We show that there exists the density functions of the solutions and then, study the asymptotic behavior of these densities.. It is proved that the densities either converges in L1to an
Trang 1DOI 10.1007/s10440-011-9628-4
Asymptotic Behavior of Predator-Prey Systems
Perturbed by White Noise
Nguyen Hai Dang · Nguyen Huu Du · Ta Viet Ton
Received: 8 November 2010 / Accepted: 1 July 2011 / Published online: 21 July 2011
© Springer Science+Business Media B.V 2011
Abstract In this paper, we develop the results in Rudnicki (Stoch Process Appl 108:93–
107,2003) to a stochastic predator-prey system where the random factor acts on the coeffi-cients of environment We show that there exists the density functions of the solutions and then, study the asymptotic behavior of these densities It is proved that the densities either
converges in L1to an invariant density or converges weakly to a singular measure on the boundary
Keywords Prey-predator model· Diffusion process · Markov semigroups · Asymptotic stability
Mathematics Subject Classification (2000) 34C12· 47D07 · 60H10 · 92D25
1 Introduction
The stable predator-prey Lotka-Volterra equation
˙X t = (αXt − βXt Y t − μX2
t ),
˙Y t = (−γ Yt + δXt Y t − νY2
t ),
(1.1)
where X t and Y trepresent, respectively, the quantities of prey and the predator populations;
α, β, γ , δ, μ and ν are positive constants, has attracted a lot of attention It has been proved
that the solution of (1.1) is asymptotically stable
For the stochastic predator-prey Lotka-Volterra equation, we have to mention one of the first attempts in this direction, the very interesting paper of Arnold et al [1] where the
N.H Dang · N.H Du ()
Faculty of Mathematics, Mechanics and Informatics, Hanoi University of Science, Hanoi, Vietnam e-mail: dunh@vnu.edu.vn
T.V Ton
Department of Applied Physics, Graduate School of Engineering, Osaka University, Osaka, Japan
Trang 2authors used the theory of Brownian motion and the related white noise models to study the sample paths of the equation
dX t = (αXt − βXt Y t − μX2
t )dt + σXt dW t ,
dY t = (−γ Yt + δXt Y t − νY2
where Wt is the one dimensional Brownian motion defined on the complete probability
space ( , F ,{F t}t≥0, P) with the filtration { F t}t≥0satisfying the usual conditions, i.e., it is increasing and right continuous whileF0contains allP−null sets The positive numbers σ, ρ
are the coefficients of the effect of environmental stochastic perturbation on the prey and on the predator population respectively In this model, the random factor makes influences on the intrinsic growth rates of prey and predator
In continuing this study, in [10–12], the authors showed that the distribution of each solution starting at a point in intR2
+ has the density which either converges in L1 to an
invariant density or converges weakly to a singular measure on the boundary (0, ∞) × {0}.
This paper studies a stochastic predator-prey system where the intrinsic growth rate of
the prey, μ, and the one of the predator, ν, are perturbed stochastically μ → μ + σ ˙ W tand
ν → ν + ρ ˙ W t This means that we consider the following stochastic equation
dX t = (αXt − βXt Y t − μX2
t )dt + σX2
t dW t ,
dY t = (−γ Yt + δXt Y t − νY2
t )dt + ρY2
where σ and ρ are positive constants The existence and uniqueness of the positive solution
of (1.3) has been considered by X Mao et al [7] Moreover, the estimates of upper growth and lower growth of the sample paths of its solution were also given in [7] and [2] The aim of this paper is to study further the asymptotic behavior of the system (1.3)
by considering the convergence of the density of the solution Meanwhile the models (1.2) and (1.3) are rather similar but the calculation is much more complicated and the dynamic behavior of two these systems are different Moreover, it seems that the Khasminskii func-tion method dealt with in [11] is not suitable to this model (see the comment in Sect.4) Therefore, in order to study the existence of a stationary distribution and show its attractiv-ity, we have to use the method of analyzing the boundary distributions
The organization of the paper is as follows In Sect.2we get an outline of the singular cases where either prey or predator is absent Section3deals with asymptotic properties of Markov operators related to the dynamics of the solutions The theorem concerning with asymptotic stability and sweeping allows us to formulate the Foguel alternative This al-ternative says that under suitable conditions a Markov semigroup is either asymptotically stable or sweeping The last section gives some comments on the construction of Khasmin-skii functions
2 Singular Cases
By putting X t = e ξ t ; Y t = e η t and substituting this transformation into (1.3) we obtain
dξ t = (α − βe η t − μe ξ t−σ2
2e 2ξ t )dt + σe ξ t dW t ,
dη t = (−γ + δe ξ t − νe η t −ρ2e 2η t )dt + ρe η t dW t
(2.1)
Trang 3f1(x, y) = α − μe x − βe y−σ2
2e
2x ,
f2(x, y) = −γ + δe x − νe y−ρ2
2 e
2y ,
we can rewrite (2.1) under the form
dξ t = f1(ξ1, ηt )dt + σe ξ t dW t ,
dη t = f2(ξ1, ηt )dt + ρe η t dW t ,
or, under the Stratonovich equation
dξ t = (α − βe η t − μe ξ t − σ2e 2ξ t )dt + σe ξ t ◦ dWt ,
dη t = (−γ + δe ξ t − νe η t − ρ2e 2η t )dt + ρe η t ◦ dWt
(2.2)
The infinitesimal operator of (2.1) is
2σ
2e 2x ∂
2v
∂x2+1
2ρ
2e 2y ∂
2v
∂y2+ σρe x +y ∂2v
∂x∂y + f1
∂v
∂x + f2
∂v
∂y·
The density of the random variable (ξ t , η t ), if it exists and is smooth, can be found from the Fokker-Planck equation:
∂u
2σ
2∂2(e 2x v)
2ρ
2∂2(e 2y v)
∂y2 + σρ ∂2(e x+y v)
The behavior of the solutions for two boundary equations is easy to study If the prey is
absent, the quantity Y t = e η t of the predator at the time t satisfies the equation
dη t=
−γ − νe η t −ρ2
2e
2η t
Let
x
0
exp
−
y
0
2( −γ − νe u−ρ2
2e 2u )
dy
=
x
0
exp
2ν + γ
ρ2e 2y− 2ν
ρ2e y + y
dy.
We see that limx→∞ s1(x)= ∞ and limx→−∞s2(x) >−∞ By [3, Chap 5, Theorem 3.1],
we get
lim
t→∞ η t= −∞ or equivalently lim
t→∞ Y t = 0 a.s.
This means that without the preys, the predators die with probability one
Similarly, in the absence of the predators, the quantity Xt = e ξ t of prey at the time t
satisfies the equation
dξ t=
α − μe ξ t−σ2
2 e
2ξ t
Trang 4
x
0
exp
−
y
0
2(α − μe u−σ2
2e 2u )
dy
=
x
0
exp
2μ − α
σ2e 2y − 2μ
σ2e y + y
dy.
It is easy to see that limx→∞ s1(x)= ∞ and limx→−∞s2(x)= −∞ Then, by [3, Theo-rem 3.1] we also get lim supt→∞ ξ t= ∞; lim inft→∞ξ t= −∞ or equivalently
lim sup
t→∞ X t = ∞, lim inf
This means that, without the predators, the quantity of the preys oscillates between 0 and
∞ Further, (2.5) has a unique stationary distribution with the density f∗(x)satisfying the Fokker-Planck equation
1
2
d2
dx2
σ2e 2x f∗(x)
− d
2 e
2x )f∗(x)
The general solution of (2.7) is
y(x)= exp
−3x + 2μ
σ2e −x− α
σ2e −2x c
+ d
exp
σ2e −x+ α
σ2e −2x
dx
,
where c, d are two constants The conditions y(x)≥ 0 and
∞
imply d= 0 and
1
∞
−∞
exp
− 3x + 2μ
σ2e −x− α
σ2e −2x
dx
=
∞
0
y2exp
2μ
σ2y− α
σ2y2
dy.
It is easy to see that
1
c= exp
μ2
ασ2
σ
2α√
2α
∞
−√2μ
σ√α
σ u+
√
2μ
√
α
2
exp
−u2 2
du
=μσ2
2α2 +
√
π σ ( 2μ2+ ασ2)
2α2√
−√2μ
α
exp
μ2
ασ2
,
where (x) is the distribution function of a standard normal random variable N (0, 1) Thus,
f∗(x) = c exp
−3x + 2μ
σ2e −x− α
σ2e −2x
.
Trang 5If ξ tis a solution of (2.5) then by ergodic theorem
lim
t→∞
1
t
t
0
g(ξ s ) ds=
∞
−∞
for any g that is f∗-integrable Moreover, ξt converges in distribution to f∗as t→ ∞ (see [13, Theorems 16 and 17].) Let
+∞
−∞ e
x f∗(x)dx = c
∞
0
yexp
2μ
σ2y− α
σ2y2
It is obvious that
m=cσ2
2α +
√
π cμσ
−
√
2μ
α
exp
μ2
ασ2
√
ασ2+ 2√π αμσ
1− (−√2μ
σ√
α ) exp μ2
ασ2
√
αμσ2+√π σ ( 2μ2+ ασ2)
1− − 2μ
σ√α exp μ2
ασ2
Lemma 2.1 Denote by ¯ ξ t the solution of
d ¯ξ t=
α − μe ¯ξ t−σ2
2 e
2 ¯ξ t
dt + σe ¯ξ t dW t
with the initial condition ¯ ξ0 = ξ0and denote by ¯ηt the solution of
d ¯ηt=
−γ − νe ¯η t −ρ2
2e
2¯η t
dt + ρe ¯η t dW t
with the initial condition ¯η0= η0 Then with probability 1, ¯ ξ t ≥ ξt and ¯ηt ≤ ηt for all t≥ 0
Proof Put Z t = e −ξ t , ¯ Z t = e − ¯ξ t By Itô formula,
dZ t = (μ − αZt + βZt e η t + σ2Z t−1)dt − σdWt ,
d ¯ Z t = (μ − α ¯ Z t + σ2Z¯t−1
)dt − σdWt
Hence, by the comparison theorem (see Theorem 1.1: p 352 in [3]), Z t≥ ¯Z t for all t≥ 0 a.s It means thatP{ ¯ξt ≥ ξt} = 1 for all t ≥ 0 The second assertion ¯ηt ≤ ηt a.s for all t≥ 0
Theorem 2.2 The following assertions are true:
( a) lim sup t→∞
1
t
t
0
δe ξ s − νe η s−ρ2
2 e
2η s
ds ≤ γ a.s.
( b) lim inf t→∞
1
t
t
0
δe ξ s − νe η s−ρ2
2 e
2η s
ds ≥ 0 a.s.
( c) lim inf
t→∞
1
t
t
0
βe η s + μe ξ s+σ2
2 e
2ξ s
ds ≥ α a.s.
Trang 6Proof From [2, Theorem 2.4] we get lim supt→∞ η t
ln t ≤ 1 a.s Therefore,
−γ + lim sup
t→∞
1
t
t
0
δe ξ s − νe η s−ρ2
2e
2η s
ds + ρ
t
0
e η s dW s
y= lim sup
t→∞
η t
For any ε > 0, applying the exponential martingale inequality [6, Theorem 7.4, p 44] it yields,
P
sup
0≤t≤k
t
0
−ρe η s dW s−ε
2
t
0
ρ2e 2η s ds
> 2 ln k ε
≤ 1
k2.
By Borel-Cantelli lemma, for almost all ω, there exists a number k = k(ω) such that for all
n > kand 0≤ t ≤ n,
t
0
−ρe η s dW s−ε
2
t
0
ρ2e 2η s ds < 2 ln k
which implies that for k − 1 ≤ t ≤ k
1
t
t
0
−ρe η s dW s−ε
2
t
0
ρ2e 2η s ds
≤ 2 ln k
(k − 1)ε .
As a result,
lim inf
t→∞
1
t
t
0
ρe η s dW (s)+ε
2
t
0
ρ2e 2η s ds
From (2.10) and (2.11), it follows that
lim sup
t→∞
1
t
t
0
δe ξ s − νe η s−ρ2(1+ ε)
2η s
On the other hand, it follows from Lemma2.1that,
lim sup
t→∞
1
t
t
0
δe ξ s ds ≤ δ · lim sup
t→∞
1
t
t
0
e ¯ξ s ds = δm < ∞ a.s.
Therefore,
lim sup
t→∞
1
t
t
0
δe ξ s − νe η s−ρ2
2 e
2η s
ds
1+ εlim supt→∞
1
t
t
0
δe ξ s − νe η s−ρ2(1+ ε)
2η s
ds
1+ εlim supt→∞
1
t
t
0
δe ξ s − νe η s
1+ ε γ+
ε
1+ ε δm.
Letting ε→ 0 gets
lim sup
t→∞
1
t
t
δe ξ s − νe η s−ρ2
2e
2η s
Trang 7
Thus, item a) has been proved Further, by Lemma2.1
η t − ¯ηt=
t
0
δe ξ s − νe η s−ρ2
2 e
2η s
ds + ρ
t
0
e η s dW s
+
t
0
νe ¯η s+ρ2
2e
2¯η s
ds − ρ
t
0
e ¯η s dW s ≥ 0. (2.12)
By the same argument as the proof of the first assertion,
lim sup
t→∞
1
t
t
0
ρe η s dW s−ε
2
t
0
ρ2e 2η s ds
≤ 0,
and
lim sup
t→∞
1
t
t
0
−ρe ¯η s dW s−ε
2
t
0
ρ2e2¯η s ds
≤ 0.
Combining these inequalities and (2.12) gets
lim inf
t→∞
1
t
t
0
δe ξ s − νe η s−ρ2(1− ε)
2η s
−
νe ¯η s+ρ(1+ ε)
2¯η s
ds ≥ 0.
Since ¯ηt → −∞ as t → ∞, limt→∞1
t
t
0(νe ¯η s+ρ(1+ε)
2 e2¯η s )ds= 0 Hence,
lim inf
t→∞
1
t
t
0
δe ξ s − νe η s−ρ2(1− ε)
2η s
ds ≥ 0.
Note that if lim supt→∞1
t
t
0e 2η s = ∞ then for ε < 1,
0≤ lim inf
t→∞
1
t
t
0
δe ξ s − νe η s−ρ2(1− ε)
2η s
ds
≤ lim sup
t→∞
1
t
t
0
δe ξ s ds+ lim inft→∞ 1
t
t
0
−ρ2(1− ε)
2η s
ds
≤ δm − ρ2(1− ε)
2 · lim sup
t→∞
1
t
t
0
e 2η s ds = −∞.
That is a contradiction Thus, lim supt→∞1tt
0e 2η s <∞ Hence,
0≤ lim inf
t→∞
1
t
t
0
δe ξ s − νe η s−ρ2(1− ε)
2η s
ds
≤ lim inf
t→∞
1
t
t
0
δe ξ s − νe η s−ρ2
2e
2η s
ds + ε · lim sup
t→∞
1
t
t
0
ρ2
2e
2η s ds.
Let ε→ 0, we obtain item (b) The proof of item (c) is similar to the one of item (a)
3 Asymptotic Stability
Let the triple (X, , m) be a σ -finite measure space Denote by D the subset of the space
L1= L1(X, , m), consisting of the densities, i.e D = {f ∈ L1
Trang 8mapping P : L1→ L1is called a Markov operator if P (D) ⊂ D The Markov operator P
is called an integral or kernel operator if there exists a measurable function k : X × X → [0, ∞) such that Pf (x) =X k(x, y)f (y) m(dy) for every density f ∈ D Let {P (t)}t≥0be a
semigroup of linear operators on L1 The family{P (t)}t≥0is said to be a Markov semigroup
if P (t) is a semigroup and for every t ≥ 0 and P (t) is a Markov operator {P (t)}t≥0 is
called integral, if for each t > 0, the operator P (t) is an integral Markov operator The
semigroup{P (t)}t≥0 is called asymptotically stable if there is an invariant density f∗such
semigroup{P (t)}t≥0 if P (t)f∗= f∗for each t ≥ 0) The Markov semigroup {P (t)}t≥0is
called sweeping with respect to a set A ∈ if for any f ∈ D,
lim
t→∞
A
P (t )f (x) m(dx) = 0.
If the semigroup is either asymptotically stable or sweeping with respect to compact sets then we say that the semigroup has the “Foguel alternative” It is clear that if the Markov semigroup{P (t)}t≥0has an invariant density then it is not sweeping with respect to sets of finite measure We will use the nice property to exclude the sweeping and obtain asymptotic stability in Theorem3.10 For the more details, we can refer to [11]
We now consider the space (R2, B (R2), m) where B (R2) is the σ -algebra of Borel
sub-sets ofR2and m is the Lebesgue measure on (R2, B (R2)).We shall prove the Foguel alter-native property of the semigroup{P (t)}t≥0by using the following lemmas
Lemma 3.1 Let (ξ t , η t ) be the solution of (2.1) with the initial value (ξ0, η0) =
(x0, y0)∈ R2 Then, the transition probability function P (t, x0, y0, ·) of the Markov diffu-sion process (ξ t , η t ) , i.e., P (t, x0, y0, A) = P{(ξt , η t ) ∈ A} for any A ∈ B (R2), has a density
k(t, x, y, x0, y0) with respect to m and k ∈ C∞(( 0, ∞) × R2× R2) for all t > 0.
Proof The proof is somewhat similar to the one in [11] We apply the Hormander theorem
on the existence of smooth densities of the transition probability function for the degenerate diffusion processes described by the Stratonovich equation (2.2)
If X(x) = (X1, X2)T and Y (x) = (Y1, Y2)T are vector fields onR2then the Lie bracket
[X, Y ] is a vector field given by
[X, Y ]j (x)=
X1 ∂Y j
∂x1 (x) − Y1
∂X j
∂x1 (x)
+
X2 ∂Y j
∂x2 (x) − Y2
∂X j
∂x2 (x)
, j = 1, 2.
Let
a0(x, y)=
α − βe y − μe x − σ2e 2x
−γ + δe x − νe y − ρ2e 2y
σ e x
ρe y
.
Putting a = 2ρ2β > 0, b = 2σ2δ >0, it is easy to get that
a2 = [a0, a1] =
σ e x (α − βe y + σ2e 2x ) + ρβe 2y
ρe y ( −γ + δe x + ρ2e 2y ) − σδe 2x
,
a3 = [a1,[a0, a1]] =
2σ4e 4x + ae 3y
2ρ4e 4y − be 3x
,
a4 = [a1,[a1,[a0, a1]]] =
6σ5e 5x + 3ρae 4y − σae x +3y
−3σbe 4x + 6ρ5e 5y + bρe 3x +y
.
Trang 9Now we assume that there exists a point (x, y) such that three vectors a1(x, y), a3(x, y)and
a4(x, y)do not span the spaceR2 In this case, two vectors a1, a3are parallel and so are
a1, a4 Hence,
σ e x ( 2ρ4e 4y − be 3x ) = ρe y ( 2σ4e 4x + ae 3y ),
σ e x ( −3bσe 4x + 6ρ5e 5y + bρe 3x +y ) = ρe y ( 6σ5e 5x + 3aρe 4y − aσe x+3y ). (3.1)
The first equation of (3.1) gives that
2σρ4e x+4y = 2ρσ4e 4x +y + aρe 4y + bσe 4x (3.2) Thus,
6σρ5e x+5y = 6ρ2σ4e 4x +2y + 3aρ2e 5y + 3bσρe 4x +y , σ e x < ρe y (3.3) From the second equation of (3.1) we obtain
6σρ5e x+5y + bσρe 4x +y + aσρe x+4y = 6σ5ρe 5x +y + 3aρ2e 5y + 3bσ2e 5x (3.4)
By combining (3.3) and (3.4) it yields
bσρe 4x +y + aσρe x+4y + 6σ4ρ2e 4x +2y + 3bσρe 4x +y = 6σ5ρe 5x +y + 3bσ2e 5x (3.5) The relations (3.3) imply 6σ4ρ2e 4x +2y > 6σ5ρe 5x +y , 3bσρe 4x +y > 3bσ2e 5xwhich says that the equality (3.5) is impossible
Thus, we get the Hörmander condition:
(H) For every (x, y)∈ R2vectors
a1(x, y), [ai , a j ](x, y)0≤i,j≤1 , [ai , [aj , a k]](x, y)0≤i,j,k≤1 ,
span the spaceR2
Under the condition (H), the transition probability functionP (t, x0, y0, ·) has a density
k(t, x, y, x0, y0) and k ∈ C∞(( 0, ∞) × R2× R2)(see [5,9]) The lemma is proved
We now denote by (ξt , η t )the solution of (2.1) with the initial random variable (ξ0, η0),
where the distribution of (ξ0, η0)is absolutely continuous with the density v(x, y) From the above lemma, it is known that for any t > 0, (ξ t , η t ) has the density u(t, x, y) satisfying the
Fokker-Planck equation (2.3) Further,
u(t, x, y)=
∞
−∞
∞
−∞k(t, x, y ; x1, y1)v(x1, y1)dx1dy1.
For any t ≥ 0, we consider the operator P (t) defined by
P (t )v(x, y) = u(t, x, y) =
∞
−∞
∞
−∞k(t, x, y ; x1, y1)v(x1, y1)dx1dy1
for v ∈ D By the analytic prolongation of the operator P (t) on L1(R2, B (R2), m) and
Lemma 3.1, we now get that {P (t)}t≥0 is an integral Markov semigroup with a
continu-ous kernel k.
Trang 10Now we follow the method in [11] to check the positivity of k Fix a point (x0, y0)∈ R2
and a function φ ∈ C([0, T ]; R) Consider the following system of differential equations:
x φ = α − βe y φ + (σφ(t) − μ)e x φ − σ2e 2x φ ,
y φ = −γ + δe x φ + (ρφ(t) − ν)e y φ − ρ2e 2y φ
(3.6)
Put
¯
f1(x, y) = α − βe y + (σφ(t) − μ)e x − σ2e 2x ,
¯
f2(x, y) = −γ + δe x + (ρφ(t) − ν)e y − ρ2e 2y ,
we can rewrite (3.6) under the form
x φ (t )= ¯f1(x φ (t ), y φ (t )),
y φ (t )= ¯f2(x φ (t ), y φ (t )).
Let (xφ , y φ )be the solution of (3.6) with the initial condition xφ ( 0) = x0; yφ ( 0) = y0and
F : C([0, T ], R) → R2be a mapping defined by F (h) = (xφ +h (T ), y φ +h (T )) We calculate the Frechet derivative Dx0,y0,φ of F by using means of the perturbation method for ordinary
differential equations Let f= ( ¯ f1, ¯ f2) and (t)= f (x
φ (t ), y φ (t )) Denote Q(t, s), for 0≤
s ≤ t ≤ T , the fundamental matrix of solutions of the equation
˙Y = (t)Y,
i.e., ∂Q(t,s) ∂t = (t)Q(t, s) and Q(s, s) = I Then,
D x0,y0,φ h=
T
0
Q(T , s) q(x φ (s), y φ (s)) h(s)ds,
where q(x, y)=σ e x
ρe y
Let ε ∈ (0, T ) and h(t) = 0 if 0 ≤ t ≤ T −ε and h(t) =1
ε (t −T +ε)
if T − ε ≤ t ≤ T By Taylor formula we have Q(T , s) = I − (T )(T − s) + o(T − s) as
s → T Thus,
D x0,y0,φ h=
T
T −ε
s − T + ε
ε q(x φ (s), y φ (s))ds
+ (T )
T
T −ε
s − T + ε
ε (s − T ) q(xφ (s), y φ (s))ds + o(ε2).
Put ¯x = xφ (T ), ¯y = yφ (T ), c = φ(T ) From Theorem of Mean Value Integration, we have
lim
ε→0
1
ε
T
T −ε
s − T + ε
ε q(x φ (s), y φ (s))ds=1
2q( ¯x, ¯y) and
lim
ε→0
1
ε2(T )
T
T −ε
s − T + ε
ε (s − T )q(xφ (s), y φ (s))ds= −1
6(T )q( ¯x, ¯y).
By a direct calculation we obtain
(cσ − μ)e ¯x − 2σ2e2¯x −βe ¯y
δe ¯x (cρ − ν)e ¯y − 2ρ2e2¯y
.
... The proof of item (c) is similar to the one of item (a)3 Asymptotic Stability
Let the triple (X, , m) be a σ -finite measure space Denote by D the subset of the...
Proof The proof is somewhat similar to the one in [11] We apply the Hormander theorem
on the existence of smooth densities of the transition probability function for the degenerate... )) We calculate the Frechet derivative Dx0,y0,φ of F by using means of the perturbation method for ordinary