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Some results based on maximal regularity regarding population models with age and spatial structure

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Abstract

We review some results on abstract linear and nonlinear population models with age and spatial structure. The results are mainly based on the assumption of maximal \(L_p\)-regularity of the spatial dispersion term. In particular, this property allows us to characterize completely the generator of the underlying linear semigroup and to give a simple proof of asynchronous exponential growth of the semigroup. Moreover, maximal regularity is also a powerful tool in order to establish the existence of nontrivial positive equilibrium solutions to nonlinear equations by fixed point arguments or bifurcation techniques. We illustrate the results with examples.

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Notes

  1. If \(a_m\) is finite, individuals may attain age greater than \(a_m\) but are no longer tracked in the model.

  2. More precisely, if \(\mu (a)>0\) for \(a\in (0,a_m)\) and \(\int _0^{a_m} \mu (a)\mathrm {d}a=\infty\), then \(1-\Pi (a,0)\) is a probability distribution with density \(\mu (a)\Pi (a,0)\).

  3. We shall suppress the age interval J also in the writing of other function spaces as no confusion seems likely.

  4. Recall that if E is an ordered Banach space, then \(T\in {\mathcal {L}}(E)\) is strongly positive if \(Tz\in E\) is a quasi-interior point for each \(z\in E^+{\setminus }\{0\}\), that is, if \(\langle z',Tz\rangle _{E} >0\) for every \(z'\in (E')^+{\setminus }\{0\}\).

  5. Recall that \(\mathrm {int}(E_\varsigma ^+)\not =\emptyset\) consists exactly of the quasi-interior points of \(E_\varsigma\).

  6. Similarly, one can introduce a bifurcation parameter in the death modulus to construct nontrivial equilibria [14, 60].

  7. Note that \(E_0\) is separable since \(E_1\) is separable and dense in \(E_0\). Thus, if \(E_0\) is reflexive , then \(E_0'\) is separable.

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  1. Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

    Christoph Walker

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Walker, C. Some results based on maximal regularity regarding population models with age and spatial structure. J Elliptic Parabol Equ 4, 69–105 (2018). https://doi.org/10.1007/s41808-018-0010-9

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