The Boundedness, Existence, Uniqueness and Stability of a Delayed Reaction diffusion Predator prey Model

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2017 International Conference on Mathematics, Modelling and Simulation Technologies and Applications (MMSTA 2017) ISBN: 978-1-60595-530-8

The Boundedness, Existence, Uniqueness and Stability of a Delayed

Reaction-diffusion Predator-prey Model

Yu JIANG

1,*

and Hui-ming WEI

2

1_{Department of Public Courses, Shenzhen Institute of Information Technology, Shenzhen 518172,} China

2_{China nuclear power simulation Technology Company limited, Shenzhen 518031, China} *Corresponding author

Keywords: Reaction-diffusion, Time delay, The method of upper and lower solutions, Stability.

Abstract. In this article, a delayed reaction-diffusion predator-prey model with stage structure is constructed. The boundedness, existence and uniqueness of the model is investigated. The existence and uniqueness of the global solution of the system are proved. The local and global stability of the constant equilibria are discussed by the linearization method and the method of upper and lower solutions, respectively.

Introduction

In recent years, there are many authors who are interesting in studying the dynamic behavior of the predator-prey system by taking into account the effect of reaction-diffusion[1-3]. We remark that the spacial diffusion, which is that each species natural tendency is to move from the areas of bigger population concentration to ones of smaller population concentration, is not considered in [4]. The kind of diffusion process is called free diffusion. In [5] discussed the delayed periodic ratio-dependant predator-prey model with prey dispersal and stage structure for predator. In this paper, we consider the system (1.1) with diffusion terms and the harvesting effort:

1 1 2

1

1 1 2

1 2

( , ) ( , )

1 1 1 1 1 ( , )

( , ) ( , )

2 1 1 ( , ) 2 1

2

3 2 2 3 2 2 4 2

( , )(

( , ))

,

0,

( ,

)

( , ),

,

0,

( ,

)

( , )

( , ),

,

0

V V x t u x t

t cV x t

u V x t u x t

t cV x t

u t

D V

V x t a bV x t

x

t

D u

e

u x t

u x t x

t

D u

e u x t

d u x t

Eu x t

d u x t x

t



 















 

 

 

 



  



 



 



 



 



 

1 1 1 2 2 3

,

( , )

( , 0), ( , )

( , ),

,

[ , 0].

V x t



x

u x t



x

u x t



x t x

t













  



(1.1)

1( , )

V x t denotes the density of prey, u x t1( , ), u x t2( , ) represent the immature and mature predator

densities respectively.



is the capturing rate of the mature predator, 1

 is the conversion rate of

nutrients into the reproduction of the predator (

 

 ₁ 0).



0 is the intrinsic growth rate of prey. b0 is the coefficient of intraspecific competition.

 

, , ,d d3 4 and c are all positive

constants. We assume that at any time t0, birth into the immature population is proportional to the existing mature population with proportionality constant



. Assuming that the death rate of immature population is proportional to the existing immature population with proportionality constant

 

( d3), d3 are called the death coefficient of u x t1( , ), u x t2( , ) respectively. We

assume that the death rate of mature populations are of a logistic nature, that is proportional to the square of the population with proportionality constant d4. And 0 E 1 is the effect of continues

harvesting predator.



represents a constant time to maturity.

Note that V x t1( , ) and u x t2( , ) of system (1.1) are independent of u x t1( , ) but determine the

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convenience, we still denote V x t1( , ), u x t2( , ) by u x t1( , ), u x t2( , ) the initial functions



1( , 0)x , 3( , )x t



by



1( , 0)x ,



2( , )x t , respectively, and yield the following system (1.2):

1 1 2

1 2

( , ) ( , )

1 1 1 1 1 ( , )

2

2 2 2 3 2 2 4 2

1 1 2 2

( , )(

( , ))

,

( ,

)

( , )

( , ),

,

0,

,

0,

( , 0)

( , 0), ( , )

( , ),

,

[ , 0].

u u x t u x t

t cu x t

u t

u u

n n

D u

u x t a bu x t

D u

e u x t

d u x t

Eu x t

d u x t x

t

x

t

u x

x

u x t

x t

x

t

 











 

 

    

   







 



 





 





  



(1.2)

The remaining parts of the paper are organized as follows. In next section, we prove the existence and uniqueness of the global solution of the system (1.2). In section 3, we analyze the locally asymptotical stability of the constant equilibria and obtain the conditions of stability. Finally, we give the summary of this work.

Boundedness, Existence and Uniqueness For convenience, we set:

1 2

1 1 2 1 1

1

( , ) ( , )

( , ) ( , )( ( , )) ,

1 ( , )

u x t u x t

F u u u x t a bu x t

cu x t



  

 (2.1)

2

(

2

, )

2 2

( ,

)

3 2

( , )

2

( , )

4 2

( , ),

F u u

_





e u x t



 



d u x t



Eu x t



d u x t

(2.2) then we can rewritten the system (1.2) as the following form:

1 2 1 2

1 1 1 1 2

2 2 2 2 2

1 1 2 2

( , ),

,

0

(

, ),

,

0

0,

,

0,

( , 0)

( , 0), ( , )

( , ),

,

[ , 0].

u t u t

u u

n n

D u

F u u

x

t

D u

F u u

x

t

x

t

u x

x

u x t

x t

x

t







  



 

   

 





 

 







 





  



(2.3)

Let

 

1 2 ...



n... be the eigenvalues of the operator-Δ on Ω with the homogeneous Neumann boundary condition, and denote

E

( )



_i by the eigenfunction space corresponding to



_i in C1( ) . It is well known that



i0 and the corresponding eigenfunction



10. Let





_ij| j1,2,...,dim ( )E



_i



be a group of orthogonal basis of



1 2



1 2

( ), ( , ) |T [ ( )]

i

E



X u u u u C  and



2



1 2

| ( , )T i ij

X  C



C c c R then X i1Xi

 

 and dim ( )2

1

E

i j ij

X _  X .

Lemma 2.1.If u x t v x t C( , ), ( , ) ( [0, ])T C2,1([0, ])T satisfy

2

1 3 4

2

1 3 4

( ,

)+

( , )

( ,

)+

( , )

( , ),

,

[0, ],

, ( , )

( , ),

,

[ ,0],

u t

v t

u v

n n

D u

e

u x t

d u x t

Eu x t

d u x t

D v

e

v x t

d v x t

Ev x t

d v x t x

t

T

u x t

v x t x

t

 









 



 

    

   





   





 



 



  



(2.4)

then there holds

( , ) ( , ),( , ) [0, ].

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2

1 3 4

2

3 4

( ,

) (

) ( , )

( , )

(

( ,

) (

) ( , )

( , ))

=

( ,

) (

( ( , )

( , ))) ( , ).

W

D W

e u x t

d

E u x t

d u x t

t

e

v x t

d

E v x t

d v x t

e W x t

d

E d u x t

v x t W x t



 















 



  

 









 





 

 



(2.6)

Letting c x t11( , ) e 0, ( , )b x t11 (d E d u x t v x t3 4( ( , ) ( , ))) 





       using (2.6) and Lemma 3.1 in [6], we have W x t( , ) 0 , which implies that (2.5) holds.

Theorem 2.1. Assume that 2,1

1( , ), ( , )2 ( [0, ]) ( [0, ])

u x t u x t  C T C  T is a pair of solutions to

the system (2.3), then there holds

1 1 2 2

0



u x t

( , )



M

,0



u x t

( , )



M

,( , )

x t



[0, ],

T

(2.7)

where M1max



1 ,ab and

3 4

2 max 2 ,

e d E d

M 



_    .

Proof. Let 0 



T. Before proving Theorem 2.1, we firstly consider the following system:

1 2

1 1 1 1

2

2 2 2 3 2 2 4 2

1 1 2 2

( , )(

( , )),

,

[0, ],

( ,

)

( , )

( , ),

,

[0, ],

0,

,

[0, ],

( , 0)

( , 0) 0,

( , )

( , ) 0,

,

[ , 0].

t

n n

D

x t a b

x t

x

t

T

D

e

x t

d

x t

E

x t

d

x t x

t

T

x

t

T

x

x t

x

t



 

 



















 

 



 



 





 









 



 







 





  



_(2.8)

Applying Lemma 3.1 in [6] to the system (2.8), we have

( , ) 0,( , )

[0, ],

1, 2.

i

x t

i











(2.9) From the system (2.8), we obtain that



2( , )x t is bounded in [0, ]



for all



(0 



T).

Since



2( , )x t satisfies the homogeneous Neumann boundary condition, hence, if

[0, ] 2 2

max_ _



( , )x t 



_ then there exists

( , )

x t

₀ ₀



[0, ]



such that

2( , ) maxx t0 0 [0, ] 1( , )x t 1 .



 _







_ (2.10) Thus, using (2.10) and Lemma 2.3 in [7], we have

0 0

2

2( , ) 3 2( , ) 2( , ) 4 2( , ) |( , )x t 0,

e  x t d x t E x t d x t







_{ }





_



_



_ _(2.11)

which implies that

3 4

2( , )0 0 .

e d E d

x t  



 _{ }

 (2.12) Combining (2.9) and (2.12) yields



3



4

2 2

0 ( , ) max , e d E ,( , ) [0, ].

d

x t   x t



 _{ }





   (2.13)

Similarly, there exists

( , )

x t

₀' ₀'



[0, ]



such that ' '

1( , ) maxx t0 0 [0, ] 1( , )x t



 _



and

0 0

1

( , )(

x t a b

1

( , )) |

x t

( , )x t

0.

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1 1 2 2

0





( , )

x t



M

,0





( , )

x t



M x t

,

( , )



[0, ].

T

(2.15) Using Lemma 3.1 in [6] and Lemma 2.1, we obtain

0



u x t

_i

( , )





_i

( , ),( , )

x t x t



[0, ],

T i



1,2.

(2.16) Combining (2.15) and (2.16) implies (2.7).

Theorem 2.2. If



e



  

d

₃

E



*_{, then the system (2.3) has a unique pair of nonnegative}

solutions ( , )u u1 2 for all

t



0

, where 2

* 1 _{( , 2)}

c



  , Moreover, u x t₁( , ) and u x t₂( , ) are locally bounded in _L2_{( )}_ _and_{(0, 0)}_{is not a global attractor in} _L2_{( )}_ _.

Proof. Since it is quite standard to show the local existence and uniqueness of the solutions to the system (2.3), so we omit its proof here. In order to show the global existence, it suffices to establish

L∞_{-estimates. Let} *

1 1

ˆ ( , )=

t

( , )

u x t e



u x t

_and *

2 2

ˆ ( , )=

t

( , )

u x t e



u x t

_then

1 2

ˆ ˆ

( ( , ), ( , ))u x t u x t satisfies

* *

1 1 2

* 1

* *

2 1 2

*

ˆ * 2 ˆ ˆ

1 1 1 1 1 ₁ _ˆ

ˆ * 2

2 2 2 2 3 2 2 4 2

ˆ

1 1

ˆ

,( , )

[0, ],

ˆ

( ,

)-

ˆ

,( , )

[0, ],

0, ( , )

[0, ],

ˆ ( , )

( , 0)

t t

u t e u u

t _{ce u}

u t

t

u u

n n

t

D u

u

au

be u

x t

T

D u

e

u x t

u

d u

Eu

d e u

x t

T

x t

T

u x t

e

x

 

 

   





 





 _

  



 



   









 













0, ( , )

u x t

ˆ

₂



₂

( , ) 0,

x t

x

,

t

[ , 0].

















  



(2.17)

By Lemma 2.1 and Theorem 2.1, we have









1 1 1 2 2 2

[0, ]

ˆ

[0, ]

ˆ

sup

_ _T

u x t

( , ) max





_L

,

N

,sup

_ _T

u x t

( , ) max





_L

,

N

,

(2.18)

Where ₁ a *_T*

be

N



_ _and * * 3 * 4

2 T

e d E

d e

N       _ . If



e



  

d

₃

E



*_{, we know that}

1 0

N  and

2 0

N  cannot hold simultaneously. Thus, from (2.18) it follows that there exists a constant

c



0

such that

1 2

[0, ]

ˆ

[0, ]

ˆ

sup

_ _T

u x t

( , )



c

and sup

_ _T

u x t

( , )



c

,

(2.19) which implies that

1 2

[0, ] [0, ]

sup

_ _T

u x t

( , )



c T

( ) and sup

_ _T

u x t

( , )



c T

( ).

(2.20) From (2.20), it is standard to show the global existence, which can be omitted. Multiplying the first equation of the system (2.3) by u x t₁( , ), using Lemma 3.1 in [8] and integrating over Ω, we have

2 2

2 ₂

2 * 2 2

1 1 1 1 _{( )} 1 ( ) 1

1 2

2 _*

1 _{( )} 1 1 1 _{( )}

( , )

1

( , )

( , )(

)

2

1

( , )

(

)

L L

L _L

u x t

d

u x t dt

D

u

a u

u x t b

dx

dt

cu x t

a b u

D

u







  

 _

 















(2.21)

Similarly, we have

2 ₂

2

2 _*

2 ( ) 2 2 2 2 2 _{( )}

2

3 4 2 2

1

( , ) ( ,

)

2

(

( , ))

( , ) .

L _L

d

u

e

u x t u x t

dx D

u

dt

d

E d u x t

u x t dx











 _ _







 



(2.22)

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2 2 2 ₂

2 2 ₂

2

2 2 2 _*

2 _{( )} 2 _{( )} 2 _{( )} 2 2 2 _{( )}

2

2 3 4 2

2

2 2 _*

2 ( ) 3 2 ( ) 2 2 2 _{( )}

1

( ,

)

2

( , )(

( , ))

(

)

,

2

L L L _L

L L _L

d

u

e

u

u x t

D

u

dt

u x t d

E d u x t dx

e

u

d

E u

D

u

 

















    _     _























 











(2.23)

Where



is small enough.

Combining (2.21) and (2.23), yields



2 2



2 2

1

( , )

2

( , )

1 ( ) 2 ( )

,

d

dt



_

（

u x t



u x t

）

dx





u

L 



u

L  (2.24)

Where

a b

e₂

(

d

₃

E

) 0







  







when



is small enough. Applying the Gronwall

Lemma to (2.24), we have





2 2 2 2

1

( )

( ) 2

( )

( ) 1

(0)

( ) 2

(0)

( )

t

L L L L

u t

_



u t

_



u

_



u

_

e

 (2.25)

which implies that u x t1( , ) and u x t2( , ) are locally bounded in 2_{( )}

L  and (0, 0) is not a global attractor in L2( ) .

Asymptotical Stability of Constant Equilibria

In this section, we discuss locally asymptotical stability of the nonnegative constant equilibria by the linearization method in [9]. It is easy to check that the system (1.2) only has four nonnegative

constant equilibria: 3

4

1(0,0), 2( , 0), 3(0, )

e d E a

b d

E E E    and

E c c

4

( , )

1* 2* , if

 

(

e



d

3

E

)

ad

4



 



and



e





d

₃



E

, * ( - )2 4 ( 4 ( 3 ))/ 4

1 2

ac b ac b bc ad e d E d bc

c         , 3

4

* 2

e d E d

c    . From biological

meaning, the equilibrium 3 4

3(0, )

e d E d

E    has no meaning anymore. So, in the following, the

asymptotical stability of equilibria ₁(0,0), ₂( ,0)a b

E E and

E c c

₄

( , )

₁* ₂* are only discussed, respectively. Setting u t( ) ( ( , ), ( , )) u x t u x t₁ ₂ E i_i( 1, 2,3, 4). We still denote u x t₁( , ), u x t₂( , )

by

u x t

1*

( , )

,

u x t

*2

( , )

, respectively, then, we get the linearized system:

* *

1 2 1 1 2

* 2 *

1 1

2

*

1 1 1 1 1 ₍₁ ₎ ₁

*

2 2 2 3 2 2 4 2 2

,

( ,

)

2

u c u c u

t cc cc

u t

D u

au

bc u

D u

e

u x t

d u

Eu

d c u

  





  _ _   

   









 

 





(3.1)

From [9,10], we know that the linearized equation (3.1) admits nontrivial solutions of the form

1

2

t i x

c e c      

  if and only if the determinant

* * 1 2 * * 2 1 1 2 *

1 1 ₍₁ ₎ ₁

2 *

2 3 4 2

2

0,

2

0

c c cc cc

D

a

bc

D

e

d

E

d c

   











   



 











 

which implies that

* 2 * 2 1

2 * 2 *

1 1 ₍₁ ₎ 2 3 4 2

(6)

Asymptotical Stability of Equilibrium E₁(0,0)

From (3.2), it follows that at the equilibrium E1(0,0)

2 2

1 2 3

(





D





a

)(





D







e

 

 

d

E

) 0





 

D

1



2



a

which satisfy





0

if

2 1

D





a

. From the second factor of (3.3), we deduce that

2

2 3

D

e

 

d

E



 







 

(3.4) Substituting

  

 

i

₁ into (3.4) and separating real and imaginary parts, we have

2

2 1 3

1

cos(

)

,

sin(

).

D

e

d

E

e

   















 

  







 



(3.5)

Firstly, letting



₁0,



0 and using (3.5), we have:

2 2

2

cos(

1

)

3 2 3

,

D

e

 

d

E

D

e

 

d

E



 







 



   







 

(3.6) which implies that there at least exists a positive root



0 by plotting against



the graphs of

y





and

y

 

D

₂



2





e

 

 

d

₃

E

if

D

₂



2





e

 

 

d

₃

E

. From the above analysis, we know that there at least exists a root



₀ such that

Re



₀



0

. Secondly, letting

1 0



 ,



0 and using (3.5), we have:

2

cos(

1

)

3 1

sin(

1

),

D







e

 



 

d

E and



 



e





3

1

[ , 2 ) 2

2

n n

,

1, 2...

 











(3.8) Therefore, E1(0,0) is unstable.

Asymptotical Stability of Equilibrium ₂( ,0)a b

E

From (3.2), it follows that at ₂( , 0)a b

E

2 2

1 2 3

(





D



 

a

2 )(

a





D







e

 

 

d

E

) 0



which leads to 1

D

1 2

a

,

2

D

2 2

e

d

3

E

 



 







 







 

. The proof is similar with the procession of E₁. So we omit it. Then E₂ is unstable.

Asymptotical Stability of Equilibrium

E c c

4

( , )

1* 2*

From (3.2), we have

*

2 * 2 2 *

1 1 * 2 2 3 4 2

1

(

2

+

)(

2

) 0,

(1

)

c

D

a

bc

D

e

d

E

d c

cc

 









 











 

  





(3.9)

(7)

*

2 * 2 2 *

3 1 1 * 2 4 2 3 4 2

1

2

,

2

(1

)

c

D

a

bc

D

e

d

E

d c

cc

 





 



 





 







 

  



In fact

* *

* 2 * 1

3 1 * 2 1 *

1 1

* *

* 2

1 1

1 2

* *

1 1

2

(1

)

1

(

2

(

)

4 (

),

1

c

a bc

a

bc

a

bc

cc

c

ac b

bcc

c

b ac

bc a

u

cc









 



 









 











if ad4  ( e d3 E) 

   , then



3



0

. From the second factor of (3.13), we deduce that

2

3

D

e



d

E

e

 













 





 

(3.10) We claim that all roots of (3.10) satisfy

Re





0

. Otherwise, we suppose that there exists a root

*



such that Re



* 0. Using (3.10) and the above assumption Re



* 0, we deduce that:

*

* 2

2

3

D

e



d

E

e

  

e















  



 







(3.11)

which implies that



*_{is in the circle centered at the point} 2

2 3

( (



D





2



e



 

d

E

),0)

and of radius



e

_{in the complex plane. Therefore, the inequality} _Re



*_₀_{contradicts with}

the fact

D

₂



2



2



e



  

d

₃

E



e

 (From Theorem 2.1, we know

2



e





d

₃



E

). So we have

Re





0

. So we know that



₄



0

. Therefore,

E c c

₄

( , )

₁* *₂ is locally asymptotically stable as ad₄ _ ( e _d₃_E)_{. From the above discussion, we can conclude the following} results:

Theorem 3.1.The equilibria E1(0,0) and E2( ,0)ab of the system (1.2) are unstable.

Theorem 3.2. If ad₄__{ }( e _d₃_E)_,_{then the system (1.2) has the positive equilibrium}

* *

4

( , )

1 2

E c c

; Moreover, it is locally asymptotically stable. If ad₄_ ( e _d₃_E)_{, then the} positive equilibrium

E c c

₄

( , )

₁* *₂ does not exist.

Summary

In this work, a delayed reaction-diffusion predator-prey model with stage structure and continuous harvesting for predator is constructed. The boundedness, existence and uniqueness of the model is investigated. The existence and uniqueness of the global solution of the system are proved. The local and global stability of the constant equilibria are discussed by the linearization method and the method of upper and lower solutions, respectively. By using the linearization method and the method of upper and lower solutions, the local and global stability of the constant equilibria of the system are obtained, respectively. By Theorem 3.1, one can see that the equilibria E1(0,0) and

2( ,0)ab

E of the system (1.2) are unstable. By Theorem 3.2, one can see that the system (1.2) has the positive equilibrium

E c c

4

( , )

1* *2 and it is locally asymptotically stable if ad4  ( e  d3 E)



   .

Acknowledgement

(8)

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