ello -...
Transcript of ello -...
On
atw
osp
eci
es
chem
ota
xis
syst
em
J.I
gnaci
oT
ello
Un
ivers
idad
Poli
tecn
ica
de
Mad
rid
.S
pain
Bedle
wo,
June
12th
2015
Conte
nts
1-.
Intr
odu
ctio
n
2.-
On
esp
ecie
sp
arab
olic
-elli
pti
cch
emot
axis
syst
ems
(kn
own
resu
lts)
3-.
Tw
osp
ecie
sch
emot
axis
syst
ems,
wit
hM
.W
inkl
er†
4-.
Tw
osp
ecie
sch
emot
axis
syst
em-
Aca
seof
com
pet
itiv
eex
clu
sion
,
wit
hC
.S
tin
ner‡
and
M.
Win
kler†
5-.
Tw
osp
ecie
sch
emot
axis
syst
emw
ith
non
loca
lte
rms,
wit
hM
.N
egre
anu††
6.-
Th
eP
arab
olic
-OD
Esy
stem
wit
hM
.N
egre
anu††
†P
ader
bor
nU
niv
ersi
ty.
Pad
erb
orn.
Ger
man
y
‡T
echnis
che
Univ
ersi
ttK
aise
rsla
ute
rn,
Kai
sers
laute
rn.
Ger
man
y
††U
niv
ersi
dad
Com
plu
tense
de
Mad
rid.
Mad
rid.
Spai
n.
1In
troduct
ion
Ch
em
ota
xis
isth
eab
ility
ofm
icro
orga
nis
ms
tore
spon
dto
chem
ical
sign
als
bym
ovin
gal
ong
the
grad
ient
ofth
ech
emic
alsu
bst
ance
,ei
ther
tow
ard
the
hig
her
con
cent
rati
on(p
osit
ive
taxi
s)or
away
from
it(n
egat
ive
taxi
s).
Dic
tyos
teliu
md
isco
ideu
m(B
acte
ria)
On
esp
ecie
sC
hem
otax
issy
stem
s
–K
elle
ran
dS
egel
[197
0],
[197
1]
-Ja
ger
and
Lu
ckh
aus
[199
2],
Bile
r[1
995a
][1
995b
],[1
997]
Her
rero
-Vel
azqu
ez[1
996]
,[1
997]
,V
elaz
quez
[200
2][2
004]
T.
Nag
ai,
[199
5],
T.
Nag
ai,
T.
Sen
ba
and
K.
Yos
hid
a[1
997]
A.
Fri
edm
an,
A.
Ste
ven
s,B
.H
u,
M.
Mim
ura
,M
izog
uch
i,P
h.
Sou
-
ple
tN
aito
,S
uzu
ki,
Bla
nch
et,
Dol
bea
ult
,P
erth
ame,
Fas
ano
Car
rillo
,
Mas
mou
nd
i,H
orst
man
n,
M.
Pel
etie
r,T
.T
suji
kaw
a,K
.O
saki
,A
.Y
agi,
Pai
nter
,T
.C
iesl
ak,
C.
Mor
ales
-Rod
rigo
,C
.S
tin
ner
,A
.S
uar
ez,
M.
Win
kler
,D
.W
roze
k,A
.S
teve
ns,
Oth
mer
,D
.H
orst
man
n,
A.K
ub
o,M
.
Pel
etie
r,I.
Gu
erra
,E
.E
spej
o,C
.C
onca
,D
olb
eau
lt,
H.
Zaa
g,S
emb
a,
Miz
ogu
chi,
T.
Tsu
jika
wa,
Oku
da,
Cal
vez,
Lit
canu
,W
olan
ski,
J.I.
Dıa
z
Lev
ine,
Sle
eman
,M
.N
ilsen
-Ham
ilton
,T
.H
illen
,G
aje
wsk
y,Z
ach
aria
s,
B.
Hu
,M
.D
elga
do,
I.G
ayte
,P
aint
er,
Y.
Tao
,M
.N
egre
anu
,W
ang,
Cu
i,Iz
uh
ara.
M.
Fon
telo
s,M
.V
ela
etc
....
Dic
tyos
teliu
md
isco
ideu
m(s
oil-
livin
gam
oeb
a)
Inm
un
esy
stem
(wh
ite
blo
odce
lls)
-M
acro
ph
ages
cells
-N
eutr
oph
il(g
ranu
locy
te)
Mor
ph
ogen
esis
:th
ep
roce
ssof
form
atio
nof
the
emb
ryo.
See
Mer
kin
-Nee
dh
am-S
leem
an[2
005]
,B
olle
nbac
het
al[2
007]
,
An
giog
enes
is
-G
row
thof
tum
ou
rs.
See
An
der
son
and
Ch
apla
in[1
998]
-F
orm
atio
nof
emb
ryos
-H
ealin
gof
skin
wou
nd
.
Ast
rop
hysi
csan
dgr
avit
atio
nal
inte
ract
ion
ofp
arti
cles
.
See
Bile
r[1
995]
,B
iler-
Hilh
orst
-Nad
ziej
a[1
994]
2T
wo
speci
es
chem
ota
xis
syst
em
s
We
con
sid
era
two
spec
ies
syst
em
ut
=d
1∆u
︸︷︷︸
diff
usi
on
−χ
1∇·(u∇w
)︸
︷︷︸
chem
otax
is
+µ
1u
(1−u−a
1v
)︸
︷︷︸
pro
life
rati
onan
dco
mp
etit
ion
v t=d
2∆v
︸︷︷︸
diff
usi
on
−χ
2∇·(v∇w
)︸
︷︷︸
chem
otax
is
+µ
2v
(1−a
2u−v
)︸
︷︷︸
pro
life
rati
onan
dco
mp
etit
ion
εwt
=dw
∆w
︸︷︷
︸d
iffu
sion
−λw ︸︷︷︸
deg
rad
atio
n
+g
(u,v
)︸
︷︷︸
pro
du
ctio
n
Neu
man
nb
oun
dar
yco
nd
itio
ns
and
app
rop
riat
ein
itia
ld
ata
inΩ
.
ε=
0,g
(u,v
)=u
+v,
0≤ai<
1fo
ri
=1,
2
We
assu
me: 2(χ
1+χ
2)
+a
1µ
2<µ
1an
d2(χ
1+χ
2)
+a
2µ
1<µ
2.
Th
elim
itca
se
•χ
1=χ
2=
0is
aco
mp
etit
ive
syst
emof
two
equ
atio
ns
wel
lst
ud
ied
inth
e
liter
atu
re.
See
Pao
[198
1],
Cos
ner
and
Laz
er[1
984]
ut
=d
1∆u
+u
(e1−b 1u−c 1v
)
v t=d
2∆v
+v
(e2−b 2u−c 2v
)
+N
eum
ann
bou
nd
ary
con
dit
ion
s.U
nd
eras
sum
pti
one 1 c 1>e 2 c 2,
e 2 b 2>e 1 b 1
u−→
e 1c 2−c 1e 2
b 1c 2−c 1b 2,
v−→
b 1e 2−e 1b 2
b 1c 2−c 1b 2.
Inou
rca
se,
the
assu
mp
tion
iseq
uiv
alen
tto
0<ai<
1(i
=1,
2).
•µ
1=µ
2=
0.E
spej
o-A
ren
as,
Ste
ven
san
dV
elaz
quez
[200
9]-[
2010
]
Sim
ult
aneo
us
and
non
-sim
ult
aneo
us
blo
wu
pof
bot
hsp
ecie
sd
epen
din
g
onth
ep
aram
eter
san
din
itia
lm
ass.
See
also
Esp
ejo-
Are
nas
and
Con
ca
[201
2],
Bile
r-E
spej
o-G
uer
ra[2
013]
.
Ste
ad
yst
ate
s
0=
∆u−χ
1∇·(u∇w
)+µ
1u
(1−u−a
1v
)in
Ω,
0=
∆v−χ
2∇·(v∇w
)+µ
2v
(1−a
2u−v
)in
Ω,
0=
∆w−λw
+u
+v
inΩ,
∂u∂n
=∂v∂n
=∂v∂n
=0,
in∂
Ω
Th
eu
niq
ue
pos
itiv
ean
db
oun
ded
stea
dy
stat
esis
give
nby
u?≡
1−a
1
1−a
1a
2,
v?≡
1−a
2
1−a
1a
2.
An
Au
xil
iary
Syst
em
of
OD
Es
u′=u
[µ1−
(µ1−χ
1)u−χ
1u
+χ
1v−
(χ1
+µ
1a
1)v
],t>
0,
u′=u
[µ1−χ
1u−
(µ1−χ
1)u−
(χ1
+µ
1a
1)v
+χ
1v
],t>
0,
v′=v
[µ2
+χ
2u−
(χ2
+µ
2a
2)u−
(µ2−χ
2)v−χ
2v
],t>
0,
v′=v
[µ2−
(χ2
+µ
2a
2)u
+χ
2u−χ
2v−
(µ2−χ
2)v
],t>
0,
wit
hin
itia
lco
nd
itio
ns
u(0
)=u
0,
u(0
)=u
0,
v(0
)=v
0,
and
v(0
)=v
0.
Th
est
ead
yst
ates
ofth
eO
DE
ssy
stem
are
give
nby
u?
=u?
=u?≡
1−a
1
1−a
1a
2,
v?
=v?
=v?≡
1−a
2
1−a
1a
2
We
anal
ize
the
syst
emof
OD
Es.
un
der
assu
mp
tion
s
0<u
0<u∗<u
0<∞,
0<v
0<v∗<v
0<∞,
Ste
p1a.-
0<u<u<∞
fort∈
(0,∞
);
Ste
p1b
.-0<v<v<∞
fort∈
(0,∞
);
Ste
p2a.-
0<u<u∗<u<∞
fort∈
(0,∞
);
Ste
p2b
.-0<v<v∗<v<∞
fort∈
(0,∞
);
Ste
p3.-
limt→∞|u−u|+|v−v|−→
0.
Idea
of
Ste
p3.
d dt
logu u
=ut u−ut u
=−
(µ1−
2χ1)(u−u
)+
(2χ
1+µ
1a
1)(v−v
)
and d dt
logv v
=(2χ
2+µ
2a
2)(u−u
)−
(µ2−
2χ2)(v−v
)fo
ral
lt>
0.
We
add
bot
hto
obta
ind dt
( logu u
+lo
gv v
) =(−µ
1+
2(χ
1+χ
2)
+µ
2a
2)(u−u
)+
(−µ
2+
2(χ
1+χ
2)
+µ
1a
1)(v−v
)
for
ε:=
minµ
1−
2(χ
1+χ
2)−µ
2a
2,µ
2−
2(χ
1+χ
2)−µ
1a
1
then
d dt
logu u
+lo
gv v
≤−ε(u−u
)−ε(v−v
)fo
ral
lt>
0,
Th
isfir
sten
tails
that
logu u≤
logu
0
u0
+lo
gv
0
v0
:=c 0
for
allt>
0,
and
logu∗ u≤c 0
=⇒
u≥u∗ e−c 0≥u∗u
0
u0
v0
v0>
0fo
ral
lt>
0.
Inth
esa
me
way
v≥v∗u
0
u0
v0
v0>
0fo
ral
lt>
0.
By
the
Mea
nV
alu
eT
heo
rem
u(t
)−u
(t)
=eξ
1(t
)(
logu
(t)−
logu
(t))
v(t
)−v
(t)
=eξ
2(t
)(
logv
(t)−
logv
(t)).
and
d dt
logu u
+lo
gv v
≤−ε 0
logu u
+lo
gv v
fo
ral
lt>
0
isva
lidw
ith
ε 0=εu
0
u0
v0
v0
minu∗ ,v∗ .
Aft
erro
uti
ne
com
pu
tati
ons
0<
logu u≤e−
ε 0t
logu
0v
0
u0v
0,
0<
logv v≤e−
ε 0t
logu
0v
0
u0v
0,
for
allt>
0
and
ther
eby
show
sth
at
|u(t
)−u
(t)|
+|v
(t)−v
(t)|→
0ast→∞.
wh
ich
end
sth
ean
alys
isof
the
OD
Esy
stem
.
Com
pari
son
:P
DE
syst
em
-O
DE
syst
em
Rea
ctio
nD
iffu
sion
Sys
tem
s:R
ecta
ngl
eM
eth
odse
eP
ao[1
981]
.
We
pro
veth
at,
un
der
assu
mp
tion
0<u
0≤u
0(x
)≤u
0an
d0<v
0≤v 0
(x)≤v
0fo
ral
lx∈
Ω.
we
hav
e
u≤u≤u
and
v≤v≤v.
We
con
stru
ctth
efu
nct
ion
s:
U(x,t
):=u
(x,t
)−u
(t),
U(x,t
):=u
(x,t
)−u
(t),
V(x,t
):=v
(x,t
)−v
(t),
V(x,t
):=v
(x,t
)−v
(t)
wh
ich
sati
sfy
Ut−
∆U
+χ
1∇U∇w
=U
[µ1
+(χ
1−µ
1)(u
+u
)+
(χ1−µ
1a
1)v−χ
1λw
]
+χ
1uV−µ
1a
1uV
+χ
1(u
+v−λw
).
we
mu
ltip
lybyU
+an
daf
ter
inte
grat
ion
d dt1 2
∫ ΩU
2 ++
∫ Ω|∇U
+|2
=−χ
1 2
∫ Ω∇U
2 +∇w
+∫ Ωb(x,t
)U2 +
+χ
1
∫ ΩuVU
+−µ
1a
1
∫ ΩuVU
++χ
1
∫ Ω(u
+v−λw
)U+
we
obta
insi
mila
rex
pre
ssio
ns
forU
,V
,V
.
Aft
erm
any
rou
tin
eco
mp
uta
tion
sw
eob
tain
,fo
ran
yt<T
d dt
∫ Ω
( U2 +
+U
2 −+V
2 ++V
2 +
) ≤k
(T)∫ Ω
( U2 +
+U
2 −+V
2 ++V
2 −
)
Gro
nwal
l′ sle
mm
agi
ves
U+
=U−
=V
+=V−
=0.
3.-
Com
peti
tive
Excl
usi
on
(Sti
nn
er-
T-w
inkle
r2013)
We
con
sid
erth
esy
stem
ut
=d
1∆u
︸ ︷︷︸
diff
usi
on
−χ
1∇·(u∇w
)︸
︷︷︸
chem
otax
is
+µ
1u
(1−u−a
1v
)︸
︷︷︸
pro
life
rati
onan
dco
mp
etit
ion
v t=d
2∆v
︸︷︷︸
diff
usi
on
−χ
2∇·(v∇w
)︸
︷︷︸
chem
otax
is
+µ
2v
(1−a
2u−v
)︸
︷︷︸
pro
life
rati
onan
dco
mp
etit
ion
0=dw
∆w
︸︷︷
︸d
iffu
sion
−λw ︸︷︷︸
deg
rad
atio
n
+ku
+v
︸︷︷
︸p
rod
uct
ion
Neu
man
nb
oun
dar
yco
nd
itio
ns
and
app
rop
riat
ein
itia
ld
ata
inΩ
.
We
defi
ne
the
new
par
amet
ers
q 1:=
χ1
µ1
and
q 2:=
χ2
µ2.
We
con
sid
erth
eas
sum
pti
ons
a1>
1>a
2
k,q
1an
dq 2
are
non
neg
ativ
ean
dq 1
=χ
1µ
1≤a
1,q
2=
χ2µ
2<
1 2an
d
kq 1
+m
axq
2,a
2−a
2q 2
1−
2q2,kq 2−a
2q 2
1−
2q2<
1.
Th
eas
sum
pti
ons
are
equ
ival
ent
tokq 1
+q 2<
1an
d kq 1
+(2−a
2)q
2+a
2−
2kq 1q 2<
1ifkq 2<a
2,
kq 1
+(2−a
2+k
)q2−
2kq 1q 2<
1ifkq 2≥a
2.
Th
en,
we
hav
eth
at
u−→
0,v−→
1
Not
ice
that
the
assu
mp
tion
sin
the
pro
toty
pic
alca
seχ
1=χ
2,µ
1=µ
2ar
e
red
uce
dto
χ µ<
2+k−a
2−√
(k+
2−a
2)2−
8k(1−a
2)
4kifa
2>kq
2+2k−a
2−√
(2k+
2−a
2)2−
8k
4kifa
2≤kq.
Ifw
em
oreo
ver
hav
ek
=1
then
χ µ<
4−a
2−√
8−8a
2+a
2 24
ifa
2≤q
1−a
22
ifa
2>q.
Inth
elim
itca
sek
=0,
χ µ<
1−a
2
2−a
2
Th
eb
ord
erlin
eca
sea
2=
0re
ads
χ µ<
1 2
alre
ady
fou
nd
inT
-win
kler
[200
7]
Non
-loca
lte
rms
(Neg
rean
u-T
[201
3])
ut−
∆u
=−χ
1∇·(u∇w
)+u( a 0−a
1u−a
2v−a
3
∫ Ωu−a
4
∫ Ωv) ,
v t−
∆v
=−χ
2∇·(v∇w
)+v( b 0−b 1u−b 2v−b 3
∫ Ωu−b 4
∫ Ωv) ,
−∆w
+λw
=f
+k
1u
+k
2v,
wit
hth
eh
omog
eneo
us
Neu
man
nb
oun
dar
yco
nd
itio
ns
∂u
∂ν
=∂v
∂ν
=∂w ∂ν
=0,
x∈∂
Ω,t>
0,
and
init
ial
dat
a u(x,0
)=u
0(x
),v
(x,0
)=v 0
(x),
x∈
Ω.
“G
lob
al
Com
peti
tion
”
Mem
ber
sof
one
spec
iesu
com
pet
efo
ra
limit
edre
sou
rcez
sati
sfyi
ng
ut−
∆u
+µu
=zu,
x∈
Ω,
t>
0
KP
P-F
ish
ereq
uat
ionz
=(1−u
).
We
assu
me
that
the
reso
urc
esd
iffu
sean
dd
egra
de
wit
hla
rge
diff
usi
onco
ef-
ficie
ntε−
1
−1 ε∆
z ε+α
1z ε
=1−α
2u
x∈
Ω
∂z ε ∂~n
=0.
Aft
erIn
tegr
atio
nw
eh
ave
∫ Ωz ε
=|Ω|
α1−α
2
α1
∫ Ωu.
Mu
ltip
lyin
gby
uan
daf
ter
inte
grat
ion
and
than
ksto
You
ng
ineq
ual
ity
we
hav
e∫ Ω|∇z ε|2 dx≤ε(
1+c(α
1,α
2)∫ Ωu
2)→
0asε→
0.
Th
en
z ε−→
constant
:=|Ω|
α1−α
2
α1
∫ Ωu.
the
equ
atio
n
ut−
∆u
+µu
=zu,
x∈
Ω,
t>
0
isre
pla
ceby
ut−
∆u
+µu
=µu
(1−a
3
∫ Ωu
),x∈
Ω,
t>
0.
Ifα
2=α
2(x
)th
en,
the
non
loca
lte
rm∫ Ωα
3(x
)u.
For
the
two
spec
ies
chem
otax
issy
stem
we
con
sid
er
−1 ε∆
z ε+α
1z ε
=1−α
2u−α
3v
x∈
Ω+
NBC
z ε→
α−a
3
∫ Ωu−a
4
∫ Ωv.
Un
der
assu
mp
tion
s∫ ∞ 0|s
up
x∈Ωf−
inf
x∈Ωf|≤
C0<∞.
χ1,χ
2,k
1,k
2,ai,b i>
0,fo
ri
=1,
2,
ai∈IR,b i∈IR,
fori
=3,
4
a1>
2k1(χ
1+χ
2)
+b 1
+|b
3|+|a
3|
andb 2>
2k2(χ
1+χ
2)
+a
1+|a
4|+|b
4|
we
obta
inth
eas
ymp
toti
cb
ehav
ior
for
pos
itiv
ein
itia
ld
ata
u(·,t)−→
u∗≡
a0(b
2+b 4
)−b 0
(a2
+a
4)
(b2
+b 4
)(a
1+a
3)−
(b1
+b 3
)(a
2+a
4)
v(·,t)−→
v∗≡
a0(b
1+b 3
)−b 0
(a1
+a
3)
(b1
+b 3
)(a
2+a
4)−
(b2
+b 4
)(a
1+a
3).
Tw
osp
eci
es
Para
boli
c-O
DE
chem
ota
ctic
syst
em
ut
=d
1∆u
︸︷︷︸
diff
usi
on
−∇
(uχ
1(w
)∇w
)︸
︷︷︸
chem
otax
is
v t=d
2∆v
︸ ︷︷︸
diff
usi
on
−∇
(vχ
2(w
)∇w
)︸
︷︷︸
chem
otax
is
wt
=h
(u,v,w
)
We
assu
me
χi,h∈W
1,∞
loc
(IR
2 +×IR
),χi>
0.
∂h
∂u≥ε u>
0an
d∂h
∂v≥ε v>
0
∂h
∂w<
0.
Th
ere
exis
tsw∗
such
that
h(u∗ ,v∗ ,w∗ )
=0
wh
ere
u∗
=1 |Ω|∫ Ω
u0dx,
v∗
=1 |Ω|∫ Ω
v 0dx.
Con
sequ
entl
y(u∗ ,v∗ ,w∗ )
isa
con
stan
tst
atio
nar
yso
luti
onof
the
syst
em.
Glo
bal
exis
ten
ceof
solu
tion
s.
We
assu
me
−h
(0,0,w
)≤
ki
χi(w
)fo
rso
meki>
0,
0<k
0i≤χi(w
)e
∫ w wχi(s)ds
forw>w,
fork
0i>
0,w
ithi
=1,
2.T
her
eex
istsu
andv
such
that
h(u,v,w
)≥
0,h
(u,v,w
)≤
0,fo
r0≤u≤u,
0≤v≤v,
wh
ere
u:=f 1
(w)
max
k1
(εuk
01)−
1,‖u
0‖ L∞
(Ω) ,
v:=f 2
(w)
max
k2
(εvk
02)−
1,‖v 0‖ L∞
(Ω) ,
forf i
defi
ned
by
f i(w
)=e∫ w w
χi(s)ds
i=
1,2.
Th
en,b
yu
sin
gan
iter
ativ
em
eth
od(A
likak
os-M
osh
erit
erat
ion
)w
eh
ave
glob
al
bou
nd
edn
ess.
Sta
bil
ity
of
the
hom
ogen
eou
sst
ead
yst
ate
s.
We
con
sid
er
The
reex
istsα∈
(0,1
)su
chth
at
αhw
+uhuχ
1+vhvχ
2<
0an
d2√
1−αhw
+uhvχ
1+vhuχ
2<
0.
Usi
ng
aL
yap
un
ovfu
nct
ion
al,
we
get
that
the
stea
dy
stat
eis
glob
ally
asym
p-
toti
cally
stab
le.
Not
ice
that
the
pre
viou
sas
sum
pti
ons
are
sati
sfied
,fo
rin
stan
cefo
r
h(u,v,w
)=u
+v−w,
χi(w
)=
γi
1+γiw
forγi<
1/4
Refe
ren
ces:
1.-
JIT
,M.W
inkl
er.
Sta
bili
zati
onin
atw
o-sp
ecie
sch
emot
axis
syst
emw
ith
logi
sitc
sou
rce.
Non
linea
rity
25
(201
2)14
13-1
425.
2.-
Ch
rist
ian
Sti
nn
er,
JIT
,M
ich
ael
Win
kler
.C
ompe
titi
veex
clu
sion
ina
two-
spec
ies
chem
otax
ism
odel
.J.
Mat
h.
Bio
logy
,V
olu
me
68,
Issu
e7
(201
4)p
p16
07-1
626.
3.-
Mih
aela
Neg
rean
u,
JIT
.O
na
two
spec
ies
chem
otax
ism
odel
wit
hsl
ow
chem
ical
diff
usi
on.
SIA
MJ.
Mat
h.
An
al.
46-6
(201
4),
pp
.37
61–3
781
4.-
Mih
aela
Neg
rean
u,
JIT
.A
sym
ptot
icst
abil
ity
ofa
two
spec
ies
chem
o-
taxi
ssy
stem
wit
hn
on-d
iffu
sive
chem
oatt
ract
ant.
J.D
iffer
enti
alE
qua-
tion
s258
(201
5)15
92–1
617.