VNU Journal of Science: Mathematics – Physics, Vol. 30, No. 3 (2014) 49-69

Predator-prey System with the Effect of Environmental Fluctuation
Le Hong Lan*
Faculty of Basic Sciences, Hanoi University of Communications and Transport,
Lang Thuong, Dong Da, Hanoi, Vietnam
Received 18 July 2014
Revised 27 August 2014; Accepted 15 September 2014

Abstract: In this paper we study the trajectory behavior of Lotka - Volterra predator - prey
systems with periodic coefficients under telegraph noises. We describe the ω - limit set of the
solution, give sufficient conditions for the persistence and prove the existence of a Markov
periodic solution.
Keywords: Key words and phrases, Lotka-Volterra Equation, Predator - Prey, Telegraph noise, ω
- limit set, Markov periodic solution.

1. Introduction*
The Kolmogorov equation
 x ( t ) = x f t , x ( t ) , y ( t )

 y ( t ) = y g t , x ( t ) , y ( t ) 

with the functions f t , x ( t ) , y ( t )  ; g t , x ( t ) , y ( t ) periodic in t is a strong tool to describe the
evolution of prey-predator communities depending on the changing of seasons. There is a lot of work
dealing with the asymptotic behavior of such systems as the existence of periodic solutions, the
persistence... [1-4] In particular, the classical model for a system consisting of two species in preypredator relation
 x ( t ) = x ( t )  a ( t ) − b ( t ) x ( t ) − c ( t ) y ( t ) 

 y ( t ) = y ( t ) −
 d ( t ) + e ( t ) x ( t ) − f ( t ) y ( t ) 


omega limit set of the positive solutions of Equation (1.1) with the periodic coefficients under the
telegraph noises. Also, the existence of a Markov periodic solution that attracts the other solutions of
Equation (2.4), starting in » + × » + under certain conditions is proved.
The rest of the paper is divided into three sections. Section 2 details the model. Some properties of
the solution and the set of omega limit are shown in section 3. The last section is some simulations and
discussions.

2. Preliminary
Let (Ω, F , P ) be a complete probability space and {ξ (t ) : t ≥ 0} be a continuous-time Markov
chain defined on (Ω, F , P ) , whose state space is a two-element set M = {−, +} and whose generator is
given by
q
Q =  11
 q21

q12   −α
=
q22   β

α 

−β 

with α > 0 and β > 0 . It follows that, ϖ = ( p, q ) , the stationary distribution of {ξ ( t ) :t ≥ 0}

satisfying the system of equations
ϖ Q = 0

 p + q =1
is given by

σ 1 := τ 1 − τ 0 , σ 2 := τ 2 − τ 1 , ... , σ n := τ n − τ n −1

(2.3)

It is known that the sequence {σ k }k =1 is an independent random variables in the condition of
n

{ }

given sequence ξτ k

n
k =1

(see [15, 16]). Note that if ξ 0

is given then ξτ n is constant since the

process {ξ t } takes only two values. Hence, (σ k )∞k =1 is a sequence of conditionally independent random

variables, valued in [0, + ∞ ] . Moreover, if ξ0 = + then σ 2 n +1 has the exponential density α1[0, + ∞ ) e−α t
and σ 2 n +1 has the density β 1[0, + ∞ ) e − β t . Conversely, if ξ0 = − then σ 2n has the exponential density

α1[0, + ∞ ) e−α t and σ 2 n +1 has the density β 1[0, + ∞ ) e − β t (see [15]). Here
1 , t ≥ 0
.
1[0, + ∞ ) = 
0 , t < 0

Denote ℑ0n = σ (τ k , k ≤ n ) ; ℑ∞n = σ (τ k −τ n , k > n ) . We see that ℑ0n is independent of ℑ∞n for any

(2.5)

and another
 x− ( t ) = x− ( t )  a ( −, t ) − b ( −, t ) x− ( t ) − c ( −, t ) y− ( t ) 

 y− ( t ) = y− ( t )  −d ( −, t ) + e ( −, t ) x− ( t ) − f ( −, t ) y− ( t ) 

(2.6)


L.H. Lan / VNU Journal of Science: Mathematics – Physics, Vol. 30, No. 3 (2014) 49-69

52

Thus, the relationship of these two systems will determine the trajectory behavior of Equation
(2.4).
System (2.4) without the noise {ξt } , i.e., g (ξt , t ) = g ( t ) for any g = a, b, ..., f is studied in [9].
They show that

Theorem 2.1. Consider the system
 x ( t ) = x ( t )  a ( t ) − b ( t ) x ( t ) − c ( t ) y ( t ) 

 y ( t ) = y ( t )  −d ( t ) + e ( t ) x ( t ) − f ( t ) y ( t )

(2.7)

where a, b,..., f are T-periodic functions.
a) If
a
d 

 
b

(2.11)

then the (unique) periodic solution u * ( t ) of the equation u ( t ) = u ( t )  a ( t ) − b ( t ) u ( t ) is stable and

( x ( t ) − u (t ) , y (t )) → ( 0,0 )
*

t →+∞

(2.12)

for any positive solution ( x ( t ) , y ( t ) ) to (2.7).

Figure 1. Coexistence of predator and prey.

Figure 2. Extinction of predators.


L.H. Lan / VNU Journal of Science: Mathematics – Physics, Vol. 30, No. 3 (2014) 49-69

53

Lemma 2.2. Consider the system
 x ( t ) = f ( x, y , t )

 y ( t ) = g ( x, y, t )
where f , g : » 2 × [0, + ∞ ) → » 2 × [0, + ∞ ) are T-periodic functions in t.

that if ( x0 , y0 ) ∈ Κ
and
0 ≤ s ≤ T , there is a compact set Κ ′ such that

( x ( T , s, x

0

, y0 ) , y (T , s, x0 , y0 ) ) ∈ Κ ′. Due to the periodicity of parameters, it is therefore sufficient to

verify (2.13) for s = 0 . Since ( x* ( t ) , y* ( t ) ) is stable, we can find a δ ε > 0 such that if
x − x0* + y − y0* ≤ δ ε then

x ( t ,0, x , y ) − x* ( t ) + y ( t ,0, x , y ) − y* ( t ) ≤ ε , ∀t ≥ 0

( 2.14 )

On the one hand, ( x* ( t ) , y * ( t ) ) is globally asymptotic then for every ( x0 , y0 ) ∈Κ , there exist a
T( x0 , y0 ) = k( x0 , y0 )T , k( x0 , y0 ) ∈ » such that

(

) (

)

(

)


{

The family Vx0 , y0 : ( x0 , y0 ) ∈ Κ

finite family

{V

} is an open covering of

} such that Κ ⊂ ∪ V
n

x10 , y10

,..., Vxn , y n
0

0

i =1

x0i , y0i

point ( x0 , y0 ) ∈ Κ and for all t > T , we have:
*

x ( t ,0, x0 , y0 ) − x* ( t ) + y ( t ,0, x0 , y0 ) − y * ( t ) < ε .
The proof is complete.



y = y −
 d (ξ t , t ) + e (ξ t , t ) x − f (ξt , t ) y  < y −
 d (ξ t , t ) + e (ξt , t ) M / m − f (ξ t , t ) y 
< y ( −m + M 2 / m − my ) ,
which follows that y ( t ) ≤ M 2 / m 2 − 1 , ∀ t > t1 for some t1 > t0 .
From these estimates, we also see that the rectangle ( 0, M / m ] × ( 0, M 2 / m 2 − 1 is forward
invariant. The proof is complete.

Proposition 3.2. There exists A δ 0 > 0 such that limsup x ( t ,0, x0 , y0 ) ≥ δ 0 for any ( x0 , y0 ) with
t → +∞

probability 1.

Proof. By the system (2.4), there exist δ 0 > 0, ε 0 > 0 such that
− d (ξt , t ) + e (ξt , t ) x − f (ξt , t ) y < −ε 0 ; ∀ 0 < x < δ 0 , 0 < y ≤ M 2 / m 2 − 1
and
a (ξ t , t ) − b (ξ t , t ) x − c (ξt , t ) y > ε 0 for all 0 ≤ x, y ≤ δ 0 .

( 3.1)

Assume that limsup x ( t ,0, x0 , y0 ) < δ 0 with a positive probability. Then, there is a t3 > 0 such that
t → +∞

x ( t ) < δ 0 , y ( t ) ≤ M 2 / m 2 − 1 ∀t ≥ t3 , which implies that y ( t ) < −ε 0 y ( t ) . Therefore, for some t4 > t3
, y ( t ) < δ 0 , ∀ t ≥ t4 . From (3.1) we see x ( t ) > ε 0 x ( t ) , ∀ t ≥ t4 , which follows that lim x ( t ) = ∞ .
t →+ ∞

This contradiction implies the assertion of this proposition.


− δ1 ( t − h1 )

) , ∀t ∈(h , h ) .
1

2

Put
t

(

n ( t ) = ∫ m − M ymax e

−δ1 ( t − h1 )

t

) ds ; N (t ) = ∫ e ( ) ds

h1

n s

h1

By comparison theorem we get

x (t ) ≥


process. Indeed, put
u* ( t ) =

e
t

A( t )

∫ b (ξ , s ) e
s

−∞

A( s )

ds


L.H. Lan / VNU Journal of Science: Mathematics – Physics, Vol. 30, No. 3 (2014) 49-69

56

t

where A ( t ) = ∫ a (ξ s , s ) ds . Firstly, we see that
0
t +T

∫ a ξ s (ω ), s  ds


t +T

∫ b ξ (θ ω ) , s − T  e
T

s −T

T
∫ a ξτ −T (θ ω ), τ −T  dτ
0

ds

−∞

t

T
∫ a ξ s (θ ω ), s  ds

e −T

=

0

T
∫ a ξ s (θ ω ), s  ds

e −T

b ξ s (θ T ω ) , s  ds
∫ 0
t

T

= u* (t , θ T , ω ).

−∞

Hence, by virtue of P - preserving measure property of θ , for any continuous function h , for any
t1 < t2 < ... < tn ; k ∈ » we have

{

}

Ε h ξt1 + k T , u * ( t1 + k T ) , ξt2 + k T , u * ( t2 + k T ) ,..., ξ tn + k T , u * ( tn + k T )

{

}

= Ε h ξ t1 (θ kT ) , u * ( t1 , θ kT ) , ξ t2 (θ kT ) , u * ( t2 , θ kT ) ,..., ξtn (θ kT ) , u * ( tn , θ kT ) 

{

}

= Ε h ξ t1 (.) , u * ( t1 , .) , ξt2 (.) , u * ( t2 , .) ,..., ξ tn (.) , u * ( tn , .) .

 T
0

T

∫ h  s , ξ
0

s


, u * ( s ) ds 


57

(3.3)

Proof. Put
( n +1) T

∫

Xn =

h  s , ξ s , u * ( s ) ds

nT

Since {ξt , u * ( t )} is periodic then { X n } is a stationary process. By the law of large numbers we have

ds
=
Xk =
Ε[ X 0 ]
,
,
lim
( )
∑
s
t →+ ∞ t ∫ 
t →+∞ T t / T
T
[ ] k =0
0
t

t /T

lim
=

T

1 
Ε  ∫ h  s , ξ s , u * ( s ) ds 
T 0


Where, [ x ] denotes the integer number such that [ x ] ≤ x < [ x ] + 1 . Lemma is proved.

t→+∞

a) From Equations (3.2) and (2.4) we have
ln u * ( t ) − ln u * ( 0 )
t
ln x ( t ) − ln x ( 0 )
t
t

t

=

t

1
1
a (ξ s , s ) ds − ∫ b (ξ s , s ) u * ( s ) ds
∫
t 0
t 0

t

=

(3.5)

t


t 0

t
t
1
1

≤ liminf  ∫ M y ( s ) ds − ∫ m u * ( s ) − x ( s )  ds 
t 0
t→+∞ 

t 0

Hence,
t
t
1
1 M

liminf  ∫
y ( s ) ds − ∫ u * ( s ) − x ( s )  ds  ≥ 0
t 0
t→+ ∞ 

t 0 m

Otherwise,

y (t )
y (t )

and
t

t

1
1
e (ξ s , s ) u * ( s ) − x ( s ) ds + ∫ f (ξ s , s ) y ( s ) ds =
∫
t 0
t 0
=

t
ln y ( t ) − ln y ( 0 )
1
 − d (ξ s , s ) + e (ξ s , s ) u * ( s )  ds −
∫
t 0
t

 ln y ( t ) − ln y ( 0 ) 
Moreover, y (t ) is bounded above then liminf −
 ≥ 0 and we apply the law
t
t→+∞ 

t
1
of large numbers (Lemma 3.5), lim ∫  − d (ξ s , s ) + e (ξ s , s ) u * ( s ) ds =λ , consequently,

t 0
Hence,
t
t
1
1

liminf  ∫ u * ( s ) − x ( s )  ds + ∫ y ( s ) ds  ≥
t 0
t→+ ∞ 

t 0


L.H. Lan / VNU Journal of Science: Mathematics – Physics, Vol. 30, No. 3 (2014) 49-69

59

t
t
 λ
1 f (ξ s , s )
1 e (ξ s , s ) *


liminf  ∫
u ( s ) − x ( s )  ds + ∫
y ( s ) ds  ≥

t 0 M

1 t
 λ
1
+ liminf  ∫ u * ( s ) − x ( s ) ds + ∫ y ( s ) ds  ≥
t 0
t→+ ∞ 
t 0
 M
t

then liminf
t→+ ∞

1
m
y ( s ) ds ≥
λ ≥ 0 and limsup y ( t ) > δ > 0.
∫
t 0
M ( M + m)
t→+∞

b) From the second equality of systems (2.4) and λ > 0 we have

limsup

ln y ( t ) − ln y ( 0 )
t

t→+ ∞

)
(
)
(
)
s
∫
∫


t 0
t 0


t

≤ λ − limsup
t→+ ∞

t

1
1
e (ξ s , s ) u * ( s ) − x ( s )  ds − limsup ∫ f (ξ s , s ) y ( s ) ds < 0
∫
t 0
t 0
t→+ ∞

which implies that lim y ( t ) = 0 . The proof is complete.

1  
 e ( ± , t ) 
Ε  ∫  − d (ξt , t ) + inf 
b (ξ t , t ) u * ( t ) dt 

t→+∞
T  0 
 b (ξ t , t ) 
 


=

T
 
1  
 e ( ± , t ) 
Ε  ∫  − d (ξt , t ) + inf 
a (ξt , t )  dt  +

t→+∞
T  0 
 b (ξ t , t ) 
 


T
 e ( ± , t )  1 

+ inf 


 b ( ± , t )  t → + ∞  e ( ± , t ) 

(3.9)

t→+∞

Then, λ > 0 under the condition 3.9, which is similar to (2.8).
From now on, we suppose that λ > 0 .

Lemma 3.8. With probability 1, there are infinitely many sn = sn (ω ) > 0 such that sn > sn −1 ,
lim sn = ∞ and x ( sn ) ≥ δ , y ( sn ) ≥ δ , ∀n ∈ » .

n →+ ∞

Proof. By Proposition 3.3 we can find t > 0 such that x ( t ) ≥ xmin , for all t > t . On the other
hand, there exists δ < xmin and a random sequences {sn } ↑ ∞ , sn > t such that y ( sn ) > δ , ∀ n ∈ » . The
proof is complete.
For the sake of simplicity, we suppose ξ 0 = + a.s and set xn := x (τ n , x , y ) , yn := y (τ n , x , y )
ℑ0n = σ (τ k , k ≤ n ) ; ℑ∞n = σ (τ k −τ n , k > n ) . It is clear that

( xn , yn ) is

ℑ0n measurable ℑ0n is

independent ℑ∞0 if ξ0 is given.
Hypotheses 3.9. On the quadrant int » 2+ , the system (2.5) has a stable positive T − periodic

solution ( x+* , y+* ) such that


algebra at ς is ℑς = { A ∈ ℑ∞ : A ∩ {ς ≤ t} ∈ ℑt , ∀t ∈ » + } . Fix a T1 > 0 , by Lemma 3.8, we can define
almost surely finite stopping times

η1 = inf {t > 0: x ( t ) ≥ δ , y ( t ) ≥ δ }
η2 = inf {t > η1 + T1 : x ( t ) ≥ δ , y ( t ) ≥ δ }
..........

ηn = inf {t > ηn −1 + T1 : x ( t ) ≥ δ , y ( t ) ≥ δ }


L.H. Lan / VNU Journal of Science: Mathematics – Physics, Vol. 30, No. 3 (2014) 49-69

For a stopping time ς , we write τ (ς ) for the first jump of ξ ( t )

{

61

after ς , i.e.,

}

τ (ς ) = inf {t > ς : ξ ( t ) ≠ ξ (ς )} . Let σ (ς ) = τ (ς ) − ς and Ak = σ (ηk ) < T1 , k ∈ » . Obviously, Ak is
in the σ − algebra generated by {ξ (η n + s ): s ≥ 0} and Ak ∈ ℑηk +1 also. Therefore, in view of the
strong Markov property of (ξ ( t ) , x ( t ) , y ( t ) ) and [see 15, Theorem 5, p. 59] we have
Ρ  Ak ξ (η k ) = ±  = Ρ σ ( 0 ) > T1 ξ ( 0 ) = ±  .


Hence,



ℑηk +1  ξ (η k +1 ) , x (η k +1 ) , y (η k +1

Ak +1

Ak +1

(ξ (η ) , x (η ) , y (η ) )
k +1

k +1

k +1

}

ξ (η k +1 ) , x (η k +1 ) , y (η k +1 ) 

{

= Ε 1Ak ξ (η k +1 ) , x (η k +1 ) , y (η k +1 )  Ε 1Ak +1 (ξ (η k +1 ) , x (η k +1 ) , y (η k +1 ) ) 

}

which implies that

{

} {



{

}

Ρ Ak +1 ∩ Ak ≤ p Ρ Ak ξ (η k +1 ) = + Ρ {ξ (η k +1 ) = +} + p Ρ Ak ξ (η k +1 ) = − Ρ {ξ (η k +1 ) = −} ≤ p
Continuing this way, we conclude that
 n 
 n

Ρ  ∪ Ai  = 1 − Ρ  ∩ Ai  ≤ 1 − p
 i=k 
 i =k 

( )

Consequently,
 +∞ +∞ 
Ρ  ∩∪ Ai  = 1.
 k =1 i = k 

n − k +1

2


L.H. Lan / VNU Journal of Science: Mathematics – Physics, Vol. 30, No. 3 (2014) 49-69

62


occurs then ∆ ≤ xηk , yηk , xηk +1 , yηk +1 ≤ M * . Using arguments

similar to the previous part of this proof, we can show that Bk occurs infinitely often. Consequently,
we obtain the second assertion of this lemma due to the fact that η k is odd then η k + 1 is even and
conversely.
Next, we will describe the ω − limit sets of the system (2.4). Denoted by Ω ( x, y , ω ) the ω − limit
set of the solution

( x ( t ,0, x, y ) , y ( t ,0, x, y ) ) (ω )

starting in

( x, y ) .

To simplify the notations, for

t ≥ s ≥ 0 , we denote

π t+,s ( x, y ) := ( x+ ( t , s, x, y ) , y+ ( t , s, x, y ) )

; resp π t−,s ( x, y ) := ( x− ( t , s, x, y ) , y− ( t , s, x, y ) ) ) is the

solution to the system (2.5) (resp. (2.6)) starting at ( x, y ) ∈ » 2+ at time s.
Suppose that the solution starting at γ +* ( 0 ) = ( x+* ( 0 ) , y+* ( 0 ) ) at time 0 is a periodic solution to the
system (2.5), we now describe the pathwise dynamic behavior of the solutions of system (2.4). Put

Γ=

{( x, y ) = π



1

2

(3.12)


L.H. Lan / VNU Journal of Science: Mathematics – Physics, Vol. 30, No. 3 (2014) 49-69

(
(

63

) ( ) ( ) ( )
) ( ) ( ) ( )

h ξ , t , z = a ξ , t − b ξ , t x − c ξ , t y
t
t
t
 1 t
where 
 h2 ξt , t , z = − d ξt , t + e ξt , t x − f ξt , t y


Then, with probability 1, the closure Γ of Γ is the ω - limit set Ω ( x0 , y0 , ω ) . Moreover, Γ
absorbs all positive solutions in the sense that for any initial value ( x0 , y0 ) ∈ int » 2+ , the value



where

, τηk +1

(x

η k +1

)

(

(

)

, yηk +1 = π +t + mod τ , mod τ
x ,y
= π +t − T + mod τ , 0 x , y
( ηk +1 ) ( ηk +1 ) ηk +1 ηk +1
( ηk +1 )

(x, y ) = π

+

(

T , mod τηk +1


(

, yηk ∈U ε1 γ +* ( 0 )  ; ∀t > T * , mod t + τ ηk ∈ ( −δ 2 , δ 2 ) .
ηk

2

ηk

2

*

ηk +1

*

ηk +1

Now, let δ 3 = min {δ1 , δ 2 } ; for any u > 0, δ 3 > 0, k ∈ » , put

2

, in which K ∈ »


L.H. Lan / VNU Journal of Science: Mathematics – Physics, Vol. 30, No. 3 (2014) 49-69

64

(

)

Ρ Ak = Ρ σ ηk +1 ∉ − mod τηk +1 + K T − δ 3 , − mod τηk +1 + K T + δ 3
+∞

=

∫

{

(

)}

τη k = t

{

(

)}

τηk = t , ξτη = +

{

(


∫

Ρ σ ηk +1 ∉ T − δ 3 , T + δ 3

)}

ξτη = +
k

0
+∞

{ (

= Ρ σ 3 ∉ T − δ3 , T + δ3

)} ∫ Ρ{τ

{

× Ρ τ ηk ∈ d t

} {

{

× Ρ τ ηk ∈ d t

k

)} := ϕ < 1

0

{

We now estimate Ρ Ak ∩ Ak + 2

(ξ ( t ) , x ( t ) , y ( t ) )

{

}

. Since Ak ∈ ℑηk +2 , applying the strong Markov property of

we have

}

{
}
} = Ε 1 Ε{1

{
}
= −} ≤ ϕ Ε (1 ) = ϕ


Ρ Ak ∩ Ak +1 = Ε  Ε 1A 1A

Continuing this way, we have,

{

(

(

)

(

)

}

)

Ρ {∩+k =∞1 ∪ +i =∞k Ai } = Ρ ω : σ ηk +1 ∈ − mod τηk +1 + K T − δ 3 , − mod τ ηk +1 + K T + δ 3 i. o. of n = 1 .
The even Ak occurs infinitely means that, with probability 1, for any δ1 > 0 , for any U ε1 γ +* ( 0 )  ,
there

are

infinitely

n = n (ω ) ∈ »

many


*
+

0

0

4

and mod (τ 2 n ) ∈ ( mod ( t1 − δ 4 ) , mod ( t1 + δ 4 ) ) . By continuity of solutions with


L.H. Lan / VNU Journal of Science: Mathematics – Physics, Vol. 30, No. 3 (2014) 49-69

ε3 > 0 , δ5 > 0 , δ6 > 0

respect to initial conditions, there are
∀ ( x, y ) ∈ U ε 3 γ

*
+

( 0 ) , ∀t ∈ ( t1 − δ 5 , t1 + δ 5 )

π t−,0 γ +* ( 0 ) − π t−,0 γ +* ( 0 )

ς2

3

}

ς 1 : ( x2 k +1 , y2 k +1 ) ∈ U ε γ +* ( 0 ) , mod (τ 2 k +1 ) ∈ ( −δ 6 , δ 6 )
3

}

........

{

}

ς n = inf 2k + 1 > ς n −1 : ( x2 k +1 , y2 k +1 ) ∈ U ε γ +* ( 0 ) , mod (τ 2 k +1 ) ∈ ( −δ 6 , δ 6 ) .
3

From the previous part of this proof, it follows that ς k < + ∞

{ς k = n}∈ ℑ , {ς k }

{

n
0

and lim ς k = + ∞ a.s.. Since

1

ς k +1

1

ε2

1

4

1

4

This means γ ∈ Ω ( x0 , y0 , ω ) a.s..
Lastly, by similar way and induction, we conclude that Γ is a subset of Ω ( x0 , y0 , ω ) . Because
Ω ( x0 , y0 , ω ) is a close set, we have Γ ⊂ Ω ( x0 , y0 , ω ) a.s..

(

b) We now prove the second assertion of this theorem. Let z = x , y

)

satisfying the condition

(3.12). By the existence and continuous dependence on the initial values of the solutions, there exist
two numbers a > 0

)

 x h +, x, y
x h1 − , x , y 
1
 ∂ϕ ∂ϕ 


det 
,
= det 


 ∂s ∂t  (t , t )
 y h2 + , x , y y h2 − , x , y 


 h +, z
h1 − , z 
1

 ≠ 0.
= x y det 

 h2 + , z
h2 − , z 



(


a ( s* , t * ) ∈ V = ( 0, a1 ) × ( 0, b1 )

such

that

, y0 , ω ) . Thus, there is a stopping time γ < + ∞ a.s.

such that ( x (γ ) , y ( γ ) ) ∈ U . Since Γ is a forward invariant set and U ⊂ Γ , it follows that

( x (t ) , y (t )) ∈ Γ , ∀ t > γ

with probability 1. The fact

( x (t ) , y (t )) ∈ Γ

for all t > γ implies that

Ω ( x0 , y0 , ω ) ⊂ Γ . By combining with the part a) we get Ω ( x0 , y0 , ω ) = Γ a.s..The proof is

complete.

4. Simulation and discussion
Noting that λ can be estimated by using the law of large number and formula (3.4) for an initial
concrete set. We will illustrate the above model by following numerical examples in three cases.

Example I. λ > 0 and the coexistence case presents in both states (see figure 3). It corresponds to

α = 0.6 ; β = 0.4 ; a ( + ) = 10 + sin t ; b ( + ) = 2 +

6
 2

e ( − ) = 1.2 +

1
1
 π
cos t ; f ( − ) = 2.7 − cos  t + 
4
2
 5


L.H. Lan / VNU Journal of Science: Mathematics – Physics, Vol. 30, No. 3 (2014) 49-69

67

the initial condition ( x ( 0 ) , y ( 0 ) ) = ( 2.5 ; 2.8 ) and number of switching n = 300. In this example,
the periodic T = 2π , the solution of (2.4) switches between two positive periodic orbit of the systems
(2.5) and (2.6).

Figure 3. Orbit of the system (2.4) in example I.

Example II. λ > 0 and one state is coexistence, the other is extinction of predator. The system
(2.5) with coefficients
1
a ( + ) = 12 + sin π t ; b ( + ) = 2.8 + cos π t ;
2
1

π
1
π


sin  π t +  ; d ( − ) = 6 + sin  π t +  ;
2
2
6
2



e ( − ) = 0.5 +

1
1
π

cos π t ; f ( − ) = 1.9 − cos  π t + 
4
2
5


has predator tending to 0. The number of switching n = 300 , transition intensities
α = 0.3 , β = 0.7 and initial condition ( x ( 0 ) , y ( 0 ) ) = (1.2, 3.4 ) . Since λ > 0 , the system (2.4) is
persistent (see figure 4).



Japan, 33 (1981), no. 2, pp. 335-366.
[12] M. Liu, K. Wang. Persistence, extinction and global asymptotical stability of a nonautonomous predator-prey
model with random perturbation. Appl. Math. Model, 36 (2012), no. 11, pp. 5344-5353.


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69

[13] S. S. De. Random predator-prey interactions in a varying environment: extinction or survival Bull. Math. Biol, 46
(1984), no. 1, pp. 175-184.
[14] P. Auger, N. H. Du, N. T. Hieu. Evolution of Lotka-Volterra predator-prey systems under telegraph noise. Math.
Biosci. Eng, 6 (2009), no. 4, 683-700.
[15] I.I Gihman and A.V. Skorohod. The Theory of Stochastic Processes. Springer -Verlag Berlin Heidelberg New
York 1979.
[16] R.S. Lipshter and Shyriaev. Statistics of Stochastic Processes. Nauka, Moscow 1974 (in Russian).