Large time behavior of evolutionary problemsin cell biology
Roberto NataliniConsiglio Nazionale delle Ricerche
Istituto per le Applicazioni del Calcolo
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Outline
1 A short introduction to population dynamics: stability vs. instability2 Continuous semilinear models of chemotaxis3 A quasilinear hyperbolic model in vasculogenesis4 A hybrid model for morphogenesis in zebrafish5 A simple model of collective motion under alignment and chemotaxis
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Basics in population dynamics
Malthus modelu′ = αu
α > 0, u(t)→∞
α < 0, u(t)→ 0
Logistic modelu′ = αu(1− u)
α > 0, u(t)→ 1
α < 0, u(t)→ 0
→ Lotka-Volterra model
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Basics in population dynamics
Malthus modelu′ = αu
α > 0, u(t)→∞
α < 0, u(t)→ 0
Logistic modelu′ = αu(1− u)
α > 0, u(t)→ 1
α < 0, u(t)→ 0
→ Lotka-Volterra model
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Pattern formation: spaceKPP-Fischer equation u = u(x .t) with x ∈ (0, L)
ut = D∆u + αu(1− u)
u(0, t) = u(L, t) = 0For D = 0: Logistic u(t)→ 1
For α = 0: Diffusion u(t)→ 0
What happens in general?
If αL2
D < π2 then u(t)→ 0
otherwise αL2
D > π2 then u(t)→ U
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Turing instability and pattern formationLet f (0, 0) = g(0, 0) = 0
ut = Du∆u + αf (u, v)
vt = Dv ∆v + αg(u, v)
Assume that for Du = Dv = 0 the point (0, 0) is stable.
Turing 1952: Under some conditions on the linearized system and forDu 6= Dv , the are some unstable modes, which create patterns in thenonlinear case.
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
What is chemotaxis?
Chemotaxis is the movement of cells (bacteria, human cells...) influencedby a chemical substance called chemoattractant.
Dictyostellium discoideum (Dicty)
Angiogenesis
Stem cells Fibroblasts
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Diffusive models of chemotaxis
Patlak 1953; Keller-Segel 1970
∂tu = div (Du∇u − χ(u, φ)∇φ) ,
τ∂tφ = Dc ∆φ+ f (u, φ).
• u is the density of bacteria,• φ is the density of the chemoattractant.
Many analytical and numerical results [Horstmann, 03 & 04]:existence of global solutions vs. finite time blow-up;analysis of the blow-up profile,self-similar solutions, traveling waves...
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Diffusive models of chemotaxis
Patlak 1953; Keller-Segel 1970
∂tu = div (Du∇u − χ(u, φ)∇φ) ,
τ∂tφ = Dc ∆φ+ f (u, φ).
• u is the density of bacteria,• φ is the density of the chemoattractant.
Many analytical and numerical results [Horstmann, 03 & 04]:existence of global solutions vs. finite time blow-up;analysis of the blow-up profile,self-similar solutions, traveling waves...
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Diffusive models of chemotaxis
Patlak 1953; Keller-Segel 1970
∂tu = div (Du∇u − χ(u, φ)∇φ) ,
τ∂tφ = Dc ∆φ+ f (u, φ).
• u is the density of bacteria,• φ is the density of the chemoattractant.
Many analytical and numerical results [Horstmann, 03 & 04]:existence of global solutions vs. finite time blow-up;analysis of the blow-up profile,self-similar solutions, traveling waves...
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Diffusive models of chemotaxis
Patlak 1953; Keller-Segel 1970
∂tu = div (Du∇u − χ(u, φ)∇φ) ,
τ∂tφ = Dc ∆φ+ f (u, φ).
• u is the density of bacteria,• φ is the density of the chemoattractant.
Many analytical and numerical results [Horstmann, 03 & 04]:existence of global solutions vs. finite time blow-up;analysis of the blow-up profile,self-similar solutions, traveling waves...
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Hyperbolic models of chemotaxis
Cattaneo-type model (Hillen)∂tu + div(uV ) = 0,∂t(uV ) + γ2∇u = u∇φ− uV ,∂tφ = D ∂xxφ+ f (u, φ).
Euler-type model (Preziosi)
∂tu + div(uV ) = 0,∂t(uV ) + div(uV ⊗ V ) +∇P(u) = u∇φ− uV ,∂tφ = D ∂xxφ+ f (u, φ).
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Hyperbolic models of chemotaxis
Cattaneo-type model (Hillen)∂tu + div(uV ) = 0,∂t(uV ) + γ2∇u = u∇φ− uV ,∂tφ = D ∂xxφ+ f (u, φ).
Euler-type model (Preziosi)
∂tu + div(uV ) = 0,∂t(uV ) + div(uV ⊗ V ) +∇P(u) = u∇φ− uV ,∂tφ = D ∂xxφ+ f (u, φ).
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Why hyperbolic models?
Vessel formation (left) and numerical simulations by Preziosi et al. (right)
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
The 1D case (Greenberg-Alt)
∂tu + ∂x v = 0,∂tv + γ2 ∂x u = (χ(φ)∂xφ)u − v ,∂tφ = D ∂xxφ+ au − bφ,
x ∈ [0, L], No-flux conditions:
v(0) = v(L) = ∂xφ(0) = ∂xφ(L) = 0(= ∂x u(0) = ∂x u(L)).
Conserved quantities
Mass∫
[0,L]
u(x , t) dx =
∫[0,L]
u(x , 0) dx
Symmetry (u, v , φ)(L− x , t) = (u,−v , φ)(x , t)
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
The 1D case (Greenberg-Alt)
∂tu + ∂x v = 0,∂tv + γ2 ∂x u = (χ(φ)∂xφ)u − v ,∂tφ = D ∂xxφ+ au − bφ,
x ∈ [0, L], No-flux conditions:
v(0) = v(L) = ∂xφ(0) = ∂xφ(L) = 0(= ∂x u(0) = ∂x u(L)).
Conserved quantities
Mass∫
[0,L]
u(x , t) dx =
∫[0,L]
u(x , 0) dx
Symmetry (u, v , φ)(L− x , t) = (u,−v , φ)(x , t)
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
The Neumann problem
Joint with Guarguaglini, Mascia, Ribot, DCDS-B 2009Let (U, 0,Φ) be a constant steady state of the Cattaneo model(U + u, v ,Φ + φ) perturbed solutionThe perturbation w = (u, v , φ) satisfies∂tu + ∂x v = 0,
∂tv + γ2 ∂x u − χU ∂xφ+ β v= F1(φ, ∂xφ) + F2(φ, ∂xφ) u + F3(φ, ∂xφ) v ,
∂tφ− D ∂xxφ+ bφ− a u = F4(u, φ),
v = ∂xφ = 0,for x = 0, L
where
F1(φ, ψ) := U(
g(Φ + φ, ψ)− χψ)
= O(|(φ, ψ)|2),
F2(φ, ψ) := g(Φ + φ, ψ) = O(|(φ, ψ)|),F3(φ, ψ) := β − h(Φ + φ, ψ) = O(|(φ, ψ)|),
|(φ, ψ)| → 0
F4(u, φ) := f (U + u,Φ + φ)− a u + b φ = O(|(u, φ)|2) |(u, φ)| → 0
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
The Neumann problem
Joint with Guarguaglini, Mascia, Ribot, DCDS-B 2009Let (U, 0,Φ) be a constant steady state of the Cattaneo model(U + u, v ,Φ + φ) perturbed solutionThe perturbation w = (u, v , φ) satisfies∂tu + ∂x v = 0,
∂tv + γ2 ∂x u − χU ∂xφ+ β v= F1(φ, ∂xφ) + F2(φ, ∂xφ) u + F3(φ, ∂xφ) v ,
∂tφ− D ∂xxφ+ bφ− a u = F4(u, φ),
v = ∂xφ = 0,for x = 0, L
where
F1(φ, ψ) := U(
g(Φ + φ, ψ)− χψ)
= O(|(φ, ψ)|2),
F2(φ, ψ) := g(Φ + φ, ψ) = O(|(φ, ψ)|),F3(φ, ψ) := β − h(Φ + φ, ψ) = O(|(φ, ψ)|),
|(φ, ψ)| → 0
F4(u, φ) := f (U + u,Φ + φ)− a u + b φ = O(|(u, φ)|2) |(u, φ)| → 0
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Global existence for the Neumann case
Theorem
Under the previous assumptions, let (U, 0,Φ) be a constant steady statesuch that
χ = ∂ψg(Φ, 0) > 0, β = h(Φ, 0) > 0, ∂φf (U,Φ) = −b < 0 < a = ∂uf (U,Φ)
Assume the stability condition
U <γ2
χ a
(b +
Dπ2
L2
)Let w0 = (u0, v0, φ0) ∈ H1 the perturbation (with zero mass for u0),and w the corresponding solution. Then there exists ε0 > 0 such that,if ‖w0‖H1 ≤ ε0, then
‖w‖H1 (t) ≤ C ‖w0‖H1 e−θ t . ∀ t > 0.
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Global existence for the Neumann case
Theorem
Under the previous assumptions, let (U, 0,Φ) be a constant steady statesuch that
χ = ∂ψg(Φ, 0) > 0, β = h(Φ, 0) > 0, ∂φf (U,Φ) = −b < 0 < a = ∂uf (U,Φ)
Assume the stability condition
U <γ2
χ a
(b +
Dπ2
L2
)Let w0 = (u0, v0, φ0) ∈ H1 the perturbation (with zero mass for u0),and w the corresponding solution. Then there exists ε0 > 0 such that,if ‖w0‖H1 ≤ ε0, then
‖w‖H1 (t) ≤ C ‖w0‖H1 e−θ t . ∀ t > 0.
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Stationary solutions
From ∂tu + ∂x v = 0, we have v = 0.Constant solutions
(u, v , φ) =(
U, 0, ab U
), U ≥ 0.
Nonconstant solutions
γ2 ∂x u − u∂xφ = 0,D ∂xxφ+ au − bφ = 0,∂xφ(0) = ∂xφ(L) = 0.
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Stationary solutions
From ∂tu + ∂x v = 0, we have v = 0.Constant solutions
(u, v , φ) =(
U, 0, ab U
), U ≥ 0.
Nonconstant solutions
γ2 ∂x u − u∂xφ = 0,D ∂xxφ+ au − bφ = 0,∂xφ(0) = ∂xφ(L) = 0.
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Stationary solutions
From ∂tu + ∂x v = 0, we have v = 0.Constant solutions
(u, v , φ) =(
U, 0, ab U
), U ≥ 0.
Nonconstant solutions
γ2 ∂x u − u∂xφ = 0,D ∂xxφ+ au − bφ = 0,∂xφ(0) = ∂xφ(L) = 0.
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Bifurcation diagram (for the Neumann Pb.)
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Bifurcation diagram (for the Neumann Pb.)
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Bifurcation diagram (for the Neumann Pb.)
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Bifurcation diagram (for the Neumann Pb.)
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Bifurcation diagram (for the Neumann Pb.)
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Comparison between Hyperbolic and Parabolic modelsThe constant case
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
The Quasilinear model
[Ambrosi, Gamba, Preziosi et al., 03]∂tρ+ ∂x (ρu) = 0∂t(ρu) + ∂x (ρu2) + ∂x P(ρ) = χρ∂xφ− αρu,τ∂tφ = D ∂xxφ+ aρ− bφ,
x ∈ [0, L], with ”no-flux” conditionsStability in 1D and even multi-D of constant stationary states forsmall perturbations [Di Russo, Sepe, 13]Solutions with vacuum zones!
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Stationary Solutions
[R.N., M. Ribot, M. Twarogowska]On [0, L], fix a mass M
∂tρ+ ∂x (ρu) = 0
∂t(ρu) + ∂x (ρu2) + ∂x P(ρ) = χρ∂xφ− αρu,
τ∂tφ = D ∂xxφ+ aρ− bφ.
+ Boundary conditions ⇒ ρu = 0
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Stationary Solutions
[R.N., M. Ribot, M. Twarogowska]On [0, L], fix a mass M
∂tρ+ ∂x (ρu) = 0
∂t(ρu) + ∂x (ρu2) + ∂x P(ρ) = χρ∂xφ− αρu,
τ∂tφ = D ∂xxφ+ aρ− bφ.
+ Boundary conditions ⇒ ρu = 0
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Stationary Solutions
[R.N., M. Ribot, M. Twarogowska]On [0, L], fix a mass M
∂tρ+ ∂x (ρu) = 0
∂t(ρu) + ∂x (ρu2) + ∂x P(ρ) = χρ∂xφ− αρu,
τ∂tφ = D ∂xxφ+ aρ− bφ.
+ Boundary conditions ⇒ ρu = 0
∂x P(ρ) = χρ∂xφ,
0 = D ∂xxφ+ aρ− bφ.
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Stationary Solutions
[R.N., M. Ribot, M. Twarogowska]On [0, L], fix a mass M
∂tρ+ ∂x (ρu) = 0
∂t(ρu) + ∂x (ρu2) + ∂x P(ρ) = χρ∂xφ− αρu,
τ∂tφ = D ∂xxφ+ aρ− bφ.
+ Boundary conditions ⇒ ρu = 0
κγργ−1∂xρ = χρ∂xφ,
0 = D ∂xxφ+ aρ− bφ,
with P(ρ) = κργ .
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Stationary solutions, again
κγργ−1∂xρ = χρ∂xφ,
0 = D ∂xxφ+ aρ− bφ.+ ∂xρ|0,L = ∂xφ|0,L = 0.
Constants: (ρ, φ) =
(ML ,
aMbL
)Otherwise: If γ ≥ 2, we have ρ = 0 or κγργ−2∂xρ = χ∂xφ.For γ = 2, explicit solutions
ρ = 0 et D ∂xxφ− bφ = 0.
ρ =χ
2κφ+ K et D ∂xxφ+(aχ
2κ − b)φ+ aK = 0.
⇒ Explicit solutions (for ω =1D
(aχ2κ − b
)> 0 ).
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Stationary solutions, again
κγργ−1∂xρ = χρ∂xφ,
0 = D ∂xxφ+ aρ− bφ.+ ∂xρ|0,L = ∂xφ|0,L = 0.
Constants: (ρ, φ) =
(ML ,
aMbL
)Otherwise: If γ ≥ 2, we have ρ = 0 or κγργ−2∂xρ = χ∂xφ.For γ = 2, explicit solutions
ρ = 0 et D ∂xxφ− bφ = 0.
ρ =χ
2κφ+ K et D ∂xxφ+(aχ
2κ − b)φ+ aK = 0.
⇒ Explicit solutions (for ω =1D
(aχ2κ − b
)> 0 ).
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
A zoological classification of stationary solutions for(γ = 2)
Density ρ.
With κ = 1, χ =10, D = 0.1, a =20, b = 10, L = 1 etM = 10.
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Numerical test 1, with respect to the size of the domain Land the friction coefficient α, for P(ρ) = κρ2
L\α 5 50 200
1
7
30
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Numerical test 3: hyperbolic–parabolic comparison
t = 0.1 t = 0.5 t = 3
t = 20 - 300 t = 331 t = 333κ = 1, χ = 10, D = 0.1, a = 20, b = 10, L = 1 et γ = 3.Strong difference in the asymptotic behavior between the hyperbolic andthe parabolic cases
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Numerical test 3: hyperbolic–parabolic comparison
t = 0.1 t = 0.5 t = 3
t = 20 - 300 t = 331 t = 333κ = 1, χ = 10, D = 0.1, a = 20, b = 10, L = 1 et γ = 3.Strong difference in the asymptotic behavior between the hyperbolic andthe parabolic cases
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
A hybrid mathematical model for self-organizing cells inzebrafish
A joint work with Ezio Di Costanzo and Luigi Preziosi, J. Math. Bio.2015
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Lateral lineA fundamental sensory system present in fish and amphibians.
Large variety of behaviours:detect movement and vibration inthe surrounding water;prey and predator detection;school swimming.
NeuromastsMain sensory organs of the lateral line, embedded in the body surface in arosette-shaped pattern: 1–2 sensory hair cells in the centre, surrounded by othersupport cells (8–12 cells).Neuromasts extend a ciliary bundle into the water, which detect movement in thesurrounding environment.
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Experimental observationsAn initial elongated group (80-100 cells) of mesenchymal cells(primordium), with a trailing region near the head and a leading regiontowards the future tail of the embryo.
Two primary mechanisms in the morphogenesis process:1 a collective cell migration guided by a haptotactic signal, with
constant velocity of about 69 µm h−1;2 a process of differentiation in the trailing region that induces a
mesenchymal–epithelial transition and causes the neuromastsassembly and their detachment.
Movie zebrafish (Gilmour et al, 2006).Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Collective migrationTwo main factors:
1 chemokine protein SDF-1a (stromal cell-derived factor-1a), stronglyhaptotactic and expressed by the substratum;
2 the receptor CXCR4b expressed by the primordium itself.
Neuromasts assemblyTwo main factors:
1 fibroblast growth factors FGF3–FGF10, strongly chemotactic;2 receptor FGFR.
Experimental observations on the FGF activity1 FGF3 and FGF10 are substantially equivalent (robustness of the
system);2 FGF and FGFR are mutually exclusive.
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Leader to follower differentiation
leader mesenchymal cells: produce FGF but the receptor FGFR is not activated;follower epithelial cells: activate FGFR, but do not produce FGF.
Cyclic mechanism
1 at the beginning, all cells are leader;2 leader–follower differentiation (MET
transition) produces rosette-shapedstructures (proto-neuromasts);
3 neuromasts deposition.
Three sufficient conditions for the leader–follower transition
1 a low level of SDF-1a (trailing zone is preferred for transition);2 a high level of FGF;3 a lateral inhibition effect (leader/follower transition favored by a low number of
neighboring cells).
return
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
The mathematical modelOur aim is to obtain a minimal mathematical model which is able to:
1 describe the collective cell migration, the detachment of theneuromasts, in the physical spatial and temporal scale;
2 ensure the existence and stability of the rosette structures of theneuromasts, as stationary solutions.
Request 2) provides a restriction for some parameters.
Hybrid discrete in continuous descriptiondiscrete on cellular scale (but nonlocal sensing area);continuous at molecular scale.
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
A second order modelXi (t) : position of the i-th cell;ϕi (t) : switch variable for the i-th cell (ϕi = 0, 1 resp. follower-leader);f (x, t) : concentration of FGF (equivalent FGF3 and FGF10);s(x, t) : concentration of SDF-1a;
acceleration i-th cell︷︸︸︷Xi =
haptotaxis︷ ︸︸ ︷αF1 (∇s) +
chemotaxis︷ ︸︸ ︷γ(1− ϕi )F1 (∇f ) +
alignment︷ ︸︸ ︷F2(X) +
attraction/repulsion︷ ︸︸ ︷F3(X)
−
damping︷ ︸︸ ︷[µF + (µL − µF)ϕi ] Xi ,
leader-follower state︷︸︸︷ϕi =
0, if
SDF conc.︷ ︸︸ ︷δF1(s) −
FGF conc.︷ ︸︸ ︷[kF + (kL − kF)ϕi ] F1(h(f )) +
lateral inhib.︷ ︸︸ ︷λΓ(ni ) ≤ 0,
1, otherwise,
FGF rate in time︷︸︸︷∂t f =
diffusion︷︸︸︷D∆f +
production︷ ︸︸ ︷ξF4(X) −
molecular degradation︷︸︸︷ηf ,
SDF rate in time︷︸︸︷∂ts = −
degradation︷ ︸︸ ︷σsF5(X) ,
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Numerical test of the steady solution.Dynamical simulation with initial zero velocity, a FGF concentrationgiven by the stationary equation
D∆f = ηf − ξχB(XL,R3),∂f∂n = 0, on ∂Ω,
and a zero initial concentration of SDF-1a.
Figure : A 8-rosette
0 5 10 15 20 25 30 35 40 45 500
0.5
1
1.5
2
2.5
3x 10
−3
Emax,rel
Time (h)
Figure :maxi ‖Xi(t)− Xi0‖ /R
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
7x 10
−4
Time (h)
Vmax(µm
h−1)
Figure : maxi∥∥Xi(t)
∥∥Numerical simulation
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Dynamical model
Numerical simulation
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
The Cucker-Smale modelBehaviour of a flock of birds, general phenomena where autonomousagents reach a consensus. Vi =
β
N
N∑j=1
α1
(α2 + ||Xi − Xj ||2)σ(Vj − Vi );
Xi = Vi ,
i = 1, . . . ,N.
σ: rate of decay of the influence between agents.Cucker and Smale (2007), Ha et al (2009), see also J. A. Carrillo, et al(2010) proved that:
if 0 ≤ σ ≤ 1/2 there is unconditional flocking (Movie with σ = 1/2);if σ > 1/2 there is conditional flocking (Movie with σ = 2).
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Cucker-Smale modelVi =
β
N
N∑j=1
α1(α2 + ‖Xi − Xj‖2)σ (Vj − Vi ),
Xi = Vi ,
Applications: biological field, collective dynamics of different interacting groups(Szabo et al, 2006; Sepulveda et al, 2013; Albi and Pareschi, 2013).Extensions: repulsion, individuals with preferred directions.
A simple Collective motion under alignment and chemotaxis
Vi =β
N
N∑j=1
1(1 +‖Xi −Xj‖2
R2
)σ (Vj − Vi ) + γ∇f (Xi ),
Xi = Vi ,
∂t f = D∆f + ξ
N∑j=1
χB(Xj ,R) − ηf ,
(1)
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Existence and uniqueness
Joint with E. Di Costanzo 2016
Theorem 1 (Local existence and uniqueness)System (1) has a unique solution on [0,T ], where T is suitably estimated.
Proposition 1 (Continuation of solutions)Let y(t) be a solution of system (1) on a interval [0,T ), if there is aconstant P with ‖y− y0‖ ≤ P on [0,T ), then there is a T > T suchthat y(t) can be continued to [0, T ].
Theorem 2 (Global existence and uniqueness)System (1) has a unique solution in all [0,+∞). go to
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology
Numerical simulations: asymptotic behavior
Simulation 1. Absence of chemotaxis (γ = 0).Simulation 2. Strong flocking state.Simulation 3. Absence of alignment (β = 0).
Roberto Natalini IAC–CNR Large time behavior of evolutionary problems in cell biology