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The Annals of Applied Probability2001, Vol. 11, No. 4, 12631291
ASYMPTOTIC ANALYSIS AND EXTINCTION INA STOCHASTIC LOTKAVOLTERRA MODEL
By F. C. Klebaner1 and R. Liptser1
University of Melbourne and Tel Aviv University
A stochastic LotkaVolterra model is formulated by using the semi-
martingale approach. The large deviation principle is established, and is
used to obtain a bound for the asymptotics of the time to extinction of prey
population. The bound is given in terms of past-dependent ODEs closely
related to the dynamics of the deterministic LotkaVolterra model.
1. Introduction. Main result.
1.1. Deterministic LotkaVolterra system. The LotkaVolterra system of
ordinary differential equations [Lotka (1925) and Volterra (1926)],
xt = xt xtytyt = xtyt yt
(1.1)
with positive x0 y0 and positive parameters describes a behavior ofa predatorprey system in terms of the prey and predator intensities xt andyt. Here, is the rate of increase of prey in the absence of predator, is a rateof decrease of predator in the absence of prey while the rate of decrease in
prey is proportional to the number of predators yt, and similarly the rate ofincrease in predator is proportional to the number of prey xt [see, e.g., May(1976)]. The system (1.1) is one of the simplest nonlinear systems.
Since the population numbers are discrete, a description of the predator
prey model in terms of continuous intensities xt yt is based implicitly ona natural assumption that the numbers of both populations are large, andthe intensities are obtained by a normalization of population numbers by a
large parameter K. Thus (1.1) is an approximation, an asymptotic descriptionof the interaction between the predator and prey. Although this model may
capture some essential elements in that interaction, it is not suitable to answer
questions of extinction of populations, as the extinction never occurs in the
deterministic model; see Figure 1 for the pair xt yt in the phase plane.We introduce here a probabilistic model which has as its limit the determin-
istic LotkaVolterra model, evolves in continuous time according to the same
local interactions and allows evaluating asymptotically the time for extinction
of prey species.
There is a vast amount of literature on the LotkaVolterra model, and a his-
tory of research on stochastic perturbations of this system both exact, approx-
Received June 2000; revised January 2001.1Supported in part by Australian Research Council.
AMS 2000 subject classifications. Primary 60I27, 60F10.
Key words and phrases. Predatorprey models, large deviations, semimartingales, extinction
1263
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1264 F. C. KLEBANER AND R. LIPTSER
Fig. 1. rx y when = 5, = 1, = 5, = 1.
imate and numerical; see, for example, Goel, Maitra and Montroll (1971),
Turelli (1977), Kesten and Ogura (1981), Hitchcock (1986), Watson (1987),
Roozen (1989) and references therein. We approach the problem of extinc-
tion via the theory of large deviations, thus obtaining new theoretical results,
which previously were studied numerically.
The system (1.1) possesses the first integral which is a closed orbit in the
first quadrant of phase plane x y. It is given by
rx y = cx logx + y logy + r0(1.2)where r0 is an arbitrary constant. It depends only on the initial points x0 y0(see Figure 1).
1.2. Stochastic LotkaVolterra system. In this paper, we introduce and
analyze a probabilistic model of preypredator population related to the clas-
sical LotkaVolterra equations.
Let Xt, and Yt be numbers of prey and predators at time t. We start withsimple balance equations for preypredator populations
Xt = X0 + t t Yt = Y0 + t t
(1.3)
where:t is a number of prey born up to time t.t is a number of prey killed up to time t.t is a number of predators born up to time t.t is a number of predators died up to time t.
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ANALYSIS AND EXTINCTION IN LOTKAVOLTERRA MODEL 1265
We assume that t t
t
t , are double Poisson processes with the following
random rates: Xt,
KXtYt,
KXtYt, Yt, respectively and disjoint jumps [the
latter assumption reflects the fact that in a short time interval t t + tonly one prey might be born and only one might be killed, only one predator
might be born and only one might die, with the above-mentioned intensities;
moreover all these events are disjoint in time].
The existence of such a model is not obvious; therefore in Section 2 we give
its detailed probabilistic derivation.
Assume X0 = Kx0 and Y0 = Kx0 for some fixed positive x0 y0 and alarge integer parameter K. Introduce the normalized by K prey and predatorpopulations
xKt =XtK
and yKt =YtK
In terms ofxKt and yKt the introduced intensities for double Poisson processes
can be written as
KxKt x
Kt y
Kt x
Kt y
Kt y
Kt
We justify the choice of the probabilistic model given in (1.3) by Theorem 2
which states that the solution of the LotkaVolterra equations is the limit (in
probability) for xKt yKt ,xKt y
Kt
xt yt K Such an approximation is known as the fluid approximation. Results on the
fluid approximation for Markov discontinuous processes can be found in Kurtz
(1981) and are adapted to the case considered here, despite that in our case
the two intensities do not satisfy the linear growth condition in xKt yKt .
1.3. Fomulation of the problem. Here, we are interested in evaluation of
the prey extinction, namely, the asymptotics in K of
P
TKext T inftT
yKt > 0
= 0
where TKext = inft > 0 xKt = 0 is the prey extinction time. Unfortunately,the fluid approximation does not provide much information on the extinction
time TKext. Since for x0 > 0, y0 > 0 the fluid limit xt yt remains positivefor any t > 0 (see Figure l), we have
limK
P
TKext T inftT
yKt > 0
= 0(1.4)
A historical comment onT
K
extof evaluation is due. There is a large amount of
literature on the subject of extinction mostly using some simplification of the
original problem, such as linearization, and numerical studies. For a somewhat
different model with a state-dependent noise, Hitchcock (1986) showed that
ultimate extinction is certain, and derived exact probabilities for the predators
to become eventually extinct when the prey birth rate is zero. A power series
7/30/2019 Asymptotic Analysis and Extinction In
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1266 F. C. KLEBANER AND R. LIPTSER
approximation for the extinction probabilities as well as the number of steps
to extinction in special cases were given. Numerical studies of probabilities of
extinction were also done in Smith and Mead (1979, 1980), and Watson (1987)where a rough approximation (based on the normal approximation) was also
given.
Due to (1.4), the rate of convergence in (1.4) is of interest, and is the subject
of this paper. Neither fluid nor even diffusion approximations for the stretched
differences
K
xKt xt
K
yKt yt
are effective for such analysis.
Freidlin (1998), Freidlin and Weber (1998) carried out an effective asymp-
totic analysis for randomly perturbed oscillators and other Hamiltonian
systems. Their approach is based on the approximation of the first integral
process [in our case rxKt yKt ] by a scalar diffusion. However, in our case thisapproach does not seem to be of use, since at the time of extinction the first
integral process rxKt yKt explodes; see (1.2).A large deviation (LD) type evaluation yields results in our case. The ran-
dom process xKt yKt is a vector semimartingale, so that it appears that onecan have the large deviation principle (LDP) by using a general results from
Pukhaskii (1999). However, the method from Pukhaskii does not serve the
model studied here, since the intensities of two double Poisson processes are
of quadratic form in xKt yKt . For the same reason we could not find adequate
methods for proving the (LDP) in the literature, for example, Wentzell (1989),
Dupuis and Ellis (1997), Freidlin and Wentzel (1984), Dembo and Zeitouni
(1993).
It is well known that the main helpful tool in verification of the LDP with
unbounded intensities, satisfying a linear growth condition, is the exponen-
tial negligibility for sets suptT xKt C, suptT yKt C, for large K, Cand every T > 0. In our case two from four intensities do not satisfy the lin-ear growth condition. Nevertheless, specifics of the model and the fact that
xKt yKt are nonnegative processes with bounded jumps allow establishing the
above-mentioned exponential negligibility (Lemma 2) and deriving the LDP
similarly to Liptser and Pukhalskii (1992).
Although in principle the LDP allows finding the logarithmic rate with
norming 1K
for
P
TKext T inftT
yKt > 0
a realistic procedure for determining the required rate depends heavily on the
structure of the LDP rate function. In our case we deal with purely discontin-
uous process and the rate function is extremely inconvenient for this purpose;
therefore, we restrict ourselves to finding only a lower bound for the required
rate.
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ANALYSIS AND EXTINCTION IN LOTKAVOLTERRA MODEL 1267
Fig. 2. Trajectory that attains the lower bound. Parameters are the same as for Figure 1.
1.4. Main result. To formulate the main result, we need to introduce a
system of past-dependent differential equations parameterized by q > 0,
qt = qt
qt
qe t0 qt dsqt qt
qt = qt
qt
(1.5)
subject to the initial condition q0 = y0, q0 = x0, and denote
Tq = inft > 0 qt = 0(1.6)
Theorem 1. For every T > 0,
liminfK
1
Klog P
TKext T inf
tTyKt > 0
x
20
2Tq
0 e2t0 qt dsqt qt dt
where Tq is associated with the smallest q = q for which Tq T.1.5. Example. We give here an example with = 5, = 1, = 5, = 1 and
T = 15. The trajectories for qt qt , on which the lower bound is attainable, aregiven in the phase plane (see Figure 2). Parameter q 00023 and Tq 15,while the value of the rate function defined later in (4.2) is Jq q 00018.
1.6. Remark. We can show for a stochastic LotkaVolterra model with a
different noise structure (much simpler and somewhat artificial) the exact
asymptotic relation in Theorem 1 (rather than a lower bound), as well as that
the minimum of the rate function in the LDP occurs on the solution similar
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1268 F. C. KLEBANER AND R. LIPTSER
to (1.5). Therefore, loosely speaking, (1.5) (see Figure 2) gives a likely path to
extinction in the stochastic LotkaVolterra model.
The article is organized as follows. In Section 2 we give description of thestochastic model, in Section 3 we show the fluid approximation, in Section 4
we formulate the LDP and prove the main result. The verification of the LDP,
which is quite technical, is done in the Appendix.
2. The model: description of stochastic dynamics.
2.1. Existence. In this section, we show that the random process Xt Ytis well defined by (1.3). To this end, let us introduce four independent
sequences of processes,
t =
t 1 t 2
/K
t = /Kt 1 /Kt 2 /Kt =
/Kt 1 /Kt 2
t =
t 1 t 2
Each of them is a sequence of i.i.d. Poisson processes characterized by rates
, K
, K
, , respectively. Define the processes Xt Yt by the system of Itoequations
Xt = X0 +t
0
n1
IXs n dsn t
0
n1
IXsYs n d/Ks n
Yt = Y0 + t
0 n1IXsYs n d/Ks n
t
0 n1IYs n dsn
(2.1)
governed by these Poisson processes, which obviously has a unique solution
on the time interval 0 T, whereT = inft > 0 Xt Yt =
The double Poisson processes involved in (1.3) are obtained then in the follow-
ing way:
t =t
0
n1
IXs n dsn
t =
t
0 n1IXsYs n d/Ks n
t =t
0
n1
IXsYs n d/Ks n
t =t
0
n1
IYs n dsn
(2.2)
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ANALYSIS AND EXTINCTION IN LOTKAVOLTERRA MODEL 1269
Let us show that t, t ,
t,
t defined in 22 satisfy the required prop-
erties. Since all Poisson processes are independent, their jumps are disjoint,
so that the jumps of t, t , t, t are disjoint as well. To describe structureof intensities for t,
t ,
t,
t , let us introduce a stochastic basis F =
tt0 P supplied by the filtration F generated by all Poisson processes andsatisfying the general conditions. Then, obviously, the random process
At =t
0
n1
IXs nds
is the compensator of t. On the other hand, since Xs is an integer-valuedrandom variable, we have
n1 IXs n = Xs; that is, At =
t0 Xs ds and
the intensity of t is Xt. Analogously, others compensators are seen to be
At =t
0
KXsYs ds
At =
t0
KXsYs ds
At =
t0Ys ds
and thus all other intensities have the required form.
We now show that the process Xt Ytt0 does not explode.Lemma 1.
PT = = 1Proof. Set TXn = inft > 0 Xt n, n 1 and denote by TX =
limn TXn . Due to (2.1) it holds that
EXtTXn X0 +t
0EXsTXn ds
and so, by the GronwallBellman inequality EXTXn T X0eT for every T > 0.Hence by the Fatou lemma EXTXT X0eT. Consequently,
P
TX
T =
0
T > 0
Set TY = inft Yt , 1 and denote by TY = lim TY . Due to (2.1)it holds that
EYtTXn TYt Y0 +t
0
KEXsTXn TY YsTXn TY ds
Y0 +t
0
KnEYsTXn TY ds
Hence, by the GronwallBellman inequality for every T > 0 we haveEYTTxnTy Y0enT and by the Fatou lemma,
EYTTXn TY Y0e/KnT n 1Consequently, P
TY
TXn
T
=0
T > 0, n
1 and, since TXn
,
n , we obtainPTY T = 0 T > 0
Since T = TX TY, PT T = PTX T + PTY T, and wehave PT T = 0 for any T > 0.
7/30/2019 Asymptotic Analysis and Extinction In
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1270 F. C. KLEBANER AND R. LIPTSER
Corollary 1. For Tn = inft Xt Yt n,
limn PTn T = 0 T > 0
The above description of the model allows claiming that Xt Yt is acontinuous-time pure jump Markov process with jumps of two possible sizes
in both coordinates: 1 and 1 and infinitesimal transition probabilities (ast 0,
PXt+t = Xt + 1Xt Yt = Xtt + ot
PXt+t = Xt 1Xt Yt =
KXtYtt + ot
PYt+t = Yt + 1Xt Yt =
KXtYtt + ot
PYt+t = Yt 1Xt Yt = Ytt + ot
2.2. Semimartingale description for xKt yKt . Let At, At , At, At be thecompensators of t,
t ,
t,
t defined above. Introduce martingales
Mt = t At Mt = t At Mt = t At Mt = t At and also normalized martingales
mKt =Mt Mt
Kand mKt =
Mt MtK
(2.3)
Then, from (1.3) it follows that the process xKt yKt admits the semimartin-gale decomposition
xKt = x0 +t
0
xKs Ks xKs yKs
ds + mKt (2.4)
yKt = y0 +t
0
xKs y
Ks yKs
ds + mKt (2.5)
which is a stochastic analogue (in integral form) of (1.1)
In the sequel we need quadratic variations of the martingales in (2.4) and
(2.5). It is well known [see, e.g., Liptser and Shiryaev (1998), Chapter 18 or
Klebaner (1998), Theorem 9.3] that all martingales are locally square inte-
grable and possess the predictable quadratic variations
Mt = At Mt = At and Mt = At Mt = At (2.6)and zero mutual predictable quadratic variations M Mt 0 M Mt 0, implied by the disjointness of jumps for t, t , t, t .
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ANALYSIS AND EXTINCTION IN LOTKAVOLTERRA MODEL 1271
Hence
mK
m
K
t 0mKt =
1
K
t0
xKs + xKs yKs
ds
mKt =1
K
t0
xKs y
Ks + yKs
ds
(2.7)
2.3. Stochastic exponential and cumulant function. In the large deviation
theory we use stochastic exponential, and to this end the following represen-
tation is more convenient:
xKt = x0 =tK
t
K
yKt = y0 = tK tK (2.8)
tK
,tK
,tK
,tK
are counting process with jumps of the unit size K1 and com-
pensatorsAtK
,AtK
,AtK
,AtK
, respectively. With every pair tK
,AtK
, , tK
,AtK
and a predictable process t such that for any T > 0 and K large enough,T
0et/K
xKt + yKt + xKt yKt
dt < P-a.s.
we associate nonnegative processes
zt
K = expt
0
sK
ds
es/K
1
dAs
(2.9)
zt
K
= exp
t0
sK
ds es/K 1 d AsApplying the Ito formula to zt K, we find
dzt
K
= zt
K
et/K 1 dt Atthat is, zt K is a local martingale [analogously zt K, zt K, zt K are localmartingales as well]. All these martingales are nonnegatives, so that they
are supermartingales [see Problem 1.4.4 in Liptser and Shiryayev (1989) or
Theorem 7.20 in Klebaner (1998)]. Hence for any Markov time we haveEz K 1 Ezt K 1.
Introduce also (for predictable processes and )
zKt = zt
K
zt
K
zt
K
zt
K
(2.10)
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1272 F. C. KLEBANER AND R. LIPTSER
Applying the Ito formula to zKt and taking into account that jumps oft,
t ,
t,
t are disjoint we find
dZKt = zKt et/K 1 dt At + et/K 1 dt At + et/K 1 d t At + et/K 1 d t At
so that ZKt is a positive local martingale as well as a supermartingalewith
EZK 1for any Markov time .
Set
G u v= vu + e 1 u + e 1 + uv
G u v= u v + e
1 uv + e
1 + v(2.11)
and define the so-called cumulant function,
G u v = G u v + G u v(2.12)Using the cumulant function, (2.10) can be written as
ZKt = expt
0
s dxKs +
t0
s dyKs
expt
0KG
s
K
sK
xKs yKs
ds
(2.13)
Hence an equivalent multiplicative decomposition holds for the exponential
semimartingale
expt
0
s dxKs + s dyKs = ZKt VKt
with the positive local martingale ZKt and a positive predictable process
VKt = expt
0KG
s
K
sK
xKs yKs
ds
3. Fluid approximation. In this section we justify the choice of thestochastic dynamics by showing that the LotkaVolterra equations describe
a limit (fluid approximation) for the family
xKt y
Kt
K
In what follows xt yt is a solution of the LotkaVolterra equation (1.1).Theorem 2. For any T > 0 and > 0,
limK
P
suptT
xKt xt + yKt yt
>
= 0
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ANALYSIS AND EXTINCTION IN LOTKAVOLTERRA MODEL 1273
Proof. Set
TKn =
inft xKt yKt n(3.1)Since TKn = TnK (see Corollary 1 to Lemma 1),
limn
limsupK
P
TKn T = 0(3.2)
Hence, it suffices to show that for every n 1,
limK
P
sup
tTKn T
xKt xt + yKt yt
>
= 0(3.3)
Since suptTKn TxKt yKt n+ 1, there is a constant Ln, depending on n andT, such that for t TKn T,
xKt xKt yKt xt xtyt LnxKt xt+ yKt ytxKt yKt yKt xtyt yt LnxKt xt + yKt ytThese inequalities and (1.1), (2.4) implyxKTKn T xTKn T+ yKTKn T yTKn T
2Lnt
0
xKsTKn xsTKn + yKsTKn ysTKn ds+ sup
tTKn T
mKt + suptTKn T
mKt Now, by the GronwallBellman inequality we find
suptTKn T
xKt xt + yKt yt e2LnT suptTKn T
mKt + suptTKn T
mKt
Therefore (3.3) holds, if both suptTKn T mKt and suptTKn T mKt convergein probability to zero as K . By the Doob inequality for martingales (see,e.g., Theorem 1.9.1(3) and Problem 1.9.2 in Liptser and Shiryayev (1989) the
required convergence takes place provided that both mKTKn T and mKTKn Tconverge to zero in probability as K . But by (2.7) mKTKn T constK , mKTKn T constK , and the proof is complete.
4. Formulation of the LDP and the proof of Theorem 1.
4.1. LDP. The random process xKt yKt t0 has nonnegative paths fromthe Skorokhod space 2 = 20 . Since we are going to apply the LDPfor asymptotic analysis of the extinction time on a finite interval 0 T, weexamine the LDP in 20 T. Because of the fluid approximation, the limit forxKt yKt tT is xt yttT with continuous differentiable functions xt and yt.
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1274 F. C. KLEBANER AND R. LIPTSER
This fact allows examining the LDP in the metric space 20 T suppliedby the uniform metric
x x
2
0 T
and
x x =2
j=1suptT
xtj xt j(4.1)
More exactly, since xKt 0, yKt 0 andKx = inft > 0 xKt = 0 Ky = inft > 0 yKt = 0
are also absorption points for processes xKt and yKt , respectively, let us intro-
duce a subspace 2 +0 T of nonnegative function from
20 T. It is obvious that
2 +0 T is a closed subset of
20 T in the metric .
Therefore we formulate the LDP in metric space 2 +0 T .
Theorem 3. For every T > 0, the family xK
t yK
t tT, K obeys theLDP in the metric space 2 +0 T with the rate of speed 1K and the (good)rate function
JT=
T
0sup
t +t Gtt
dt dtdtdtdt
0=x00=y0 ,
otherwise.(4.2)
The proof of the LDP is given in the Appendix.
4.2. Proof of Theorem 1. Denote by o the interior of the set
2 +0 T inf
t
T
t = 0 inft
T
t > 0Due to the LDP,
lim infK
1
KlogP
inftT
xKt = 0 inftT
yKt > 0
inf
JT (4.3)
It is clear that
inf o
JT inf o JT
where is a subset of o of absolutely continuous functions t, t with0 = x0, 0 = y0 and for any = ttT,
t = tt (4.4)To emphasize the fact that t is a solution of (4.4) we use the notation
t .
The function t remains positive for any finite time interval and moreover,
sup
t + t G t t
= sup
t G t t
= sup
t t t e 1 t e 1 + tt
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ANALYSIS AND EXTINCTION IN LOTKAVOLTERRA MODEL 1275
Taking into account that functions ts from have to be absorbed on 0 T,we choose a subset
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1276 F. C. KLEBANER AND R. LIPTSER
that is,
t
x0 pt. Now, for any T > 0, the choice p =
x0T
guarantees T T.If some ut is chosen and associated with T < T, it is obvious that ut 0
for t T T. Then, particularly, in (4.6) the integral T0
u2t dt is replacedby
T0
u2t dt. On the other hand, due to T = 0 we have
0 = eT
0 s ds
x0
T0
et
0 s dsut
t
t dt
that is, x0 =T
0e
t0
s dsut
t
t dt. Further, the CauchySchwarz
inequality implies
T
0 u
2
t dt x20T
0e2 t0 s dstt dt (4.7)
Take any positive q and ut = qet
0 s ds
t
t then the inequality given
in (4.7) becomes an equality, and only for
q = x0T0
e2t
0 s dst
t dt
(4.8)
we have T = 0.Let q Tq be a pair such that (1.6) holds with q and T replaced by q and
Tq respectively and Tq
T. Denote by
qt and
qt the solution of (1.5) on
0 Tq corresponding to the pair q Tq. Then we get the lower bound
lim infK
1
Klog P
TKext T inf
tTyKt > 0
inf x
20
2Tq
0 e2 t0 s dsutqt qt dt
where inf is taken over all Tq T .The final step of the proof uses the property
q < q x2
0Tq0 e
2 t0 qs dsqt q dt C
= (A.1)
lim0
lim supK
1
Klog sup P
supt
xKt+ xK + supt
yKt+ yK > =
(A.2)
for any > 0, where sup is taken over all stopping times T. C local LDPis valid, if for any 20 T
lim sup0 limsupK
1
K logPxK yK JT (A.3)
lim inf0
lim infK
1
KlogPxK yK JT (A.4)
Since the linear growth condition for intensities of counting processes is
lost, a verification of (A.1) requires a different technique from the one used in
Liptser and Pukhalskii (1992). We verify (A.1) in the next subsection.
A.2. Exponential negligibility of suptT xKt , suptT y
Kt .
Lemma 2. For every T > 0,
limL
lim supK
1
KlogP
suptT
xKt L =
limC
limsupK
1
KlogP
suptT
yKt C
=
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1278 F. C. KLEBANER AND R. LIPTSER
Proof. Recall that xKt = x0 + ttK
and Mt = t At; that is, dxKt dtK
=xKt dt
+dMt
Kand so xKt
et
x0
+1
K t
0es dMs
. Consequently, the
first statement of the lemma is valid provided that
limL
lim supK
1
KlogP
suptT
t0
es dMs KL
=
It is clear that with KL = inftt
0es dMs KL it suffices to establish
limL
lim supK
1
KlogP
KL T
= (A.5)
To this end, with r > 0 and s = res we define zt by the first formula in(2.9) with
sK
replaced by s and use an obvious modification of this formula,
zt = expt0 s dMs t0 es 1 s dAs(A.6)Recall that zt is a supermartingale with EzKL T 1. We use thisinequality for the next one,
1 EzKL T
IKL T(A.7)We sharpen that inequality, by evaluating below z
KL T on the set KL T,
logzKL T
= rKL T
0es dMs
KL T0
ere
s 1 resKxKs ds
rKL
KL T
0 er
1
r
KxKs ds
rKL T
0er 1 rKxK
sKLds
To continue this evaluation we find an upper bound for xKsKL
. Since
KL T0
esdMs KL + 1
(recall that jumps oft
0esdMs is bounded by 1), we claim
xKKL t
etx0 + 1 + L
Hence, with p =eT
x0
+1
+L
L , we arrive at the lower boundlogz
KL T KLr er 1 rTp
It is clear that r > 0 can be chosen so that r er 1 rTp = > 0 andtherefore logz
KL T KL > 0.
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ANALYSIS AND EXTINCTION IN LOTKAVOLTERRA MODEL 1279
Thus, (A.7) with s = res replaced by sres, implies1
K logPKL T L L
that is, (A.5) holds.
The proof of the second statement of the lemma uses (A.5) heavily. SincesuptT
yKt C
suptKL T
yKt C
KL Tand thereby
P
suptT
yKt C
2
P
sup
tKL TyKt C
P
KL T
by virtue of (A.5) it suffices to check that for every fixed L,
limC
lim supK
1K
logP suptKL T
yKt + C = (A.8)The verification of (A.8) is similar to the proof of the first statement of the
lemma. It is clear that there is a positive constant RL so that IKL txKt RL, t 0. Further, due to dyKt xKt yKt dt + 1K dMt we have
dyKtKL
IKL txKt yKt dt +1
KdM
tKL
RLyKt dt +1
KdM
tKL
and hence yKt
KL
eRLt
y0
+ tKL
0 eRLd
Ms
Now, introduce
KL C = inf
ttKL
0eRL dMs C
and note that (A.8) holds provided that
limC
lim supK
1
KlogP
KL C T
= L > 0(A.9)The proof of (A.9) is similar to the proof of (A.5), so here we give only a sketch
of it. Set = RL and for positive r take s = res. Introduce a super-martingale
zt = expt
0s d
Ms
t
0 es 1 s
d
As
Due to EzKL TKL CT
1 write 1 EzKL TKL CT
IKL C T and evaluatefrom below z
KL TKL CT on the set KL C T as
log zKL TKL CT
KCr er 1 rTp
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1280 F. C. KLEBANER AND R. LIPTSER
With r such that r er 1rTp = > 0 similarly to the proof of thefirst statement of the lemma we obtain 1
KlogPKL C T C and (A.9)
holds.
A.3. C-Exponential tightness. (A.1) is implied by Lemma 2.To verify (A.2), let us note that
xKt+ xK =1
K
t+
+ 1K
t+
yKt+ yK =1
K
t+
+ 1K
t+
Evidently, it suffices to establish the validity of (A.2) for every
1
Kt+
1
Kt+
separately.Recall that the Markov time TKn is defined in (3.1). By virtue of Lemma 2
we claim that limn lim supK1
KlogPTKn T = . Hence, with any
n 1 we have to verify (A.2) only for
limK
1
Klog sup
TP
+TKn TKn > K
= 0
limK
1
Klog sup
TP
+TKn TKn > K
= 0
The proofs for the above relations are similar. So, we give below only one of
them for t . With r > 0 set
s
=rI
TKn
+
TKn
s
. The random process
s is bounded and left continuous (and thereby predictable). Consequently[compare (2.9)], the random process
zt = expt
0s ds
t0
es 1dAs
is a supermartingale with EzT+ 1. Therefore,
1 EzT+I
+TKn TKn > K
(A.10)
Since A+TKn ATKn
Kn + 12, on the set +TKn
TKn > K,
logzT+ = r
+TKn
TKn
er 1
A+TKn A
TKn
Kr er 1n + 12Now, with small enough and r = log
n+12 ,
zT+ exp
K
log
n + 12
+ n + 12
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ANALYSIS AND EXTINCTION IN LOTKAVOLTERRA MODEL 1281
Hence
1
K log supT P+TKn TKn > K log
n + 12
+ n + 12 0
A.4. C-Local-LDP. Upper bound. Obviously, for 0 = x0 or 0 = y0 theupper bound is equal to . The proof that for 0 = x0 and 0 = y0 and 20T\20T the upper bound is equal to under the C-exponentialtightness can be found in Theorem 1.3 in Liptser and Pukhalskii (1992).
Thus, the proof is concentrated on the case 20 T with 0 = x0,0 = y0. Let ZKt K K be defined in (2.13) with piecewise constant (deter-ministic) functions
t
and
t
. Since EZK
T K K
1, write
1 EZKT
K
K
I
xK yK
(A.11)
On the set xK yK , we evaluate ZKt K K from below. Since are piecewise constant functions notations,
T0
t dtT
0t dt will be
used for tjT
tj1tj tj1 and
tiTti1ti ti1
respectively. Obviously, a positive constant c can be chosen such that
logZKT K K K
c +
T0
t dt + t dt Gt t t t dt
Therefore, this lower bound jointly with (A.11) imply
lim sup0
lim supK
1
KlogPxK yK
T
0t dt + t dt Gt t t t dt
Since the left side of this inequality is independent oft t
the required
upper bound is obtained by minimization of the right side in t t. As inLiptser and Pukhalskii [(1992), Theorem 6.1], we find that for not absolutely
continuous t or t the minimal value of the right-hand side is equal to ,while if both t t are absolutely continuous functions the minimal valuecoincides with JT .
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1282 F. C. KLEBANER AND R. LIPTSER
A.5. C-Local-LDP. Lower bound. It suffices to verify (A.4) only for JT, < . That supposes the verification of (A.4) for absolutely continuousfunctions t ttT from
2+0 T with 0 = x0 0 = y0. Moreover, to satisfy
JT < functions t t have to be absorbed on axis x = 0; y = 0(recall that xKt y
Kt are absorbed on these axes). For definiteness we denote
the class of the above-mentioned functions t t by .Set T = inft > 0 t = 0 and T = inft > 0 t = 0 inf = and
introduce
JT =
TT0
sup
t G t t dt(A.12)
JT =
TT0
sup
t G t t dt(A.13)
Then, obviously, we have JT = JT + JT . Below we givedetailed description of t = argmaxt G t t and t =argmaxt G t t for T T = ,
t= log
t2t
+
2t4t2
+ t
t= log
t2tt
+
2t4tt2
+ t
(A.14)
A.5.1. Main Lemma.
Lemma 3. Assume
(i) .(ii) T T = .
(iii) t const t const
Then there is a positive constant L , depending on L , such thatfor > 0 small enough,
lim infK
1
K
logP
xK yK
JT
L
Proof. Set K = inft xKt t and K = inft yKt t .Since jumps of xKt y
Kt are bounded by
1K
, by the triangular inequality we
have xKt t xKt t + 1K and yKt t yKt t + 1K . Consequently
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ANALYSIS AND EXTINCTION IN LOTKAVOLTERRA MODEL 1283
with K 4
the inclusion holds:
suptT xKt t + suptT yKt t
suptT
xKt t + suptT
yKt t
2
=
suptK T
xKt t + suptK T
yKt t
2
Hence, the first lower bound we use is the following:
PxK yK
P
sup
tK
T
xKt t + supt
K
T
yKt t
2
(A.15)
For K large enough set
Kt= log
t
2xKt+
2t
xKt2+ x
Kty
Kt
xKt
t K T
Kt= log
t
2xKtyKt
+
2t
2xKtyKt2yKt
xKtyKt
t K T
(A.16)
[Kt Kt are defined similarly with an obvious change]. It is clear thatfor K large enough Kt Kt are bounded and strictly positive.
We define now the positive supermartingale [compare (2.13)]
ZKt K K= exp
KtK T
0
Kt dxKt GKt xKt yKt dt
exp
K
tK T0
Kt dyKt GKt xKt yKt dt
(A.17)
We show that ZKt K KtT is the square integrable martingale withEZKT K K = 1
To prove this property we have to check that EZKT K K2 < . (A.17)implies
ZKT
K K
2
=ZKT
2K 2K
exp
K
K T0
G
2Kt xKt yKt
2GKt xKt ykt dt exp
K
K T0
G
2Kt xKt yKt
2GKt xKt ykt dt
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1284 F. C. KLEBANER AND R. LIPTSER
The process ZKT 2K 2KtT is a supermartingale as well; that is
EZ
K
T 2K
2
K
1and at the same time under assumptions of the lemma G2Kt xKt yKt ,GKt xKt yKt and G2Kt xKt yKt GKt xKt yKt are boundedfunctions on the time intervals 0 K T and 0 K T, respectively. Con-sequently, ZKT K K2 has a finite expectation.
Now, F = tt0 QK with dQK = ZKT K K dP is the stochasticbasis. Due to the positiveness of ZKT K K not only QK P but also P QK with dP = ZKT K K1 dQK. We apply this formula for the right sideof (A.15). With the notation
K =
sup
tK TxKt t + sup
tK TyKt t
2
we have
PK =
K
ZKT K K1 dQK(A.18)
The random process ZKtTK K is the martingale with respect to P andsince P QK it is a semimartingale with respect to QK [see, e.g., Liptser andShiryayev (1989), Chapter 4, Section 5]. For further analysis QK semimartin-gale description of ZKtTK K is required. To this end,we use the fact thatP-semimartingales xKt y
KT are Q
K semimartingales as well [see Liptser and
Shiryayev (1989), Chapter 4, Section 5] and find their semimartingale decom-
positions. Let us note that t t
t
t are counting processes with respect to
both measures P and QK. Recall that At
At
At At are their compensatorswith respect to P and denote by AQt AQt AQt AQt their compensators withrespect to QK. Then the P-martingale mKt = 1Kt At t At obeysthe semimartingale decomposition (P-and QK-a.s.)
mKt =1
K
t AQt
t AQt 1
K
At AQt
At AQt = mKQt
1
K
At AQt
At AQt (A.19)
with QK- square integrable martingales mK Qt with the predicatable quadratic
variation mK Qt =1
KAQ
t + AQ
t . Analogously, the QK-semimartingaledecomposition mKt = mK Qt 1KAt AQt At AQt holds with thesquare integrable martingale m
K Qt with mK Qt = 1KAQt + AQt and mK Q,
mK Qt 0 (recall that disjointness for jumps of counting processes remainsvalid with respect to QK).
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ANALYSIS AND EXTINCTION IN LOTKAVOLTERRA MODEL 1285
We give now formulas for AQt A
Qt
A
Qt
A
Qt . With
t = t t1 ,
t = t t we have
ZKt K K= XKtK K expKtt T+Ktt t
(A.20)
AQt : It is well known that AQt is defined by integral equation: for anybounded and predictable function ut ,
T0
ut dt dQK =
T0
ut dAQt dQK
Taking into account that dQK = ZKT K KP and ZKtTK K is the mar-tingale with respect to P we get
T0
ut dt dQK =
T0
ut ZKT K K dt dP
=
T0
ut ZKt K K dt dP
Since jumps of counting processes are disjoint the right side of the above equal-
ity is equal to [see (A.20)]
T0
ut ZKtK KeKtt dtdP or, what is
the same,
T0
ut ZKtK KeKt dt dP. Now, since Z
KtK K is
the predictable process, the latter integral coincides with
T0 ut ZKtK KeKt dAt dPwhich, due to the martingale property of ZKt K K, is equal to
T0
ut ZKT K KeKt dAt dP =
T0
ut eKt dAt dQK
Consequently,
dAQt = e
Kt dAt
and dAQt = eKt dAt d
AQt = eKt d
At d A
Qt = eKtd
At are
obtained in the same way.Hence
dmKt = dmKQt +
eKt 1xKt eKt 1xKt yKt dt
dmKt = dmKQt +
eKt 1xKt yKt eKt 1yKt dt
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1286 F. C. KLEBANER AND R. LIPTSER
and thereby we find the semimartingale decomposition with respect to QK:
dxKt= xKt yKt +eKt 1xKt eKt 1xKt yKt dt + dmKQt=
G
Kt xKt yKt
dt + dmKQt = t dt + dmKQt
dyKt =
yKtxKt
+eKt 1xKt yKt eKt 1yKt dt + dmKQt
=
G
Kt xKt yKt
dt + dmKQt = t dt + dmKQt
(A.21)
The use of this decomposition allows to obtain a description for 1K
log ZKT K,K with respect to QK:
1K
logZKT K K
=K T
0Kt dmKQt +
K T0
Ktt GKt xKt yKt
dt
+K T
0Kt dmKQt +
K T0
Ktt GKt xKt yKt
dt
Let us estimate above the right side of this equality on the set K . There are
positive constants c and c so that
suptK T
Kt t + suptK T
GKt xKt yKt Gt t t c
suptK T
K
t
t +
suptK T
G
K
t
xK
t yK
t G
t
t
t c
Further, since
tt Gt t t = sup
t G t t
tt Gt t t = sup
t G t t
and
JT =K T
0sup
t G t t dt
+
K T
0sup
t G t t dt
we find
1
KlogZKT K K
K T0
Kt dmKQt+ K T
0Kt dmKQt
+c + cT + JT
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ANALYSIS AND EXTINCTION IN LOTKAVOLTERRA MODEL 1287
This upper bound implies the lower one,
ZK
T KK
exp
K
J
T
+ c
+c
T
exp
K
K T0
Kt dmKQtKK T
0Kt dmKQt
which, being applied in (A.18), gives
PK expKJ + c + cT
K
exp
K K
0Kt dmKQt
K
K T0
Kt mKQt
dQK
(A.22)
(A.22) can be sharpened as
PK eKJT+c+cT+2QK
K K K
where
K =
K T0
Kt dmKQt < and K = K T
0Kt dmKQt
< Hence
liminfK
1
Klog PK JT c + cT 2
+ lim infK
1
Klog QK
K K K
The latter inequality jointly with (A.15) imply the statement of the lemma
provided that for fixed and it holds lim infK QKK K K = 1.We derive that property from
limK
QK
\K = 0 lim
KQK
\K
= 0 limK
QK
\K = 0(A.23)
The first part of (A.23) follows from the Chebyshev inequality (EK is theexpectation with respect to QK),
QK
\K = QK K T
0Kt dmKQt
1
2EK
K T
0 K
t
2 d
mKQ
t
1K22
EKK T
0Kt2eKt dAt + e
Kt dAt
constK2
0 K
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1288 F. C. KLEBANER AND R. LIPTSER
The validity of the second part of (A.23) is verified similarly. The third part
in (A.23) is verified similarly to the first and second ones since by virtue of
(A.21),
\K =
suptK T
mKQt + suptK T
mKQt 2
A.5.2. The lower bound under T T = .
Lemma 4. Assume
(i) .(ii) T T = .Then
lim inf0 lim infK1
K logPxK yK JT Proof. Set nt = x0 +
r0
sIs n ds. SinceT
0t dt < , it holds
limn
suptT
t nt T
0It > nt dt = 0
Similarly, for nt = y0+r
0 sIs n ds we have limn suptT tnt = 0. Letus choose n0 so that for n n0 it holds suptT t nt + suptT t nt 2 .Since by the triangular inequality,
supt
T
xKt t+ sup
t
T
yKt t+ sup
t
T
xKt nt+ sup
t
T
yKt nt+
2
for any n n0 we get
PxK yK P
xK yK n n 2
Therefore, by Lemma 3,
lim infK
1
KlogP
xK yK n n
2
JTn n Ln n
Since inftT t > 0 and inftT t > 0 and t t and approximated by nt
nt
uniformly in t T, for n large enough a majorant L for Ln n can bechosen, so that
lim infK
1K
logPxK yK 2 JTn n L
We show that
lim supn
JTn n JT (A.24)
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ANALYSIS AND EXTINCTION IN LOTKAVOLTERRA MODEL 1289
Since for n large enough inftT nt > 0, inftT
nt > 0, introduce
nt= log nt2nt + n2t4nt 2 + nt t= log
nt
2nt nt
+
n2t4nt nt 2
+ nt
(A.25)
and note that
JTn n =T
0It n
ntt G
nt nt nt
dt
+T
0It > n
G
nt nt nt
dt
+ T0
It nntt Gnt nt nt dt+
T0
It > nGnt n ndt
JT +T
0
Gt t t Gnt nt nt dt+
T0
Gt t t Gnt nt nt dtThe second and third terms in the right side of the above inequality converge
to zero, as n , by virtue of the uniform (in t T) convergence n n
and T0 nt nt tt dt 0 and T0 nt nt tt dt 0;that is, (A.24) holds.
Thus
lim infK
1
KlogP
xK yK
2
JT L (A.26)
and the statement of the lemma holds.
A.5.3. The lower bound under T T < . For and h smallenough we have
lim inf
0
lim infK
1
KlogPxK yK
= lim inf0
liminfK
P
suptTT
xKt t+ suptTT
yKt t lim inf
0liminf
KP
sup
tThT
xKt t+ suptThT
yKt t
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1290 F. C. KLEBANER AND R. LIPTSER
On the other side, for ht t, t T h, ht t, t T h, and fort > T h, t > T h,
ht = ht ht ht = ht ht respectively, we have
liminf0
lim infK
P
sup
tThT
xKt t + suptThT
yKt t = lim inf
0liminf
KPxK yK h h JTh h
where the latter inequality follows from Lemma 4.
Finally,
JYh h =
ThT
0sup
t G t t dt
+ ThT0
sup
t G t t dt
TT
0sup
t G t t dt
+TT
0sup
t G t t dt
= JT Now combining the obtained results above we arrive at the required lower
bound,
lim inf0 lim infK 1K logPxK yK JT
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Department of Mathematics
and Statistics
University of Melbourne
Parkville, Victoria 3052
AustraliaE-mail: [email protected]
Department of Electrical
Engineering Systems
Tel Aviv Univeristy
69978 Tel Aviv
IsraelE-mail: [email protected]