Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 150, pp. 1–13.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
MULTIPLE POSITIVE PERIODIC SOLUTIONS FOR A
FOOD-LIMITED TWO-SPECIES RATIO-DEPENDENT
PREDATOR-PREY PATCH SYSTEM WITH DELAY AND
HARVESTING
HUI FANG
Abstract. By using Mawhin’s coincidence degree theory, this paper establishes some sufficient conditions on the existence of four positive periodic solutions for a food-limited two-species ratio-dependent predator-prey patch system with delay and harvesting. Some novel techniques are employed to obtain
the appropriate a priori estimates. An example is given to illustrate our results.
1. Introduction
Since the exploitation of biological resources and the harvest of population
species are related to the optimal management of renewable resources (see [5]),
many researchers pay much attention to the study of population dynamics with
harvesting in mathematical bioeconomics. For example, Brauer and Soudack [1, 2]
analyzed the global behaviour of some predator-prey systems under constant rate
harvesting and/or stocking of both species. Xiao and Jennings [15] studied the
Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting. But all the coefficients in the system they studied are constants. Recently,
some researchers studied the existence of multiple positive periodic solutions for
some predator-prey systems under the assumption of periodicity of the parameters
by using Mawhin’s coincidence degree theory (see [4, 7, 14, 17]). These papers only
focused on predator-prey systems without diffusion. However, due to the spatial
heterogeneity and unbalanced food resources, the migration phenomena of biological species can often occur between heterogeneous spatial environments or patches.
On the other hand, as pointed out by Huusko and Hyvarinen in [12]: ”the dynamics of exploited populations are clearly affected by recruitment and harvesting,
and the changes in harvesting induced a tendency to generation cycling in the dynamics of a freshwater fish population”. So, it is very important to describe the
potential role of harvesting as an external factor in producing and maintaining the
2000 Mathematics Subject Classification. 34C25, 92D25.
Key words and phrases. Coincidence degree; periodic solution; food-limited supply;
predator-prey patch system; harvesting rate.
c
2012
Texas State University - San Marcos.
Submitted April 18, 2012. Published August 29, 2012.
Supported by grants 10971085 and 11061016 from the National Natural Science Foundation
of China.
1
2
H. FANG
EJDE-2012/150
periodic fluctuation of population of species. Mathematically, this is equivalent to
the investigation of harvesting induced periodic solutions.
In this paper, we consider the following food-limited two-species ratio-dependent
predator-prey patch system with delay and harvesting:
h
a13 (t)x1 (t)y(t) i
x1 (t)
a1 (t) − a11 (t)x1 (t) −
k1 (t) + c1 (t)x1 (t)
m(t)y 2 (t) + x21 (t)
+ D1 (t)[x2 (t) − x1 (t)] − H1 (t),
x2 (t)
a2 (t) − a22 (t)x2 (t) + D2 (t)[x1 (t) − x2 (t)] − H2 (t),
x02 (t) =
k2 (t) + c2 (t)x2 (t)
h
i
a31 (t)x21 (t − τ )
y 0 (t) = y(t) − a3 (t) +
,
m(t)y 2 (t − τ ) + x21 (t − τ )
(1.1)
where x1 and y are the population numbers of prey species x and predator species y
in patch 1, and x2 is the population number of species x in patch 2. Prey dispersion
is included in the model to allow for prey refuge from predation. So, the predator
species y is confined to patch 1, while the prey species x can diffuse between two
patches. For biological relevance of allowing for prey refuge from predation, see
Yakubu [16]. a1 (t)(a2 (t)) is the natural growth rate of prey species x in patch 1
(patch 2) in the absence of predation, a3 (t) is the natural death rate of predator
species y in the absence of food, a13 (t) is the death rate per encounter of prey
species x due to predation, a31 (t) is the efficiency of turning predated prey species
x into predator species y. ki (t) (i = 1, 2) are the population numbers of prey species
in patch 1 (patch 2) at saturation, respectively.
ai (t)
When ci (t) 6= 0(i = 1, 2), ki (t)c
(i = 1, 2) are the rate of replacement of mass
i (t)
in the population at saturation (including the replacement of metabolic loss and
of dead organisms). The effect of delay τ refers to the dynamics of a predator
being related to the predation in the past. m(t) is the half capturing saturation
coefficient. aii (t) (i = 1, 2) are the density-dependent coefficients. Di (t) (i = 1, 2)
are diffusion coefficients of species x. Hi (t) (i = 1, 2) denote the harvesting rates.
x01 (t) =
a13 (t)x1 (t)y(t)
,
m(t)y 2 (t) + x21 (t)
a31 (t)x21 (t − τ )
m(t)y 2 (t − τ ) + x21 (t − τ )
are ratio-dependent Holling type III functional responses. When ci (t) ≡ 0 (i = 1, 2),
Hi (t) ≡ 0 (i = 1, 2), Dong and Ge in [6] showed that (1.1) has at least one positive
periodic solution under the appropriate conditions. When ci (t) 6= 0 (i = 1, 2),
system (1.1) is a food-limited population model. For other food-limited population
models, we refer to [9, 10, 11, 13] and the references cited therein.
To our knowledge, a few papers have been published on the existence of multiple
periodic solutions for (1.1). In this paper, we study the existence of multiple positive
periodic solutions of (1.1) by using Mawhin’s coincidence degree. Since system
(1.1) involves the diffusion terms and the rates of replacement, the methods used in
[4, 7, 14, 17] are not available to system (1.1). Our method of defining the operator
N (u, λ) facilitates the computation of Brouwer degree degB (JQN (·, 0)|ker L , Ω ∩
ker L, 0).
EJDE-2012/150
MULTIPLE POSITIVE PERIODIC SOLUTIONS
3
2. Existence of four positive periodic solutions
To give the proof of the main results, we first make the following preparations
(see [8]).
Let X, Z be normed vector spaces, L : dom L ⊂ X → Z a linear mapping, N :
X × [0, 1] → Z is a continuous mapping. The mapping L will be 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, there then exist continuous projectors
P : X → X and Q : Z → Z such that ImP = ker L, Im L = ker Q = Im(I − Q).
If we define LP : dom L ∩ ker P → Im L as the restriction L|dom L∩ker P of L to
dom L ∩ ker P , then LP is invertible. We denote the inverse of that map by KP .
If Ω is an open bounded subset of X, the mapping N will be called L-compact on
Ω̄ × [0, 1] if QN (Ω̄ × [0, 1]) is bounded and KP (I − Q)N : Ω̄ × [0, 1] → X is compact;
i.e., continuous and such that KP (I − Q)N (Ω̄ × [0, 1]) is relatively compact. Since
Im Q is isomorphic to ker L, there exists isomorphism J : Im Q → ker L.
For convenience, we introduce Mawhin’s continuation theorem [8, p.29] as follows.
Lemma 2.1. Let L be a Fredholm mapping of index zero and let N : Ω̄ × [0, 1] → Z
be L-compact on Ω̄ × [0, 1]. Suppose
(a) Lu 6= λN (u, λ) for every u ∈ dom L ∩ ∂Ω and every λ ∈ (0, 1);
(b) QN (u, 0) 6= 0 for every u ∈ ∂Ω ∩ ker L;
(c) Brouwer degree degB (JQN (·, 0)|ker L , Ω ∩ ker L, 0) 6= 0.
Then Lu = N (u, 1) has at least one solution in dom L ∩ Ω̄.
For the sake of convenience and simplicity, we denote
Z
1 T
g(t)dt, g l = min g(t), g u = max g(t),
ḡ =
T 0
t∈[0,T ]
t∈[0,T ]
where g is a nonnegative continuous T -periodic function. Set
( ak13
)u
1
b1 = √
, b2 = 0.
2 ml
For the rest of this article, we assume the following:
(A1) τ > 0, ki (t) (i = 1, 2), ai (t) (i = 1, 2, 3), aii (t) (i = 1, 2), a13 (t), a31 (t),
m(t), Di (t) (i = 1, 2), Hi (t) (i = 1, 2) are positive continuous T -periodic
functions, ci (t) (i = 1, 2) are nonnegative continuous T -periodic functions.
(A2) al31 − ā3 > 0.
q
a1 u a2 u ) ,(
) ,
N1 = max (
a11
a22
(A3)
kil
( akii )l > Diu + bi
kil +cu
i N1
Hil > Diu N1 (i = 1, 2).
+2
( akiii )u Hiu (i = 1, 2).
(A4)
For further convenience, we introduce the 12 positive numbers:
r
4kl
ai u
( ki ) ± [( akii )u ]2 − kl +cui N1 ( akiii )l (Hil − Diu N1 )
i
i
li± =
,
2kil
aii l
(
)
l
u
k
ki +ci N1
i
r
l
l
k
k
[ kl +ciu N1 ( akii )l − Diu − bi ] ± [ kl +ciu N1 ( akii )l − Diu − bi ]2 − 4( akiii )u Hiu
i
i
i
i
±
ui =
,
aii u
2( ki )
4
H. FANG
x±
i =
( akii ) ±
EJDE-2012/150
q
[( akii )]2 − 4( akiii )H̄i
2( akiii )
, i = 1, 2.
It is not difficult to show that
−
+
+
+
li− < x−
i < ui < ui < xi < li
(i = 1, 2)
(2.1)
Now, we are ready to state the main result of this article.
Theorem 2.2. Assume that (A1)-(A4) hold, then (1.1) has at least four positive
T -periodic solutions.
Proof. Since we are concerning with positive solutions of (1.1), we make the change
of variables,
xj (t) = euj (t) (j = 1, 2), y(t) = eu3 (t) .
Then (1.1) is rewritten as
h
a13 (t)eu1 (t) eu3 (t) i
1
u1 (t)
a
(t)
−
a
(t)e
−
u01 (t) =
1
11
k1 (t) + c1 (t)eu1 (t)
m(t)e2u3 (t) + e2u1 (t)
H1 (t)
eu2 (t)
− 1] − u (t) ,
u
(t)
1
e
e 1
(2.2)
1
eu1 (t)
H2 (t)
0
u2 (t)
u2 (t) =
[a2 (t) − a22 (t)e
] + D2 (t)[ u (t) − 1] − u (t) ,
k2 (t) + c2 (t)eu2 (t)
e 2
e 2
+ D1 (t)[
u03 (t) = −a3 (t) +
a31 (t)e2u1 (t−τ )
.
m(t)e2u3 (t−τ ) + e2u1 (t−τ )
Take
X = Z = {u = (u1 , u2 , u3 )T ∈ C(R, R3 ) : ui (t + T ) = ui (t), i = 1, 2, 3}
and define
kuk = max |u1 (t)| + max |u2 (t)| + max |u3 (t)|,
t∈[0,T ]
t∈[0,T ]
t∈[0,T ]
T
for u = (u1 , u2 , u3 ) in X or in Z. Equipped with the above norm k · k, it is easy
to verify that X and Z are Banach spaces.
Set
h k (t) + (1 − λ)c (t)eu1 (t) ih a (t) a (t)eu1 (t)
1
1
1
11
∆1 (u, t, λ) =
−
k1 (t)
k1 (t)
k1 (t) + c1 (t)eu1 (t)
−1
u1 (t) u3 (t) i
H1 (t)
λk1 (t) a13 (t)e
e
eu2 (t)
−
+ λD1 (t) u (t) − 1 − u (t) ,
2u
(t)
2u
(t)
3
1
1
m(t)e
+e
e
e 1
h k (t) + (1 − λ)c (t)eu2 (t) ih a (t) a (t)eu2 (t) i
22
2
2
2
−
∆2 (u, t, λ) =
k2 (t)
k2 (t)
k2 (t) + c2 (t)eu2 (t)
u
(t)
1
e
H2 (t)
+ λD2 (t) u (t) − 1 − u (t) ,
e 2
e 2
a31 (t)e2u1 (t−τ )
∆3 (u, t, λ) = −a3 (t) +
.
2u
m(t)e 3 (t−τ ) + λe2u1 (t−τ )
For any u ∈ X, because of the periodicity, we can easily check that ∆i (u, t, λ) ∈
C(R2 , R) (i = 1, 2, 3) are T -periodic in t.
Let
L : dom L = {u ∈ X : u ∈ C(R, R3 )} 3 u 7→ u0 ∈ Z,
EJDE-2012/150
MULTIPLE POSITIVE PERIODIC SOLUTIONS
T
1
T
Z
1
Q : Z 3 u 7→
T
Z
P : X 3 u 7→
5
u(t)dt ∈ X,
0
T
u(t)dt ∈ Z,
0
N : X × [0, 1] 3 (u, λ) 7→ (∆1 (u, t, λ), ∆2 (u, t, λ), ∆3 (u, t, λ))T ∈ Z.
Here, for any k ∈ R3 , we also identify it as the constant function in X or Z with
the constant value k. It is easy to see that
Z T
ker L = R3 , Im L = {u ∈ X :
ui (t)dt = 0, i = 1, 2, 3}
0
is closed in Z, dim ker L = codim Im L = 3, and P, Q are continuous projectors such
that
ImP = ker L, Im L = ker Q = Im(I − Q).
Therefore, L is a Fredholm mapping of index zero. On the other hand, Kp : Im L 7→
dom L ∩ KerP has the form
Z Z
Z t
1 T t
u(s) ds dt.
Kp (u) =
u(s)ds −
T 0 0
0
Thus,
Z
Z
T
1 Z T
1 T
1 T
∆1 (u, t, λ)dt,
∆2 (u, t, λ)dt,
∆3 (u, t, λ)dt ,
QN (u, λ) =
T 0
T 0
T 0
Kp (I − Q)N (u, λ) = (Φ1 (u, t, λ), Φ2 (u, t, λ), Φ3 (u, t, λ))T ,
where
Z
t
∆j (u, s, λ)ds −
Φj (u, t, λ) =
0
t
1
−( − )
T
2
Z
1
T
Z
T
Z
t
∆j (u, s, λ) ds dt
0
0
T
∆j (u, s, λ)ds,
j = 1, 2, 3.
0
Obviously, QN and Kp (I − Q)N are continuous. By the Arzela-Ascoli theorem, it
is not difficult to show that the closure of Kp (I − Q)N (Ω × [0, 1]) is compact for
any open bounded set Ω ⊂ X. Moreover, QN (Ω̄ × [0, 1]) is bounded. Thus N is
L-compact on Ω̄ × [0, 1] with any open bounded set Ω ⊂ X.
To apply Lemma 2.1, we need to find at least four appropriate open, bounded
subsets Ω1 , Ω2 , Ω3 , Ω4 in X.
Corresponding to the operator equation Lu = λN (u, λ), λ ∈ (0, 1), we have
h k (t) + (1 − λ)c (t)eu1 (t) ih a (t) a (t)eu1 (t)
1
1
1
11
u01 (t) = λ
−
k1 (t)
k1 (t)
k1 (t) + c1 (t)eu1 (t)
(2.3)
−1
u1 (t) u3 (t) i
λH1 (t)
λk1 (t) a13 (t)e
e
eu2 (t)
2
+ λ D1 (t) u (t) − 1 − u (t) ,
−
m(t)e2u3 (t) + e2u1 (t)
e 1
e 1
h k (t) + (1 − λ)c (t)eu2 (t) ih a (t) a (t)eu2 (t) i
2
2
2
22
u02 (t) = λ
−
k2 (t)
k2 (t)
k2 (t) + c2 (t)eu2 (t)
(2.4)
h eu1 (t)
i λH (t)
2
2
+ λ D2 (t) u (t) − 1 − u (t) ,
e 2
e 2
h
i
a31 (t)e2u1 (t−τ )
u03 (t) = λ − a3 (t) +
.
(2.5)
m(t)e2u3 (t−τ ) + λe2u1 (t−τ )
6
H. FANG
EJDE-2012/150
Suppose that (u1 (t), u2 (t), u3 (t))T is a T -periodic solution of (2.3), (2.4) and
m
(2.5) for some λ ∈ (0, 1). Choose tM
i , ti ∈ [0, T ], i = 1, 2, 3, such that
ui (tM
i ) = max ui (t),
t∈[0,T ]
ui (tm
i ) = min ui (t),
i = 1, 2, 3.
t∈[0,T ]
Then, it is clear that
u0i (tM
i ) = 0,
u0i (tm
i ) = 0,
i = 1, 2, 3.
From this and (2.3), (2.4), we obtain that
ih a (tM ) − a (tM )eu1 (tM
h k (tM ) + (1 − λ)c (tM )eu1 (tM
1 )
1 )
1 1
11 1
1 1
1 1
0=
M u1 (tM
1 )
k1 (tM
k1 (tM
1 )
1 ) + c1 (t1 )e
M
M i
M
M −1
M u1 (t1 ) u3 (t1 )
H1 (tM
eu2 (t1 )
λk1 (t1 ) a13 (t1 )e
)
e
M
−
−
1
− u (t1M ) ,
+
λD
(t
)
1
M
M
M
1
M
2u
(t
)
2u
(t
)
u
(t
)
3
1
1
1
1
1
1
1
m(t1 )e
+e
e
e
(2.6)
h k (tM ) + (1 − λ)c (tM )eu2 (tM
ih a (tM ) a (tM )eu2 (tM
i
2 )
2 )
2 2
2 2
2 2
22 2
0=
−
M u2 (tM
2 )
k2 (tM
k2 (tM
k2 (tM
2 )
2 )
2 ) + c2 (t2 )e
(2.7)
M
H2 (tM
eu1 (t2 )
2 )
M
+ λD2 (t2 ) u (tM ) − 1 − u (tM ) ,
e 2 2
e 2 2
h k (tm ) + (1 − λ)c (tm )eu1 (tm
ih a (tm ) − a (tm )eu1 (tm
1 )
1 )
11 1
1 1
1 1
1 1
0=
m)
m
m
m
u
(t
1
k1 (t1 )
k1 (t1 ) + c1 (t1 )e 1
m i
m
m u1 (tm
)
u
(t
)
m −1
3
u2 (t1 )
H1 (tm
λk1 (t1 ) a13 (t1 )e 1 e 1
)
m e
−
+
λD
(t
)
−
1
− u (t1m ) ,
1
m
m
m
1
2u3 (t1 ) + e2u1 (t1 )
u1 (t1 )
1 1
m(tm
e
e
)e
1
(2.8)
i
ih a (tm ) a (tm )eu2 (tm
h k (tm ) + (1 − λ)c (tm )eu2 (tm
2 )
2 )
2 2
2 2
2 2
22 2
0=
m) −
m )eu2 (tm
2 )
k
(t
k2 (tm
)
+
c
(t
k2 (tm
2
2
2
2 )
2
2
(2.9)
m
m
u1 (t2 )
H2 (t2 )
m e
+ λD2 (t2 ) u (tm ) − 1 − u (tm ) .
e 2 2
e 2 2
For u(tM
)
(i
=
1,
2),
there
are
two
cases
to consider.
i
M
M
)
≥
u
(t
),
then u1 (tM
Case 1. Assume that u1 (tM
2 2
1
1 ) ≥ u2 (t1 ). From this and
(2.6), we have
M u1 (tM
1 ) > 0,
a1 (tM
1 ) − a11 (t1 )e
which implies
M
a1 u
a1 (tM
1 )
) ≤ N1 .
eu1 (t1 ) <
≤(
M
a
a11 (t1 )
11
That is
a1 u
M
u2 (tM
) ≤ ln N1 .
2 ) ≤ u1 (t1 ) < ln(
a11
M
M
M
Case 2. Assume that u1 (tM
1 ) < u2 (t2 ), then u2 (t2 ) > u1 (t2 ). From this and
(2.7), we have
M u2 (tM
2 ) > 0,
a2 (tM
2 ) − a22 (t2 )e
which implies
M
a2 u
a2 (tM
2 )
≤(
) ≤ N1 .
eu2 (t2 ) <
a
a22 (tM
)
22
2
That is,
a2 u
M
u1 (tM
) ≤ ln N1 .
1 ) < u2 (t2 ) < ln(
a22
EJDE-2012/150
MULTIPLE POSITIVE PERIODIC SOLUTIONS
7
Therefore,
M
max{u1 (tM
1 ), u2 (t2 )} < ln N1 .
(2.10)
It follows from (2.6) that
ih a (tM ) a (tM )eu1 (tM
h k (tM ) + (1 − λ)c (tM )eu1 (tM
i H (tM )
1 )
1 )
1 1
1 1
1 1
1
11 1
− u (t1M )
−
M)
M)
M )eu1 (tM
1 1
1 )
k
(t
k
(t
k1 (tM
)
+
c
(t
e
1
1
1 1
1
1
1
M
M
h k (tM ) + (1 − λ)c (tM )eu1 (tM
i
M
−1
M
u
(t
)
)
u
(t
1
k1 (t1 ) a13 (t1 )e 1 1 e 3 1 )
1 1
1 1
=λ
M
M u1 (tM
2u3 (tM
1 )
1 ) + e2u1 (t1 )
k1 (tM
m(tM
1 ) + c1 (t1 )e
1 )e
−
u2 (tM
1 )
M e
λD1 (t1 ) u (tM )
1
e
1
+ λD1 (tM
1 ).
Therefore,
h k (tM ) + (1 − λ)c (tM )eu1 (tM
ih a
i
1 )
a11 u u1 (tM
H1u
1 1
1 1
1 l
1 ) −
)
−
(
)
e
(
M
M)
M
M
u
u
(t
k1
k1
k1 (t1 ) + c1 (t1 )e 1 1
e 1 (t1 )
h k (tM ) + (1 − λ)c (tM )eu1 (tM
i
1 )
)
a13 (tM
1 1
1 1
p1
<
+ D1 (tM
M)
1 ),
M
M
u
(t
M
1
k1 (t1 ) + c1 (t1 )e 1
2k1 (t1 ) m(tM
)
1
which implies
i
ih a
h k (tM ) + (1 − λ)c (tM )eu1 (tM
1 )
a11 u u1 (tM
H1u
1 1
1 1
1 l
1 ) −
)
−
(
)
e
(
M)
M
M
M
u
(t
u
k1
k1
k1 (t1 ) + c1 (t1 )e 1 1
e 1 (t1 )
i ( a13 )u
h k (tM ) + (1 − λ)c (tM )eu1 (tM
1 )
1 1
1 1
k1
√
+ D1u .
<
M )eu1 (tM
l
1 )
)
+
c
(t
k1 (tM
2
m
1
1
1
From this and noticing that
M
M u1 (t1
k1 (tM
k1 (tM
1 )
1 ) + (1 − λ)c1 (t1 )e
≤
M u1 (tM
M u1 (tM
1 )
1 )
k1 (tM
k1 (tM
1 ) + c1 (t1 )e
1 ) + c1 (t1 )e
)
≤ 1,
we have
( ak13
)u
a11 u u1 (tM
H1u
k1 (tM
a1 l
1 )
1
1 ) −
√
)
−
(
)
e
+ D1u .
(
<
M )eu1 (tM
u1 (tM
l
k
1 ) k1
1 )
k1 (tM
)
+
c
(t
e
1
2
m
1 1
1
Therefore, we have
( ak13
)u
k1l
a1 l
a11 u u1 (tM
H1u
1
1 ) −
√
(
)
−
(
)
e
<
+ D1u ;
M
k1
k1l + cu1 N1 k1
eu1 (t1 )
2 ml
that is,
(
h
i
M
k1l
a1
a11 u 2u1 (tM
1 ) −
) e
( )l − D1u − b1 eu1 (t1 ) + H1u > 0.
l
u
k1
k1 + c1 N1 k1
From (A3) and the above inequality, we have
+
u1 (tM
1 ) > ln u1
−
or u1 (tM
1 ) < ln u1
(2.11)
Similarly, we have
+
−
u1 (tm
or u1 (tm
1 ) > ln u1
1 ) < ln u1
By a similar argument, from (2.7), it follows that
h
i
M
k2l
a2
a22 u 2u2 (tM
2 ) −
) e
( )l − D2u − b2 eu2 (t2 ) + H2u > 0.
(
l
u
k2
k2 + c2 N1 k2
(2.12)
8
H. FANG
EJDE-2012/150
By (A3) and the above inequality, we have
+
u2 (tM
2 ) > ln u2
−
or u2 (tM
2 ) < ln u2 .
(2.13)
+
u2 (tm
2 ) > ln u2
−
or u2 (tm
2 ) < ln u2 .
(2.14)
Similarly, we have
Again, from (2.6) it follows that
ih a (tM ) a (tM )eu1 (tM
i
h k (tM ) + (1 − λ)c (tM )eu1 (tM
1 )
1 )
1 1
1 1
11 1
1 1
−
M
M u1 (t1 )
k1 (tM
k1 (tM
k1 (tM
1 )
1 )
1 ) + c1 (t1 )e
M
u2 (t1 )
H1 (tM
1 )
M e
−
D
(t
)
1
M
M .
1
eu1 (t1 )
eu1 (t1 )
Hence, we have
>
k1l
k1l
a11 (tM
a1 (tM
1 ) 2u1 (tM
1 ) u1 (tM
M u2 (tM
1 ) −
1 ) < 0,
e
e 1 ) + H1 (tM
1 ) − D1 (t1 )e
M
M
u
k1 (t1 )
+ c1 N1 k1 (t1 )
which implies
k1l
a1 u u1 (tM
a11 l 2u1 (tM
k1l
1 ) − (
)e
) e 1 ) + H1l − D1u N1 < 0.
(
k1
+ cu1 N1 k1
From (A3), (A4) and the above inequality, we have
+
ln l1− < u1 (tM
1 ) < ln l1 .
(2.15)
+
ln l1− < u1 (tm
1 ) < ln l1 .
(2.16)
Similarly, we have
By a similar argument, it follows from (2.7) that
k2l
a22 l 2u2 (tM
a2 u u2 (tM
2 ) − (
(
)e
) e 2 ) + H2l − D2u N1 < 0.
u
l
k
k2
k2 + c2 N1 2
From (A3), (A4) and the above inequality, we have
+
ln l2− < u2 (tM
2 ) < ln l2 .
(2.17)
+
ln l2− < u2 (tm
2 ) < ln l2 .
(2.18)
Similarly, we have
It follows from (2.11), (2.12), (2.15), (2.16) that
−
−
+
+
u1 (tM
1 ) ∈ (ln l1 , ln u1 ) ∪ (ln u1 , ln l1 ),
−
−
+
+
u1 (tm
1 ) ∈ (ln l1 , ln u1 ) ∪ (ln u1 , ln l1 ).
It follows from (2.13), (2.14), (2.17), (2.18) that
−
−
+
+
u2 (tM
2 ) ∈ (ln l2 , ln u2 ) ∪ (ln u2 , ln l2 ),
−
−
+
+
u2 (tm
2 ) ∈ (ln l2 , ln u2 ) ∪ (ln u2 , ln l2 ).
From (2.5), we have
Z T
Z
a3 (t)dt =
0
0
T
a31 (t)e2u1 (t−τ )
dt.
m(t)e2u3 (t−τ ) + λe2u1 (t−τ )
(2.19)
EJDE-2012/150
Therefore,
Z T
MULTIPLE POSITIVE PERIODIC SOLUTIONS
|u03 (t)|dt <
0
T
Z
Z
T
a3 (t)dt +
0
0
T
Z
=
Z
0
a31 (t)e2u1 (t−τ )
dt
m(t)e2u3 (t−τ ) + λe2u1 (t−τ )
T
a3 (t)dt +
9
a3 (t)dt
(2.20)
0
< 2T ā3 := d3 .
By (2.19), there must be η ∈ (0, T ) such that
ā3 =
a31 (η)e2u1 (η−τ )
.
m(η)e2u3 (η−τ ) + λe2u1 (η−τ )
(2.21)
From (2.21), we have
e2u3 (η−τ ) =
a31 (η) − λā3 2u1 (η−τ )
e
.
ā3 m(η)
Therefore,
au31 2
N := p,
ā3 ml 1
al − ā3
> 31 u (l1− )2 := q.
ā3 m
e2u3 (η−τ ) <
e2u3 (η−τ )
(2.22)
(2.23)
Since u3 (t) is a T -periodic function, there exists η1 ∈ [0, T ] such that
u3 (η1 ) = u3 (η − τ ).
Again, it follows from (2.20), (2.22), (2.23) that for any t ∈ [0, T ], we have
Z t
1
u3 (t) = u3 (η1 ) +
u03 (s)ds < ln p + d3 ,
2
η1
Z t
1
u3 (t) = u3 (η1 ) +
u03 (s)ds > ln q − d3 .
2
η1
Set
1
1
H = max{| ln p + d3 |, | ln q − d3 |}.
2
2
Then
max |u3 (t)| < H.
(2.24)
t∈[0,T ]
Clearly, li± , u±
i (i = 1, 2), H are independent of λ. Now, let us consider QN (u, 0)
with u = (u1 , u2 , u3 )T ∈ R3 . Note that


H̄1
u1
( ak11 ) − ( ak11
)e
−
u1
e
1


QN (u, 0) = ( ak22 ) − ( ak22
)eu2 − eH̄u22  .
2
2u1
31 e
−ā3 + ( am
) e2u3
Letting QN (u, 0) = 0, we have
a1
a11 u1 H̄1
)−(
)e − u1 = 0,
k1
k1
e
a2
a22 u2 H̄2
( )−(
)e − u2 = 0,
k2
k2
e
(
(2.25)
(2.26)
10
H. FANG
−ā3 + (
EJDE-2012/150
a31 e2u1
)
= 0.
m e2u3
(2.27)
From (2.25), we have
a11 2u1
a1
)e
− ( )eu1 + H̄1 = 0,
(2.28)
k1
k1
which implies that (2.28) has two distinct roots ln x±
1 . From (2.26), we have
a22 2u2
a2 u2
(
)e
− ( )e + H̄2 = 0,
(2.29)
k2
k2
(
which implies that (2.29) has two distinct roots ln x±
2 . Again, it follows from (2.27)
that
s
31
( am
) ±
x .
x±
=
3
ā3 1
Therefore, QN (u, 0) = 0 has four distinct solutions.
+
+ T
ũ1 = (ln x+
1 , ln x2 , ln x3 ) ,
−
+ T
ũ2 = (ln x+
1 , ln x2 , ln x3 ) ,
+
− T
ũ3 = (ln x−
1 , ln x2 , ln x3 ) ,
−
− T
ũ4 = (ln x−
1 , ln x2 , ln x3 ) .
Choose C > 0 such that
s
n
C > max | ln
31
( am
) +
x |, | ln
ā3 1
s
31
( am
) −o
x | .
ā3 1
Let
n
+
Ω1 = u = (u1 , u2 , u3 )T ∈ X : max u1 (t) ∈ (ln u+
1 , ln l1 ),
t∈[0,T ]
+
max u2 (t) ∈ (ln u+
2 , ln l2 ),
o
+
min u2 (t) ∈ (ln u+
2 , ln l2 ), max |u3 (t)| < H + C ,
min u1 (t) ∈
t∈[0,T ]
+
(ln u+
1 , ln l1 ),
t∈[0,T ]
t∈[0,T ]
t∈[0,T ]
n
+
Ω2 = u = (u1 , u2 , u3 )T ∈ X : max u1 (t) ∈ (ln u+
1 , ln l1 ),
t∈[0,T ]
max u2 (t) ∈ (ln l2− , ln u−
2 ),
o
min u2 (t) ∈ (ln l2− , ln u−
2 ), max |u3 (t)| < H + C ,
min u1 (t) ∈
t∈[0,T ]
+
(ln u+
1 , ln l1 ),
t∈[0,T ]
t∈[0,T ]
t∈[0,T ]
n
Ω3 = u = (u1 , u2 , u3 )T ∈ X : max u1 (t) ∈ (ln l1− , ln u−
1 ),
t∈[0,T ]
(ln l1− , ln u−
1 ),
+
max u2 (t) ∈ (ln u+
2 , ln l2 ),
o
+
min u2 (t) ∈ (ln u+
,
ln
l
),
max
|u
(t)|
<
H
+
C
,
3
2
2
min u1 (t) ∈
t∈[0,T ]
t∈[0,T ]
t∈[0,T ]
t∈[0,T ]
n
Ω4 = u = (u1 , u2 , u3 )T ∈ X : max u1 (t) ∈ (ln l1− , ln u−
1 ),
t∈[0,T ]
max u2 (t) ∈ (ln l2− , ln u−
2 ),
o
min u2 (t) ∈ (ln l2− , ln u−
2 ), max |u3 (t)| < H + C .
min u1 (t) ∈
t∈[0,T ]
t∈[0,T ]
(ln l1− , ln u−
1 ),
t∈[0,T ]
t∈[0,T ]
(2.30)
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MULTIPLE POSITIVE PERIODIC SOLUTIONS
11
Then Ω1 , . . . , Ω4 are bounded open subsets of X. It follows from (2.1) and (2.30)
that ũi ∈ Ωi (i = 1, 2, 3, 4). It is easy to see that Ω̄i ∩ Ω̄j = ∅ (i, j = 1, 2, 3, 4, i 6= j)
and Ωi satisfies (a) in Lemma 2.1 for i = 1, 2, 3, 4. Moreover, QN (u, 0) 6= 0 for
u ∈ ∂Ωi ∩ ker L.
Set
c
h(x) = b − ax − , x ∈ (0, +∞),
x
√
where a, b, c are positive constants. When b > 2 ac, it is easy to see that h(x) = 0
has exactly two positive solutions
√
√
b − b2 − 4ac
b + b2 − 4ac
−
+
x =
, x =
2a
2a
0 −
0 +
such that h (x ) > 0, h (x ) < 0. From this, a direct computation gives
deg{JQN (·, 0), Ω1 ∩ ker L, 0} = −1,
deg{JQN (·, 0), Ω3 ∩ ker L, 0} = 1,
deg{JQN (·, 0), Ω2 ∩ ker L, 0} = 1,
deg{JQN (·, 0), Ω4 ∩ ker L, 0} = −1.
Here, J is taken as the identity mapping since Im Q = ker L. So far we have proved
that Ωi satisfies all the assumptions in Lemma 2.1. Hence, (2.2) has at least four T periodic solutions (ui1 (t), ui2 (t), ui3 (t))T (i = 1, 2, 3, 4) and (ui1 , ui2 , ui3 )T ∈ dom L∩Ω̄i .
i
Obviously, (ui1 , ui2 , ui3 )T (i = 1, 2, 3, 4) are different. Let xij (t) = euj (t) (j = 1, 2),
i
y i (t) = eu3 (t) (i = 1, 2, 3, 4). Then (xi1 (t), xi2 (t), y i (t))T (i = 1, 2, 3, 4) are four
different positive T -periodic solutions of (1.1). The proof is complete.
Theorem 2.3. In addition to (A1), (A2), (A4), assume further that (1.1) satisfies
kl
1
i
(A3)* Hiu < NN
( ai )l − ( akiii )u − bi (i = 1, 2).
u
l
1 +1 k +c N1 ki
i
i
Then (1.1) has at least four positive T -periodic solutions.
Proof. It suffices to verify (A3) in Theorem 2.2. Indeed, it follows from (A3)* and
(A4) that
r
ai
aii
Hl
aii
kl
Diu + bi + 2 ( )u Hiu < i + bi + ( )u + Hiu < l iu ( )l ,
ki
N1
ki
ki + ci N1 ki
for i = 1, 2.
Finally, we describe the biological meaning of (A1)–(A4) and Theorem 2.2. In
realistic world, the environment is always varying periodically with time. This
motivates us to consider system (1.1) under the condition (A1). (A2) indicates that
the efficiency of turning predated prey species x into predator species y is higher
than the natural death rate of predator species y,qwhich is the necessary condition
for the survival of predator species y. Note that ( akiii )u Hiu is the geometric mean
of ( akiii )u and Hiu (i = 1, 2), which describes the mean effect of the intra-species
competition and harvesting. Hence, (A3) indicates that the natural growth rate of
prey species x with a food-limited supply in patch 1 is higher than the decay rate
due to the intra-species competition, predation, dispersion and harvesting, and that
the natural growth rate of prey species x with a limited food supply in patch 2 is
higher than the decay rate due to the intra-species competition, the dispersion and
harvesting. Since N1 is the environmental carrying capacity of prey species x, (A4)
indicates that the effect of harvesting on populations is stronger than the effect of
the dispersion during each time period. From the ecological viewpoint, Theorem
12
H. FANG
EJDE-2012/150
2.2 indicates that a food-limited two-species ratio-dependent predator-prey patch
systems with delay and harvesting can lead to four different periodic fluctuations
in a periodic environment if the regulated harvesting on prey populations is made
according to (A1)–(A4).
3. An Example
Take τ = 2, T = 2, k1 (t) = 4 + sin(πt), k2 (t) = 3 + sin(πt), ci (t) = 0.2 +
0.05 sin(πt) (i = 1, 2),
1 + sin2 (πt)
1 + sin2 (πt)
, D1 (t) = D2 (t) =
,
10
200
(4 + sin(πt))2
(4 + sin(πt))2
a1 (t) = (4 + sin(πt))2 , a11 (t) =
, a13 (t) =
,
4
4
(3 + sin(πt))2
,
m(t) = 1 + sin2 (πt), a2 (t) = (3 + sin(πt))2 , a22 (t) =
4
a31 (t) = 4 + sin(πt), a3 (t) = 2 + sin(πt).
H1 (t) =
1 + sin2 (πt)
,
24
H2 (t) =
Then we have k1l = 3, k2l = 2, cui = 0.25 (i = 1, 2),
1
1
1
1
1
H1u =
, H2u = , H1l =
, H2l =
, D1u = D2u =
,
12
5
24
10
100
a1
a11 u
5
a13 u
5
N1 = 4, ( )l = 3, (
) = , (
) = , ml = 1,
k1
k1
4
k1
4
a22 u
5
a2 l
) = 1, al31 = 3, ā3 = 2.
b1 = , ( ) = 2, (
8
k2
k2
It is easy to see that the conditions in Theorem 2.3 are satisfied. By Theorem 2.3,
system (1.1) has at least four positive 2-periodic solutions.
References
[1] F. Brauer, A. Soudack; Coexistence properties of some predator-prey systems under constant
rate harvesting and stocking, J. Math. Biol. 12 (1981) 101-114.
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systems, J. Theoret. Biol. 95 (1982) 247-252.
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responses, Nonlinear Anal. Real World Appl. 5 (2004) 45-53.
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[9] K. Gopalsamy, M. Kulenovic, G. Ladas; Environmental periodicity and time delays in a
food-limited population model, J. Math. Anal. Appl. 147 (1990) 545-555.
[10] S. Gourley, M. Chaplain; Travelling fronts in a food-limited population model with time delay,
Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 132A (2002) 75-89.
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13
[13] Y. Li; Periodic solutions of periodic generalized food limited model, Int. J. Math. Math. Sci.
25 (2001) 265-271.
[14] Y. Xia, J. Cao, S. Cheng; Multiple periodic solutions of a delayed stage-structured predatorprey model with non-monotone functional responses, Appl. Math. Modelling 31 (2007) 19471959.
[15] D. Xiao and L. Jennings; Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math. 65 (2005) 737-753.
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Hui Fang
Department of Mathematics, Kunming University of Science and Technology, Kunming,
Yunnan 650500, China
E-mail address: kmustfanghui@hotmail.com