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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 asynchrono
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Mathematical Tools
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 infinitedimensional 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.
10.1 Basics for Semigroup Approach In comparison with the classical integral equation approach, the abstract differential (and integral) equation approach can be a more powerful tool for treating nonlinear structured population models, because this formulation can make use of mathematical methods developed for infinite-dimensional dynamical systems theory. In fact, the theoretical developments in structured population dynamics since the 1980s have been led by the functional analytic approach. For classical results, readers are referred to the work of Metz and Diekmann [75] and Webb [108, 111]. For more elaborated modern methods, readers are referred to [34–36, 94]. Here, we only sketch an introduction for the semigroup approach to age-dependent population models.
10.1.1 Linear Problems As a most simple linear case, we now consider the semigroup solution of the scalartype stable population model. Desch and Schappacher [31], Prüss [85], Song et al. [95–98], and Webb [107, 108] all considered semigroup approaches for the stable © Springer Science+Business Media Singapore 2017 H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, DOI 10.1007/978-981-10-0188-8_10
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10 Mathematical Tools
population model in the early 1980s. Readers are referred to [54, 55] for the case of multistate populations. The stable population model (1.25) is formulated as an abstract Cauchy problem (initial value problem) on a Banach space L 1 (0, ω): dp(t) = Ap(t), dt
p(0) = p0 ,
(10.1)
where A denotes the population operator introduced in Sect. 1.2: (Aφ)(a) := −
dφ(a) − μ(a)φ(a), da
the domain of which is given by D(A) = φ ∈ L 1 (0, ω) : φ(0) =
ω
β(a)φ(a)da, Aφ ∈ L 1 (0, ω) ,
0 1 where we assume that β ∈ L 1+ (0, ω)∩L ∞ + (0, ω), μ ∈ L loc,+ (0, ω) and ∞.
ω 0
μ(σ )dσ =
Remark 10.1 Note that if we choose an age a† ∈ (β2 , ω) where β2 is the maximum reproductive age, we can assume that μ ∈ L ∞ + (0, a† ) and the term −μφ can be treated as a bounded perturbation in L 1 . If individuals with age a > a† do not have any influence on individuals in the age interval [0, a† ], as in the linear ca
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