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Age-Structured Population Dynamics in Demography and Epidemiology pp 503–549Cite as

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Abstract

This last chapter is a collection of mathematical tools to treat population equations. First, we explain a semigroup approach to consider the stable population model. Using the semigroup setting, the idea of strong ergodicity is extended as the asynchronous exponential growth of the semigroup, which can be applied to certain nonlinear problems. We then briefly discuss functional analytic methods for nonlinear population problems. Next, we introduce some results for the infinite-dimensional Perron–Frobenius theorem, the contraction mapping principle given by the Hilbert pseudometric, Birkhoff’s linear multiplicative process theory, and some properties of nonlinear positive operators, which are useful tools for studying population dynamics. Finally, we summarize some basic results about the Laplace transformation and Volterra integral equation that are used in the previous chapters.

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Notes

  1. 1.

    \(\lambda \in \sigma (A)\) is an essential spectrum if either \(R(\lambda ,A)\) is not closed, \(\lambda \) is a limit point of \(\sigma (A)\), or the generalized eigenspace \(N_\lambda (A)\) is infinite-dimensional.

  2. 2.

    The essential growth bound of T(t) is defined by \(\omega _1(A):=\lim _{t \rightarrow \infty }t^{-1}\log (\alpha [T(t)])\) where \(\alpha [A]\) is the measure of non-compactness of a bounded linear operator A.

  3. 3.

    Let \(X^*\) be the adjoint space of X. \(x \in X_+\) is called a quasi-interior point if \(\langle x^*,x \rangle >0\) for any \(x^* \in X^*_+\setminus \{0\}\).

  4. 4.

    Note that in older papers such as Birkhoff’s, a positive operator is called nonnegative and a strictly positive operator is called positive.

  5. 5.

    \(T \in B(E)\) is power compact if there is a positive integer n such that \(T^n\) is compact.

  6. 6.

    Although, for simplicity, we restrict our argument to strictly positive processes, we can relax this condition [16, 17].

  7. 7.

    \(P_{\sigma }(A)\) denotes the point spectrum of an operator A.

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    Hisashi Inaba

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Inaba, H. (2017). Mathematical Tools. In: Age-Structured Population Dynamics in Demography and Epidemiology. Springer, Singapore. https://doi.org/10.1007/978-981-10-0188-8_10

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