Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 162, pp. 1–17.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
ASYMPTOTIC BEHAVIOR OF STOCHASTIC GILPIN-AYALA
MUTUALISM MODEL WITH JUMPS
XINHONG ZHANG, KE WANG
Abstract. This article concerns the study of stochastic Gilpin-Ayala mutualism models with white noise and Poisson jumps. Firstly, an explicit solution
for one-dimensional Gilpin-Ayala mutualism model with jumps is obtained and
the asymptotic pathwise behavior is analyzed. Then, sufficient conditions for
the existence of global positive solutions, stochastically ultimate boundedness
and stochastic permanence are established for the n-dimensional model. Asymptotic pathwise behavior of n-dimensional Gilpin-Ayala mutualism model
with jumps is also discussed. Finally numerical examples are introduced to
illustrate the results developed.
1. Introduction
In nature, mutualism is a usual phenomena. Rhinos and tick birds are an example
of a mutualism relationship. Tick birds eat the ticks on a rhino while the rhino
loses annoying parasites, the relationship is positive for both the rhino and the tick
birds because they both get what they want. Therefore it is important to study
the mutualism models for multi-species. As is well known now, the most important
model among several cooperative models is the following Lotka-Volterra mutualism
system:
d
h
i
X
dxi (t) = xi (t) ri −
aij xj (t) dt, 1 ≤ i ≤ n,
j=1
where xi (t) is the population size of species i, ri is the intrinsic growth rate of species
i, aij (i 6= j) represents the effect of species j upon the growth rate of species i,
aii stands for the intraspecies interaction, aii > 0, aij < 0, i 6= j. There is an
extensive literature concerned this model, for example, [1, 9, 10, 13, 21, 24, 26, 28].
However, in the practical case, population systems are often subject to various
stochastic small perturbation. The growth rates, interaction coefficients and so on
may be influenced by environmental noise. In recent years, stochastic differential
equations have received much attention, many results have been derived to reveal
how environmental noise affects the population systems. In particular, Mao, Marion
and Renshaw [22] revealed that the environmental noise can suppress a potential
2000 Mathematics Subject Classification. 34F05, 92D25, 60H10, 60H20.
Key words and phrases. Gilpin-Ayala mutualism model; jumps; moment boundedness;
asymptotic behavior; stochastic permanence.
c
2013
Texas State University - San Marcos.
Submitted May 17, 2013. Published July 19, 2013.
1
2
X. ZHANG, K. WANG
EJDE-2013/162
population
Pm explosion. Suppose the growth rates are perturbed by white noise ri →
ri + j=1 σij Ḃj (t), here Ḃj (t) is a white noise, i.e., (B1 (t), . . . , Bm (t)) is an m2
dimensional Brownian motion defined on a complete probability space (Ω, F, P), σij
stands for the intensity of the noise. Then the stochastic Lotka-Volterra mutualism
system becomes
d
m
h
i
X
X
dxi (t) = xi (t) (ri −
aij xj (t))dt +
σij dBj (t) ,
j=1
1 ≤ i ≤ n.
j=1
Various forms of cooperative Lotka-Volterra system have been extensively studied
and we here mention Hung [11], Cheng[6], Liu and Wang [19, 20] and the references
cited therein. The key method used in our paper is motivated by them.
Unfortunately, in the Lotka-Volterra model, the rate of change of the size of
each species is linear function of sizes of the interacting species [16, 7]. However
in complex ecosystem this is almost impossible. Therefore, to meet the practical
situations, in 1973, Gilpin and Ayala [8] provided a modification for Lotka-Volterra
model, called Gilpin-Ayala model. For various forms about the Gilpin-Ayala system
readers can see [5, 17, 29, 30] and references therein for details. But here are few
works about stochastic Gilpin-Ayala mutualism model.
On the other hand, the population systems may suffer sudden environmental
perturbations, that is, some jump type stochastic perturbations; e.g., earthquakes,
hurricanes, epidemics and so on [3, 4]. These phenomena can not be described by
stochastic integrals driven only by Brownian motion. So it is feasible to introduce
a jump process into the underlying population system. SDEs with jumps have
received considerable attention in the past few years. We here mention Applebaum
[2], Situ [27], Bao et al [3, 4]. Particularly, the books by Applebaum [2] and Situ
[27] are good references in this area. To the best of the authors knowledge, to
this day, n-dimensional Gilpin-Ayala mutualism model with jumps has not been
studied. Motivated by these, the following n-dimensional Gilpin-Ayala mutualism
model with jumps is considered in this article:
d
m
nh
i
X
X
dxi (t) = xi (t− ) ri − aii xθi i (t− ) −
aij xj (t− ) dt +
σij dBj (t)
j6=i
Z
+
j=1
(1.1)
o
γi (u)N (dt, du) ,
Y
for 1 ≤ i ≤ n, where x(t− ) is the left limit of x(t), θi ≥ 1 is the parameter to modify
the classical Lotka-Volterra model, γi (u) > −1 is a bounded function, i = 1, . . . , n,
N is a Poisson counting measure with characteristic measure ν on a measurable
e (dt, du) := N (dt, du) − ν(du)dt. We
subset Y of (0, ∞) with ν(Y) < ∞, and N
assume Brownian motion and N are independent.
Throughout this article Rn+ := {x ∈ Rn : xi > 0 for i = 1, . . . , n}, Ā = (aij ), ĀT
Pn
1
denotes the transposepof Ā. If x ∈ Rn , its norm is denoted by |x| = ( i=1 x2i ) 2 . If
Q is a matrix, |Q| = trace(QT Q) represents its trace norm. If Q = (qij )n×n is a
symmetric matrix, then λ+
xT Qx. K is a positive constant
max (Q) = supx∈Rn
+ ,|x|=1
and may be different at different places. We impose the following assumptions:
EJDE-2013/162
ASYMPTOTIC BEHAVIOR
3
(A1) There is a positive constant κ such that
Z
(| ln(1 + γ(y))| ∨ | ln(1 + γ(y))|2 )ν(dy) < κ.
Y
(A2) There are positive constants p1 , . . . , pn such that
T
−2τ := λ+
max (−P̄ Ā − Ā P̄ ) < 0,
where P̄ = diag(p1 , . . . , pn ).
From Assumption (A2), it is easy to see that τ ≤ aii pi for i = 1, . . . , n.
The aim of our work is to study the properties of n-dimensional stochastic GilpinAyala mutualism model with jumps. The significance of this paper is mainly: (1)
Gilpin-Ayala system is more suitable for the real situations than Lotka-Volterra
system, but more complicated; (2) The white noise and Poisson jumps are taken
into account. The remaining part of this paper is organized as follows. In section 2,
we provide an explicit solution for one-dimensional Gilpin-Ayala model with jumps
and study its asymptotic pathwise behavior. In section 3, we show that (1.1) will
have a unique global positive solution under certain conditions. Section 4 and 5
deal with the asymptotic moment properties and asymptotic pathwise behavior of
the solution, respectively. In section 6, we show that the system is stochastically
permanent if the white noise and Poisson jumps satisfy our conditions. Finally we
introduce some simulation figures to illustrate our main results.
2. One-dimensional Gilpin-Ayala model
As the single population is the basic unit of the whole ecological system, the
establishment and theoretical analysis of the single population model can help us
to understand the overall structure of the complex model. So we firstly analyze
one-dimensional Gilpin-Ayala model with jumps.
Lemma 2.1 ([3]). Consider the following system of equations with jumps:
Z
h
i
−
−
e (dt, du) ,
dYi (t) = Yi (t ) (ai − bii Yi (t ))dt + σi dB(t) +
ci (u)N
Y
where ai > 0, bii > 0, ci (u) > −1, B(t) is a one-dimensional Brownian motion.
Then for any initial value Yi (0) ∈ Rn+ , this equation admits a unique positive solution Yi (t), t ≥ 0, which is global and admits the explicit formula
Yi (t) =
1
Yi (0)
+
ϕi (t)
,
Rt
b ϕ (s)ds
0 ii i
where
Z th
1
ai − σi2 +
2
Z
Z t
i
(ln(1 + ci (u)) − ci (u))ν(du) ds +
σi dB(s)
ϕi (t) := exp
0
Y
Z tZ
e (dt, du) .
+
ln(1 + ci (u))N
0
0
Y
Remark 2.2. In general, the intrinsic growth rate ai is positive, but the above
explicit solution holds for ai ≤ 0.
4
X. ZHANG, K. WANG
EJDE-2013/162
The one-dimensional Gilpin-Ayala model with jumps is
dyi (t) = yi (t− )(ri − aii yiθi (t− ))dt + yi
m
X
σij dBj (t)
j=1
Z
+
yi (t− )γi (u)N (dt, du),
(2.1)
θi > 1,
Y
yi (0) = xi (0).
Set zi = yiθi , by Itô’s formula, we have
Z
m
h
θi (θi − 1) X 2
dzi (t) = zi (t− ) θi ri +
σij + ((1 + γi (u))θi − 1)ν(du)
2
Y
j=1
m
i
X
− θi aii zi (t− ) dt + zi (t− )θi
σij dBj (t)
(2.2)
j=1
Z
+
e (dt, du),
zi (t− )((1 + γi (u))θi − 1)N
θi > 1,
Y
zi (0) = xθi i (0).
From Lemma 2.1 and Remark 2.2, it follows that (2.2) has an explicit solution
zi (t) =
1
θ
xi i (0)
Φi (t)
,
Rt
+ 0 aii θi Φi (s)ds
where
Z
m
i
h
1X 2
σij +
ln(1 + γi (u))ν(du) ds
Φi (t) := exp
θi ri −
2 j=1
Y
0
Z t X
Z tZ
m
o
e (dt, du)
+
θi
σij dBj (s) +
θi ln(1 + γi (u))N
nZ
0
t
0
j=1
Y
Combining [3, Lemma 4.4 and Theorem 4.4], we can deduce the following lemma.
Pm 2 R
Lemma 2.3. Assume (A1) holds. If ci := ri − 21 j=1 σij
+ Y ln(1+γi (u))ν(du) ≥
0, then for i = 1, . . . , n, we have
lim
t→∞
ln zi (t)
= 0 a.s.
t
(2.3)
Remark 2.4. Based on the above analysis, (2.1) has a unique positive solution
yi (t) for any value yi (0) = xi (0) > 0 which is global and represented by
1/θi
Φi (t)
yi (t) =
,
R
t
1
+ 0 aii θi Φi (s)ds
θi
xi (0)
where Φi (t) is defined as above. Under the conditions of Lemma 2.3, we obtain
limt→∞ ln yti (t) = 0 a.s.
Theorem 2.5. Suppose that yi (t) is a positive solution of (2.1). If ci ≥ 0, then
Pm 2 R
Z
ri − 12 j=1 σij
+ Y ln(1 + γi (u))ν(du)
1 t θi
lim
yi (s)ds =
a.s.
t→∞ t 0
aii
EJDE-2013/162
ASYMPTOTIC BEHAVIOR
5
Proof. Applying Itô’s formula to ln yi (t) results in
Z
m
1X 2
d ln yi (t) = (ri −
σij + ln(1 + γi (u))ν(du) − aii yiθi )dt
2 j=1
Y
Z
m
X
e (dt, du).
+
σij dBj (t) +
ln(1 + γi (u))N
Y
j=1
Integrating from 0 to t yields
Z
m
1X 2
σij +
ln(1 + γi (u))ν(du) t
ln yi (t) − ln yi (0) = ri −
2 j=1
Y
Z t
− aii
yiθi (s)ds + M1 (t) + M2 (t),
0
R t Pm
RtR
e (ds, du) are real
where M1 (t) = 0 j=1 σij dBj (s), M2 (t) = 0 Y ln(1 + γi (u))N
valued local martingales vanishing at t = 0. Hence
Z
m
1X 2
ln yi (t) ln yi (0) −
= ri −
σij +
ln(1 + γi (u))ν(du)
t
t
2 j=1
Y
(2.4)
Z t
aii
M1 (t) M2 (t)
θi
−
y (s)ds +
+
.
t 0 i
t
t
Then by [14, Proposition 2.4],
hM1 i(t) =
Z tX
m
0 j=1
Z tZ
hM2 i(t) =
0
2
σij
ds =
m
X
j=1
[ln(1 + γi (u))]2 ν(du)ds = t
Y
2
σij
t,
Z
[ln(1 + γi (u))]2 ν(du),
Y
where hM i(t) := hM, M i is Meyer’s angle bracket process. We have
Z t
t
1
ds =
< ∞,
2
(1
+
s)
t
+
1
0
by the strong law of large numbers for local martingales [18], we then obtain
M1 (t)
M2 (t)
= 0 a.s., lim
= 0 a.s.
t→∞
t
t
Taking limits on both sides of (2.4) and combining Remark 2.4 lead to
Pm 2 R
Z
+ Y ln(1 + γi (u))ν(du)
ri − 12 j=1 σij
1 t θi
lim
yi (s)ds =
a.s.
t→∞ t 0
aii
This completes the proof.
lim
t→∞
3. Global positive solutions of (1.1)
As xi (t) in (1.1) denotes the size of species i, it should be nonnegative. To
guarantee that the stochastic differential equations (SDEs) have a unique global
solution for any given initial value, the coefficients of the equation are generally
required to satisfy both the linear growth condition and the local Lipschitz condition
(see e.g.[23, 12]). But we can find that the coefficients of (1.1) are locally Lipschitz
continuous, and they do not satisfy the linear growth condition. So the solution of
6
X. ZHANG, K. WANG
EJDE-2013/162
(1.1) may explode in a finite time. The following theorem gives sufficient condition
for global positive solutions.
Theorem 3.1. Let (A1), (A2) hold and θi ≥ 1, i = 1, . . . , n. Then for any initial
value x0 ∈ Rn+ , Equation (1.1) has a unique global solution x(t) ∈ Rn+ for all t ≥ 0
almost surely.
Proof. Our proof is motivated by Mao, Marion and Renshaw [22]. Clearly, the
coefficients of (1.1) are locally Lipschitz continuous, so for any initial value x0 ∈ Rn+
Equation (1.1) has a unique maximal local solution x(t) on t ∈ [0, τe ), where τe is
the explosion time. If we show that τe = ∞ a.s., then the solution is global. Now
let k0 be big enough for every component of x0 lying within the interval [1/k0 , k0 ].
For any integer k ≥ k0 , define the stopping time
τk = inf{t ∈ [0, τe ) : xi (t) ∈
/ (1/k, k) for some i = 1, . . . , n},
where we set inf ∅ = ∞. Obviously, τk is increasing as k → ∞. Set τ∞ = limk→∞ τk ,
whence τ∞ ≤ τe a.s. Now all we need to show is τ∞ = ∞ a.s. If this assertion is
false, then there is a pair of constants T > 0 and ∈ (0, 1) such that P{τ∞ ≤ T } > .
Therefore, there is an integer k1 ≥ k0 such that
P{τk ≤ T } ≥ for all k ≥ k1 .
(3.1)
Define a C 2 -function V : Rn+ → R+ by
V (x) =
n
X
pi (xi − 1 − ln xi ).
i=1
If x(t) ∈ Rn+ , Itô’s formula shows that
dV (x(t))
n n h
m
n
io
X
X
X
2
σij
=
pi (xi − 1)(ri − aii xθi i −
aij xj ) + 0.5
dt
i=1
n
X
+
=
pi (xi − 1)
i=1
n n
X
m
X
j6=i
n Z
X
σij dBj (t) +
j=1
h
pi (xi − 1)(ri − aii xθi i −
i=1
i=1
n
X
pi [xi γi (u) − ln(1 + γi (u))]N (dt, du)
Y
aij xj ) + 0.5
m
X
+
2
σij
j=1
j6=i
Z
n
m
io
X
X
(xi γi (u) − ln(1 + γi (u)))ν(du) dt +
pi (xi − 1)
σij dBj (t)
Y
i=1
n Z
X
+
j=1
i=1
j=1
e (dt, du),
pi [xi γi (u) − ln(1 + γi (u))]N
Y
where we drop t− from x(t− ). Using (A1) and (A2), we get that there exists a
positive constant K such that
n
n
m
h
X
X
X
2
pi (xi − 1)(ri − aii xθi i −
aij xj ) + 0.5
σij
i=1
j6=i
Z
+
Y
i
(xi γi (u) − ln(1 + γi (u)))ν(du)
j=1
EJDE-2013/162
≤
n
X
ASYMPTOTIC BEHAVIOR
Z
h
pi (ri +
7
γi (u)ν(du))xi + aii x2i − aii xθ+1
+ aii xθi − ri + 0.5
i
Y
i=1
Z
m
X
2
σij
j=1
i
ln(1 + γi (u))ν(du) + 0.5xT (−P̄ Ā − ĀT P̄ )x
−
Y
≤
n n
X
Z
− aii pi xθi i +1 − (τ − aii pi )x2i + pi aii xθi i + pi ri + γi (u)ν(du) xi
Y
i=1
+ pi − ri + 0.5
m
X
2
σij
−
τk ∧T
Z
dV (x(t)) ≤
0
o
ln(1 + γi (u))ν(du)
≤ K.
Y
j=1
Therefore,
Z τk ∧T
Z
Z
τk ∧T
Kdt +
0
0
τk ∧T
Z
+
0
pi (xi (t) − 1)
i=1
Z X
n Z
Y i=1
n
X
m
X
σij dBj (t)
j=1
e (dt, du).
pi [xi γi (u) − ln(1 + γi (u))]N
Y
Taking expectations on both sides results in
EV (x(τk ∧ T )) ≤ V (x0 ) + KE(τk ∧ T ) ≤ V (x0 ) + KT.
(3.2)
Set Ωk = {τk ≤ T } for k ≥ k1 , from (3.1) we have P(Ωk ) ≥ . Note that for every
ω ∈ Ωk , there is some i such that xi (τk , ω) equals either k or 1/k, hence
V (x(τk , ω)) ≥ pi [k − 1 − ln(k)] ∧ pi [1/k − 1 − ln(1/k)].
Using (3.2), yields
V (x0 ) + KT ≥ E (IΩk V (x(τk , ω))) ≥ (pi [k − 1 − ln(k)] ∧ pi [1/k − 1 − ln(1/k)]) ,
where IΩk is the indicator function of Ωk . When k → ∞ we obtain
∞ > V (x0 ) + KT = ∞,
it results in τ∞ = ∞ a.s. The proof is complete.
4. Ultimate boundedness
In the previous section, we saw that (1.1) has a unique global solution x(t) ∈
Rn+ for any t ≥ 0 almost surely. Based on this fundamental theorem, we discuss
the ultimate boundedness and asymptotic boundedness in any pth moment of the
solutions.
Theorem 4.1. Under the assumptions of Theorem 3.1, for any initial value x0 ∈
Rn+ , the solution of (1.1) satisfies
lim sup E|x(t)| ≤ K.
t→∞
Proof. Define the Lyapunov function
V (x) :=
n
X
i=1
pi xi , x ∈ Rn+ .
8
X. ZHANG, K. WANG
EJDE-2013/162
Applying Itô’s formula, we obtain
dV (x(t)) =
n
X
pi xi (ri − aii xθi i −
i=1
+
n
X
aij xj )dt +
n
X
i=1
j6=i
Z X
n
pi xi
m
X
σij dBj (t)
j=1
pi xi γi (u)N (dt, du)
(4.1)
Y i=1
= LV (x)dt +
n
X
pi xi
i=1
m
X
σij dBj (t) +
Z X
n
e (dt, du),
pi xi γi (u)N
Y i=1
j=1
where we write x(t− ) = x, and
Z
n
n
X
X
θi
LV (x) =
pi ri − aii xi −
aij xj + γi (u)ν(du) xi
i=1
=
n
X
Y
j6=i
pi ri − aii xθi i + aii xi +
=
≤
γi (u)ν(du) xi + xT (−P̄ Ā)x
Y
i=1
n
X
Z
pi ri − aii xθi i + aii xi +
i=1
n h
X
Z
γi (u)ν(du) xi + 0.5xT (−P̄ Ā − ĀT P̄ )x
Y
−
aii pi xθi i +1
− (τ −
aii pi )x2i
Z
i
γi (u)ν(du) xi .
+ pi ri +
Y
i=1
For arbitrary α > 0, making use of the conditions of this Theorem, applying Itô’s
formula once again yields
d(eαt V (x(t)))
= αeαt V (x(t))dt + eαt dV (x(t))
≤ eαt
+e
n
X
[−aii pi xθi i +1 − (τ − aii pi )x2i + pi (α + ri +
i=1
n
X
αt
Z
γi (u)ν(du))xi ]dt
Y
p i xi
i=1
αt
≤ K0 e dt + e
m
X
σij dBj (t) + eαt
j=1
n
X
αt
Z X
n
e (dt, du)
pi xi γi (u)N
Y i=1
pi xi
i=1
m
X
σij dBj (t) + e
αt
Z X
n
e (dt, du),
pi xi γi (u)N
Y i=1
j=1
where K0 is a positive constant. Therefore,
E(eαt V (x(t))) ≤ V (x0 ) +
K0 αt
(e − 1);
α
that is to say
lim sup EV (x(t)) ≤
t→∞
Noting that |x(t)| ≤
Pn
i=1
xi (t) ≤
V (x(t))
min1≤i≤n pi ,
lim sup E|x(t)| ≤
t→∞
This completes the proof.
K0
.
α
(4.2)
we obtain
K0
=: K.
α min1≤i≤n pi
EJDE-2013/162
ASYMPTOTIC BEHAVIOR
9
Definition 4.2 ([15]). The solution of(1.1) is said to be stochastically ultimately
bounded if for any ∈ (0, 1), there is a constant H = H() such that for any
x0 ∈ Rn+ ,
lim sup P {|x(t)| > H} < .
t→∞
As an application of Theorem 4.1, together with the Chebyshev inequality, we
have the following corollary.
Corollary 4.3. Under the conditions of Theorem 4.1, the solution of (1.1) is
stochastically ultimately bounded.
Furthermore, we can get the following property.
Theorem 4.4. Assume (A1), (A2) hold and θi > 1, i = 1, . . . , n. Then for p > 0,
there exists a positive constant K = K(p), for any initial value x0 ∈ Rn+ , the
solution of (1.1) has the property
lim sup Expi (t) ≤ K(p),
t ≥ 0, i = 1, . . . , n.
t→∞
Proof. Define the Lyapunov function
n
X
V (x, t) :=
et xpi ,
x ∈ Rn+ .
i=1
Applying Itô’s formula, we obtain
dV (x(t), t) = LV (x(t))dt + et p
n
X
i=1
+ et
n
X
xpi
Z
xpi
m
X
σij dBj (t)
j=1
e (dt, du),
[(1 + γi (u))p − 1]N
Y
i=1
−
where we write x(t ) = x, and
Z
m
n
i
nh 1
X
p−1X 2
1
t
+ ri +
σij +
[(1 + γi (u))p − 1]ν(du) xpi
LV (x) = e
p
p
2 j=1
p Y
i=1
− aii xip+θi −
n
X
aij xpi xj
o
j6=i
n
X
Z
m
nh 1
i
p−1X 2
1
≤ et
p
+ ri +
σij +
[(1 + γi (u))p − 1]ν(du) xpi
p
2 j=1
p Y
i=1
io
xp+1
j
p+1
p+1
j6=i
Z
n
m
nh 1
i
X
p−1X 2
1
+ ri +
[(1 + γi (u))p − 1]ν(du) xpi
≤ et
p
σij +
p
2 j=1
p Y
i=1
− aii xip+θi −
− aii xip+θi −
n
X
aij
≤ et
i=1
Ki (p),
i
+
n hX
i
o
p
1
(aij ) +
(aji ) xp+1
i
p+1
p+1
j6=i
n
X
h pxp+1
10
X. ZHANG, K. WANG
EJDE-2013/162
where Ki (p) is a positive constant. Hence
Z t X
n
n
n
n
n
X
X
X
X
p
p
p
t
s
e E[
e
Ki (p)ds =
xi (t)] ≤
xi (0) + E
xi (0) +
Ki (p)(et − 1).
i=1
0
i=1
i=1
i=1
i=1
It is not difficult to derive that
n
n
X
X
lim sup E[
xpi (t)] ≤
Ki (p) =: K(p).
t→∞
i=1
i=1
The required assertion follows immediately.
5. Pathwise estimation
In this section we consider the asymptotic pathwise estimation of the solution
to (1.1).
Theorem 5.1. For θi > 1, i = 1, . . . , n, under Assumptions (A1), (A2), for any
initial value x0 ∈ Rn+ , the solution of (1.1) has the property
lim sup
t→∞
ln xi (t)
≤1
ln t
a.s., i = 1, . . . , n.
Proof. Here we adopt the same notation as in the proof of Theorem 4.1. From
(4.1), by simple manipulation, one has
E
sup V (x(u))
t≤u≤t+1
Z
≤ E(V (x(t))) + E
sup
t≤u≤t+1
Z
+ pi ri +
n h
uX
t
− aii pi xθi i +1 (s) − (τ − aii pi )x2i (s)
i=1
i γi (u)ν(du) xi (s) ds + E
Z
u
sup
t≤u≤t+1
Z X
n
t
e (ds, du)
pi xi (s)γi (u)N
Z
sup
t≤u≤t+1
Z
+ pi ri +
n
uX
t
Y i=1
≤ E(V (x(t))) + E
sup
t≤u≤t+1
Y
+E
Z
n h
uX
t
aii pi x2i (s)
i=1
i |γi (u)|ν(du) xi (s) ds
Y
+E
Z
sup
t≤u≤t+1
+E
n
uX
t
i=1
u
Z
sup
|
t≤u≤t+1
pi xi (s)
t
m
X
≤ E(V (x(t))) + q1
e (ds, du)|
pi xi (s)γi (u)N
Z
Z
t≤u≤t+1
t
n
uX
i=1
E|x(s)|2 ds
t
pi xi (s)
m
X
j=1
t+1
E|x(s)|ds + q2
t
sup
j=1
Z X
n
Y i=1
t+1
Z
+E
σij dBj (s)
σij dBj (s)
i=1
pi xi (s)
m
X
j=1
σij dBj (s)
EJDE-2013/162
ASYMPTOTIC BEHAVIOR
u
Z
+E
|
sup
t≤u≤t+1
Z X
n
t
11
e (ds, du)| ,
pi xi (s)γi (u)N
(5.1)
Y i=1
R
√
where q1 = n max1≤i≤n {pi (ri + Y |γi (u)|ν(du))}, q2 = max1≤i≤n {aii pi }. By the
Burkholder-Davis-Gundy inequality for local martingale (see, e.g., [23, 2]) and the
Hölder inequality, we obtain that
Z uX
n
m
Z t+1
1/2
X
E
sup
pi xi (s)
σij dBj (s) ≤ 3E
|xT P̄ σ|2 ds
t≤u≤t+1
t
i=1
t
j=1
t+1
Z
≤ 3|P̄ σ| E
1/2
|x(s)|2 ds
,
t
where σ = (σij ), and
E
sup
Z
|
t≤u≤t+1
≤
≤
n
X
i=1
n
X
t
=
Z X
n
t
i=1
n
X
≤J
1/2
x2i (s)γi2 (u)N (ds, du)
1/2
x2i (s)γi2 (u)N (ds, du)
Y
γi2 (u)ν(du)E
Z
t+1
Z
t+1
1/2 Z
γi2 (u)ν(du)
E
1/2
|x(s)|2 ds
,
t
Y
i=1
1/2
x2i (s)ds
t
Y
pi
Y
t
Z
Z
t+1 Z
Z
pi J E
pi J
e (ds, du)|
pi xi (s)γi (u)N
Y i=1
t+1
Z
pi JE
i=1
n
X
u
where J is a positive constant. Moreover, we can derive fromRTheorem 4.4 that
t+1
there exists positive constants K1 , K2 such that lim supt→∞ E t |x(s)|ds ≤ K1
R t+1
and lim supt→∞ E t |x(s)|2 ds ≤ K2 . Substituting the above inequalities into
(5.1) and combining (4.2), we can see that
lim sup E
sup V (x(u))
t→∞
t≤u≤t+1
n
≤
X K0
+ q1 K1 + q2 K2 + 3|p̄σ| + J
pi
α
i=1
Z
1/2 1/2
γi2 (u)ν(du)
K2 .
Y
Hence there is a positive constant M such that
E
sup |x(t)| ≤ M, n = 1, 2, . . . .
n≤t≤n+1
Let ε > 0 be arbitrary, by the Chebyshev inequality, we have
M
P
sup |x(t)| > n1+ε ≤ 1+ε n = 1, 2, . . . .
n
n≤t≤n+1
P∞ M
Since the series n=1 n1+ε converges, then from the Borel-Cantella lemma [23] that
there exists a n0 := n0 (ω) such that for almost all ω ∈ Ω, whenever n ≥ n0 and
n ≤ t ≤ n + 1, we have
sup |x(t)| ≤ n1+ε .
n≤t≤n+1
12
X. ZHANG, K. WANG
EJDE-2013/162
So
ln |x(t)|
≤ 1 + ε.
ln t
t→∞
Letting ε → 0 leads to the desired assertion.
lim sup
Remark 5.2. Noting that the limit limt→∞ lnt t = 0, under the conditions of Theorem 5.1, we obtain lim supt→∞ ln xti (t) ≤ 0, a.s., i = 1, . . . , n.
On the other hand, by the positivity of solution of (1.1) and the comparison
theorem [25, Theorem 3.1], we obtain that
xi (t) ≥ yi (t),
i = 1, . . . , n,
where yi (t) is the solution of (2.1). According to the analysis for (2.1) in section 2,
we obtain the following results.
Pm 2
+
Theorem 5.3. Let (A1), (A2) hold and θi > 1, i = 1, . . . , n. If ri − 21 j=1 σij
R
n
,
the
solution
of
(1.1)
ln(1
+
γ
(u))ν(du)
≥
0,
then
for
any
initial
value
x
∈
R
i
0
+
Y
satisfies
Z
Z
1 t θi
1 t θi
lim inf
xi (s)ds ≥ lim
yi (s)ds
t→∞ t 0
t→∞ t 0
Pm 2 R
ri − 21 j=1 σij
+ Y ln(1 + γi (u))ν(du)
=
,
aii
ln xi (t)
lim inf
≥ 0 a.s., i = 1, . . . , n.
t→∞
t
Now combining Remark 5.2 and Theorem 5.3 leads to the following theorem.
Theorem 5.4. Under the conditions of Theorem 5.3, for each i = 1, . . . , n,
lim
t→∞
ln xi (t)
=0
t
a.s.
6. Stochastic permanence
Stochastic permanence is one of the most interesting and important topics. In
this section, stochastic permanence is studied based on the results in Section 4. We
firstly introduce the definition of stochastic permanence.
Definition 6.1 ([20]). If for arbitrary ε ∈ (0, 1), there are two positive constants
β1 and β2 such that for any initial data x0 ∈ Rn+ , the solution x(t) of Eq.(1.1) has
the property that
lim inf P{xi (t) ≥ β1 } ≥ 1 − ε,
t→∞
lim inf P{xi (t) ≤ β2 } ≥ 1 − ε,
t→∞
1 ≤ i ≤ n,
then (1.1) is said to be stochastically permanent.
Theorem 6.2. Under the conditions of Theorem 3.1, if there exists a positive
constant α, such that
Z h
m
3+α X 2
1
1 i
ri −
σij −
−
ν(du) > 0,
2+α
2 j=1
2+α
Y (2 + α)(1 + γi (u))
then, for the case 1 ≤ θi ≤ 2 + α, Equation (1.1) is stochastically permanent.
EJDE-2013/162
ASYMPTOTIC BEHAVIOR
Proof. Define yi = 1/xi for xi > 0, applying Itô’s formula, we obtain
dyi2+α = d(
1 2+α
)
xi
n
= −(2 + α)(
X
1 α+3
)
xi (ri − aii xθi i −
aij xj )dt
xi
j6=i
m
X
m
2+α
1 α+4 2
1 α+3 X
2
+
(α + 3)( )
xi
σij dt − (2 + α)( )
xi
σij dBj
2
xi
xi
j=1
j=1
Z
1
1
− 2+α ]N (dt, du)
+ [
2+α
(x
+
x
γ
(u))
x
i
i i
Y
i
Z
m
n
h
1
α+3X 2
1
1
≤ (2 + α)( )α − 2 ri −
σij − (
2+α
xi
xi
2 j=1
Y (2 + α)(1 + γi (u))
m
i
X
aii o
1
1
)ν(du) + 2−θi dt − (2 + α)( )α+2
σij dBj
2+α
xi
xi
j=1
Z
1 e
1
− 2+α ]N
(dt, du)
+ [
2+α
(x
+
x
γ
(u))
xi
i
i i
Y
Z
m
n
h
α+3X 2
1
= (2 + α)yiα − yi2 ri −
σij − (
2 j=1
(2
+
α)(1
+
γi (u))2+α
Y
−
m
i
o
X
1
σij dBj
)ν(du) + aii yi2−θi dt − (2 + α)yiα+2
2+α
j=1
Z
1
e (dt, du).
+
yi2+α [
− 1]N
(1 + γi (u))2+α
Y
−
Choose a sufficiently small positive ζ such that
m
ri −
α+3X 2
σ −
2 j=1 ij
Z Y
1
1 ζ
−
ν(du) >
.
(2 + α)(1 + γi (u))2+α
2+α
2+α
Define V = eζt yi2+α , using Itô’s formula results in
Z
m
n
h
α+3X 2
1
dV ≤ (2 + α)eζt yiα − yi2 ri −
σij − (
2 j=1
(2
+
α)(1
+
γi (u))2+α
Y
i
o
1
)ν(du) + aii yi2−θi dt + ζeζt yi2+α dt
−
2+α
Z
m
X
1
e (dt, du)
− (2 + α)eζt yiα+2
σij dBj + eζt yi2+α [
− 1]N
2+α
(1
+
γ
(u))
i
Y
j=1
=: eζt F (yi )dt − (2 + α)eζt yi2+α
m
X
σij dBj
j=1
+e
ζt
Z
Y
yi2+α [
1
e (dt, du),
− 1]N
(1 + γi (u))2+α
13
14
X. ZHANG, K. WANG
EJDE-2013/162
where
Z
m
h
α+3X 2
1
F (yi ) = (2 +
−
ri −
σ − (
2 j=1 ij
(2
+
α)(1
+
γi (u))2+α
Y
o
1
ζ i
−
+ aii yi2−θi
)ν(du) −
2+α
2+α
has an upper positive bound, say K. Integrating from 0 to t and taking expectations, we obtain
K
E[eζt yi2+α ] ≤ yi2+α (0) + (eζt − 1).
ζ
Therefore
K
−(α+2)
=: K0 .
(t)] ≤
lim sup E[xi
ζ
t→∞
α)yiα
n
yi2
1/(2+α)
For any given ε > 0, set β1 = ε1/(2+α) /K0
, by the Chebyshev inequality, we
obtain
−(2+α)
−(2+α) lim sup P xi (t) < β1 = lim sup P xi
(t) > β1
t→∞
t→∞
−(2+α)
≤ lim sup βi2+α E[xi
(t)] = ε.
t→∞
Hence lim inf t→∞ P{xi (t) ≥ β1 } ≥ 1 − ε, i = 1, . . . , n.
On the other hand, as an application of Theorem 4.1, and the Chebyshev inequality, we can easily show that for arbitrary ε ∈ (0, 1), there is a positive constant β2
such that for any initial data x0 ∈ Rn+ , lim inf t→∞ P{xi (t) ≤ β2 } ≥ 1 − ε, 1 ≤ i ≤ n.
Therefore, (1.1) is stochastically permanent.
7. Example and numerical simulations
Consider the two-species stochastic Gilpin-Ayala mutualism system with jumps
Z
dx1 = x1 (r1 − a11 xθ11 − a12 x2 )dt + σ11 dB1 (t) + σ12 dB2 (t) +
γ1 (u)x1 N (dt, du),
ZY
dx2 = x2 (r2 − a22 xθ22 − a21 x1 )dt + σ21 dB1 (t) + σ22 dB2 (t) +
γ2 (u)x2 N (dt, du).
Y
(7.1)
In Figure 1, we choose r1 = 0.06, r2 = 0.05, a11 = 0.08, a22 = 0.04, a12 = a21 =
−0.005, θ1 = θ2 = 1.01, σ11 = 0.2, σ22 = 0.05, σ12 = σ21 = 0, γ1 (u) = 0.2, γ2 (u) =
0.24, x1 (0) = 1.1, x2 (0) = 1.5, Y = (0, ∞), λ(Y) = 1. Since a11 a22 − a12 a21 > 0,
then (A1) and (A2) hold, so (7.1) has a unique global positive solution for any
positive initial value by Theorem 3.1 and pth moment of the solution of (7.1) is
asymptotic bounded, see Figure 1. Moreover,
Z
2
2
+ σ12
σ11
+
ln(1 + γ1 (u))ν(du) = 0.22 > 0,
r1 −
2
Y
Z
2
σ 2 + σ22
r2 − 21
+
ln(1 + γ2 (u))ν(du) = 0.26 > 0.
2
Y
Then in view of Theorem 5.3, we obtain ln xti (t) → 0, i = 1, 2, Figure 1 confirms
these.
In Figure 2, we choose r1 = 0.5, r2 = 0.2, a11 = a22 = 0.9, a12 = a21 = −0.05,
θ1 = θ2 = 1.01, σ11 = 0.05, σ22 = 0.1, σ12 = σ21 = 0, γ1 (u) = 0.2, γ2 (u) = 0.12,
EJDE-2013/162
ASYMPTOTIC BEHAVIOR
15
x1(t)
3.5
x2(t)
(1/(t+1))log x1(t)
3
(1/(t+1))log x2(t)
Ex21(t)
2.5
Ex22(t)
2
1.5
1
0.5
0
5
10
15
20
25
30
35
Time t
Figure 1. Solutions of (7.1) for r1 = 0.06, r2 = 0.05, a11 = 0.08,
a22 = 0.04, a12 = a21 = −0.005, θ1 = θ2 = 1.01, σ11 = 0.2,
σ22 = 0.05, σ12 = σ21 = 0 , γ1 (u) = 0.2, γ2 (u) = 0.24, x1 (0) = 1.1,
x2 (0) = 1.5, Y = (0, ∞), λ(Y) = 1
x1 (0) = 0.1, x2 (0) = 0.8, Y = (0, ∞), λ(Y) = 1. Since a11 a22 − a12 a21 > 0, then
(A1) and (A2) hold. In Theorem 6.2, choose α = 1, by simple calculation, we have
r1 −
r2 −
3+α X 2
σ −
2 j=1 1j
m
Z h
m
X
Z h
3+α
2
j=1
2
σ2j
−
Y
Y
1
1 i
−
ν(du) = 0.64 > 0,
2+α
(2 + α)(1 + γ1 (u))
2+α
1
1 i
−
ν(du) = 0.28 > 0 .
(2 + α)(1 + γ2 (u))2+α
2+α
Theorem 6.2 tells us that (7.1) is stochastically permanent, and Figure 2 confirms
this.
Conclusions. An stochastic Gilpin-Ayala mutualism model with jumps has been
studied in this article. The high nonlinearity of Gilpin-Ayala model and Poisson
jumps make the problem difficult. Sufficient criteria for the existence of global
positive solution, stochastically ultimate boundedness and stochastic permanence
are derived for the n-dimensional model by analysis of Lyapunov functions which has
been used by many authors. We also investigate asymptotic pathwise estimation.
The simulation results verify the effectiveness of the proposed results.
Acknowledgements. The authors want thank the editor and referee for their
important and valuable comments. The authors also want thank the National
Natural Science Foundation of China for their economic support through the grants:
11171081, 11171056, 11126219, 11001032, 11226254, and 11101183.
16
X. ZHANG, K. WANG
EJDE-2013/162
0.9
x1(t)
0.8
x2(t)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
Time t
40
50
60
Figure 2. Solutions of (7.1) for r1 = 0.5, r2 = 0.2, a11 = a22 =
0.9, a12 = a21 = −0.05, θ1 = θ2 = 1.01, σ11 = 0.05, σ22 = 0.1,
σ12 = σ21 = 0, γ1 (u) = 0.2, γ2 (u) = 0.12, x1 (0) = 0.1, x2 (0) = 0.8,
Y = (0, ∞), λ(Y) = 1
References
[1] J. F. Addicott, H. I. Freedman; On the structure and stability of mutualistic systems: analysis
of predatorprey and competition models as modified by the action of a slow-growing mutualist,
Theor. Popul. Biol. 26 (1984), 320-339.
[2] D. Applebaum; Lévy Processes and Stochastic Calculus, 2nd ed., Cambridge University Press,
2009.
[3] J. Bao, X. Mao, G. Yin, C. Yuan; Competitive Lotka-Volterra population dynamics with
jumps, Nonlinear Anal. 74 (2011), 6601-6616.
[4] J. Bao, C. Yuan; Stochastic population dynamics driven by Lévy noise, J. Math. Anal. Appl.
391 (2012), 363-375.
[5] F. Chen, C. Shi; Global attractivity in an almost periodic multi-species nonlinear ecological
model, Appl. Math. Comput. 180 (2006), 376-392.
[6] S. Cheng; Stochastic population systems, Stoch. Anal. Appl. 27 (2009), 854-874.
[7] M. Fan, K. Wang; Global periodic solutions of a generalized n-species Gilpin-Ayala competition model, Comput. Math. Appl. 40 (2000), 1141-1151.
[8] M. E. Gilpin, F. J. Ayala; Global models of growth and competition, Proc. Nat. Acad. Sci.
USA 70 (1973), 3590-3593.
[9] B. S. Goh; Stability in models of mutualism, Amer. Natural. 113 (1979), 261-275.
[10] K. Golpalsamy; Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992.
[11] L. Hung; Stochastic delay population systems, Appl. Anal. 88 (2009), 1303-1320.
[12] R. Z. Khasminskii; Stochastic Stability of Differential Equations, Sijthoff and Noordhoff,
Alphen aan den Rijn, 1980.
[13] Y. Kuang; Delay Differential Equations with Applications in Population Dynamics, Academic
Press, Boston, 1993.
[14] H. Kunita; Itô’s stochastic calculus: Its surprising power for applications, Stochastic Processses and their Applications, 120 (2010), 622-652.
[15] X. Li, X. Mao; Population dynamical behavior of non-autonomous Lotka-Volterra competitive
system with random perturbation, Discrete Contin. Dyn. Syst. 24 (2009), 523-545.
[16] B. Lian, S. Hu; Asymptotic behaviour of the stochastic Gilpin-Ayala competition models, J.
Math. Anal. Appl. 339 (2008), 419-428.
EJDE-2013/162
ASYMPTOTIC BEHAVIOR
17
[17] X. Liao, S. Zhou, Y. Chen; On permanence and global stability in a general Gilpin-Ayala
competition predator-prey discrete system, Appl. Math. Comput. 190 (2007), 500-509.
[18] R. Lipster; A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217-228.
[19] M. Liu, K. Wang; Population dynamical behavior of Lotka-Volterra cooperative systems with
random perturbations, Discrete Contin. Dyn. Syst. 33 (2013), 2495-2522.
[20] M. Liu, K. Wang; Analysis of a stochastic autonomous mutualism model, J. Math. Anal.
Appl. 402 (2013), 392-403.
[21] Z. Lu, Y. Takeuchi; Permanence and global stability for cooperative Lotka-Volterra diffusion
systems, Nonlinear Anal. 19 (1992), 963-975.
[22] X. Mao, G. Marion, E. Renshaw; Environmental Brownian noise suppresses explosions in
population dynamics, Stoch. Proc. Appl. 97 (2002), 95-110.
[23] X. Mao, C. Yuan; Stochastic Differential Equations With Markovian Switching, Imperial
College Press, London, 2006.
[24] J. Pan, Z. Jin, Z. Ma; Thresholds of survival for an n-dimensional Volterra mutualistic system
in a polluted environment, J. Math. Anal. Appl. 252 (2000), 519-531.
[25] S. Peng, X. Zhu; Necessary and sufficient condition for comparison theorem of 1-dimensional
stochastic differential equations, Stoch. Proc. Appl. 116 (2006), 370-380.
[26] H. L. Smith; On the asymptotic behavior of a class of deterministic models of cooperating
species, SIAM J. Math. Anal. 46 (1986), 368-375.
[27] R. Situ; Theory of Stochastic Differential Equations with Jumps and Applications, Springer,
2005.
[28] A. Tineo; On the asymptotic behavior of some population models, II, J. Math. Anal. Appl.
197 (1996), 249-258.
[29] M. Vasilova, M. Jovanović; Dynamics of Gilpin-Ayala competition model with random perturbation, Filomat 24 (2010), 101-113.
[30] M. Vasilova, M. Jovanović; Stochastic Gilpin-Ayala competition model with infinite delay,
Appl. Math. Comput. 217 (2011), 4944-4959.
Xinhong Zhang
Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China.
College of Science, China University of Petroleum, Qingdao 266555, China
E-mail address: zhxinhong@163.com
Ke Wang
Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China.
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024,
China
E-mail address: wangke@hitwh.edu.cn, Tel:+86 06315687086