2012 2nd International Conference on Information Communication and Management (ICICM 2012)
IPCSIT vol. 55 (2012) © (2012) IACSIT Press, Singapore
DOI: 10.7763/IPCSIT.2012.V55.21
Stability Of An N-Patch Predator-Prey System
With Stochastic Perturbation
Yanqiu Li 1,2, Jizhang Fan1
1
Research Institute of Synthetic Information Mineral Resources Prediction, Jilin University,
Changchun 130026, Jilin, P. R. China
2
Applied Sciences, Jilin Teacher's Institute of Engineering and Technology,
Changchun 130052, Jilin, P. R. China.
Abstract. This paper discusses a randomized predator-prey dynamical system which is composed of
several patches connected by linear diffusion. We show that the positive solution of the associated stochastic
differential equation does not explode to infinity in a finite time. In addition, we show the positive
equilibrium is global-stability under simple assumptions.
Keywords: Linear diffusion, Ito formula, Explodes, Global stability.
1. Introduction
Recently, many authors consider the effect of spatial distributions of species over the range of the habitat
in population dynamics. Allen[1],by using a comparison theorem, obtained a partial answer to the persistent
and extinction problem for single-species discrete diffusion systems. Kuang[2]discussed predator-prey
dynamics in models of prey dispersal in two-path environments. Lu[3]discussed global asymptotic behavior
in single-species discrete diffusion systems.
Li[4] considered the following predator-prey system :
n
⎧
x
x
(
r
d
x
e
y
)
α ij ( x j − xi ),
=
−
−
+
∑
i
i
i i
i i
⎪ i
j =1
⎪
⎨
n
⎪ y = y ( − γ − δ y + ε x ) +
β ij ( y j − y i ), i = 1" n.
∑
i
i
i i
i i
⎪⎩
j =1
(1)
Here xi , yi denote the densities of preys and predators on the patch i , α ij , β ij is the dispersal rate of preys
and predators from patch j to patch i .The parameters in the system are nonnegative.
On the other hand, special population systems are often subject to environmental noise. Suppose
that ri , γ i are stochastically perturbed, with ri → ri + ( xi − x *i )W i (t ) , γ i → γ i + ( yi − y *i )W i (t ) .Where Wi (t )
are independent white noises with Wi (0) = 0 , T ≥ 0 .Then this environmentally perturbed system may be
described by the Ito equation
n
⎧
x
x
(
r
d
x
e
y
)
α ij ( x j − xi ) + xi x *i ( xi − x *i )W i ( t ),
=
−
−
+
∑
i
i
i i
i i
⎪ i
j =1
⎪
⎨
n
⎪ y = y ( − γ − δ y + ε x ) +
β ij ( y j − y i ) + y i y *i ( y i − y *i )W i ( t ), i = 1" n.
∑
i
i
i i
i i
⎪⎩
j =1
113
(2)
The parameters in the system are nonnegative. Our theory shows that the solution of Eq (2) is not only
positive but also will not explode to infinity at any finite time, see Sections II. We show that the global
asymptotic stability of a positive equilibrium if following conditions hold in III:
(A1) system (1.1) has a positive equilibrium ( x 1* , y 1* , x *2 , y *2 , " , x *n , y *n ) ;
(A2) d i > 1 x * 2 and δ i > 1 y * 2 ;
2
2
i
i
(A3) α ij ε i x * = λβ ij ei y * .
j
j
2. Positive and global solutions
Wi (t ) denote the independent standard Brownian motions defined on this
R 2 n = {x, y ∈ R+n } .
probability space,, We introduce the notation +
Throughout this paper,
Theorem 2.1. Let the initial data
R+2 n (0) , then there exists a unique solution ( x, y ) to Eq. (2) on
2n
t ≥ 0 and the solution will remain in R+ with probability 1 under (A1) (A2).
Proof. Since the coefficients of the equation are locally Lipschitz continuous, for any given the initial
( x(0), y (0)) ∈ R+2 n , where τ e is the explosion time. To show this Solution is global; we need to show
τ = ∞ a.s. Let be sufficiently large for every component of {(x, y)∈Rn : xi > 0, yi > 0} lying within the
that e
data
interval
[
τ
1
, k0 ]
k0
k
k ≥ k0 , define the stopping time
. For each integer
= in f{t ∈ (0 ,τ e ) : xi (t ) ∉ (
1
1
, k ) , y i ( t ) ∉ ( , k ) , f o r s o m e i}
k
k
τ
Where throughout this paper we set inf Φ = ∞ .Clearly, k is increasing as k → ∞ ,Set
τ ∞ = lim τ k
k →∞
,
τ ∞ = ∞ a.s., then τ ∞ = ∞ a.s. and x(t ) ∈ R a.s. for all t ≥ 0 . In other
τ = ∞ a.s. For if this statement is false, then there
words, to complete the proof all we need to show is that ∞
n
+
whence a.s. If we can show that
P (τ ∞ ≤ T ) > ε Hence there is an integer k1 ≥ k0 such
is a pair of constants T > 0 and ε ∈ (0,1) Such that
that
P (τ ∞ ≤ T ) ≥ ε for all k > k1
Define a C function V : R
2
2n
+
→ R + by
n
∑ [ε
V ( x, y) =
i =1
i
(3)
( x i − 1 − lo g ( x i ) ) + e i ( y i − 1 − lo g ( y i ) ) ]
The nonnegative of this function can be seen from u−1−log(u) ≥0 on u > 0 .If
show that
( x(t ), y (t )) ∈ R+2 n , we
n
d [V ( x ( t ), y ( t ))] = F ( x , y ) dt + ∑ ε i x *i ( x i − 1)( x i − x *i ) dW i ( t )
i =1
+
n
∑ey
i =1
i
*
i
( y i − 1)( y i − y i ) dW i ( t )
*
Where
F ( x, y ) =
n
∑ ε [( x
i
i =1
+
n
∑ e [( y
i =1
i
n
i
− 1)( ri − d i xi − ei y i + ∑ α ij (
j =1
n
i
xj
n
1
− 1))] + ∑ ε i xi*2 [ ( xi − xi* ) 2 ]
xi
2
i =1
− 1)( − γ i − δ i y i + ε i xi + ∑ β ij (
j =1
yj
n
1
− 1))] + ∑ ei y i*2 [ ( y i − y i* ) 2 ]
yi
2
i =1
≤K
E (V ( x (τ k ^ T ), y (τ k ^ T ))) ≤ V ( x (0), y (0)) + KE (τ k ^ T )
114
(4)
Here, and in the sequel, E( f ) shall mean the mathematical expectation of f .
E (V ( x (τ k ^ T ), y (τ k ^ T ))) ≤ V ( x (0), y (0)) + KT
Ω k = {τ k ≤ T } for k ≥ k1 and, by (3), P (Ω k ) ≥ ε Note that for every ω ∈ Ω k there is some i
x (τ , ω ) equals either k or 1 / k and hence V ( x (τ k , ω ), y (τ k , ω )) is no less than
such that i k
either k − 1 − log(k ) or 1 / k − 1 − lo g (1 / k ) .
Set
Consequently, V ( x (τ k , ω ) , y (τ k , ω ) ) ≥ [ k − 1 − lo g ( k )] ^ [1 / k − 1 − lo g (1 / k )]
It then follows from (4) that
V ( x (0 ), y (0 )) + K T ≥ E [1 Ω k V ( x (τ k , ω ), y (τ k , ω ))]
≥ ε ([ k − 1 − lo g ( k )] ^ [1 / k − 1 − lo g (1 / k )])
Where
1Ωk
is the indicator function of
Ωk
.Letting k → ∞ leads to the contradiction
∞ > V ( x(0), y (0)) + KT = ∞ So we have τ ∞ = ∞ a.s.
3. Global stability
In this section, we will prove system (2) is global asymptotic stability under some assumptions.
Theorem 3.1 Assume (A1),( A2), (A3) hold, then the trivial solution of (2) is stochastically stable and
stochastically asymptotically stable.
Proof. If
of(2).Define
( x1* , y1* , x*2 , y*2 , " , x*n , y*n )
is a positive equilibrium of(1),it also is a positive equilibrium
xi
y
) + e i ( y i − y *i + y *i ln *i )
*
xi
yi
V i ( x i , y i ) = ε i ( x i − x *i + x *i ln
By the Ito formula, we have
dVi ( xi , yi ) = LVi ( xi , yi ) dt + ε i x*i ( xi − x*i ) 2 dWi (t ) + ei y *i ( yi − y *i ) 2 dWi (t )
Where LV : R+n × R+n → R is defined by
n
xj
j =1
xi
LVi ( xi , y i ) = ε i ( xi − x *i )( ri − d i xi − ei y i + ∑ α ij
n
− ∑ α ij )
j =1
n
yj
j =1
yi
+ ei ( y i − y *i )( − γ i − δ i y i + ε i xi + ∑ β ij
n
− ∑ β ij )
j =1
1
1
( xi − x *i ) 2 + ei y *2
( y i − y *i ) 2
ε i x *2
i
i
2
2
1
1
)( y i − y *i ) 2
= − ε i ( d i − x *i 2 )( x i − x *i ) 2 − ei (δ i − y *2
2
2 i
n
n
x j x*
y j y*
x
y
x
y
+ ∑ α ij ε i x *j ( *j − *i + 1 − * i ) + ∑ β ij ei y *j ( *j − *i + 1 − * i )
x j xi
x j xi
yj
yi
y j yi
j =1
j =1
+
LV i ( x i , y i ) ≤
n
∑α
j =1
ij
ε i x * (1 −
j
n
+ ∑ β ij ei y *j (1 −
j =1
≤
n
∑α
j =1
=
n
∑α
j =1
=
ij
j =1
j
xj
x
*
j
xj
*
x j xi
y j y *i
*
y j yi
+ ln
+ ln
+ ln
xj
*
xj
xj
−
x j x *i
*
x j xi
y j y *i
*
y j yi
n
xj
j =1
x *j
) + ∑ α ij ε i x *j (
n
) + ∑ β ij ei y *j (
j =1
ij
ε i x * [ G j ( x j , y j ) − G i ( x i , y i )]
x
*
j
+ ln
*
xj
+λ
yj
ε i x * [(
j
yj
y
*
j
+ ln
+ ln
xj
x *j
yj
y *j
−
−
xi
x
+ ln *i )
x i*
xi
yi
y
+ ln *i )
y i*
yi
n
yj
yj
xi
x
y
y
+ ln *i ) + ∑ β ij ei y *j ( * + ln * − *i + ln *i )
*
xi
xi
y
y
y
y
j =1
j
i
i
j
ij
n
∑α
ε i x* (
x j x *i
y
*
j
+ λ ln
j
115
yj
*
yj
)−(
xi
x
y
y
+ ln *i + λ *i + λ ln *i )]
*
xi
xi
yi
yi
(5)
and G i ( xi , y i ) and α ij , β ij satisfy the assumption of reference[5],then LVi ( xi , y i ) ≤ 0 .Therefore
LV (x, y) =
n
∑
i=1
L Vi ( xi , yi ) ≤ 0
Especially, LV ( x, y ) = 0 if and only if xi = x*i , yi = y*i , which means the solution of (2) is
stochastically stable and stochastically asymptotically stable, see [6].
4. Summary
In this paper, we show the positive solution exists of randomized predator-prey dynamical systems which
are composed of several patches connected by linear diffusion. In addition, we show that the positive
equilibrium is global-stability under simple assumptions.
5. References
[1] Allen,L.J.S. Persistence and extinction in single-species reaction-diffusion models . Bull.Math.Biol.1983,45:209227.
[2] Y.Kuang,Y.Takeuchi. Predator-prey dynamics in models of prey dispersal in two-path environments. Math.Biosci.
1994, 120: 77-98.
[3] Z.Lu and Y.Takeuchi. Global asymptotic behavior in single-species discrete diffusion systems. J.Math Biol. 1993,
32:67-77.
[4] Yanqiu Li. Global Asymptotic Stability of An n-patch Predator-prey system(In press).
[5] H.Guo,M.Y.Li,and Z.Shuai. Global stability of the endemic equilibrium of multi-group SIR epidemic models.
Can.Appl.Math.Q. 2006, 14:259-284.
[6] X. Mao. Stochastic Differential Equations and Applications. Harwood, Chi Chester, 1997.
Yanqiu Li(1982-) is born in Gongzhuling, Jilin, China, Bachelor of Science,
Department
of
Mathematical,
North
East
Normal
University,Changchun,Jilin,China,1999-2003. Master of Science, Department of
Mathematical, North East Normal University ,Changchun,Jilin,China,2003-2006.
PhD candidate, Research Institute of Synthetic Information Mineral
Resources Prediction, Jilin University, Changchun, China,2010-.
Jizhang Fan(1945-) is born in Changchun, Jilin, China, Bachelor of engineering,
Faculty
of
GeoExploration
Science
and
Technology
,Jilin
University ,Changchun,Jilin,China,1963-1967. Master of engineering, Faculty of
GeoExploration
Science
and
Technology,
Jilin
University ,Changchun,Jilin,China,1982-1985. PhD, Research Institute of Synthetic
Information Mineral. Resources Prediction, Jilin University, Changchun ,
China,1996-2000.
116