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Topics in Mathematical Biology pp 265–299Cite as

Coupled Movements

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Abstract

In Sect. 1.3 we have seen that a reaction diffusion equation is the limit of diffusively coupled diffusion and reaction equations. In other models particles move on straight lines, the direction of the line and the time for a change of direction are governed by random processes, e.g., by choosing from a given distribution or by diffusion on the space of directions. The correlated random walks and the related telegrapher equation is one example. Here we look at other examples where modes of movement in space and reactions are coupled and we try to find limiting equations, usually in the form of diffusion equations. But first we look, for comparison, at invariants and boundary value problems for reaction diffusion equations.

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Notes

  1. 1.

    Huygens principle for three space dimensions says that signals are transported exactly along characteristics. The sound that is produced from a source at position 0 at time 0 can be heard at position x at time t = | x | ∕c, not before and not later. In two dimensions sound is reverberant.

  2. 2.

    The telegraph equation has been derived from Maxwell’s equations by Lord Kelvin, it has been studied by Heaviside, Poincaré, Picard, and many others. These studies were important for the design of transoceanic cables, in particular for the “pupinisation” (to increase of self induction), see [189] for the history and [289] the mathematics.

    Let R, L, C, A be the Ohm resistance, self induction, the capacity, and leakage, per unit of length, of a transmission line of two parallel wires. The voltage v and the cross current satisfy v x + Li t + Ri = 0, i x + Cv t + Av = 0. Eliminating i leads to \(v_{tt} + \frac{AL+RC} {LC} v_{t} = \frac{1} {LC}v_{xx} -\frac{RA} {LV }v\) and the same equation for i.

  3. 3.

    Cattaneo’s idea is used frequently in physics where wave-like heat transfer (second sound) is modeled (see [153]) or whenever one wants to avoid the effects of infinitely fast propagation, see an application to thermoelasticity [73] and the references therein.

  4. 4.

    In physics transport equations occur in the form of Boltzmann equations. Then instead of a linear turning operator there is a quadratic nonlinearity which describes collisions of particles which between collisions move with constant velocity. Standard references on the Boltzmann equation are the monographs [42, 43]. Equations of the form (7.95) arise from linearizing the Boltzmann equation.

  5. 5.

    Here we integrate over the unit sphere rather than the sphere of radius γ.

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  1. FB Biologie, Universität Tübingen, Tübingen, Germany

    Karl-Peter Hadeler

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Hadeler, KP. (2017). Coupled Movements. In: Topics in Mathematical Biology. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-65621-2_7

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