Derivation of macroscopic equations for individual
cell-based models: A formal approach
Bodnar and Velasquez, 2005
November 10, 2015
Veronica Ciocanel
Derivation of macroscopic equations
Notations and a few assumptions
Notations
N = number of cells
V: potential
Xk (t): position of the center of cell k at time t
ξk (t): uncorrelated white noises
d: average distance between cells
R: range of interactions of potentials
Veronica Ciocanel
Derivation of macroscopic equations
Notations and a few assumptions
Assumptions
ρ(x, t) =
the number of cells in [x,x+∆x] at time t
N
d ∼ 1/N
d ∆x l
P
I.e., consider XN (t) = N1 N
k=1 δXkN (t) and consider the weak
limit XN (t) → ρ(x, t)dx as N → ∞ .
R ≈ d: short range interactions
R d: long range interactions
Veronica Ciocanel
Derivation of macroscopic equations
Main model equation
Model equation:
Scaling of potential:
VN (x) = N β V1 (N βx ) ,
Veronica Ciocanel
0<β≤ .
Derivation of macroscopic equations
Summary of results
Veronica Ciocanel
Derivation of macroscopic equations