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Coupling and Quiescence

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Abstract

Suppose we have two vector fields \(f,g: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) and the differential equations

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Notes

  1. 1.

    This situation corresponds to an ultimate discretization of the diffusion equation, with only two grid points.

  2. 2.

    We replace p, q by p∕ε, q∕ε and let ε → 0.

  3. 3.

    There are other notations for coupled systems. Hale [126] starts from systems \(\dot{z}_{j} = f_{j}(z_{j})\), \(j \in \mathbb{Z}\), in \(\mathbb{R}^{n}\), collects the z j in a “row” vector z, and the f j into f j (z) = ( f 1(z 1), …, f m (z m ), and then, with a symmetric tridiagonal coupling matrix A, writes the system as \(\dot{z} = Az + f(z)\).

  4. 4.

    Notice that in this “additive” equation and in the “multiplicative” equation \(\dot{V } = AV P\) the operators V → AV and V + VP commute. Thus, if μ i , λ j are the eigenvalues of A, P, respectively, then μ i + λ j and μ i λ j are the eigenvalues of the operators V → AV + VP and AV P, respectively.

  5. 5.

    It suffices that each matrix has a unique positive Perron vector.

  6. 6.

    An example may be the European starling which was introduced in North America without effect until flocks of sixty to one hundred birds were settled in New York Central Park around 1890. Now there are an estimated 200 million starlings in North America.

  7. 7.

    In the thermodynamic interpretation of the heat equation the diffusion coefficient is k∕(cρ) where k is heat conductivity, c is the specific heat capacity, and ρ the specific mass.

  8. 8.

    This assumption is called the first Fickian law, the diffusion equation itself the second Fickian law. In an experimental setting, the first Fickian law does not hold for very small gradients. This assumption leads to the unwanted effect of infinitely fast propagation in the heat equation (its parabolic character as opposed to hyperbolic).

  9. 9.

    The transition from (1.40) to (1.42) is called Kac’ trick [156]. Notice that the transition is not invertible, solutions “without mass” (0, v 0exp(−μt)) of the system (1.41) are mapped into the zero solution of Eq. (1.42).

  10. 10.

    The Schur–Cohn criterion, from 1914/1922, see [205], often attributed to E.I. Jury 1964,is the analogue of the Routh–Hurwitz criterion for discrete time systems.

  11. 11.

    These inequalities imply detA < 1.

  12. 12.

    We use the formula for the real part of the square root of a complex number \(\mathfrak{R}\sqrt{\xi +i\eta } = [\xi +\sqrt{\xi ^{2 } +\eta ^{2}}]^{1/2}/\sqrt{2}\).

  13. 13.

    We underline that here \(\mathbb{R}_{+}^{n}\) is the positive cone in the tangent space. It could be that also the underlying system for x preserves positivity. But such property would not be relevant here.

  14. 14.

    This result was not widely known in 1974 as can be seen from [140]. But in 1982 M. Hirsch published the first paper of a series of six on cooperative and competitive systems [139].

  15. 15.

    In [126], on coupled systems, it is assumed that the small system and the coupled systems have compact global attractors.

  16. 16.

    This section is almost identical with the paper [105].

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  1. FB Biologie, Universität Tübingen, Tübingen, Germany

    Karl-Peter Hadeler

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Hadeler, KP. (2017). Coupling and Quiescence. 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_1

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