Positive Periodic Solution for a Two-Species Predator-Prey System ()
1. Introduction
Dynamical systems generated by predator-prey models have long been the topic of research interest of many biomathematical scholars, and there have been vast studies to investigate the dynamics of predator-prey models, see e.g., Refs. [1] -[12] and references therein. In 1975, Beddington [13] and DeAngelis [14] proposed the predator-prey system with the Beddington-DeAngelis functional response as follows.
(1.1)
In the last years, some experts have studied the system [15] -[21] . Recently, Li and Takeuchi [22] proposed the following model with both Beddington-DeAngelis functional response and density dependent predator
(1.2)
and discussed the dynamic behaviors of the model. In this paper, we consider the following nonautonomous two-species predator-prey system with diffusion and time delays.
(1.3)
where 
represents the prey population in the ith patch
, and 
represents the predator population. 
denotes the dispersal rate of the prey in the ith patch
. We always make the following fundamental assumptions for system (1.3): 
is positive constant and
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
re positive continuous 
-periodic functions.
The main purpose of this paper is, by using the coincidence degree theory to derive the sufficient conditions for the existence of periodic solution of (1.3).
2. Preliminaries
The method to be used in this paper involves the applications of the continuation theorem of coincidence degree. we shall use some concepts and results from the book by Gaines and Mawhin [23] .
Let X, Z be real Banach spaces, 
be a linear mapping, and 
be a continuous mapping. The mapping L is called a Fredholm mapping of index zero if
and 
is closed in Z. If L is a Fredholm mapping of index zero and there exist continuous projectors 
and 
such that
, 
, then the restriction LP of L to 
is invertible. Denote the inverse of LP by
. If 
is an open bounded subset of X, the mapping N will be called L-compact on 
if 
is bounded and 
is compact. Since 
is isomorphic to
, there exists isomorphism
.
Lemma 2.1 (Continuation theorem [23] ) Let 
be an open bounded set, L be a Fredholm mapping of index zero and N be L-compact on
. Assume 1) for each 
,
;
2) for each 
3) 
Then 
has at least one solution in
.
Throughout this paper, we adopt the notations
, 
, 
where 
is an 
-periodic continuous function.
3. Main Result
Theorem 3.1 Assume that 1)
;
2)
;
3)
;
4)
.
Then system (1.3) has at least one positive 
-periodic solution.
Proof. Let
, 
, 
, 
, 
then (1.3) can be rewritten as follows:
(3.1)
where all function are defined as ones in system (1.3). It is easy to know that if (3.1) has one 
-periodic solution
, then 
is a positive 
-periodic solution of system (1.3) Therefore, to complete the proof , it suffices to show that system (3.1) has one 
-periodic solution.
Take 
and
, then X and Z are Banach space with the norm
.
Set
, 
, 
, 
,
. Obviously, 
,
is closed in Z and
. Therefore, L is a Fredholm mapping of index zero. Through an easy computation we find that the inverse 
of 
has the form
,
. Clearly, QN and 
are continuous. By using Arzela-Ascoli theorem, it is not difficult to prove that 
is compact for any open bounded set
. Moreover, 
is bounded. Therefore, N is L-compact on 
with any open bounded set
.
Corresponding to the operator equation
, 
, we have
. (3.2)
Suppose that 
is a solution of (3.2) for an appropriate
. Integrating (3.2) over the interval 
leads to
(3.3)
, (3.4)
. (3.5)
From (3.2)-(3.5), we have
(3.6)
(3.7)
(3.8)
Multiplying the first equation of (3.2) by 
and integrating over 
gives.
which implies
(3.9)
By using the inequalities
.
It follows from (3.9) that
, (3.10)
This yields
. (3.11)
By using the inequalities
.
It follows from (3.10) that
. (3.12)
Multiplying the second equation of (3.2) by 
and integrating over
, similarly, we can obtain
. (3.13)
Substitute (3.13) to (3.12), which leads to
So, there exist a positive constant 
such that
. (3.14)
It follows from (3.13) and (3.14) that there exist a positive constant 
such that
. (3.15)
Substitute (3.14), (3.15) to (3.6) and (3.7), which leads to
, (3.16)
. (3.17)
From (3.3) we have
(3.18)
From (3.4) we have
. (3.19)
It follows from (3.14), (3.15), (3.18) and (3.19) that there exist 
such that
, (3.20)
, (3.21)
. (3.22)
From (3.16), (3.17) and (3.20)-(3.22) we have
,
.
So, for 
we have
, (3.23)
. (3.24)
From (3.5) we have
So, there exist 
such that
. (3.25)
From (3.5) we also have
. (3.26)
It follows from (3.26) that there exist 
such that
.
So
. (3.27)
It follows from (3.8), (3.25) and (3.27) that for 
we have
.
So 
we have
.
Clearly, 
are independent of
. On other hand, we consider the following algebraic equation
(3.28)
Take
, where 
is large enough such that the solution 
of (3.28) satisfies
.
Let
, then 
satisfies the condition (1) in Lemma 2.1. When
, u is a constant vector in R3 and
. It follows from the definition of 
that
, so the condition (2) in Lemma 2.1 is satisfied. In order to verify the condition (3) in Lemma 2.1, we define 
by
where 
is a parameter. When
, u is a constant vector in 
and
. It is easy to obtain that
, then
. So, 
is a Homotopy mapping, due to homogoy invariance theorem of topology degree, we have
It is not difficult to see that the following algebraic equation
has a unique solution
Thus
By now we have proved the condition (3) in Lemma 2.1. This completes the proof of Theorem 3.1.
NOTES
*Corresponding author.