Kinetic models for chemotaxis: Hydrodynamic limits and spatio-temporal mechanisms

Abstract

We study kinetic models for chemotaxis, incorporating the ability of cells to assess temporal changes of the chemoattractant concentration as well as its spatial variations. For prescribed smooth chemoattractant density, the macroscopic limit is carried out rigorously. It leads to a drift equation with a chemotactic sensitivity depending on the time derivative of the chemoattractant density. As an application it is shown by numerical experiments that the new model can resolve the chemotactic wave paradox. For this purpose, the macroscopic equation is coupled to a simple activation-inhibition model for the chemoattractant which produces the chemoattractant waves typical for the slime mold Dictyostelium discoideum.

Taxis equations for amoeboid cells

“…We showed earlier that for non-interacting walkers the internal dynamics can be incorporated in the transport equation as follows [17]. Let p(x, v, y, t) be the density of individuals in a (2N + m)−dimensional phase space with coordinates [x, v, y], where x ∈ R N is the position of a cell, v ∈ V ⊂ R N is its velocity and y ∈ Y ⊂ R m is its internal state, which evolves according to (12). The evolution of p is governed by the transport equation…”

A well-balanced scheme for kinetic models of chemotaxis derived from one-dimensional local forward–backward problems

“…(1) General kinetic models of chemotaxis [16,18]: another widely used mass-preserving model, different from (23), reads ∂ t f (t, z, v) + v∂ z f = They don't lead to transfer matrices which are rank-one perturbations; however, their inverse may be computed by means of the Woodbury formula (see again [33]). (2) Inclusion of the time-derivative ∂ t S: it can be considered interesting to include inside the kinetic model an information about the time-evolution of the chemo-attractant substance at the location z.…”

“…This is precisely the kind of asymptotic states one expects for (9) according to Remark 4.2 in [42] and this is the reason why few differences appear between both numerical methods. In particular, they admit very similar time-asymptotic (constant) states, see also [18]. [11,50]).…”

“…However, the parabolic formulation has the drawback of inducing an infinite propagation speed of information, which can be fairly considered unphysical. Hyperbolic models eliminate this undesirable feature, see [18,22,43]; discrete velocity kinetic models can be handled by means of similar techniques [27,28,40]. Advances in numerical analysis allow to simulate efficiently much more detailed kinetic models [2,16,23,41,42,48]: consult for instance [11,50].…”

“…This equation is referred to as the Stroock's model [48] (it appears also in [16,18,43,42]), see also (36) without birth/death term in [41]; it can be deduced from Alt's model by assuming that the run times τ follow a Poisson distribution [23]. This type of motion belongs to the class of velocity-jump processes because individuals run in given directions, but at random instants, they stop in order to choose a new velocity, and the time spent in the latter stage is small compared to the run length, thus this tumbling step can be considered instantaneous.…”