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.

Th

an

kyou

very

mu

chfo

ryou

ratt

enti

on

!

.