Nonlinear Hyperbolic Systems of Generalized Navier-Stokes Type for Interactive Motion in Biology
We review two modelling approaches to obtain genuinely nonlinear systems of one hyperbolic transport equation (for density) accompanied by parabolic or elliptic equations (for mean velocity and, eventually, pressure), namely generalized Navier-Stokes or (
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Summary. We review two modelling approaches to obtain genuinely nonlinear systems of one hyperbolic transport equation (for density) accompanied by parabolic or elliptic equations (for mean velocity and, eventually, pressure), namely generalized Navier-Stokes or (pseudostationary) Stokes equations. Background and applications are related to models of interactive biological motion, namely for contractile polymer networks in intra-cellular motility, for cell movement and tissue formation during wound healing as well as for cohorts of migrating birds. One approach is to derive, after suitable scaling, a formal continuum limit of (stochastic) Hamiltonian equations for 'visco-elastic' multi-particle networks with specific interaction laws. The other consists in the derivation of (highly) viscous two-phase flow equations by minimization of a corresponding energy-loss functional. In both procedures there remain convergence or existence problems to be solved analytically. Some results and a few numerical simulations are shown, particularly for the I-dimensional case. For further results, technical details and for comparison with other methods we give corresponding references.
1 'Visco-Elastic' Multi-particle Networks in Biological Systems 1.1 Model Setup Let us describe the Hamiltonian dynamics of N 'particles' constituting a deformable network in ~rn(meant are biochemical or biological entities as 'polymers', 'cells' or 'birds') by standard (stochastic) evolution equations for positions Xi(t) €~rn and velocities V'i(t) €~rn, i = 1, ... ,N:
dXi = V'i . dt dV'i = (Ai
+ .L Aij) . dt + fii . de;
(1)
(2)
#i
where Ai resp. Aij denote the deterministic acceleration vectors for the i- th 'particle' , either induc~d by its own activity or by interaction with other 'particles' ,respectively, and where f3i measures the amplitude of small stochastic perturbations, e.g. due to 'thermodynamics' or individual arbitrariness, represented as mutually independent Brownian increments de; € ~rn during infinitesimally small time steps dt. The first deterministic acceleration term in equation (2) might describe a selfinduced adjustment to a given preferred speed 8*, common for all 'particles', say
Ai = cl>(V'i) =
1'(11 V'i II) . (8* / II
V'i
II
-1) . Vi(t).
S. Hildebrandt et al. (eds.), Geometric Analysis and Nonlinear Partial Differential Equations © Springer-Verlag Berlin Heidelberg 2003
(3)
432
W. Alt
with an adjustment rate "'( = "'(Si) that could depend on the 'particle's actual speed Si = II Vi II· In case of S* = 0, the vector Ai = -"'( . Vi could just describe physical friction force divided by mass. Analogously, the acceleration term Aij could describe the physical interaction force, which the j-th 'particle' exerts onto the i-th one, divided by its body mass. However, for polymer networks, and even more for biological entities like cells or birds, which physiologically respond to interactions with neighbors and then adjust their moving velocity in an active manner, a simple linear superposition in Lj,.oi Aij is quest
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