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On a nonlocal equation arising
in
population dynamics
by
Jérôme Coville (Paris 6)
Workshop ACI
Equations aux dérivées partielles non linéaires et applications
17 & 18 June
Rouen
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Nonlocal reaction-diffusion equation
We were interested in analyzing the following equation
ut − (J ? u − u) = f (u)
on Rn × R+ ,
u(t = 0, x) = u0 (x)
(1)
(2)
where f is a given monostable nonlinearity and J is a given kernel.
Recall that f ∈ C 1 ((0, 1)) is monostable if it satisfies
• f (0) = f (1) = 0 and f 0 (1) < 0
• f (s) > 0 for s ∈ (0, 1).
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From Modelling...
• For K := δ0 , then equation (1) comes as a reduction of the following
problem:
∂p(x, t)
− (J ? p − p) = f (p, K ? p)
∂t
on Rn × R+ .
(3)
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 Méléard (03),
derive and analyze via probabilistic methods a similar equation
∂p
= J ? p − p + p(λ − K ? p)
∂t
on Rn × R+ .
(4)
• With K = J, (4) was recently analyze by Perthame-Souganidis using
viscosity solution technics.
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Some Remarks
• Equation (1) was first introduced by Kolmogorov-Pretrovskii and
Piskunov to derive the usual Fisher Equation.
∂p(x, t)
= ¢p + f (p)
∂t
on Rn × R+ .
(5)
Take J² (x) := 1² “( x² ) where “ is even, smooth, with a compact
support. Then
J² ? u − u = d²2 ¢u + o(²2 ).
• Ising Model and Neural Network see (Presutti, Triolo, Orlandi,
Souganidis, Bates, Fife,...)
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Regularization of discrete diffusion operator
For example, let J² (x) :=
discretization parameter.
1
2
(“² (x + h) + “² (x − h)) where h is a
8
J_{eps}(x)
eps=0.2
eps=0.1
eps=0.025
6
4
2
0
-2
J² ? u − u →
&
-1
0
1
2
1
((u(x − h) + u(x + h) − 2u(x)) =: h2 ¢h u
2
as ² → 0.
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Study of Travelling fronts
Travelling Fronts: solutions u(x, t) := φ(x.e + ct) where e ∈ S n−1 and
c ∈ R.
New unknowns: c and φ which are related by

0

+ f (φ) = 0 on R
J
?
φ
−
φ
−
cφ


(6)
φ(x) → 0
as x → −∞



φ(x) → 1
as x → +∞
1
φ
0.5
&
0
-10
-5
0
5
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Natural Questions ?
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• Existence of front (φ, c)?
• Uniqueness of the speed c?
• Uniqueness of the profile φ ?
• Regularity, monotonicity and asymptotic behavior of φ?
• Is there any explicit formula for the speed c?
Known results
• Schumacher (80)
• Weinberger (82)
• Zinner-Harris – Hudson (93)
• Perthame – Souganidis (03)
• Carr – Chmaj (03)
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Existence of travelling fronts
Let’s assume that J ∈ C (R), J(z) ≥ 0, R J(z)dz = 1 and
Z +∞
J(z)eλz dz < ∞.
∀λ > 0
0
R
(H1)
0
Theorem 1 C. & Dupaigne (03)
Assume f ∈ C 1 (R) monostable and assume further that J is even. Then there
exists real c∗ > 0, such that ∀c ≥ c∗ , there exists a increasing travelling front
(ψ, c) solution of (6). Furthermore for all speed c < c ∗ no monotone travelling
front exists.
Theorem 2 C. & Dupaigne (04)
Assume f ∈ C 1 (R) monostable. Then there exists two reals c∗∗ ≥ c∗ , such that
∀c ≥ c∗∗ , there exists a increasing travelling front (ψ, c) solution of (6).
Furthermore for all speed c < c∗ no monotone travelling front exists.
Conjecture almost proved c∗∗ = c∗ .
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Exponentials behaviors & Formulae
• Assume f monostable then ∃C, µ0 , µ, ν > 0 so that
C
−1 −µx
e
≤ 1 − φ ≤ Ce
and
−µ0 x
C −1 eνx ≤ φ
as x → +∞
as x → −∞.
• If furthermore f 0 (0) > 0 then ∃C, ν 0 so that
φ ≤ Ceν
• Variational formula:
c∗ = min sup
w∈X x∈R

0
x
as x → −∞
J ? w(x) − w(x) + f (w(x))
w0 (x)
ff
where X = {w|w 0 > 0, w(−∞) = 0, w(+∞) = 1}.
f (s)
:
s
minλ>0 { λ1 (λ2
∗
• Exact formula when f 0 (0) >
– KPP:
c∗KP P
=
– C.& Dupaigne (03):
p
+ f (0))} = 2f 0 (0).
R
1
c ≤ minλ>0 { λ ( R J(z)eλz dz − 1 + f 0 (0))} = γ.
0
– Carr & Chmaj (03): If J has compact support and f = s(1 − s) then
c∗ = γ.
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Ideas to obtain fronts solution of (6)
• Analyzing the approximated problem below
 00
0
²φ
+
J
?
φ
−
φ
−
cφ
+ f (φ) = 0 on R



φ(x) → 0 as x → −∞


 φ(x) → 1 as x → +∞ ,
(7)
Theorem 3
Assume that ² > 0, J is even and satisfies (H1), then there exists a real
c∗ (²) > 0 such that ∀c ≥ c∗ (²) there exists a monotone, smooth, travelling
front, denoted φc² solution of (7). Moreover ∀c < c∗ (²) no smooth monotone
travelling front solution exists.
• Study the singular limit as ² → 0.
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Construction of solutions of (7):
• We first study the following problem for a c ∈ R fixed:
Z +∞

00
0


²φ
+
J(x
−
y)φ(y)dy
−
φ
−
cφ
+ f (φ) = −hr (x) on (r, +∞)


r
φ(r) = θ




φ→1
x → +∞,
where r ∈ R, ² > 0, θ ∈ (0, 1), and hr (x) = θ
• For ² > 0 and r fixed, on can show:
Rr
−∞
J(x − y)dy.
– Existence and uniqueness of a solution φθr
– φθr is monotone increasing
– ∀ c ∈ [κ, +∞), ∃θr ∈ [0, 1) so that φθrr (0) = 12 .
• r → −∞ + Helly + a priori estimates⇒ ∀c ∈ [κ, +∞), there exists a
solution of (7).
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Perspectives
• Convergence in shape of a solution of (1) to a travelling wave
• Analysing
∂u
− (J ? u − u) = f (u, K ? u)
∂t
u(t = 0, x) = u0
for positive measure K 6= δ.
• Generalization of these results to nonlinear integro-differential
equations of the form
Z
∂u
−
J(x − y)S(u(y))dy = f (u)
∂t
R
with S increasing.
• Multi D and existence of other type of fronts (e.g. pulsating fronts).
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