Existence of Solutions for Front-Type Forced Waves of Leslie-Gower Predator-Prey Model in Shifting Habitats

i i

Let P P1 , P2 , 1 , 2 P1 , 2 . Then,

P : Y X2 and s(z) (z) F (z), z .

i i i

We define

Thus, a fixed point of P is a solution of (4).

Let 0, be a constant and we define the norm

s

d* J (y)eydy 1

sup max (z) , (z) e z , Y.

2 2 1

z 1 2

() .

Moreover, the set

i

A 1, 2 Y : i i 0, i 1, 2

is a non-empty convex, closed, and bounded set in Y, .

d* z

(zy)

Then, we show that P maps A into A. Let A.

1 J1 (y ) 1 () 1 () e e de s dy

s

By using (5), we can get

F1 (z) F1 , 2 (z),

*

1 1

d

J (y)e s

z

y dy e y e

(zy) s

dy e z

1 1

1

and

1 z (zy) 2

1

and hence, P1 (z) P1 , 2 (z) for all z .

From (1), we have

Thus, we have

e y e s

s

dy e z .

s

z

, (z) 1

( yz)

e s

F , (y) dy

s

1 1 2

1 1 2

P1 1, 2 (z) P1 1, 2 (z)

L1 1 1

L2 2 2

( yz) F , (y)

where

s 1 1 2 2

lim

e dy

L d*

J (y)e y dy d* r 2 k(1 h) ,

0

,z\ Uzi E(zi , zi ) s

1 s 1

1 1 L ( )

( yz) c (y) (y) 2k

lim inf

e s 1 1 dy

L2 .

0

, z\ Uzi E(zi , zi ) s

s

(z),z .

1

Similarly, we have

Thus, P , (z) (z). Similarly, we have

P2 1, 2 (z) P2 1, 2 (z)

1 1 2 1

P , (z) P , (z) e z

(z) P ,

(z).

2 1 2 2 1 2

1 1 1 2

* z (zy)

s

By using the choice of and the definition of super and sub-

d2

J

(y) J

(y)e s

dy e z

solutions, we can calculate

L ( )

*

2 2 2 2

(1 h)

d2 r 2 z

(zy)

1 P1

1, P1 P1 , 2 ,

h (y) (y)e s dy e z

2 1 1

s 2 2

P , P P , .

2

2 2

1 2

2 2

1 2 2

(1 h)

s

p

z

(y) (y)e

(zy) s

dy e z

Therefore, P(A) A.

1 1

Next, we show that the mapping P : A A is completely

M M

,

continuous with respect to the norm . We give some

1 1 1

where

2 2 2

details and show the continuity of P on A .

For any 1 1, 2 A and 2 1,2 A , we have

(1 h)2

p

P , (z) P , (z)

M1 s ,

1 1 2 1 1 2

P , (z) P , (z) e z

J (y)e dy d* r

2 .

1 1 2 1 1 2

2 y

(1 h)

M

d*

P ,

(z) P ,

(z)

2 s 2

2 2 L ( ) h

1 1 2 1 1 2

d*

1

s

z

*

J1 1 (y) J1 1 (y)e

(zy) s

dy e z

Therefore, there exists a positive constant C such that

P1 P2 C 1 2 .

d1

r 2 k(1 h)

L ( )

s

z

1 (y) 1 (y)e

(zy) s

dy e z

Hence, P is continuous with respect to the norm .

k z (zy)

z

Now, we will prove P in A is compact with respect

to the norm . For any , A and n , we define

s

2 (y) 2 (y)e s

dy e .

1 2

P , ( n), z , n,

1 2

Pn , (z) P , (z), z n, n,

We note that

1 2 1 2

d

* z

1

s

J1 1 (y) J1 1 (y)e

(zy) s

dy e z

P1 , 2 (n), z n, .

1 2

Pn , (z)

is equicontinuous and uniform bounded in

n

s (n) (0) d*

J (z) dz

Y, .

1 1 1

0

n

1 1 1

Thus, there exists a constant N such that

Pn , (z) P , (z) e z Nen .

1(z) r( z) 1(z) k2 (z)dz

0

1 2 1 2

1 1

n 1

1

Then,

Pn , (z) converges to

P1, 2 (z)

as n .

d* J (y)

n

(z ry)(y)dr dz dy

0 0

1 2

Thus, P , (z) is compact and then

Pn , (z) is

(z) r( z) (z) k (z)dz.

1 2 [n,n]

compact. We verify that

P1, 2 (z)

1 2

is precompact. Hence,

1 1 2

0

by Schauder s fixed-point theorem, the proof completes.

1. Theorem

   Hence,

   1 1 1

   s (n) (0) d*

   1

   J1 (y)y

   0

   1(n ry) 1( ry)dr dy

   (7)

   Assume that 1 , 2

   and ,

   1 2

   are a pair of upper and

   n

   (z)(r( z) (z) k (z)dz.

   lower solutions of (4) satisfying

   1 1 2

   0

   0 1, 0 1 h

   in ,

   By taking a sufficiently large positive constant M and

   1 1

   1 1 2 2

   (0,1) so that (z) and r( z) 0 for all z M,

   and admits a solution , such that (z) (z) (z)

   we get

   1 2 i i i

   for all z for i 1, 2, then 1,2 0,0

   nonnegative solution 1, 2 of (4).

   Proof:

   for any

   Thus,

   (z)r(z) (z) k (z) ( )2 0.

   1 1 2 1

   n

   1 1

   For contradiction, we assume that lim sup (z) 0.

   lim 1(z) r( z) 1(z) k2 (z)dz ,

   z

   When 1 is oscillatory near z , there is a maximal

   n

   0

   which contradicts the boundedness of (7).

   sequence z of such that z and (z ) as

   n 1 n

   n .

   From the first equation of (4), we get

   1 n 1

   Thus, 1( ) 0.

   For the contradiction, we set that

   limsup (z) 0.

   2 2

   z

   When 2

   is oscillatory near z , we have a maximal

   1 1

   1 1 n 1 n

   (6)

   sequence zn of 2 such that zn and 2 (zn ) 2 as

   0 s(z) d*

   J (y) (z y)dy (z )

   n .

   1(zn ) r zn 1(zn ) k2 (zn ).

   By letting n , it follows from Fatou s lemma that

   From the 2 equation of (4), we get

   0 *

   2 (zn )

   * lim sup d2 J2 (y) 2 (zn y) dy 2 (zn ) 2 (zn ) r( zn ) (z ) h

   limsup d1 J1(y) 1(zn y) 1(zn )dy 0

   and

   n

   1 n

   n

   r( ) 2

   0.

   2

   h

   lim sup (z ) r z (z ) k (z ) 1

   n

   1 n n 1 n 2 n

   This is a contradiction. When 2 is monotone ultimately at

   r( ) k lim inf (z ) 0.

   z the process is similarly to the proof of 1 .

   1

   1

   n

   2 n

   Hence, 2 ( ) 0.

   From r( ) 0, it contradicts the equatio (6). On the other hand, suppose that 1 is monotone ultimately at z , then

   1 1

   ( ) .

   By integrating the first equation of (4) from 0 to n, we get

   Thus, 1,2 () 0,0,0

   1, 2 of (4).

   for any nonnegative solution

   IV EXISTENCE OF FRONT-TYPE FORCED WAVES

   Since , we have

   i

   i

   for i 1, 2, where 1 k(1 h),

   h.

   In this section, the existence of front type forced waves i i

   connecting E* to 0, 0 is shown. Since r is non-monotonic,

   1 2 1

   we consider the following problem

   c(z) d(J )(z) (z)r( z) (z), z

   ,

   4. Theorem

   Assume that

   k 1

   1 h

   . Let 1 , 2

   be a solution of (4)

   ( ) r(),

   ( ) 0.

   (8)

   obtained from Theorem 3. Then, 1, 2 E*

   Proof:

   v*,* .

   A positive function satisfying

   1

   s d* (J (z) (z)) (z) r( z) k(1 h) (z), z ,

   For 0,1, we define the following functions

   1

   m v* 1k ,

   m * 1 r ,

   1 1 1

   1 1 1

   1 2 2 1

   ( ) r() k(1 h),

   ( ) 0.

   M v* 11 ,

   1 1

   Similarly, there exists a function satisfying

   2

   1

   2 2

   M * 11 h r ,

   where

   (z)

   r 1 ,

   1 r2

   1 ,

   0 min 1

   , 2 ,

   k2

   .

   (9)

   1

   s d* (J

   (z) (z)) (z) r( z) 2 , z k k

   r1

   kr1 1

   2 2 2

   2 2 2

   (z) h

   and

   2

   lim (z) r()r() k(1 h) h 0,

   lim (z) 0.

   Let A [0,1) m () M (),i 1, 2.

   i i i i

   From Theorem 3, it is obvious that

   i i i i

   m () M ()

   z 2

   z 2

   is true for 0 and i 1, 2. Thus, A 0. In addition, we

2. Theorem

Suppose that k

1

1 h

. Then, for each s 0 there exists

know v* 1, * 1 h. Meanwhile, *, * satisfies

v* 1 k*, * v* h.

a positive solution , of (4) such that 1

and

Then, we get that

1 2 1 1

1 h in .

v* 1 k1 h , * h .

2 2

Proof:

1 1 2

We denote 1 , 2 (1,1 h).

By the definition of , ,

1 2

Then, we obtain that mi (Mi , i 1, 2) is a monotone

we get Li (z) 0, i 1, 2.

increasing function of 0,1 such that

m , m 1 M , M 1 v*, * .

Since r(z) 1 for z , there are

1 2 1 2

Thus, it is sufficient to show that sup A 1.

d* (J (z) (z)) s (z) (z) r(z) (z) k (z)

1 1 1 1 1 1

r( z) 1 0 ,

1 2

We argue by a contradiction and suppose that

sup A 0 0,1. By taking the limit, we can get

and

d* (J

(z) (z)) s (z)

z)

i 0 i i i 0

2 (z)

0.

m M ,

i 1, 2,

2 2 2 2 2 2 (z) r(

(z) h

It should be pointed out that at least one of the following

1

equalities holds

Therefore, 1 , 2

and ,

1 2

are a pair of upper and

m ,

M , i 1, 2,

lower solutions of system (4). Hence, by Theorem 1, the proof

i i 0 i i 0

completes.

according to the definition of 0

mi and Mi .

and the continuity of

To proceed further, we set

lim sup z ,

lim inf z , i 1, 2.

Now, we consider the case

m .

If 1 is

i i i i

1 0 1

z

z

eventually monotone, then 1() m1 (0 ).

By integrating the first equation of (4) from n

to 0, we

It is contradiction. That is, m . Additionally, the

can get

s1 (0) 1 (n)

0 0

other cases are similar to the discussion above by applying the following inequalities:

1 1 0

1 1 0

(i) M ,

d* J (z)dz (z) r( z) (z) k (z)dz

1 1 1 1

n

0 1

1 1 2

• n

  lim sup r( zn ) 1 (zn ) k2 (zn )

  n

  * *

  1 M1 (0 ) km2 (0 )

  d*

  J (y)

  (z y)(y)ddzdy

  1 1 1

  1 v 1 1 k 1 r

  • n 0

    0 0

    0 0 2 1

    0

    (z) r(z) (z) k (z)dz.

    1 v* k * 1 1 k1 r

• n

Hence,

1 1 2

1

0 0 0 0 2 1

10 11 k2 kr1

1 0 kr1 1 k2

1 1 1

s (0) ( n) d*

0

J1 (y)y

0

1(y) 1( n y)ddy

0;

1. m

   ,

   1(z) r(z) 1(z) k2 (z)dz.

   • n

     (10)

     2 2 0

     2 (zn )

     lim inf r( zn ) (z ) h

     We note that

     n

     1

     1 n

     m2 (0 )

     lim inf r(z) 1 (z) k2 (z)

     z

     1 m1 (0 ) kM2 (0 )

     1 v* 1 k * 1 1 h r

     1

     m1 (0 ) h

     0 0 2 1

     * 1 r

     0 0 1

     v* 1 h

     * *

     0 0 1 0 0 2

     0v 1 0 1 h 0 1 0 2 r1

     0 0 0 1 0 2

     1 v* k * 1 k1 1 h r

     10 10 1 k10 1 h r2

     10 11 k hk kr2

     v* 1 h

     0 0 1

     0 0 1 2 1

     (v* * ) h 1 r

     0 0 1

     v* 1 h

     10 11 k hk k hk kr2

     h0 h 1 0 1 2 r1

     1 1 kr 0,

     v* 1 h

     0 2

     0 0 1

     by using (9) and 1.

     1 0 h 1 2 r1

     v

     *

     0

     Thus,

     0 1 0 1 h

     0

     n

     lim 1(z) r( z) 1(z) k2 (z)dz .

     • n

       This contradicts the boundedness of left side of (10).

       1 0 1 1

       0 0 1

       v* 1 h

       • 0;

         On the other hand, we assume 1 is oscillatory at

         nN

         • . Then, we can choose a sequence zn of minimal point of 1 with zn as n so that

2. M ,

2 2 0

(z )

lim z m .

limsup r(z ) 2 n

n

1 n 1 0

n

1 (zn ) h

n

We note that 1 zn 0 and the Fatou s lemma gives that

lim inf J1 1 1 zn 0.

n

1 M2 (0 )

M1 (0 ) h

* 1 1 h r

From the first equation of (4), we have

1

0 0 2

0 0

v* 1 1 h

0 sliminf (z ) liminf (z ) r(z ) (z ) k (z)

v* 1 1 h * 1 1 h r

1 n

n

n

1 n n 1 n 2

0 0 0 0 2

v* 1 1 h

m1(0 )1 m1(0 ) kM2 (0 ) 0 ,

0 0

0 0 2

(v* * ) 1 1 h 1 1 h r

0 0

v* 1 1 h

h0 h 1 0 1 1 h r2

0 0

v* 1 1 h

1 0 h h r2

0 0

v* 1 1 h

1 0 1 r2 0,

0 0

v* 1 1 h

nN

in which the sequence zn is the corresponding minimal or maximal point of i (i 1, 2). The proof completes.

CONCLUSION

We calculate the existence of nonnegative solution for Leslie-Gower predator-prey model by constructing appropriate upper-lower solution and employing fixed-point theorems. The conditions for the existence of front type the Leslie- Gower formulation are established. Our analysis demonstrate that climate change speed models as a shifting environment. Specifically, it is shown that: Front-type forced waves emerge when populations respond to the environmental shifts by forming monotone traveling wave profiles, capturing the invasion-extinction transition.

.

ACKNOWLEDGMENT

The author would like to express unlimited gratitude to his foremost, respectable and noble teacher, Associate Professor Dr. Aung Zaw Myint, Department of Mathematics, University of Mandalay for his encouragements on doing research. The author acknowledges the unknown reviewers for their suggestions and patient review on this papere.

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