) has at
least one positive ω-periodic solution. This completes the proof.
Remark. In this paper, by borrowing Gaines and Mawhin’s continuation theorem
of coincidence degree theory, we have established sufficient conditions for the existence of positive periodic solutions to system (1.1) with initial conditions (1.2).
We would like to mention here that it is interesting but challenging to discuss the
existence of positive periodic solutions of (1.1) when we incorporate time delays to
the self-regulated terms of the prey in n-patch environments. We leave this for our
future work.
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Qiming Liu
Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang
050003, China
E-mail address: llqm2002@yahoo.com.cn
Shijie Dong
Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang
050003, China
E-mail address: j ds@sina.com