1
Multi-species systems in biology:
cross-diffusion and hidden gradient-flow
structure
Ansgar Jüngel
Vienna University of Technology, Austria
www.jungel.at.vu
Joint work with L. Chen (Mannheim); X. Chen (Beijing); E. Daus, I. Stelzer (Vienna)
•
•
•
•
Introduction
Some examples
Boundedness-by-entropy method
Examples revisited
Tumor cells in breast tissue
Introduction
What are multi-species systems?
Examples:
• Wildlife populations
• Tumor growth
• Gas mixtures
• Lithium-ion batteries
• Population herding
→ Nature is composed of
multi-species systems
+
oxygen
separator
Li+
Al
Li+
Number of publications
graphite
–
Cu
2
Introduction
How to model multi-species systems?
Microscopic models:
• Discrete-time Markov chains:
matrix-based models
• Continuous-time Markov chains:
species move to neighboring cells
with transition rate p±
j (ui)
Continuum models:
• Stochastic differential equations: Brownian motion
represents erratic particle motion
• Kinetic differential equations: distribution function
depends also on trait parameters (age, size, maturity, etc.)
• Diffusive differential equations: deterministic dynamics for
particle densities ← Focus of this talk
3
Introduction
From microscopic to diffusive models: single species
Simplification: Single species, one-dimensional
• Particle number u(xi) at ith cell, transition rate p > 0
• Master equation: time variation = incoming − outgoing
∂tu(xi) = p(u(xi−1) + u(xi+1)) − 2pu(xi)
• Taylor expansion: (h = grid size)
u(xi±1) − u(xi) = ±h∂xu(xi) + 21 h2∂x2u(xi) + O(h3)
• Diffusion scaling: t 7→ t/h2 ⇒ ∂t
h2 ∂ t
h2∂tu(xi) = p(u(xi−1) − u(xi)) + p(u(xi+1) − u(xi))
= ph2∂x2u(xi) + O(h3)
• Limit h → 0 gives ∂tu(x) = p∂x2u(x) (heat equation)
Rigoros limits: De Masi, Lebowitz, Sinai, Spohn, ... > 1980
4
Introduction
From microscopic to diffusive models: multiple species
Generalizations:
• Master equation for particle number uj (xi) at ith cell:
−
+
−
+
p
u
(x
)
−
(p
u
(x
)
+
p
∂tuj (xi) = p+
j,i)uj (xi)
j,i
j,i+1 j i+1
j,i j i−1
• Taylor expansion, diffusion scaling and limit h → 0 leads
P
to system of diffusion equations ∂tuj = ∂x( k Ajk (u)∂xuk )
• Multi-dimensional case analogous
Example: Two-species system
• Transition rates pj (u) = aj0 + aj1u1 + aj2u2, j = 1, 2
• Diffusionmatrix A = (Ajk (u))
a10 + 2a11u1 + a12u2
a12u1
A=
a21u2
a20 + a21u1 + 2a22u2
Main aim: Analyze systems ∂tu − div (A(u)∇u) = f (u)
5
Overview
• Introduction
• Some examples
• Boundedness-by-entropy method
• Examples revisited
6
7
Some examples
∂tu − div (A(u)∇u) = f (u), x ∈ Ω, t > 0
Example 1: Population dynamics
a10 + 2a11u1 + a12u2
a12u1
A=
a21u2
a20 + a21u1 + a22u2
+ no-flux boundary and initial conditions for u1 and u2
case(a)
• Shigesada-Kawasaki-Teramoto 1979
• Derivation from lattice model
• Population densities: u1, u2,
Lotka-Volterra term: f (u)
• Cross-diffusion induces segregation
• Also includes drift terms
→ A(u) generally not symmetric
positive definite
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Species 1:
cross diffusion large− u(x)
30
−− v(x)
0
cross diffusion
small
25
0.1
10
20
Species 2:
cross diffusion
large
15
10
10
0.1
cross diffusion
small
0
5
0
0.5
1
1.5
2
2.5
3
Some examples
∂tu − div (A(u)∇u) = f (u), x ∈ Ω, t > 0
Example 2 : Tumor growth
2
−2βu1u2(1 + θu1)
2u1(1 − u1) − βθu1u2
A(u) =
−2u1u2 + βθu22(1 − u2) 2βu2(1 − u2)(1 + θu1)
• Derived by Jackson-Byrne 2002
• Derivation from mass balance and force
balance equations (avascular growth)
• Volume fractions of tumor cells u1,
extracellular matrix (ECM) u2,
water u3 = 1 − u1 − u2
• Symmetry assumption: x ∈ Ω = (0, 1)
• Pressure parameters: β ≥ 0, θ ≥ 0
→ A(u) gener. not pos. definite! Expect that 0 ≤ u1, u2 ≤ 1
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Some examples
∂tu − div (A(u)∇u) = f (u), x ∈ Ω, t > 0
Example 3 : Multicomponent gas mixtures
X
∂tui − div Ji = fi(u), ∇ui =
dij (uj Ji − uiJj )
j6=i
• Proposed by Maxwell 1866 & Stefan 1871
• Derivation from Boltzmann eq. for simple
mixtures: Boudin-Grec-Salvarani 2015
• Ideal mixture of N + 1 gas components
• Molar fractions u = (u1, . . . , uN +1) with
PN +1
total molar fraction i=1 ui = 1
→ (dij ) generally not positive definite, inversion ∇ui ↔ Ji
necessary (and nontrivial); expect that 0 ≤ ui ≤ 1
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Some examples
∂tu − div (A(u)∇u) = f (u), x ∈ Ω, t > 0
Main features:
• Cross-diffusion: Diffusion matrix A(u) non-diagonal
• Matrix A(u) may be neither symmetric nor pos. definite
• Variables ui expected to be bounded from below/above
Objectives:
• Global-in-time existence of weak solutions
• Positivity and boundedness of solution if physically expected
• Large-time behavior, design of stable numerical schemes
Mathematical difficulties:
• No general theory for diffusion systems
• Generally no maximum principle, no regularity theory
• Lack of positive definiteness ⇒ global existence nontrivial
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Some examples
∂tu − div (A(u)∇u) = 0, x ∈ Ω ⊂ Rd, t > 0
Some previous results: Global existence if . . .
• Growth conditions on nonlinearities (Ladyženskaya ... 1988)
• Control on W 1,p norm with p > d (Amann 1989)
• Invariance principle holds (Redlinger 1989, Küfner 1996)
• Positivity, mass control, diagonal A(u) (Pierre-Schmitt ’97)
Unexpected behavior:
• Finite-time blow-up of Hölder solutions (Stará-John 1995)
• Weak solutions may exist after L∞ blow-up (Pierre 2003)
• Cross-diffusion may lead to pattern formation (instability)
or may avoid finite-time blow-up (Hittmeir-A.J. 2011)
Special structure needed for global existence theory:
gradient-flow or entropy structure
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Some examples
Entropy: Measure of molecular disorder or energy dispersal
• Introduced by Clausius (1865) in thermodynamics
• Boltzmann, Gibbs, Maxwell: statistical interpretation
• Shannon (1948): concept of information entropy
Entropy in mathematics: ∼ convex Lyapunov functional
• Hyperbolic conservation laws (Lax), kinetic theory (Lions)
• Relations to stochastic processes (Bakry, Emery) and
optimal transportation (Carrillo, Otto, Villani)
Gradient flow: ∂tu = −gradH|u on differential manifold
• Example: Rd with Euclidean structure ⇒ ∂tu = −H ′(u)
• Gradient flow of entropy
w.r.t. Wasserstein distance (Otto)
R
• Entropy H(u) = u log udx: ∂tu = div (u∇H ′(u)) = ∆u
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Overview
• Introduction
• Some examples
• Boundedness-by-entropy method
• Examples revisited
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Boundedness-by-entropy method
Main assumption: ∂tu − div (A(u)∇u) = f (u) possesses
formal gradient-flow structure
δH
= f (u),
∂tu − div B∇
δu
R
where B positive semi-definite, H(u) = Ω h(u)dx entropy
′
Equivalent formulation: δH
≃
h
(u) =: w (entropy variable)
δu
′′
−1
∂tu − div (B∇w) = f (u), B = A(u)h (u)
Consequences:
• H is Lyapunov
Z functional if f =Z0:
dH
=
∂tu · |h′{z
(u)} dx = − ∇w : B∇wdx ≤ 0
dt
Ω
Ω
=w
• L∞ bounds for u: Let h′ : D → Rn (D ⊂ Rn) be invertible
⇒ u = (h′)−1(w) ∈ D (no maximum principle needed!)
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Boundedness-by-entropy method
Example: no volume filling
• Mass densities u1, u2 satisfying ui > 0
P2
• Entropy: h(u) = i=1 ui(log ui − 1), u ∈ D = (0, ∞)2
• Entropy variable: w = h′(u) or u = (h′)−1(w)
∂h
wi =
= log ui, ui = ewi > 0
∂ui
Example: volume filling
• Mass fractions ui satisfying u1 + u2 + u3 = 1
P3
• Entropy: h(u) = i=1 ui(log ui − 1), u3 = 1 − u1 − u2,
u ∈ D = {(u1, u2) : u1, u2 > 0, u1 + u2 < 1}
• Entropy variable: w = h′(u) or u = (h′)−1(w)
ui
ewi
∂h
= log , ui =
wi =
∈ (0, 1)
w
w
∂ui
u3
1+e 1+e 2
→ Thermodynamics: wi = chemical potential
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Boundedness-by-entropy method
General global existence result
∂tu − div (A(u)∇u) = f (u) in Ω, ∇u · ν|∂Ω = 0, u(0) = u0
R
Entropy-dissipation
identity for entropy ZH(u) = Ω h(u)dx:
Z
dH
+ ∇u : h′′(u)A(u)∇udx =
f (u) · h′(u)dx
dt
Ω
Ω
Assumptions: Let D be bounded.
(H1) ∃ h ∈ C 2(D ; [0, ∞)) with invertible h′ : D → Rn
(H2) ∀u: h′′(u)A(u) ≥ diag(ai(ui)), where ai(ui) ∼ ui2mi−2
P
mi 2
′′
and mi ≥ 0 (yields ∇u : h A∇u ∼ i |∇ui | )
(H3) A continuous, ∀u: f (u) · h′(u) ≤ C(1 + h(u))
Theorem: (A.J., Nonlinearity 2015)
Let (H1)-(H3), u0 ∈ L1 ∩ D . Then ∃ global weak solution
u(x, t) ∈ D , u ∈ L2loc(0, ∞; H 1), ∂tu ∈ L2loc(0, ∞; (H 1)′)
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Overview
• Introduction
• Some examples
• Boundedness-by-entropy method
• Examples revisited
17
Example 3 : Multicomponent gas mixtures
∂tui − div Ji = fi(u),
∇ui =
P
j6=i dij (uj Ji
− ui J j )
• Write ∇u = A(u)J, u ∈ RN +1 but not invertible
PN
• Replace uN +1 = 1 − i=1 ui ⇒ u∗ = A0(u∗)J ∗, u∗ ∈ RN
−1
∗
∗
∗
• Invert J ∗ = A−1
∇u
⇒
∂
u
−
div
(A
∇u
) = f (u)
t
0
0
PN +1
∗
(H1) Entropy functional: h(u ) = i=1 ui(log ui − 1)
PN
∗
∗
N
u ∈ D = {u ∈ R : 0 < ui < 1,
i=1 ui < 1}
(H2) Entropy-dissipation inequality:
Z
N
+1 Z
X
√ 2
d
∗
|∇ ui| dx ≤ C2
h(u )dx + C1
dt Ω
i=1 Ω
Theorem: (A.J.-Stelzer, SIMA 2014)
PN +1
Assume (dij ) symm., i=1 fi(u) log ui ≤ C. Then
√
∃ weak solution ui ∈ L2loc(0, ∞; H 1(Ω)), ui ∈ D
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Example 1 : Population dynamics model
∂tu − div
(A(u)∇u)
=
f
(u)
+
homog.
Neumann
b.c.
d1 + 2α1u1 + u2
u1
A(u) =
u2
d2 + 2α2u2 + u1
(H1) Entropy
functional:
Z
Z
H(u) =
h(u)dx =
u1(log u1 − 1) + u2(log u2 − 1) dx
Ω
Ω
h′(u) = (log u1, log u2) on unbounded D = (0, ∞)2
(H2) Entropy-dissipation inequality: Let di > 0, αi ≥ 0
Z
Z
2
2
X
X
√ 2
dH
fi log uidx
(2di|∇ ui| + αi|∇ui|2)dx ≤
+2
dt
i=1 Ω
i=1 Ω
Theorem: (L. Chen-A.J., SIMA 2004)
√
∃ global nonnegative weak solution ui ∈ L2loc(0, ∞; H 1)
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Example 1 : Population dynamics model
Generalization: nonlinear pi(u)
Macroscopic limit of random-walk on lattice:
!
∂p1
∂p1
p1(u) + u1 ∂u
(u)
u
(u)
1
∂u
1
2
A(u) =
∂p2
∂p2
u2 ∂u1 (u)
p2(u) + u2 ∂u2 (u)
• pi linear: L. Chen-A.J. 2004
• pi sublinear: Desvillettes-Lepoutre-Moussa 2014
• pi superlinear: pi(u) = ai0 + ai1us1 + ai2us2 (i = 1, 2),
entropy density: h(u) = a21us1 + a12us2
Theorem: (A.J., Nonlinearity 2015)
Let 1 < s < 4 and (1 − 1s )a12a21 ≤ a11a22, H(u0) < ∞.
s/2
Then ∃ nonnegative weak solution ui ∈ L2loc(0, ∞; H 1(Ω))
• pi superlinear, s > 1: Desvillettes-Lepoutre-Moussa-Trescases 2015
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Example 1 : Population dynamics model
Generalization: more than two species
∂tu − div (A(u)∇u) = f (u) + homog. Neumann b.c.
Aij (u) = (ai0 + ai1u1 + · · · + ainun)δij + aij ui
R
R Pn
• Entropy: H[u] = Ω h(u)dx = Ω i=1 πiui(log ui − 1)
• Key assumption: πiaij = πj aji (detailed balance), πi > 0
• Detailed balance ⇔ (πi) reversible measure ⇔
h′′(u)A(u) symmetric ⇒ entropy H[u(t)] decreases ∀t
• Detailed balance not satisfied: aii “large” ⇒
H[u(t)] decreases, otherwise ∃ u(0) s.t. H[u(t)] increases
Theorem: (X. Chen-Daus-A.J., in progress 2016)
Let aij > 0 and detailed balance hold. Then ∃ nonnegative
1/2
weak solution ui ∈ L2loc(0, ∞; H 1(Ω)), i = 1, . . . , n
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Example 2 : Tumor-growth model
∂tu − ∂x
(A(u)∂xu) = f (u) + homog. Neumann b.c. 2u1(1 − u1) − βθu1u22
−2βu1u2(1 + θu1)
A(u) =
−2u1u2 + βθu22(1 − u2) 2βu2(1 − u2)(1 + θu1)
(H1) Entropy
Z functional:
Z u ∈ D = {(u1, u2) ∈ (0, 1) : u1 + u2 < 1}
H=
h(u)dx =
Ω
[u1(log u1 − 1) + u2(log u2 − 1)
Ω
u2)(log(1
wi
+ (1 − u1 −
− u1 − u2) − 1)]dx
→ wi = ∂h/∂ui or ui = e /(1 + ew1 + ew2 ) ∈ D
√
(H2) Entropy-dissipation
Z inequality: if θ < 4/ β then
dH
2
2
(u1)x + (u2)x dx ≤ const.
+ Cθ
dt
Ω
Theorem: (A.J.-Stelzer, M3AS 2012)
√
Let θ < 4/ β, H(u0) < ∞ ⇒ ∃ weak solution 0 ≤ ui ≤ 1
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Example 2 : Tumor-growth model
√
Fact: global existence if θ < 4/ β
Question: What happens for “large” θ?
Answer: Numerical results show “peaks” in ECM fraction
0.25
0.6
Extracellular matrix
Tumor cells
0.2
0.15
0.1
0.05
0
0
0.5
0.4
0.3
0.2
0.1
0.2
0.4
0.6
Spatial position
0.8
1
0
0
0.2
0.4
0.6
Spatial position
0.8
1
• Tumor front spreads from left to right (production f = 0 )
• Tumor causes increase of ECM (encapsulation)
Summary
Global existence analysis of cross-diffusion systems
∂tu − div (A(u)∇u) = f (u), x ∈ Ω, t > 0
What did we learn?
• Cross-diffusion systems are highly relevant for applications
• New tool: boundedness-by-entropy method allows for proof
of global weak solutions
Pro: n species possible, proof of L∞ bounds possible,
physical interpretation, very flexible method
Con: entropy functional may be not easy to find
Work in progress:
• Derivation of cross-diffusion from kinetic models
• Analyze discrete entropy structure (stable numer. methods)
• Coupled Maxwell-Stefan and fluiddynamical models
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