# limit theorems for the reaction-diffusion equations with ecological

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```					Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

limit theorems for the reaction-diffusion
equations with ecological applications

Speaker: S. Molchanov(UNCC)

In collaboration with:
L.Bogachev(Leeds,UK)                            G.Derfel (Israel)
Y.Feng(UNCC)                                    J.Whitmeyer(UNCC)
In Scientiﬁc contacts with:
Yu. Kondratiev(Bielefeld,Germany)               S. Pirogov (MSU,Russia)
J.Ockondon(Oxford,UK)                           G.C.Wake(Auckland,New Zealand)

February 3, 2010
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

Outline
1    Introduction
2    Model 1:Supercritical reaction-diffusion process on
Zd , d ≥ 1
Model description
The generating function and the moments
The mass process and its asymptotic behavior
Main theorem in model 1
3    Model 2 Continous contact model
correlation functions
Main results
4    Model 3: Critical reaction-diffusion process on Zd , d ≥ 3
5    Model 4: Critical reaction-diffusion process on
Zd , d = 1, 2
6    Model 5: Random media (RM) models in the population
dynamics
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

Motivations and central problems

My studies on "mathematical ecology" were initiated by the workshop
on Stochastic Population Dynamics, Summer 2009, Edinburgh, UK (
which took place in the memorial house for Maxwell).

How to explain with appropriate models of the following
empirical facts?
The stationarity in time and space of biopopulations ( at least
those not strongly supressed by civilization).
Strong deviations of the spatial distribution of the species from a
Poissonian point ﬁeld ( patches ).
The non-Gaussian distribution of parameters such as size,
mass, etc ( at least for one-cell populations ,plankton etc ).
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

The typical picture of the corresponding point ﬁeld

Figure: Point ﬁeld (patches)
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

We’ll discuss isolated populations which are not involved in
complex multispecies interaction ( predator-pray schemes, etc).
The main ideas are based on the FKPP(Fisher-Kolmogorov
-Petrovsky-Piscunov) type model of the evolution of a new gene
( after mutation ). Our central object will be, as in FKPP
theory,the branching process with random dynamics in the
space. We’ll exclude the direct interaction between particles ,
but the birth-death processes will create, however, some type
of attractive mean ﬁeld interaction.
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

Notations and Assumption
n(t, x) = (particles at the moment t ≥ 0 and site x ∈ Zd )
The evolution of each particle will be independent and will include
death with the rate µ ≥ 0
the birth of a new particle ( splitting ) with the rate β ≥ 0
random displacement x → x + z with the rate q(z),
q(z) = K > 0
z=0

The generator of the random walk x(t) is
Lf (x) =            (f (x + z) − f (x))q(z), f ∈ L∞ (Zd )
z=0

Remark:In the simplest case , the particles jump only to the nearest
neighbours : x → x , x − x = 1, the generator is the usual lattice
Laplacian:
Lf (x) = κ∆f (x) = κ                             (f (x ) − f (x))
x : x −x =1
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

Model description

Main differenc with classic FKPP Model
The presence of an extra parameter mass for each particle

Assumption:β > µ, and n(0, x) = δ0 (x), x ∈ Zd ,the single initial
particle has mass m at t = 0 and x ∈ Zd . For t > 0, the particle
performs a random walk with generator L = κ∆
its mass increases linearly: m → m(t) = m + vt, v > 0,
i.e the underlying Markow process (x(t), m(t)) in Zd × R1 has the
+
generator
∂
L = κ∆ + v
∂m
At the moment of the ﬁrst transformation τ1 (exponentially distributed
with parameter β + µ), the particle either dies or splits into two
particles at the point x(τ1 ), (β + µ) is the rate of "transformations".
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

Model description

The mass m1 = m + v τ1 of the particle at the moment of splitting is
distributed between two offspring in random proportion (the
phenomenon of mitosis):
law
m2 = m1 θ,          m2 = m1 (1 − θ),               θ = 1−θ
θ has distribution q(dθ). The offspring evolve independently with the
same rules as the initial particle.

m1 = θmτ 1
′

(m0 , x0 )       (mτ1 , xτ1 )

′
m 2 = (1 − θ ) mτ 1

Figure: The evolution of the particles
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

The generating function and the moments

The generating functions

Consider the following random ﬁeld on Zd × R1 , where nΓ (t, x) =
+
particles at the moment t ≥ 0 and the site x ∈ Zd , and the masses of
the particles belong to the set Γ ⊂ R1 . Put
+

n (t,x1 )          nΓ (t,xl )
u(t, x, m; x1 , · · · , xl ; z1 , · · · , zl ) = Ex z1 Γ               · · · zl

This generating function satisﬁes the following non-linear
integral-differential FKPP backward equation:
 ∂u              ∂u
 ∂t = κ∆u + ∂m · v − (β + µ)u + µ
1

+β 0 u(t, x, θm) · u(t, x, (1 − θ)m)q(dθ)


 u(0, x, m; x , · · · , x ; z , · · · , z ) =
                                              IΓ (m), x ∈ x1 , · · · , xl
/
1          l   1           l
zi IΓ (m), x = xi , i = 1, · · · , l

(1)
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

The generating function and the moments

The ﬁrst moments

Differentiation of (1) over z1 , · · · , zl provides the equations for the
moments
ml,Γ (t, x1 , · · · , xl ) = Ex nΓ (t, x1 ) · · · nΓ (t, xl )
For example, the equation of the ﬁrst moment m1 (t, x1 , x, m) has the
form:
 ∂m1,Γ                   ∂m1,Γ
 ∂t = κ∆x m1,Γ + ∂m · v + (β − µ)m1,Γ
1
                      +2β 0 m1,Γ (t, x1 , x, θm) − m1,Γ (t, x1 , x, m)q(dθ)
m1,Γ (0, x1 , x, m) = δx1 (x)IΓ (m)
(2)
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

The generating function and the moments

The higher order moments

In general,the higher order moments ml,Γ are the solutions of the
triangular system of equations (given inductively)
1
∂ml,Γ            ∂ml,Γ
= κ∆ml,Γ +       v +(β−µ)ml,Γ +2β                                 ml,Γ (t, x, θm)−ml,Γ (t, x, m)q(dθ)
∂t               ∂m                                                0

+ source containing a non-linear (in our case quadratic) combination
of the moments of order ≤ l − 1.
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

The mass process and its asymptotic behavior

∂
The linear part of (2) includes the Laplacian κ∆ + v ∂m generator of
the underlying space-mass process (x(t), m(t)) , the creative
potential (β − µ) > 0 and an integral term that corresponds to mitosis.
Formula (2) contains the generator of the "mass process"
1
∂φ
Lm (φ) = v            + 2β            [φ(mθ) − φ(m)]qd(θ)
∂m             0

It has a Poissonian jump with intensity 2β, the linear growth of mass
with coefﬁcient v between jumps, and the transition m(τ ) → θm(τ ) at
the moment τ of the jump (See ﬁgure below).Here the τi , i ≥ 1 are
i.i.d exponential random variables.
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

The mass process and its asymptotic behavior

m2 = m1 + vτ 1
m1 = m0 + vτ 0

θ1m1
m0

τ0      τ 0 + τ1

Figure: mass process
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

The mass process and its asymptotic behavior

The transition probabilities of the mass process ρ(t, m, m ), i.e the
fundamental solution of
∂ρ(t,m,m )
∂t   = Lm ρ(t, m, m )
ρ(0, m, m ) = δm (m)

have a limit
Π(m ) = lim ρ(t, m, m )
t→∞

Calculations show that the limiting (invariant) density Π(m ) has same
distribution as the random geometric progression

ξ = v τ0 + v τ1 θ1 + · · · + v τn θ1 · · · θn + · · ·
v
It converges, since Eξ = 2vEτ =                       β.
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

The mass process and its asymptotic behavior

Let’s formulate several analytic results about the invariant density
1
Π(m). Assume that Suppθ = [a, 1 − a], 0 < a ≤ 2 , then

Π(m) ∼ e−(1−a)m L(m), L(m) − − → 0
−−
m→∞

and L(m) depends on the distribution q(dθ) near the maximum
point (1 − a) (or symmetric point a).
If m → 0, then ln Π(m) ∼ c1 ln2 ( m ).
1

Π(m) ∈ C ∞ and Π(k ) (m) → 0 as m → 0 for any k = 0, 1, · · · . I.e,
invariant density is not an analytic function of m.
Conjecture: Π(m) is an one-vertex density (see ﬁgure).
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

The mass process and its asymptotic behavior

Π ( m)
1
− c1 ln 2
m
e
e − c0 m

m

Figure: Asymptotic behavior of Π(m)
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

The mass process and its asymptotic behavior

Note that nΓ (t) =                 nΓ (t, x) is the total number of particles with
x∈Zd
masses m ∈ Γ. If Γ = R1 , it is the total number of particles N(t).
+

Lemma
Using EN(t) = e(β−µ)t for the normalization of N(t), we get P-a.s that
there exists
N(t)
p(        < a) −→ G(a)
EN(t)
µ
The limiting distribution G(a) has an atom α =                             β   at 0 and an
exponential density for a > 0. Formally

dG
= αδ0 (a) + (1 − α)λe−λa , a ≥ 0
da
β−µ
Here λ = 1 − α =              β .

Remark: The presence of the atom αδ0 (a) in the limiting distribution
G(a) indicates the possibility of the degeneracy of the population at
an early stage of its evolution.
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

Main theorem in model 1

Main results in model 1

Theorem 1
3
Let n(0, x) = δ0 (x), x = O(t 4 ),Γ ∈ R1 , then
+

x 2
e−    4kt   e(β−µ)t
EnΓ (t, x) ∼                      d      Π(Γ).
(4πkt) 2

nΓ (t, x)
p(              < a) −→ G(a)
EnΓ (t, x)

This means that for each site in the lattice, the mass distribution of
the particles is given by the invariant distribution of the process m(t)
with generator Lm . Let’s recall that e(β−µ)t = EN(t)
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

Main theorem in model 1

Let’s note that the front of the population can be deﬁned by the
formula:
Ft = {x : p0 (t, 0, x)e(β−µ)t ≥ 1}
where p0 (t, x, y ) is the transition probability of the process xt with
generator κ∆. The Front Ft propagates linearly in t but it is not
spherical (due to non-isotropic large deviations for
p0 (t, 0, x), x = O(t)). Theorem (1) shows that
varnΓ (t, x) = O(EnΓ (t, x))2 , i.e. the higher moments of
nΓ (t, x)increase very fast. This can be considered as an indicator of
the intermittency (strong local non-uniformity) of the plankton colony.
However this is only an illusion. The cause of the bad behavior of the
local moments is the strong correlation between nΓ (t, x) and N(t).
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

Main theorem in model 1

Conditional distribution

Consider the conditional distribution of the random variable
3
nΓ (t, x), x = O(t 4 ) under the condition Hc0 = {N(t) = c0 e(β−µ)t }.

Theorem 2

x 2
(β−µ)t           e−    4κt
E[nΓ (t, x)|Hc0 ] ∼ c0 e                   π(Γ)                d   , t →∞
(4πκt) 2

x 2
(β−µ)t           e−    4κt
var [nΓ (t, x)|Hc0 ] ∼ c1 e                  π(Γ)                   d   , t →∞
(4πκt) 2
Note that the conditional variance is comparable to the
conditional expectation but not to its square.
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

Main theorem in model 1

Continuation of Theorem 2

x
nΓ (t, x) − E[nΓ |Hc0 ]                        1                         z2
p                           <x                 −− √
−→                       e    2   dz
var [nΓ |Hc0 ]                      n→∞  2π              −∞

Under condition Hc0 , the ﬁeld nΓ (t, x) is close to its (conditional)
mean plus Gaussian ﬂuctuations of much smaller order.

Compare with Theorem 1!
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

Model 2 Continous contact model (critical case)
This model was introduced by Yu. Kondratiev, this work also contains
existence theorems for the limiting distribution in dimensions d ≥ 3.
Consider the initial Possonian ﬁeld in Rd . The death rate µ of the
particles is equal to the birth rate β. At the moment τ :
1
p{τ > s} = e−2βs , a particle either dies with probability 2 or produces
a new seed, which is randomly distributed with density
a(z), z ∈ Rd .After a birth at the point x, the tree remains in x but the
seed produces new tree at x + ξ ∈ Rd . This is a simpliﬁed model of a
forest.

p{ξ ∈ ( z , z + dz )} = a( z )dz

x +ξ

x
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

correlation functions

The central object: correlation functions
Correlation function

k (l) (t; x1 , · · · , xl )dx1 · · · dxl = p{to ﬁnd l particles at the neighbourhoods

x1 + dx1 , · · · , xl + dxl }, l = 1, 2, · · ·
The system of integral equations for k (l) (.) has the form:

∂k (l) (t; x1 , · · · , xl )
= −lβk (l) (t; x1 , · · · , xl )
∂t
l
+β          k (l−1) (t, x1 , · · · , xi−1 , xi+1 , · · · , xl )         a(xi − xj )
i=1                                                         i=j
l
+β            a(xi − z)k (l) (t, x1 , · · · , xi−1 , z, xi+1 , · · · , xl ) dz
i=1    Rd

The initial condition is trivial:

k (l) (0; x1 , · · · , xl ) = ρl0 , l ≥ 1
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

Main results

Leting t → ∞ in dimension d ≥ 3 and using the Carleman’s
estimation k (l) ≤ c l l for some c > 1 provides the existence and
uniqueness of the limiting point ﬁeld.

Theorem 3
The Fourier transform of the limiting correlation function has a cluster
structure, similar to the expansion of the resolvent of the multiparticle
¨
Schrodinger operator. The general (a bit complicated) formula is clear
from the presentation of the ﬁrst three moments.

ˆ
k (1) (λ) = (2π)d ρ0 δ0 (λ) ⇒ k1 (x) = ρ0

ˆ                                                        ˆ
a(λ)δ0 (λ1 + λ2 )
k (2) (λ1 , λ2 ) = (2π)2d ρ2 δ0 (λ1 )δ0 (λ2 ) + (2π)d ρ0
0                                     ˆ
1 − a(λ)
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

Main results

Continuation of Theorem 3
ˆ
k (3) (λ1 , λ2 , λ3 ) = (2π)3d ρ3 δ0 (λ1 )δ0 (λ2 )δ0 (λ3 )
0

ˆ        ˆ
(a(λ1 ) + a(λ2 ))δ0 (λ1 + λ2 )δ0 (λ3 )
+(2π)2d ρ2 [
0                                               +two similar terms]
ˆ         ˆ
2 − a(λ1 ) − a(λ2
cluster containing (x1 ,x2 ), x3 is separated

ˆ        ˆ      ˆ
(a(λ1 ) + a(λ2 ))a(λ1 + λ2 )δ0 (λ1 + λ2 + λ3 )
+(2π)d ρ0 [
ˆ        ˆ         ˆ
3 − a(λ1 ) − a(λ2 ) − a(λ3 )
+ two similar terms]

Remark:It is a typical cluster expansion.
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

Main results

Continuation of Theorem 3
ˆ
k (n) (λ1 , · · · , λn ) = (2π)nd ρn δ0 (λ1 ) · · · δ0 (λn )
0

n                                                  ˆ
(a(λi ) + a(λj ))k (n−1) (λi + λj , λi1 · · · , λin−3 )
ˆ        ˆ
+                                  [                                                           ]
ˆ               ˆ
n − a(λ1 ) − · · · − a(λn )
j=1             i=j
i1 <i2 <···<in−3 ,=i,j

Theorem 4
All terms in the cluster expansion of k (l) (x1 , · · · , xl ) are positive. It is
the manifestation of the FKG-property of the limiting ﬁeld.
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

Model 3: Critical reaction-diffusion process on Zd , d ≥ 3

Here β = µ > 0,L = κ∆ (the generator of the underlying random
walk.) Put
m(l) (t, x1 , · · · , xl ) = En(t, x1 ) · · · n(t, xl )
The initial ﬁeld n(0, x) is the system of i.i.d Poissonian random
variables with parameter ρ0 > 0 (the density of the population). In
contrast to the FKPP model where we used the backward equations
with respect to the position of the single particle, we’ll use the forward
equations. They have the following form (β = µ > 0!)

∂m1 (t,x)
∂t      = κ∆m1
m1 (0, x) = ρ0

⇒ m1 (t, x) = ρ0
for any t ≥ 0, x ∈ Zd
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

For the second moment we have
∂m2 (t, x1 , x2 )
= κ(∆x1 + ∆x2 )m2 + 2(β + 2dκ)ρ0 δ(x1 − x2 )
∂t
−2κδ( x1 − x2 = 1)
ρ2 , x1 = x2
0
m2 (0, x1 , x2 )       =
ρ2 + ρ0 , x1 = x2
0

In general
∂ml
∂t = κ(∆x1 + · · · + ∆xl )ml + source depending non-linearly on
the moments ms , s < l + Poissionian initial conditions.
Analysis of the moment equations is similar to the technique
used in the Model 2 (Fourier transformation over the space
variables, resolution of the moment equations, using their
triangular structure).
Theorem 5
In model 3, for d ≥ 3, there exists a limiting distribution n(t, .) → n (.).
The ﬁeld n (.) has the FKG-property.
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

Model 4: Critical reaction-diffusion process on Zd , d = 1, 2

Problem :The previous model in the low dimensions d = 1, 2 has
no limit!
m(2) (t, x1 , x2 ) − − +∞
−→
t→∞

Explanation:
Assume that κ = 0 (no diffusion). The Poissonian set of families
that survive has concentration ρ0 ct0 = ρt . The typical distance
1
between such families will be O(t d ) and each such family will
1
have radius O(t d ), the density ρ0 will be preserved. If κ > 0,
1
O(t d ) . Deﬁnitely, for d = 1, the clusterization will tend to inﬁnity.
The case d = 2 is a the borderline situation, however
m(2) (t, .) → ∞(no limiting distribution for n(t, .)).
How to improve?
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

To improve the situation, one can use long jumps spatial dynamics,
i.e the generator of underlying random variable x(t) has the form

L(f ) =           (f (x + z) − f (x))q(z),                    q(z) = k
z=0                                        z∈Zd

Assume that                                        z
L( |z| )
q(z) ∼                 , 0<α<2                                         (3)
|z|2+α
In this situation
x(st)      law
1      −→
− − η(s), s ∈ [0, 1]
t            α      t→∞

where η(s) is a stable process with parameter α < 2. The random
variable x(t) is transient on Z2 for any α < 2 and even on Z1 but for
α < 1.
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

Theorem 6
For the critical reaction-diffusion equation on Z2 with condition
z
L( |z| )
q(z) ∼                  , 0<α<2
|z|2+α

on the underlying random variable , all results of the previous
theorems are valid (the existence of a limiting distribution for n(t, .)
with Carleman estimation and the FKG-property).

More realistic model of such type can include not only a migration of
particles but also an immigration ( depending on the local
conﬁguration).
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

Let’s discuss brieﬂy the relationships between the limiting
distributions in Models 2-4 and Poissonian statistics.
Theorem 7
If for ﬁxed diffusivity κ (or in the more general setting, ﬁxed
q(z) = k ),If the density of the population is increasing, the limiting
z=0
distribution (the law of π (x)) is close to Poissonian one. More
precisely

π (x) − ρ0 law
π (x) =          √       −−
− − → ξ(x)
ρ0    ρ0 →∞

where ξ(x) is an independent Gaussian N(0,1) random variable.

In contrast to ρ0 → 0, the limiting ﬁeld π (x) has a patches structure.
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

Model 5: Random media (RM) models in the population dynamics

Let’s consider on Zd , d ≥ 1, the reaction-diffusion equation under
condition that the mortality and the birth rate (µ, β)(x, ωm ) are
homogenous ergodic ﬁelds, ωm ∈ (Ωm , pm ). In the simplest case,
(µ, β)(x, ωm ), x ∈ Zd are i.i.d random vectors with

Eµ = Eβ( criticality )

v (x, ωm ) = β(x, ωm ) − µ(x, ωm ) = 0
For the ﬁrst quenched moment m1 (t, x, ωm ) = En(t, x, ωm ), we have
the Anderson parabolic equation:
∂m1 t,x
= κ∆x m1 t, x          + v (x, .)m1 (t, x) − Hm1 (t, x)
∂t                                                                                    (4)
m1 0, x = ρ0 > 0

Since Sp(H) = [−4dk , 0] ⊕ Suppv (x, ωm ) has a positive part, then the
ﬁrst moment and the population itself are growing exponentially. The
spatial distribution of the particles here is highly irregular(
intermittency).
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

If β, µ are time independent (stationary random media), the
limiting distribution exists only in the degenerated case
v (x, ωm ) = 0.
The case of nonstationary (time-dependent) random media is
more interesting and biologically realistic. Consider the following
model with renewal time T :
v (t, x) = v1 (x), t ∈ [0, T ]
v (t, x) = v2 (x), t ∈ [T , 2T ]
······
d
where vi (x), x ∈ Z , i = 1, 2, · · · are i.i.d random variable. For
appropriate conditions on κ and the distribution of vi (x, ωm ), one
can prove the uniform boundness of the annealed moments
< m1 >, < m2 > in time t, which is the indication that the limiting
distribution for n(t, .), t → ∞ exisit (compared with old works by
Carmona-Molchanov on the non-stationary Anderson model with
white noise potential). This work is in the progress.
Introduction Model 1:Supercritical reaction-diffusion process on Zd , d ≥ 1 Model 2 Continous contact model Model 3: Critical rea

Thank you!

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