Permanence and Extinction of Periodic Delay Predator-Prey System with Two
Predators and Stage Structure for Prey
Wei-wei Zheng, Er-dong Han
School of Science, Xi’an Polytechnic University, Xi’an, Shaanxi, 710048, PR CHINA
(e-mail: zww@nwpu.edu.cn)


Abstract – In this paper, a periodic predator-prey delay
system with Beddington-DeAngelis and Holling IV functional response is proposed and analyzed, where prey has stage
structure and all three species are density dependent. Using
the comparison theorem and analytical method, sufficient
conditions of the permanence and extinction of the predators
and prey species are obtained. In addition, sufficient
conditions are derived for the existence of positive periodic
solutions of the system. According to the conclusions of the
theorems, two examples are given to check the correctness of
the main results.
here we let R0  ( x1 , x2 , x3 , x4 ) xi  0, i  1,2,3,4 .
Keywords – Beddington-DeAngelis functional response,
Delay, Extinction, Permanence, Stage Structure
According to the analysis of the above, we get the
following system (1.3).
h1 (t ) x1 (t )

 x1 (t )  a(t ) x2 (t )  b(t ) x1 (t )  B(t ) x2 (t  1 )   (t )  x 2 (t ) y1 (t ),
1
1

h
(
t
)
x
(
t
)

2
2
2
 x2 (t )  B(t ) x2 (t  1 )  c(t ) x2 (t )   (t )   (t ) x (t )   (t ) y (t ) y2 (t ),
2
2
2

(1.3)



 y1 (t )  y1 (t )  q1 (t )  p1 (t ) x1 (t   2 )  g1 (t ) y1 (t )  ,

1 (t )  x12 (t   2 )





p2 (t ) x2 (t   3 )

 y2 (t )  y2 (t )  q2 (t )   (t )   (t ) x (t   )   (t ) y (t   )  g 2 (t ) y2 (t )  .
2
2
3
2
3



The dynamic behavior of the predator-prey system
with delay and stage structure for prey has been long
discussed (see [1-4]). Recently, Wang [5] study the model
of this type, by using software Maple, the authors got the
corresponding numeric results of the conclusions. We can
refer to [6-14] to have in-depth understanding of more
research achievement of those models. Furthermore, the
research concerning the stage structure while being time
delays predator-prey periodic systems are quite rare.
Indeed, recently, Kar [15], Huang [16] and Chen [17]
project those systems on permanence and extinction. To
keep the biological variety of ecosystem, the dynamic
behavior of biotic population is a significant and
comprehensive problem in biomathematics. So it is
meaningful to investigate the system (1.3).
In the next section, we state the main results of this
paper. Sufficient conditions of the permanence and
extinction of the system (1.3) are proved in Section III.
The conclusions we obtain further promote the analysis
technique of Huang [16] and Chen [17].
I. INTRODUCTION
The aim of this paper is to investigate the permanence
and extinction of the following periodic delay three species
predator-prey system with Holling IV and BeddingtonDeAngelis functional response and stage-structure for prey
  b ( s )ds

h1 (t ) x1 (t )
t 
y1 (t ),
 x1(t )  a(t ) x2 (t )  b(t ) x1 (t )  a(t  1 )e 1 x2 (t  1 ) 
2

1 (t )  x1 (t )

t

 b ( s )ds
h2 (t ) x2 (t )
 x2 (t )  a(t  1 )e t1 x2 (t  1 )  c(t ) x22 (t ) 
y2 (t ),
 2 (t )   (t ) x2 (t )   (t ) y2 (t )

(1.1)

 y(t )  y (t )  q (t )  p1 (t ) x1 (t   2 )  g (t ) y (t )  ,
1
1
1 
 1
1
1 (t )  x12 (t   2 )






p2 (t ) x2 (t   3 )
 g 2 (t ) y2 (t )  ,
 y2 (t )  y2 (t )  q2 (t ) 
 2 (t )   (t ) x2 (t   3 )   (t ) y2 (t   3 )



t
x1 (t ) and x2 (t ) denote the densities of immature
and mature prey species at time t , respectively; y1 (t ) and
y2 (t ) denote the densities of the predators that prey on
immature and mature prey at time t , respectively;
a(t ), b(t ), c(t ), gi (t ), hi (t ), pi (t ), qi (t ), i (t ), i  1,2,  (t ),  (t )
are all continuous positive - periodic functions;  i ,
i  1, 2,3 are positive constants.
where
We can refer to [1-2] to get biological significance of
all parameters and assumptions explanation of (1.1).
The initial conditions for system (1.1) is as follow
xi ( s)  i ( s), yi ( s)   i ( s)  0, i (0)  0, i (0)  0,
i  1, 2, s  [ , 0].

(1.2)

where   max  1 , 2 , 3  ,   1 , 2 , 3 , 4   C [ ,0], R0 ,
4
4
For the convenient of the following discussion, to a
continuous - periodic function f (t ) , we set
m( f ) 
1 
f (t )dt .
 0
Meanwhile, we add definition
def

t

B(t )  a(t   1 ) exp   b( s)ds .
t 1
II. STATEMENT OF THE MAIN RESULTS
Theorem 2.1 Suppose that

p (t ) x1* (t   2 ) 
m   q1 (t )  1
  0,
1 (t )  x1*2 (t   2 ) 


(2.2)


p2 (t ) x (t   3 )
m   q2 (t ) 
  0,
*
 2 (t )   (t ) x2 (t   3 ) 

*
*
hold, where  x1 (t ), x2 (t )  is the unique positive periodic
*
2
solution of system (2.1) given by Lemma 2.2 (see [16]).
Then system (1.3) is permanent.
Theorem 2.2 Assume the condition (2.2) hold, there
is at least a positive  -periodic solution of system (1.3).
Theorem 2.3 Suppose that

p (t ) x1* (t   2 ) 
m  q1 (t )  1
  0,
1 (t )  x1*2 (t   2 ) 



p2 (t ) x2* (t   3 )
m  q2 (t ) 
  0,
*
 2 (t )   (t ) x2 (t   3 ) 

(2.3)
hold, then any solutions of system (1.3) with initial
condition (1.2) satisfies
lim yi (t )  0, i  1, 2 .
t 
every given  (0    1), there exists a
t 
for all solutions of system (1.3) with initial condition
(1.2).
Proof. Let ( x1 , x2 , y1 , y2 ) be a solution of system
(1.3) with initial conditions (1.2), so we have
 x1(t )  a(t ) x2 (t )  b(t ) x1 (t )  B(t ) x2 (t  1 ),

2
 x2 (t )  B(t ) x2 (t  1 )  c(t ) x2 (t ).
By using Lemma 2.2, the following auxiliary equation:
u1(t )  a(t )u2 (t )  b(t )u1 (t )  B(t )u2 (t  1 ),
(3.1)

2
u2 (t )  B(t )u2 (t  1 )  c(t )u2 (t ).
has a globally asymptotically stable positive - periodic

solution x1 (t ), x2 (t ) . Let  u1 (t ), u2 (t )  be the solution of
On the basis of (3.2) and (3.4), we have
xi (t )  xi* (t )   , t  T1 .
According to the comparison theorem (see [6]), we have
(3.2)
xi (t )  ui (t )(i  1, 2), t  0 .
 0 which is sufficiently small,

pi (t )  x (t )    
  0.
m  qi (t ) 


 i (t )


*
1

(3.5)

Let M x  max xi (t )   , i  1, 2 , we have
t[0, ]
*
lim sup xi (t )  M x , i  1, 2 .
t 
For t
 T1 , from system (1.3) and (3.5), we can obtain


p (t ) x (t   2 )
yi(t )  yi (t )  qi (t )  i i
 gi (t ) yi (t ) 
 i (t )




p (t )
 yi (t )  qi (t )  i  ui* (t   2 )     gi (t ) yi (t )  .
 i (t )


Consider the following equation:


p (t )
vi(t )  vi (t ) qi (t )  i  ui* (t   2 )     gi (t )vi (t )  . (3.6)
i (t )


analysis, there exists a T2  T1 , such that for the above  ,
we have
yi (t )  yi* (t )   , t  T2 .
t[0, ]

lim sup yi (t )  M y , i  1, 2 .
t 
To the same argument of Lemma 3.1, we can easily
get Lemma 3.2.
Lemma 3.2 there exists positive constant ix  M x ,
i  1, 2 , such that
lim inf xi (t )  ix , i  1, 2 .
t 
Lemma 3.3 Assumed that (2.2) holds, then there
exists two positives constants iy , i  1, 2 , such that any
solutions  x1 (t ), x2 (t ), y1 (t ), y2 (t )  of system (1.3) with initial
condition (1.2) satisfies
lim sup yi (t )  iy , i  1, 2 .
t 
(3.7)
Proof. Assume that condition (2.2) is establish, there
exists a constant  0
that
(3.3)

Let M y  max yi* (t )   , i  1, 2 , then
(3.1) with initial condition  u1 (0), u2 (0)    x1 (0), x2 (0)  ,
By (2.2), there exists a 
such that
(3.4)
*
lim sup xi (t )  M x , lim sup y (t )  M y , i  1, 2.

T1  0, such that
ui (t )  xi* (t )   , t  T1 .
M y , such that
*

periodic solution yi (t )  0, i  1, 2 . Similarly to the above
We need the Lemma3.1-3.4 to proof Theorem 2.1.
Lemma 3.1 There exist positive constants M x and
*
*
From the Lemma 2.2 of [16], (3.6) has a unique  -
III. PROOF OF THE MAIN RESULTS
t 
*
Thus, from the global attractive of x1 (t ), x2 (t ) , for

 0 , and  0 
1
min  xi* (t ) , such
t
2 [0, ]



m  0 (t )  0, m   0 (t )  0 ,
where
(3.8)
0 (t )  q1 (t ) 
p1 (t )  x1* (t   2 )   0 
1 (t )   x1* (t   2 )   0 
 0 (t )  q2 (t ) 
 q1 (t ) 0 ,
p2 (t )  x2* (t   3 )   0 
 q2 (t ) 0 .
 2 (t )   (t )  x2* (t   3 )   0    (t ) 0
Take the equation below with a parameter e  0 into
account:


h1 (t ) 
 x1(t )  a(t ) x2 (t )   b(t )  2e
 x1 (t )  B(t ) x2 (t  1 ),
1 (t ) 


(3.9)



 x (t )  B(t ) x (t   )  c(t )  2e h2 (t ) x 2 (t ).

 2
2
1
 2
 2 (t ) 


By Lemma 2.2, system (3.9) has a unique positive
- periodic

*
*

h (t ) 
x2 (t )  B(t ) x2 (t  1 ,  )   c(t )  2e1 2  x22 (t ,  ).
 2 (t ) 

Let  u1 (t ), u2 (t )  be the solution of (3.9), with e  e1
and  x1 (T6 ,  ), x2 (T6 ,  )  , then
xi (t ,  )  ui (t ), i =1,2, t  T6 .

we let  
xie (t )  xie (t ) 
, t  T4 .
4
*
We have xie (t )  xi (t ) in T4 , T4    , as e  0 .Then,
for  0
 0 , such that

xie (t )  xi* (t )  0 , t  T4 , T4    , 0  e  e0 .
4
So, we can get
xie* (t )  xi* (t )  xie (t )  xie* (t )  xie* (t )  xi* (t ) 
*
*
Since xie (t ), xi (t ) are all
xie* (t )  xi* (t ) 
0
0
.
2
 -periodic, hence
, i  1, 2, t  0, 0  e  e0 .
2
Choosing a constant e1 (0  e1  e0 , 2e1   0 ) , we have
xie* (t )  xi* (t ) 
0
2
, there exists T7
0
2
 T6 , such that
, i  1, 2, t  T7 .
xi (t ,  )  ui (t )  xie*1 (t ) 
0
, i =1,2, t  T7 .
2
Hence, by (3.10), we can obtain
xi (t ,  )  xi* (t )   0 , i  1, 2, t  T7 .
*
 0 , there exists a sufficiently large T4  T3 , such that
0
*
2
So, we have

initial condition xie (0)  xi (0), i  1, 2 . Then, for the above
0
ui (t )  xie*1 (t ) 
solution x1e (t ), x2 e (t ) , which is global
attractive. Let  x1e (t ), x2e (t )  be the solution of (3.9) with
Therefore, by (3.11) and (3.12), for t
y1(t ,  )
(3.10)
Assuming (3.7) is false, then there exists   R , such
4

that, under the initial condition  x1 ( ), x2 ( ), y1 ( ), y2 ( )    ,
   ,0 . We have lim sup yi (t ,  )  e1 , i  1, 2 .
t 
 T4 , such that
yi (t ,  )  2e1   0 , t  T5 .
(3.11)
By using (3.11), from system (1.3), for all t  T6  T5  1 ,
So, there exists T5
we can obtain

h (t ) 
x1(t )  a(t ) x2 (t ,  )   b(t )  2e1 1  x1 (t ,  )  B(t ) x2 (t  1,  ),
1 (t ) 

(3.12)
 T7   2 , such that


p1 (t )  x1* (t   2 )   0 
 y1 (t ,  )  q1 (t ) 

g
(
t
)

1
0
1 (t )   x1* (t   2 )   0 


  0 (t ) y1 (t ,  );
y2 (t ,  )


p2 (t )  x2* (t   3 )   0 

 y2 (t ,  )  q2 (t ) 

g
(
t
)

2
0
 2 (t )   (t )  x2* (t   3 )   0    (t ) 0


  0 (t ) y2 (t ,  ).
(3.13)
  2 to t and
Integrating both sides of (3.13) from T7
from
T7   3 to t , respectively, so we can get
y1 (t ,  )  y1 (T7   2 ,  ) exp 
t
T7  2
, i =1,2, t  0 .

From the global attractive of x1e1 (t ), x2 e1 (t ) , here
y2 (t ,  )  y2 (T7   3 ,  ) exp 
t
T7  3
Thus, from (3.11), we obtain
 (t )dt ,
0
  (t )dt.
0
yi (t ,  )  , i  1, 2,
t   . It is a contradiction of the Lemma 3.1. So the
proof of the theorem 3.3 is complete.
Lemma 3.4 Under the condition (2.2), there exist
positive constants  iy , i  1, 2 , such that any solutions of
system (1.3) with initial condition (1.2) satisfies
lim inf yi (t )   iy , i  1, 2 .
t 
(3.14)
Proof of Theorem 2.1 By Lemma 3.2 and 3.3,
system (1.3) is uniform weak persistent. Further, from the
Lemma 3.1 and 3.4, system (1.3) is persistent.
Proof of Theorem 2.2 From the proof of Lemma 3.1-
3.4 in Theorem 2.1, using the same method, we can proof
the Theorem 2.2. Here we omit the detail of certificate
process.
Proof of Theorem 2.3 Actually, by (2.3), for any
given positive constant  (  1), there exist 1  0
(0  1   ) and  0  0 , we get the following (3.15).


p1 (t )  x1* (t   2 )  1 


m q1 (t ) 
 g1 (t )    m  q1 (t )    0 ,
2

 2
1 (t )   x1* (t   2 )  1 


Since
x1(t )  a(t ) x2 (t )  b(t ) x1 (t )  B(t ) x2 (t   1 ),
x2 (t )  B(t ) x2 (t   1 )  c(t ) x22 (t ).
(1)
 0 , such that
xi (t )  x (t )  1 , t  T (1) .
*
i
It
follows
from
(3.15)
and
t  max T   2 , T   3 ,
(1)
(3.16)
(3.16)
that
for
(2)


p (t ) x (t   )
m  q1 (t )  1 1 2 2  g1 (t )    0 ,
1 (t )  x1 (t   2 )


(2)
(3.17)
)   (i  1, 2) . Otherwise, by (3.17), we
have
  y1 (t )
 t 
 
h (t ) x (s   2 )
 y1 (T (1)   2 )exp  (1)  q1 (s)  1 1
 q1 (s)  ds 
1 (s)  x1 (t   2 )
 T  2 
 


 y1 (T   2 )exp  0  t  T   2   0.
(1)
(1)
  . Similarly, we can get
  y2 (t )  y2 (T (1)   3 )exp  0  t  T (1)   3   0, t   .
as t


So we have   0 , which is contradictions.
Second, we will prove that
yi (t )   exp M ( ) , i  1, 2, t  T (2) ,
(3.18)
where


p (t ) x (t   j )
M ( )  max  qi (t )  i 1
 gi (t ) , i, j  2,3, i  j  ,

t[0, ] 
 i (t )  x1 (t   j )


is a bounded constant for   0,1 . Otherwise, then there
exists a T
(3)
 T (2) , we can obtain
yi (t ) , then there must exists
T (4)  T (2) , T (3)  , such that yi (T (4) )   and yi (t )   ,

 . Let P1 be the nonnegative integer such
(3)
(4)
(3)
that T  T  P1 , T  ( P1  1)  . From (3.17), we have
for t  T , T
(4)
(3)
 
h (t ) x (t   )
 T (3) 
 y1 (T (3) )  y1 (T (4) )exp   (4)  q1 (t )  1 1 2 2  g1 (t )  dt 
T
1 (t )  x1 (t   2 )
 
 




h1 (t ) x1 (t   2 )
 g1 (t )  dt
 q1 (t ) 
2
1 (t )  x1 (t   2 )


 T (3) 
 
h (t ) x (t   )
  exp   (4)  d 2 (t )  1 1 2 2  q(t )  dt 
1 (t )  x1 (t   2 )
 T  P1 
 
  exp M ( ) .
  exp
T (4) +P1
T (4)
T (3)
  (4)
T  P1
We can see that this is a contradiction. Similarly, from
the second equation of (3.17), we have
 exp M ( )  y2 (T (3) )   exp M ( ) .


p2 (t ) x2 (t   3 )
m  q2 (t ) 
 g 2 (t )    0 .
 2 (t )   (t ) x2 (t   3 )


(2)
(1)
(2)
First, there exists a T  max T   2 , T   3  ,
such that yi (T
By the continuity of
 exp M ( )

 
p2 (t )  x1* (t   3 )  1 
   m  q2 (t )    0 .
m  q2 (t ) 

g
(
t
)

2
*

 2

(
t
)


(
t
)
x
(
t


)




2
1
2
1


For the above  1 , there exists a T
yi (T (3) )   exp M ( ) , i  1, 2 .
Which is also contradiction, so (3.18) holds. By the
random of the parameter  , we know yi (t )  0, i  1,2,
t   . So we complete the proof of Theorem 2.3.
IV. EXAMPLES AND CONCLUSION
Example 1 From system (1.3), cause a(t )  4, b(t )  2 / 3,
c(t )  9exp0.3 1  exp0.3 , q1(t )  1/ 4  cos t, q2 (t)  1/ 5  cos t,
h1 (t )  5, h2 (t )  6, p1 (t )  2  cos t, p2 (t )  3  cos t, 1 (t )  1,
2 (t )  1 4 1  exp0.3 ,  (t )  7  2cos t,  (t)  1,1   2   3  0.6,
gi (t ), i  1,2 are any arbitrary nonnegative continuous
2 -periodic functions.
The above parameters conditions satisfy Theorem 2.1,
so system (1.3) is permanent and admits at least a positive
2 - periodic solution. From Fig. 1, we can see that the
density restriction of the predators have a major impact on
the stability of the predator-prey system. When predator
species have no crowding effect, the predator species is at
high density; and with crowding effect, the predator
species is at low density.
Example 2 Assuming that the conditions of example
1 are established. Causing q1 (t )  5/ 4  cos t , q2 (t )  1/ 2  cos t ,
those parameters satisfy the Theorem 2.3. So any positive
solution of system (1.3) satisfies
lim yi (t )  0, i  1,2 .
t 
Fig. 2 shows that two predators are extinction and the
immature and mature preys are permanent.
[2]
[3]
[4]
[5]
[6]
Fig.1 The growth curve of system (1.3) with initial condition
 x1 ( ), x2 ( ), y1 ( ), y2 ( )   1  exp 0.5,0.5,1, 2  , g1(t)  2  sin t, g2 (t)  2  sin t ,
0  t  50, 0.6    0.
[7]
[8]
[9]
[10]
[11]
Fig.2 The growth curve of system (1.3) with initial condition
 x1 ( ), x2 ( ), y1 ( ), y2 ( )   1  exp0.5,0.5,1,2 ,0  t  50, 0.6    0.
From the Theorem 2.1-2.3, we can get a conclusion:
the death rate and the density restriction of the two
predator population have a great extent influence on the
dynamic behavior of the system.
[12]
ACKNOWLEDGMENT
[14]
This paper is supported by the Natural Science Fund of
Shaanxi Provincial Education Administration Bureau
(Grant No.11JK0502) and the Doctor’s Research Fund of
Xi’an Polytechnic University.
[15]
[13]
[16]
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