On a nonlocal equation arising in population dynamics

Jerome Coville

Description

& $ % On a nonlocal equation arising in population dynamics by Jerome Coville (Paris 6) Workshop ACI Equations aux derivees partielles non lineaires et applications 17 & 18 June Rouen

• perthame-souganidis using viscosity


• nonlocal reaction-diffusion equation


• souganidis


• diffusion operator


• equations aux derivees partielles


• following equation


• analyze via probabilistic methods

Sujets

• Coville
• Équation aux dérivées partielles

Informations

• Publié par: pefav
• Nombre de lectures: 17
• Langue: English

Extrait

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On a nonlocal equation arising
in
population dynamics
by
Jer· ome? Coville (Paris 6)
Workshop ACI
Equations aux deriv· ees· partielles non lineaires· et applications
17 & 18 June
Rouen
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Nonlocal reaction-diffusion equation
We were interested in analyzing the following equation
n +
u ¡ (J?u¡u) =f(u) onR £R ; (1)
t
u(t = 0;x) =u (x) (2)
0
wheref is a given monostable nonlinearity andJ is a given kernel.
1
Recall thatf 2C ((0; 1)) is monostable if it satis es
0
† f(0) =f(1) = 0 andf (1)< 0
† f(s)> 0 fors2 (0; 1).
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From Modelling...
† ForK :=– , then equation (1) comes as a reduction of the following
0
problem:
@p(x;t)
n +
¡ (J?p¡p) =f(p;K?p) onR £R : (3)
@t
Equation (3) is commonly used in various models of spatio-temporel
development of populations (see Kendall, Dieckmann, Schumacher,
Weinberger ...).
† From a model of Bolker and Pacala (97), Fournier and Mel· ear· d (03),
derive and analyze via probabilistic methods a similar equation
@p
n +
=J?p¡p +p(‚¡K?p) onR £R : (4)
@t
† WithK =J, (4) was recently analyze by Perthame-Souganidis using
viscosity solution technics.
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Some Remarks
† Equation (1) was rst introduced by Kolmogorov-Pretrovskii and
Piskunov to derive the usual Fisher Equation.
@p(x;t)
n +
= ¢p +f(p) onR £R : (5)
@t
1 x
TakeJ (x) := “( ) where “ is even, smooth, with a compact
†
† †
support. Then
2 2
J ?u¡u =d† ¢u +o(† ):
†
† Ising Model and Neural Network see (Presutti, Triolo, Orlandi,
Souganidis, Bates, Fife,...)
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Regularization of discrete diffusion operator
1
For example, letJ (x) := (“ (x +h) + “ (x¡h)) whereh is a
† † †
2
discretization parameter.
8
J_{eps}(x)
eps=0.2
eps=0.1
eps=0.025
6
4
2
0
-2 -1 0 1 2
1
2
J ?u¡u! ((u(x¡h) +u(x +h)¡ 2u(x)) =:h ¢ u as†! 0:
† h
2
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Study of Travelling fronts
n¡1
Travelling Fronts: solutionsu(x;t) :=`(x:e +ct) wheree2S and
c2R.
New unknowns:c and` which are related by
8
0
>
J?`¡`¡c` +f(`) = 0 onR
<
(6)
`(x)! 0 asx!¡1
>
:
`(x)! 1 asx! +1
1
f
0.5
0
-10 -5 0 5 10
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Natural Questions ?
† Existence of front (`;c)?
† Uniqueness of the speedc?
† of the pro le` ?
† Regularity, monotonicity and asymptotic behavior of`?
† Is there any explicit formula for the speedc?
Known results
† Schumacher (80)
† Weinberger (82)
† Zinner-Harris ? Hudson (93)
† Perthame ? Souganidis (03)
† Carr ? Chmaj (03)
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Existence of travelling fronts
R
0
Let’s assume thatJ 2C (R);J(z)‚ 0; J(z)dz = 1 and
R
Z
+1
‚z
8‚> 0 J(z)e dz<1: (H1)
0
Theorem 1 C. & Dupaigne (03)
1
Assumef 2C (R) monostable and assume further thatJ is even. Then there
⁄ ⁄
exists realc > 0, such that8c‚c , there exists a increasing travelling front
⁄
(ˆ;c) solution of (6). Furthermore for all speedc<c no monotone travelling
front exists.
Theorem 2 C. & Dupaigne (04)
1 ⁄⁄ ⁄
Assumef 2C (R) monostable. Then there exists two realsc ‚c , such that
⁄⁄
8c‚c , there exists a increasing travelling front (ˆ;c) solution of (6).
⁄
Furthermore for all speedc<c no monotone travelling front exists.
⁄⁄ ⁄
Conjecture almost provedc =c .
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Exponentials behaviors & Formulae
0
† Assume f monostable then9C; „ ; „; ” > 0 so that
0
¡1 ¡„x ¡„ x
C e • 1¡ `• Ce as x! +1
¡1 ”x
and C e • ` as x!¡1:
0 0
† If furthermore f (0) > 0 then9C; ” so that
0
” x
`• Ce as x!¡1
† Variational formula:
‰ ?
J ? w(x)¡ w(x) + f(w(x))
⁄
c = min sup
0
w2X
w (x)
x2R
0
where X =fwjw > 0;w(¡1) = 0;w(+1) = 1g.
f(s)
0
† Exact formula when f (0) > :
s
p
⁄ 1 2 0
0
? KPP: c = min f (‚ + f (0))g = 2f (0):
‚>0
KPP
‚
R
⁄ 1 ‚z 0
? C.& Dupaigne (03): c • min f ( J(z)e dz¡ 1 + f (0))g = ?:
‚>0
‚ R
? Carr & Chmaj (03): If J has compact support and f = s(1¡ s) then
⁄
c = ?.
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Ideas to obtain fronts solution of (6)
† Analyzing the approximated problem below
8
00 0
†` +J?`¡`¡c` +f(`) = 0 onR
>
<
`(x)! 0 asx!¡1
(7)
>
:
`(x)! 1 asx! +1;
Theorem 3
Assume that†> 0,J is even and satis es (H1), then there exists a real
⁄ ⁄
c (†)> 0 such that8c‚c (†) there exists a monotone, smooth, travelling
c ⁄
front, denoted` solution of (7). Moreover8c<c (†) no smooth monotone
†
travelling front exists.
† Study the singular limit as†! 0.
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