Dynamics of a Discrete Predator-Prey System with Beddington-DeAngelis Function Response ()
1. Introduction
Since the end of the 19th century, many biological models have been established to illustrate the evolutionary of species, among them, predator-prey models attracted more and more attention of biologists and mathematicians. There are many different kinds of predator-prey models in the literature. In 1975, Beddington [1] and DeAngelis [2] proposed the predator-prey system with the Beddington-DeAngelis functional response as follows
(1.1)
Recently, Li and Takeuchi [3] proposed the following model with both Beddington-DeAngelis functional response and density dependent predator
(1.2)
and discussed the dynamic behaviors of the model.
On the other hand, when the size of the population is rarely small or the population has non-overlaping generation, the discrete time models are more appropriate than the continuous ones. Discrete time models can also provide efficient computational models of continuous models for numerical simulations.
In [4], Qin and Liu studied the dynamic behavior of the following discrete time competitive system
(1.3)
In [5], Wu and Li considered the following discrete time predator-prey system with hassell-varley type functional response
(1.4)
some sufficient conditions for the permanence and global attractivity of system (1.4) are obtained. For more work on this direction, one could refer to [6-14].
Based on the above discussion, in this paper, we consider the discrete analogous of (1.2), one can easily derive the discrete analogue of system (1.2), which takes the form of
(1.5)
In this paper, we always assume that
, 
,
, 
, 
, 
, 
, 
,
are all positive bounded sequences and
, 
,
, 
,
, 
,
, 
Here, for any bounded sequence
,
,
.
From the view point of biology, we will focus our discussion on the positive solutions of system (1.4). So it is assumed that the initial conditions of (1.4) are of the form
. (1.6)
It is easily to see that the solutions of (1.4) with the initial condition (1.5) are defined and remain positive for all
.
2. Permanence
DEFINITION 2.1. System (1.5) is said to be permanent, if there are positive constants
, 
such that each positive solution 
of system (1.5) satisfies
,
.
LEMMA 2.1. [6] Assume that 
satisfies 
and
for all
, where
, 
are positive sequences. Then
.
LEMMA 2.2. [6] Assume that 
satisfies
and
where
, 
are positive sequences. Then
.
LEMMA 2.3. Assume that 
holds, then for any positive solution 
of system (1.4)one has
and
.
Proof. Let 
be any positive solution of system (1.5), from the first equation of (1.5), it follows that
By Lemma 2.1, we obtain
.
Similarly, from the second equation of (1.5), it follows that
Under the assumption
, by Lemma 2.1, we obtain
.
This completes the proof of Lemma 2.3.
LEMMA 2.4. Assume that
. Then for any positive solution 
of system (1.5), one has
where
,
,
.
Proof. Let 
be any positive solution of system (1.5), from the first equation of (1.5), it follows that
Under the assumption
, By Lemma 2.2 and Lemma 2.3, we obtain
.
Similarly, from the second equation of (1.5) and Lemma 2.3, it follows that
By Lemma 2.2 and Lemma 2.3, we have
.
From Lemma 2.3 and Lemma 2.4, we obtain the following theorem.
THEOREM 2.1. Assume that
(2.1)
(2.2)
hold, then system (1.5) is permanent.
3. Global Attractivity
This section devotes to study the global attractivity of the positive solution of system (1.5).
DEFINITION 3.1. A positive solution 
of system (1.5) is said to be globally attractive if each other positive solution 
of (1.5) satisfies
.
THEOREM 3.1. In addition to (2.1) and (2.2), assume further that there exist positive constants
, 
and 
such that
(3.1)
and
(3.2)
Then the positive solution of system (1.5)is globally attractive.
Proof. From (3.1) and (3.2), there exists an enough small positive constant 
such that
(3.3)
and
(3.4)
For any positive solutions 
and 
of system (1.4), it follows from Lemma 2.3 and Lemma 2.4 that
(3.5)
In view of (3.5), for above
, there exists an integer 
such that, for all
,
(3.6)
Let
,
,
.
From the first equation of system (1.5), we have
By the mean value theorem, we have
(3.7)
where 
lies between 
and
. It follows from (3.7) that
and so, for 
(3.8)
Let
.
From the second equation of system (1.5), we have
By the mean value theorem, we have
(3.9)
where 
lies between 
and
. It follows from (3.9) that
and so, for 
(3.10)
Now we define a Lyapunov function as follows:
.
Calculating the difference of 
along the solution of system (1.5), for
, it follows from (3.8) and (3.10) that
It follows from (3.3) and (3.4) that
Summating both sides of the above inequalities from 
to
, we have
.
Which implies
.
Then
.
Therefore,
.
That is
.
This completes the proof of Theorem 3.1.
NOTES